# A Partial Channel Reciprocity-based Codebook for Wideband FDD Massive MIMO

Haifan Yin and David Gesbert,  This work was supported by National Natural
Science Foundation of China under grant No. 62071191. The corresponding author
is Haifan Yin. H. Yin is with Huazhong University of Science and Technology,
430074 Wuhan, China (e-mail: yin@hust.edu.cn). D. Gesbert is with EURECOM,
06410 Biot, France (e-mail: gesbert@eurecom.fr).

###### Abstract

The acquisition of channel state information (CSI) in Frequency Division
Duplex (FDD) massive MIMO has been a formidable challenge. In this paper, we
address this problem with a novel CSI feedback framework enabled by the
partial reciprocity of uplink and downlink channels in the wideband regime. We
first derive the closed-form expression of the rank of the wideband massive
MIMO channel covariance matrix for a given angle-delay distribution. A low-
rankness property is identified, which generalizes the well-known result of
the narrow-band uniform linear array setting. Then we propose a partial
channel reciprocity (PCR) codebook, inspired by the low-rankness behavior and
the fact that the uplink and downlink channels have similar angle-delay
distributions. Compared to the latest codebook in 5G, the proposed PCR
codebook scheme achieves higher performance, lower complexity at the user
side, and requires a smaller amount of feedback. We derive the feedback
overhead necessary to achieve asymptotically error-free CSI feedback. Two low-
complexity alternatives are also proposed to further reduce the complexity at
the base station side. Simulations with the practical 3GPP channel model show
the significant gains over the latest 5G codebook, which prove that our
proposed methods are practical solutions for 5G and beyond.

###### Index Terms:

Massive MIMO, FDD, reciprocity, covariance matrix, 5G

##  I Introduction

The large-scale commercialization of 5G cellular systems gradually brings the
concept of massive multiple-input multiple-output (MIMO) [1] to reality. With
a large number of antennas at the base station, massive MIMO leads to much
higher spectral and energy efficiencies compared with traditional MIMO. The
stronger pair-wise orthogonality between the channel vectors of different User
Equipments (UEs) allows for higher interference rejection capability for the
base stations. In other words, the narrow beam of the massive antenna array
reduces the leakage of the transmitted signal towards unwanted UEs, which is
one of the main features of massive MIMO. Nevertheless, the performance of
massive MIMO is primarily determined by the accuracy of the CSI. In practical
cellular systems, there are mainly three reasons of inaccurate CSI, which are
pilot contamination [2], the mobility problem [3], and the CSI feedback with
limited overhead in FDD [1].

The uplink (UL) and downlink (DL) in FDD occupy different frequency bands,
which are separated by around 100 MHz in current 5G setups - a gap much larger
than the channel coherence bandwidth. Therefore, unlike in Time Division
Duplex (TDD) mode, the UL and DL channels in FDD are typically non-reciprocal.
As a result, the DL CSI is mainly obtained by closed-loop feedback methods:
The base station (BS) first transmits reference signals, e.g., Channel State
Information Reference Signal (CSI-RS) as in 5G, then the UEs estimate their
individual channels, and finally the UEs send back the quantized channel
information to the base station. This traditional CSI feedback scheme suffers
from two problems, 1) the time-frequency resource spent on the reference
signals increases quickly with the number of base station antennas, leaving
less resource for data transmission; and 2) the CSI feedback is always
corrupted by quantization errors.

In the literature, there have been many proposals to enhance the performance
of FDD massive MIMO. The works in [4] and [5] indicated that the spatial
channel covariance matrix of the UE channels exhibits a certain low-rankness
property, where the rank is governed by the angular spread of the multipath.
Such a property helps to reject interference from non-orthogonal training
signals of other UEs which have non-overlapping angular distribution with the
desired UE, thus reducing the training overhead. In some other related works,
an idea of covariance shaping was proposed and studied [6] [7] [8] [9], where
the effective channels with low-dimensionality were manually created and
exploited through precoding. Some papers proposed to adopt Deep Learning for
CSI compression and reconstruction [10, 11, 12]. However these learning-based
method generally require training with large datasets.

The possibility of exploiting the reciprocity in angular domain has been
studied in [13, 14, 15, 16]. In particular, [13] proposed a scalable CSI
feedback method which exploited the angle and delay reciprocity of the FDD
channel. A Fast Fourier Transform (FFT)-based pilots were used and the UE
computed its low-dimensional feedback based on the DL channel statistics
measured by itself or instructed by its serving base station. The authors
found that only a scalar per each DL channel path was needed for the base
station to reconstruct the DL channel. [14] proposed a dictionary learning
approach which relied on the reciprocity and the sparsity of the multipath
angles in UL and DL channels. [15] proposed a directional training method
which was also motivated by the angular reciprocity. The authors in [16]
utilized the angular reciprocity in designing an angle-adaptive codebook and
proved that the amount of feedback scales with the number of paths, which was
assumed much smaller than the number of base station antennas.

In Release-16 (Rel-16) of the 5G standards, the Enhanced Type II Codebook is
adopted [17]. Such a codebook capitalizes on the sparsity in both spatial
domain and frequency domain by introducing a 2D Discrete Fourier Transform
(DFT) operation on the wideband channel matrix. Briefly speaking, the base
station first broadcasts non-precoded CSI-RS to its serving UEs. The UE then
estimates its wideband channel matrix. Afterwards, the UE performs 2D DFT on
the wideband precoding matrix, where each column of this matrix is a precoder
for a certain frequency band. Finally the UE sends back the non-negligible
elements (quantized) in the transformed matrix to the base station along with
their corresponding positions in the transformed matrix.

In this paper, we propose a novel CSI feedback framework based on an idea of
joint spatial-frequency domain precoded reference signal. This framework is
enabled by the partial UL/DL reciprocity of angle-delay distributions, which
was recently validated by measurements [18]. In the traditional methods, the
precoding is always performed in spatial domain. However in this paper, the
frequency-domain precoding is made possible by our scheme, which entails a
joint operation of the BS and the UE. The basic idea is to exploit the channel
sparsity in angle-delay domain by finding the non-negligible spatial-
frequency-domain projections from the wideband uplink channel estimation, and
then design the downlink joint precoder based on the positions of these
projected values. In this way, we only sample the non-negligible coefficients
of the sparse representation of the channel, which leads to significant
training overhead reduction. In practice, the projection is realized by either
the eigenvectors of the wideband channel covariance matrix or the 2D DFT
matrix.

We show that the amount of feedback coefficients can be greatly reduced, which
is always smaller than the total number of multipaths. In the meantime, the
computational complexity for the UE is quite low - only several summations of
the estimates of the effective channels will suffice. This is well-aligned
with the commercial interest to simplify the implementation of UEs. Different
from [13], the UEs do not need any information about the DL channel statistics
or path delays, nor do they need to perform path aligning process, and the
complexity of our method is much lower.

Compared with the latest Rel-16 codebook, i.e., the Enhanced Type II Codebook,
our proposed PCR codebook scheme has five main advantages.

  * •

PCR scheme achieves better performance in terms of the CSI feedback accuracy.
It is because of the exploitation of the second-order channel statistics.
Compared to the DFT-based methods, e.g., the Rel-16 codebook, the dimension of
the signal subspace of a channel covariance matrix is smaller than the number
of non-negligible DFT projections. This is because of the spectrum leakage of
DFT projections, particularly when the size of the DFT matrix is relatively
small.

  * •

The PCR method has lower computational complexity. In fact the Enhanced Type
II Codebook entails a 2D DFT operation, and in some cases, the singular value
decomposition (SVD). While our method only requires some scalar additions
instead of matrix operations.

  * •

The proposed method has a smaller feedback overhead. There are two reasons of
the feedback reduction. 1) The eigenvector based compression is more effective
than the DFT based compression; and 2) In Rel-16 codebook, apart from feeding
back the quantized complex coefficients, the UE has to feed back the indices
of the corresponding DFT vectors as well. While in our proposed feedback
scheme, the UE only needs to feedback the quantized coefficients.

  * •

The proposed method has a better quality of channel estimation at the UE side.
In the Enhanced Type II Codebook, the BS broadcasts non-precoded CSI-RS to all
UEs. No beamforming is performed at the base station. In such cases, the
Signal to Noise Ratio (SNR) of the received CSI-RS might be low. In contrast,
our proposed codebook scheme enables precoded CSI-RS. In other words, the base
station transmits beamformed reference signals to UEs, which is similar to the
dominant eigen-beamforming. As a result, the receive SNR of the reference
signal is higher.

  * •

The proposed scheme relies on no assumption of the topology of the BS antenna
array. In fact, the Enhanced Type II Codebook in Rel-16 makes the explicit
assumption that the base station is equipped with a Uniform Planar Array
(UPA). When UE performs 2D DFT transformations, the size of the DFT matrix is
determined by the numbers of columns and rows of antenna elements on the
antenna panel. A mismatch will result in obvious performance degradations.
However, our propose method works with arbitrary antenna topologies, and is
therefore more general.

The contributions of our paper are summarized below:

  * •

We generalize the known low-rankness property of channel covariance matrix for
a narrowband large-scale uniform linear array (ULA) to the narrowband UPA
array, and to the wideband UPA array. Closed-form expressions of the ranks are
derived.

  * •

We propose a practical and scalable PCR codebook scheme, which is a very
competitive candidate for the standards of 5G and beyond. This scheme exploits
the UL and DL channel reciprocity of the statistical signal subspace in joint
spatial-frequency domains, as well as the proved low-rankness property of the
wideband channel covariance matrix.

  * •

We make performance analysis and give the upper-bound of the feedback overhead
to achieve asymptotically error-free CSI feedback. We show that the feedback
overhead is smaller than the number of paths.

  * •

Two low-complexity alternatives named PCR-E and PCR-D are proposed, which
further reduce the complexity at the base station side at the cost of mild
performance loss.

We evaluate the performances of our proposed method under the realistic
channel model of 3GPP, which is widely adopted in industry and the de facto
model used for 5G standardization. Simulation results show that in the
wideband massive MIMO regime, the UE only needs to send back tens of scalar
values to the base station to achieve near optimal performance, even in the
presence of hundreds of multipath.

The organization of this paper is as follows: Sec. II introduces the channel
model of 3GPP and the channel reciprocity modelling for FDD [19]. Sec. III
derives the rank of the channel covariance matrices for a given angle-delay
distribution. Sec. IV shows the proposed PCR codebook scheme and the
asymptotic performance analysis. Sec. V describes the low-complexity
alternatives. The simulation results are shown in Sec. VI. Finally, the
conclusions are drawn in Sec. VII.

Notations: We use boldface to denote matrices and vectors. Specifically,
${\mathbf{I}}$ denotes the identity matrix. ${({\mathbf{X}})^{T}}$,
${({\mathbf{X}})^{*}}$, and ${({\mathbf{X}})^{H}}$ denote the transpose,
conjugate, and conjugate transpose of a matrix ${\mathbf{X}}$ respectively.
${({\mathbf{X}})^{\dagger}}$ is the Moore-Penrose pseudoinverse of
${\mathbf{X}}$. $\operatorname{tr}\left\\{\cdot\right\\}$ denotes the trace of
a square matrix. ${\left\|\cdot\right\|_{2}}$ denotes the $\ell^{2}$ norm of a
vector. ${\left\|\cdot\right\|_{F}}$ stands for the Frobenius norm.
$\mathbb{E}\left\\{\cdot\right\\}$ denotes the expectation.
${\bf{X}}\otimes{\bf{Y}}$ is the Kronecker product of ${\bf{X}}$ and
${\bf{Y}}$. ${\bf{X}}\odot{\bf{Y}}$ denotes the Hadamard product of ${\bf{X}}$
and ${\bf{Y}}$. $\operatorname{vec}({\bf X})$ is the vectorization of the
matrix ${\bf X}$, and $\operatorname{unvec}({\bf x})$ is the corresponding
inverse operation. ${\mathop{\rm diag}\nolimits}\\{{{{\bf a}_{1},...,{\bf
a}_{N}}}\\}$ denotes a diagonal matrix or a block diagonal matrix with ${{\bf
a}_{1},...,{\bf a}_{N}}$ at the main diagonal. $\triangleq$ is used for
definition. $\mathbb{N}$ and $\mathbb{N}^{+}$ are the set of non-negative and
positive integers respectively.

##  II Channel Models

We consider an arbitrary UE in a certain cell. The base station is equipped
with a uniform planar array (UPA) of cross-polarized antenna elements. The
total number of multipaths is $M$. For a certain path $m$, we denote the
Zenith angle Of Arrival (ZOA) as $\theta_{\text{m,ZOA}}$, the Azimuth angle Of
Arrival (AOA) as $\varphi_{\text{m,AOA}}$, the Zenith angle Of Departure (ZOD)
as $\theta_{\text{m,ZOD}}$, and the Azimuth angle Of Departure (AOD) as
$\varphi_{\text{m,AOD}}$. The channel impulse response between the $s$-th BS
antenna and the $u$-th UE antenna at time $t$ is shown in Eq. (9).

