Methods and Apparatus for Multi-Domain Conversions of High Dimensional Channel Statistics

Abstract

A method includes calculating first high dimensional channel characteristics of a first frequency band in accordance with measurements of the first frequency band; estimating a power angle delay spectrum (PADS) in accordance with the first high dimensional channel characteristics of the first frequency band; determining a domain conversion matrix for a second frequency band in accordance with the PADS; and generating second high dimensional channel characteristics of the second frequency band in accordance with the domain conversion matrix and the first high dimensional channel characteristics of the first frequency band.

Claims

1. A method, the method comprising: calculating, by a device, first high dimensional channel characteristics of a first frequency band in accordance with measurements of the first frequency band; estimating, by the device, a power angle delay spectrum (PADS) in accordance with the first high dimensional channel characteristics of the first frequency band; determining, by the device, a domain conversion matrix for a second frequency band in accordance with the PADS; and generating, by the device, second high dimensional channel characteristics of the second frequency band in accordance with the domain conversion matrix and the first high dimensional channel characteristics of the first frequency band.

2. The method of claim 1, wherein the determining the domain conversion matrix comprises: determining the domain conversion matrix in accordance with at least one of an antenna structure of the device, a first carrier frequency of the first frequency band, a second carrier frequency of the second frequency band, and configured frequency spacing of the first and second frequency bands.

3. The method of claim 2, wherein estimating the PADS comprises: estimating the PADS in accordance with the antenna structure of the device and the carrier frequency of the first frequency band.

4. The method of claim 3, wherein the estimating the PADS comprises: projecting the PADS onto a subspace of a Hilbert space defined by basis functions in accordance with the antenna structure of the device and the first carrier frequency of the first frequency band.

5. The method of claim 3, wherein estimating the PADS comprises: generating a set of weights of a set of basis functions defined in accordance with the antenna structure of the device and the first carrier frequency of the first frequency band.

6. The method of claim 1, wherein the generating the second high dimensional channel characteristics of the second frequency band comprises: obtaining, by the device, a product of the domain conversion matrix and the PADS; and reforming, by the device, the second high dimensional channel characteristics of the second frequency band from the product.

7. The method of claim 1, wherein determining the domain conversion matrix comprises: determining a discrete domain conversion matrix for each configured antenna lag.

8. The method of claim 1, wherein the generating the second high dimensional channel characteristics of the second frequency band comprises: obtaining, by the device, subblock matrices for each configured frequency lag as a product of the domain conversion matrix and the PADS; reforming, by the device, subblocks of the second high dimensional channel characteristics of the second frequency band from the subblock matrices; and reconstructing, by the device, the second high dimensional channel characteristics of the second frequency band from the subblocks.

9. The method of claim 1, wherein the first high dimensional channel characteristics comprises first space-frequency covariance, or wherein the second high dimensional channel characteristics comprises second space-frequency covariance.

10. The method of claim 1, wherein the first frequency band and the second frequency band are identical.

11. The method of claim 1, wherein the first frequency band and the second frequency band are different.

12. The method of claim 1, further comprising: transmitting, by the device, uplink transmissions using the first frequency band; and receiving, by the device, downlink transmissions using the second frequency band.

13. A device comprising: one or more processors; and a non-transitory memory storage comprising instructions that, when executed by the one or more processors, cause the device to: calculate first high dimensional channel characteristics of a first frequency band in accordance with measurements of the first frequency band; estimate a power angle delay spectrum (PADS) in accordance with the first high dimensional channel characteristics of the first frequency band; determine a domain conversion matrix for a second frequency band in accordance with the PADS; and generate second high dimensional channel characteristics of the second frequency band in accordance with the domain conversion matrix and the first high dimensional channel characteristics of the first frequency band.

14. The device of claim 13, wherein determination of the domain conversion matrix is in accordance with at least one of an antenna structure of the device, a first carrier frequency of the first frequency band, a second carrier frequency of the second frequency band, and configured frequency spacing of the first and second frequency bands.

15. The device of claim 14, wherein the PADS is estimated in accordance with the antenna structure of the device and the first carrier frequency of the first frequency band.

16. The device of claim 13, wherein the instructions that cause the device to generate the second high dimensional channel characteristics include instructions to cause the device to: obtain a product of the domain conversion matrix and the PADS; and reform the second high dimensional channel characteristics of the second frequency band from the product.

17. The device of claim 13, wherein the instructions that cause the device to determine the domain conversion matrix include instructions to cause the device to: determine a discrete domain conversion matrix for each configured antenna lag.

18. The device of claim 13, wherein the instructions that cause the device to generate the second high dimensional channel characteristics include instructions to cause the device to: obtain subblock matrices for each configured frequency lag as a product of the domain conversion matrix and the PADS; reform subblocks of the second high dimensional channel characteristics of the second frequency band from the subblock matrices; and reconstruct the second high dimensional channel characteristics of the second frequency band from the reformed subblocks.

19. The device of claim 13, wherein the first high dimensional channel characteristics comprises first space-frequency covariance, or wherein the second high dimensional channel characteristics comprises second space-frequency covariance.

20. A non-transitory computer-readable media storing computer instructions for communicating over multiple frequency bands, that when executed by one or more processors, cause the one or more processors to perform operations of: calculating first high dimensional channel characteristics of a first frequency band in accordance with measurements of the first frequency band; estimating a power angle delay spectrum (PADS) in accordance with the high dimensional channel characteristics of the first frequency band; determining a domain conversion matrix for a second frequency band in accordance with the PADS; and generating second high dimensional channel characteristics of the second frequency band in accordance with the domain conversion matrix and the first high dimensional channel characteristics of the first frequency band.

21. The non-transitory computer-readable media of claim 20, the estimating the PADS comprising: projecting the PADS onto a subspace of a Hilbert space defined by basis functions in accordance with an antenna structure and a first carrier frequency of the first frequency band.

22. The non-transitory computer-readable media of claim 20, the generating the second high dimensional channel characteristics of the second frequency band comprising: obtaining a product of the domain conversion matrix and the PADS; and reforming the high dimensional channel characteristics of the second frequency band from the product.

23. The non-transitory computer-readable media of claim 20, the determining the domain conversion matrix: determining a discrete domain conversion matrix for each configured antenna lag.

24. The non-transitory computer-readable media of claim 20, the generating the second high dimensional channel characteristics of the second frequency band comprising: obtaining subblock matrices for each configured frequency lag as a product of the domain conversion matrix and the PADS; reforming subblocks of the second high dimensional channel characteristics of the second frequency band from the subblock matrices; and reconstructing the second high dimensional channel characteristics of the second frequency band from the subblocks.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0049] For a more complete understanding of the present disclosure, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

[0050] FIG. 1 illustrates an example communications system;

[0051] FIG. 2A illustrates a communication system used in presenting a signal model utilized in the discussion presented herein;

[0052] FIG. 2B illustrates a diagram of a 3-D vector of a channel path;

[0053] FIG. 3 illustrates an example process for generating the spatial-frequency covariance of a communication system;

[0054] FIG. 4 illustrates an exemplary space-frequency frame structure of a communication system as a function of time and the illustration of space-frequency covariance;

[0055] FIG. 5 illustrates a flow diagram of example operations occurring in a device converting spatial-frequency covariance R.sub.SF according to example embodiments presented herein;

[0056] FIG. 6A illustrates a flow diagram of example operations occurring in a device estimating the power angle delay spectrum (PADS) for the uplink according to example embodiments presented herein;

