EXPLICIT CHANNEL INFORMATION FEEDBACK BASED ON HIGH-ORDER PCA DECOMPOSITION OR PCA COMPOSITION

Abstract

A communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system includes a transceiver to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals including the reference signal configuration, and a processor. The processor estimates the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructs a frequency-domain channel tensor using the CSI estimate, performs a higher-order principal component analysis, HO-PCA, on the channel tensor, identifies a plurality of dominant principal components of the channel tensor, thereby obtaining a compressed channel tensor, and reports to the transmitter the explicit CSI including the dominant principal components of the channel tensor.

Claims

1. A communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, and a processor configured to estimate the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, construct a frequency-domain channel tensor using the CSI estimate, perform a higher-order principal component analysis, HO-PCA, on the channel tensor, identify a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and report to the transmitter the explicit CSI comprising the dominant principal components of the channel tensor.

2. The communication device of claim 1, wherein the communication device is configured to receive from the transmitter an explicit CSI report configuration comprising a CSI channel-type information, CSI-Ind, indicator for the CSI report, wherein the CSI-Ind indicator is associated with a channel-type configuration, the channel tensor is a three-dimensional, 3D, channel tensor, or is represented by a 3D channel covariance tensor, a 3D beamformed-channel tensor, or a 3D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and wherein the plurality of dominant principal components of the compressed 3D channel tensor of dimension N.sub.rN.sub.tS comprise: a first set of r.sub.1 basis vectors comprised by a matrix .sub.R=[u.sub.R,1, . . . , u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1; a second set of r.sub.2 basis vectors comprised by a matrix .sub.T=[u.sub.T,1, . . . , u.sub.T,r.sub.2].sup.N.sup.t.sup.r.sup.2; a third set of r.sub.3 basis vectors comprised by a matrix .sub.S=[u.sub.S,1, . . . , u.sub.S,r.sub.3].sup.Sr.sup.3; and r.sub.1r.sub.2r.sub.3associated high-order singular values s.sub.ijk, sorted such that .sub.j,k|s.sub.i,j,k|.sup.2.sub.j,k|s.sub.i+1,j,k|.sup.2, .sub.i,k|s.sub.i,j,k|.sup.2.sub.i,k|s.sub.i,j+1,k|.sup.2, and .sub.i,j|s.sub.i,j,k|.sup.2.sub.i,j|s.sub.i,j,k+1|.sup.2 for all i, j, k.

3. The communication device of claim 2, wherein the values of r.sub.1, r.sub.2, and r.sub.3 representing the number of dominant principal components with respect to the first, second and third dimension of the compressed 3D channel tensor, respectively, are configured via the CSI report configuration by the transmitter, or reported by the communication device in the CSI report, or pre-determined and known at the communication device.

4. The communication device of claim 2, wherein the processor is configured to quantize the coefficients in the vectors u.sub.R,i, u.sub.T,j, and u.sub.S,k and the HO singular values s.sub.ijk of the 3D channel tensor using a codebook approach, the number of complex coefficients to be quantized being given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3 for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r.sub.1r.sub.2r.sub.3 for the higher-order singular values, respectively.

5. The communication device of claim 1, wherein the channel tensor is a four-dimensional, 4D, channel tensor, or represented either by a 4D channel covariance tensor, a 4D beamformed-channel tensor, or a 4D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and wherein the plurality of dominant principal components of the compressed 4D channel tensor of dimension N.sub.rN.sub.tSD comprise: a first set of r.sub.1 basis vectors comprised by a matrix .sub.R=[u.sub.R,1, . . . , u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1; a second set of r.sub.2 basis vectors comprised by a matrix .sub.T=[u.sub.T,1, . . . , u.sub.T,r.sub.2].sup.N.sup.t.sup.r.sup.2; a third set of r.sub.3 basis vectors comprised by a matrix .sub.S=[u.sub.S,1, . . . , u.sub.S,r.sub.3].sup.Sr.sup.3; a fourth set of r.sub.4 basis vectors comprised by a matrix .sub.D=[u.sub.D,1, . . . ,u.sub.D,r.sub.4].sup.Dr.sup.4; and r.sub.1r.sub.2r.sub.3r.sub.4 associated high-order singular values s.sub.ijkl, sorted such that .sub.j,k,l|s.sub.i,j,k,l|.sup.2.sub.j,k,l|s.sub.i+1,j,k,l|.sup.2, .sub.i,k,l|s.sub.i,j,k,l|.sup.2.sub.i,k,l|s.sub.i,j+1,k,l|.sup.2, .sub.i,j,l|s.sub.i,j,k,l|.sup.2.sub.i,j,l|s.sub.i,j,k+1,l|.sup.2, and .sub.i,j,k|s.sub.i,j,k,l|.sup.2.sub.i,j,k|s.sub.i,j,k,l+1|.sup.2 for all i, j, k, l.

6. The communication device of claim 5, wherein the values of r.sub.1,r.sub.2, r.sub.3, and r.sub.4 representing the number of dominant principal components of the 4D channel tensor are configured via the CSI report configuration by the transmitter, or they are reported by the communication device in the CSI report, or they are pre-determined and known at the communication device.

7. The communication device of claim 5, wherein the processor is configured to quantize the coefficients of the vectors u.sub.R,i, u.sub.T,j, u.sub.S,k, u.sub.D,l and the singular values s.sub.ijkl using a codebook approach, a number of complex coefficients to be quantized being given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3+Dr.sub.4 for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r.sub.1r.sub.2r.sub.3r.sub.4 for the higher-order singular values, respectively.

8. The communication device of claim 1, wherein the explicit CSI comprises a delay-domain CSI for the higher-order singular-value matrix .sub.S, and wherein the processor is configured to calculate a reduced-sized delay-domain higher-order singular-matrix .sub.S from the frequency-domain higher-order singular-matrix .sub.S, wherein the delay-domain higher-order singular-matrix is given by
.sub.SF.sub.S.sub.S, where F.sub.S.sup.SL contains L vectors of size S1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed delay-domain matrix .sub.S is given by Lr.sub.3, and L is the number of delays, and the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.SO.sub.f.sub.1], where $d_i={\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.S}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(S-1\right)}{O_f.Math.S}}\right]}^T{}^{S1},$ i{0, . . . , SO.sub.f1}, and O.sub.f{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix; quantize the coefficients in vectors .sub.S=[.sub.S,1, . . . , .sub.S,r.sub.3].sup.Lr.sup.3 using a codebook approach; report to the transmitter the explicit CSI comprising the coefficients of .sub.S instead of .sub.S, along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook .

9. The communication device of claim 1, wherein the explicit CSI comprises a Doppler-frequency domain CSI for the higher-order singular-value matrices .sub.D, and wherein the processor is configured to calculate a reduced-sized Doppler-frequency domain higher-order singular-matrix .sub.D from the time-domain higher-order singular-matrix .sub.D, wherein the Doppler-frequency domain higher-order singular-matrix is given by
.sub.DF.sub.D.sub.D, where F.sub.D.sup.DG contains G vectors of size D1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed Doppler-frequency domain matrix .sub.D is given by Gr.sub.4, and G is the number of Doppler-frequency components, and the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.DO.sub.t.sub.1], where $d_i={\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_t.Math.D}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(D-1\right)}{O_t.Math.D}}\right]}^T{}^{D1},$ i{0, . . . , O.sub.tD1}, and O.sub.t{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; quantize the coefficients in vectors .sub.D[.sub.D,1, . . . , .sub.D,r.sub.4].sup.Gr.sup.4 using a codebook approach; or report to the transmitter the explicit CSI report comprising the coefficients of .sub.D instead of .sub.D, along with the G Doppler-frequency components, represented by a set of indices that correspond to the selected DFT-vectors in the codebook .

10. A communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and a processor configured to estimate the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, construct a frequency-domain channel tensor using the CSI estimate, calculate a transformed channel tensor using the channel tensor, rewrite the transformed channel tensor to a transformed channel matrix, perform a standard principal component analysis, PCA, on the transformed channel matrix, identify a plurality of dominant principal components of the transformed channel matrix, thereby acquiring a transformed/compressed channel matrix, and report to the transmitter the explicit CSI comprising the plurality of dominant principal components of the transformed/compressed channel tensor.

11. The communication device of claim 10, wherein the communication device is configured to receive from the transmitter an explicit CSI report configuration comprising a CSI channel-type information, CSI-Ind, indicator for the CSI report, wherein the CSI-Ind indicator is associated with a channel-type configuration, the channel tensor is a three-dimensional, 3D, channel tensor, or is represented either by a 3D channel covariance tensor, a 3D beamformed-channel tensor, or a 3D beamformed channel covariance tensor as indicated by the CSI-Ind indicator, or the channel tensor is a four-dimensional, 4D, channel tensor, or is represented either by a 4D channel covariance tensor, a 4D beamformed-channel tensor, or a 4D beamformed channel covariance tensor as indicated by the CSI-Ind indicator, and wherein the 3D transformed channel tensor of dimension N.sub.rN.sub.tS, or the 4D transformed channel tensor of dimension N.sub.rN.sub.tSD, is rewritten to a transformed channel matrix of dimension N.sub.tN.sub.rSD, where D=1 for the 3D transformed channel tensor and D>1 for the 4D transformed channel tensor, respectively, and the plurality of dominant principal components of the transformed channel matrix comprise: a first set of r basis vectors comprised by a matrix =[u.sub.1, u.sub.2, . . . , u.sub.r].sup.N.sup.t.sup.N.sup.r.sup.r; a second set of r basis vectors comprised by a matrix V=[v.sub.1, v.sub.2, . . . , v.sub.r].sup.SDr; a set of r coefficients comprised by a diagonal matrix =diag(s.sub.1, s.sub.2, . . . , s.sub.r).sup.rr with ordered singular values s.sub.i (s.sub.1s.sub.2 . . . s.sub.r) on its diagonal.

12. The communication device of claim 11, wherein the value of r representing the number of dominant principal components of the transformed channel matrix is configured via the CSI report configuration by the transmitter, or reported by the communication device in the CSI report, or pre-determined and known at the communication device.

13. The communication device of claim 11, wherein the processor is configured to quantize the coefficients of the basis vectors u.sub.i, v.sub.i, and the singular values s.sub.i of the transformed channel matrix using a codebook approach.

14. The communication device of claim 1, wherein the processor is configured to apply, after the construction of the 3D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the 3D channel tensor, or the frequency dimension of the 3D channel tensor, or both the frequency and space dimensions of the 3D channel tensor, or after the construction of the 4D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the channel tensor, or the frequency dimension of the channel tensor, or the time dimension of the channel tensor, or both the frequency and time dimensions of the channel tensor, so as to exploit a sparse or nearly-sparse representation of the 3D or 4D channel tensor in one or more dimensions.

15. The communication device of claim 14, wherein the processor is configured to apply, after the construction of the 3D channel tensor, a transformation/compression with respect to all three dimensions of the 3D channel tensor of dimension N.sub.rN.sub.tS represented by a (column-wise) Kronecker product as
=.sub.(1,2,3)() $\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2,3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math..Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math..Math.{}_{n_r,n_t,s}.Math..Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$ where b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the 3D channel tensor , selected from a codebook matrix .sub.1; b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.t1 with respect to the second dimension of the 3D channel tensor , selected from a codebook matrix .sub.2; b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the 3D channel tensor , selected from a codebook matrix .sub.3; .sub.n.sub.r.sub.,n.sub.t.sub.,s is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s, and b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s, and N.sub.r, N.sub.t, and S (N.sub.rN.sub.r, N.sub.tN.sub.t, SS) represents the value of the first, second and third dimension of the transformed/compressed 3D channel tensor , respectively, or the space dimensions of the 3D channel tensor, represented by $.Math.={}_{\left(1,2\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math..Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math..Math.{}_{n_r,n_t,s}.Math..Math.b_{3,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$ where b.sub.3,s is a transformation vector of all zeros with the s-th element being one, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a vector of size N.sub.t1 with respect to the second dimension of the 3D channel tensor , selected from a codebook matrix .sub.2, and b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the 3D channel tensor , selected from a codebook matrix .sub.1, N.sub.rN.sub.r, N.sub.tN.sub.t, S=S, or the frequency dimension of the 3D channel tensor, represented by
=.sub.(3)(), $\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math..Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math..Math.{}_{n_r,n_t,s}.Math..Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r}.Math.b_{1,n_t},$ where b.sub.2,n.sub.r is a transformation vector of all zeros with the n.sub.r-th element being one, b.sub.1,n.sub.t is a vector of all zeros with the n.sub.t-th element being one, and b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the 3D channel tensor , selected from a codebook matrix .sub.3, N.sub.r=N.sub.r and N.sub.t=N.sub.t, SS.

16. The communication device of any one of claim 15, wherein the 3D transformation function .sub.(1,2,3)(.Math.) is given by a two-dimensional Discrete Cosine transformation (2D-DCT) and a 1D-DFT transformation with respect to the space and frequency dimension of the channel tensor, respectively, the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices, and the codebook matrix .sub.3 is given by an oversampled DFT matrix, or the 3D transformation function .sub.(1,2,3)(.Math.) is given by a 3D-DFT transformation and the codebook matrices .sub.n, n=1,2,3 are given by oversampled DFT matrices, or the 2D transformation function .sub.(1,2)(.Math.) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices, or the 1D transformation function .sub.(3)(.Math.) is given by a 1D-DFT transformation and the codebook matrix .sub.3 is given by an oversampled DFT matrix.

17. The communication device of claim 14, wherein the processor is configured to apply, after the construction of the 4D channel tensor a transformation/compression with respect to all four dimensions of the channel tensor, represented by a (column-wise) Kronecker product as
=.sub.(1,2,3,4)(), $\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2,3,4\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math..Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math..Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math..Math.{}_{n_r,n_t,s,d}.Math..Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$ where b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1; b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2; b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3; b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor , selected from a codebook matrix .sub.4; .sub.n.sub.r.sub.,n.sub.t.sub.,s,d is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d, and b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d and N.sub.r, N.sub.t, S and D (N.sub.rN.sub.r, N.sub.tN.sub.t, SS, DD) represents the value of the first, second, third and fourth dimension of the transformed/compressed channel tensor , respectively, or the space dimensions of the 4D channel tensor, represented by $.Math.={}_{\left(1,2\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math..Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math..Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math..Math.{}_{n_r,n_t,s,d}.Math..Math.b_{4,d}.Math.b_{3,s}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$ where b.sub.4,d is a transformation vector of all zeros with the d-th element being one, b.sub.3,s is a transformation vector of all zeros with the s-th element being one, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2, and b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1 and N.sub.rN.sub.r, N.sub.tN.sub.t, S=S and D=D, or the frequency and time dimensions of the 4D channel tensor, represented by $.Math.={}_{\left(3,4\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(3,4\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math..Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math..Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math..Math.{}_{n_r,n_t,s,d}.Math..Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r}.Math.b_{1,n_t},$ where b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor ,selected from a codebook matrix .sub.4, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3, b.sub.2,n.sub.r is a transformation vector of all zeros with the n.sub.r-th element being one, and b.sub.1,n.sub.t is a transformation vector of all zeros with the n.sub.t-th element being one and N.sub.r=N.sub.r N.sub.t=N.sub.t, SS and DD.

