Multi-beam codebooks with further optimized overhead

Abstract

A method, wireless device and network node for determining an indication of a precoder are provided. According to one aspect, a method in a wireless device includes determining and the indication of the precoder from a codebook, the indication comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and a second beam respectively. The first beam phase parameter takes on one of a first integer number of phase values and the second beam phase parameter takes on one of a second integer number of phase values. At least one of the following conditions apply: the second integer number of phase values is less than the first number of phase values, and the second frequency-granularity is greater than the first frequency-granularity. The method includes transmitting the determined indication of a precoder to the network node.

Claims

1. A method in a base station for receiving an indication of a precoder from a user equipment, the method comprising: receiving the indication of the precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively, each of the first and second beam phase parameters corresponding to one of a plurality of frequency subbands; the first beam phase parameter taking on one of a first integer number of phase values; and the second beam phase parameter taking on one of a second integer number of phase values, the second beam having a lesser power than the first beam and the second integer number of phase values being less than the first integer number of phase values; determining the precoder based on the received indication of the precoder; and generating the first and second beams based on the precoder.

2. The method of claim 1, further comprising receiving the first and second integer number of phase values.

3. The method of claim 1, wherein a frequency-granularity of each beam is determined to be a multiple of a subband size.

4. The method of claim 1, wherein the first and second integer number of phase values are respective numbers of values attainable in a phase shift keyed, PSK, constellation.

5. The method of claim 1, wherein each of the first and second beams is a kth beam, d(k), that has associated a set of complex numbers and has index pair (l.sub.k,m.sub.k), each element of the set of complex numbers being characterized by at least one complex phase shift such that: d.sub.n(k)=d.sub.i(k)α.sub.i,ne.sup.j2π(pΔ.sup.1,k.sup.+qΔ.sup.2,k.sup.); d.sub.n(k), and d.sub.i(k) are an i.sup.th and n.sup.th elements of d(k), respectively; α.sub.i,n is a real number corresponding to the i.sup.th and n.sup.th elements of d(k); p and q are integers; beam directions Δ.sub.1,k and Δ.sub.2,k are real numbers corresponding to beams with index pair (l.sub.k,m.sub.k) that determine complex phase shifts e.sup.j2πΔ.sup.1,k and e.sup.j2πΔ.sup.2,k respectively; and each of the first and second beam phase parameters is a complex coefficient c.sub.k for d(k) used to adjust at least the phase of the i.sup.th element of d(k) according to c.sub.kd.sub.i(k).

6. The method of claim 1, wherein the first and second integer number of phase values are respective granularities of respective co-phasing factors for the first and second beams.

7. The method of claim 1, wherein at least one of the first plurality and second plurality of first and second beam phase parameters are parametrically encoded, wherein the at least one of the first plurality and second plurality of beam phase parameters are coefficients within a predetermined function over frequency.

8. A base station for receiving an indication of a precoder from a user equipment, the method comprising: processing circuitry configured to: receive the indication of the precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively, each of the first and second beam phase parameters corresponding to one of a plurality of frequency subbands; the first beam phase parameter taking on one of a first integer number of phase values; and the second beam phase parameter taking on one of a second integer number of phase values, the second beam having a lesser power than the first beam and the second integer number of phase values being less than the first integer number of phase values; determine the precoder based on the received indication of the precoder; and generate the first and second beams based on the precoder.

9. The base station of claim 8, further comprising receiving the first and second integer number of phase values.

10. The base station of claim 8, wherein a frequency-granularity of each beam is determined to be a multiple of a subband size.

11. The base station of claim 8, wherein the first and second integer number of phase values are respective numbers of values attainable in a phase shift keyed, PSK, constellation.

12. The base station of claim 8, wherein each of the first and second beams is a k.sup.th beam, d(k), that has associated a set of complex numbers and has index pair (l.sub.k,m.sub.k), each element of the set of complex numbers being characterized by at least one complex phase shift such that: d.sub.n(k)=d.sub.i(k)α.sub.i,ne.sup.j2π(pΔ.sup.1,k.sup.+qΔ.sup.2,k.sup.); d.sub.n(k), and d.sub.i(k) are an i.sup.th and n.sup.th elements of d(k), respectively; α.sub.i,n is a real number corresponding to the i.sup.th and n.sup.th elements of d(k); p and q are integers; beam directions Δ.sub.1,k and Δ.sub.2,k are real numbers corresponding to beams with index pair (l.sub.k,m.sub.k) that determine complex phase shifts e.sup.j2πΔ.sup.1,k and e.sup.j2πΔ.sup.2,k respectively; and each of the first and second beam phase parameters is a complex coefficient c.sub.k for d(k) used to adjust at least the phase of the i.sup.th element of d(k) according to c.sub.kd.sub.i(k).

13. The base station of claim 8, wherein the first and second integer number of phase values are respective granularities of respective co-phasing factors for the first and second beams.

14. The base station of claim 8, wherein at least one of the first plurality and second plurality of first and second beam phase parameters are parametrically encoded, wherein the at least one of the first plurality and second plurality of beam phase parameters are coefficients within a predetermined function over frequency.

15. A method for a user equipment to transmit an indication of a precoder to a base station, the method comprising: determining from a codebook an indication of a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively; the first beam phase parameter taking on one of a first integer number of phase values; and the second beam phase parameter taking on one of a second integer number of phase values, the second beam having a lesser power than the first beam and the second integer number of phase values being less than the first integer number of phase values, each of the first and second beam phase parameters corresponding to one of a plurality of frequency subbands; and reporting the determined indication of the precoder to a base station.

16. The method of claim 15, further comprising determining the first and second integer number of phase values.

17. The method of claim 15, wherein the first and second integer number of phase values are respective numbers of values attainable in a phase shift keyed, PSK, constellation, wherein the PSK constellation is 8 PSK for beams having a beam strength above a first threshold and the PSK constellation is quadrature PSK (QPSK) for beams having a beam strength below a second threshold.

18. The method of claim 15, wherein the first and second integer number of phase values are respective granularities of respective co-phasing factors for the first and second beams.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) A more complete understanding of the present embodiments, and the attendant advantages and features thereof, will be more readily understood by reference to the following detailed description when considered in conjunction with the accompanying drawings wherein:

(2) FIG. 1 is a block diagram of a system for spatial multiplexing;

(3) FIG. 2 is a diagram of partitioning of system bandwidth;

(4) FIG. 3 is 4×4 array of dual-polarized antennas;

(5) FIG. 4 is a grid of DFT beams;

(6) FIG. 5 illustrates mapping of antenna ports;

(7) FIG. 6 is a block diagram of a wireless communication system constructed in accordance with principles set forth herein;

(8) FIG. 7 is a block diagram of a network node;

(9) FIG. 8 is a block diagram of an alternative embodiment of a network node;

(10) FIG. 9 is a block diagram of a wireless device;

(11) FIG. 10 is a block diagram of an alternative embodiment of a wireless device;

(12) FIG. 11 is a block diagram of another alternative embodiment of a wireless device;

(13) FIG. 12 is a block diagram of yet another alternative embodiment of a wireless device;

(14) FIG. 13 is a flowchart of an exemplary process for configuring a wireless device;

(15) FIG. 14 is a flowchart of an exemplary process for determining a precoder;

(16) FIG. 15 is a flowchart of an alternative exemplary process for determining a precoder;

(17) FIG. 16 illustrates different frequency granularities;

(18) FIG. 17 is a plurality of graphs of beam phases;

(19) FIG. 18 illustrates a sum of vectors for determining phase error; and

(20) FIG. 19 illustrates a binary tree for encoding phase changes.

DETAILED DESCRIPTION

(21) The term wireless device (WD) used herein may refer to any type of wireless device communicating with a network node and/or with another wireless device in a cellular or mobile communication system. Examples of a wireless device are a user equipment (UE), target device, device to device (D2D) wireless device, machine type wireless device or wireless device capable of machine to machine (M2M) communication, PDA, iPAD, Tablet, mobile terminals, smart phone, laptop embedded equipped (LEE), laptop mounted equipment (LME), USB dongles etc.

(22) The term “network node” used herein may refer to a radio network node or another network node, e.g., a core network node, MSC, MME, O&M, OSS, SON, positioning node (e.g. E-SMLC), MDT node, etc.

(23) The term “radio network node” used herein can be any kind of network node comprised in a radio network which may further comprise any of base station (BS), radio base station, base transceiver station (BTS), base station controller (BSC), radio network controller (RNC), evolved Node B (eNB or eNodeB), or 3GPP New Radio Node B, known as gNB, multi-standard radio (MSR) radio node such as MSR BS, relay node, donor node controlling relay, radio access point (AP), transmission points, transmission nodes, Remote Radio Unit (RRU) Remote Radio Head (RRH), nodes in distributed antenna system (DAS) etc.

(24) Note further that functions described herein as being performed by a wireless device or a network node may be distributed over a plurality of wireless devices and/or network nodes.

(25) Before describing in detail exemplary embodiments, it is noted that the embodiments reside primarily in combinations of apparatus components and processing steps related to multi-beam codebooks with optimized overhead. Accordingly, components have been represented where appropriate by conventional symbols in the drawings, showing only those specific details that are pertinent to understanding the embodiments so as not to obscure the disclosure with details that will be readily apparent to those of ordinary skill in the art having the benefit of the description herein.

(26) As used herein, relational terms, such as “first” and “second,” “top” and “bottom,” and the like, may be used solely to distinguish one entity or element from another entity or element without necessarily requiring or implying any physical or logical relationship or order between such entities or elements.

(27) Some implementations use two dimensional antenna arrays. Such antenna arrays may be (partly) described by the number of antenna columns corresponding to the horizontal dimension N.sub.h, the number of antenna rows corresponding to the vertical dimension N.sub.v, and the number of dimensions corresponding to different polarizations N.sub.p. The total number of antennas is thus N=N.sub.hN.sub.vN.sub.p. It should be pointed out that the concept of an antenna is non-limiting in the sense that it can refer to any virtualization (e.g., linear mapping) of the physical antenna elements. For example, pairs of physical sub-elements could be fed the same signal, and hence share the same virtualized antenna port.

(28) An example of a 4×4 (i.e. four rows by four columns) array with cross-polarized antenna elements is shown in FIG. 3.