[0057] FIG. 6B illustrates a flow diagram of example operations occurring in a device converting the spatial-frequency covariance for the uplink R.sub.SF.sup.UL to the spatial-frequency covariance for the downlink R.sub.SF.sup.DL according to example embodiments presented herein;

[0058] FIG. 7A illustrates a flow diagram of example operations occurring in a device estimating the PADS for the uplink with asymptotic solutions according to example embodiments presented herein;

[0059] FIG. 7B illustrates a flow diagram of example operations occurring in a device converting the spatial-frequency covariance for the uplink R.sub.SF.sup.UL to the spatial-frequency covariance for the downlink R.sub.SF.sup.DL, with asymptotic solutions according to example embodiments presented herein;

[0060] FIGS. 8A and 8B illustrate antenna panels with the antenna panels serving different frequency bands according to example embodiments presented herein;

[0061] FIG. 8C illustrate a third antenna panel serving two different frequency bands according to example embodiments presented herein;

[0062] FIGS. 9A and 9B illustrate antenna panels with different steering directions according to example embodiments presented herein;

[0063] FIG. 10 illustrates a flow diagram of example operations occurring in a device converting spatial-frequency covariance where the device has two antenna panels serving different frequency bands according to example embodiments presented herein;

[0064] FIG. 11A illustrates a flow diagram of example operations occurring in a device converting spatial-frequency covariance in the spatial and frequency domains for two antenna panels with different locations according to example embodiments presented herein;

[0065] FIG. 11B illustrates a flow diagram of example operations occurring in a device converting spatial-frequency covariance in the spatial and frequency domains for two antenna panels with different locations, with asymptotic solutions, according to example embodiments presented herein;

[0066] FIG. 12A illustrates a flow diagram of example operations occurring in a device converting spatial-frequency covariance in the spatial and frequency domains for two antenna panels with different steering directions according to example embodiments presented herein;

[0067] FIG. 12B illustrates a flow diagram of example operations occurring in a device converting spatial-frequency covariance in the spatial and frequency domains for two antenna panels with different steering directions, with asymptotic solutions, according to example embodiments presented herein;

[0068] FIG. 13A illustrates a diagram of two ULA antenna arrays with different antenna spacings according to example embodiments presented herein;

[0069] FIG. 13B illustrates a diagram of two ULA antenna arrays with different steering directions according to example embodiments presented herein;

[0070] FIG. 14 illustrates an example communication system according to example embodiments presented herein;

[0071] FIGS. 15A and 15B illustrate example devices that may implement the methods and teachings according to this disclosure; and

[0072] FIG. 16 is a block diagram of a computing system that may be used for implementing the devices and methods disclosed herein.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

[0073] The structure and use of disclosed embodiments are discussed in detail below. It should be appreciated, however, that the present disclosure provides many applicable concepts that can be embodied in a wide variety of specific contexts. The specific embodiments discussed are merely illustrative of specific structure and use of embodiments, and do not limit the scope of the disclosure.

[0074] FIG. 1 illustrates an example communications system 100. Communications system 100 includes an access node 105 serving user equipments (UEs), such as UEs 120. In a first operating mode, communications to and from a UE passes through access node 105. In a second operating mode, communications to and from a UE do not pass through access node 105, however, access node 105 typically allocates resources used by the UE to communicate when specific conditions are met. Communication between a UE and access node pair occur over uni-directional communication links, where the communication links between the UE and the access node are referred to as uplinks 130, and the communication links between the access node and UE is referred to as downlinks 135.

[0075] Access nodes may also be commonly referred to as Node Bs, evolved Node Bs (eNBs), next generation (NG) Node Bs (gNBs), master eNBs (MeNBs), secondary eNBs (SeNBs), master gNBs (MgNBs), secondary gNBs (SgNBs), network controllers, control nodes, base stations, access points, transmission points (TPs), transmission-reception points (TRPs), cells, carriers, macro cells, femtocells, pico cells, and so on, while UEs may also be commonly referred to as mobile stations, mobiles, terminals, tablets, laptop PCs, users, subscribers, stations, and the like. Access nodes may provide wireless access in accordance with one or more wireless communication protocols, e.g., the Third Generation Partnership Project (3GPP) long term evolution (LTE), LTE advanced (LTE-A), 5G, 5G LTE, 5G NR, sixth generation (6G), High Speed Packet Access (HSPA), the IEEE 802.11 family of standards, such as 802.11a/b/g/n/ac/ad/ax/ay/be, etc. While it is understood that communications systems may employ multiple access nodes capable of communicating with a number of UEs, only one access node and two UEs are illustrated for simplicity.

[0076] As discussed previously, in time division duplex (TDD) communication systems, channel information for a first channel may be obtained through channel measurements of a second channel due to channel reciprocity. As an example, channel information for a downlink channel may be obtained through channel measurements of a corresponding uplink channel due to channel reciprocity. However, the reciprocity that applies for instantaneous channel state information (CSI) does not hold in frequency division duplex (FDD) communication systems where training and measurements of the first channel, along with information feedback is required.

[0077] Due to the large number of antennas present in massive multiple input multiple output (MIMO) communication systems, the computational complexity associated with downlink channel training and the feedback overhead associated with information feedback is very large. The computational complexity is further exacerbated in computationally limited UEs, which may be incapable of providing computational resources needed to train, or generate the channel information and feedback information.

[0078] Furthermore, in the case of wideband transmissions, inter-symbol interference (ISI) becomes a critical issue because, due to multipath channel fading, the transmitted signal arrives at the receiver at different times on different paths. Orthogonal frequency division multiplexing (OFDM) has been adopted in 4G and 5G communication systems to combat ISI and simplify receiver design. Because multipath fading introduces frequency selectivity, there is significant channel variance on different frequency subbands. With only spatial channel covariance, channel training has to be processed independently along with the frequency domain, which leads to increased training complexity and overhead. Because the channel spatial covariance only includes channel statistics related to the spatial (or equivalently, angular) domain, a key component of channel characteristics is missed, namely, delays over different channel paths.

[0079] According to an example embodiment, methods and apparatus are provided that utilize high dimensional (HiDi) channel characteristics to capture both spatial (angular) and delay information of a channel. As an example, a spatial-frequency covariance matrix is transformed from the power angle delay spectrum rather than the power angle spectrum for the spatial channel covariance matrix. Information regarding the radio environment of a gNB or the coverage of the gNB may help improve the network performance (e.g., spectrum access, user handover, radio resource management, and so on). HiDi provides an improved radio environment map because it captures multi-dimensional channel characteristics.

[0080] In an embodiment, the conversion of the spatial-frequency covariance on the downlink based on the uplink observation allows for improving the downlink channel training or cross subband channel filtering in FDD communication systems. However, besides FDD communication systems, the conversion of the spatial-frequency covariance may be applied to any communication system that operates on multiple frequency bands with the same antenna array aperture, as long as the carrier frequency spacing between any two bands is not too great. Additionally, assumption that the power angle delay spectrum on these frequency bands is the same (or approximately the same) remains valid. The ability to obtain HiDi channel statistics of a first frequency band by converting HiDi channel statistics of a second frequency band benefits the system design on the first frequency band by enabling efficient wideband channel training, beam management, and so on, without requiring the determination of the HiDi channel statistics of the first frequency band through training.