18. The communication device of claim 17, wherein the 4D transformation function .sub.(1,2,3,4)(.Math.) is given by a 4D-DFT transformation and the codebook matrices .sub.n, n=1,2,3,4 are given by oversampled DFT matrices, or the 2D transformation/compression function .sub.(3,4)(.Math.) is given by a 2D-DFT and the codebook matrices .sub.n, n=3,4 are given by oversampled DFT matrices, or the 2D transformation function .sub.(1,2)(.Math.) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices, or the 1D transformation function F.sub.(3)(.Math.) is given by a 1D-DFT transformation and the codebook matrix .sub.3 is given by an oversampled DFT matrix, or the 1D transformation function F.sub.(4)(.Math.) is given by a 1D-DFT transformation and the codebook matrix .sub.4 is given by an oversampled DFT matrix.

19. The communication device of claim 15, wherein the processor is configured to select the transformation vectors from codebook matrices .sub.n, and store the selected indices in a set of g-tuples, where g refers to the number of transformed dimensions of the channel tensor, and report the set as a part of the CSI report to the transmitter.

20. The communication device of claim 8, wherein the oversampling factors of the codebooks are configured via the CSI report configuration, or via higher layer or physical layer by the transmitter, pre-determined and known at the communication device.

21. A communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and a processor configured to estimate the CSI using measurements on the downlink reference signals of the radio channel, construct a frequency-domain channel matrix using the CSI estimate, perform a standard principal component analysis, PCA, on the channel matrix, identify the dominant principal components of the channel matrix, the channel matrix comprising a first set of r basis vectors comprised by a matrix =[u.sub.1, u.sub.2, . . . , u.sub.r].sup.N.sup.t.sup.N.sup.r.sup.r; a second set of r basis vectors comprised by a matrix V=[v.sub.1, v.sub.2, . . . , v.sub.r].sup.Sr; and a third set of r coefficients comprised by a diagonal matrix =diag(s.sub.1, s.sub.2, . . . , s.sub.r).sup.rr with ordered singular values s.sub.i (s.sub.1s.sub.2 . . . s.sub.r) on its diagonal; calculate a reduced-sized delay-domain matrix {tilde over (V)} from the frequency-domain matrix V, wherein the delay-domain matrix, comprising r basis vectors, is given by
VF.sub.V{tilde over (V)}, where F.sub.V.sup.SL contains L vectors of size S1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed delay-domain matrix {tilde over (V)}=[{tilde over (v)}.sub.1, {tilde over (v)}.sub.2, . . . , {tilde over (v)}.sub.r] is given by Lr, and L is the number of delays, and the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.SO.sub.f.sub.1], where ${\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.S}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(S-1\right)}{O_f.Math.S}}\right]}^T{}^{S1},$ i(0, . . . , SO.sub.f1), and O.sub.f(1,2,3, . . . ), denotes the oversampling factor of the DFT-codebook matrix; and report to the transmitter the explicit CSI comprising the identified first set of r basis vectors, the second reduced-sized delay-domain set of r basis vectors, along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook .

22. The communication device of claim 21, wherein the value of r representing the number of dominant principal components of the transformed channel matrix is configured via the CSI report configuration by the transmitter, or it is reported by the communication device in the CSI report, or it is pre-determined and known at the communication device.

23. The communication device of claim 21, wherein the processor is configured to quantize the coefficients of the basis vectors u.sub.i, {tilde over (v)}.sub.i, and the singular values s.sub.i of the channel matrix using a codebook approach.

24. The communication device of claim 1, wherein the communication device is configured with one or more scalar codebooks for the quantization of each entry of each basis vector of the plurality of dominant principal components of the channel tensor or the compressed channel tensor and the singular values or high order singular values, or with one or more unit-norm vector codebooks for the quantization of each basis vector of the plurality of dominant principal components of the channel tensor or the compressed channel tensor, and wherein the communication device selects for each basis vector a codebook vector to represent the basis vector, and the communication device is configured to report the indices corresponding to the selected entries in the scalar or vector codebook as a part of the CSI report to the transmitter.

25. A transmitter in a wireless communication system, the transmitter comprising: an antenna array comprising a plurality of antennas for a wireless communication with one or more communication devices of claim 1 for providing a channel state information, CSI, feedback to the transmitter; and a precoder connected to the antenna array, the precoder to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams, a transceiver configured to transmit, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS, and downlink signals comprising the CSI-RS configuration; and receive uplink signals comprising a plurality of CSI reports comprising an explicit CSI from the communication device; and a processor configured to construct a precoder matrix applied on the antenna ports using the explicit CSI.

26. A wireless communication network, comprising: at least one communication device for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising: a transceiver configured to receive, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, and a processor configured to estimate the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, construct a frequency-domain channel tensor using the CSI estimate, perform a higher-order principal component analysis, HO-PCA, on the channel tensor, identify a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and report to the transmitter the explicit CSI comprising the dominant principal components of the channel tensor, and at least one transmitter of claim 25.

27. The wireless communication network of claim 26, wherein the communication device and the transmitter comprises one or more of: a mobile terminal, or stationary terminal, or cellular IoT-UE, or an IoT device, or a ground based vehicle, or an aerial vehicle, or a drone, or a moving base station, or road side unit, or a building, or a macro cell base station, or a small cell base station, or a road side unit, or a UE, or a remote radio head, or an AMF, or an SMF, or a core network entity, or a network slice as in the NR or 5G core context, or any transmission/reception point (TRP) enabling an item or a device to communicate using the wireless communication network, the item or device being provided with network connectivity to communicate using the wireless communication network.

28. A method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising: receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, performing a high-order principal component analysis, HO-PCA, on the channel tensor, identifying a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and reporting the explicit CSI comprising the dominant principal components of the channel tensor from the communication device to the transmitter.

29. A method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising: receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, calculating a transformed channel tensor using the channel tensor, rewriting the transformed channel tensor to a transformed channel matrix, performing a standard principal component analysis, PCA, on the transformed channel matrix, identifying a plurality of dominant principal components of the transformed channel matrix, thereby acquiring a transformed/compressed channel matrix, and reporting the explicit CSI comprising the plurality of dominant principal components of the transformed/compressed channel tensor from the communication device to the transmitter.

30. A method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising: receiving, from a transmitter, a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and estimating the CSI using measurements on the downlink reference signals of the radio channel, constructing a frequency-domain channel matrix using the CSI estimate, performing a standard principal component analysis, PCA, on the channel matrix, identifying the dominant principal components of the channel matrix, the channel matrix comprising a first set of r basis vectors comprised by a matrix =[u.sub.1, u.sub.2, . . . , u.sub.r].sup.N.sup.t.sup.N.sup.r.sup.r; a second set of r basis vectors comprised by a matrix V=[v.sub.1, v.sub.2, . . . , v.sub.r].sup.Sr; and a third set of r coefficients comprised by a diagonal matrix =diag(s.sub.1, s.sub.2, . . . , s.sub.r).sup.rr with ordered singular values s.sub.i (s.sub.1s.sub.2 . . . s.sub.R) on its diagonal; calculating a reduced-sized delay-domain matrix {tilde over (V)} from the frequency-domain matrix V, wherein the delay-domain matrix, comprising r basis vectors, is given by
VF.sub.V{tilde over (V)}, where F.sub.V.sup.SL contains L vectors of size S1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed delay-domain matrix {tilde over (V)}=[{tilde over (v)}.sub.1, {tilde over (v)}.sub.2, . . . , {tilde over (v)}.sub.r] is given by Lr, and L is the number of delays, and the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.SO.sub.f.sub.1], where ${\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.S}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(S-1\right)}{O_f.Math.S}}\right]}^T{}^{S1},$ i{0, . . . , SO.sub.f1}, and O.sub.f{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; and reporting the explicit CSI comprising the identified first set of r basis vectors, the second reduced-sized delay-domain set of r basis vectors, along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook from the communication device to the transmitter.

31. A method for transmitting in a wireless communication system comprising a communication device a communication device of claim 1 and a transmitter, the method comprising: transmitting, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS configuration, and downlink signals comprising the CSI-RS configuration; receiving, at the transmitter, uplink signals comprising a plurality of CSI reports comprising an explicit CSI from the communication device; constructing a precoder matrix for a precoder connected to an antenna array comprising a plurality of antennas; applying the precoder matrix on antenna ports using the explicit CSI so as to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams.

32. A non-transitory digital storage medium having a computer program stored thereon to perform the method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising: receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal comprising downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, constructing a frequency-domain channel tensor using the CSI estimate, performing a high-order principal component analysis, HO-PCA, on the channel tensor, identifying a plurality of dominant principal components of the channel tensor, thereby acquiring a compressed channel tensor, and reporting the explicit CSI comprising the dominant principal components of the channel tensor from the communication device to the transmitter; when said computer program is run by a computer.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Embodiments of the present invention will be detailed subsequently referring to the appended drawings, in which:

[0062] FIG. 1 shows a schematic representation of an example of a wireless communication system;

[0063] FIG. 2 shows a block-based model of a MIMO DL transmission using codebook-based-precoding in accordance with LTE release 8;

[0064] FIG. 3 is a schematic representation of a wireless communication system for communicating information between a transmitter, which may operate in accordance with the inventive teachings described herein, and a plurality of receivers, which may operate in accordance with the inventive teachings described herein;

[0065] FIG. 4 is a flow diagram illustrating a HO-PCA decomposition/compression of a channel tensor, the reporting at a UE and the reconstruction of the channel tensor at the gNB in accordance with an embodiment of the present invention;

[0066] FIG. 5 illustrates a frequency-domain channel tensor (three-dimensional array) of dimension N.sub.rN.sub.tS;

[0067] FIG. 6 is a flow diagram illustrating a HO-PCA decomposition/compression of a channel tensor in combination with delay-domain compression, the reporting at the UE and the reconstruction of the channel tensor at gNB in accordance with an embodiment of the present invention;

[0068] FIG. 7 is a flow diagram illustrating a PCA decomposition/compression of a channel matrix, the reporting at a UE and the reconstruction of the channel matrix at the gNB in accordance with an embodiment of the present invention;

[0069] FIG. 8 illustrates a frequency-domain channel matrix (two-dimensional array) H of dimension NS, where N=N.sub.tN.sub.r;

[0070] FIG. 9 is a flow diagram illustrating a transformation/compression of a channel tensor in combination with a HO-PCA decomposition, the reporting at the UE and the reconstruction of the channel tensor at gNB in accordance with an embodiment of the present invention;

[0071] FIG. 10 is a flow diagram illustrating a transformation/compression of a channel matrix in combination with a non-HO PCA decomposition, the reporting at the UE and the reconstruction of channel tensor at the gNB in accordance with an embodiment of the present invention;

[0072] FIG. 11 is a flow diagram illustrating a HO-PCA decomposition/compression of a four-dimensional channel tensor, the reporting at the UE and the reconstruction of channel tensor at the gNB in accordance with an embodiment of the present invention;

[0073] FIG. 12 illustrates a frequency domain channel matrix of size NSD, where N=N.sub.tN.sub.r;

[0074] FIG. 13 is a flow diagram illustrating a HO-PCA decomposition/compression of a four-dimensional channel tensor in combination with a delay- or time/doppler domain compression, the reporting at the UE and the reconstruction the of channel tensor at the gNB in accordance with an embodiment of the present invention;

[0075] FIG. 14 is a flow diagram illustrating a transformation/compression of a four-dimensional channel tensor in addition to a HO-PCA decomposition, the reporting at the UE and the reconstruction of the channel tensor at the gNB in accordance with an embodiment of the present invention;

[0076] FIG. 15 is a flow diagram illustrating a transformation/compression of a four-dimensional channel tensor in addition to non-HO-PCA decomposition, the reporting at the UE and the reconstruction of the channel matrix at the gNB in accordance with an embodiment of the present invention; and

[0077] FIG. 16(a) illustrates a CSI-RS with a periodicity of 10 slots and no repetition (CSI-RS-BurstDuration not configured or CSI-RS-BurstDuration=0);

[0078] FIG. 16(b) illustrates a CSI-RS with a periodicity of 10 slots and repetition of 4 slots (CSI-RS-BurstDuration=4);

[0079] FIG. 17 illustrates a CSI-RS-BurstDuration information element in accordance with an embodiment;

[0080] FIG. 18 illustrates an example of a computer system on which units or modules as well as the steps of the methods described in accordance with the inventive approach may execute.

DETAILED DESCRIPTION OF THE INVENTION

[0081] In the following, embodiments of the present invention are described in further detail with reference to the enclosed drawings in which elements having the same or similar function are referenced by the same reference signs.

[0082] Embodiments of the present invention may be implemented in a wireless communication system or network as depicted in FIG. 1 or FIG. 2 including transmitters or transceivers, like base stations, and receivers or users, like mobile or stationary terminals or IoT devices, as mentioned above. FIG. 3 is a schematic representation of a wireless communication system for communicating information between a transmitter 200, like a base station, and a plurality of receivers 202.sub.1 to 202.sub.n, like UEs, which are served by the base station 200. The base station 200 and the UEs 202 may communicate via a wireless communication link or channel 204, like a radio link. The base station 200 includes one or more antennas ANT.sub.T or an antenna array having a plurality of antenna elements, and a signal processor 200a. The UEs 202 include one or more antennas ANT.sub.R or an antenna array having a plurality of antennas, a signal processor 202a.sub.1, 202a.sub.n, and a transceiver 202b.sub.1, 202b.sub.n. The base station 200 and the respective UEs 202 may operate in accordance with the inventive teachings described herein.

Communication Device

[0083] The present invention provides a communication device 202 for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising a transceiver 202b configured to receive, from a transmitter 200 a radio signal via a radio time-variant frequency MIMO channel 204, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, and a processor 202a configured to [0084] estimate the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, [0085] construct a frequency-domain channel tensor using the CSI estimate, [0086] perform a higher-order principal component analysis, HO-PCA, on the channel tensor, [0087] identify a plurality of dominant principal components of the channel tensor, thereby obtaining a compressed channel tensor, and [0088] report to the transmitter 200 the explicit CSI comprising the dominant principal components of the channel tensor.