(29) Precoding may be interpreted as multiplying the signal with different beamforming weights for each antenna prior to transmission. A typical approach is to tailor the precoder to the antenna form factor, i.e. taking into account N.sub.h, N.sub.v, and N.sub.p when designing the precoder codebook. A common type of precoding is to use a DFT-precoder, where the precoder vector used to precode a single-layer transmission using a single-polarized uniform linear array (ULA) with N.sub.1 antennas is defined as

(30) $\begin{array}{cc}{w}_{1D}\left(l,N_1,O_1\right)=\frac{1}{\sqrt{N_1}}\left[\begin{array}{c}{e}^{j2\pi .Math.0.Math.\frac{l}{O_1{N}_1}}\\ e^{j2\pi .Math.1.Math.\frac{l}{O_1{N}_1}}\\ \mathrm{.Math.}\\ e^{j2\pi .Math.\left(N_1-1\right).Math.\frac{l}{O_1{N}_1}}\end{array}\right]& \mathrm{Equation}4\end{array}$
where l=0,1, . . . O.sub.1N.sub.1−1 is the precoder index and O.sub.1 is an integer oversampling factor. A precoder for a dual-polarized uniform linear array (ULA) with N.sub.1 antennas per polarization (and so 2N.sub.1 antennas in total) can be similarly defined as

(31) $\begin{array}{cc}{w}_{1D,\mathrm{DP}}\left(l,N_1,O_1\right)=\left[\begin{array}{c}{w}_{1D}\left(l\right)\\ e^{j\varphi }{w}_{1D}\left(l\right)\end{array}\right]=\left[\begin{array}{cc}{w}_{1D}\left(l\right)& 0\\ 0& w_{1D}\left(l\right)\end{array}\right]\left[\begin{array}{c}1\\ e^{j\varphi }\end{array}\right]& \mathrm{Equation}5\end{array}$
where e.sup.jϕ is a co-phasing factor between the two polarizations that may for instance be selected from a QPSK alphabet

(32) $\varphi \in \left\{0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}\right\}.$

(33) A corresponding precoder vector for a two-dimensional uniform planar array (UPA) with N.sub.1×N.sub.2 antennas can be created by taking the Kronecker product of two precoder vectors as w.sub.2D(l,m)=w.sub.1D(l,N.sub.1,O.sub.1).Math.w.sub.1D(m,N.sub.2,O.sub.2), where O.sub.2 is an integer oversampling factor in the N.sub.2 dimension. Each precoder w.sub.2D (l,m) forms a 2D DFT beam, all the precoders {w.sub.2D(l,m), l=0 . . . , N.sub.1O.sub.1−1; m=0, . . . , N.sub.2O.sub.2−1} form a grid of discrete Fourier transform (DFT) beams. An example is shown in FIG. 4 where (N.sub.1,N.sub.2)=(4,2) and (O.sub.1,O.sub.2)=(4,4). Each of the grid of DFT beams points to a spatial direction which can be described by an azimuth and elevation. For simplicity, throughout the following sections, the terms ‘DFT beams’ and ‘DFT precoders’ are used interchangeably, although ‘precoders’ are used to form ‘beams’.

(34) More generally, a beam with an index pair (l,m) can be identified by the direction in which the greatest energy is transmitted when precoding weights w.sub.2D (l,m) are used in the transmission. Also, a magnitude taper can be used with DFT beams to lower the beam's sidelobes, the beam pattern at directions away from the main beam. A 1D DFT precoder along N.sub.1 and N.sub.2 dimensions with magnitude tapering can be expressed as

(35) $w_{1D}\left(l,N_1,O_1,\beta \right)=\frac{1}{\sqrt{N_1}}\left[\begin{array}{c}{\beta }_0{e}^{j2\pi .Math.0.Math.\frac{l}{O_1{N}_1}}\\ {\beta }_1{e}^{j2\pi .Math.1.Math.\frac{l}{O_1{N}_1}}\\ \mathrm{.Math.}\\ {\beta }_{N_1-1}{e}^{j2\pi .Math.\left(N_1-1\right).Math.\frac{l}{O_1{N}_1}}\end{array}\right],\mathrm{and}$$w_{1D}\left(m,N_2,O_2,\gamma \right)=\frac{1}{\sqrt{N_2}}\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}{\gamma }_0{e}^{j2\pi .Math.0.Math.\frac{m}{O_2{N}_2}}\\ {\gamma }_1{e}^{j2\pi .Math.1.Math.\frac{m}{O_2{N}_2}}\end{array}\\ \mathrm{.Math.}\end{array}\\ {\gamma }_{N_2-1}{e}^{j2\pi .Math.\left(N_2-1\right).Math.\frac{m}{O_2{N}_2}}\end{array}\right]$

(36) Where 0<β.sub.i,γ.sub.k≤1 (i=0,1, . . . , N.sub.1−1; k=0,1, . . . , N.sub.2−1) are amplitude scaling factors. β.sub.i=1, γ.sub.k=1 (i=0,1, . . . , N.sub.1−1; k=0,1, . . . , N.sub.2−1) correspond to no tapering. DFT beams (with or without a magnitude taper) have a linear phase shift between elements along each of the two dimensions. Without loss of generality, one can assume that the elements of w(l,m) are ordered according to w(l,m)=w.sub.1D(l,N.sub.1,O.sub.1,β).Math.w.sub.1,D(m,N.sub.2,O.sub.2, γ) such that adjacent elements correspond to adjacent antenna elements along dimension N.sub.2, and elements of w(l,m) spaced N.sub.2 apart correspond to adjacent antenna elements along dimension N.sub.1. Then the phase shift between two elements w.sub.s.sub.1(l,m) and w.sub.s.sub.2 (l,m) of w(l,m) can be expressed as:

(37) $w_{s_2}\left(l,m\right)=w_{s_1}\left(l,m\right).Math.\left(\frac{{\alpha }_{s_2}}{{\alpha }_{s_1}}\right).Math.e^{j2\pi \left(\left(k_1-i_1\right){\Delta }_1+\left(k_2-i_2\right){\Delta }_2\right)}$

(38) Where s.sub.1=i.sub.1N.sub.2+i.sub.2 and s.sub.2=k.sub.1N.sub.2+k.sub.2 (with 0≤i.sub.2<N.sub.2, 0≤i.sub.1<N.sub.1, 0≤k.sub.2<N.sub.2, and 0≤k.sub.1<N.sub.1) are integers identifying two entries of the beam w(l,m) so that (i.sub.1, i.sub.2) indicates to a first entry of beam w(l,m) that is mapped to a first antenna element (or port) and (k.sub.1, k.sub.2) indicates to a second entry of beam w(l,m) that is mapped to a second antenna element (or port). α.sub.s.sub.1=β.sub.i.sub.1γ.sub.i.sub.2 and α.sub.s.sub.2=β.sub.k.sub.1γ.sub.k.sub.2 are real numbers. α.sub.i≠1 (i=s.sub.1,s.sub.2) if magnitude tapering is used; otherwise α.sub.i=1.

(39) ${\Delta }_1=\frac{l}{O_1{N}_1}$ is a phase shift corresponding to a direction along an axis, e.g. the horizontal axis (‘azimuth’).

(40) ${\Delta }_2=\frac{m}{O_2{N}_2}$ is a phase slim corresponding to direction along an axis, e.g. the vertical axis (‘elevation’).

(41) Therefore, a k.sup.th beam d(k) formed with precoder w(l.sub.k, m.sub.k) can also be referred to, for simplicity, by the corresponding precoder w(l.sub.k,m.sub.k), i.e. d(k)=w(l.sub.k,m.sub.k), although a ‘precoder’ is used to form a ‘beam’. Thus, when referring to the precoder used to form it, in this disclosure, a beam d(k) can in this disclosure be also described as a set of complex numbers, each element of the set being characterized by at least one complex phase shift such that an element of the beam is related to any other element of the beam where d.sub.n(k)=d.sub.i(k)α.sub.i,ne.sup.j2π(pΔ.sup.1,k.sup.+qΔ.sup.2,k.sup.)=d.sub.i(k)α.sub.i,n(e.sup.j2πΔ.sup.1,k).sup.p(e.sup.j2πΔ.sup.2,k).sup.q, where d.sub.i (k) is the i.sup.th element of a beam d(k), α.sub.i,n is a real number corresponding to the i.sup.th and n.sup.th elements of the beam d(k); p and q are integers; and Δ.sub.1,k and Δ.sub.2,k are real numbers corresponding to a beam with index pair (l.sub.k, m.sub.k) that determine the complex phase shifts e.sup.j2πΔ.sup.1,k and e.sup.j2πΔ.sup.2,k, respectively. Index pair (l.sub.k, m.sub.k) corresponds to a direction of arrival or departure of a plane wave when beam d(k) is used for transmission or reception in a UPA or ULA. A beam d(k) can be identified with a single index k′ where=l.sub.k+N.sub.1O.sub.1m.sub.k, i.e., along vertical or N.sub.2 dimension first, or alternatively k′=N.sub.2O.sub.2l.sub.k+m.sub.k, i.e. along horizontal or N.sub.1 dimension first.

(42) An example of precoder elements of a beam w(l,m) to antenna ports mapping is shown in FIG. 5 where a single polarization 2D antenna with (N1, N2)=(4,2) is illustrated. w.sub.i(l,m) is applied on the transmit (Tx) signal to port i (i=1, 2, . . . , 8) to antenna elements E1-E8. There is a constant phase shift between any two precoder elements associated with two adjacent antenna ports along each dimension. For example, with Δ.sub.2 defined as above, the phase shift between w.sub.1(l,m) and w.sub.2(l,m) is e.sup.j2πΔ.sup.2, which is the same as the phase shift between w.sub.7(l,m) and w.sub.8(l,m). Similarly, with Δ.sub.1 defined as above, the phase shift between w.sub.2(l,m) and w.sub.7(l,m) is e.sup.j2πΔ.sup.1, which is the same as the phase shift between w.sub.5(l,m) and w.sub.7(l,m).

(43) Extending the precoder for a dual-polarized ULA may then be done as

(44) $\begin{array}{cc}\begin{array}{c}{w}_{2D,\mathrm{DP}}\left(l,m,\varphi \right)=\left[\begin{array}{c}1\\ e^{j\varphi }\end{array}\right].Math.w_{2D}\left(l,m\right)\\ =\left[\begin{array}{c}{w}_{2D}\left(l,m\right)\\ e^{j\varphi }{w}_{2D}\left(l,m\right)\end{array}\right]\\ =\left[\begin{array}{cc}{w}_{2D}\left(l,m\right)& 0\\ 0& w_{2D}\left(l,m\right)\end{array}\right]\left[\begin{array}{c}1\\ e^{j\varphi }\end{array}\right]\end{array}& \mathrm{Equation}6\end{array}$

(45) A precoder matrix W.sub.2D,DP for multi-layer transmission may be created by appending columns of DFT precoder vectors as
W.sub.2D,DP.sup.(R)[w.sub.2D,DP(l.sub.1,m.sub.1,ϕ.sub.1)w.sub.2D,DP(l.sub.2,m.sub.2,ϕ.sub.2) . . . w.sub.2D,DP(l.sub.R,m.sub.R,ϕ.sub.R)]
where R is the number of transmission layers, i.e. the transmission rank. In a special case for a rank-2 DFT precoder, m.sub.1=m.sub.2=m and l.sub.1=l.sub.2=l, we have

(46) 0$\begin{array}{cc}\begin{array}{c}{W}_{2D,\mathrm{DP}}^{\left(2\right)}\left(l,m,{\varphi }_1,{\varphi }_2\right)=\left[\begin{array}{cc}{W}_{2D,\mathrm{DP}}\left(l,m,{\varphi }_1\right)& W_{2D,\mathrm{DP}}\left(l,m,{\varphi }_2\right)\end{array}\right]\\ =\left[\begin{array}{cc}{w}_{2D}\left(l,m\right)& 0\\ 0& W_{2D}\left(l,m\right)\end{array}\right]\left[\begin{array}{cc}1& 1\\ e^{j{\varphi }_1}& e^{j{\varphi }_2}\end{array}\right]\end{array}& \mathrm{Equation}7\end{array}$

(47) For each rank, all the precoder candidates form a ‘precoder codebook’ or a ‘codebook’. A wireless device can first determine the rank of the estimated downlink wideband channel based on CSI-RS. After the rank is identified, for each subband the wireless device then searches through all the precoder candidates in a codebook for the determined rank to find the best precoder for the subband. For example, in case of rank=1, the wireless device would search through w.sub.2D,DP(k,l,ϕ) for all the possible (k,l, ϕ) values. In case of rank=2, the wireless device would search through w.sub.2D,DP.sup.(2)(k, l, ϕ.sub.1, ϕ.sub.2) for all the possible (k, l, ϕ.sub.1, ϕ.sub.2) values.