[0081] FIG. 2A illustrates a communication system 200 used in presenting a signal model utilized in the discussion presented herein. Communication system 200 includes an access node 205 and a UE 210. Access node 205 includes an antenna array with M antennas 207, while UE 210 includes a single receive antenna 212. For a discrete spatial channel model, assuming that there are L channel paths, the channel response at the antenna array (denoted h(t)=[h.sub.i(t), . . . , h.sub.M(t)].sup.T) may be expressed as

h(t)=Σ.sub.l=1.sup.Lc.sub.l(t)a(θ.sub.l,ϕ.sub.l)δ(t−τl),  (1)

where a(θ.sub.l, ϕ.sub.l)=[a.sub.1(θ.sub.l, ϕ.sub.l), . . . , a.sub.M(θ.sub.l, ϕ.sub.l)].sup.T is a normalized beam vector on the channel from the angle of arrival (AoA) θ based on the antenna location and

[00001]$a_m\left(\theta ,\varphi \right)=e^{-\frac{j2\pi }{\lambda }{k\left(\theta ,\varphi \right)}^T{u}_m},$

k(θ, ϕ)=[sin(θ)sin(ϕ), cos(θ)sin(ϕ), cos(ϕ)].sup.T, θ∈[−π, π], ϕ∈[0, π], and u.sub.m is in the three-dimensional (3-D) spatial coordinates of antenna in.

[0082] FIG. 2B illustrates a diagram 250 of a 3-D vector of a channel path 255. Channel path 255 may be represented in angular form with angles θ 260 and ϕ 262 or in Cartesian coordinate form as sin θ sin ϕ 270 (in the X axis), cos θ sin ϕ 272 (in the Y axis), and cos ϕ 274 (in the Z axis).

[0083] If a two-dimensional (2-D) antenna array is considered, then k(θ)=[sin(θ), cos(θ)].sup.T and u.sub.m is the 2-D spatial coordinates of antenna in. Additionally, in Equation (1), c.sub.l(t), θ.sub.l, ϕ.sub.l, and τ.sub.l are the instantaneous complex channel gain, the horizontal and vertical AoA, and the delay of the l-th path, respectively, and δ(⋅) is the delta function with δ(0)=1 and 0 otherwise. Then {c.sub.l(t)}=p.sub.l. The channel statistics can be specified as the power angle delay profiles, i.e., {p.sub.l, θ.sub.l, ϕ.sub.l, τ.sub.l}.sub.l=1.sup.L. Without including the large scale channel parameters, e.g., pathloss or receive signal-to-noise ratio (SNR) in general, it is assumed that Σ.sub.lp.sub.l=1.

[0084] In the discussion presented herein, the focus is on 2-D systems. However, the example embodiments presented are operable with 3-D systems for any planar antenna array. Therefore, the focus on 2-D systems should not be construed as being limiting to the spirit of the example embodiments.

[0085] Considering wideband systems with OFDM modulation, after FFT at the receiver, the fading channel in the frequency domain on subcarrier k is expressible as

[00002]$\begin{array}{cc}h\left[k\right]\left(t\right)=Fh\left(t\right)=\sqrt{\frac{1}{N_F}}{.Math.}_{l=1}^L{c}_l\left(t\right)a\left({\theta }_l\right)e^{-j2\pi k{f}_{sc}{\tau }_{\mathrm{.Math.}}},& \left(2\right)\end{array}$

where f.sub.sc is the subcarrier spacing and N.sub.F is the Fast Fourier Transfer (FFT) size.

[0086] Given the instantaneous channel in the frequency domain, channel measurements may be taken at different subcarriers k.sub.1, k.sub.2, . . . , k.sub.N, i.e., h [k.sub.1], . . . , h [k.sub.N]. The channel measurements may be vectorized as

[00003]$\begin{array}{cc}h\stackrel{\Delta }{=}{\left[{h\left[k_1\right]}^T,.Math.,{h\left[k_N\right]}^T\right]}^T.& \left(3\right)\end{array}$

[0087] A first version of the spatial-frequency covariance (a form of HiDi channel statistics) is expressible as

[00004]$\begin{array}{cc}{R}_{\mathrm{SF}}=𝔼\left\{\tilde{h}{\tilde{h}}^H\right\}=\left[\begin{array}{cccc}𝔼\left\{h\left[k_1\right]{h\left[k_1\right]}^H\right\}& 𝔼\left\{h\left[k_1\right]{h\left[k_2\right]}^H\right\}& .Math.& 𝔼\left\{h\left[k_1\right]{h\left[k_N\right]}^H\right\}\\ 𝔼\left\{h\left[k_2\right]{h\left[k_1\right]}^H\right\}& 𝔼\left\{h\left[k_2\right]{h\left[k_2\right]}^H\right\}& .Math.& 𝔼\left\{h\left[k_2\right]{h\left[k_N\right]}^H\right\}\\ .Math.& .Math.& \ddots & .Math.\\ 𝔼\left\{h\left[k_N\right]{h\left[k_1\right]}^H\right\}& 𝔼\left\{h\left[k_N\right]{h\left[k_2\right]}^H\right\}& .Math.& 𝔼\left\{h\left[k_N\right]{h\left[k_N\right]}^H\right\}\end{array}\right].& \left(4\right)\end{array}$

[0088] From Equation (2), each subblock {h[k.sub.i]h[k.sub.j].sup.H} in Equation (4) is expressible as

[00005]$\begin{array}{cc}\begin{array}{c}𝔼\left\{h\left[k_i\right]{h\left[k_j\right]}^H\right\}=𝔼\left\{{.Math.}_{l=1}^L{c}_{l_1}a\left({\theta }_{l_l}\right)e^{-j2\pi k_i{f}_{sc}{\tau }_{l_1}}{.Math.}_{l_2=1}^L{c}_{l_2}^{*}{a\left({\theta }_{l_2}\right)}^H{e}^{j2\pi k_j{f}_{sc}{\tau }_{l_2}}\right\}\\ =𝔼\left\{{.Math.}_{l=1}^L{.Math.c_l.Math.}^{2}a\left({\theta }_l\right){a\left({\theta }_l\right)}^H{e}^{-j2\pi \left(k_i-k_j\right)f_{sc}{\tau }_l}\right\}\\ ={.Math.}_{l=1}^L{p}_{l}a\left({\theta }_l\right){a\left({\theta }_l\right)}^H{e}^{-j2\pi \left(k_i-k_j\right)f_{sc}{\tau }_l},\end{array}& \left(5\right)\end{array}$

where the second equality follows the assumption that the complex channel gain of each path c.sub.l is independent.

[0089] From Equation (5), {h[k.sub.i]h[k.sub.j].sup.H} depends only on the difference of the subcarriers, i.e., Δ.sub.k.sub.ij=k.sub.i−k.sub.j, instead of the absolute values of k.sub.i or k.sub.j. Hence, when k.sub.i=k.sub.j, Δ.sub.k.sub.ij=0. Then,

{h[k.sub.i]h[k.sub.i].sup.H}=Σ.sub.l=1.sup.Lp.sub.la(θ.sub.l)a(θ.sub.l).sup.H, ∀i,  (6)

which reduces to the wideband spatial covariance R.

[0090] Based on the above observation, it is possible to rewrite Equation (4) and define a new version of spatial-frequency covariance. The channel on a frequency subcarrier k,

[00006]$k+{\Delta }_{k_1},.Math.,k_{{\Delta }_{k_{N_{{\mathrm{SF}}^{-1^{\prime }}}}}},$

i.e., h[k], h[k+Δ.sub.k.sub.1], . . . ,

[00007]$h\left[k\right],h\left[k+{\Delta }_{k_1}\right],.Math.,h\left[k+{\Delta }_{k_{N_{\mathrm{SF}}-1}}\right],$

is selected, where {Δ.sub.k.sub.i} are the frequency lags considered in the spatial-frequency covariance. Furthermore, the following are set: Δ.sub.k.sub.i≠Δ.sub.k.sub.j and Δ.sub.k.sub.i≠0, ∀ i, j.