$\displaystyle\begin{array}[]{l}H_{u,s,m}(t)={\left[{\begin{array}[]{*{20}{c}}{{F_{\text{rx,}u,\theta}}\left({{\theta_{m\text{,ZOA}}},{\varphi_{m\text{,AOA}}}}\right)}\\\
{{F_{\text{rx,}u,\varphi}}\left({{\theta_{m\text{,ZOA}}},{\varphi_{m\text{,AOA}}}}\right)}\end{array}}\right]^{T}}\left[{\begin{array}[]{*{20}{c}}{\exp\left({j\Phi_{m}^{\theta\theta}}\right)}&{\sqrt{{\kappa^{-1}_{m}}}\exp\left({j\Phi_{m}^{\theta\varphi}}\right)}\\\
{\sqrt{{\kappa_{m}}^{-1}}\exp\left({j\Phi_{m}^{\varphi\theta}}\right)}&{\exp\left({j\Phi_{m}^{\varphi\varphi}}\right)}\end{array}}\right]\\\
\left[{\begin{array}[]{*{20}{c}}{{F_{\text{tx,s},\theta}}\left({{\theta_{m\text{,ZOD}}},{\varphi_{m\text{,AOD}}}}\right)}\\\
{{F_{\text{tx,s},\varphi}}\left({{\theta_{m\text{,ZOD}}},{\varphi_{m\text{,AOD}}}}\right)}\end{array}}\right]\exp\left({j2\pi\frac{{\hat{\bf
r}_{\text{rx,}m}^{T}{{\bar{\bf
d}}_{\text{rx,}u}}}}{{{\lambda_{0}}}}}\right)\exp\left({j2\pi\frac{{\hat{\bf
r}_{\text{tx,}m}^{T}{{\bar{\bf
d}}_{\text{tx,}s}}}}{{{\lambda_{0}}}}}\right)\exp\left({j2\pi\frac{{\hat{\bf
r}_{\text{rx,}m}^{T}\bar{\bf v}}}{{{\lambda_{0}}}}t}\right)\end{array}$ (9)

${{F_{{\text{rx,}u},\theta}}\left({{\theta_{m\text{,ZOA}}},{\varphi_{m\text{,AOA}}}}\right)}$,
${{F_{{\text{rx,}u},\varphi}}\left({{\theta_{m\text{,ZOA}}},{\varphi_{m\text{,AOA}}}}\right)}$,
${{F_{{\text{tx,}s},\theta}}\left({{\theta_{m\text{,ZOD}}},{\varphi_{m\text{,AOD}}}}\right)}$,
and
${{F_{{\text{tx,}s},\varphi}}\left({{\theta_{m\text{,ZOD}}},{\varphi_{m\text{,AOD}}}}\right)}$
are the field patterns defined in Section 7.3 of [20]. $\kappa_{m}$ is the
cross polarization ratio (XPR), which follows a log-Normal distribution.
$\left\\{{\Phi_{m}^{\theta\theta},\Phi_{m}^{\theta\varphi},\Phi_{m}^{\varphi\theta},\Phi_{m}^{\varphi\varphi}}\right\\}$
are the random initial phases for four different polarization combinations.
These random phases are uniformly distributed within $(-\pi,\pi)$.
$\lambda_{0}$ is the wavelength of the center frequency. $\hat{{\bf
r}}_{\text{rx},m}$ is the spherical unit vector with Azimuth arrival angle
$\varphi_{m,\text{AOA}}$ and Zenith arrival angle $\theta_{m,\text{ZOA}}$:

$\hat{{\bf
r}}_{\text{rx},m}\triangleq\left[{\begin{array}[]{*{20}{c}}{\sin{\theta_{m,\text{ZOA}}}\cos{\varphi_{m,\text{AOA}}}}\\\
{\sin{\theta_{m,\text{ZOA}}}\sin{\varphi_{m,\text{AOA}}}}\\\
{\cos{\theta_{m,\text{ZOA}}}}\end{array}}\right].$ (10)

Likewise, $\hat{{\bf r}}_{\text{tx},m}$ is the spherical unit vector defined
as:

$\hat{{\bf
r}}_{\text{tx},m}\triangleq\left[{\begin{array}[]{*{20}{c}}{\sin{\theta_{m,\text{ZOD}}}\cos{\varphi_{m,\text{AOD}}}}\\\
{\sin{\theta_{m,\text{ZOD}}}\sin{\varphi_{m,\text{AOD}}}}\\\
{\cos{\theta_{m,\text{ZOD}}}}\end{array}}\right].$ (11)

$\bar{{\bf d}}_{\text{rx},u}$ is the 3D Cartesian coordinate of the $u$-th UE
antenna, and $\bar{{\bf d}}_{\text{tx},s}$ is the 3D Cartesian coordinate of
the $s$-th base station antenna. The exponential term $e^{j({\hat{{\bf
r}}_{\text{rx},p}^{T}\bar{{\bf v}}}/{\lambda_{0}})t}$ is the Doppler, where
$\bar{{\bf v}}$ is the UE velocity vector with speed $v$, travel azimuth angle
$\varphi_{v}$, and travel Zenith angle $\theta_{v}$:

$\bar{{\bf v}}\triangleq
v{[\begin{array}[]{*{20}{c}}{\sin{\theta_{v}}\cos{\varphi_{v}}}&{\sin{\theta_{v}}\sin{\varphi_{v}}}&{\cos{\theta_{v}}}\end{array}]^{T}}.$
(12)

Channel reciprocity is modeled according to [19] and [21], where the following
small-scale parameters are reciprocal:

  * •

The angles
$\theta_{m,\text{ZOA}},\theta_{m,\text{ZOD}},\varphi_{m,\text{AOA}}$, and
$\varphi_{m,\text{AOD}}$.

  * •

The delay and the power of the path.

  * •

The cross polarization ratio $\kappa_{m}$.

The non-reciprocal channel parameters are listed below.

  * •

The wavelength of the center frequency $\lambda_{0}$ and the related phase
terms in Eq. (9).

  * •

Path loss factors for UL and DL.

  * •

The initial phases
$\left\\{{\Phi_{m}^{\theta\theta},\Phi_{m}^{\theta\varphi},\Phi_{m}^{\varphi\theta},\Phi_{m}^{\varphi\varphi}}\right\\}$,
which are independent for UL and DL channels.

The topology of antennas at the base station is a UPA with $N_{v}$ rows and
$N_{h}$ columns of antenna elements. The horizontal and vertical antenna
spacings are denoted by ${D_{h}}$ and ${D_{v}}$ respectively. The total number
of base station antennas is $N_{t}=N_{v}N_{h}N_{p}$, where $N_{p}=\\{1,2\\}$
is the number of polarizations. For ease of exposition, we assume the antenna
panel is in YZ plane of our coordinate system, with the first antenna element
on the origin. The indices of the antennas are arranged the same way as in
[3], i.e., we order the antennas column by column. According to the channel
model Eq. (9), we introduce a 3D steering vector, which is corresponding to a
certain path with Azimuth departure angle $\varphi_{\text{AOD}}$ and Zenith
departure angle $\theta_{\text{ZOD}}$.

${\bf{a}}({\theta}_{\text{ZOD}},{\varphi}_{\text{AOD}})={{\bf{a}}_{h}}({\theta}_{\text{ZOD}},{\varphi}_{\text{AOD}})\otimes{{\bf{a}}_{v}}({\theta}_{\text{ZOD}}),$
(13)

where

$\displaystyle{{\bf{a}}_{h}}({\theta_{{\rm{ZOD}}}},{\varphi_{{\rm{AOD}}}})=\left[{\begin{array}[]{*{20}{c}}1\\\
{{e^{j2\pi{{D_{h}}\sin({\theta_{{\rm{ZOD}}}})\sin({\varphi_{{\rm{AOD}}}})}/\lambda_{0}}}}\\\
\vdots\\\
{{e^{j2\pi{({N_{h}}-1){D_{h}}\sin({\theta_{{\rm{ZOD}}}})\sin({\varphi_{{\rm{AOD}}}})}/{{\lambda_{0}}}}}}\end{array}}\right],$
(18)

and

${{\bf{a}}_{v}}({\theta_{{\rm{ZOD}}}})=\left[{\begin{array}[]{*{20}{c}}1\\\
{{e^{j2\pi\frac{{{D_{v}}\cos({\theta_{{\rm{ZOD}}}})}}{{{\lambda_{0}}}}}}}\\\
\vdots\\\
{{e^{j2\pi\frac{{({N_{v}}-1){D_{v}}\cos({\theta_{{\rm{ZOD}}}})}}{{{\lambda_{0}}}}}}}\end{array}}\right].$
(19)

For notational simplicity, we temporarily assume the UE has one antenna.
However the generalization from single-antenna UE to multiple-antenna UE is
straightforward. In fact, the simulations in Sec. VI are mostly carried out
with multiple-antenna UEs. Denote the number of subcarriers in UL and DL as
$N_{f}^{\rm{(U)}}$ and $N_{f}^{\rm{(D)}}$ respectively. Let ${\bf
h}^{\text{(U)}}(f,t)\in{\mathbb{C}^{N_{t}\times 1}}$ and ${\bf
h}^{\text{(D)}}(f,t)\in{\mathbb{C}^{N_{t}\times 1}}$ denote respectively the
UL and DL channels between all base station antennas and the UE antenna at
time $t$ and frequency $f$. We write the UL and DL channels at all subcarriers
in a matrix form:

$\displaystyle{{\bf{H}}^{\text{(U)}}}(t)$
$\displaystyle\triangleq[\begin{array}[]{*{20}{c}}{{\bf
h}^{\text{(U)}}(f_{1},t)}&{{\bf h}^{\text{(U)}}(f_{2},t)}&\cdots&{{{\bf
h}^{\text{(U)}}(f_{N_{f}^{\rm{(U)}}},t)}}\end{array}],$ (21)
$\displaystyle{{\bf{H}}^{\text{(D)}}}(t)$
$\displaystyle\triangleq[\begin{array}[]{*{20}{c}}{{\bf
h}^{\text{(D)}}(f_{1},t)}&{{\bf h}^{\text{(D)}}(f_{2},t)}&\cdots&{{{\bf
h}^{\text{(D)}}(f_{N_{f}^{\rm{(D)}}},t)}}\end{array}],$ (23)

where $f_{i}$ is the frequency of the $i$-th subcarrier of either UL or DL
channel.

Depending on the context, a superscript $\text{X}=\\{\text{U},\text{D}\\}$ is
introduced to simplify the presentation. According to the multipath model in
Eq. (9), we may further write

${{\bf{H}}^{(\text{X})}}(t)={\bf{A}_{\text{blk}}^{(\text{X})}}{{\bf{C}}^{(\text{X})}}(t){{\bf{B}}^{(\text{X})}},$
(24)

where ${\bf{A}^{(\text{X})}_{\text{blk}}}={\mathop{\rm
diag}\nolimits}\\{{\bf{A}^{(\text{X})},...,\bf{A}^{(\text{X})}}\\}\in{\mathbb{C}^{N_{t}\times
N_{p}M}}$ is a block diagonal matrix with ${\bf
A}^{(\text{X})}\in{\mathbb{C}^{N_{h}N_{v}\times M}}$ composed of $M$ 3D
steering vectors:

${\bf{A}}^{\rm{(X)}}\buildrel\Delta\over{=}\left[{\begin{array}[]{*{20}{c}}{{\bf{a}}^{\rm{(X)}}({\theta_{1,{\rm{ZOD}}}},{\phi_{1,{\rm{AOD}}}})}&\cdots&{{\bf{a}}^{\rm{(X)}}({\theta_{M,{\rm{ZOD}}}},{\phi_{M,{\rm{AOD}}}})}\end{array}}\right],$
(25)

and

${\bf{B}}^{\rm{(X)}}\buildrel\Delta\over{=}\left[{\begin{array}[]{*{20}{c}}{{\bf{b}}^{\rm{(X)}}({\tau_{1}})}&{{\bf{b}}^{\rm{(X)}}({\tau_{2}})}&\cdots&{{\bf{b}}^{\rm{(X)}}({\tau_{M}})}\end{array}}\right]^{T},$
(26)

with ${{\bf{b}}^{\rm{(X)}}({\tau_{m}})}$, $(m=1,\cdots,M)$ being the delay
response vector of the $m$-th path, which is defined as

$\mbox{\small$\displaystyle{\bf{b}}^{\rm{(X)}}({\tau_{m}})=\left[\begin{array}[]{*{20}{c}}{{e^{-j2\pi{f_{1}}{\tau_{m}}}}}&{{e^{-j2\pi{f_{2}}{\tau_{m}}}}}&\cdots&{{e^{-j2\pi{f{{N_{f}^{\text{(X)}}}}{\tau_{m}}}}}}\end{array}\right]^{T}$}.$
(27)