(48) With multi-user MIMO, two or more users in the same cell are co-scheduled on the same time-frequency resource. That is, two or more independent data streams are transmitted to different wireless devices at the same time, and the spatial domain is used to separate the respective streams. By transmitting several streams simultaneously, the capacity of the system can be increased. This however, comes at the cost of reducing the SINR per stream, as the power has to be shared between streams and the streams will interfere with each other.

(49) When increasing the antenna array size, the increased beamforming gain will lead to higher SINR, however, as the user throughput depends only logarithmically on the SINR (for large SINRs), it is instead beneficial to trade the gains in SINR for a multiplexing gain, which increases linearly with the number of multiplexed users.

(50) Accurate CSI is required in order to perform appropriate null forming between co-scheduled users. In the current LTE Rel.13 standard, no special CSI mode for MU-MIMO exists and thus, MU-MIMO scheduling and precoder construction has to be based on the existing CSI reporting designed for single-user MIMO (that is, a PMI indicating a DFT-based precoder, a RI and a CQI). This may prove quite challenging for MU-MIMO, as the reported precoder only contains information about the strongest channel direction for a user and may thus not contain enough information to do proper null forming, which may lead to a large amount of interference between co-scheduled users, reducing the benefit of MU-MIMO.

(51) Advanced codebooks comprising precoders with multiple beams have been shown to improve MU-MIMO performance due to enhanced null forming capabilities. Such multi-beam precoders may be defined as follows. We first define D.sub.N as a size N×N DFT matrix, i.e. the elements of D.sub.N are defined as

(52) ${\left[D_N\right]}_{k,l}=\frac{1}{\sqrt{N}}{e}^{\frac{j2\pi kl}{N}}.$
Further we define

(53) $R_N\left(q\right)=\mathrm{diag}\left(\left[\begin{array}{cccc}{e}^{j2\pi .Math.0.Math.\frac{q}{N}}& e^{j2\pi .Math.1.Math.\frac{q}{N}}& \mathrm{.Math.}& e^{j2\pi .Math.\left(N-1\right).Math.\frac{q}{N}}\end{array}\right]\right)$
to be a size N×N rotation matrix, defined for 0≤q<1. Multiplying D.sub.N with R.sub.N(q) from the left creates a rotated DFT matrix with entries

(54) ${\left[R_N\left(q\right)D_N\right]}_{k,l}=\frac{1}{\sqrt{N}}{e}^{\frac{j2\pi k\left(l+q\right)}{2}}.$
The rotated DFT matrix R.sub.N(q)D.sub.N=[d.sub.1 d.sub.2 . . . d.sub.N] consists of normalized orthogonal column vectors {d.sub.i}.sub.i=1.sup.N which furthermore span the vector space .sup.N. That is, the columns of R.sub.N(q)D.sub.N, for any q, is an orthonormal basis of .sup.N.

(55) We begin with extending the (rotated) DFT matrices that were appropriate transforms for a single-polarized ULA as discussed above to also fit the more general case of dual-polarized 2D uniform planar arrays (UPAs).

(56) We define a rotated 2D DFT matrix as D.sub.N.sub.V.sub.,N.sub.H(q.sub.V, q.sub.H)=(R.sub.N.sub.H(q.sub.H)D.sub.N.sub.H).Math.(R.sub.N.sub.V(q.sub.V)D.sub.N.sub.V)=[d.sub.1 d.sub.2 . . . d.sub.N.sub.V.sub.N.sub.H]. The columns {d.sub.i}.sub.i=1.sup.N.sup.DP of D.sub.N.sub.V.sub.,N.sub.H (q.sub.V, q.sub.H) constitutes an orthonormal basis of the vector space .sup.N.sup.V.sup.N.sup.H. Such a column d.sub.i is henceforth denoted a (DFT) beam, and we note that it fulfills the earlier definition of a beam given above.

(57) Consider now a dual-polarized UPA, where the channel matrix H=[H.sub.pol1 H.sub.pol2]. Create a dual-polarized beam space transformation matrix

(58) $\begin{array}{c}{B}_{N_V,N_H}\left(q_V,q_H\right)=I_2.Math.D_{N_V,N_H}\left(q_V,q_H\right)\\ =\left[\begin{array}{cc}{D}_{N_V,N_H}\left(q_V,q_H\right)& 0\\ 0& D_{N_V,N_H}\left(q_V,q_H\right)\end{array}\right]\\ =\left[\begin{array}{cccccccc}{d}_1& d_2& \mathrm{.Math.}& d_{N_V{N}_H}& 0& 0& \mathrm{.Math.}& 0\\ 0& 0& \mathrm{.Math.}& 0& d_1& d_2& \mathrm{.Math.}& d_{N_V{N}_H}\end{array}\right]\\ =\left[\begin{array}{cccc}{b}_1& b_2& \mathrm{.Math.}& b_{2N_V{N}_H}\end{array}\right].\end{array}\hspace{1em}$
The columns {b.sub.i}.sub.i=1.sup.2N.sup.V.sup.N.sup.H of B.sub.N.sub.V.sub.,N.sub.H (q.sub.V, q.sub.H) constitutes an orthonormal basis of the vector space .sup.2N.sup.V.sup.N.sup.H. Such a column b.sub.i is henceforth denoted a single-polarized beam (SP-beam) as it is constructed by a beam d transmitted on a single polarization

(59) $\left(i.e.b=\left[\begin{array}{c}d\\ 0\end{array}\right]\mathrm{or}b=\left[\begin{array}{c}0\\ d\end{array}\right]\right).$
We also introduce a notation dual-polarized beam to refer to a beam transmitted on both polarizations (co-phased with an (arbitrary) co-phasing factor e.sup.jα, i.e.

(60) $b_{\mathrm{DP}}=\left[\begin{array}{c}d\\ e^{j\alpha }d\end{array}\right]).$

(61) Utilizing the assumption that the channel is somewhat sparse, we can capture sufficiently much of the channel energy by only selecting a column subset of B.sub.N.sub.V.sub.,N.sub.H (q.sub.v, q.sub.H). That is, it is sufficient to describe a couple of the SP-beams, which keeps down the feedback overhead. So, we can select a column subset l.sub.S consisting of N.sub.SP columns of B.sub.N.sub.V.sub.,N.sub.H (q.sub.v, Q.sub.h), to create a reduced beam space transformation matrix B.sub.i.sub.S=[b.sub.I.sub.S.sub.(1) b.sub.I.sub.S.sub.(2) . . . b.sub.I.sub.S.sub.(N.sub.SP.sub.)]. E.g., one can select columns number I.sub.S=[1 5 10 25] to create the reduced beam space transformation matrix B.sub.I.sub.S=[b.sub.1 b.sub.5 b.sub.10 b.sub.25].

(62) The most general precoder structure for precoding of a single layer is given as:

(63) $\begin{array}{c}w=B_{I_S}\left[\begin{array}{c}{c}_1\\ c_2\\ \mathrm{.Math.}\\ c_{N_{\mathrm{SP}}}\end{array}\right]\\ =\left[\begin{array}{cc}{b}_{I_S\left(1\right)}& \begin{array}{cc}{b}_{I_S\left(2\right)}& \begin{array}{cc}\mathrm{.Math.}& b_{I_S\left(N_{\mathrm{SP}}\right)}\end{array}\end{array}\end{array}\right]\left[\begin{array}{c}{c}_1\\ c_2\\ \mathrm{.Math.}\\ c_{N_{\mathrm{SP}}}\end{array}\right]\\ =\underset{i=1}{\overset{N_{\mathrm{SP}}}{.Math.}}{c}_i{b}_{I_S\left(i\right)},\end{array}\hspace{1em}$

(64) where {c.sub.i}.sub.i=1.sup.N.sup.SP are complex coefficients. A more refined multi-beam precoder structure is achieved by separating the complex coefficients in a power (or amplitude) and a phase part as

(65) $\begin{array}{c}w=B_{I_S}\left[\begin{array}{c}{c}_1\\ c_2\\ \mathrm{.Math.}\\ c_{N_{\mathrm{SP}}}\end{array}\right]\\ =B_{I_S}\left[\begin{array}{c}\sqrt{p_1}{e}^{j{\alpha }_1}\\ \sqrt{p_2}{e}^{j{\alpha }_2}\\ \mathrm{.Math.}\\ \sqrt{p_{N_{\mathrm{SP}}}}{e}^{j{\alpha }_{N_{\mathrm{SP}}}}\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& & \\ 0& \sqrt{p_2}& & \ddots \\ & & \ddots & 0\\ & \ddots & 0& \sqrt{p_{N_{\mathrm{SP}}}}\end{array}\right]\left[\begin{array}{c}{e}^{j{\alpha }_1}\\ e^{j{\alpha }_2}\\ \mathrm{.Math.}\\ e^{j{\alpha }_{N_{\mathrm{SP}}}}\end{array}\right]\\ =B_{I_S}\sqrt{P}\left[\begin{array}{c}{e}^{j{\alpha }_1}\\ e^{j{\alpha }_2}\\ \mathrm{.Math.}\\ e^{j{\alpha }_{N_{\mathrm{SP}}}}\end{array}\right]\end{array}\hspace{1em}$

(66) The precoder vector may then be expressed as

(67) $w=\underset{\underset{=W_1}{︸}}{B_{I_S}\sqrt{P}}\underset{\underset{=W_2}{︸}}{\left[\begin{array}{c}{e}^{j{\alpha }_1}\\ e^{j{\alpha }_2}\\ \mathrm{.Math.}\\ e^{j{\alpha }_{N_{\mathrm{SP}}}}\end{array}\right]}=W_1{W}_2.$
The selection of W.sub.1 may then be made on a wideband basis while the selection of W.sub.2 may be made on a subband basis. The precoder vector for subband l may be expressed as w.sub.l=W.sub.1W.sub.2 (l). That is, only W.sub.2 is a function of the subband index l.

(68) As multiplying the precoder vector w with a complex constant C does not change its beamforming properties (as only the phase and amplitude relative to the other single-polarized beams is of importance), one may without loss of generality assume that the coefficients corresponding to e.g. SP-beam 1 is fixed to p.sub.1=1 and e.sup.jα.sup.1=1, so that parameters for one less beam needs to be signaled from the wireless device to the base station. Furthermore, the precoder may be further assumed to be multiplied with a normalization factor, so that, e.g., a sum power constraint is fulfilled, i.e., that is ∥w∥.sup.2=1. Any such normalization factor is omitted from the equations herein for clarity.