[0091] The spatial channels on these subcarriers with different frequency lags are vectorized to obtain

[00008]$\begin{array}{cc}\breve{h}\stackrel{\Delta }{=}{\left[{h\left[k\right]}^T,{h\left[k+{\Delta }_{k_1}\right]}^T.Math.,{h\left[k+{\Delta }_{k_{N_{SF}-1}}\right]}^T\right]}^T.& \left(7\right)\end{array}$

[0092] The new spatial-frequency covariance matrix, defined as

[00009]$\begin{array}{cc}{R}_{\mathrm{SF}}=𝔼\left\{\breve{h}{\breve{h}}^H\right\}=\left[\begin{array}{cccc}𝔼\left\{h\left[k\right]{h\left[k\right]}^H\right\}& 𝔼\left\{h\left[k\right]{h\left[k+{\Delta }_{k_1}\right]}^H\right\}& .Math.& 𝔼\left\{h\left[k\right]{h\left[k+{\Delta }_{k_{N_{\mathrm{SF}}}-1}\right]}^H\right\}\\ 𝔼\left\{h\left[k+{\Delta }_{k_1}\right]{h\left[k\right]}^H\right\}& 𝔼\left\{h\left[k+{\Delta }_{k_1}\right]{h\left[k+{\Delta }_{k_1}\right]}^H\right\}& .Math.& 𝔼\left\{h\left[k+{\Delta }_{k_1}\right]{h[k+{\Delta }_{k_{N_{\mathrm{SF}}}-1}]}^H\right\}\\ .Math.& .Math.& \ddots & .Math.\\ 𝔼\left\{h\left[k+{\Delta }_{k_{N_{\mathrm{SF}}}-1}\right]{h\left[k\right]}^H\right\}& 𝔼\left\{h\left[k+{\Delta }_{k_{N_{\mathrm{SF}}}-1}\right]{h\left[k+{\Delta }_{k_1}\right]}^H\right\}& .Math.& 𝔼\left\{h\left[k+{\Delta }_{k_{N_{\mathrm{SF}}}-1}\right]{h\left[k+{\Delta }_{k_{N_{\mathrm{SF}}}-1}\right]}^H\right\}\end{array}\right]=\left[\begin{array}{cccc}{R}_0& R_{{\Delta }_{k_1}}^H& .Math.& R_{{\Delta }_{k_{N_{\mathrm{SF}}}-1}}^H\\ R_{{\Delta }_{k_1}}& R_0& .Math.& R_{{\Delta }_{k_{N_{\mathrm{SF}}}-2}}^H\\ .Math.& .Math.& \ddots & .Math.\\ R_{{\Delta }_{k_{N_{\mathrm{SF}}}-1}}& R_{{\Delta }_{k_{N_{\mathrm{SF}}}-2}}& .Math.& R_0\end{array}\right],& \left(8\right)\end{array}$

where N.sub.SF is the number of frequency lags in the covariance including the zero lag and

[00010]$\begin{array}{cc}{R}_{{\Delta }_{k_i}}=\left\{h\left[k+{\Delta }_{k_i}\right]{h\left[k\right]}^H\right\}={.Math.}_{l=1}^L{p}_{l}a\left({\theta }_l\right){a\left({\theta }_l\right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{sc}{\tau }_l}.& \left(9\right)\end{array}$

[0093] From Equation (9), the new spatial-frequency covariance matrix is not dependent on an arbitrary subcarrier k, but on all frequency lags Δ.sub.k.sub.i, with the first frequency lag being zero lag.

[0094] Also the path power p.sub.l is a function of the path angle and delay. For the continuous case, a continuous function representing power angle delay spectrum ρ(θ, τ) is used. Then the block entries of spatial-frequency covariance matrix as the function of ρ(θ, τ) are expressible as

[00011]$\begin{array}{cc}{R}_{{\Delta }_{k_i}}={\int }_0^{+\infty }{\int }_{-\pi }^{\pi }\rho \left(\theta ,\tau \right)a\left(\theta \right){a\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{\mathrm{sc}}\tau }d\theta d\tau ,i=0,.Math.,N_{\mathrm{SF}}-1.& \left(10\right)\end{array}$

[0095] Naturally, ∫.sub.0.sup.+∞ ∫.sub.−π.sup.πρ(θ, τ)dθdτ<+∞ due to the total channel power constraint. In practice, it is assumed that there is a largest delay spread threshold on the channel delay, denoted as τ.sub.DS. Later, an asymptotic result is obtained by takin τ.sub.DS to ∞. For a given τ.sub.DS,

[00012]$\begin{array}{cc}{R}_{{\Delta }_{k_i}}={\int }_0^{{\tau }_{\mathrm{DS}}}{\int }_{-\pi }^{\pi }\rho \left(\theta ,\tau \right)a\left(\theta \right){a\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{\mathrm{sc}}\tau }d\theta d\tau .& \left(11\right)\end{array}$

R.sub.SF is a Hermitian. For the uplink (UL) and the downlink (DL) in FDD communication systems,

[00013]$\begin{array}{cc}{R}_{{\Delta }_{k_i}}^{UL}={\int }_0^{{\tau }_{DS}}{\int }_{-\pi }^{\pi }\rho \left(\theta ,\tau \right)a^{UL}\left(\theta \right){a^{UL}\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{sc}\tau }d\theta d\tau & \left(12\right)\end{array}$$\begin{array}{cc}{R}_{{\Delta }_{k_i}}^{\mathrm{DL}}={\int }_0^{{\tau }_{\mathrm{DS}}}{\int }_{-\pi }^{\pi }\rho \left(\theta ,\tau \right)a^{\mathrm{DL}}\left(\theta \right){a^{\mathrm{DL}}\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{\mathrm{sc}}\tau }d\theta d\pi ,& \left(13\right)\end{array}$

where the entries for a.sup.UL(θ) and a.sup.DL(θ) are expressible as

[00014]$a_m^{UL}\left(\theta \right)=e^{-\frac{j2\pi }{{\lambda }^{UL}}{k\left(\theta \right)}^{T_{u_m}}}$$\mathrm{and}$$a_m^{DL}\left(\theta \right)=e^{-\frac{j2\pi }{{\lambda }^{DL}}{k\left(\theta \right)}^{T_{u_m}}},$

respectively.

[0096] According to an example embodiment, methods and apparatus for obtaining the spatial-frequency covariance matrix R.sub.SF of a FDD communication system are provided. The spatial-frequency covariance matrix may be determined in accordance with measurements of the channels made at difference subcarriers.

[0097] FIG. 3 illustrates an example process 300 for generating the spatial-frequency covariance of a communication system. Process 300 may be used in FDD or multiband communication systems. Process 300 may also be used as input for TDD systems. Process 300 includes a frequency domain channel estimation unit 305, a time-frequency offset compensation unit 310, a spatial-frequency vectorization unit 315, and a spatial-frequency correlation unit 320. Frequency domain channel estimation unit 305 is configured to derive channel estimates (in the frequency domain) of the channels of the communication system from received signals. Time-frequency offset compensation unit 310 is configured to provide compensation for the time offsets present in the channel estimates. Spatial-frequency vectorization unit 315 is configured to vectorize the time compensated channel estimates. Spatial-frequency correlation unit 320 is configured to correlate the vectorized channel estimates to determine the spatial-frequency covariance.