The matrix ${\bf{C}}^{(\text{X})}(t)\in\mathbb{C}^{N_{p}M\times M}$ is defined
as

$\displaystyle{{\bf{C}}^{({\rm{X}})}}(t)={\left[{\begin{array}[]{*{20}{c}}{{{\bf{C}}_{1}}^{({\rm{X}})}(t)}&\cdots&{{{\bf{C}}_{{N_{p}}}}^{({\rm{X}})}(t)}\end{array}}\right]^{T}},$
(29)

where ${{\bf{C}}_{i}}^{({\rm{X}})}(t),i=1,\cdots,N_{p}$ is a diagonal matrix

$\displaystyle{{{\bf{C}}_{i}}^{({\rm{X}})}(t)}={\mathop{\rm
diag}\nolimits}\\{{c^{(\text{X})}_{i,1}(t),...,{c^{(\text{X})}_{i,M}}(t)}\\},$
(30)

with its $m$-th diagonal element defined as

$\displaystyle
c^{(\text{X})}_{i,m}(t)={\left[{\begin{array}[]{*{20}{c}}{{F_{{\rm{rx,}}\theta}}\left({{\theta_{m{\rm{,ZOA}}}},{\varphi_{m{\rm{,AOA}}}}}\right)}\\\
{{F_{{\rm{rx,}}\varphi}}\left({{\theta_{m{\rm{,ZOA}}}},{\varphi_{m{\rm{,AOA}}}}}\right)}\end{array}}\right]^{T}}$
(33)
$\displaystyle\left[{\begin{array}[]{*{20}{c}}{\exp\left({j\Phi_{m}^{\theta\theta}}\right)}&{\sqrt{\kappa_{m}^{-1}}\exp\left({j\Phi_{m}^{\theta\varphi}}\right)}\\\
{\sqrt{{\kappa_{m}}^{-1}}\exp\left({j\Phi_{m}^{\varphi\theta}}\right)}&{\exp\left({j\Phi_{m}^{\varphi\varphi}}\right)}\end{array}}\right]$
(36)
$\displaystyle\left[{\begin{array}[]{*{20}{c}}{{F_{{\rm{tx}},\theta}}\left({{\theta_{m{\rm{,ZOD}}}},{\varphi_{m{\rm{,AOD}}}}}\right)}\\\
{{F_{{\rm{tx}},\varphi}}\left({{\theta_{m{\rm{,ZOD}}}},{\varphi_{m{\rm{,AOD}}}}}\right)}\end{array}}\right]\exp\left({j2\pi\frac{{\hat{r}_{{\rm{rx,}}m}^{T}\bar{v}}}{{{\lambda_{0}}}}t}\right).$
(39)

The vectorized channel representation of Eq. (24) is written as

$\displaystyle{\underline{\bf{h}}^{(\text{X})}}(t)\triangleq\operatorname{vec}\left({{\bf{H}}^{(\text{X})}}(t)\right)={{\bf{B}}^{(\text{X})T}}\otimes{\bf{A}_{\text{blk}}^{(\text{X})}}\operatorname{vec}\left({{\bf{C}}^{(\text{X})}}(t)\right).$
(40)

Denote the channel covariance matrix in spatial domain and frequency domain as

$\displaystyle{{\bf{R}}^{{\rm{(X,S)}}}}=\mathbb{E}\left\\{{{\bf{H}}^{\text{(X)}}}(t){{\left({{\bf{H}}^{\text{(X)}}}(t)\right)}^{H}}\right\\},$
(41)

and

$\displaystyle{{\bf{R}}^{{\rm{(X,F)}}}}=\mathbb{E}\left\\{{{\left({{\bf{H}}^{\text{(X)}}}(t)\right)}^{T}}\left({{\bf{H}}^{\text{(X)}}}(t)\right)^{*}\right\\}.$
(42)

respectively, where the expectation is taken over time. We also denote the
joint spatial-frequency channel covariance matrix as

$\displaystyle{{\bf{R}}^{{\rm{(X,J)}}}}=\mathbb{E}\left\\{{{\left({\underline{\bf{h}}^{\text{(X)}}}(t)\right)}}\left({\underline{\bf{h}}^{\text{(X)}}}(t)\right)^{H}\right\\}.$
(43)

Note that the base station may obtain the DL channel covariance matrix by some
existing methods, e.g., the projection method in a Hilbert space by exploiting
the angular reciprocity of UL and DL channels [22].

##  III The Rank of the Covariance Matrix

In this section, we investigate the ranks of the channel covariance matrices
of a UPA, for the cases of 1) the narrow-band spatial covariance matrix, 2)
the frequency covariance matrix, and 3) the wideband joint spatial-frequency
covariance matrix. we demonstrate that the channel covariance matrix of a
large-scale UPA with antenna spacing no larger than half wavelength always
exhibits a low-rankness property no matter what the angular distribution is.
This property shows that the spatial channel vector lives in a reduced space
and thus can be compressed using the second-order channel statistics. For ease
of exposition, we derive the ranks for a single polarization. The results can
be readily generalized to cross-polarization setting by simply multiplying the
expressions of the ranks by 2. We start with a simple setting where the
multipath angles are randomly distributed within a closed region. For
notational simplicity, we drop the subscripts “ZOD” and “AOD” and do not
specify UL or DL in this section. The Zenith angle of departure $\theta$ and
the Azimuth angle of departure $\varphi$ are distributed with an arbitrary
probability density function that is non-zero inside the bounded and convex
support $\zeta(\theta,\varphi)$. The boundary is defined as follows:
$\theta\in[\theta^{\text{min}},\theta^{\text{max}}]$, and for a given Zenith
angle $\theta$, the bound of the Azimuth angle is
$[\varphi^{\text{min}}(\theta),\varphi^{\text{max}}(\theta)]$. Consider a
certain carrier frequency with wavelength $\lambda$. The rank of the spatial
covariance matrix ${{\bf{R}}^{{\rm{(S)}}}}$, is characterized in Theorem 1 for
massive MIMO regime.

###### Theorem 1

When $N_{h}$ and $N_{v}$ are large, the rank of the spatial channel covariance
matrix ${{\bf{R}}^{{\rm{(S)}}}}$ is given by

$\frac{\text{rank}\\{{{\bf{R}}^{{\rm{(S)}}}}\\}}{N_{h}N_{v}}=\rho^{{\rm{(S)}}},$
(44)

where the ratio $\rho^{{\rm{(S)}}}$ is defined as

$\displaystyle\rho^{{\rm{(S)}}}{\triangleq}\frac{{{D_{h}}{D_{v}}}}{{{\lambda^{2}}}}\int\limits_{{\theta^{\min}}}^{{\theta^{\max}}}{{{\sin}^{2}}(\theta)\left({\sin\left({{\varphi^{\max}}(\theta)}\right)-\sin\left({{\varphi^{\min}}(\theta)}\right)}\right)d\theta}.$
(45)

_Proof:_ The proof can be found in Appendix -A.

For the special case where the ZOD and AOD are mutually independent and
distributed within their own intervals, i.e.,
$\theta\in[\theta^{\text{min}},\theta^{\text{max}}]$,
$\varphi\in[\varphi^{\text{min}},\varphi^{\text{max}}]$, as shown in Fig. 1,
we may readily obtain the rank ratio of the spatial covariance matrix:

$\displaystyle\rho^{{\rm{(S)}}}$
$\displaystyle=\frac{{{D_{h}}{D_{v}}}}{{{\lambda^{2}}}}\left({\sin({\varphi^{\max}})-\sin({\varphi^{\min}})}\right)$
$\displaystyle\left({\frac{1}{2}({\theta^{\max}}-{\theta^{\min}})-\frac{1}{4}\left({\sin(2{\theta^{\max}})-\sin(2{\theta^{\min}})}\right)}\right).$
(46)

Figure 1: An example of the bounded angular support.

Furthermore, if the ZOD and AOD spread the full angular range, e.g., $\theta$
and $\varphi$ are distributed with non-zero probability in all directions,
then the rank ratio of the spatial channel covariance matrix is upper-bounded
by

$\displaystyle\rho^{{\rm{(S)}}}$
$\displaystyle=\frac{{{D_{h}}{D_{v}}}}{{{\lambda^{2}}}}\pi,$ (47)

or equivalently

$\displaystyle{\text{rank}\\{{{\bf{R}}^{{\rm{(S)}}}}\\}}=L_{h}L_{v}\pi,$ (48)

where $L_{h}$ and $L_{v}$ denote the normalized apertures of the UPA in
horizontal and vertical directions respectively:

$\displaystyle L_{h}{\triangleq}D_{h}N_{h}/\lambda,\quad
L_{v}{\triangleq}D_{v}N_{v}/\lambda.$ (49)

Eq. (48) indicates that the rank of the spatial covariance matrix is
determined by the area of the ellipse circumscribing the rectangle of size
$L_{v}\times L_{h}$. In the practical cases that the antenna spacing is no
larger than half-wavelength, such an observation means
${{\bf{R}}^{{\rm{(S)}}}}$ always have low-rankness property, and the rank
ratio will never exceed $\pi/4$ under arbitrary angular distributions.

Interestingly, Eq. (47) coincides with the Degrees of Freedom (DoF) in [23]
for a UPA in an isotropic scattering environment. In fact, Theorem 1 is a
generalized result of the DoF for the case that the angles have a certain
bounded support instead of unbounded.

Note that Theorem 1 describes the dimensionality of the signal subspace for a
simple setting that the multipath angles exists inside one bounded support.
Corollary 1 is an extension to the case that the angles are distributed within
several disjoint angular sub-supports, which is more realistic. Denote the
number of disjoint sub-supports as $Q$, and the boundary of the $q$-th sub-
support as

$\displaystyle\zeta_{q}{\triangleq}\left\\{(\theta,\varphi)|\theta\in[\theta_{q}^{\text{min}},\theta_{q}^{\text{max}}],\varphi\in[\varphi_{q}^{\text{min}}(\theta),\varphi_{q}^{\text{max}}(\theta)]\right\\}.$
(50)

The sub-supports are mutually disjoint, i.e.,
$\zeta_{i}\cap\zeta_{j}=\emptyset,\forall i\neq j$. Consider a certain UE with
the angular support $\zeta_{u}$ being the union of all $Q$ sub-supports, i.e.,
$\zeta_{u}=\zeta_{1}\cup\zeta_{2}\cup,\cdots,\cup\zeta_{Q}$, and we assume the
angular power spectrum (APS) is non-zero over the support $\zeta_{u}$.

###### Corollary 1

When $N_{h}$ and $N_{v}$ are large, the rank of the spatial channel covariance
matrix ${{\bf{R}}^{{\rm{(S)}}}}$ is given by

$\frac{\text{rank}\\{{{\bf{R}}^{{\rm{(S)}}}}\\}}{N_{h}N_{v}}=\rho^{{\rm{(S)}}},$
(51)

where the ratio $\rho^{{\rm{(S)}}}$ is defined as

$\displaystyle\rho^{{\rm{(S)}}}{\triangleq}\frac{{{D_{h}}{D_{v}}}}{{{\lambda^{2}}}}\sum\limits_{q=1}^{Q}{\int\limits_{\theta_{q}^{\min}}^{\theta_{q}^{\max}}{{{\sin}^{2}}(\theta)\left({\sin\left({\varphi_{q}^{\max}(\theta)}\right)-\sin\left({\varphi_{q}^{\min}(\theta)}\right)}\right)d\theta}}.$
(52)

_Proof:_ The proof is a simple extension of Appendix -A and thus omitted.

Theorem 1 and Corollary 1 show that the spatial covariance matrix has a
reduced subspace and only a limited number of eigenvectors have corresponding
non-negligible eigenvalues. As a result, the channel matrix is compressible
using the dominant eigenvectors of ${\bf{R}}^{{\rm{(S)}}}$.

We then show the frequency covariance matrix has a similar low-rankness
property, which can be exploited to reduce the dimension of the channel in
frequency domain. Suppose the power delay profile (PDP) is strictly non-zero
inside $Q$ delay intervals $\xi_{1},\xi_{2},\cdots,\xi_{Q}$. The range of the
$q$-th interval is $[\tau_{q}^{\text{min}},\tau_{q}^{\text{max}}]$,
$q=1,\cdots,Q$. Denote the subcarrier spacing as $\Delta f$ and the number of
subcarriers as $N_{f}$. Proposition 1 describes the rank of the corresponding
spatial covariance matrix ${\bf R}^{\text{(F)}}$.

###### Proposition 1

When $N_{f}$ is large, the rank of the frequency channel covariance matrix
${\bf R}^{\text{(F)}}$ is given by

$\frac{\text{rank}\\{{\bf R}^{\text{(F)}}\\}}{N_{f}}=\rho^{{\rm{(F)}}},$ (53)

where the ratio $\rho^{{\rm{(F)}}}$ is defined as

$\displaystyle\rho^{{\rm{(F)}}}{\triangleq}\min\left\\{\sum\limits_{q=1}^{Q}\Delta
f{\left({\tau_{q}^{\max}-\tau_{q}^{\min}}\right)},1\right\\}.$ (54)

_Proof:_ The proof is readily obtained by applying some modifications of Lemma
2 in [4], which was originally dedicated to the rank of the spatial covariance
matrix for large-scale ULA. Thus we skip the detailed proof here.