(69) What needs to be fed back by the wireless device to the base station is thus The chosen columns of B.sub.N.sub.V.sub.,N.sub.H (q.sub.V,q.sub.H), i.e. the N.sub.SP single-polarized beams. This requires at most N.sub.SP.Math.log.sub.2 2N.sub.VN.sub.H bits;

(70) The vertical and horizontal DFT basis rotation factors q.sub.V and q.sub.H. For instance, the

(71) 0$q\left(i\right)=\frac{i}{Q},$
i=0,1, . . . , Q−1, for some value of Q. The corresponding overhead would then be 2.Math.log.sub.2 Q bits; The (relative) power levels {p.sub.2, p.sub.3, . . . p.sub.N.sub.SP} of the SP-beams. If L is the number of possible discrete power levels, (N.sub.SP−1).Math.log.sub.2 L bits are needed to feed back the SP-beam power levels; and The co-phasing factors {e.sup.jα.sup.2, e.sup.jα3, . . . , e.sup.jαN.sup.SP} of the SP-beams. For instance,

(72) $\alpha \left(k\right)=\frac{2\pi k}{K},$
k=0,1, . . . K−1, for some value of K. The corresponding overhead would be, (N.sub.SP−1) log.sub.2K bits per rank per W2 report.

(73) In some implementations, the phases of the SP-beams may be quantized across frequency. We assume that a multi-beam precoder vector w.sub.f for each PRB f=0,1, . . . , N.sub.RB−1 should be quantized and fed back and that the multi-beam precoder vector is a function of the SP-beam phases as

(74) $w_f=B_{I_S}\sqrt{P}\left[\begin{array}{c}{e}^{j{\alpha }_1\left(f\right)}\\ e^{j{\alpha }_2\left(f\right)}\\ \mathrm{.Math.}\\ e^{j{\alpha }_{N_{\mathrm{SP}}\left(f\right)}}\end{array}\right].$
Note here again that one may set e.sup.jα.sup.1.sup.(f)=1 as only the relative phases are important. We are interested in characterizing the phase change over frequency for each SP-beam, that is, the vectors ϕ.sub.i=[e.sup.jα.sup.i.sup.(0) e.sup.jα.sup.i.sup.(1) . . . e.sup.jα.sup.i.sup.(N.sup.RB.sup.−1)].sup.T, i=2,3, . . . , N.sub.SP.

(75) In some such implementations, the phases ϕ.sub.i of each SP-beam are approximated as a polynomial function over frequency. That is,
ϕ.sub.i(f)=e.sup.JΣ.sup.m=0.sup.M.sup.a.sup.m.sup.f.sup.m,

(76) Where {a.sub.m}.sub.m=0.sup.M are a set of real-valued coefficients. Instead of quantizing and feeding back a selection of the actual phases for each SP-beam and frequency, the parametrized real-valued coefficients are quantized and fed back as part of the precoder feedback. This may significantly reduce the feedback overhead required to convey the selection of SP-beam phases, especially if the channel bandwidth is large and the order of the polynomial is small.

(77) The coefficients a.sub.m may then be selected from a set of possible polynomial phase coefficients. For example, the order M of the polynomial may be equal to 1, so that the phase change over frequency is approximated as a linear function. In this case, only two coefficients per beam, a.sub.0 and a.sub.1, need to be estimated, quantized and fed back.

(78) Some embodiments include optimization of the feedback overhead associated with a multi-beam precoder codebook by using different quantization granularity and/or frequency granularity of different beam components.

(79) Given that there is a certain “budget” of feedback bits to spend on quantizing the frequency-selective phases for each beam, one may spend more bits on quantizing the stronger beam components. Beams with weaker power level will to a lesser extent contribute to the overall precoding performance. Therefore, if one wishes to minimize the overall absolute quantization error, one may allow that the weaker beams have a larger relative quantization error than the stronger beams and thus assign less bits for phase quantization to the weaker beams than the stronger beams.

(80) The frequency-selectivity of the phases of each beam component in a multi-beam precoder may be different. Thus, performance may be increased if the frequency-granularity of the reporting of each constituent beam component may be set independently.

(81) Since the channel delay spread within each beam is relatively small, the phase change over frequency (or subbands) is generally also relatively small (e.g., within 180 degrees). Therefore, the phase differences between adjacent subbands may be encoded with less number of bits than encoding the absolute phase of each subband.

(82) By allowing the quantization and/or frequency granularity of the phases of each constituent beam component in the multi-beam precoder codebook to be set individually, the precoding performance can be improved while maintaining the same or decreasing the feedback overhead.

(83) Returning to the drawing figures, FIG. 6 is a block diagram of a wireless communication network configured according to principles set forth herein. The wireless communication network 10 includes a cloud 12 which may include the Internet and/or the public switched telephone network (PSTN). Cloud 12 may also serve as a backhaul network of the wireless communication network 10. The wireless communication network 10 includes one or more network nodes 14A and 14B, which may communicate directly, for example via an X2 interface in LTE embodiments, and are referred to collectively as network nodes 14. The network nodes 14 may serve wireless devices 16A and 16B, referred to collectively herein as wireless devices 16. Note that, although only two wireless devices 16 and two network nodes 14 are shown for convenience, the wireless communication network 10 may typically include many more wireless devices (WDs) 16 and network nodes 14. Further, in some embodiments, WDs 16 may communicate directly using what is sometimes referred to as a side link connection.

(84) The network node 14 has a granularity determiner or controller 18 configured to configure the wireless device to transmit a precoder comprising a first and second beam phase parameter, where the frequency granularity of the beam phase parameters is different. A number of phase values used for the second beam phase parameter may be less than a number of phase values used for the first beam phase parameter. Alternatively, or additionally, a frequency-granularity used for the second beam phase parameter may be greater than a frequency-granularity used for the first beam phase parameter. Similarly, the wireless device 16 has a CSI reporter with configurable granularity 20 configured to transmit a precoder comprising a first and a second beam phase parameter, wherein the granularity of the beam phase parameters is different. A number of phase values used for the second beam phase parameter may be less than a number of phase values used for the first beam phase parameter. Alternatively, or additionally, a frequency-granularity used for the second beam phase parameter may be greater than a frequency-granularity used for the first beam phase parameter.

(85) FIG. 7 is a block diagram of a network node 14 configured to determine a precoder using a multi-beam precoder codebook. The network node 14 has processing circuitry 22. In some embodiments, the processing circuitry may include a memory 24 and processor 26, the memory 24 containing instructions which, when executed by the processor 26, configure processor 26 to perform the one or more functions described herein, including those relating to determining a configuration of a wireless device 16. In addition to a traditional processor and memory, processing circuitry 22 may include integrated circuitry for processing and/or control, e.g., one or more processors and/or processor cores and/or FPGAs (Field Programmable Gate Array) and/or ASICs (Application Specific Integrated Circuitry).

(86) Processing circuitry 22 may comprise and/or be connected to and/or be configured for accessing (e.g., writing to and/or reading from) memory 24, which may comprise any kind of volatile and/or non-volatile memory, e.g., cache and/or buffer memory and/or RAM (Random Access Memory) and/or ROM (Read-Only Memory) and/or optical memory and/or EPROM (Erasable Programmable Read-Only Memory). Such memory 24 may be configured to store code executable by control circuitry and/or other data, e.g., data pertaining to communication, e.g., configuration and/or address data of nodes, etc. Processing circuitry 22 may be configured to control any of the methods described herein and/or to cause such methods to be performed, e.g., by processor 26. Corresponding instructions may be stored in the memory 24, which may be readable and/or readably connected to the processing circuitry 22. In other words, processing circuitry 22 may include a controller, which may include a microprocessor and/or microcontroller and/or FPGA (Field-Programmable Gate Array) device and/or ASIC (Application Specific Integrated Circuit) device. It may be considered that processing circuitry 22 includes or may be connected or connectable to memory, which may be configured to be accessible for reading and/or writing by the controller and/or processing circuitry 22.

(87) The memory 24 is configured to store a granularity 30 of a co-phasing factor for each of a plurality of beams. The memory 24 is further configured to store co-phasing factors 32 for each of a plurality of beams. In some embodiments, the processor 26 is configured to determine a granularity of a co-phasing factor for each of a plurality of beams via a granularity determiner 18. The processor 26 is configured to determine a precoder using a first and second co-phasing factors. A transceiver 28 is configured to receive the first co-phasing factor determined for a first beam with a first granularity, and receive the second co-phasing factor determined for a second beam with a second granularity, the second granularity being greater than the first granularity. In some embodiments, the transceiver 28 is further configured to transmit the granularities to a wireless device 16. Of note, although the term “transceiver” is used herein, it is understood that this term is used for convenience and should not be construed as limiting implementations to a single communication element, e.g., an integrated circuit, that includes both a transmitter and a receiver. It is understood that a physically separate transmitter and receiver can be used.

(88) FIG. 8 is a block diagram of an alternative embodiment of the network node 14, including a memory module 25 configured to store a granularity 30 of a co-phasing factor for each of a plurality of beams and to store co-phasing factors 32. A granularity determiner module 19 is configured to determine a granularity of a co-phasing factor for each of a plurality of beams. A precoder module 35 is configured to determine a precoder using the co-phasing factors. The network node 14 also includes a transceiver module 29 configured to receive co-phasing factors and, in some embodiments, to transmit the granularities to a wireless device 16.

(89) FIG. 9 is a block diagram of a wireless device 16 configured to determine a precoder from a multi-beam precoder codebook. The wireless device 16 includes processing circuitry 42 including a memory 44 and a processor 46. In some embodiments, the processing circuitry may include a memory 44 and processor 46, the memory 44 containing instructions which, when executed by the processor 46, configure processor 46 to perform the one or more functions described herein, including those relating to determining a configuration of a wireless device 16. In addition to a traditional processor and memory, processing circuitry 42 may include integrated circuitry for processing and/or control, e.g., one or more processors and/or processor cores and/or FPGAs (Field Programmable Gate Array) and/or ASICs (Application Specific Integrated Circuitry).

(90) Processing circuitry 42 may comprise and/or be connected to and/or be configured for accessing (e.g., writing to and/or reading from) memory 44, which may comprise any kind of volatile and/or non-volatile memory, e.g., cache and/or buffer memory and/or RAM (Random Access Memory) and/or ROM (Read-Only Memory) and/or optical memory and/or EPROM (Erasable Programmable Read-Only Memory). Such memory 44 may be configured to store code executable by control circuitry and/or other data, e.g., data pertaining to communication, e.g., configuration and/or address data of nodes, etc. Processing circuitry 42 may be configured to control any of the methods described herein and/or to cause such methods to be performed, e.g., by processor 46. Corresponding instructions may be stored in the memory 44, which may be readable and/or readably connected to the processing circuitry 42. In other words, processing circuitry 42 may include a controller, which may include a microprocessor and/or microcontroller and/or FPGA (Field-Programmable Gate Array) device and/or ASIC (Application Specific Integrated Circuit) device. It may be considered that processing circuitry 42 includes or may be connected or connectable to memory, which may be configured to be accessible for reading and/or writing by the controller and/or processing circuitry 42.