[0098] FIG. 4 illustrates an exemplary space-frequency frame structure 400 of a communication system as a function of time and frequency. FIG. 4 also illustrates space-frequency covariance. Space-frequency diagram 400 displays MT distinct time-frequency frames 405-407, one for each of the MT transmit antennas. Also illustrated in FIG. 4 are time-frequency resources used for estimating the channels, such as resources 410-413 of time-frequency frame 407, as well as similar resources of time-frequency frames 405 and 406. The space-frequency covariance, including the cross correlations, for the resources on different subcarriers and different antennas are illustrated in FIG. 4 as arrowed lines (for example, arrowed lines 420 and similar arrowed lines of time-frequency frames 405 and 406). The resources used for estimating the channels are pilot signals that are usually separated by a constant number of subcarriers. The space-frequency covariance depends on the frequency difference between subcarriers and not arbitrary subcarrier locations. As an example resource 410 is spaced one subcarrier away from resource 411, which is spaced one subcarrier away from resource 412, and so on.

[0099] According to an embodiment, a conversion of the spatial-frequency covariance for a first frequency band to the spatial-frequency covariance for a second frequency band is provided. The ability to convert an existing spatial-frequency covariance to another spatial-frequency covariance may eliminate training requirements, thereby reducing computational requirements and communication overhead, and improve overall communication performance. As an example, in a FDD communication system, the spatial-frequency covariance for an uplink channel may be obtained using uplink transmissions and computations. However, because the instantaneous channel for the downlink is not known, UE measurements and feedback are needed to determine the spatial-frequency covariance in the downlink. Being able to convert the spatial-frequency covariance for the uplink channel from uplink measurement into the spatial-frequency covariance for the downlink channel without UE measurement and feedback would facilitate the efficient channel training and reduce training and feedback overhead. As another example, the spatial-frequency covariance for a first frequency band may be converted from the spatial-frequency covariance for a second frequency band, again with similar savings in computational resources and communication overhead.

[0100] Although the discussion focuses on spatial and frequency domain conversion, the example embodiments presented herein are also operable with single domain conversion. As an example, in a situation where the first frequency band and the second frequency band are the same, then spatial-frequency domain conversion becomes only spatial domain conversion (because the frequency domain does not change). Therefore, the discussion of spatial-frequency domain conversion should not be construed as being limiting to the scope of the example embodiments.

[0101] FIG. 5 illustrates a flow diagram of example operations 500 occurring in a device converting spatial-frequency covariance R.sub.SF. Operations 500 may be indicative of operations occurring in a device as the device converts the spatial-frequency covariance. The device may be an access node. The device may also be a UE (or other similar devices). The device may alternatively be a separate device configured for generating spatial-frequency covariance for communicating devices utilizing information provided by the communicating devices. As an example, the device may be a device located in the communication system dedicated to receiving channel information from communicating devices and converting spatial-frequency covariance for the communicating devices.

[0102] Operations 500 begin with the device obtaining a spatial-frequency covariance of the uplink R.sub.SF.sup.UL (block 505). The device may make channel measurements and generate the spatial-frequency covariance based on transmissions occurring in the uplink. As an example, the device may include an implementation of process 300 of FIG. 3 to generate spatial-frequency covariance. Alternatively, the device may receive channel measurements or frequency domain channel estimates from a communicating device and the device generates the spatial-frequency covariance based on the channel measurements or the frequency domain channel estimates. As an example, the device may include an implementation of process 300 of FIG. 3 to generate spatial-frequency covariance from the received channel measurements or the frequency domain channel estimates.

[0103] The device estimates the power angle delay spectrum (PADS) for the uplink (block 507). In an embodiment, a technique based on a projection onto the Hilbert space is utilized to estimate the PADS. Without specifying any one particular carrier frequency band, the spatial-frequency covariance is vectorized and is expressible as

r=vec([{{R.sub.SF},{R.sub.SF}]),

where vec (X) denotes a vectorizing operation that vectorizes the matrix X into a vector. The n-th entry of r, r.sub.n, is then expressible as

r.sub.n=∫.sub.0.sup.τ.sup.DS∫.sub.−π.sup.πρ(θ,τ)g.sub.n(θ,τ)dθdτ, n=1, . . . ,N,  (14)

where g.sub.n(θ, τ) is the entry of vectorized angular and delay term

[00015]$ca\left(\theta \right){a\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{sc}\tau }$

corresponds to r.sub.n, with real and imaginary parts separated, i.e., the mapping from [−π, π]×[0, τ.sub.DS].fwdarw., c is the normalization factor which is defined as , and N is total number of entries r.sup.UL, i.e., N=2 (M.sup.2−M)N.sub.SF. The normalization is a scaling factor, which may not affect the results for any given τ.sub.DS because the normalization simply scales the PADS with a common factor. The normalization will be scaled back with the conversion. However, the normalization is important to form the solution when τ.sub.DS goes to infinity.

[0104] A Hilbert space of a real function in L.sup.2 space with inner product is defined, and is expressible as

(f,g=∫.sub.0.sup.τ.sup.DS∫.sub.−π.sup.πf(θ,τ)g(θ,τ)dθdτ.  (15)

The L−2 norm is then defined as

[00016]$\begin{array}{cc}{.Math.f.Math.}^2\stackrel{\Delta }{=}.Math.f,f.Math.={\int }_0^{{\tau }_{DS}}{\int }_{-\pi }^{\pi }{f}^2\left(\theta ,\tau \right)d\theta d\tau .& \left(16\right)\end{array}$

Equation (14) may be rewritten as

r.sub.n=ρ,g.sub.n, n=1, . . . ,N.  (17)

Then, for the uplink,

r.sub.nuL=ρ,g.sub.n.sup.UL, n=1, . . . ,N  (18)

where g.sub.n.sup.UL is the entry of

[00017]${\mathrm{ca}}^{\mathrm{UL}}\left(\theta \right){a^{\mathrm{UL}}\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{\mathrm{sc}}\tau }$

corresponding to r.sub.n.sup.UL.

[0105] Therefore, the problem becomes: for a given {r.sub.n.sup.UL} and {g.sub.n.sup.UL (θ, τ)}, the PADS ρ(θ, T) is estimated. The solution involves solving a linear equation array if the PADS ρ is a discrete vector of the quantized pair (θ, τ). However, {ρ(θ, τ)} is a continuous function of the 2-D variables (θ, τ), and the problem becomes intractable with conventional methods.

[0106] In order to solve the PADS estimation in Equation (18), the problem may be reformulated and solve the problem with the projection method in Hilbert space. First, without loss of generality, it is assumed that g.sub.n.sup.UL(θ, τ), n=1, . . . , N′ are the linearly independent vector subset of all g.sub.n.sup.UL(θ, τ), n=1, . . . , N. Then the subspace V of ρ∈ specified by {g.sub.n.sup.UL(θ, τ)}.sub.n=1.sup.N′ is defined as

[00018]$\begin{array}{cc}V\stackrel{△}{=}{.Math.}_{n=1}^N{V}_n{V}_n\stackrel{△}{=}\left\{\rho \in \mathscr{H}:.Math.\rho ,g_n^{\mathrm{UL}}.Math.=r_n^{\mathrm{UL}}\right\},n=1.Math.,N^{\prime }.& \left(19\right)\end{array}$

[0107] To obtain the estimation of PADS ρ(θ, T), the minimum norm criterion is considered, i.e.,

[00019]$\begin{array}{cc}\hat{\rho }\left(\theta ,\tau \right)=\arg \underset{\rho \in V}{\min }{.Math.\rho .Math.}^2,& \left(20\right)\end{array}$

where the subspace V is defined in Equation (19). The solution of Equation (20) will be the orthogonal projection of PADS ρ(θ, τ) on the orthogonal subspace of the linear variety V, which will be the projection to the subspace V. The estimate of PADS is then expressible as