[0089] In accordance with embodiments, the communication device is configured to receive from the transmitter an explicit CSI report configuration containing a CSI channel-type information, CSI-Ind, indicator for the CSI report, wherein the CSI-Ind indicator is associated with a channel-type configuration, the channel tensor is a three-dimensional, 3D, channel tensor, or is represented by a 3D channel covariance tensor, a 3D beamformed-channel tensor, or a 3D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and wherein the plurality of dominant principal components of the compressed 3D channel tensor of dimension N.sub.rN.sub.tS comprise: [0090] a first set of r.sub.1 basis vectors contained in a matrix .sub.R=[u.sub.R,1, . . . , u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1; [0091] a second set of r.sub.2 basis vectors contained in a matrix .sub.T=[u.sub.T,1, . . . , u.sub.T,r.sub.2].sup.N.sup.t.sup.r.sup.2; [0092] a third set of r.sub.3 basis vectors contained in a matrix .sub.S=[u.sub.S,1, . . . , u.sub.S,r.sub.3].sup.Sr.sup.3; and [0093] r.sub.1r.sub.2r.sub.3associated high-order singular values s.sub.ijk, sorted such that .sub.j,k|s.sub.i,j,k|.sup.2.sub.j,k|s.sub.i+1,j,k|.sup.2, .sub.i,k|s.sub.i,j,k|.sup.2.sub.i,k|s.sub.i,j+1,k|.sup.2, and .sub.i,j|.sup.2.sub.i,j|s.sub.i,j,k+1|.sup.2 for all i,j,k.

[0094] In accordance with embodiments, the values of r.sub.1, r.sub.2, and r.sub.3 representing the number of dominant principal components with respect to the first, second and third dimension of the compressed 3D channel tensor, respectively, are [0095] configured via the CSI report configuration by the transmitter, or [0096] reported by the communication device in the CSI report, or [0097] pre-determined and known at the communication device.

[0098] In accordance with embodiments, the processor is configured to quantize the coefficients in the vectors u.sub.R,i, u.sub.T,j, and u.sub.S,k and the HO singular values s.sub.ijk of the 3D channel tensor using a codebook approach, the number of complex coefficients to be quantized being given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3 for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r.sub.1r.sub.2r.sub.3 for the higher-order singular values, respectively.

[0099] In accordance with embodiments, the channel tensor is a four-dimensional, 4D, channel tensor, or represented either by a 4D channel covariance tensor, a 4D beamformed-channel tensor, or a 4D beamformed channel covariance tensor, as indicated by the CSI-Ind indicator, and the plurality of dominant principal components of the compressed 4D channel tensor of dimension N.sub.rN.sub.tSD comprise: [0100] a first set of r.sub.1 basis vectors contained in a matrix .sub.R [u.sub.R,1, . . . , u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1; [0101] a second set of r.sub.2 basis vectors contained in a matrix .sub.T=[u.sub.T,1, . . . , u.sub.T,r.sub.2].sup.N.sup.t.sup.r.sup.2; [0102] a third set of r.sub.3 basis vectors contained in a matrix .sub.S=[u.sub.S,1, . . . , u.sub.S,r.sub.3].sup.Sr.sup.3; [0103] a fourth set of r.sub.4 basis vectors contained in a matrix .sub.D=[u.sub.D,1, . . . , u.sub.D,r.sub.4].sup.Dr.sup.4; and [0104] r.sub.1r.sub.2r.sub.3r.sub.4 associated high-order singular values s.sub.ijkl, sorted such that .sub.j,k,l|s.sub.i,j,k,l|.sup.2.sub.j,k,l|s.sub.i+1,j,k,l|.sup.2,|s.sub.i,j,k,l|.sup.2.sub.i,k,l|.sup.2,.sub.i,j,l|s.sub.i,j,k,l|.sup.2.sub.i,j,l|s.sub.i,j,k+1,l|.sup.2, and .sub.i,j,k|.sup.2.sub.i,j,k|s.sub.i,j,k,l+1|.sup.2 for all i, j, k, l.

[0105] In accordance with embodiments, the values of r.sub.1, r.sub.2, r.sub.3, and r.sub.4 representing the number of dominant principal components of the 4D channel tensor are configured via the CSI report configuration by the transmitter, or they are reported by the communication device in the CSI report, or they are pre-determined and known at the communication device.

[0106] In accordance with embodiments, the processor is configured to quantize the coefficients of the vectors u.sub.R,i, u.sub.T,j, u.sub.S,k, u.sub.D,l and the singular values s.sub.ijkl using a codebook approach, a number of complex coefficients to be quantized being given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3+Dr.sub.4 for the higher-order singular vectors, and a number of real coefficients to be quantized being given by r.sub.1r.sub.2r.sub.3r.sub.4 for the higher-order singular values, respectively.

[0107] In accordance with embodiments, the explicit CSI comprises a delay-domain CSI for the higher-order singular-value matrix .sub.S, and wherein the processor is configured to calculate a reduced-sized delay-domain higher-order singular-matrix .sub.S from the frequency-domain higher-order singular-matrix .sub.S, wherein the delay-domain higher-order singular-matrix is given by

.sub.SF.sub.SS,

where [0108] F.sub.S.sup.SL contains L vectors of size S1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed delay-domain matrix .sub.S is given by Lr.sub.3, and L is the number of delays, and [0109] the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.SO.sub.f.sub.1], where d.sub.i=

[00003]${\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.S}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(S-1\right)}{O_f.Math.S}}\right]}^T{}^{S1},$

i{0, . . . , SO.sub.f1}, and O.sub.f{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix;
quantize the coefficients in vectors .sub.S=[.sub.S,1, . . . , .sub.S,r.sub.3].sup.Lr.sup.3 using a codebook approach;
report to the transmitter the explicit CSI containing the coefficients of .sub.S instead of .sub.S, along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook .

[0110] In accordance with embodiments, the explicit CSI comprises a Doppler-frequency domain CSI for the higher-order singular-value matrices .sub.D, and wherein the processor is configured to

calculate a reduced-sized Doppler-frequency domain higher-order singular-matrix U from the time-domain higher-order singular-matrix .sub.D, wherein the Doppler-frequency domain higher-order singular-matrix is given by

.sub.DF.sub.D.sub.D,

where [0111] F.sub.D.sup.DG contains G vectors of size D1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed Doppler-frequency domain matrix .sub.D is given by Gr.sub.4, and G is the number of Doppler-frequency components, and [0112] the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.DO.sub.t.sub.1], where d.sub.i=

[00004]${\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.D}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(D-1\right)}{O_f.Math.D}}\right]}^T{}^{D1},$

i{0, . . . , O.sub.tD1}, and O.sub.t{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix;
quantize the coefficients in vectors .sub.D=[.sub.D,1, . . . , .sub.D,r.sub.4].sup.Gr.sup.4 using a codebook approach; or
report to the transmitter the explicit CSI report containing the coefficients of .sub.D instead of .sub.D, along with the G Doppler-frequency components, represented by a set of indices that correspond to the selected DFT-vectors in the codebook .

[0113] The present invention provides a communication device 202 for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising a transceiver 202b configured to receive, from a transmitter 200 a radio signal via a radio time-variant frequency MIMO channel 204, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and a processor 202a configured to [0114] estimate the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots, [0115] construct a frequency-domain channel tensor using the CSI estimate, [0116] calculate a transformed channel tensor using the channel tensor, [0117] rewrite the transformed channel tensor to a transformed channel matrix, [0118] perform a standard principal component analysis, PCA, on the transformed channel matrix, [0119] identify a plurality of dominant principal components of the transformed channel matrix, [0120] thereby obtaining a transformed/compressed channel matrix, and [0121] report to the transmitter the explicit CSI comprising the plurality of dominant principal components of the transformed/compressed channel tensor.

[0122] In accordance with embodiments, the communication device is configured to receive from the transmitter an explicit CSI report configuration containing a CSI channel-type information, CSI-Ind, indicator for the CSI report, wherein the CSI-Ind indicator is associated with a channel-type configuration, the channel tensor is a three-dimensional, 3D, channel tensor, or is represented either by a 3D channel covariance tensor, a 3D beamformed-channel tensor, or a 3D beamformed channel covariance tensor as indicated by the CSI-Ind indicator, or the channel tensor is a four-dimensional, 4D, channel tensor, or is represented either by a 4D channel covariance tensor, a 4D beamformed-channel tensor, or a 4D beamformed channel covariance tensor as indicated by the CSI-Ind indicator, and the 3D transformed channel tensor of dimension N.sub.rN.sub.tS, or the 4D transformed channel tensor of dimension N.sub.tN.sub.tSD, is rewritten to a transformed channel matrix of dimension N.sub.tN.sub.rSD, where D=1 for the 3D transformed channel tensor and D>1 for the 4D transformed channel tensor, respectively, and the plurality of dominant principal components of the transformed channel matrix comprise: [0123] a first set of r basis vectors contained in a matrix =[u.sub.1, u.sub.2, . . . , u.sub.r].sup.N.sup.t.sup.N.sup.r.sup.r; [0124] a second set of r basis vectors contained in a matrix V=[v.sub.1, v.sub.2, . . . , v.sub.r].sup.SDr. [0125] a set of r coefficients contained in a diagonal matrix =diag(s.sub.1, s.sub.2, . . . , s.sub.r) .sup.rr with ordered singular values s.sub.i (s.sub.1s.sub.2, . . . , s.sub.r) on its diagonal.

[0126] In accordance with embodiments, the value of r representing the number of dominant principal components of the transformed channel matrix is [0127] configured via the CSI report configuration by the transmitter, or [0128] reported by the communication device in the CSI report, or [0129] pre-determined and known at the communication device.

[0130] In accordance with embodiments, the processor is configured to quantize the coefficients of the basis vectors u.sub.i, v.sub.i, and the singular values s.sub.i of the transformed channel matrix using a codebook approach.

[0131] In accordance with embodiments, the processor is configured to apply,

after the construction of the 3D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the 3D channel tensor, or the frequency dimension of the 3D channel tensor, or both the frequency and space dimensions of the 3D channel tensor, or
after the construction of the 4D channel tensor, a one-dimensional, a two-dimensional, or multi-dimensional transformation/compression of the channel tensor with respect to the space dimension of the channel tensor, or the frequency dimension of the channel tensor, or the time dimension of the channel tensor, or both the frequency and time dimensions of the channel tensor,
so as to exploit a sparse or nearly-sparse representation of the 3D or 4D channel tensor in one or more dimensions.

[0132] In accordance with embodiments, the processor is configured to apply, after the construction of the 3D channel tensor, a transformation/compression with respect to

all three dimensions of the 3D channel tensor of dimension N.sub.rN.sub.tS represented by a (column-wise) Kronecker product as

[00005]$.Math.={}_{\left(1,2,3\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2,3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$

where [0133] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the 3D channel tensor , selected from a codebook matrix .sub.1; [0134] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.t1 with respect to the second dimension of the 3D channel tensor , selected from a codebook matrix .sub.2; [0135] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the 3D channel tensor , selected from a codebook matrix .sub.3; [0136] .sub.n.sub.r.sub.,n.sub.t.sub.,s is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s, and b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s, and [0137] N.sub.r, N.sub.t, and S (N.sub.rN.sub.r, N.sub.tN.sub.t, SS) represents the value of the first, second and third dimension of the transformed/compressed 3D channel tensor , respectively, or
the space dimensions of the 3D channel tensor, represented by

[00006]$.Math.={}_{\left(1,2\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$

where [0138] b.sub.3,s is a transformation vector of all zeros with the s-th element being one, [0139] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a vector of size N.sub.t1 with respect to the second dimension of the 3D channel tensor , selected from a codebook matrix .sub.2, and [0140] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the 3D channel tensor , selected from a codebook matrix .sub.1, N.sub.rN.sub.r, N.sub.tN.sub.t, S=S, or
the frequency dimension of the 3D channel tensor, represented by

[00007]$.Math.={}_{\left(3\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r}.Math.b_{1,n_t},$

where [0141] b.sub.2,n.sub.r is a transformation vector of all zeros with the n.sub.r-th element being one, b.sub.1,n.sub.t is a vector of all zeros with the n.sub.t-th element being one, and [0142] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the 3D channel tensor , selected from a codebook matrix .sub.3, N.sub.r=N.sub.r and N.sub.t=N.sub.t, SS.

[0143] In accordance with embodiments, (i) the 3D transformation function .sub.(1,2,3))(.Math.) is given by a two-dimensional Discrete Cosine transformation (2D-DCT) and a 1D-DFT transformation with respect to the space and frequency dimension of the channel tensor, respectively, the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices, and the codebook matrix .sub.3 is given by an oversampled DFT matrix, or (ii) the 3D transformation function .sub.(1,2,3)(.Math.) is given by a 3D-DFT transformation and the codebook matrices .sub.n, n=1,2,3 are given by oversampled DFT matrices, or (iii) the 2D transformation function .sub.(1,2)(.Math.) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices, or (iv) the 1D transformation function .sub.(3)(.Math.) is given by a 1D-DFT transformation and the codebook matrix .sub.3 is given by an oversampled DFT matrix.

[0144] In accordance with embodiments, the processor is configured to apply, after the construction of the 4D channel tensor a transformation/compression with respect to

all four dimensions of the channel tensor, represented by a (column-wise) Kronecker product as =.sub.(1,2,3,4)(),

[00008]$\mathrm{vec}\left(\right)=\mathrm{ec}\left({}_{\left(1,2,3,4\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{.Math.}_{n_r,n_t,s,d}.Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$

where [0145] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1; [0146] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2; [0147] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3; [0148] b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor , selected from a codebook matrix .sub.4; [0149] .sub.n.sub.r.sub.,n.sub.t.sub.,s,d is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d, and b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d and [0150] N.sub.r, N.sub.t, S and D (N.sub.rN.sub.r, N.sub.tN.sub.t, SS, DD) represents the value of the first, second, third and fourth dimension of the transformed/compressed channel tensor , respectively, or
the space dimensions of the 4D channel tensor, represented by =.sub.(1,2)(),

[00009]$\mathrm{vec}\left(\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{.Math.}_{n_r,n_t,s,d}.Math.b_{4,d}.Math.b_{3,s}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$

where [0151] b.sub.4,d is a transformation vector of all zeros with the d-th element being one, [0152] b.sub.3,s is a transformation vector of all zeros with the s-th element being one, [0153] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2, and [0154] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1 and N.sub.rN.sub.r, N.sub.tN.sub.t, S=S and D=D, or
the frequency and time dimensions of the 4D channel tensor, represented by =.sub.(3,4)(),

[00010]$\mathrm{vec}\left(\right)=\mathrm{ec}\left({}_{\left(3,4\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{.Math.}_{n_r,n_t,s,d}.Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r}.Math.b_{1,n_t},$

where [0155] b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor ,selected from a codebook matrix .sub.4, [0156] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3, [0157] b.sub.2,n.sub.r is a transformation vector of all zeros with the n.sub.r-th element being one, and [0158] b.sub.1,n.sub.t is a transformation vector of all zeros with the n.sub.t-th element being one and N.sub.r=N.sub.r N.sub.t=N.sub.t, SS and DD.