Proposition 1 indicates that as the bandwidth increases, the frequency
covariance matrix is also rank-deficient due to the limited delay
distributions. As a result, the channel matrix can be compressed in frequency
domain using the dominant eigenvectors of the covariance matrix ${\bf
R}^{\text{(F)}}$.

We future study the rank of the joint spatial-frequency covariance matrix
${\bf R}^{\text{(J)}}$. Denoting now the number of angle-delay sub-supports as
$Q$, the boundary of the $q$-th angle-delay sub-support is

$\displaystyle\eta_{q}$
$\displaystyle{\triangleq}\left\\{(\theta,\varphi,\tau)|\theta\in[\theta_{q}^{\text{min}},\theta_{q}^{\text{max}}],\varphi\in[\varphi_{q}^{\text{min}}(\theta),\varphi_{q}^{\text{max}}(\theta)],\right.$
$\displaystyle\quad\left.\tau\in[\tau_{q}^{\text{min}}(\theta,\varphi),\tau_{q}^{\text{max}}(\theta,\varphi)]\right\\}.$
(55)

The union of all the sub-supports is defined as

$\displaystyle\eta_{u}=\eta_{1}\cup\eta_{2}\cup,\cdots,\cup\eta_{Q}.$ (56)

The probability density function of the multipath is uniformly non-zero within
each sub-support, which is assumed convex for ease of exposition. We also make
an assumption that the longest delay $\tau^{\text{max}}$ satisfies
$\tau^{\text{max}}\leq 1/{\Delta f}$. Taking a 15 kHz subcarrier spacing for
example, it means the paths are not longer than 20 km, which is reasonable.
The rank of ${\bf R}^{\text{(J)}}$ is shown in Theorem 2.

###### Theorem 2

When $N_{h}$, $N_{v}$, and $N_{f}$ are large, the rank of the joint spatial-
frequency channel covariance matrix ${{\bf{R}}^{{\rm{(J)}}}}$ is given by

$\frac{\text{rank}\\{{{\bf{R}}^{{\rm{(J)}}}}\\}}{N_{h}N_{v}N_{f}}=\rho^{{\rm{(J)}}},$
(57)

where the ratio $\rho^{{\rm{(J)}}}$ is defined as

$\displaystyle{\rho^{({\rm{J}})}}$ $\displaystyle=\frac{{{D_{h}}{D_{v}}\Delta
f}}{{{\lambda^{2}}}}\sum\limits_{q=1}^{Q}\int\limits_{\theta_{q}^{\min}}^{\theta_{q}^{\max}}\int\limits_{\varphi_{q}^{\min}(\theta)}^{\varphi_{q}^{\max}(\theta)}\left\\{{\sin}^{2}(\theta)\cos(\varphi)\right.$
$\displaystyle\quad\left.\left({{\tau_{q}^{\max}}(\theta,\varphi)-{\tau_{q}^{\min}}(\theta,\varphi)}\right)\right\\}d\varphi
d\theta.$ (58)

_Proof:_ The proof can be found in Appendix -B.

Note that the rank expressions for the spatial and frequency covariance
matrices may serve as an upper-bound. In practice, the multipath do not have
strictly non-zero APS or PDP over one or several intervals. In some settings,
the angles and delays are modeled as discrete values that are constant over a
relatively long time period, as in the clustered delay line (CDL) model of
3GPP [20] for example. In such cases, the rank ratio ${\rho^{({\rm{S}})}}$,
${\rho^{({\rm{F}})}}$, and ${\rho^{({\rm{J}})}}$ converge to zero for any
finite number of multipaths.

##  IV A Partial Channel Reciprocity-based Codebook

In this section, we show the details of our proposed partial channel
reciprocity-based codebook. Since the multipath angle-delay distribution of
the UL and DL are reciprocal [19] [18], there exists a partial reciprocity in
terms of the signal subspace for UL and DL wideband channel covariance
matrices. Our method makes use of such a partial reciprocity and the low-
dimensional structure of the statistical channel information in both spatial
and frequency domains. The DL channel sounding is designed based on the prior
knowledge of the channel covariance matrices in such a way that only the
signal space corresponding to the non-negligible eigenvectors of the joint
spatial-frequency covariance matrices is captured and measured by the UE,
while the null space of the covariance matrices is automatically ignored. This
will help reduce the feedback overhead and the complexity at the UE side. More
specifically, the training signal is sent according to the dominant
eigenvectors of the covariance matrices.

For ease of exposition, we drop the superscript (U) or (D) and consider the
downlink by default, unless otherwise notified. Denote the eigen-value
decomposition of the joint spatial-frequency channel covariance matrices as

$\displaystyle{{\bf{R}}^{{\rm{(J)}}}}$ $\displaystyle={{\bf
U}}^{{\rm{(J)}}}{\mathbf{\Sigma}^{{\rm{(J)}}}}{{\bf U}}^{{{\rm{(J)}}}H},$ (59)

where ${{\bf U}}^{{\rm{(J)}}}$ contains the eigen-vectors of
${{\bf{R}}^{{\rm{(J)}}}}$:

$\displaystyle{{\bf U}}^{{\rm{(J)}}}$ $\displaystyle=\left[{{\bf
u}}_{1}^{{\rm{(J)}}},{{\bf u}}_{2}^{{\rm{(J)}}},\cdots,{{\bf
u}}_{N_{t}N_{f}}^{{\rm{(J)}}}\right].$ (60)

${\mathbf{\Sigma}^{{\rm{(J)}}}}$ is a diagonal matrix with its diagonal
entries (arranged in non-increasing order) being the eigenvalues of
${{\bf{R}}^{{\rm{(J)}}}}$. As shown in Sec. III, ${{\bf{R}}^{{\rm{(J)}}}}$ has
a low-rankness property and only a small number of its eigenvalues are
significant, while the rest are negligible. We exploit this property and
propose the joint spatial-frequency precoder for CSI-RS, which is aligned with
the dominant eigenvectors of the covariance matrix ${{\bf{R}}^{{\rm{(J)}}}}$.
Denote the total number of antenna ports for CSI-RS by $N_{a}$. The joint
spatial-frequency precoder ${\bf w}_{n}\in{\mathbb{C}^{N_{t}N_{f}\times 1}}$
for port $n$ of CSI-RS is

$\displaystyle{\bf w}_{n}{\triangleq}\left({{\bf
u}}_{n}^{\rm{(J)}}\right)^{*},n=1,2,\cdots,N_{a}.$ (61)

Note that the antenna port here is a generalized concept of the 5G [24]. The
antenna port $n$ ($n=1,\cdots,N_{a}$) for CSI-RS corresponds to the specific
reference signals with the $n$-th spatial-frequency precoder. In order to
better illustrate the idea of our method, we show the joint operation of the
base station and the UE step by step. We introduce here the subband concept,
which means a group of consecutive subcarriers in frequency domain. Depending
on implementation, the bandwidth of a subband is equal to the width of one or
several Resource Blocks (RBs), and it is smaller than the coherence bandwidth.
In other words, the frequency response of the channel is flat within this
interval and only one channel estimate is needed for each subband. With some
misuse of notation, $N_{f}$ denotes either the number of subcarriers or the
number of subbands.

For a certain port $n$, the precoder ${\bf w}_{n}$ can be unvectorized into
$N_{f}$ vectors as below:

$\displaystyle{\bf W}_{n}{\triangleq}[{\bf w}_{n,1},{\bf w}_{n,2},\cdots,{\bf
w}_{n,N_{f}}]=\text{unvec}\\{{\bf w}_{n}\\},$ (62)

where ${\bf w}_{n,k}\in\mathbb{C}^{N_{t}\times 1}$, $k=1,\cdots,N_{f}$, is the
precoder of CSI-RS for the $k$-th subband. For notational simplicity, we
assume the training sequences for each CSI-RS port is identical in different
subbands. Denote the reference signal for the $n$-th CSI-RS port as
${{\bf{x}}_{n}}\in\mathcal{C}^{N_{x}\times 1}$, which is distributed in a set
of $N_{x}$ neighboring Resource Elements (REs). The reference signals of
different antenna ports may share the same set of REs by code-domain sharing
(CDM) with orthogonal patterns, or Frequency-domain sharing (FDM) with
different subcarriers, or Time-domain sharing (TDM) with different OFDM
symbols. For antenna port $n$, $1\leq n\leq N_{a}$, the transmitted jointly
precoded reference signal is ${{\bf{x}}_{n}}{\bf w}^{T}_{n,k}$ at the $k$-th
subband. The corresponding received signal of this port at the $k$-th subband
is

$\displaystyle{{\bf{y}}_{n,k}}(t)={{\bf{x}}_{n}}{{{{\bf
w}}}^{T}_{n,k}}{{\bf{h}}}({f_{k}},t)+{{\bf{n}}_{n,k}}(t),$ (63)

where ${{\bf{h}}}({f_{k}},t)\in\mathbb{C}^{N_{t}\times 1}$ denotes the DL
channel at subband $k$, and ${{\bf{n}}_{n,k}}(t)\in\mathbb{C}^{N_{x}\times 1}$
is the noise. Note that till now we have only completed part of the precoding,
which is the operation ${{{{\bf w}}}^{T}_{n,k}}{{\bf{h}}}({f_{k}},t)$. The
joint spatial-frequency precoding needs a further action from UE side, which
is a summation in frequency domain shown below:

$\displaystyle{{\bf{y}}_{n}}(t)$
$\displaystyle=\sum\limits_{k=1}^{N_{f}}{{\bf{y}}_{n,k}}(t)$ (64)
$\displaystyle={{\bf{x}}_{n}}{\bf
w}_{n}^{T}{\underline{\bf{h}}}(t)+{{\bf{n}}_{n}}(t),$ (65)

where

$\displaystyle{{\bf{n}}_{n}}(t)=\sum\limits_{k=1}^{N_{f}}{{{\bf{n}}_{n,k}}(t)}\in\mathcal{C}^{N_{x}\times
1}.$ (66)

${\underline{\bf{h}}}(t)$ is the vectorized wideband channel as defined in Eq.
(40). The $\sum$ operation in Eq. (64) is done by the UE. Rewriting the
received signal for all $N_{a}$ ports in matrix form, we have

$\displaystyle{\bf{Y}}(t)={\bf{Xg}}(t)+{\bf{n}}(t),$ (67)

where ${\bf{X}}\in\mathcal{C}^{N_{a}N_{x}\times N_{a}}$ is a block matrix
containing all the training sequences

$\displaystyle{\bf{X}}{\triangleq}\left[{\begin{array}[]{*{20}{c}}{{{\bf{x}}_{1}}}&{}\hfil&{}\hfil\\\
{}\hfil&\ddots&{}\hfil\\\
{}\hfil&{}\hfil&{{{\bf{x}}_{{N_{a}}}}}\end{array}}\right].$ (71)

${\bf{g}}(t)\in\mathcal{C}^{N_{a}\times 1}$ is the spatial-frequency jointly
precoded effective channel:

$\displaystyle{\bf{g}}(t){\triangleq}{\left[{\begin{array}[]{*{20}{c}}{{{{g}}_{1}}(t)}&{{{{g}}_{2}}(t)}&\cdots&{{{{g}}_{{N_{a}}}}(t)}\end{array}}\right]^{T}},$
(73)

with the effective scalar channel for port $n$ defined as

$\displaystyle g_{n}(t){\triangleq}{\bf w}_{n}^{T}{\underline{\bf{h}}}(t).$
(74)

Note that $g_{n}(t)$ can be regarded as a projection of the wideband channel
onto the linear space determined by the eigen-mode ${{\bf
u}}_{n}^{{\rm{(J)}}}$ of the joint spatial-frequency covariance matrix
${{\bf{R}}^{{\rm{(J)}}}}$. The UE then obtains an estimate of the projection
$\hat{g}_{n}(t)$ based on the known pilot sequence ${\bf x}_{n}$ by simply
performing an estimation of the effective channel ${{{{\bf
w}}}^{T}_{n,k}}{{\bf{h}}}({f_{k}},t)$ of each subband $k$ and summing them up
over all $N_{f}$ subbands. Alternatively, the UE may first perform a summation
of the received reference signal as in Eq. (64), and then estimate the
effective scalar channel $g_{n}(t)$, provided that the training sequence for a
certain port is identical for all subbands.