{circumflex over (ρ)}(θ,τ)=Σ.sub.n=1.sup.N′α.sub.ng.sub.n.sup.UL(θ,τ),  (21)

where {α.sub.n} are the solution of linear equations, which are expressible as

[00020]$\begin{array}{cc}\begin{array}{ccc}.Math.g_1^{UL},g_1^{UL}.Math.{\alpha }_1+.Math.g_1^{UL},g_2^{UL}.Math.{\alpha }_2+.Math.+.Math.g_1^{UL},g_N^{UL}.Math.{\alpha }_{N^{\prime }}& =& r_1^{UL}\\ .Math.& .Math.& \\ .Math.g_{N^{\prime }}^{UL},g_1^{UL}.Math.{\alpha }_1+.Math.g_{N^{\prime }}^{UL},g_2^{UL}.Math.{\alpha }_2+.Math.+.Math.g_{N^{\prime }}^{UL},g_{N^{\prime }}^{UL}.Math.{\alpha }_{N^{\prime }}& =& r_{N^{\prime }}^{UL}\end{array}.& \left(22\right)\end{array}$

[0108] If α=[α.sub.1, . . . , α.sub.N′].sup.T, the linear array of Equation (22) may be rewritten as

Gα=r.sup.UL,  (23)

where G is defined as

[00021]$\begin{array}{cc}G=\left[\begin{array}{ccc}.Math.g_1^{\mathrm{UL}},g_1^{\mathrm{UL}}.Math.& .Math.& .Math.g_1^{\mathrm{UL}},g_{N^{\prime }}^{\mathrm{UL}}.Math.\\ .Math.& \ddots & .Math.\\ .Math.g_{N^{\prime }}^{\mathrm{UL}},g_1^{\mathrm{UL}}.Math.& .Math.& .Math.g_{N^{\prime }}^{\mathrm{UL}},g_{N^{\prime }}^{\mathrm{UL}}.Math.\end{array}\right].& \left(24\right)\end{array}$

[0109] The above linear equations (Equation (24)) are guaranteed to have at least one solution. From Equation (21), it is seen that the PADS can be represented as the weighted sum of independent basis functions of angles and delays, g.sub.n.sup.UL(θ, τ), with weights α.sub.n, n=1, . . . , N′. In mathematics, a basis function is an element of a particular basis for a function space. The basis functions may enable the representation of a more complex function based on the simpler basis functions. Every continuous function in the function space may be represented as a linear function of basis functions, just as every vector in a vector space may be represented as a linear combination of basis vectors. From equations (22)-(24), estimating PADS becomes estimating the weights α As explained before, g.sub.n(θ, τ) is the entry of vectorized angular and delay function of

[00022]$\mathrm{ca}\left(\theta \right){a\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{\mathrm{sc}}\tau }$

with real and imaginary parts separated. Therefore, the basis functions g.sub.n(θ, τ), n=1, . . . , N′, depend on the antenna structure, e.g. antenna positions, the carry frequency of the frequency band, and frequency lags Δ.sub.k.sub.i. In general, the basis functions do not change over the UEs. However, when the UE changes or the channel statistics for the UE changes, the power angle delay spectrum of the UE may still be represented as a linear combination in Equation (21) with the same basis functions g.sub.n(θ, T) but with different coefficients or weights α.sub.n.

[0110] The device determines a frequency domain conversion for the spatial-frequency covariance (block 509). In order to obtain the spatial-frequency covariance in the downlink, it is assumed that the same PADS 92 (θ, τ) applies. After obtaining the estimated PADS {circumflex over (ρ)}(θ, τ), by using Equation (21), for example, the following is obtained for the downlink

{circumflex over (r)}.sub.n.sup.DL={circumflex over (ρ)},g.sub.n.sup.DL=Σ.sub.n′−1.sup.N′α.sub.n′g.sub.n′.sup.UL,g.sub.n.sup.DL,  (25)

where {circumflex over (r)}.sub.n.sup.DL is the estimate of n-th entry of r.sup.DL and r.sup.DL=vec ([{R.sub.SF}, {R.sub.SF}]). Then, the vectorized spatial-frequency covariance in the downlink is expressible as

{circumflex over (r)}.sup.DL=Qα,  (26)

where Q is the frequency domain conversion for the HiDi channel statistics and is expressible as

[00023]$\begin{array}{cc}Q=\left[\begin{array}{ccc}.Math.g_1^{\mathrm{UL}},g_1^{\mathrm{DL}}.Math.& .Math.& .Math.g_{N^{\prime }}^{\mathrm{UL}},g_1^{\mathrm{DL}}.Math.\\ .Math.& \ddots & .Math.\\ .Math.g_{N^{\prime }}^{\mathrm{UL}},g_N^{\mathrm{DL}}.Math.& .Math.& .Math.g_{N^{\prime }}^{\mathrm{UL}},g_N^{\mathrm{DL}}.Math.\end{array}\right].& \left(27\right)\end{array}$

[0111] The device converts the spatial-frequency covariance of the uplink to spatial-frequency covariance of the downlink (block 511). The device may convert the spatial-frequency covariance of the uplink to the spatial-frequency covariance of the downlink by reforming the spatial-frequency covariance of the downlink from the vectorized spatial-frequency covariance in the uplink. As an example, the reforming of the spatial-frequency covariance in the downlink may proceed in a manner that is in reverse of the way that the vectorized spatial-frequency covariance in the uplink is formed. The device communicates in accordance with the spatial-frequency covariance of the downlink (block 513). As an example, the device uses the spatial-frequency covariance of the downlink to perform training and facilitate downlink transmissions to other devices.

[0112] Although the discussion presented herein focuses on converting the spatial-frequency covariance in the downlink from the spatial-frequency covariance in the uplink, the example embodiments presented herein are operable for converting the spatial-frequency covariance of a first frequency band to the spatial-frequency covariance of a second frequency band. The conversion of the spatial-frequency covariance from the uplink spatial-frequency covariance to the downlink spatial-frequency covariance may be viewed as a special case of the first frequency band and the second frequency band, where the first frequency band is the uplink and the second frequency band is the downlink. Therefore, the discussion of uplink and downlink should not be construed as being limiting to the scope of the example embodiments.

[0113] FIG. 6A illustrates a flow diagram of example operations 600 occurring in a device estimating the PADS for the uplink. Operations 600 may be indicative of operations occurring in a device as the device estimates the PADS for the uplink. Operations 600 may be an example implementation of block 507 of FIG. 5. Although the discussion focuses on uplink and downlink, the example embodiments are operable for any two frequency bands. In such a situation, the uplink would be a first frequency band and the downlink would be a second frequency band.

[0114] Input to operations 600 may include the spatial-frequency covariance R.sub.SF.sup.UL, which may be determined in accordance with measurements of the uplink (block 602). Operations 600 begin with the device selecting non-repeating entries of R.sub.SF.sup.UL and vectorizing R.sub.SF.sup.UL to form a real vector r.sub.SF.sup.UL (block 605). The selection may be based on hermitian or other data structure, such as Toeplitz or block Teoplitz structures. As an example, the real vector r.sub.SF.sup.UL is expressible as

r.sub.SF.sup.UL=Vec{}}R.sub.SF.sup.UL},{R.sub.SF.sup.UL}}.