[0159] In accordance with embodiments, (i) the 4D transformation function .sub.(1,2,3,4)(.Math.) is given by a 4D-DFT transformation and the codebook matrices .sub.n, n=1,2,3,4 are given by oversampled DFT matrices, or (ii) the 2D transformation/compression function .sub.(3,4)(.Math.) is given by a 2D-DFT and the codebook matrices .sub.n, n=3,4 are given by oversampled DFT matrices, or (iii) the 2D transformation function .sub.(1,2)(.Math.) is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices, or (iv) the 1D transformation function .sub.(3)(.Math.) is given by a 1D-DFT transformation and the codebook matrix .sub.3 is given by an oversampled DFT matrix, or (v) the 1D transformation function .sub.(4)(.Math.) is given by a 1D-DFT transformation and the codebook matrix .sub.4 is given by an oversampled DFT matrix.

[0160] In accordance with embodiments, the processor is configured to

select the transformation vectors from codebook matrices Q.sub.n, and store the selected indices in a set of g-tuples, where g refers to the number of transformed dimensions of the channel tensor, and
report the set as a part of the CSI report to the transmitter.

[0161] In accordance with embodiments, the oversampling factors of the codebooks are [0162] configured via the CSI report configuration, or via higher layer or physical layer by the transmitter, [0163] pre-determined and known at the communication device.

[0164] The present invention provides a communication device 202 for providing an explicit channel state information, CSI, feedback in a wireless communication system, the communication device comprising a transceiver 202b configured to receive, from a transmitter 200 a radio signal via a radio time-variant frequency MIMO channel 204, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and a processor 202a configured to [0165] estimate the CSI using measurements on the downlink reference signals of the radio channel, [0166] construct a frequency-domain channel matrix using the CSI estimate, [0167] perform a standard principal component analysis, PCA, on the channel matrix, [0168] identify the dominant principal components of the channel matrix, the channel matrix comprising [0169] a first set of r basis vectors contained in a matrix =[u.sub.1, u.sub.2, . . . , u.sub.r].sup.N.sup.t.sup.N.sup.r.sup.r; [0170] a second set of r basis vectors contained in a matrix V=[v.sub.1, v.sub.2, . . . , v.sub.r].sup.Sr; and [0171] a third set of r coefficients contained in a diagonal matrix =diag(s.sub.1, s.sub.2, . . . , s.sub.r).sup.rr with ordered singular values s.sub.i (s.sub.1s.sub.2 . . . s.sub.r) on its diagonal; [0172] calculate a reduced-sized delay-domain matrix {tilde over (V)} from the frequency-domain matrix V, wherein the delay-domain matrix, comprising r basis vectors, is given by

VF.sub.V{tilde over (V)},

where [0173] F.sub.V.sup.SL contains L vectors of size S1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed delay-domain matrix {tilde over (V)}=[{tilde over (v)}.sub.1, {tilde over (v)}.sub.2, . . . , {tilde over (v)}.sub.r] is given by Lr, and L is the number of delays, and [0174] the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.SO.sub.f.sub.1], where

[00011]${\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.S}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(S-1\right)}{O_f.Math.S}}\right]}^T{}^{S1},$

i{0, . . . , SO.sub.f1}, and O.sub.f{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; and [0175] report to the transmitter the explicit CSI containing the identified first set of r basis vectors, the second reduced-sized delay-domain set of r basis vectors, along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook .

[0176] In accordance with embodiments, the value of r representing the number of dominant principal components of the transformed channel matrix is configured via the CSI report configuration by the transmitter, or it is reported by the communication device in the CSI report, or it is pre-determined and known at the communication device.

[0177] In accordance with embodiments, the processor is configured to quantize the coefficients of the basis vectors u.sub.i, {tilde over (v)}.sub.i, and the singular values s.sub.i of the channel matrix using a codebook approach.

[0178] In accordance with embodiments, the communication device is configured with [0179] one or more scalar codebooks for the quantization of each entry of each basis vector of the plurality of dominant principal components of the channel tensor or the compressed channel tensor and the singular values or high order singular values, or [0180] with one or more unit-norm vector codebooks for the quantization of each basis vector of the plurality of dominant principal components of the channel tensor or the compressed channel tensor, and wherein the communication device selects for each basis vector a codebook vector to represent the basis vector, and
the communication device is configured to report the indices corresponding to the selected entries in the scalar or vector codebook as a part of the CSI report to the transmitter.

Transmitter

[0181] The present invention provides a transmitter 200 in a wireless communication system, the transmitter comprising: [0182] an antenna array ANT.sub.T having a plurality of antennas for a wireless communication with one or more of the inventive communication devices 202a, 202b for providing a channel state information, CSI, feedback to the transmitter 200; and a precoder 206 connected to the antenna array ANT.sub.T, the precoder 206 to apply a set of beamforming weights to one or more antennas of the antenna array ANT.sub.T to form, by the antenna array ANT.sub.T, one or more transmit beams or one or more receive beams,
a transceiver 200b configured to [0183] transmit, to the communication device 202a, 202b, downlink reference signals (CSI-RS) according to a CSI-RS, and downlink signals comprising the CSI-RS configuration; and [0184] receive uplink signals comprising a plurality of CSI reports including an explicit CSI from the communication device 202a, 202b; and
a processor 200a configured to construct a precoder matrix applied on the antenna ports using the explicit CSI.

System

[0185] The present invention provides a wireless communication network, comprising at least one of the inventive communication devices 202a, 202b, and at least one of the inventive transmitters 200.

[0186] In accordance with embodiments, the communication device and the transmitter comprises one or more of a mobile terminal, or stationary terminal, or cellular IoT-UE, or an IoT device, or a ground based vehicle, or an aerial vehicle, or a drone, or a moving base station, or road side unit, or a building, or a macro cell base station, or a small cell base station, or a road side unit, or a UE, or a remote radio head, or an AMF, or a SMF, or a core network entity, or a network slice as in the NR or 5G core context, or any transmission/reception point (TRP) enabling an item or a device to communicate using the wireless communication network, the item or device being provided with network connectivity to communicate using the wireless communication network.

[0187] Methods

[0188] The present invention provides a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising:

receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration, estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots,
constructing a frequency-domain channel tensor using the CSI estimate,
performing a high-order principal component analysis, HO-PCA, on the channel tensor,
identifying a plurality of dominant principal components of the channel tensor, thereby obtaining a compressed channel tensor, and
reporting the explicit CSI comprising the dominant principal components of the channel tensor from the communication device to the transmitter.

[0189] The present invention provides a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising:

receiving, from a transmitter a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration,
estimating the CSI using measurements on the downlink reference signals of the radio channel according to the reference signal configuration over one or more time instants/slots,
constructing a frequency-domain channel tensor using the CSI estimate,
calculating a transformed channel tensor using the channel tensor,
rewriting the transformed channel tensor to a transformed channel matrix,
performing a standard principal component analysis, PCA, on the transformed channel matrix,
identifying a plurality of dominant principal components of the transformed channel matrix, thereby obtaining a transformed/compressed channel matrix, and
reporting the explicit CSI comprising the plurality of dominant principal components of the transformed/compressed channel tensor from the communication device to the transmitter.

[0190] The present invention provides a method for providing by a communication device in a wireless communication system an explicit channel state information, CSI, feedback, the method comprising:

receiving, from a transmitter, a radio signal via a radio time-variant frequency MIMO channel, the radio signal including downlink reference signals according to a reference signal configuration, and downlink signals comprising the reference signal configuration; and
estimating the CSI using measurements on the downlink reference signals of the radio channel,
constructing a frequency-domain channel matrix using the CSI estimate,
performing a standard principal component analysis, PCA, on the channel matrix,
identifying the dominant principal components of the channel matrix, the channel matrix comprising [0191] a first set of r basis vectors contained in a matrix =[u.sub.1,u.sub.2, . . . ,u.sub.r].sup.N.sup.t.sup.N.sup.r.sup.r; [0192] a second set of r basis vectors contained in a matrix V=[v.sub.1, v.sub.2, . . . , v.sub.r].sup.Sr; and [0193] a third set of r coefficients contained in a diagonal matrix =diag(s.sub.1, s.sub.2, . . . , s.sub.r).sup.rr with ordered singular values s.sub.i (s.sub.1s.sub.2 . . . s.sub.R) on its diagonal; [0194] calculating a reduced-sized delay-domain matrix {tilde over (V)} from the frequency-domain matrix V, wherein the delay-domain matrix, comprising r basis vectors, is given by

VF.sub.V{tilde over (V)}, [0195] where [0196] F.sub.V.sup.SL contains L vectors of size S1, selected from a discrete Fourier transform, DFT, codebook , the size of the compressed delay-domain matrix {tilde over (V)}=[{tilde over (v)}.sub.1, {tilde over (v)}.sub.2, . . . , {tilde over (v)}.sub.r] is given by Lr, and L is the number of delays, and [0197] the oversampled codebook matrix is given by =[d.sub.0, d.sub.1, . . . , d.sub.SO.sub.f.sub.1], where

[00012]${\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.S}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(S-1\right)}{O_f.Math.S}}\right]}^T{}^{S1},$

i{0, . . . , SO.sub.f1}, and O.sub.f{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix; and [0198] reporting the explicit CSI containing the identified first set of r basis vectors, the second reduced-sized delay-domain set of r basis vectors, along with the L delays, represented by a set of indices that correspond to the selected DFT-vectors in the codebook from the communication device to the transmitter.

[0199] The present invention provides a method for transmitting in a wireless communication system including a communication device a communication device of any one of the claims 1 to 14 and a transmitter, the method comprising:

transmitting, to the communication device, downlink reference signals (CSI-RS) according to a CSI-RS configuration, and downlink signals comprising the CSI-RS configuration;
receiving, at the transmitter, uplink signals comprising a plurality of CSI reports including an explicit CSI from the communication device;
constructing a precoder matrix for a precoder connected to an antenna array having a plurality of antennas;
applying the precoder matrix on antenna ports using the explicit CSI so as to apply a set of beamforming weights to one or more antennas of the antenna array to form, by the antenna array, one or more transmit beams or one or more receive beams.

Computer Program Product

[0200] The present invention provides a computer program product comprising instructions which, when the program is executed by a computer, causes the computer to carry out one or more methods in accordance with the present invention.

[0201] Thus, the present invention provides several low feedback overhead approaches for explicit CSI reporting based on channel transformations and compression techniques, and embodiments relate to wireless communications systems and, more specifically, to frequency-domain, delay-domain, time domain or mixed frequency/delay and time/Doppler-frequency domain explicit CSI feedback to represent a downlink channel between a gNB and a single UE in the form of a channel tensor or matrix, a beamformed channel tensor or matrix, a covariance channel tensor or matrix, dominant eigenvectors of a channel tensor or matrix or dominant eigenvectors of a beamformed channel tensor or matrix. In the following, several embodiments of low feedback overhead approaches for explicit (frequency-domain, delay-domain, time domain or mixed frequency/delay and time/Doppler-frequency domain) CSI reporting based on combinations of channel transformations and compression techniques will be described.

High Order PCA on Frequency-Domain Channel Tensor

[0202] In accordance with a first embodiment 1, a UE is configured to report explicit CSI Type I that represents a compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed channel covariance tensor over the configured subbands (SB), PRBs or subcarriers, according to the following sub-embodiments. Here, an SB corresponds to a set of consecutive PRBs. For example, for a bandwidth part of 10 MHz, 6 subbands each having 8 PRBs are configured.

[0203] The compressed CSI is based on a high-order principal component analysis (HO-PCA) of the channel tensor to exploit the correlations of the channel tensor in the space- and frequency-domains.

[0204] An illustration of this approach is shown in FIG. 4. The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a three-dimensional (3D) frequency-domain channel tensor (a three-dimensional array) .sup.N.sup.r.sup.N.sup.t.sup.s of dimension N.sub.rN.sub.tS, where S is the number of subbands, PRBs, or subcarriers (see FIG. 5). The definition of N.sub.t and N.sub.r is dependent on the configuration of the CSI type: [0205] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 for CSI type configuration channel tensor, N.sub.t=2N.sub.1N.sub.2, and N.sub.r is the number of UE receive antenna ports; [0206] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 at the gNB, N.sub.t=2N.sub.1N.sub.2, N.sub.r=2N.sub.1N.sub.2, for CSI type configuration channel covariance tensor; [0207] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of UE receive antenna ports for CSI type configuration beamformed-channel tensor; [0208] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U beams 2U and N.sub.r is the number of beamformed antenna ports/beams N.sub.t=2U beams 2U for CSI type configuration beamformed-channel covariance tensor;

[0209] Then, the UE performs a HO-PCA on the channel tensor , such that is represented by

[00013]$=\underset{i=1}{\overset{N_r}{.Math.}}.Math.\underset{j=1}{\overset{N_t}{.Math.}}.Math.\underset{k=1}{\overset{S}{.Math.}}.Math.s_{\mathrm{ijk}}\left(u_{R,i}{u}_{T,j}{u}_{S,k}\right),$

where [0210] U.sub.R=[U.sub.R,1, . . . ,U.sub.R,N.sub.r].sup.N.sup.r.sup.N.sup.r is matrix containing the higher order singular vectors with respect to the first dimension of the channel tensor ; [0211] U.sub.T=[u.sub.T,1, . . . ,u.sub.T,N.sub.t].sup.N.sup.t.sup.N.sup.t is matrix containing the higher order singular vectors with respect to the second dimension of the channel tensor ; [0212] U.sub.S=[u.sub.S,1, . . . , u.sub.S,R].sup.SR is a matrix containing the higher order singular vectors with respect to the frequency dimension (third dimension of the channel tensor ) with R=min(S, N.sub.tN.sub.r); [0213] S.sub.ijk are the higher order singular values, sorted as s.sub.ijks.sub.ijk with ii, jj, kk.

[0214] Moreover, the symbol represents the outer product operator which is the generalization of the outer product of two vectors (giving rise to a matrix that contains the pairwise products of all the elements of the two vectors) to multi-way matrices/tensors. The outer product of an R-way tensor (i.e., a matrix that is indexed by R indices and a P-way tensor is a (R+P)-way tensor containing all pairwise products of all the elements of and . Note that vectors and matrices may be seen as 1-way and 2-way tensors respectively. Therefore, the outer product between two vectors is a matrix, the outer product between a vector and a matrix is a 3-way tensor, and so on.