Finally, the UE feeds back the projections $\hat{g}_{n}(t),n=1,\cdots,N_{a}$
after a proper quantization, e.g., elementwise quantization [17] or more
sophisticated schemes like random vector quantization (with higher
complexity), to the base station. The DL channel is reconstructed as

$\displaystyle{\underline{\bf{\hat{h}}}}(t)=\sum\limits_{n=1}^{{N_{a}}}{{{\hat{g}}_{n}}(t)}{\bf{w}}_{n}^{*}.$
(75)

We point out the fundamental difference between the classical precoding and
the proposed spatial-frequency joint precoding here. The classical precoding
is to apply a beamforming weight vector (i.e., the precoder) on the base
station antennas while they are transmitting signals. The UE receives the
superimposed signals of all transmit antennas. Such a process is done by the
base station alone. On the contrary, the joint spatial-frequency precoding
functions in a different manner, as it entails a joint processing of the base
station and the UE. More precisely, the base station performs an elementwise
multiplication, while the UE makes a summation. This is because in an OFDM
system, the signal transmitted in all subcarriers are orthogonal to one
another, which prohibits the summation of signals with orthogonal frequencies
over the air as in the spatial-domain precoding case. As a result, the process
consists of three steps, 1) Each base station antenna multiplies the CSI-RS at
the pilot-carrying resource elements by the corresponding entry of ${\bf
w}_{n}$; 2) Upon receiving the precoded CSI-RS, the UE estimates the effective
channels for all pilot-carrying subcarriers; 3) the UE sums up these effective
channels over all pilot-carrying subcarriers.

The proposed PCR codebook scheme is summarized in Algorithm 1.

Algorithm 1 Partial Channel Reciprocity based Codebook Scheme

1: The base station obtains the joint spatial-frequency precoders ${\bf
w}_{n}$ $n=1,\cdots,N_{a}$ by eigenvalue decomposition of the covariance
matrix ${\bf R}^{\text{(J)}}$;

2: The base station applies the precoders when transmitting CSI-RS;

3: For each antenna port, the UE estimates the effective channels in all
$N_{f}$ subbands and sum them up to obtain an estimate of $g_{n}(t)$;

4: The UE feeds back the scalar values $g_{n}(t),n=1,\cdots,N_{a}$;

5: The base station reconstructs the DL CSI with Eq. (75);

Note that a prerequisite of this algorithm is that the DL channel covariance
matrix ${\bf R}^{\text{(J)}}$ is known by the base station. In practice, the
UL channel covariance matrices are easily obtained by channel estimation using
UL pilot (SRS, Sounding Reference Signal). The DL channel covariance matrix
may be obtained by transforming the UL covariance matrix to DL using the
Hilbert space projection method in [22]. Since the second-order statistics are
slow-varying, their eigenvectors may not have to be updated very frequently.
What is more, it is also possible to replace the DL channel covariance
matrices with the UL ones without any transformation. This only leads to minor
performance drops, which will be shown in Sec. VI.

Overall, this proposed training and feedback scheme greatly reduces the
computation burden at the UE side, as the spatial and frequency-domain
precoders are computed and applied by the base station alone. Such an
operation is transparent to the UE, which does not have to know the precoders
themselves. In order to obtain coefficients $\hat{\bf{g}}(t)$ for feedback,
the UE only has to perform an estimation of the precoded effective channels
and an addition in frequency domain.

The complexity of this scheme is now analyzed. The main operation for the UE
is the CSI-RS based channel estimation and a summation. Since the channel
estimation is done by correlating the received signal with the pilot sequence
of length $N_{x}$ for all $N_{f}$ subbands and $N_{a}$ ports, it requires
$N_{x}N_{f}N_{a}$ floating point operations (FLOPs). The summation in
frequency domain takes $N_{f}N_{a}$ FLOPs. Therefore, the total complexity
order for the UE is $\mathcal{O}(N_{x}N_{f}N_{a})$. Given that the orders of
magnitude for $N_{x}$, $N_{f}$, and $N_{a}$ are all ten for practical 5G
systems, the complexity of our proposed method is quite low for UEs.

We now analyze the asymptotic performance of our proposed PCR method. We
derive the amount of feedback required by PCR method to achieve asymptotically
error-free CSI feedback. Note that the number of antenna ports for CSI-RS,
$N_{a}$, is also the number of scalar coefficients needed for feedback in our
scheme for the case of single-antenna UE. The boundaries of the angle and
delay distribution is the same as in Eq. (III) and Eq. (56). The result is
shown in Theorem 3.

###### Theorem 3

The quantization error of the proposed PCR scheme yields

$\mathop{\lim}\limits_{{N_{h}},{N_{v}},{N_{f}}\to\infty}\frac{{\left\|{{\bf{\hat{H}}}-{\bf{H}}}\right\|_{F}^{2}}}{{\left\|{\bf{H}}\right\|_{F}^{2}}}=0,$
(76)

under the condition that the number of scalar coefficients to feedback
satisfies

$\displaystyle N_{a}\geq\rho^{{\rm{(J)}}}N_{h}N_{v}N_{f},$ (77)

where ${\bf{H}}$ is the wideband channel matrix and ${\bf{\hat{H}}}$ is the
reconstructed channel matrix based on the feedback from the UE.
$\rho^{{\rm{(J)}}}$ is the rank ratio defined in Eq. (2).

_Proof:_ The proof can be found in Appendix -C.

Theorem 3 demonstrates that our PCR scheme only requires a small amount of
feedback, since the rank ratio $\rho^{\rm{(J)}}\ll 1$. In fact,
$\rho^{\rm{(J)}}$ is also the CSI compression ratio of our scheme compared to
the method of feeding back the full channel matrix ${\bf{H}}$. For the case
that angles and delays are modeled as constant over a long period as in the
CDL model, the number of feedback coefficients $N_{a}$ does not scale with the
number of antennas or the bandwidth. It is in fact a finite value that is
always smaller than the number of paths. This is because in a practical model
such as CDL, the multipaths exist in the form of clusters. And the path angles
inside a certain cluster are close to each other, since they are reflected by
one or several neighboring scatterers. Some examples in Sec. VI will show that
although the total number of paths is up to several hundreds, the number of
feedback coefficients can be as small as 64 for the whole wideband massive
MIMO channel.

##  V Low Complexity Alternatives

The PCR codebook scheme in Algorithm 1 of Sec. IV requires the eigen-
decomposition of a large matrix ${\bf R}^{\text{(J)}}$. Although this
operation is not done in a real-time manner, the complexity is still high for
a base station. In this section, we propose two low-complexity alternatives
which have mild performance losses compared with the original PCR codebook
scheme. Note that all the proposed schemes have the same complexity at the UE
side.

###  V-A PCR codebook with spatial/frequency eigen basis

We now propose an alternative - the PCR codebook with spatial/frequency eigen
basis (PCR-E) to mitigate this impediment of high complexity at the base
station. It is done by selecting the eigenvectors of the covariance matrices
${\bf R}^{\text{(S)}}\in\mathbb{C}^{N_{t}\times N_{t}}$ and ${\bf
R}^{\text{(F)}}\in\mathbb{C}^{N_{f}\times N_{f}}$ separately and then form a
joint spatial-frequency precoder. The design of a joint spatial-frequency
precoder breaks down to a spatial-domain precoder and a frequency-domain
precoder.

The eigen-value decomposition of the spatial and frequency channel covariance
matrices are written as

$\displaystyle{{\bf{R}}^{{\rm{(S)}}}}$ $\displaystyle={{\bf
U}}^{{\rm{(S)}}}{\mathbf{\Sigma}^{{\rm{(S)}}}}{{\bf U}}^{{{\rm{(S)}}}H},$ (78)
$\displaystyle{{\bf{R}}^{{\rm{(F)}}}}$ $\displaystyle={{\bf
U}}^{{\rm{(F)}}}{\mathbf{\Sigma}^{{\rm{(F)}}}}{{\bf U}}^{{{\rm{(F)}}}H},$ (79)

where ${{\bf U}}^{{\rm{(S)}}}$ and ${{\bf U}}^{{\rm{(F)}}}$ contain the eigen-
vectors of ${{\bf{R}}^{{\rm{(S)}}}}$ and ${{\bf{R}}^{{\rm{(F)}}}}$
respectively:

$\displaystyle{{\bf U}}^{{\rm{(S)}}}$ $\displaystyle=\left[{{\bf
u}}_{1}^{{\rm{(S)}}},{{\bf u}}_{2}^{{\rm{(S)}}},\cdots,{{\bf
u}}_{N_{t}}^{{\rm{(S)}}}\right],$ (80) $\displaystyle{{\bf U}}^{{\rm{(F)}}}$
$\displaystyle=\left[{{\bf u}}_{1}^{{\rm{(F)}}},{{\bf
u}}_{2}^{{\rm{(F)}}},\cdots,{{\bf u}}_{N_{f}}^{{\rm{(F)}}}\right].$ (81)

${\mathbf{\Sigma}^{{\rm{(S)}}}}$ and ${\mathbf{\Sigma}^{{\rm{(F)}}}}$ are
diagonal matrices with their diagonal entries (arranged in non-increasing
order) being the eigenvalues of ${{\bf{R}}^{{\rm{(S)}}}}$ and
${{\bf{R}}^{{\rm{(F)}}}}$ respectively. We once again exploit the low-rankness
properties of ${{\bf{R}}^{{\rm{(S)}}}}$ and ${{\bf{R}}^{{\rm{(F)}}}}$ in this
PCR-E codebook scheme. We build the spatial-domain precoder of the CSI-RS such
that it is aligned with the dominant eigenvectors of the spatial-domain
covariance matrix ${{\bf{R}}^{{\rm{(S)}}}}$. Meanwhile, the frequency-domain
precoder is aligned with the dominant eigenvectors of
${{\bf{R}}^{{\rm{(F)}}}}$. The joint spatial-frequency precoder ${\bf
w}_{n}\in{\mathbb{C}^{1\times N_{t}N_{f}}}$ for port $n$ of CSI-RS is given as

$\displaystyle{\bf w}_{n}{\triangleq}{\bf f}_{n}\otimes{\bf s}_{n},$ (82)

where ${\bf f}_{n}\in{\mathbb{C}^{1\times N_{f}}}$ is the conjugate transpose
of a certain eigenvector selected from ${{\bf U}}^{{\rm{(F)}}}$ and ${\bf
s}_{n}\in{\mathbb{C}^{1\times N_{t}}}$ is the conjugate transpose of a certain
eigenvector from ${{\bf U}}^{{\rm{(S)}}}$. One simple eigenvector selection
method is to choose the eigenvectors corresponding to the greatest eigenvalues
of ${{\bf{R}}^{{\rm{(S)}}}}$ and ${{\bf{R}}^{{\rm{(F)}}}}$. However, this
method may not lead to the best performance. The reason is that the multipath
distribution of angles is not independent to the distribution of delays in
most cases, e.g., when the paths are distributed in the form of clusters.
${{\bf{R}}^{{\rm{(S)}}}}$ only contains the multipath distribution information
in angular domain, while ${{\bf{R}}^{{\rm{(F)}}}}$ only reflects the multipath
delay distribution. If we choose a set of spatial precoders independently with
the set of frequency precoders and construct the joint precoders by all the
combinations of the two sets, it will result in excessive consumption of
training and feedback overhead. This is due to the fact that some constructed
joint precoders may lead to very small coefficients, which are not necessary
for feedback. For the sake of overhead reduction, we propose a selection
criterion which selects the spatial precoders and frequency precoders jointly
based on recent UL channel samples
${{\bf{H}}^{{\rm{(U)}}}}({t_{1}}),{{\bf{H}}^{{\rm{(U)}}}}({t_{2}}),\cdots,{{\bf{H}}^{{\rm{(U)}}}}({t_{N_{c}}})$.