[0115] The device forms an estimation matrix G (block 607). The estimation matrix G may be formed in accordance with parameters of the communication system, including the antenna structure (such as aperture, spacing, polarization, etc.), carrier frequency of the uplink, configured frequency spacings or lags, and so on). As an example, estimation matrix G may be formed as shown in Equation (24).

[0116] The device estimates the PADS (block 609). The estimation of the PADS may be performed in accordance with the real vector r.sub.SF.sup.UL and the estimation matrix G, and may include solving Equation (23), for example. The estimated PADS is as expressed as Equation (21). The device outputs the estimation output α (block 611). The device outputs the weights α of the estimated PADS, for example.

[0117] FIG. 6B illustrates a flow diagram of example operations 650 occurring in a device converting the spatial-frequency covariance for the uplink RD to the spatial-frequency covariance for the downlink R.sub.SF.sup.DL. Operations 65o may be indicative of operations occurring in a device as the device converts the spatial-frequency covariance for the uplink R.sub.SF.sup.DL to the spatial-frequency covariance for the downlink RD. Operations 650 may be an example implementation of blocks 509 and 511 of FIG. 5. Although the discussion focuses on uplink and downlink, the example embodiments are operable for any two frequency bands. In such a situation, the uplink would be a first frequency band and the downlink would be a second frequency band.

[0118] Input to operations 65o may include the estimation output α, such as produced by a device performing block 611 of FIG. 6A (block 652). Operations 650 begin with the device forming a conversion matrix Q (block 655). The conversion matrix may be formed in accordance with parameters of the communication system, including the antenna structure (such as aperture, spacing, polarization, etc.), carrier frequency of the uplink and the downlink, configured frequency spacings or lags, and so on). As an example, the conversion matrix Q may be formed as shown in Equation (27).

[0119] The device obtains the vectorized form of the spatial-frequency covariance for the downlink r.sub.SF.sup.DL (block 657). The vectorized form of the spatial-frequency covariance being obtained by multiplying the estimation output α (provided as input to operations 650) and the conversion matrix Q formed in block 655, and may be expressible as r.sup.DL=Qα, for example. The device reforms the spatial-frequency covariance matrix R.sub.SF.sup.DL (block 659). The R.sub.SF.sup.DL may be reformed from the vectorized form of the spatial-frequency covariance for the downlink r.sub.SF.sup.DL. The R.sub.SF.sup.DL may be formed in a reverse manner from forming the spatial-frequency covariance for the uplink r.sub.SF.sup.UL with r.sub.SF.sup.UL=Vec{{{R.sub.SF.sup.DL},{R.sub.SF.sup.UL}}, such as in block 605 of FIG. 6A, for example. As an example, RD may be formed using a reversal of the process used to determine r.sub.SF.sup.UL=Vec{{{R.sub.SF.sup.UL}, {R.sub.SF.sup.UL}}. In other words, given that r.sub.SF.sup.UL is known, R.sub.SF.sup.DL may be determined.

[0120] It is also possible to obtain solutions for general array configurations. If it is defined that

[00024]${{\phi }_{m,m}}^{\prime }\left(\theta \right)\stackrel{△}{=}{k\left(\theta \right)}^T{u}_m-{k\left(\theta \right)}^T{u}_{m^{\prime }},$

men g.sub.n.sup.UL(θ, τ) as an entry of

[00025]$\mathrm{vec}\left(\left\{\mathrm{ca}\left(\theta \right){a\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{sc}\tau }\right\},\left\{\mathrm{ca}\left(\theta \right){a\left(\theta \right)}^H{e}^{-j2\pi {\Delta }_{k_i}{f}_{sc}\tau }\right\}\right)$

may be represented as

[00026]$\begin{array}{cc}{g}_n^{\mathrm{UL}}\left(\theta ,\tau \right)=\frac{1}{\sqrt{2\pi {\tau }_{DS}}}\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m,m^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_i}{f}_{sc}\tau \right)& \left(28\right)\end{array}$

for the real term, and

[00027]$\begin{array}{cc}{g}_n^{\mathrm{UL}}\left(\theta ,\tau \right)=-\frac{1}{\sqrt{2\pi {\tau }_{\mathrm{DS}}}}\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m,m^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_i}{f}_{sc}\tau \right)& \left(29\right)\end{array}$

for the imaginary term.

[0121] So there is one-one mapping between n and four tuple {m, m′, i, I.sub.R/I} where I.sub.R/I indicates whether it is an entry corresponding to the real term or imaginary term. An explicit expression for the mapping is not defined, rather it is simply defined as n=M(m, m′, i, I.sub.R/I) for the purpose of general presentation. Also for any given finite τ.sub.DS, the normalization factor does not impact the results as it just scales every entry of G and Q matrices with the same value. However, as will be seen below, it is critical to obtain the asymptotic results when the case τ.sub.DS.fwdarw.∞ is considered.

[0122] Due to separation of the real and imaginary parts, the matrix G in Equation (24) can be expressed as

[00028]$\begin{array}{cc}G=\left[\begin{array}{cc}{G}_{\mathrm{ℛℛ}}& G_{\mathrm{ℛ𝒥}}\\ G_{\mathrm{𝒥ℛ}}& G_{\mathrm{𝒥𝒥}}\end{array}\right].& \left(30\right)\end{array}$

[0123] The entries in are expressible as

[00029]$\begin{array}{cc}{\left[G_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}=.Math.g_{n_1}^{\mathrm{UL}},g_{n_2}^{\mathrm{UL}}.Math.=\frac{1}{2\pi {\tau }_{DS}}{\int }_0^{{\tau }_{DS}}{\int }_{-\pi }^{\pi }{\left[F_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)d\theta d\tau ,\mathrm{where}{\left[F_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)=\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_1}}{f}_{\mathrm{sc}}\tau \right)\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_2}}{f}_{sc}\tau \right)=\frac{1}{2}\left\{\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{sc}\tau \right)\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right)-\sin \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{sc}\tau \right)\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right)+\cos \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{sc}\tau \right)\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right)-\sin \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{sc}\tau \right)\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+{\phi }_{m_2,m_2^{\prime }}2\left(\theta \right)\right)\right)\right\}.& \left(31\right)\end{array}$

Therefore, the entries of are expressible as

[00030]$\begin{array}{cc}{\left[G_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}=\frac{1}{2}\left\{\sin c\left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{sc}{\tau }_{DS}\right)F_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c-}\left({\lambda }^{\mathrm{UL}}\right)+\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{sc}{\tau }_{DS}\right)}{2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{sc}{\tau }_{DS}}{F}_{m_1,m_1^{\prime },m_2,m_2^{\prime },}^{s-}\left({\lambda }^{\mathrm{UL}}\right)+\sin c\left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{sc}{\tau }_{DS}\right)F_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c+}\left({\lambda }^{UL}\right)+\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{sc}{\tau }_{DS}\right)}{2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{sc}{\tau }_{DS}}{F}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s+}\left({\lambda }^{\mathrm{UL}}\right)\right\},& \left(33\right)\end{array}$$$\begin{gathered} \begin{array}{cc}\mathrm{where}{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c-}\left(\lambda \right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\cos \left(\frac{2\pi }{\lambda }\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right)d\theta ,{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s-}\left(\lambda \right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\sin \left(\frac{2\pi }{\lambda }\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right)d\theta ,{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c+}\left(\lambda \right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\cos \left(\frac{2\pi }{\lambda }\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right)d\theta , \\ {ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s+}\left(\lambda \right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\sin \left(\frac{2\pi }{\lambda }\left({\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right)d\theta .& \left(33\right)\end{array} \end{gathered}$$