[0215] To reduce the number of channel coefficients, the channel tensor i is approximated by (r.sub.1,r.sub.2,r.sub.3), (1r.sub.1N.sub.r, 1r.sub.2N.sub.t, 1r.sub.3S) dominant principal components with respect to the first, second and third dimensions and the corresponding left, right and lateral singular matrices, respectively. The compressed explicit frequency-domain channel tensor (explicit CSI) is given by

[00014]${}_c=\underset{i=1}{\overset{r_1}{.Math.}}.Math.\underset{j=1}{\overset{r_2}{.Math.}}.Math.\underset{k=1}{\overset{r_3}{.Math.}}.Math.s_{\mathrm{ijk}}\left(u_{R,i}{u}_{T,j}{u}_{S,k}\right).$

where

.sub.R=[u.sub.R,1, . . . ,u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1,

.sub.T=[u.sub.T,1, . . . ,u.sub.T,r.sub.2].sup.N.sup.t.sup.r.sup.2,

.sub.S=[u.sub.S,1, . . . ,u.sub.S,r.sub.3].sup.Sr.sup.3.

[0216] To report the compressed frequency-domain channel tensor (explicit CSI) from the UE to the gNB, the UE quantizes the coefficients of the vectors u.sub.R,i, u.sub.T,j, and u.sub.S,k and the HO singular values s.sub.ijk using a codebook approach.

[0217] The gNB reconstructs the compressed channel tensor as

[00015]${}_c=\underset{i=1}{\overset{r_1}{.Math.}}.Math.\underset{j=1}{\overset{r_2}{.Math.}}.Math.\underset{k=1}{\overset{r_3}{.Math.}}.Math.s_{\mathrm{ijk}}\left(u_{R,i}{u}_{T,j}{u}_{S,k}\right).$

[0218] The number of complex coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3 for the higher-order singular vectors and number of real coefficients that need to be quantized for the frequency-domain HO-PCA approach is r.sub.1r.sub.2r.sub.3, for the higher-order singular values, respectively. In comparison, for the standard (non-high order) PCA, it is needed to quantize N.sub.rN.sub.tr+Sr+r values for the singular vectors and the singular values with r=min(N.sub.rN.sub.t, S). For small values of (r.sub.1, r.sub.2, r.sub.3) (low rank approximation of the channel tensor), the compression achieved by the HO-PCA is higher than the compression achieved by the standard non-HO PCA approach.

[0219] In one method, the values of (r.sub.1, r.sub.2, r.sub.3), representing the number of dominant principal components with respect to the first, second and third dimension of the channel tensor, respectively, are configured via higher layer (e.g., RRC, or MAC-CE) signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r.sub.1,r.sub.2,r.sub.3) as a part of the CSI report, or they are known at the UE.

[0220] In accordance with a sub-embodiment 1-1 of the first embodiment 1, as illustrated in FIG. 6, a UE is configured to report explicit CSI Type I with delay-domain CSI for the higher-order singular-value matrix Us In this configuration, the UE calculates an approximated reduced-sized (compressed) delay-domain higher-order singular-matrix .sub.S from the frequency-domain higher-order singular-matrix .sub.S. The delay-domain higher-order singular-matrix is given by

.sub.SF.sub.S.sub.S,

where F.sub.S.sup.SL is non-square, or square matrix of size SL consisting of L discrete Fourier transform (DFT) vectors. The size of the compressed delay-domain matrix .sub.S is given by Lr.sub.3. A compression is achieved when L<S.

[0221] The DFT vectors in F.sub.S are selected from a oversampled DFT-codebook matrix =[d.sub.0, d.sub.1, . . . , d.sub.SO.sub.f.sub.1], where

[00016]$d_i={\left[1.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i}{O_f.Math.S}}.Math..Math.\mathrm{.Math.}.Math..Math.e^{\frac{-j.Math..Math.2.Math..Math..Math.i\left(S-1\right)}{O_f.Math.S}}\right]}^T{}^{S1},$

i=0, . . . , O.sub.fS1. Here, 0.sub.f{1,2,3, . . . }denotes the oversampling factor of the DFT-codebook matrix. The indices of the vectors in F.sub.S selected from the codebook are stored in a set =(i.sub.1, i.sub.2, . . . , i.sub.L).

[0222] The UE quantizes the coefficients of the vectors in .sub.R=[u.sub.R,1, . . . ,u.sub.R,r.sub.1], .sub.T=[u.sub.T,1, . . . ,u.sub.T,r.sub.2], and .sub.S=[.sub.S,1, . . . , .sub.S,r.sub.3].sup.Lr.sup.3 and the HO singular values s.sub.ijk using a codebook approach, and reports them along with the L delays, represented by a set of indices that correspond to the selected DFT vectors in the codebook , to the gNB. The gNB reconstructs the frequency-domain channel tensor .sub.c according to embodiment 1, where .sub.S is calculated as .sub.S=[u.sub.S,1, . . . , u.sub.S,r.sub.3]=F.sub.S.sub.S.

[0223] In one method, the number of delays L is configured via higher layer (e.g., RRC, or MAC) signaling from the gNB to the UE. In another method, the UE reports the preferred value of L as a part of the CSI report, or it is known at the UE.

[0224] The oversampled factor O.sub.f of the DFT codebook matrix is configured via higher layer (e.g., RRC, or MAC) or physical layer (via the downlink control indicator (DCI)) signaling from the gNB to the UE, or it is known at the UE.

Standard (Non-High-Order) PCA of Frequency-Domain Channel Matrix with Delay-Domain Compression

[0225] In accordance with a second embodiment 2, a UE is configured to report explicit CS Type II that represents a compressed form of a channel matrix, or a beam-formed channel matrix, or a channel covariance matrix, or beam-formed channel covariance matrix over the configured subbands (SB), PRBs or subcarriers, according to the following sub-embodiments.

[0226] The compressed CSI performs a standard non-high-order principal component analysis (non-HO-PCA) on a channel matrix combined with a delay-domain transformation and compression of the channel matrix.

[0227] An illustration of this approach is shown in FIG. 7. The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 2D frequency-domain channel matrix H.sup.N.sup.t.sup.N.sup.r.sup.S of dimension N.sub.tN.sub.rS, where S is the number of subbands, PRBs, or subcarriers (see FIG. 8). The definition of N.sub.t and N.sub.r is dependent on the configuration of the CSI type: [0228] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 for CSI type configuration channel matrix, N.sub.t=2N.sub.1N.sub.2, and N.sub.r is the number of UE receive antenna ports; [0229] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 at the gNB, N.sub.t=2N.sub.1N.sub.2, N.sub.r=2N.sub.1N.sub.2, for CSI type configuration channel covariance matrix; [0230] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of UE receive antenna ports for CSI type configuration beamformed-channel matrix; [0231] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of beamformed antenna ports/beams N.sub.r=2U for CSI type configuration beamformed-channel covariance matrix;

[0232] The UE applies a standard PCA-decomposition to the frequency-domain channel matrix H, represented by

[00017]$H=U.Math..Math.V^H=\underset{i=1}{\overset{R}{.Math.}}.Math.s_i.Math.u_i.Math.v_i^H,$

where [0233] U=[u.sub.1, u.sub.2, . . . , u.sub.R] is the N.sub.tN.sub.rR left-singular matrix; [0234] V=[v.sub.1, v.sub.2, . . . , v.sub.R] is the SR right-singular matrix; [0235] is a RR diagonal matrix with ordered singular values s.sub.i (s.sub.1s.sub.2 . . . S.sub.R) on its main diagonal, and R=min(S, N.sub.tN.sub.t).

[0236] To reduce the number of channel coefficients, the channel matrix H is approximated by r, 1rR dominant principal components. The compressed channel matrix H.sub.c is given by

H.sub.c=V.sup.H

where =[u.sub.1, u.sub.2, . . . , u.sub.r], V=[v.sub.1, v.sub.2, . . . v.sub.r], and =diag(s.sub.1, s.sub.2, . . . , s.sub.r).

[0237] Furthermore, the UE calculates from the frequency-domain right-singular matrix V the corresponding compressed delay-domain right singular matrix. The delay-domain right-singular matrix is approximated by

VF.sub.V{tilde over (V)},

where F.sub.V.sup.SL is a squared, or non-squared DFT matrix of size SL. The size of the compressed delay-domain left-singular matrix is given by Lr. A compression is achieved for L<S.

[0238] The columns of the transformation/compression matrix F.sub.V are selected from a DFT codebook matrix () of dimension SSO.sub.f, where O.sub.f denotes the oversampling factor of the DFT codebook-matrix. The indices of the selected vectors in F.sub.V from the codebook are stored in a set =(i.sub.1, i.sub.2, . . . , i.sub.L).

[0239] The UE quantizes the frequency-domain left singular-matrix , the delay-domain right singular-matrix {tilde over (V)} and the singular values s.sub.1, s.sub.2, . . . , s.sub.r using a codebook approach, and then reports them along with the L delays, represented by the index set to the gNB.

[0240] The gNB reconstructs the explicit CSI as

H.sub.c=(F{tilde over (V)}).sup.H.

[0241] In one method, the number of delays L is configured via higher layer (e.g., RRC, or MAC) signaling, or physical layer (via the downlink control indicator (DCI)) signaling from the gNB to the UE. In another method, the UE reports the preferred value of L as a part of the CSI report, or it is known at the UE.

[0242] In one method, the value of r, representing the number of dominant principal components of the channel matrix is configured via higher layer (e.g., RRC, or MAC-CE) signaling from the gNB to the UE. In another method, the UE reports the preferred value of r as a part of the CSI report, or it is known at the UE.

[0243] The oversampled factor O.sub.f of the DFT codebook matrix is configured via higher layer (e.g., RRC, or MAC) or via DCI physical signaling from the gNB to the UE, or it is known at the UE.

Transformation/Compression of Channel Tensor in Combination with HO-PCA

[0244] In accordance with a third embodiment 3, a UE is configured to report explicit CSI Type III that represents a transformed and compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, or space and frequency dimension of the channel tensor. The CSI combines channel tensor transformation with data compression by exploiting the sparse representation in the delay domain and the correlations of the channel coefficients in the spatial and frequency/delay domains.

[0245] An illustration of this approach is shown in FIG. 9. The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 3D frequency-domain channel tensor .sup.N.sup.r.sup.N.sup.t.sup.S of dimension N.sub.rN.sub.tS, where S is the number of subbands, PRBs, or subcarriers. The definition of N.sub.t and N.sub.r is dependent on the configuration of the CSI type: [0246] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 for CSI type configuration channel tensor, N.sub.t=2N.sub.1N.sub.2, and N.sub.r is the number of UE receive antenna ports; [0247] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 at the gNB, N.sub.t=2N.sub.1N.sub.2, N.sub.r=2N.sub.1N.sub.2, for CSI type configuration channel covariance tensor; [0248] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of UE receive antenna ports for CSI type configuration beamformed-channel tensor; [0249] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of beamformed antenna ports/beams N.sub.r=2U for CSI type configuration beamformed-channel covariance tensor;

[0250] After the construction of the frequency-domain channel tensor , a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) transformation of the channel tensor is applied with respect to the space, frequency, or frequency and space dimensions of the channel tensor. The aim of the transformation is to obtain a sparse or nearly-sparse representation of the channel tensor in one, two, or three dimensions. After the transformation and compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions of the channel tensor is achieved. For example, a transformation/compression with respect to all three dimensions of the channel tensor is represented by a (column-wise) Kronecker product as

[00018]$.Math.={}_{\left(1,2,3\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2,3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$

where [0251] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1; [0252] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2; [0253] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3; [0254] .sub.n.sub.r.sub.,n.sub.t.sub.,s is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s, and b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s, and [0255] N.sub.r, N.sub.t, and S represents the value of the first, second and third dimension of the transformed/compressed channel tensor , respectively.

[0256] The transformed/compressed channel coefficients .sub.n.sub.r.sub.,n.sub.t.sub.,s are used to form the transformed/compressed channel tensor of dimension N.sub.rN.sub.tS, where N.sub.rN.sub.r, N.sub.tN.sub.t, SS.

[0257] For example, a transformation/compression with respect to the two space dimensions of the channel tensor is represented by

[00019]$.Math.={}_{\left(1,2\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$

where b.sub.3,s is a vector of all zeros with the s-th element being one, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2, b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1 and N.sub.rN.sub.rN.sub.tN.sub.t, S=S.

[0258] For example, a transformation/compression with respect to the frequency dimension of the channel tensor is represented by

[00020]$.Math.={}_{\left(3\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r}.Math.b_{1,n_t},$

where b.sub.2,n.sub.r is a vector of all zeros with the n.sub.r-th element being one, b.sub.1,n.sub.t is a vector of all zeros with the n.sub.t-th element being one, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3, and N.sub.r=N.sub.r, N.sub.t=N.sub.t and SS.

[0259] The indices of the selected vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s and b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s from the codebook matrices .sub.n, n=1,2,3 are stored in a set of g-tuples, where g refers to the number of transformed dimensions.

[0260] For example, for g=1, the set is represented by ={i.sub.1, i.sub.2, i.sub.T}, where T denotes the number of selected vectors with respect to the transformed/compressed dimension of the channel tensor . For example, in the case of a transformation/compression with respect to the frequency dimension, T=S.

[0261] For example, for g=3, the set is represented by 3-tuples (i.sub.1,n.sub.r,i.sub.2,n.sub.t,i.sub.3,s), where i.sub.1,n.sub.r is the index associated with vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, i.sub.2,n.sub.t, is the index associated with vector b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s, and i.sub.3,s is the index associated with vector b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s. The set is given by

={(i.sub.1,0,i.sub.2,0,i.sub.3,0), . . . ,(i.sub.1,N.sub.r.sub.1,i.sub.2,N.sub.t.sub.1,i.sub.3,S1)}

[0262] In one method, the size of the set is configured via higher layer (e.g., RRC, or MAC) or physical layer signaling from the gNB to the UE. In another method, the UE reports the preferred size of the set as a part of the CSI report or it is known at the UE.

[0263] The codebook matrices .sub.n are given by matrices .sub.n=[d.sub.n,0, d.sub.n,1, . . . , d.sub.n,TO.sub.n.sub.1], where the parameter O.sub.n denotes the oversampling factor with respect to the n-th dimension where T=N.sub.r for n=1, T=N.sub.t for n=2, and T=S for n=3.

[0264] The oversampled factors O.sub.f,n of the codebook matrices are configured via higher layer (e.g., RRC, or MAC) or via DCI physical layer signaling from the gNB to the UE, or they are known at the UE.

[0265] As an example, the selection of the transformation/compression vectors and transformed/compressed channel coefficients for a transformation/compression with respect to the second and third dimension, can be calculated by

minvec()vec(.sub.(2,3)()).sub.2.sup.2.

The optimization problem may be solved by standard algorithms such as orthogonal matching pursuit. As a result, the indices of the vectors in the transformation matrices selected from the codebooks and the transformed channel coefficients associated with each domain are known.