Define the joint spatial-frequency eigen projection as

$\displaystyle{\bf
G}(t){\triangleq}{{\left({\bf{U}}^{({\rm{S}})}\right)^{H}}{{\bf{H}}^{{\rm{(U)}}}}({t})\left({{\bf{U}}^{(F)}}\right)^{*}}.$
(83)

Then, the accumulated element-wise power matrix of the projection above is

$\displaystyle\bm{\mathcal{G}}{\triangleq}\sum\limits_{i=1}^{{N_{c}}}{\bf
G}(t_{i})\odot\left({\bf G}(t_{i})\right)^{*}.$ (84)

We select the positions of the greatest $N_{a}$ values in $\bm{\mathcal{G}}$
and denote the row index and column index of the $n$-th position as $r_{n}$
and $c_{n}$ respectively. Then, the spatial-domain precoder for the $n$-th
CSI-RS port is the conjugate of the $r_{n}$-th eigenvector of
${{\bf{R}}^{{\rm{(S)}}}}$:

$\displaystyle{\bf s}_{n}=\left({{\bf u}}_{r_{n}}^{{\rm{(S)}}}\right)^{*}.$
(85)

Similarly, the frequency-domain precoder for this port is

$\displaystyle{\bf f}_{n}=\left({{\bf u}}_{c_{n}}^{{\rm{(F)}}}\right)^{*}.$
(86)

Note that among all $N_{a}$ ports, there are always some duplicates of the
spatial-domain precoder or frequency-domain precoder for different ports. In
other words, the following circumstance may take place:

$\displaystyle\exists i\neq j,\text{ s.t. }r_{i}=r_{j}\text{ or
}c_{i}=c_{j},\text{ for }i,j=1,\cdots,N_{a}.$ (87)

This is due to the fact that the accumulated power matrix $\bm{\mathcal{G}}$
may have some significant entries distributed in the same row or column.
However, two different CSI-RS ports never have exactly the same spatial
precoder and frequency precoder at the same time. Regarding the antenna port
$n$, $1\leq n\leq N_{a}$, the transmitted jointly precoded reference signal is
${{\bf{x}}_{n}}\left({{{\bf{f}}_{n}}\otimes{{\bf{s}}_{n}}}\right)$. The
corresponding received signal at frequency $f_{k}$, a.k.a., the $k$-th
subband, is

$\displaystyle{{\bf{y}}_{n,k}}(t)={{\bf{x}}_{n}}{{{f}}_{n,k}}{{\bf{s}}_{n}}{\bf{h}}({f_{k}},t)+{{\bf{n}}_{n,k}}(t),$
(88)

where ${{{f}}_{n,k}}$ is the $k$-th row of the frequency-domain precoder
${{\bf{f}}_{n}}$. After the frequency-domain summation at the UE side, the
combined received signal is

$\displaystyle{{\bf{y}}_{n}}(t)$
$\displaystyle=\sum\limits_{k=1}^{N_{f}}{\left({{{\bf{x}}_{n}}{{{f}}_{n,k}}{{\bf{s}}_{n}}{{\bf{h}}}({f_{k}},t)+{{\bf{n}}_{n,k}}(t)}\right)}$
(89)
$\displaystyle={{\bf{x}}_{n}}\left({{{\bf{f}}_{n}}\otimes{{\bf{s}}_{n}}}\right){\underline{\bf{h}}}(t)+{{\bf{n}}_{n}}(t),$
(90)

where

$\displaystyle{{\bf{n}}_{n}}(t)=\sum\limits_{k=1}^{N_{f}}{{{\bf{n}}_{n,k}}(t)}\in\mathcal{C}^{N_{x}\times
1}.$ (91)

The $\sum$ operation in Eq. (89) is done by the UE. Based on the training
sequence ${\bf x}_{n}$, we may obtain the spatial-frequency jointly precoded
effective channel:

$\displaystyle{\bf{g}}(t){\triangleq}{\left[{\begin{array}[]{*{20}{c}}{{{{g}}_{1}}(t)}&{{{{g}}_{2}}(t)}&\cdots&{{{{g}}_{{N_{a}}}}(t)}\end{array}}\right]^{T}},$
(93)

with the effective scalar channel for port $n$ defined as

$\displaystyle
g_{n}(t){\triangleq}{{\bf{f}}_{n}}\otimes{{\bf{s}}_{n}}{\underline{\bf{h}}}(t).$
(94)

Then, the UE feeds back the quantized scalars
$\hat{g}_{n}(t),n=1,\cdots,N_{a}$ to the base station. Finally the DL channel
is reconstructed as

$\displaystyle{{\bf{\widehat{H}}}}(t)=\sum\limits_{n=1}^{{N_{a}}}{{{\hat{g}}_{n}}(t)}{\bf{s}}_{n}^{H}{\bf{f}}_{n}^{*}.$
(95)

The proposed low-complexity alternative, i.e., PCR-E, is summarized in
Algorithm 2.

Algorithm 2 PCR Codebook scheme with spatial/frequency eigen basis

1: The base station obtains the precoders ${\bf s}_{n}$ and ${\bf
f}_{n},n=1,\cdots,N_{a}$ by finding the positions of the $N_{a}$ greatest
values in Eq. (84);

2: The base station applies the precoders when transmitting CSI-RS;

3: For each antenna port, the UE estimates the effective channels in all
$N_{f}$ subbands and sum them up to obtain $g_{n}(t)$;

4: The UE feeds back the quantized $g_{n}(t),n=1,\cdots,N_{a}$;

5: The base station reconstructs the DL CSI with Eq. (95);

Note that Algorithm 2 assumes the eigenvectors ${{\bf U}}^{{\rm{(S)}}}$ and
${{\bf U}}^{{\rm{(F)}}}$ of the DL channel covariance matrices are known by
the base station. This can be achieved with the UL-DL channel covariance
transformation method in [22].

The complexity of this scheme at UE side is exactly the same as in the PCR
scheme of Sec. IV. As for the base station, the main complexity comes from the
searching of the spatial and frequency eigenmodes, which has the complexity
order of $\mathcal{O}(N_{c}(N_{t}^{2}N_{f}+N_{t}(N_{f})^{2})$. Nevertheless,
the searching can be made faster by considering only the dominant
eigenvectors, or even skipped at the cost of some mild performance loss.

###  V-B PCR codebook with DFT basis

The two schemes mentioned above both needs the eigenvalue decomposition of
channel covariance matrix. In this section, we propose an alternative named
PCR codebook with DFT basis (PCR-D) that circumvents such an operation and
thus has even lower complexity at the base station side. The key ingredient of
this scheme is the exploitation of the channel reciprocity of the multipath
angles and delays. Similar to PCR-E, the joint spatial-frequency precoder
again consists of two components, the spatial precoder and the frequency
precoder, which are DFT-based vectors instead of eigenvectors.

Denote a DFT matrix of size $K\times K$ as

$\displaystyle{\bf
E}(K)\triangleq\frac{1}{\sqrt{K}}\begin{bmatrix}\omega^{0\cdot
0}&\omega^{0\cdot 1}&\cdots&\omega^{0(K-1)}\\\ \omega^{1\cdot
0}&\omega^{1\cdot 1}&\cdots&\omega^{1(K-1)}\\\ \vdots&\vdots&\ddots&\vdots\\\
\omega^{(K-1)\cdot 0}&\omega^{(K-1)\cdot
1}&\cdots&\omega^{(K-1)(K-1)}\end{bmatrix},$

where $\omega\triangleq e^{-2\pi j/K}$. According to the UPA antenna array
structure, we construct a DFT-based spatial orthogonal basis as ${\bf
S}{\triangleq}{\bf E}(N_{h})\otimes{\bf E}(N_{v})$. Note that for the case of
dual-polarized antennas, there are two options: we may either construct the
spatial basis as

$\displaystyle{\bf S}=\text{diag}\\{{\bf E}(N_{h})\otimes{\bf E}(N_{v}),{\bf
E}(N_{h})\otimes{\bf E}(N_{v})\\},$ (96)

or process the two polarizations one by one. These two options are equivalent.
Similarly, a frequency basis is defined as ${\bf F}{\triangleq}{\bf
E}(N_{f})$.

The spatial precoders are selected from the column vectors in ${\bf S}$, while
the frequency precoders are from ${\bf F}$. The selection method is similar to
PCR-E scheme, which relies on the projections of the UL channel samples
${{\bf{H}}^{{\rm{(U)}}}}({t_{1}}),{{\bf{H}}^{{\rm{(U)}}}}({t_{2}}),\cdots,{{\bf{H}}^{{\rm{(U)}}}}({t_{N_{c}}})$
on the spatial and frequency basis:

$\displaystyle\bm{\mathcal{G}}{\triangleq}\sum\limits_{i=1}^{{N_{c}}}\left({{\bf
S}^{H}}{{\bf{H}}^{{\rm{(U)}}}}({t_{i}}){\bf F}\right)\odot\left({{\bf
S}^{H}}{{\bf{H}}^{{\rm{(U)}}}}({t_{i}}){\bf F}\right)^{*}.$ (97)

Denote the row index and column index of the $n$-th greatest value in
$\bm{\mathcal{G}}$ as $r_{n}$ and $c_{n}$ respectively. For the $n$-th CSI-RS
port, The spatial-domain precoder ${\bf s}_{n}$ is

$\displaystyle{\bf s}_{n}=\left({\bf S}_{r_{n}}\right)^{H},$ (98)

where ${\bf S}_{r_{n}}$ is the $r_{n}$-th column of ${\bf S}$. The
corresponding frequency-domain precoder ${\bf f}_{n}$ is the $c_{n}$-th row of
${\bf F}$. The joint spatial-frequency precoder is ${\bf f}_{n}\otimes{\bf
s}_{n}$. The remaining operations are the same as in the PCR-E scheme and thus
omitted. This scheme is summarized in Algorithm 3.

Algorithm 3 PCR Codebook scheme with DFT basis

1: The base station obtains the precoders ${\bf s}_{n}$ and ${\bf
f}_{n},n=1,\cdots,N_{a}$ by finding the positions of the $N_{a}$ greatest
values in Eq. (97);

2: The base station applies the precoders when transmitting CSI-RS;

3: For each antenna port, the UE estimates the effective channels in all
$N_{f}$ subbands and sum them up to obtain an estimate of $g_{n}(t)$;

4: The UE feeds back the quantized $g_{n}(t),n=1,\cdots,N_{a}$;

5: The base station reconstructs the DL CSI by Eq. (95);

##  VI Numerical Results

In this section, we show the simulation results of our proposed schemes,
primarily under the practical model of 3GPP. The main parameters are listed in
Table I. The configuration of the antennas, including the base station and the
UE, is expressed by a tuple $(\underline{M},\underline{N},\underline{P})$,
where $\underline{M},\underline{N},\underline{P}$ denote the number of rows,
columns, and polarizations of the antenna elements respectively. We consider a
typical parameter set of 5G with the center frequency at 3.5 GHz and the
subcarrier spacing of 30 kHz. We adopt the CDL-A and CDL-D models in the
evaluations. In CDL-A model, the total number of multipaths is 460, i.e., each
UE has 23 multipath clusters and each cluster contains 20 paths. No Line-of-
Sight (LoS) path exists. While in CDL-D channel model, there are totally 273
paths with one LoS component. The distributions of the paths in CDL-A model
and CDL-D model are defined in Table 7.7.1-1 and Table 7.7.1-4 of [20]
respectively.

The UEs have $N_{r}=2$ antennas in simulations, and for each UE antenna, a set
of $N_{a}$ coefficients are fed back to the BS. Therefore, the number of
feedback coefficients is $2N_{a}$ for the schemes of PCR, PCR-E, and PCR-D.
The maximum number of streams for each UE is two.

TABLE I: Basic simulation parameters Scenario |  3D Urban Macro (3D UMa)
---|---
DL center frequency |  3.5 GHz
UL center frequency |  3.4 GHz
Subcarrier spacing |  30 kHz
Bandwidth |  20 MHz (51 RBs)
Number of UEs |  8
BS antenna configuration |  $(\underline{M},\underline{N},\underline{P})=(2,8,2)/(4,8,2)$, $(D_{h},D_{v})=(0.5,0.8)\lambda$, the polarization angles are $\pm 45^{\circ}$
UE antenna configuration |  $(\underline{M},\underline{N},\underline{P})=(1,1,2,1,1)$, the polarization angles are $0^{\circ}$ and $90^{\circ}$
Channel model |  CDL-A, CDL-D
Delay spread |  300 ns
DL precoder |  EZF
UE receiver |  MMSE-IRC
Number of paths |  460, 271

We first visualize the effects of the dimension reduction of the three
proposed methods in Fig. 2.

Figure 2: The element-wise amplitude of the wideband channel matrix and its
projections (a) The channel matrix; (b) the projections onto the bases of the
PCR scheme; (c) the projections onto the bases of the PCR-E scheme; (d) the
projections onto the bases of the PCR-D scheme.

One snapshot of the wideband channel ${\bf H}$ between the base station and UE
is taken as an example. The antenna configuration at the base station is
$(\underline{M},\underline{N},\underline{P})=(4,8,2)$, and the number of
subband is 51, as shown in Table I. Fig. 2 (a) shows the amplitude of the
wideband channel coefficient of ${\bf H}\in\mathbb{C}^{N_{t}\times N_{f}}$
where $N_{t}=64,N_{f}=51$. Fig. 2 (b) is the amplitude of the projected
channel onto the orthogonal basis Eq. (60) of the proposed PCR scheme in Sec.
IV. We may observe that only a very small portion of the projected values are
significant. Such a high sparsity allows us to reconstructed the full channel
matrix with only a small number of non-negligible scalar coefficients. Fig. 2
(c) is the amplitude of the projection onto the basis of Eq. (80) and Eq. (81)
as in the low-complexity alternative PCR-E scheme. Fig. 2 (d) is amplitude
when projected on to the 2D-DFT basis as in the PCR-D scheme. From (c) and (d)
we also observe that the projected channel matrix is sparse, and the eigen-
mode based projection is better than the DFT-based projection in terms of the
sparse representation of the channel matrix.

Figure 3: The spectral efficiency vs. SNR, $N_{t}=32$, $N_{a}=32$, CDL-A
model.
Figure 4: The spectral efficiency vs. SNR, $N_{t}=64$, $N_{a}=32$, CDL-A
model.