Similarly,

[0124] [00031]$\begin{array}{cc}{\left[{ℱ}_{\mathrm{ℛ𝒥}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)\stackrel{△}{=}-\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+2{\mathrm{\pi \Delta }}_{k_{i_1}}{f}_{\mathrm{sc}}\tau \right)\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_2}}{f}_{\mathrm{sc}}\tau \right)& \left(34\right)\end{array}$$\begin{array}{cc}{\left[{ℱ}_{\mathrm{𝒥ℛ}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)\stackrel{△}{=}-\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_1}}{f}_{\mathrm{sc}}\tau \right)\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_2}}{f}_{\mathrm{sc}}\tau \right)& \left(35\right)\end{array}$$\begin{array}{cc}{\left[{ℱ}_{\mathrm{𝒥𝒥}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)\stackrel{△}{=}\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_1}}{f}_{\mathrm{sc}}\tau \right)\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_2}}{f}_{\mathrm{sc}}\tau \right).& \left(36\right)\end{array}$$\begin{array}{cc}\mathrm{Then}{\left[G_{\mathrm{ℛ𝒥}}\right]}_{n_1,n_2}=\frac{1}{2\pi {\tau }_{\mathrm{DS}}}{\int }_0^{{\tau }_{\mathrm{DS}}}{\int }_{-\pi }^{\pi }{\left[{ℱ}_{\mathrm{ℛ𝒥}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)d\theta d\tau =\frac{1}{2}\left\{-\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c-}\left({\lambda }^{\mathrm{UL}}\right)+\sin c\left(2\pi \left({\Sigma }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s-}\left({\lambda }^{\mathrm{UL}}\right)\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c+}\left({\lambda }^{\mathrm{UL}}\right)-\sin c\left(2\pi \left({\Delta }_{k_{i_1}+{\Delta }_{k_{i_2}}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s+}\left({\lambda }^{\mathrm{UL}}\right)\right\},{\left[G_{\mathrm{𝒥ℛ}}\right]}_{n_1,n_2}=\frac{1}{2\pi {\tau }_{\mathrm{DS}}}{\int }_{-\pi }^{\pi }{\left[{ℱ}_{\mathrm{𝒥ℛ}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)d\theta d\tau =\frac{1}{2}\left\{\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c-}\left({\lambda }^{\mathrm{UL}}\right)-\sin c\left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s-}\left({\lambda }^{\mathrm{UL}}\right)\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c+}\left({\lambda }^{\mathrm{UL}}\right)-\sin c\left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s+}\left({\lambda }^{\mathrm{UL}}\right)\right\},{\left[G_{\mathrm{𝒥𝒥}}\right]}_{n_1,n_2}=\frac{1}{2\pi {\tau }_{\mathrm{DS}}}{\int }_0^{{\tau }_{\mathrm{DS}}}{\int }_{-\pi }^{\pi }{\left[{ℱ}_{\mathrm{𝒥𝒥}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)d\theta d\tau =\frac{1}{2}\left\{\sin c\left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c-}\left({\lambda }^{\mathrm{UL}}\right)+\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s-}\left({\lambda }^{\mathrm{UL}}\right)-\sin c\left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c+}\left({\lambda }^{\mathrm{UL}}\right)-\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}+D_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{ℱ}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s+}\left({\lambda }^{\mathrm{UL}}\right)\right\}.& \left(37\right)\end{array}$

For some special entries when

[00032]${\Delta }_{k_{i_1}}={\Delta }_{k_{i_2}},$

the following are valid

[00033]$\begin{array}{cc}\sin c\left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)=1,& \left(38\right)\end{array}$$\begin{array}{cc}\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}=0,& \left(39\right)\end{array}$

and when

[00034]${\Delta }_{k_{i_1}}={\Delta }_{k_{i_2}}=0,$

the following are valid

[00035]$\begin{array}{cc}\sin c\left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)=1,& \left(40\right)\end{array}$$\begin{array}{cc}\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}=0.& \left(41\right)\end{array}$

[0125] In order to obtain the conversion matrix Q, again due to separation of the real and imaginary components, the block matrix is rewritten as

[00036]$\begin{array}{cc}Q=\left[\begin{array}{cc}{Q}_{\mathrm{ℛℛ}}& Q_{\mathrm{ℛ𝒥}}\\ Q_{\mathrm{𝒥ℛ}}& Q_{\mathrm{𝒥𝒥}}\end{array}\right].& \left(42\right)\end{array}$

The first entries in are expressible as

[00037]$\begin{array}{cc}{\left[Q_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}=.Math.g_{n_1}^{\mathrm{UL}},g_{n_2}^{\mathrm{DL}}.Math.=\frac{1}{2\pi {\tau }_{\mathrm{DS}}}{\int }_0^{{\tau }_{\mathrm{DS}}}{\int }_{-\pi }^{\pi }{\left[{\tilde{F}}_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)d\theta d\tau ,\mathrm{where}{\left[{\tilde{F}}_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}\left(\theta ,\tau \right)=\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_1}}{f}_{\mathrm{sc}}\tau \right)\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{DL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)+2\pi {\Delta }_{k_{i_2}}{f}_{\mathrm{sc}}\tau \right)=\frac{1}{2}\left\{\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}\tau \right)\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-\frac{2\pi }{{\lambda }^{\mathrm{DL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)-\sin \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}\tau \right)\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)+\cos \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}\tau \right)\cos \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)-\sin \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}\tau \right)\sin \left(\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+\frac{2\pi }{{\lambda }^{\mathrm{UL}}}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)\right\}.& \left(43\right)\end{array}$

[0126] The entries in are expressible as

[00038]${\left[Q_{\mathrm{ℛℛ}}\right]}_{n_1,n_2}=\frac{1}{2}\left\{\sin c\left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c-}\left({\lambda }^{\mathrm{UL}},{\lambda }^{\mathrm{DL}}\right)+\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}-{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s-}\left({\lambda }^{\mathrm{UL}},{\lambda }^{\mathrm{DL}},\right)+\sin c\left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right){𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c+}\left({\lambda }^{\mathrm{UL}},{\lambda }^{\mathrm{DL}}\right)+\frac{1-\cos \left(2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}\right)}{2\pi \left({\Delta }_{k_{i_1}}+{\Delta }_{k_{i_2}}\right)f_{\mathrm{sc}}{\tau }_{\mathrm{DS}}}{𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s+}\left({\lambda }^{\mathrm{UL}},{\lambda }^{\mathrm{DL}}\right)\right\},$$\begin{array}{cc}\mathrm{where}{𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c-}\left({\lambda }_1,{\lambda }_2\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\cos \left(\frac{2\pi }{{\lambda }_1}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-\frac{2\pi }{{\lambda }_2}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)d\theta ,{𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s-}\left({\lambda }_1,{\lambda }_2\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\sin \left(\frac{2\pi }{{\lambda }_1}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)-\frac{2\pi }{{\lambda }_2}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)d\theta ,{𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{c+}\left({\lambda }_1,{\lambda }_2\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\cos \left(\frac{2\pi }{{\lambda }_1}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+\frac{2\pi }{{\lambda }_2}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)d\theta ,{𝒬}_{m_1,m_1^{\prime },m_2,m_2^{\prime }}^{s+}\left({\lambda }_1,{\lambda }_2\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\sin \left(\frac{2\pi }{{\lambda }_1}{\phi }_{m_1,m_1^{\prime }}\left(\theta \right)+\frac{2\pi }{{\lambda }_2}{\phi }_{m_2,m_2^{\prime }}\left(\theta \right)\right)d\theta .& \left(45\right)\end{array}$