[0266] After the channel transformation/compression, the UE performs a HO-PCA on the transformed/compressed channel tensor , such that is represented by

[00021]$\hat{}=\underset{i=1}{\overset{N_r^{}}{.Math.}}.Math.\underset{j=1}{\overset{N_t^{}}{.Math.}}.Math.\underset{k=1}{\overset{S^{}}{.Math.}}.Math.s_{\mathrm{ijk}}\left(u_{R,i}{u}_{T,j}{u}_{S,k}\right),$

where [0267] U.sub.R=[u.sub.R,1, . . . , u.sub.R,N.sub.r].sup.N.sup.r.sup.N.sup.r is a matrix containing the high-order singular vectors with respect to the first dimension of the transformed/compressed channel tensor ; [0268] U.sub.T=[u.sub.T,1, . . . , U.sub.T,N.sub.t].sup.N.sup.t.sup.N.sup.t is matrix containing the high-order singular vectors with respect to the second dimension of the transformed/compressed channel tensor ; [0269] U.sub.S=[u.sub.S,1, . . . , U.sub.S,R].sup.SR is a matrix containing the high-order singular vectors with respect to the third dimension of the channel tensor with R=min(S,N.sub.tN.sub.r) [0270] S.sub.ijk are the high-order singular values, sorted as s.sub.ijks.sub.ijk, with ii, jj, kk.

[0271] To further compress the number of channel coefficients, the transformed/compressed channel tensor is approximated by (r.sub.1,r.sub.2,r.sub.3), (1r.sub.1N.sub.r, 1r.sub.2N.sub.t, 1r.sub.3S) dominant principal components with respect to the first, second and third dimensions and the corresponding left, right and lateral singular matrices. The transformed/compressed explicit channel tensor (explicit CSI) is then given by

[00022]${\hat{}}_c=\underset{i=1}{\overset{r_1}{.Math.}}.Math.\underset{j=1}{\overset{r_2}{.Math.}}.Math.\underset{k=1}{\overset{r_3}{.Math.}}.Math.s_{\mathrm{ijk}}\left(u_{R,i}{u}_{T,j}{u}_{S,k}\right).$

where

.sub.R=[u.sub.R,1, . . . ,u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1,

.sub.T=[u.sub.T,1, . . . ,u.sub.T,r.sub.1].sup.N.sup.t.sup.r.sup.2,

.sub.S=[u.sub.S,1, . . . ,u.sub.S,r.sub.3].sup.Sr.sup.3.

[0272] The UE quantizes the coefficients of the vectors U.sub.R,i, U.sub.T,j, U.sub.S,k and the singular values s.sub.ijkl using a codebook approach. The quantized vectors u.sub.R,i, U.sub.T,j, U.sub.S,k and the quantized singular values s.sub.ijk along with the index set are reported to the gNB.

[0273] The gNB reconstructs first the transformed/compressed channel tensor, as

[00023]${\hat{}}_c=\underset{i=1}{\overset{r_1}{.Math.}}.Math.\underset{j=1}{\overset{r_2}{.Math.}}.Math.\underset{k=1}{\overset{r_3}{.Math.}}.Math.s_{\mathrm{ijk}}\left(u_{R,i}{u}_{T,j}{u}_{S,k}\right).$

[0274] Then, based on the transformed/compressed channel tensor .sub.c and the signaled index set , the frequency-domain channel tensor is reconstructed as

vec()(.sub.(1,2,3)(.sub.c)(three-dimensional transformation/compression);

vec().sub.(1,2)(.sub.c)(two-dimensional transformation/compression);

vec().sub.(3)(.sub.c)(one-dimensional transformation/compression).

[0275] The number of coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3+r.sub.1r.sub.2r.sub.3, for the higher-order singular vectors and the higher-order singular values.

[0276] In one method, the values of (r.sub.1,r.sub.2,r.sub.3), representing the number of dominant principal components with respect to the first, second and third dimension of the transformed/compressed channel tensor .sub.c, respectively, are configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r.sub.1,r.sub.2,r.sub.3) as a part of the CSI report or they are known at the UE.

[0277] In accordance with a sub-embodiment 3 1 of the third embodiment 3, the 3D transformation/compression function .sub.(1,2,3)() is given by a two-dimensional Discrete Cosine transformation (2D-DCT) with respect to the space dimensions and a 1D-DFT transformation with respect to the frequency dimension of the channel tensor. The codebook matrices .sub.n, n=1,2 are given by oversampled discrete cosine transform (DCT) matrices. The codebook matrix .sub.3 is given by an oversampled DFT matrix.

[0278] In accordance with a sub-embodiment 3 2 of the third embodiment 3, the 3D transformation/compression function .sub.(1,2,3)() is given by a 3D-DFT transformation and the codebook matrices .sub.n, n=1,2,3 are given by oversampled DFT matrices.

[0279] In accordance with a sub-embodiment 3 3 of the third embodiment 3, the 2D transformation/compression unction .sub.(1,2)() is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices.

[0280] In accordance with a sub-embodiment 3 4 of the third embodiment 3, the 1D transformation/compression function .sub.(3)() is given by a 1D-DFT transformation and the codebook matrix .sub.3 is given by an oversampled DFT matrix.

Transformation/Compression of Channel Matrix in Combination with Standard Non-HO-PCA

[0281] In accordance with a fourth embodiment 4, a UE is configured to report explicit CSI Type IV that represents a transformed and compressed form of a channel matrix, or a beam-formed channel tensor, or a channel covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, or space and frequency dimension of the channel matrix. The CSI combines channel tensor transformation with data compression by exploiting the sparse representation in the delay domain and the correlations of the channel coefficients in the spatial and frequency/delay domains.

[0282] An illustration of this approach is shown in FIG. 10. The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a three-dimensional (3D) frequency-domain channel tensor (a three-dimensional array) .sup.N.sup.r.sup.N.sup.t.sup.S of dimension N.sub.rN.sub.tS, where S is the number of subbands, PRBs, or subcarriers. The definition of N.sub.t and N.sub.r is dependent on the configuration of the CSI type: [0283] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 for CSI type configuration channel tensor, N.sub.t=2N.sub.1N.sub.2, and N.sub.r is the number of UE receive antenna ports; [0284] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 at the gNB, N.sub.t=2N.sub.1N.sub.2, N.sub.r=2N.sub.1N.sub.2, for CSI type configuration channel covariance tensor; [0285] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of UE receive antenna ports for CSI type configuration beamformed-channel tensor; [0286] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of beamformed antenna ports/beams N.sub.r=2U for CSI type configuration beamformed-channel covariance tensor;

[0287] After the construction of the frequency-domain channel tensor , a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) transformation and compression of the channel tensor is applied with respect to the space, frequency, or frequency and space dimensions of the channel tensor. The aim of the transformation is to obtain a sparse or nearly-sparse representation of the channel tensor in one, two, or three dimensions and to extract the dominant coefficients having the highest energy. After the transformation/compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions of the channel tensor is achieved.

[0288] For example, a transformation/compression with respect to all three dimensions of the channel tensor is represented by a (column-wise) Kronecker product as

[00024]$.Math.={}_{\left(1,2,3\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2,3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$

where [0289] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1; [0290] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2; [0291] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3; [0292] .sub.n.sub.r.sub.,n.sub.t.sub.,s is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s, and b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s, and [0293] N.sub.r, N.sub.t, and S represents the value of the first, second and third dimension of the transformed/compressed channel tensor , respectively.

[0294] The transformed/compressed channel coefficients .sub.n.sub.r.sub.,n.sub.t.sub.,s are used to form the transformed/compressed channel tensor of dimension N.sub.rN.sub.tS.

[0295] For example, a transformation/compression with respect to the two space dimensions of the channel tensor is represented by

[00025]$.Math.={}_{\left(1,2\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(1,2\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,s}.Math.b_{2,n_r,n_t,s}.Math.b_{1,n_r,n_t,s},$

where b.sub.3,s is a vector of all zeros with the s-th element being one, S=S, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s is a vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2, b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1, N.sub.rN.sub.r,N.sub.tN.sub.t and S=S.

[0296] For example, a transformation/compression with respect to the frequency dimension of the channel tensor is represented by

[00026]$.Math.={}_{\left(3\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=\mathrm{vec}\left({}_{\left(3\right)}\left(\hat{}\right)\right)=\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s}.Math.b_{3,n_r,n_t,s}.Math.b_{2,n_r}.Math.b_{1,n_t},$

where b.sub.2,n.sub.r is a vector of all zeros with the n.sub.r-th element being one, b.sub.1,n.sub.t is a vector of all zeros with the n.sub.t-th element being one, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3, N.sub.r=N.sub.r, N.sub.t=N.sub.t and SS.

[0297] As an example, the selection of the transformation/compression vectors and transformed/compressed channel coefficients for a transformation/compression with respect to the second and third dimension, can be calculated by

minvec()vec(.sub.(2,3)()).sub.2.sup.2.

The optimization problem may be solved by standard algorithms such as orthogonal matching pursuit. As a result, the indices of the vectors in the transformation matrices selected from the codebooks and the transformed channel coefficients associated with each domain are known.

[0298] The indices of the selected vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s and b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s from the codebook matrices .sub.n, n=1,2,3 are stored in a set of g-tuples, where g refers to the number of transformed dimensions.

[0299] For example, for g=1, the set is represented by ={i.sub.1, i.sub.2, . . . , i.sub.T}, where T denotes the number of selected vectors with respect to the transformed/compressed dimension of the channel tensor . For example, in the case of a transformation/compression with respect to the frequency dimension, T=S.

[0300] For example, for g=3, the set is represented by 3-tuples (i.sub.1,n.sub.r, i.sub.2,n.sub.t, i.sub.3,s), where i.sub.1,n.sub.r is the index associated with vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s, i.sub.2,n.sub.t, is the index associated with vector b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s, and i.sub.3,s is the index associated with vector b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s. The set is given by

={(i.sub.1,0,i.sub.2,0,i.sub.3,0), . . . ,(i.sub.1,N.sub.r.sub.1,i.sub.2,N.sub.t.sub.1,i.sub.3,S1)}

In one method, the size of the set is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred size of the set as a part of the CSI report or it is known at the UE.

[0301] The codebook matrices n are given by matrices .sub.n=[d.sub.n,0, d.sub.n,1, . . . , d.sub.n,TO.sub.n.sub.1], where the parameter O.sub.f,n denotes the oversampling factor with respect to the n-th dimension where T=N.sub.r for n=1, T=N.sub.t for n=2, and T=S for n=3.

[0302] The oversampled factors O.sub.f,n of the codebook matrices are configured via higher layer or via DCI physical layer signaling from the gNB to the UE, or they are known at the UE.

[0303] After the channel transformation/compression, the UE rewrites the transformed/compressed channel tensor to a transformed/compressed channel matrix , and applies a standard PCA decomposition, represented by

[00027]$\hat{H}=U.Math..Math..Math..Math.V^H=\underset{i=1}{\overset{R}{.Math.}}.Math..Math.s_i.Math.u_i.Math.v_i^H,$

where [0304] U=[u.sub.1,u.sub.2, . . . ,u.sub.R] is the N.sub.tN.sub.rR left-singular matrix; [0305] V=[v.sub.1, v.sub.2, . . . ,v.sub.R] is the SR right-singular matrix; [0306] is a RR diagonal matrix with ordered singular values s.sub.i (s.sub.1s.sub.2 . . . S.sub.R) on its main diagonal, and R=min(S, N.sub.tN.sub.r).

[0307] The transformed/compressed channel matrix .sub.c is then constructed using r, 1rR dominant principal components as

.sub.c=V.sup.H,

where =[u.sub.1, u.sub.2, . . . , u.sub.r] and =diag(s.sub.1, s.sub.2, . . . , s.sub.r), and V=[v.sub.1, v.sub.2, . . . , v.sub.r].

[0308] The UE quantizes U, V and the singular values s.sub.1, s.sub.2, . . . , s.sub.r using a codebook approach, and then reports them along with the index set , to the gNB.

[0309] The gNB reconstructs first the transformed/compressed channel matrix .sub.c as

.sub.c=V.sup.H.

[0310] Then, based on the transformed/compressed channel matrix, the gNB constructs the transformed/compressed channel tensor .sub.c. Using the signaled index set , the frequency-domain channel tensor is reconstructed as

vec().sub.(1,2,3)(.sub.c)(three-dimensional transformation/compression);

vec().sub.(1,2)(.sub.c)(two-dimensional transformation/compression);

vec().sub.(3)(.sub.c)(one-dimensional transformation/compression).

[0311] In one method, the value of r representing the number of dominant principal components of the transformed/compressed channel matrix .sub.c is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r.sub.1, r.sub.2, r.sub.3) as a part of the CSI report or they are known at the UE.

[0312] In accordance with a sub-embodiment 4 1 of the fourth embodiment 4, the 3D transformation/compression function .sub.(1,2,3)() is given by a 2D-DCT transformation with respect of the space dimensions and a 1D-DFT transformation with respect to the frequency dimension of the channel tensor. The codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices. The codebook matrix .sub.3 is given by an oversampled DFT matrix.

[0313] In accordance with a sub-embodiment 4 2 of the fourth embodiment 4, the 3D transformation/compression function .sub.(1,2,3)() is given by a 3D DFT transformation and the codebook matrices .sub.n, n=1,2,3 are given by oversampled DFT matrices.

[0314] In accordance with a sub-embodiment 4 3 of the fourth embodiment 4, the 2D transformation/compression function .sub.(1,2)() is given by a 2D Discrete Cosine transformation (DCT) and the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices.

[0315] In accordance with a sub-embodiment 4 4 of the fourth embodiment 4, the 1D transformation/compression function .sub.(3)() is given by a 1D DFT transformation and the codebook matrix .sub.3 is given by an oversampled DFT matrix.

Extension to Doppler Frequency Domain: High Order PCA on Four-Dimensional Frequency-Domain Channel Tensor

[0316] In accordance with a fifth embodiment 5, a UE is configured to report explicit CSI Type V that represents a compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed channel covariance tensor over the configured subbands (SB), PRBs or subcarriers according to the following sub-embodiments.

[0317] The compressed CSI is based on a high-order principal component analysis (HO-PCA) of the four-dimensional channel tensor to exploit the correlations of the channel tensor in the space-, frequency- and time/Doppler-frequency domains.