(91) The memory 44 is configured to store co-phasing factors. The processor 46 is configured to determine, for each beam, via a granularity determiner 20 a granularity of a co-phasing factor and to determine via a co-phase factor determiner 52, a co-phasing factor for each beam with the determined granularity. The wireless device 16 also includes a transceiver 48 configured to transmit the granularities and co-phasing factors to a network node 14.

(92) FIG. 10 is a block diagram of an alternative embodiment of the wireless device 16. A memory module 45 is configured to store co-phasing factors 50. A granularity determiner module 21 is configured to determine, for each beam, a granularity of a co-phasing factor. A co-phase factor determiner module 53 is configured to determine a co-phasing factor for each beam with the determined granularity. A transceiver module 49 is configured to transmit the granularities and co-phasing factors to a network node 14.

(93) FIG. 11 is a block diagram of an alternative embodiment of the wireless device 16 configured to determine a precoder from a multi-beam precoder codebook. The wireless device 16 includes processing circuitry 62 including a memory 64 and a processor 66. In some embodiments, the processing circuitry may include a memory 64 and processor 66, the memory 64 containing instructions which, when executed by the processor 66, configure processor 66 to perform the one or more functions described herein, including those relating to determining a configuration of a wireless device 16. In addition to a traditional processor and memory, processing circuitry 62 may include integrated circuitry for processing and/or control, e.g., one or more processors and/or processor cores and/or FPGAs (Field Programmable Gate Array) and/or ASICs (Application Specific Integrated Circuitry).

(94) Processing circuitry 62 may comprise and/or be connected to and/or be configured for accessing (e.g., writing to and/or reading from) memory 64, which may comprise any kind of volatile and/or non-volatile memory, e.g., cache and/or buffer memory and/or RAM (Random Access Memory) and/or ROM (Read-Only Memory) and/or optical memory and/or EPROM (Erasable Programmable Read-Only Memory). Such memory 64 may be configured to store code executable by control circuitry and/or other data, e.g., data pertaining to communication, e.g., configuration and/or address data of nodes, etc. Processing circuitry 62 may be configured to control any of the methods described herein and/or to cause such methods to be performed, e.g., by processor 66. Corresponding instructions may be stored in the memory 64, which may be readable and/or readably connected to the processing circuitry 62. In other words, processing circuitry 62 may include a controller, which may include a microprocessor and/or microcontroller and/or FPGA (Field-Programmable Gate Array) device and/or ASIC (Application Specific Integrated Circuit) device. It may be considered that processing circuitry 62 includes or may be connected or connectable to memory, which may be configured to be accessible for reading and/or writing by the controller and/or processing circuitry 62.

(95) The memory 64 is configured to store beam phase parameters 70. The processor 66 is configured to implement a beam phase parameter determiner 72 configured to determine from a codebook an indication of a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam. In some embodiments, the first beam phase parameter takes on one of a first integer number of phase values. The second beam phase parameter takes on one of a second integer number of phase values, the second beam having a lesser power than the first beam and the second integer number of phase values being less than the first integer number of phase values. The transceiver 68 is configured to transmit the selected precoder to a network node 14.

(96) FIG. 12 is a block diagram of an alternative embodiment of the wireless device 16. A memory module 65 is configured to store beam phase parameters 70. A beam phase parameter determiner module 73 may be implemented as software executed by a processor to determine from a codebook an indication of a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam. The transceiver module 69 may be implemented in part by software executed by a processor and is configured to transmit the selected precoder to a network node 14.

(97) FIG. 13 is a flowchart of an exemplary process in a network node 14 for determining a precoder using a multi-beam precoder codebook. The process includes receiving, via the transceiver 28, a first co-phasing factor determined for a first beam with a first granularity (block S100). The process further includes receiving, via the transceiver 28, a second co-phasing factor determined for a second beam with a second granularity, the second granularity being greater than the first granularity (block S102). The process further includes determining, via the processor 26, a precoder using the first and second co-phasing factors (block S104).

(98) FIG. 14 is a flowchart of an exemplary process of determining a precoder from a multi-beam precoder codebook. The process includes determining via a granularity determiner 18, for each beam, a granularity of a co-phasing factor, a granularity of a co-phasing factor for a beam being based on a beam strength, a weaker beam having a lower granularity than a stronger beam (block S106) and determining, via the co-phasing factor determiner 52, a co-phasing factor for each beam with the determined granularity (block S108). The co-phasing factors are transmitted to a network node 14 (block S110).

(99) FIG. 15 is a flowchart of an exemplary process for a wireless device to determine a precoder from a multi-beam precoder codebook. The process includes determining, via a beam phase parameter 72 from a codebook an indication of a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively (block S112). The first beam phase parameter takes on one of a first integer number of phase values and corresponds to a first frequency-granularity. The second beam phase parameter takes on one of a second integer number of phase values and corresponds to a second frequency-granularity (block S112). The process also includes reporting, via the transceiver 68, the determined indication of the precoder to the network node 14 (block S114).

(100) Consider a multi-beam precoder codebook as described above, where the precoders W may be constructed by linearly combining a set of beam components and where at least the phases of each beam may be selected. The rank-1 precoders in such a codebook may be expressed as, for instance,

(101) $\begin{array}{c}W=\left[\begin{array}{cccc}{b}_{I_S\left(1\right)}& b_{I_S\left(2\right)}& \mathrm{.Math.}& b_{I_S\left(N_{\mathrm{SP}}\right)}\end{array}\right]\\ \left[\begin{array}{cccc}1& 0& & \\ 0& \sqrt{p_2}& & \ddots \\ & & \ddots & 0\\ & \ddots & 0& \sqrt{p_{N_{\mathrm{SP}}}}\end{array}\right]\left[\begin{array}{c}1\\ e^{j{\alpha }_2}\\ \mathrm{.Math.}\\ e^{j{\alpha }_{N_{\mathrm{SP}}}}\end{array}\right]\\ =B_{I_S}\sqrt{P}{W}_2.\end{array}\hspace{1em}$

(102) In some embodiments, the precoder structure may not include beam power allocation so that=B.sub.I.sub.SW.sub.2. In other embodiments, the beam power level may be included in W.sub.2. It is noted that there are many equivalent ways of expressing a precoder codebook structure comprising constructing the precoders by linearly combining a set of beam components and where at least the phases of each beam may be selected, the specific codebook structure used herein should be regarded as an example of such a codebook structure but some principles of the disclosure are applicable to other codebook structures as well. On a similar note, only rank-1 precoders are used in the examples and explanations herein, but some principles of the disclosure are applicable to codebooks with precoders of any rank.

(103) Some embodiments employ a (possibly) different quantization granularity for each SP-beam component of the precoder W.

(104) In a first set of embodiments, different frequency-granularities of the phases of each beam component is used. This may be described by letting

(105) $W_2\left(f\right)=\left[\begin{array}{c}1\\ {\varphi }_2\left(.Math.\frac{f}{G_2}.Math.\right)\\ \mathrm{.Math.}\\ {\varphi }_{N_{\mathrm{Sp}}}\left(.Math.\frac{f}{G_{N_{\mathrm{SP}}}}.Math.\right)\end{array}\right],$
where G.sub.i denotes the frequency-granularity of beam i and ϕ.sub.i(k) denotes a phase corresponding to beam i. Thus, the same value of ϕ.sub.i(k) is used for G.sub.i consecutive PRBs in frequency which implies that fewer parameters needs to be fed back as part of the W.sub.2 report. An example of different frequency-granularities is illustrated in FIG. 15. Thus, for a bandwidth of N.sub.RB PRBs and a frequency-granularity of G.sub.i PRBs, the values

(106) ${\varphi }_i\left(1\right),\mathrm{.Math.},{\varphi }_i\left(.Math.\frac{N_{\mathrm{RB}}}{G_i}.Math.\right)$
needs to be ted back for each beam. The resulting W.sub.2 report would require

(107) $.Math.\frac{N_{\mathrm{RB}}}{G_i}.Math..Math.{\log }_{2}K\mathrm{bits}$
per beam and rank and thus

(108) ${.Math.}_{i=2}^{N_{\mathrm{SP}}}.Math.\frac{N_{\mathrm{RB}}}{G_i}.Math..Math.{\log }_{2}K$
bits per rank in total.

(109) In some embodiments, the frequency-granularity of each beam is selected to be a multiple of the subband size. For instance, given a system bandwidth of N.sub.RB=50 in Table 1, the subband size is defined as k.sub.sub=6 PRBs. Hence, the frequency-granularity G.sub.i associated with beam i is chosen as an integer multiple of 6 PRBs. Written more generally, the frequency-granularity G.sub.i associated with beam i is chosen as G.sub.i=m.sub.ik.sub.sub, wherein m.sub.i is the integer multiple associated with beam i. In some embodiments, the network node 14 may semi-statically configure the integer multiple values {m.sub.i} to a wireless device 16. In some further embodiments, the integer multiples m.sub.2, m.sub.3, . . . , m.sub.N.sub.SP may be RRC configured by the network node 14 to the wireless device 16 and the leading beam (i.e., beam i=1) is assumed to have a frequency-granularity of one subband.

(110) In other, similar, embodiments, the frequency-granularity of each beam is selected to be either an integer multiple or a fractional multiple of the subband size. In other words, the multiplicity factor m.sub.i is allowed to be e.g. 0.5.

(111) In some embodiments, the frequency-granularity of each beam is selected from a fixed set of possible values, for instance G.sub.i∈{2,6,10,25} PRBs; and directly encoded with a number of bits.

(112) In such embodiments, the granularity of each beam may be selected by the wireless device 16 and signaled to the network node 14 as part of the CSI report. In some such embodiments, the granularity selection may be semi-persistent so that the same granularity is used for a number of subsequent CSI reports in order to reduce feedback overhead. For instance, the beam frequency-granularities may be considered part of the W.sub.1 report and be transmitted separately from the W.sub.2 report.

(113) In other such embodiments, the granularity is selected by the network node 14 and signaled to the wireless device 16 as for example part of a CSI report trigger contained in a DCI.

(114) A motivation for using different frequency-granularities for different beams is now given. The beam components i are co-phased with the “leading beam” (i.e. beam i=1) in

(115) $W_2=\left[\begin{array}{c}1\\ e^{j{\alpha }_2}\\ \mathrm{.Math.}\\ e^{j{\alpha }_{N_{SP}}}\end{array}\right].$
The optimal phase selection of a beam component i (for i>1), α.sub.i(f), changes as a function of frequency f and will depend on the difference in average delay between beam i and the leading beam (i.e., beam 1). That is, if the average delay of the leading beam and beam i is similar, the phase α.sub.i(f) may change rather slowly over frequency and thus α.sub.i(f) may be reported with a coarser frequency-granularity. If, on the other hand, the difference in average delay is large between beam i and the leading beam, α.sub.i(f) may be reported with a finer frequency-granularity. An illustration of this is given in FIG. 16 where the optimal phase selection of each beam as a function of frequency is shown.