Fig. 3 and Fig. 4 shows the spectral efficiency performance of the proposed
codebook schemes for the settings that the base station is equipped with 32
antennas and 64 antennas, respectively. The spectral efficiency is computed as
$\sum_{n=1}^{{N_{u}}}\log_{2}(1+\text{SINR}_{n})$ averaged over time and
frequency, where $\text{SINR}_{n}$ is the Signal and Interference to Noise
Ratio for the $n$-th UE, and $N_{u}=8$ is the total number of the active UEs.
The number of coefficients to feed back is $2N_{a}=64$ for each of the schemes
proposed by this paper. The benchmark of Enhanced Type II codebook in Rel-16
also requires $2N_{a}$ feedback coefficients, i.e., two sets of $N_{a}$ 2D DFT
projections corresponding to the two UE antennas respectively. However, the
Rel-16 codebook requires the feedback of indices apart from the $2N_{a}$
scalar coefficients, to notify the positions of the selected $N_{a}$
projections of the channel matrix on the 2D-DFT basis. The curve labeled with
“CRNet” is the Deep Learning-based CSI compression and reconstruction scheme
proposed by [12]. The CRNet consists of an encoder and decoder, where the
encoder performs compression at the UE and the decoder reconstructs the
original CSI at the BS. The training, validation, and test datasets contain
80000, 20000 and 20000 wideband channel samples respectively. The number of
feedback coefficients for CRNet is also 64. We may notice that our proposed
PCR scheme is very close to the ideal case that perfect CSI is known at the
base station. It is interesting to note that the wideband massive MIMO
channel, although very large in size, can be effectively represented by only
$2N_{a}=64$ scalar coefficients using our PCR codebook scheme. The low-
complexity alternative of PCR-E has some mild performance loss compared to the
PCR scheme. Nevertheless it has significant gains over the Enhanced Type II
Codebook in the latest Rel-16 standard of 5G. The reason is that there are
more power leakages under DFT basis than under eigen-basis, which is
demonstrated in Fig. 2. We may also observe that the low-complexity
alternative of PCR-D scheme has lower performance than the Rel-16 codebook of
5G when the numbers of feedback coefficients are equal. However, PCR-D does
not need to feedback indices, and the total amount of overhead is thus
smaller. The complexity of PCR-D is also much lower. In practice, there is
always a trade-off between complexity and performance, and a substantial
complexity reduction can be achieved at the cost of feeding back several more
scalar coefficients.

TABLE II: Comparisons of the schemes Schemes |  PCR /PCR-E | PCR-D | Rel-16
---|---|---|---
UE complexity | $\mathcal{O}(N_{x}N_{f}N_{a})$ | $\mathcal{O}(N_{x}N_{f}N_{a})$ |  $\mathcal{O}(N_{x}N_{f}N_{a})+\mathcal{O}$ $(N_{t}N_{f}\log(N_{t}N_{f}))$
Overhead | 2 $N_{a}$ scalars | 2 $N_{a}$ scalars |  2 $N_{a}$ scalars + $N_{a}$ indices
# of ports | $N_{a}$ | $N_{a}$ | $N_{t}$
Generality | high | UPA only | UPA only

Table II is the comparison of the proposed schemes and the Enhanced Type II
codebook of the Rel-16 in terms of UE complexity, feedback overhead, required
number of antenna ports, and generality. Since the Enhanced Type II codebook
adopts non-precoded CSI-RS, the number of antenna ports is equal to the number
of BS antennas $N_{t}$. The PCR-D and Enhanced Type II codebook both rely on
2D DFT for CSI compression, which limits their applicability to UPA topology
of BS antennas. However, the PCR and PCR-E codebooks are more general and
applicable to other BS antenna topologies as well.

Figure 5: The spectral efficiency vs. SNR, PCR with different feedback
overhead, $N_{t}=64$, CDL-A model.

Then we show the impact of the number of antenna ports $N_{a}$ on the PCR
scheme in Fig. 5. When the value of $N_{a}$ increases from 10 to 40, the
performance grows monotonically and eventually saturates near the ideal case
of perfect CSI.

Figure 6: The spectral efficiency vs. SNR, PCR-E with different feedback
overhead, $N_{t}=32$, CDL-A model.

Now we show in Fig. 6 the performance of the PCR-E scheme with different
feedback overheads, i.e., $N_{a}=32,16,8$. In order to demonstrate the
robustness of our scheme to inaccurate channel covariance matrices, e.g.,
${\bf R}^{\text{(S)}}$ and ${\bf R}^{\text{(F)}}$, we also show the cases
where the DL channel covariance matrices are replaced by the UL ones. In other
words, the curves labeled with “PCR-E with UL covariance” means the
eigenvectors of the UL channel covariance matrices are used when computing the
precoders. The transformations of the UL covariance to the DL version are not
performed, which further reduces the computational complexity for the base
station. It is interesting to note that even with such non-ideal channel
covariance matrices, the degradation of the performance is tolerable.
Moreover, the Rel-16 codebook with 64 feedback coefficients are also plotted
as a benchmark. One may observe that the PCR-E scheme with much fewer feedback
coefficients, i.e., 16 ($N_{a}=8$), is comparable with this benchmark.

Figure 7: The spectral efficiency vs. SNR, $N_{t}=64$, $N_{a}=20$, CDL-D
model.

Then, the CDL-D model is considered and the performance curves are shown in
Fig. 7. Since there are less paths and one of them is an LoS component in
CDL-D, the channel matrix has more sparsity in terms of angles and delays
compared to CDL-A. It leads to higher correlation between antennas and
subcarriers, and therefore makes it possible to reconstruct the DL channel
with less feedback coefficients. As we may observe in the figure, the PCR
scheme with $N_{a}=20$ is already very close to the perfect CSI setting.

Figure 8: The spectral efficiency vs. SNR, $N_{t}=32$, $N_{a}=32$, 8 UEs,
COST-2100 model.

Finally, the proposed schemes are evaluated under the COST-2100 channel model
[25] in Fig. 8. The BS antenna topology is ULA with 32 antenna elements, and
all 8 UEs have single antenna. The bandwidth is 20 MHz, which contains 51 RBs.
The total number of feedback coefficients is 32 for all the schemes. One may
observe that the proposed PCR and PCR-E codebooks outperforms the Enhanced
Type II codebook under COST-2100 channel model.

##  VII Conclusions

In this paper we dealt with the challenge of CSI acquisition in FDD massive
MIMO. We first derive the ranks of the channel covariance matrices of the
wideband large-scale MIMO channel for a given angle and delay distribution.
The closed-form expression of the rank indicates that a low-rankness property
is always valid for a UPA with half-wavelength or smaller antenna spacing,
regardless of the richness of the scattering environment. We then proposed a
PCR codebook scheme which exploits such a property and the partial reciprocity
of UL/DL channels. It outperforms the latest 5G codebook of Rel-16 at
performance, UE complexity, feedback overhead, received SNR of CSI-RS, and
generality. We also proposed two alternatives named PCR-E and PCR-D codebook
schemes, aiming at further reducing the complexity at the BS side at the cost
of mild performance loss. Simulation results showed that the proposed methods
are very close to the ideal case of perfect CSI even with only tens of scalar
feedback coefficients for the whole wideband massive MIMO channel.

###  -A Proof of Theorem 1:

In order to prove Theorem 1, we need to first prove that non-overlapping
angular supports lead to additive dimensionality of the signal subspace of the
spatial channel covariance matrix.

###### Lemma 1

Consider two angular supports $\zeta_{1}$ and $\zeta_{2}$. The boundaries are
defined as

$\displaystyle\zeta_{1}=\left\\{(\theta,\varphi)|\theta\in[\theta_{1}^{\text{min}},\theta_{1}^{\text{max}}],\varphi\in[\varphi_{1}^{\text{min}}(\theta),\varphi_{1}^{\text{max}}(\theta)]\right\\},$
(99)
$\displaystyle\zeta_{2}=\left\\{(\theta,\varphi)|\theta\in[\theta_{2}^{\text{min}},\theta_{2}^{\text{max}}],\varphi\in[\varphi_{2}^{\text{min}}(\theta),\varphi_{2}^{\text{max}}(\theta)]\right\\}.$
(100)

The spatial covariance matrices corresponding to the angular supports are
denoted as ${\bf R}_{1}$ and ${\bf R}_{2}$ respectively. Define the linear
space spanned by the 2-D steering vectors with corresponding angles inside the
angular support $\zeta$ as

$\displaystyle\mathcal{B}{\triangleq}\text{span}\left\\{{\frac{{{\bf{a}}(\theta,\varphi)}}{\sqrt{{{N_{h}}{N_{v}}}}},\left({\theta,\varphi}\right)\in\zeta}\right\\}.$
(101)

If the two angular supports are non-overlapping, i.e.,
$\zeta_{1}\cap\zeta_{2}=\emptyset$, then the null space of ${\bf R}_{1}$
includes the linear space $\mathcal{B}_{2}$ when $N_{h}$ and $N_{v}$ are
large:

$\displaystyle\text{null}({{\bf R}_{1}})\supset\mathcal{B}_{2},\text{ as
}N_{h},N_{v}\rightarrow\infty.$ (102)

_Proof:_ We denote the joint angular power spectrum of the multipath as
${p_{i}}(\theta,\varphi)$, which is finite and uniformly non-zero inside the
angular support $\zeta_{i}$, for $i=1,2$. Without loss of generality, we
normalize the total power to 1, such that

$\displaystyle{\left({\int_{\theta_{i}^{\min}}^{\theta_{i}^{\max}}{\int_{\varphi_{i}^{\min}(\theta)}^{\varphi_{i}^{\max}(\theta)}{p_{i}}(\theta,\varphi){d\varphi
d\theta}}}\right)=1}.$ (103)

For ease of exposition, we use ${p_{i}}$ to represent
${p_{i}}(\theta,\varphi)$ in the proof below.

The covariance matrix corresponding to the angular support $\zeta_{i},i=1,2$
is written as

$\displaystyle{{\bf R}_{i}}$
$\displaystyle=\mathbb{E}\\{{\bf{a}}({\theta_{i}},{\varphi_{i}}){\left({{\bf{a}}({\theta_{i}},{\varphi_{i}})}\right)^{H}}\\}$
(104)
$\displaystyle=\int_{\theta_{i}^{\min}}^{\theta_{i}^{\max}}{\int_{\varphi_{i}^{\min}(\theta)}^{\varphi_{i}^{\max}(\theta)}{{\bf{a}}(\theta,\varphi){{\left({{\bf{a}}(\theta,\varphi)}\right)}^{H}}{p_{i}}d\varphi
d\theta}}.$ (105)

Taking an arbitrary angle $(\theta_{2},\varphi_{2})\in\zeta_{2}$, then we have

$\displaystyle\frac{{{{\left({{\bf{a}}({\theta_{2}},{\varphi_{2}})}\right)}^{H}}}}{\sqrt{{N_{h}}{N_{v}}}}{{\bf{R}}_{1}}\frac{{{\bf{a}}({\theta_{2}},{\varphi_{2}})}}{\sqrt{{N_{h}}{N_{v}}}}$
$\displaystyle=\frac{{1}}{{{N_{h}}{N_{v}}}}\iint{p_{1}}{{{{\left({{\bf{a}}({\theta_{2}},{\varphi_{2}})}\right)}^{H}}}{\bf{a}}(\theta,\varphi){{\left({{\bf{a}}(\theta,\varphi)}\right)}^{H}}{{\bf{a}}({\theta_{2}},{\varphi_{2}})}d\varphi
d\theta}$
$\displaystyle=\frac{{1}}{{{N_{h}}{N_{v}}}}\iint{p_{1}}{\left({{{\left({{{\bf{a}}_{h}}({\theta_{2}},{\varphi_{2}})\otimes{{\bf{a}}_{v}}({\theta_{2}})}\right)}^{H}}\left({{{\bf{a}}_{h}}({\theta},{\varphi})\otimes{{\bf{a}}_{v}}({\theta})}\right)}\right)^{2}}d\varphi
d\theta$
$\displaystyle=\frac{{1}}{{{N_{h}}{N_{v}}}}\iint{p_{1}}{\left({{{\left({{{\bf{a}}_{h}}({\theta_{2}},{\varphi_{2}})}\right)}^{{}^{H}}}{{\bf{a}}_{h}}(\theta,\varphi){{\left({{{\bf{a}}_{v}}({\theta_{2}})}\right)}^{H}}{{\bf{a}}_{v}}(\theta)}\right)^{2}}d\varphi
d\theta$
$\displaystyle=\frac{{1}}{{{N_{h}}{N_{v}}}}\iint{p_{1}}\left({\left({\sum\limits_{n=0}^{{N_{h}-1}}{{e^{\frac{{j2\pi
n{D_{h}}}}{\lambda}\left({\sin(\theta)\sin(\varphi)-\sin({\theta_{2}})\sin({\varphi_{2}})}\right)}}}}\right.}\right)$
$\displaystyle\qquad\qquad{\left.{\left({\sum\limits_{n=0}^{{N_{v}-1}}{{e^{\frac{{j2\pi
n{D_{v}}}}{\lambda}\left({\cos(\theta)-\cos({\theta_{2}})}\right)}}}}\right)}\right)^{2}}d\varphi
d\theta.$ (106)

Since the angle pair $({\theta_{2}},{\varphi_{2}})$ is outside the boundary of
the double integral $\zeta_{1}$, we readily have that the terms
${\sin(\theta)\sin(\varphi)-\sin({\theta_{2}})\sin({\varphi_{2}})}$ and
${\left({\cos(\theta)-\cos({\theta_{2}})}\right)}$ never take the value zero
at the same time. Thus the expression (106) converges to zero as $N_{h}$ and
$N_{v}$ go to infinity. Therefore Lemma 1 is proved.