[0318] An illustration of this approach is shown in FIG. 11. The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 4D frequency-domain channel .sup.N.sup.r.sup.N.sup.t.sup.SD of dimension N.sub.rN.sub.tSD, where S is the number of subbands, PRBs, or subcarriers and D is number of snapshots of the channel measured at D consecutive time instants/slots. The definition of N.sub.t and N.sub.r is dependent on the configuration of the CSI type: [0319] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 for CSI type configuration channel tensor, N.sub.t=2N.sub.1N.sub.2, and N.sub.r is the number of UE receive antenna ports; [0320] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 at the gNB, N.sub.t=2N.sub.1N.sub.2, N.sub.r=2N.sub.1N.sub.2 for CSI type configuration channel covariance tensor; [0321] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of UE receive antenna ports for CSI type configuration beamformed-channel tensor; [0322] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of beamformed antenna ports/beams N.sub.r=2U for CSI type configuration beamformed-channel covariance tensor;

[0323] Then, the UE performs a HO-PCA on the four-dimensional channel tensor , such that is represented by

[00028]$=\underset{i=1}{\overset{N_r}{.Math.}}.Math..Math.\underset{j=1}{\overset{N_t}{.Math.}}.Math..Math.\underset{k=1}{\overset{S}{.Math.}}.Math..Math.\underset{l=1}{\overset{D}{.Math.}}.Math..Math.S_{\mathrm{ijkl}}\left(u_{R,i}.Math.u_{T,j}.Math.u_{S,k}.Math.u_{D,l}\right).$

where [0324] U.sub.R=[U.sub.R,1, . . . ,u.sub.R,N.sub.r].sup.N.sup.r.sup.N.sup.r is matrix containing the higher order singular vectors with respect to the receive antennas (first dimension of the channel tensor ); [0325] U.sub.T=[u.sub.T,1, . . . , u.sub.T,N.sub.t].sup.N.sup.t.sup.N.sup.t is matrix containing the higher order singular vectors with respect to the transmit antennas (second dimension of the channel tensor ); [0326] U.sub.S=[u.sub.S,1, . . . , u.sub.S,S].sup.SS is a matrix containing the higher order singular vectors with respect to the frequency dimension (third dimension of the channel tensor ); [0327] U.sub.D=[u.sub.D,1, . . . , u.sub.D,R].sup.DR is a matrix containing the higher order singular vectors with respect to the time/channel snapshot dimension (fourth dimension of the channel tensor ), where R is the rank of the channel tensor given by R=min{N.sub.rN.sub.tS, D}; [0328] s.sub.ijkl are the higher order singular values, sorted as s.sub.ijkls.sub.ijkl with ii, jj, kk, ll.

[0329] To reduce the number of channel coefficients, the channel tensor is approximated by (r.sub.1,r.sub.2,r.sub.3,r.sub.4), (1r.sub.1N.sub.r, 1r.sub.2N.sub.t, 1r.sub.3S, 1r.sub.4D) dominant principal components with respect to the first, second, third and fourth dimensions and the corresponding 1-mode (left), 2-mode (right) and 3-mode (lateral) and 4-mode singular matrices. The compressed explicit frequency-domain channel tensor (explicit CSI) is given by

.sub.c=.sub.1=1.sup.r.sup.i.sub.j=1.sup.r.sup.2.sub.k=1.sup.r.sup.3.sub.i=1.sup.r.sup.4s.sub.ijkl(u.sub.R,iu.sub.T,ju.sub.S,ku.sub.D,l),

where

.sub.R=[u.sub.R,1, . . . ,u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1,

.sub.T=[u.sub.T,1, . . . ,u.sub.T,r.sub.2].sup.N.sup.t.sup.r.sup.2,

.sub.S=[u.sub.S,1, . . . ,u.sub.S,r.sub.3].sup.Sr.sup.3,

.sub.D=[u.sub.D,1, . . . ,u.sub.D,r.sub.4].sup.Dr.sup.4.

[0330] To report the compressed frequency-domain channel tensor (explicit CSI) from the UE to the gNB, the UE quantizes the coefficients of the vectors u.sub.R,i, u.sub.T,j, U.sub.S,k, U.sub.D,l and the singular values s.sub.ijkl using a codebook approach.

[0331] The gNB reconstructs the compressed channel tensor as

[00029]${}_c=\underset{i=1}{\overset{r_1}{.Math.}}.Math..Math.\underset{j=1}{\overset{r_2}{.Math.}}.Math..Math.\underset{k=1}{\overset{r_3}{.Math.}}.Math..Math.\underset{l=1}{\overset{r_4}{.Math.}}.Math..Math.s_{\mathrm{ijkl}}\left(u_{R,i}.Math.u_{T,j}.Math.u_{S,k}.Math.u_{D,l}\right).$

[0332] The number of complex coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3+Dr.sub.4 for the higher-order singular vectors, and the number of real coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by r.sub.1r.sub.2r.sub.3r.sub.4 for the higher-order singular values, respectively. In comparison, for the standard (non-high order) PCA (see FIG. 12), it is needed to quantize N.sub.rN.sub.tr+SDr+r values for the singular vectors and the singular values with r=min(N.sub.rN.sub.t, SD). For small values of (r.sub.1, r.sub.2, r.sub.3, r.sub.4) (low rank approximation of the channel tensor), the compression achieved by the HO-PCA is higher than the compression achieved by the standard non-HO PCA approach.

[0333] In one method, the values of (r.sub.1, r.sub.2, r.sub.3, r.sub.4), representing the number of dominant principal components with respect to the first, second, third and fourth dimension of the channel tensor, respectively, are configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r.sub.1, r.sub.2, r.sub.3, r.sub.4) as a part of the CSI report or they are known at the UE.

[0334] In accordance with a sub-embodiment 5 1 of the fifth embodiment 5, a UE is configured to report explicit CSI Type V with delay-domain CSI for the higher-order singular-value matrix .sub.S. In this configuration, the UE calculates an approximated reduced-sized (compressed) delay-domain higher-order singular-matrix .sub.S from the frequency-domain higher-order singular-matrix .sub.S. The delay-domain higher-order singular-matrix is given by

.sub.SF.sub.S.sub.S,

where F.sub.SE.sup.SL is an DFT matrix of size SL. The size of the compressed delay-domain matrix .sub.S is given by Lr.sub.3. A compression is achieved when L<S.

[0335] The DFT vectors in F.sub.S are selected from an oversampled DFT-codebook matrix of dimension SSO.sub.f. Here, O.sub.f{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix. The indices of the selected vectors in F.sub.S from the codebook are stored in a set =(i.sub.1, i.sub.2, . . . , i.sub.L).

[0336] The UE quantizes the coefficients of the vectors in .Math..sub.R=[u.sub.R,1, . . . , u.sub.R,r.sub.1].sub.T=[u.sub.T,1, . . . ,u.sub.T,r.sub.2], .sub.S=[.sub.S,1, . . . , .sub.S,r.sub.3].sup.Lr.sup.3 and .sub.D=[u.sub.D,1, . . . , u.sub.D,r.sub.4] and the HO singular values s.sub.ijkl using a codebook approach, and reports them along with the L delays, represented by a set of indices that correspond to the selected DFT vectors in the codebook , to the gNB.

[0337] The gNB reconstructs the frequency-domain channel tensor .sub.c according to this embodiment where .sub.S is calculated as .sub.S=[u.sub.S,1, . . . , u.sub.S,r.sub.3]=F.sub.S.sub.S.

[0338] An illustration of this approach is shown in FIG. 13.

[0339] In one method, the number of delays L is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred value of L as a part of the CSI report or it is known at the UE

[0340] The oversampled factor O.sub.f of the DFT codebook matrix is configured via higher layer or via DCI physical layer signaling from the gNB to the UE or it is known at the UE.

[0341] In accordance with a sub-embodiment 5 2 of the fifth embodiment 5, a UE is configured to report explicit CSI Type V with Doppler-frequency domain CSI for the higher-order singular-value matrix .sub.D. In this configuration, the UE calculates an approximated reduced-sized (compressed) Doppler-frequency domain higher-order singular-matrix .sub.D from the time domain higher-order singular-matrix .sub.D.

[0342] The Doppler-frequency domain higher-order singular-matrix is given by

.sub.DF.sub.D.sub.D,

where F.sub.D.sup.DG is an DFT matrix of size DG. The size of the compressed doppler-domain matrix .sub.D is given by Gr.sub.4. A compression is achieved when G<D.

[0343] The DFT vectors in F.sub.D are selected from an oversampled DFT-codebook matrix of dimension DDO.sub.t, where O.sub.t{1,2,3, . . . } denotes the oversampling factor of the DFT-codebook matrix. The indices of the selected vectors in F.sub.D from the codebook are stored in a set =(i.sub.1, i.sub.2, . . . , i.sub.G).

[0344] The UE quantizes the coefficients of the vectors in .sub.R=[u.sub.R,1, . . . , u.sub.R,r.sub.1], .sub.T=[u.sub.T,1, . . . , u.sub.T,r.sub.2], .sub.S=[u.sub.S,1, . . . , u.sub.S,r.sub.3] and .sub.D=[.sub.D,1, . . . , .sub.D,r.sub.4].sup.Gr.sup.4 and the HO singular values s.sub.ijkl using a codebook approach, and reports them along with the G Doppler-frequency values, represented by a set of indices that correspond to the selected DFT vectors in the codebook , to the gNB.

[0345] The gNB reconstructs the frequency-domain channel tensor .sub.c according to this embodiment, where .sub.D is calculated as .sub.D=[u.sub.D,1, . . . , u.sub.D,r.sub.4]=F.sub.D.sub.D.

[0346] An illustration of this approach is shown in FIG. 13.

[0347] In one method, the number of Doppler-frequency values G is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred value of G as a part of the CSI report or it is known at the UE.

[0348] The oversampled factor O.sub.t of the DFT codebook matrix is configured via higher layer or via physical layer signaling (via DCI) from the gNB to the UE or it is known at the UE.

Extension to Doppler Frequency Domain: Compression of Four-Dimensional Channel Tensor in Combination with HO-PCA

[0349] In accordance with a sixth embodiment 6, a UE is configured to report explicit CSI Type VI that represents a transformed and compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, time or frequency and space, or frequency and time, or space and time of the channel tensor. The CSI combines channel tensor transformation with data compression by exploiting the correlations of the channel coefficients in the spatial, frequency, delay and time/channel snapshot domain.

[0350] An illustration of this approach is shown in FIG. 14. The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 4D frequency-domain channel tensor .sup.N.sup.r.sup.N.sup.t.sup.SD of dimension N.sub.rN.sub.tSD, where S is the number of subbands, PRBs or subcarriers and D is number of snapshots of the channel measured at D consecutive time instants. The definition of N.sub.t and N.sub.r is dependent on the configuration of the CSI type: [0351] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 for CSI type configuration channel tensor, N.sub.t=2N.sub.1N.sub.2, and N.sub.r is the number of UE receive antenna ports; N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 at the gNB, N.sub.t=2N.sub.1N.sub.2, N.sub.r=2N.sub.1N.sub.2 for CSI type configuration channel covariance tensor; [0352] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of UE receive antenna ports for CSI type configuration beamformed-channel tensor; [0353] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of beamformed antenna ports/beams N.sub.r=2U for CSI type configuration beamformed-channel covariance tensor;

[0354] After the construction of the frequency-domain channel tensor , a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) or four-dimensional (4D) transformation of the channel tensor is applied with respect to the space, frequency, or time dimensions of the channel tensor. The aim of the transformation is to obtain a sparse or nearly-sparse representation of the channel tensor in one, two, three or four dimensions. After the transformation and compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions or four dimensions of the channel tensor is achieved.

[0355] For example, a transformation/compression with respect to all four dimensions of the channel tensor is represented by a (column-wise) Kronecker product as

[00030]$.Math.={}_{\left(1,2,3,4\right)}\left(\hat{}\right),.Math.\mathrm{vec}.Math.\left(\right)=.Math.\mathrm{vec}\left({}_{\left(1,2,3,4\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$

where [0356] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1; [0357] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2; [0358] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d transformation is a vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3; [0359] b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor , selected from a codebook matrix .sub.4; [0360] .sub.n.sub.r.sub.,n.sub.t.sub.,s,d is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,db.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d, and b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d and [0361] N.sub.r, N.sub.t, S and D represents the value of the first, second, third and fourth dimension of the transformed/compressed channel tensor , respectively.

[0362] The transformed/compressed channel coefficients .sub.n.sub.r.sub.,n.sub.t.sub.,s,d are used to form the transformed/compressed channel tensor of dimension N.sub.rN.sub.tSD, where N.sub.rN.sub.r, N.sub.tN.sub.t, SS, DD.

[0363] For example, a transformation/compression with respect to the two space dimensions of the channel tensor is represented by

[00031]$.Math.={}_{\left(1,2\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=.Math.\mathrm{vec}\left({}_{\left(1,2\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,d}.Math.b_{3,s}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$

where b.sub.4,d is a vector of all zeros with the d-th element being one, b.sub.3,s is a vector of all zeros with the s-th element being one, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2, b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1 and N.sub.rN.sub.r, N.sub.tN.sub.t, S=S and D=D.

[0364] For example, a transformation/compression with respect to the frequency and time dimension of the channel tensor is represented by

[00032]$.Math.={}_{\left(3,4\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=.Math.\mathrm{vec}\left({}_{\left(3,4\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r}.Math.b_{1,n_t},$

[0365] Where b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor , selected from a codebook matrix .sub.4, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3, b.sub.2,n.sub.r is a vector of all zeros with the n.sub.r-th element being one, b.sub.1,n.sub.t is a vector of all zeros with the n.sub.t-th element being one and N.sub.r=N.sub.r N.sub.t=N.sub.t, SS and DD.

[0366] For example, a transformation/compression with respect to the frequency dimension of the channel tensor is represented by

[00033]$.Math.={}_{\left(3\right)}\left(\hat{}\right),.Math.\mathrm{vec}\left(\right)=.Math.\mathrm{vec}\left({}_{\left(3\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r}.Math.b_{1,n_t},$

where b.sub.4,d is a vector of all zeros with the d-th element being one, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3, b.sub.2,n.sub.r is a vector of all zeros with the n.sub.r-th element being one, b.sub.1,n.sub.t is a vector of all zeros with the n.sub.t-th element being one and N.sub.r=N.sub.r N.sub.t=N.sub.t, SS and D=D.

[0367] For example, a transformation/compression with respect to the time dimension of the channel tensor is represented by

=.sub.(4)(),

[00034]$\mathrm{vec}\left(\right)=.Math.\mathrm{vec}\left({}_{\left(4\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,n_r,n_t,s,d}.Math.b_{3,s}.Math.b_{2,n_r}.Math.b_{1,n_t},$

where b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor , selected from a codebook matrix .sub.4, b.sub.3,s is a vector of all zeros with the s-th element being one, b.sub.2,n.sub.r, is a vector of all zeros with the n.sub.r-th element being one, b.sub.1,n.sub.t is a vector of all zeros with the n.sub.t-th element being one and N.sub.r=N.sub.r N.sub.t=N.sub.t, S=S and DD.

[0368] As an example, the selection of the transformation/compression vectors and transformed/compressed channel coefficients for a transformation/compression with respect to the second and third dimension, can be calculated by

minvec(vec()vec().sub.(2,3)()).sub.2.sup.2.