(116) In some embodiments, the wireless device 16 may select the leading beam (i.e., beam i=1) in such a way that it minimizes the frequency-selectivity of the phases of the other beams in the multi-beam precoder. This may for example be done by comparing the average delays τ.sub.i of each beam component and selecting the leading beam as the one that minimizes e.g., the squared difference of the average delays Σ.sub.i(τ.sub.i−τ.sub.1){circumflex over ( )}2. By selecting the leading beam in such a way, the phases of the beams may be selected with a coarser frequency-granularity, resulting in a reduction in feedback overhead.

(117) In another set of embodiments, the phase of each beam component (as a function of frequency, “f”), is factorized into two components as

(118) $e^{j{\alpha }_i\left(f\right)}={\varphi }_{avg,i}\left(f\right).Math.{\varphi }_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)$

(119) where ϕ.sub.avg,i(f)=e.sup.j(a+b.Math.f) is a linear phase shift as a function of frequency that tries to capture the phase shift in frequency due to the difference in average delay of beam i compared to the leading beam, and

(120) 0${\varphi }_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)$
is a remainder phase that is quantized with a certain frequency-granularity as per the earlier embodiments. The idea is that if one compensates for phase shift due to the average delay difference, the remainder phase may be quantized with a coarser frequency-granularity, and thus, the resulting overhead may be reduced. As a motivation for this embodiment, refer to, e.g., “Beam 3 (pol B)” in FIG. 17, where the phase varies rapidly with frequency but with an approximately linear rate. By factoring out the linear phase change due to the average delay, the remainder is less frequency-selective. In FIG. 17, graph 60 is the leading beam, graph 62 is the second beam and graph 64 is the third beam.

(121) In some embodiments, the linear phase coefficient b is set based on an estimated average delay of the beams as b=−2π(τ.sub.i−τ.sub.1)/N.sub.RB. In some embodiments, the linear phase coefficient a is “baked in” with

(122) ${\varphi }_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)$
and set to zero. In other embodiments, it is selected from a PSK alphabet

(123) $a=\frac{2\mathrm{nk}}{K},$
k=0,1, . . . K−1, for some value of K.

(124) If the factorization of the average phase as a linear function of frequency in ϕ.sub.avg,i(f) is “successful”, the remainder phases

(125) ${\varphi }_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)$
may be close to one (i.e.

(126) ${\varphi }_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)\approx 1$
or equivalently.

(127) ${\mathrm{\angle \varphi }}_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)\approx 0).$
It would then make sense to not quantize

(128) ${\varphi }_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)$
in the entire phase range of [−π, +π] but instead quantize in a smaller range, for instance

(129) $\left[-\frac{\pi }{2},+\frac{\pi }{2}\right]$
so that a higher level of detail can be captured with the same amount of overhead. Additionally or alternatively, the remainder phases

(130) ${\varphi }_{rem,i}\left(.Math.\frac{f}{G_i}.Math.\right)$
may be quantized using non-uniform sampling, for instance Gaussian sampling, so that sampling is more dense around 0 radians than for larger phase values.

(131) In another set of embodiments, different quantization granularity of each beam component i is used. For instance, the phases e.sup.jα.sup.i may be selected from a PSK alphabet as

(132) ${\alpha }_i\left(k\right)=\frac{2\pi k}{K_i},$
k=0,1, . . . K.sub.i−1, where the size of the PSK alphabet, K.sub.i, is set separately for each beam component. Beam components with a lower selected power level p.sub.i in W.sub.1 will to a lesser extent contribute to the overall precoding performance. Therefore, if one wishes to minimize to overall absolute quantization error, one may allow that the weaker beams have a larger relative quantization error than the stronger beams and thus assign less bits for phase quantization to the weaker beams (with small selected p.sub.i) than the stronger beams (with large selected p.sub.i).

(133) To illustrate this, the normalized received power from precoding with a multi-beam precoder may be approximated as
∥HW∥.sup.2≈C.Math.∥1+Σ.sub.i=2.sup.N.sup.SPp.sub.ie.sup.j(α.sup.i.sup.−β.sup.i.sup.)∥.sup.2,

(134) where C is a normalization constant and β.sub.i is the actual channel phase of beam component i. The received power may thus be interpreted as the length of a sum of vectors, each with length p.sub.i and angle e.sup.j(α.sup.i.sup.−β.sup.i.sup.) as is illustrated in FIG. 18. Thus, the phase error is magnified with the beam power p.sub.i, so that a small phase error for a strong beam may have a larger impact than a larger phase error for a weak beam.

(135) The size of the PSK alphabet used for each beam, K.sub.i, and thus the number of bits assigned for phase quantization of the beam may depend solely on the beam power level p.sub.i. In an exemplary embodiment, beams with 0.75≤p.sub.i≤1 uses K.sub.i=8 while beams with 0≤p.sub.i<0.75 use K.sub.i=4.

(136) In a further embodiment, the network node 14 semi-statically configures a wireless device 16 (e.g., via RRC signaling) with two different PSK alphabet sizes K.sub.1 and K.sub.2, where K.sub.1>K.sub.2 The wireless device 16 applies the PSK alphabet K.sub.1 to beams with associated power levels that exceed a preconfigured power threshold P.sub.Th (That is, PSK alphabet K.sub.1 is used for beams having associated power level p.sub.i>P.sub.Th). The wireless device 16 applies the PSK alphabet K.sub.2 for beams with associated power levels below the power threshold P.sub.Th.

(137) Let

(138) 0$W_2\left(f_i\right)=\left[\begin{array}{c}1\\ e^{j{\alpha }_2\left(f_i\right)}\\ \mathrm{.Math.}\\ e^{j{\alpha }_{N_{SP}}\left(f_i\right)}\end{array}\right]$
be the W.sub.2 on subband with frequency f.sub.i(i=1,2, . . . , M), where M is the total number of subbands. Instead of quantizing and encoding the phase α.sub.k(f.sub.i), the phase differences between adjacent subbands, i.e. Δα.sub.k(f.sub.i)=α.sub.k(f.sub.i)−α.sub.k(f.sub.i-1) (with Δα.sub.k(f.sub.1)=α.sub.k(f.sub.1)) may be encoded and fed back to the network node 14. Since the channel delay spread within each beam is relatively small, the phase change over frequency (or subbands) is also small. Therefore, Δα.sub.k(f.sub.i) may be encoded with less number of bits. An example is shown in Table 3, where 3 bits are used for encoding the phase of the first subband and 2 bits for rest of the subbands. Total 3+2(M−1) bits are needed in the example instead of 3M bits if 3 bits are used for phase encoding in each subband, a saving of M−2 bits or about 33% with M=6.

(139) TABLE-US-00004 TABLE 3 Subband Number of quantization bits index Phase difference (K) Quantized phases 1 Δα.sub.k(f.sub.1) = α.sub.k(f.sub.1) 3 $\frac{2\pi k}{K},\left(k=0,1,\mathrm{.Math.},K-1\right)$ 2 Δα.sub.k(f.sub.2) = α.sub.k(f.sub.2) − α.sub.k(f.sub.1) 2 $\left(-\frac{\pi }{2},-\frac{\pi }{4},0,+\frac{\pi }{4},+\frac{\pi }{2}\right)$ . . . . . . . . . . . . M Δα.sub.k(f.sub.M) = α.sub.k(f.sub.M) − α.sub.k(f.sub.M−1) 2 $\left(-\frac{\pi }{2},-\frac{\pi }{4},0,+\frac{\pi }{4},+\frac{\pi }{2}\right)$

(140) The phase α.sub.k(f.sub.i) of each subband can be restored at the network node 14 after receiving the feedback of {Δα.sub.k(f.sub.i), i=1,2, . . . , M} as follows:
α.sub.k(f.sub.1)=Δα.sub.k(f.sub.1)
α.sub.k(f.sub.i)=Δα.sub.k(f.sub.i)+α.sub.k(f.sub.i-1), i=2, . . . ,M.

(141) In another embodiment, the phase differences between adjacent subbands Δα.sub.k(f.sub.i)=α.sub.k(f.sub.i)−α.sub.k(f.sub.i-1) are encoded with a variable length code, such as a Huffman code, so that small phase changes (that are more probable) are encoded with a small number of bits while larger, less likely, phase changes are encoded with a larger number of bits. An example of such a code is given in FIG. 19. FIG. 19, illustrated as a binary tree with the encoded value as leaves and the bit string representing it on the branches. As seen, a phase change of 0 radians is encoded with the bit string “1” while a phase change of

(142) $-\frac{\pi }{2}$
radians is encoded with “0010” and so forth. Thus, the encoding of the phase changes

(143) $0,0,\frac{\pi }{2},0,-\frac{\pi }{4},0,\frac{\pi }{4},0$
is done with the bit string “110011101010111”, requiring 15 bits to encode 8 values. With this approach, the average number of bits required to encode the phase changes between subbands can be significantly reduced, if the phase varies slowly across frequency.

(144) Principles of the disclosure may also be applied in the case where parametric phase encoding over frequency, as described above is used. That is, the phase e.sup.jα.sup.i.sup.(f) of each beam i is parametrically encoded over frequency as a polynomial phase function, i.e.
e.sup.jα.sup.i.sup.(f)=e.sup.jΣ.sup.m=0.sup.M.sup.i.sup.α.sup.m,i.sup.f.sup.m.

(145) Thus, in an embodiment of the disclosure, the order of the polynomial function M.sub.i for each beam may be set separately. For beams that illuminate a small number of multi-path components in the channel and thus likely experiences a low delay spread, a low polynomial order may be chosen, such as M.sub.i=0 or M.sub.i=1 corresponding to a constant phase function and linear phase function over frequency, respectively. The corresponding overhead for those beams is thus low since only 1 or 2 coefficients are needed to parametrize the phase change over frequency. For other beams that illuminate a larger number of multi-path components in the channel and thus likely experiences a larger delay spread, a higher polynomial order may be used, e.g. M.sub.i=5, requiring feedback of 6 polynomial coefficients.

(146) The polynomial orders of the beams M.sub.0, M.sub.1, . . . may be decided by either the network node 14 or the wireless device 16 and may be signaled in a similar fashion as the embodiments discussed above.

(147) Thus, some embodiments advantageously provide a method, wireless device and network node for determining a granularity for precoder overhead optimization. According to one aspect, a method in a wireless device 16 of determining a precoder from a multi-beam precoder codebook is provided. The method includes determining via the granularity determiner 20, for each beam, a granularity of a co-phasing factor, a granularity of a co-phasing factor for a beam being based on a beam strength, a weaker beam having a lower granularity than a stronger beam. The method also includes determining, via the co-phasing factor determiner 52, a co-phasing factor for each beam with the determined granularity. The method further includes transmitting, via the transceiver 48, co-phasing factors to a network node 14.