Note that the linear space $\mathcal{B}_{i}$ defined in Eq. (101) is
equivalent to the signal subspace of the spatial channel covariance matrix
${\bf R}_{i},i=1,2$, i.e.,

$\displaystyle\mathcal{B}_{i}=\text{span}\left\\{{{\bf
u}_{n}^{(i)}:n=1,\cdots,{r_{i}}}\right\\},$ (107)

where ${\bf u}_{n}^{(i)}$ is the eigenvector of ${\bf R}_{i}$ corresponding to
its $n$-th greatest eigenvalue and $r_{i}$ denotes the rank of ${\bf R}_{i}$.
According to Lemma 1, the signal subspaces of ${\bf R}_{1}$ and ${\bf R}_{2}$
are asymptotically orthogonal when $N_{h}$ and $N_{v}$ are large:

$\displaystyle\mathcal{B}_{1}\bot\mathcal{B}_{2},\text{ as
}N_{h},N_{v}\rightarrow\infty.$ (108)

We define the angular support $\zeta_{u}$ as the union of $\zeta_{1}$ and
$\zeta_{2}$:

$\displaystyle\zeta_{u}{\triangleq}\zeta_{1}\cup\zeta_{2},$ (109)

and the spatial channel covariance matrix ${\bf R}_{u}$ corresponding to the
multipath angular support $\zeta_{u}$. Then the rank of the spatial channel
covariance ${\bf R}_{u}$, denoted by $r_{u}$, satisfies

$\displaystyle r_{u}=r_{1}+r_{2},\text{ as }N_{h},N_{v}\rightarrow\infty.$
(110)

The above observation implies that the rank of the spatial channel covariance
matrix can be computed by summing up the contributions of all non-overlapping
angular sub-supports that forms the entire range. More precisely, consider a
spatial covariance ${\bf R}_{u}$ of a certain user that has an angular support
$\zeta_{u}$. Such a support is composed of $K$ non-overlapping sub-supports:

$\displaystyle\zeta_{u}$
$\displaystyle=\zeta_{1}\cup\zeta_{2}\cdots\cup\zeta_{K},$ (111)
$\displaystyle\zeta_{i}$
$\displaystyle\cap\zeta_{j}=\emptyset,i,j,=1,2,\cdots,K,\forall i\neq j.$
(112)

Then the rank of ${\bf R}_{u}$ yields:

$\displaystyle r_{u}=\sum\limits_{i=1}^{K}{{r_{i}}},\text{ as
}N_{h},N_{v}\rightarrow\infty,$ (113)

where ${r_{i}}$ is the rank of the spatial covariance matrix with angular
support $\zeta_{i}$. Therefore, finding the rank of the spatial covariance
matrix is equivalent to computing the size of the angular support $\zeta_{u}$,
which could be done by a double integral over the boundary of the angular
support, that is

$\displaystyle\mathop{\lim}\limits_{{N_{h}},{N_{v}}\to\infty}\frac{{{\rm{rank}}\\{{{\bf{R}}^{({\rm{S}})}}\\}}}{{{N_{h}}{N_{v}}}}$
(114)
$\displaystyle=\int\limits_{\cos({\theta^{\max}})}^{\cos({\theta^{\min}})}{\int\limits_{\sin({\varphi^{\min}}(\theta))}^{\sin({\varphi^{\max}}(\theta))}{\frac{{{D_{h}}{D_{v}}}}{{{\lambda^{2}}}}\sin(\theta)d\left({\sin(\varphi)}\right)d\left({\cos(\theta)}\right)}}$
$\displaystyle=\frac{{{D_{h}}{D_{v}}}}{{{\lambda^{2}}}}\int\limits_{{\theta^{\min}}}^{{\theta^{\max}}}{\int\limits_{{\varphi^{\min}}(\theta)}^{{\varphi^{\max}}(\theta)}{{{\left({\sin(\theta)}\right)}^{2}}\cos(\varphi)d\varphi
d\theta}}$
$\displaystyle=\frac{{{D_{h}}{D_{v}}}}{{{\lambda^{2}}}}\int\limits_{{\theta^{\min}}}^{{\theta^{\max}}}{{{\sin}^{2}}(\theta)\left({\sin\left({{\varphi^{\max}}(\theta)}\right)-\sin\left({{\varphi^{\min}}(\theta)}\right)}\right)d\theta}.$

Thus, Theorem 1 is proved.

###  -B Proof of Theorem 2:

Based on the additive dimensionality of the signal subspaces proved in Lemma
1, we only derive the rank of the joint spatial-frequency channel covariance
matrix ${\bf R}_{q}^{\text{(J)}}$ with a certain sub-support $\eta_{q}$. In a
way similar to prove Theorem 1, we may compute the rank by a triple integral
over the three-dimensional range of the sub-support.

$\displaystyle\mathop{\lim}\limits_{{N_{h}},{N_{v}},N_{f}\to\infty}\frac{{{\rm{rank}}\\{{{\bf{R}}_{q}^{({\rm{J}})}}\\}}}{{{N_{h}}{N_{v}}{N_{f}}}}=$
(115) $\displaystyle\frac{{{D_{h}}{D_{v}}\Delta
f}}{{{\lambda^{2}}}}\int\limits_{\cos({\theta^{\max}})}^{\cos({\theta^{\min}})}\int\limits_{\sin({\varphi^{\min}}(\theta))}^{\sin({\varphi^{\max}}(\theta))}\int\limits_{{\tau^{\min}}(\theta,\varphi)}^{{\tau^{\max}}(\theta,\varphi)}\sin(\theta)$
$\displaystyle\qquad
d\left({\sin(\varphi)}\right)d\left({\cos(\theta)}\right)d\tau$
$\displaystyle=\frac{{{D_{h}}{D_{v}}\Delta
f}}{{{\lambda^{2}}}}\int\limits_{\theta_{q}^{\min}}^{\theta_{q}^{\max}}{\int\limits_{\varphi_{q}^{\min}(\theta)}^{\varphi_{q}^{\max}(\theta)}{\int\limits_{\tau_{q}^{\min}(\theta,\varphi)}^{\tau_{q}^{\max}(\theta,\varphi)}{{{\sin}^{2}}(\theta)\cos(\varphi)}}}d\varphi
d\theta d\tau$ $\displaystyle=\frac{{{D_{h}}{D_{v}}\Delta
f}}{{{\lambda^{2}}}}\int\limits_{\theta_{q}^{\min}}^{\theta_{q}^{\max}}\int\limits_{\varphi_{q}^{\min}(\theta)}^{\varphi_{q}^{\max}(\theta)}{{\sin}^{2}}(\theta)\cos(\varphi)$
$\displaystyle\qquad\left({\tau_{q}^{\max}(\theta,\varphi)-\tau_{q}^{\min}(\theta,\varphi)}\right)d\varphi
d\theta$

Thus, Theorem 2 is proved.

###  -C Proof of Theorem 3:

The operation of the joint spatial-frequency beamforming with $N_{a}$ dominant
eigen-vectors of ${\bf R}^{\text{(J)}}$ is equivalent to the following matrix
multiplication

$\displaystyle{\bf c}_{s}=({\bf U}_{s}^{\text{(J)}})^{H}{\underline{\bf{h}}},$
(116)

where the columns of ${\bf U}_{s}^{\text{(J)}}$ are the dominant eigenvectors
corresponding to the $N_{a}$ largest eigenvalues of ${\bf R}^{\text{(J)}}$:

$\displaystyle{\bf
U}_{s}^{\text{(J)}}=\left[{\begin{array}[]{*{20}{c}}{{\bf{u}}_{1}^{(J)}}&{{\bf{u}}_{2}^{(J)}}&\cdots&{{\bf{u}}_{{N_{a}}}^{(J)}}\end{array}}\right],$
(118)

where ${\bf c}_{s}\in\mathbb{C}^{N_{a}\times 1}$ are the coefficients that UE
feeds back to the base station. Thus the reconstructed CSI at the base station
side is

$\displaystyle{\underline{\widehat{\bf{h}}}}={\bf{U}}_{s}^{{\rm{(J)}}}{({\bf{U}}_{s}^{{\rm{(J)}}})^{H}}{\underline{\bf{h}}}.$
(119)

The expression above is equivalent to an orthogonal project the vector
${\underline{\bf{h}}}$ to the column space of ${\bf U}_{s}^{\text{(J)}}$. As
proved in Theorem 2, the rank of ${\bf R}^{\text{(J)}}$, denoted by
$r^{\text{(}J)}$, is no larger than $\rho^{{\rm{(J)}}}N_{v}N_{h}N_{f}$. We
have that ${\underline{\bf{h}}}$ lives in the signal subspace of ${\bf
R}^{\text{(J)}}$:

$\displaystyle{\underline{\bf{h}}}\in\text{span}\left\\{{{\bf
u}_{n}^{(J)}:n=1,\cdots,{r^{\text{(J)}}}}\right\\}.$ (120)

As a result, when the number of feedback coefficients
$N_{a}\geq{r^{\text{(J)}}}$, we have

$\displaystyle\mathop{\lim}\limits_{{N_{h}},{N_{v}},{N_{f}}\to\infty}\frac{{\left\|{{{\underline{{\bf{\hat{h}}}}}}-{\underline{\bf{h}}}}\right\|_{F}^{2}}}{{\left\|{{{\underline{\bf{h}}}}}\right\|_{F}^{2}}}=0,$
(121)

which proves Theorem 3.

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|  Haifan Yin received the Ph.D. degree from Télécom ParisTech in 2015. He
received the B.Sc. degree in Electrical and Electronic Engineering and the
M.Sc. degree in Electronics and Information Engineering from Huazhong
University of Science and Technology, Wuhan, China, in 2009 and 2012
respectively. From 2009 to 2011, he has been with Wuhan National Laboratory
for Optoelectronics, China, working on the implementation of TD-LTE systems as
an R&D engineer. From 2016 to 2017, he has been a DSP engineer in Sequans
Communications - an IoT chipmaker based in Paris, France. From 2017 to 2019,
he has been a senior research engineer working on 5G standardization in
Shanghai Huawei Technologies Co., Ltd., where he made substantial
contributions to 5G standards, particularly the 5G codebooks. Since May 2019,
he has joined the School of Electronic Information and Communications at
Huazhong University of Science and Technology as a full professor. His current
research interests include 5G and 6G networks, signal processing, machine
learning, and massive MIMO systems. H. Yin was the national champion of 2021
High Potential Innovation Prize awarded by Chinese Academy of Engineering, a
winner of 2020 Academic Advances of HUST, and a recipient of the 2015 Chinese
Government Award for Outstanding Self-financed Students Abroad.
---|---
|  David Gesbert (IEEE Fellow) is Professor and Director of EURECOM. He
obtained the Ph.D. degree from Ecole Nationale Superieure des
Telecommunications, France, in 1997. From 1997 to 1999 he has been with the
Information Systems Laboratory, Stanford University. He was then a founding
engineer of Iospan Wireless Inc, a Stanford spin off pioneering MIMO-OFDM (now
Intel). Before joining EURECOM in 2004, he has been with the Department of
Informatics, University of Oslo as an adjunct professor. D. Gesbert has
published about 300 papers and 25 patents, some of them winning 2019 ICC Best
Paper Award, 2015 IEEE Best Tutorial Paper Award (Communications Society),
2012 SPS Signal Processing Magazine Best Paper Award, 2004 IEEE Best Tutorial
Paper Award (Communications Society), 2005 Young Author Best Paper Award for
Signal Proc. Society journals, and paper awards at conferences 2011 IEEE
SPAWC, 2004 ACM MSWiM. He has been a Technical Program Co-chair for ICC2017.
He was named a Thomson-Reuters Highly Cited Researchers in Computer Science.
Since 2015, he holds the ERC Advanced grant ”PERFUME” on the topic of smart
device Communications in future wireless networks. He is a Board member for
the OpenAirInterface (OAI) Software Alliance. Since early 2019, he heads the
Huawei-funded Chair on Advanced Wireless Systems Towards 6G Networks. He sits
on the Advisory Board of HUAWEI European Research Institute.
---|---