[0369] The optimization problem may be solved by standard algorithms such as orthogonal matching pursuit. As a result, the indices of the vectors in the transformation matrices selected from the codebooks and the transformed channel coefficients associated with each domain are known.

[0370] The indices of the selected vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d and b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d from the codebook matrices .sub.n, n=1,2,3,4 are stored in a set of g-tuples, where g refers to the number of transformed dimensions.

[0371] For example, for g=1, the set is represented by ={i.sub.1, i.sub.2, . . . ,i.sub.T}, where T denotes the number of selected vectors with respect to the transformed/compressed dimension of the channel tensor . For example, in the case of a transformation/compression with respect to the frequency dimension, T=S.

[0372] For example, for g=4, the set is represented by 4-tuples (i.sub.1,n.sub.r, i.sub.2,n.sub.t, i.sub.3,s, i.sub.4,d), where i.sub.1,n.sub.r is the index associated with vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d, i.sub.2,n.sub.t is the index associated with vector b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d, i.sub.3,s is the index associated with vector b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d and i.sub.4,d is the index associated with vector b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d. The set is given by

={(i.sub.1,0,i.sub.2,0,i.sub.3,0,i.sub.4,0), . . . ,(i.sub.1,N.sub.r,i.sub.2,N.sub.t,i.sub.3,S,i.sub.4,D)}

[0373] In one method, the size of the set is configured via higher layer signaling from the gNB to the UE. In another method, the UE reports the preferred size of the set as a part of the CSI report or it is known at the UE.

[0374] The codebook matrices .sub.n, are given by matrices .sub.n=[d.sub.n,0, d.sub.n,1, . . . , d.sub.n,TO.sub.n.sub.1], where the parameter O.sub.f,n denotes the oversampling factor with respect to the n-th dimension where T=N.sub.r for n=1, T=N.sub.t for n=2, T=S for n=3 and T=D for n=4.

[0375] The oversampled factors O.sub.f,n of the codebook matrices are configured via higher layer (e.g., RRC, or MAC) or via DCI physical layer signaling from the gNB to the UE, or they are known at the UE or it is known at the UE.

[0376] After the channel transformation/compression, the UE performs a HO-PCA on the transformed/compressed channel tensor , such that is represented by

[00035]$\hat{}=\underset{i=1}{\overset{N_r^{}}{.Math.}}.Math..Math.\underset{j=1}{\overset{N_t^{}}{.Math.}}.Math.\underset{k=1}{\overset{S^{}}{.Math.}}.Math.\underset{l=1}{\overset{D^{}}{.Math.}}.Math.S_{\mathrm{ijkl}}\left(u_{R,i}.Math.u_{T,j}.Math.u_{S,k}.Math.u_{D,l}\right),$

where [0377] U.sub.R=[u.sub.R,1, . . . , u.sub.R,N.sub.r].sup.N.sup.r.sup.N.sup.r is a matrix containing the high-order singular vectors with respect to the first dimension of the transformed/compressed channel tensor ; [0378] U.sub.T=[u.sub.T,1, . . . , u.sub.T,N.sub.t].sup.N.sup.t.sup.N.sup.t is matrix containing the high-order singular vectors with respect to the second dimension of the transformed/compressed channel tensor ; [0379] U.sub.S=[u.sub.S,1, . . . , u.sub.S,S].sup.SS is a matrix containing the high-order singular vectors with respect to the third dimension of the channel tensor . [0380] U.sub.D=[u.sub.D,1, . . . , u.sub.D,R].sup.DR is a matrix containing the higher order singular vectors with respect to the time/channel snapshot dimension (fourth dimension of the channel tensor , where R is the rank of the channel tensor given by R=min{N.sub.rN.sub.tS, D}; [0381] s.sub.ijkl are the higher order singular values, sorted as s.sub.ijkls.sub.ijkl with ijj, kk, ll.

[0382] To reduce the number of channel coefficients, the channel tensor is approximated by (r.sub.1, r.sub.2, r.sub.3, r.sub.4), (1r.sub.1N.sub.r, 1N.sub.t, 1r.sub.2N.sub.t, 1r.sub.3S, 1r.sub.4D) dominant principal components with respect to the first, second, third and fourth dimensions and the corresponding 1-mode (left), 2-mode (right) and 3-mode (lateral) and 4-mode singular matrices. The compressed explicit frequency-domain channel tensor (explicit CSI) is given by

.sub.c=.sub.i=1.sup.r.sup.1.sub.j=1.sup.r.sup.2.sub.k=1.sup.r.sup.3.sub.l=1.sup.r.sup.4s.sub.ijkl(u.sub.R,iu.sub.T,ju.sub.S,ku.sub.D,l),

where

.sub.R=[u.sub.R,1, . . . ,u.sub.R,r.sub.1].sup.N.sup.r.sup.r.sup.1,

T=[u.sub.T,1, . . . ,u.sub.T,r.sub.2].sup.N.sup.t.sup.r.sup.2,

.sub.S=[u.sub.S,1, . . . ,u.sub.S,r.sub.3]E.sup.Sr.sup.3.

.sub.D=[u.sub.D,1, . . . ,u.sub.D,r.sub.4].sup.Dr.sup.4

[0383] The UE quantizes the coefficients of the vectors u.sub.R,i, u.sub.T,j, u.sub.S,k, u.sub.D,l and the singular values s.sub.ijkl using a codebook approach. The quantized vectors u.sub.R,i, u.sub.T,j, u.sub.S,k, u.sub.D,l and the quantized singular values s.sub.ijkl along with the index set are reported to the gNB.

[0384] The gNB reconstructs first the transformed/compressed channel tensor .sub.c as

[00036]${\hat{}}_c=\underset{i=1}{\overset{r_1}{.Math.}}.Math..Math.\underset{j=1}{\overset{r_2}{.Math.}}.Math.\underset{k=1}{\overset{r_3}{.Math.}}.Math.\underset{l=1}{\overset{r_4}{.Math.}}.Math.S_{\mathrm{ijkl}}\left(u_{R,i}.Math.u_{T,j}.Math.u_{S,k}.Math.u_{D,l}\right).$

[0385] Then, based on the transformed/compressed channel tensor .sub.c and the signaled index set , the frequency-domain channel tensor is reconstructed as

vec().sub.(1,2,3,4)(.sub.c)(four-dimensional transformation/compression);

vec().sub.(1,2)(.sub.c)(two-dimensional transformation/compression);

vec().sub.(3,4)(.sub.c)(two-dimensional transformation/compression);

vec().sub.(3)(.sub.c)(one-dimensional transformation/compression).

vec().sub.(4)(.sub.c)(one-dimensional transformation/compression).

[0386] The number of complex coefficients that need to be quantized for the frequency-domain HO-PCA approach is given by N.sub.rr.sub.1+N.sub.tr.sub.2+Sr.sub.3+Dr.sub.4+r.sub.1r.sub.2r.sub.3r.sub.4, for the higher-order singular vectors and the higher-order singular values.

[0387] In one method, the values of (r.sub.1, r.sub.2, r.sub.3, r.sub.4), representing the number of dominant principal components with respect to the first, second, third and fourth dimension of the transformed/compressed channel tensor .sub.c, respectively, are configured via higher layer (e.g., RRC, or MAC-CE) signaling from the gNB to the UE. In another method, the UE reports the preferred values of (r.sub.1, r.sub.2, r.sub.3, r.sub.4) as a part of the CSI report or they are known at the UE.

[0388] In accordance with a sub-embodiment 6 1 of the sixth embodiment 6, the 4D transformation/compression function .sub.(1,2,3,4)() is given by a two dimensional (2D-DCT) with respect to the space dimensions and a 2D-DFT transformation with respect to the frequency and time dimensions of the channel tensor. The codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices and the codebook matrix .sub.3 and .sub.4 are given by an oversampled DFT matrices.

[0389] In accordance with a sub-embodiment 6 2 of the sixth embodiment 6, the 4D transformation/compression function .sub.(1,2,3,4)() is given by a 4D-DFT transformation and the codebook matrices .sub.n, n=1,2,3,4 are given by oversampled DFT matrices.

[0390] In accordance with a sub-embodiment 6 3 of the sixth embodiment 6, the 2D transformation/compression function .sub.(1,2)() is given by a 2D-Discrete Cosine transformation (DCT) and the codebook matrices .sub.n, n=1,2 are given by oversampled DCT matrices.

[0391] In accordance with a sub-embodiment 6 4 of the sixth embodiment 6, the 2D transformation/compression function .sub.(3,4)() is given by a 2D-Discrete Fourier transformation (DFT) and the codebook matrices .sub.n, n=3,4 are given by oversampled DFT matrices.

[0392] In accordance with a sub-embodiment 6 5 of the sixth embodiment 6, the 1D transformation/compression function .sub.(3)() is given by a 1D-DFT transformation and the codebook matrix .sub.3 is given by an oversampled DFT matrix.

[0393] In accordance with a sub-embodiment 6 6 of the sixth embodiment 6, the 1D transformation/compression function .sub.(4)() is given by a 1D-DFT transformation and the codebook matrix .sub.4 is given by an oversampled DFT matrix.

Extension to Doppler Frequency Domain: Compression of Four-Dimensional Channel Tensor in Combination with Non-HO-PCA (Standard PCA)

[0394] In accordance with a seventh embodiment 7, a UE is configured to report explicit CSI Type VII that represents a transformed and compressed form of a channel tensor, or a beam-formed channel tensor, or a channel covariance tensor, or a beam-formed covariance tensor over the configured subbands (SB), PRBs or subcarriers with respect to the space, frequency, time or frequency and space, or frequency and time, or space and time of the channel tensor. The CSI combines channel tensor transformation with data compression by exploiting the sparse representation in the delay domain and time domain and the correlations of the channel coefficients in the spatial and frequency/delay/time domains.

[0395] An illustration of this approach is shown in FIG. 15. The UE estimates in a first step the un-quantized explicit CSI using measurements on downlink reference signals (such as CSI-RS) in the frequency domain, and then constructs a 4D frequency-domain channel .sup.N.sup.r.sup.N.sup.t.sup.SD of dimension N.sub.rN.sub.tSD, where S is the number of subbands, PRBs or subcarriers and D is number of CSI-RS channel measurements over D consecutive time instants/slots. The definition of N.sub.t and N.sub.r is dependent on the configuration of the CSI type: [0396] N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 for CSI type configuration channel tensor, N.sub.t=2N.sub.1N.sub.2, and N.sub.r is the number of UE receive antenna ports; N.sub.t is the number of transmit antenna ports 2N.sub.1N.sub.2 at the gNB, N.sub.t=2N.sub.1N.sub.2, N.sub.r=2N.sub.1N.sub.2, for CSI type configuration channel covariance tensor; [0397] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of UE receive antenna ports for CSI type configuration beamformed-channel tensor; [0398] N.sub.t is the number of beamformed antenna ports/beams N.sub.t=2U and N.sub.r is the number of beamformed antenna ports/beams N.sub.r=2U for CSI type configuration beamformed-channel covariance tensor;

[0399] After the construction of the frequency-domain channel tensor , a one-dimensional (1D), two-dimensional (2D) or three-dimensional (3D) or four-dimensional (4D) transformation of the channel tensor is applied with respect to the space, frequency, and/or time dimensions of the channel tensor. After the transformation and compression step, the size of the channel tensor is reduced and a compression with respect to one, two or three dimensions or four dimensions of the channel tensor is achieved.

[0400] For example, a transformation/compression with respect to all four dimensions of the channel tensor is represented by a (column-wise) Kronecker product as

[00037]$.Math.={}_{\left(1,2,3,4\right)}\left(\hat{}\right),.Math.\mathrm{vec}.Math.\left(\right)=.Math.\mathrm{vec}\left({}_{\left(1,2,3,4\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$

where [0401] b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1; [0402] b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2; [0403] b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size S1 with respect to the third dimension of the channel tensor , selected from a codebook matrix .sub.3; [0404] b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size D1 with respect to the fourth dimension of the channel tensor , selected from a codebook matrix .sub.4; [0405] .sub.n.sub.r.sub.,n.sub.t.sub.,s,d is the transformed/compressed channel coefficient associated with the vectors b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d, b.sub.3,n.sub.r.sub.,n.sub.t.sub.,s,d, and b.sub.4,n.sub.r.sub.,n.sub.t.sub.,s,d and [0406] N.sub.r, N.sub.t, S and D represents the value of the first, second, third and fourth dimension of the transformed/compressed channel tensor , respectively.

[0407] The transformed/compressed channel coefficients .sub.n.sub.r.sub.,n.sub.t.sub.,s,d are used to form the transformed/compressed channel tensor of dimension N.sub.rN.sub.tSD, where N.sub.rN.sub.r, N.sub.tN.sub.t, SS, DD.

[0408] For example, a transformation/compression with respect to the two space dimensions of the channel tensor is represented by

[00038]$.Math.={}_{\left(1,2\right)}\left(\hat{}\right),.Math.\mathrm{vec}.Math.\left(\right)=.Math.\mathrm{vec}\left({}_{\left(1,2\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,d}.Math.b_{3,s}.Math.b_{2,n_r,n_t,s,d}.Math.b_{1,n_r,n_t,s,d},$

where b.sub.4,d is a vector of all zeros with the d-th element being one, b.sub.3,s is a vector of all zeros with the s-th element being one, b.sub.2,n.sub.r.sub.,n.sub.t.sub.,s,d is a vector of size N.sub.t1 with respect to the second dimension of the channel tensor , selected from a codebook matrix .sub.2, b.sub.1,n.sub.r.sub.,n.sub.t.sub.,s,d is a transformation vector of size N.sub.r1 with respect to the first dimension of the channel tensor , selected from a codebook matrix .sub.1 and N.sub.rN.sub.r, N.sub.tN.sub.t, S=S and D=D

[0409] For example, a transformation/compression with respect to the frequency and time dimension of the channel tensor is represented by

[00039]$.Math.={}_{\left(3,4\right)}\left(\hat{}\right),.Math.\mathrm{vec}.Math.\left(\right)=.Math.\mathrm{vec}\left({}_{\left(3,4\right)}\left(\hat{H}\right)\right)=.Math.\underset{n_r=0}{\overset{N_r^{}-1}{.Math.}}.Math..Math.\underset{n_t=0}{\overset{N_t^{}-1}{.Math.}}.Math.\underset{s=0}{\overset{S^{}-1}{.Math.}}.Math.\underset{d=0}{\overset{D^{}-1}{.Math.}}.Math.{}_{n_r,n_t,s,d}.Math.b_{4,n_r,n_t,s,d}.Math.b_{3,n_r,n_t,s,d}.Math.b_{2,n_r}.Math.b_{1,n_t},$