(148) In some embodiments, the method further includes transmitting the granularities to the network node 14. In some embodiments, a granularity of a co-phasing factor for a beam is based on a beam strength, a weaker beam having a lower granularity than a stronger beam. In some embodiments, the method further includes determining, via the granularity determiner 20, a frequency granularity of each beam to be a multiple of a subband size. In some embodiments, the method further includes determining, via the granularity determiner 20, a granularity of a phase shift keyed, PSK, constellation. In some embodiments, the PSK constellation is 8 PSK for beams having a beam strength above a first threshold and is quadrature PSK (QPSK) for beams having a beam strength below a second threshold. In some embodiments, the first and second thresholds are the same. In some embodiments, the method further includes differentially encoding a phase of each a plurality of frequency subbands of a beam. In some embodiments, the method further includes comprising parametrically encoding a phase of a beam versus frequency of the beam.

(149) In some embodiments, a wireless device 16 is configured to determine a precoder from a multi-beam precoder codebook. The wireless device 16 includes processing circuitry 42 including a memory 44 and a processor 46. The memory 44 is configured to store co-phasing factors. The processor 46 is configured to determine, for each beam, a granularity of a co-phasing factor, a granularity of a co-phasing factor for a beam being based on a beam strength, a weaker beam having a lower granularity than a stronger beam. The processor 46 is also configured to determine a co-phasing factor for each beam with the determined granularity. The wireless device 16 also includes a transceiver 48 configured to transmit the co-phasing factors to a network node 14.

(150) In some embodiments, the processor 46 is further configured to transmit the determined granularities to the network node 14. In some embodiments, the processor 46 is further configured to determine a frequency granularity of each beam to be a multiple of a subband size.

(151) In some embodiments, the processor 46 is further configured to determine a granularity of a phase shift keyed, PSK, constellation. In some embodiments, the PSK constellation is 8 PSK for beams having a beam strength above a first threshold and is quadrature PSK (QPSK) for beams having a beam strength below a second threshold. In some embodiments, the first and second thresholds are the same. In some embodiments, the processor 46 is further configured to differentially encode a phase of each of a plurality of frequency subbands of a beam.

(152) In some embodiments, a wireless device 16 is configured to determine a precoder from a multi-beam precoder codebook. The wireless device 16 includes a memory module 45 configured to store co-phasing factors, a granularity determiner module 21 configured to determine, for each beam, a granularity of a co-phasing factor, a granularity of a co-phasing factor, a granularity of a co-phasing factor for a beam being based on a beam strength, a weaker beam having a lower granularity than a stronger beam. The wireless device 16 also includes a co-phasing factor determiner module 53 configured to determine a co-phasing factor for each beam with the determined granularity. The wireless device 16 also includes a transceiver module 49 configured to transmit the co-phasing factors to a network node 14.

(153) In some embodiments, a method for a wireless device 16 of reporting a precoder to a network node 14 is provided. The method includes determining from a codebook an indication of a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively. The first beam phase parameter takes on one of a first integer number of phase values and corresponds to a first frequency-granularity S112. The second beam phase parameter takes on one of a second integer number of phase values and corresponds to a second frequency-granularity. Further, at least one of the following conditions applies: the second integer number of phase values is less than the first number of phase values, and the second frequency-granularity is greater than the first frequency-granularity. The method includes determined indication of the precoder transmitting the determined indication of the precoder to the network node. In some embodiments, the second beam has a lesser power than the first beam S114.

(154) In some embodiments, a method for a wireless device 16 of reporting a precoder to a network node 14 is provided. The method includes determining from a codebook a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively S112. The first beam phase parameter takes on one of a first integer number of phase values. The second beam phase parameter takes on one of a second integer number of phase values. The second beam has a lesser power than the first beam and the second integer number of phase values is less than the first integer number of phase values. The method also includes reporting the selected precoder to a network node S114.

(155) In some embodiments, the method also includes determining the first and second integer number of phase values, and, optionally, transmitting the first and second integer number of phase values to the network node. In some embodiments, the method also includes determining a frequency-granularity of each beam to be a multiple of a subband size. In some embodiments, the first and second integer number of phase values are respective number of values attainable in a phase shift keyed, PSK, constellation. In some embodiments, each of the first and second beams is a k.sup.th beam, d(k), that has associated a set of complex numbers and has index pair (l.sub.k,m.sub.k), each element of the set of complex numbers being characterized by at least one complex phase shift such that: d.sub.n(k)=d.sub.i(k)α.sub.i,ne.sup.j2π(pΔ.sup.1,k.sup.+qΔ.sup.2,k.sup.); d.sub.n(k), and d.sub.i(k) are the i.sup.th and n.sup.th elements of d(k), respectively; α.sub.i,n is a real number corresponding to the i.sup.th and n.sup.th elements of d(k); p and q are integers; and beam directions Δ.sub.1,k and Δ.sub.2,k are real numbers corresponding to beams with index pair (l.sub.k,m.sub.k) that determine complex phase shifts e.sup.j2πΔ.sup.1,k and e.sup.j2πΔ.sup.2,k respectively; and each of the first and second beam phase parameters is a complex coefficient c.sub.k for d(k) used to adjust at least the phase of the i.sup.th element of d(k) according to c.sub.kd.sub.i(k).

(156) In some embodiments, the first and second integer number of phase values are respective granularities of respective co-phasing factors for the first and second beams. In some embodiments, the PSK constellation is 8 PSK for beams having a beam strength above a first threshold and the PSK constellation is quadrature PSK (QPSK) for beams having a beam strength below a second threshold. In some embodiments, the first and second thresholds are the same. In some embodiments, the method further includes differentially encoding at least one of the first and second beam phase parameters, wherein each of the first and second beam phase parameters corresponds to a plurality of frequency subbands. In some embodiments, a first plurality of first beam phase parameters and a second plurality of second beam phase parameters correspond to the first beam and second beam, respectively. Further the method may include parametrically encoding at least one of the first plurality and second plurality of first and second beam phase parameters, where the at least one of the first plurality and second plurality of beam phase parameters are coefficients within a predetermined function over frequency.

(157) In some embodiments, a wireless device 16 is configured to transmit a precoder to a network node 14. The wireless device 16 includes processing circuitry 62 including a memory 64 and a processor 66. In some embodiments, the memory 64 is configured to store beam phase parameters. The processor 66 is configured to determine from a codebook an indication of a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively. The first beam phase parameter takes on one of a first integer number of phase values and corresponds to a first frequency-granularity. The second beam phase parameter takes on one of a second integer number of phase values and corresponds to a second frequency-granularity, where at least one of the following conditions apply: the second integer number of phase values is less than the first number of phase values, and the second frequency-granularity is greater than the first frequency-granularity. The wireless device 16 also includes a transceiver 68 configured to transmit the determined indication of the precoder to the network node. In some embodiments, the second beam has a lesser power than the first beam.

(158) In some embodiments, a wireless device 16 is configured to transmit a precoder to a network node 14. The wireless device 16 includes processing circuitry 62 including a memory 64 and a processor 66. The memory 64 is configured to store beam phase parameters. The processor 66 is configured to implement a beam phase parameter determiner 72 to determine from a codebook an indication of a precoder comprising a first beam phase parameter and a second beam phase parameter corresponding to a first beam and second beam, respectively. The first beam phase parameter takes on one of a first integer number of phase values. The second beam phase parameter takes on one of a second integer number of phase values, the second beam having a lesser power than the first beam and the second integer number of phase values being less than the first integer number of phase values. The wireless device 16 also includes a transceiver 68 configured to transmit the selected precoder to a network node 14.

(159) In some embodiments, the processor 66 is further configured to determine the first and second integer number of phase values, and transmitting the first and second integer number of phase values to the network node. In some embodiments, the processor 66 is further configured to determine a frequency-granularity of each beam to be a multiple of a subband size.

(160) In some embodiments, the first and second integer number of phase values are respective number of values attainable in a phase shift keyed, PSK, constellation. In some embodiments, each of the first and second beams is a k.sup.th beam, d(k), that has associated a set of complex numbers and has index pair (l.sub.k, m.sub.k), each element of the set of complex numbers being characterized by at least one complex phase shift such that: d.sub.n(k)=d.sub.i(k)α.sub.i,ne.sup.j2π(pΔ.sup.1,k.sup.+qΔ.sup.2,k.sup.); d.sub.n(k), and d.sub.i(k) are the i.sup.th and n.sup.th elements of d(k), respectively; α.sub.i,n is a real number corresponding to the i.sup.th and n.sup.th elements of d(k); p and q are integers; and beam directions Δ.sub.1,k and Δ.sub.2,k are real numbers corresponding to beams with index pair (l.sub.k, m.sub.k) that determine complex phase shifts e.sup.j2πΔ.sup.1,k and e.sup.j2πΔ.sup.2,k respectively; and each of the first and second beam phase parameters is a complex coefficient c.sub.k for d(k) used to adjust at least the phase of the i.sup.th element of d(k) according to c.sub.kd.sub.i(k).

(161) In some embodiments, the first and second integer number of phase values are respective granularities of respective co-phasing factors for the first and second beams. In some embodiments, the PSK constellation is 8 PSK for beams having a beam strength above a first threshold and the PSK constellation is quadrature PSK (QPSK) for beams having a beam strength below a second threshold. In some embodiments, the first and second thresholds are the same. In some embodiments, the processor 66 is further configured to differentially encode at least one of the first and second beam phase parameters, wherein each of the first and second beam phase parameters corresponds to a plurality of frequency subbands. In some embodiments, a first plurality of first beam phase parameters and a second plurality of second beam phase parameters correspond to the first beam and second beam, respectively. The processor 66 is further configured to parametrically encode at least one of the first plurality and second plurality of first and second beam phase parameters, where the at least one of the first plurality and second plurality of beam phase parameters are coefficients within a predetermined function over frequency.

(162) In some embodiments, a wireless device 16 includes a memory module 65 configured to store beam phase parameters 70. The wireless device 16 also includes a beam phase determiner module 73 configured to determine first and second beam phase parameters corresponding to first and second beams, respectively. The first beam phase parameter takes on one of a first integer number of phase values. The second beam phase parameter taking on one of a second integer number of phase values, the second beam having a lesser power than the first beam and the second integer number of phase values being less than the first integer number of phase values. The wireless device 16 also includes a transceiver module 69 configured to transmit the selected precoder to a network node 14.

(163) In some embodiments, a method in a network node 14 for determining a precoder using a multi-beam precoder codebook is provided. The method includes receiving via the transceiver 28, a first co-phasing factor determined for a first beam with a first frequency granularity, receiving a second co-phasing factor determined for a second beam with a second frequency granularity, the second frequency granularity being greater than the first frequency granularity, and determining a precoder 34 using the first and second co-phasing factors.

(164) In some embodiments, the method includes determining, via a granularity determiner 18, a frequency granularity of a co-phasing factor for each of a plurality of beams, a frequency granularity of a co-phasing factor for a beam being based on a beam strength, a weaker beam having a lower frequency granularity than a stronger beam, and transmitting the frequency granularities to the wireless device 16. According to this aspect, in some embodiments, a frequency granularity of a co-phasing factor for a beam is based on a beam strength, a weaker beam having a lower granularity than a stronger beam. In some embodiments, the method includes determining, via the granularity determiner 18, a frequency granularity of each beam to be a multiple of a subband size. In some embodiments, the method further includes determining, via the granularity determiner 18, a frequency granularity of a phase shift keyed, PSK, constellation. In some embodiments, the PSK constellation is 8 PSK for beams having a beam strength above a threshold and is quadrature PSK (QPSK) for beams having a beam strength below the threshold.

(165) In some embodiments, a network node 14 for determining a precoder using a multi-beam precoder codebook is provided. The network node 14 includes processing circuitry 22 including a memory 24 and a processor 26. The memory 24 is configured to store co-phasing factors for each of a plurality of beams. The processor 26 is configured to receive a first co-phasing factor determined for a first beam with a first frequency granularity and receive a second co-phasing factor determined for a second beam with a second frequency granularity, the second frequency granularity being greater than the first frequency granularity. The processor 26 is further configured to determine a precoder using the first and second co-phasing factors.

(166) In some embodiments, the processor 26 is configured to determine a granularity of a co-phasing factor for each of a plurality of beams, a frequency granularity of a co-phasing factor for a beam being based on a beam strength, a weaker beam having a lower frequency granularity than a stronger beam, and. The network node 14 further includes a transceiver 28 configured to transmit the frequency granularities to a wireless device 16. According to this aspect, in some embodiments, the processor 26 is further configured to determine a frequency granularity of each beam to be a multiple of a subband size. In some embodiments, the processor 26 is further configured to determine a granularity of a phase shift keyed, PSK, constellation. In some embodiments, the PSK constellation is 8 PSK for beams having a beam strength above a threshold and is quadrature PSK (QPSK) for beams having a beam strength below the threshold. In some embodiments, the processor 26 is further configured to differentially encode a phase of each subband.

(167) In some embodiments, a network node 14 is configured to determine a precoder using a multi-beam precoder codebook. The network node 14 includes a transceiver module 29 configured to receive a first co-phasing factor determined for a first beam with a first granularity and receive a second co-phasing factor determined for a second beam with a second granularity, the second granularity being greater than the first granularity. The network node 14 further includes a precoder module 35 configured to determine a precoder using the first and second co-phasing factors Some embodiments include:

(168) Embodiment 1. A method of determining a precoder from a multi-beam precoder codebook, the method comprising: determining, for each beam, a granularity of a co-phasing factor; determining a co-phasing factor for each beam with the determined granularity; and transmitting the granularities and co-phasing factors to a network node.

(169) Embodiment 2. The method of Embodiment 1, wherein a granularity of a co-phasing factor for a beam is based on a beam strength, a weaker beam having a lower granularity than a stronger beam.

(170) Embodiment 3. The method of Embodiment 1, further comprising determining a frequency granularity of each beam to be a multiple of a subband size.

(171) Embodiment 4. The method of Embodiment 1, further comprising determining a granularity of a phase shift keyed, PSK, constellation.

(172) Embodiment 5. The method of Embodiment 1, further comprising differentially encoding a phase of each subband.

(173) Embodiment 6. A wireless device configured to determine a precoder from a multi-beam precoder codebook, the wireless device comprising: processing circuitry including a memory and a processor; the memory configured to store co-phasing factors; and the processor configured to: determine, for each beam, a granularity of a co-phasing factor; and determine a co-phasing factor for each beam with the determined granularity; and a transceiver configured to transmit the granularities and co-phasing factors to a network node.

(174) Embodiment 7. The wireless device of Embodiment 6, wherein a granularity of a co-phasing factor for a beam is based on a beam strength, a weaker beam having a lower granularity than a stronger beam.

(175) Embodiment 8. The wireless device of Embodiment 6, further comprising determining a frequency granularity of each beam to be a multiple of a subband size.

(176) Embodiment 9. The wireless device of Embodiment 6, further comprising determining a granularity of a phase shift keyed, PSK, constellation.

(177) Embodiment 10. The wireless device of Embodiment 6, further comprising differentially encoding a phase of each subband.

(178) Embodiment 11. A wireless device configured to determine a precoder from a multi-beam precoder codebook, the wireless device comprising: a memory module configured to store co-phasing factors; a granularity determiner module configured to determine, for each beam, a granularity of a co-phasing factor; and a co-phase factor determiner module configured to determine a co-phasing factor for each beam with the determined granularity; and a transceiver module configured to transmit the granularities and co-phasing factors to a network node.

(179) Embodiment 12. A method of configuring a wireless device via a network node, the method comprising: determining a granularity of a co-phasing factor for each of a plurality of beams; and transmitting the granularities to the wireless device.

(180) Embodiment 13. The method of Embodiment 12, wherein a granularity of a co-phasing factor for a beam is based on a beam strength, a weaker beam having a lower granularity than a stronger beam.

(181) Embodiment 14. The method of Embodiment 12, further comprising determining a frequency granularity of each beam to be a multiple of a subband size.

(182) Embodiment 15. The method of Embodiment 12, further comprising determining a granularity of a phase shift keyed, PSK, constellation.

(183) Embodiment 16. The method of Embodiment 12, further comprising differentially encoding a phase of each subband.

(184) Embodiment 17. A network node configured to determine a configuration of a wireless device, the network node comprising: processing circuitry including a memory and a processor: the memory configured to store a granularity of a co-phasing factor for each of a plurality of beams; and the processor configured to determine a granularity of a co-phasing factor for each of a plurality of beams; and a transceiver configured to transmit the granularities to a wireless device.

(185) Embodiment 18. The network node of Embodiment 17, wherein a granularity of a co-phasing factor for a beam is based on a beam strength, a weaker beam having a lower granularity than a stronger beam.

(186) Embodiment 19. The network node of Embodiment 17, further comprising determining a frequency granularity of each beam to be a multiple of a subband size.

(187) Embodiment 20. The network node of Embodiment 17, further comprising determining a granularity of a phase shift keyed, PSK, constellation.

(188) Embodiment 21. The network node of Embodiment 17, further comprising differentially encoding a phase of each subband.

(189) Embodiment 22. A network node configured to determine a configuration of a wireless device, the network node comprising: a memory module configured to store a granularity of a co-phasing factor for each of a plurality of beams; a granularity determiner module configured to determine a granularity of a co-phasing factor for each of a plurality of beams; and a transceiver module configured to transmit the granularities to a wireless device.

(190) As will be appreciated by one of skill in the art, the concepts described herein may be embodied as a method, data processing system, and/or computer program product. Accordingly, the concepts described herein may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects all generally referred to herein as a “circuit” or “module.” Furthermore, the disclosure may take the form of a computer program product on a tangible computer usable storage medium having computer program code embodied in the medium that can be executed by a computer. Any suitable tangible computer readable medium may be utilized including hard disks, CD-ROMs, electronic storage devices, optical storage devices, or magnetic storage devices.

(191) Some embodiments are described herein with reference to flowchart illustrations and/or block diagrams of methods, systems and computer program products. It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer (to thereby create a special purpose computer), special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks.

(192) These computer program instructions may also be stored in a computer readable memory or storage medium that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer readable memory produce an article of manufacture including instruction means which implement the function/act specified in the flowchart and/or block diagram block or blocks.

(193) The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks.

(194) It is to be understood that the functions/acts noted in the blocks may occur out of the order noted in the operational illustrations. For example, two blocks shown in succession may in fact be executed substantially concurrently or the blocks may sometimes be executed in the reverse order, depending upon the functionality/acts involved. Although some of the diagrams include arrows on communication paths to show a primary direction of communication, it is to be understood that communication may occur in the opposite direction to the depicted arrows.

(195) Computer program code for carrying out operations of the concepts described herein may be written in an object oriented programming language such as Java® or C++. However, the computer program code for carrying out operations of the disclosure may also be written in conventional procedural programming languages, such as the “C” programming language. The program code may execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer. In the latter scenario, the remote computer may be connected to the user's computer through a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider).

(196) Many different embodiments have been disclosed herein, in connection with the above description and the drawings. It will be understood that it would be unduly repetitious and obfuscating to literally describe and illustrate every combination and subcombination of these embodiments. Accordingly, all embodiments can be combined in any way and/or combination, and the present specification, including the drawings, shall be construed to constitute a complete written description of all combinations and subcombinations of the embodiments described herein, and of the manner and process of making and using them, and shall support claims to any such combination or subcombination.

(197) Abbreviations used in the preceding description include: 1D One dimensional 2D Two-Dimensional 3GPP Third Generation Partnership Project 5G Fifth Generation ACK Acknowledgement ASIC Application Specific Integrated Circuit ARQ Automatic Retransmission Request CA Carrier Aggregation CB Codebook CDMA Code Division Multiple Access CFAI CSI Feedback Accuracy Indicator CFI Control Information Indicator CP Cyclic Prefix CPU Central Processing Unit CQI Channel Quality Indicators CRS Common Reference Symbol/Signal CSI Channel State Information CSI-RS Channel State Information Reference Symbol/Signal dB Decibel DCI Downlink Control Information DFT Discrete Fourier Transform DL Downlink eNB Enhanced or Evolved Node B DP Dual Polarization EPC Evolved Packet Core EPDCCH Enhanced Physical Downlink Control Channel EPRE Energy per Resource Element E-UTRAN Evolved or Enhanced Universal Terrestrial Radio Access Network FDD Frequency Division Duplexing FD-MIMO Full Dimension MIMO FFT Fast Fourier Transform FPGA Field Programmable Gate Array GSM Global System for Mobile Communications HARQ Hybrid ARQ ID Identifier IFFT Inverse FFT LSB Least Significant Bit LTE Long Term Evolution M2M Machine-to-Machine MCS Modulation and Coding Scheme (or State) MIMO Multiple Input Multiple Output MME Mobility Management Entity MSB Most Significant Bit MU-MIMO Multi-User MIMO NAK Non-Acknowledgement NZP Non-Zero Power OCC Orthogonal Cover Code OFDM Orthogonal Frequency Division Multiplexing PCFICH Physical Control Format Indicator Channel PDA Personal Data Assistance PDCCH Physical Downlink Control Channel PDSCH Physical Downlink Shared Channel PRB Physical Resource Block PMI Precoder Matrix Indicator PSK Phase Shift Keying PUCCH Physical Uplink Control Channel PUSCH Physical Uplink Shared Channel QPSK Quadrature Phase Shift Keying RB Resource Block RE Resource Element Rel Release RI Rank Indicator RRC Radio Resource Control SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SP Single Polarization SR Scheduling Request SU-MIMO Single User MIMO TDD Time Division Duplexing TFRE Time/Frequency Resource Element TP Transmission Point TS Technical Specification Tx Transmit UE User Equipment UL Uplink ULA Uniform Linear Array UMB Ultra Mobile Broadband UPA Uniform Planar Array WCDMA Wideband Code Division Multiple Access ZP Zero Power

(198) It will be appreciated by persons skilled in the art that the embodiments described herein are not limited to what has been particularly shown and described herein above. In addition, unless mention was made above to the contrary, it should be noted that all of the accompanying drawings are not to scale. A variety of modifications and variations are possible in light of the above teachings without departing from the scope of the following claims.