# Channel-Adaptive Pilot Design for FDD-MIMO Systems Utilizing Gaussian
Mixture Models

Nurettin Turan, Benedikt Fesl, Benedikt Böck, Michael Joham, and Wolfgang
Utschick
The authors acknowledge the financial support by the Federal Ministry of
Education and Research of Germany in the program of “Souverän. Digital.
Vernetzt.”. Joint project 6G-life, project identification number: 16KISK002.
©This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be
accessible. TUM School of Computation, Information and Technology, Technical
University of Munich, Germany
Email: {nurettin.turan, benedikt.fesl, benedikt.boeck, joham<EMAIL_ADDRESS>
###### Abstract

In this work, we propose to utilize Gaussian mixture models (GMMs) to design
pilots for downlink (DL) channel estimation in frequency division duplex (FDD)
systems. The GMM captures prior information during training that is leveraged
to design a codebook of pilot matrices in an initial offline phase. Once
shared with the mobile terminal (MT), the GMM is utilized to determine a
feedback index at the MT in the online phase. This index selects a pilot
matrix from a codebook, eliminating the need for online pilot optimization.
The GMM is further used for DL channel estimation at the MT via observation-
dependent linear minimum mean square error (LMMSE) filters, parametrized by
the GMM. The analytic representation of the GMM allows adaptation to any
signal-to-noise ratio (SNR) level and pilot configuration without re-training.
With extensive simulations, we demonstrate the superior performance of the
proposed GMM-based pilot scheme compared to state-of-the-art approaches.

###### Index Terms:

Gaussian mixture models, machine learning, pilot design, FDD-MIMO systems.

##  I Introduction

In multiple-input multiple-output (MIMO) communication systems, obtaining
channel state information (CSI) at the base station (BS) needs to occur in
regular time intervals. In FDD systems, both the BS and the MT transmit at the
same time but on different frequencies, which breaks the reciprocity between
the instantaneous uplink (UL) CSI and DL CSI. Consequently, acquiring accurate
DL CSI in FDD systems is challenging [1] and thus relies on feedback of the
estimated channel from the MT. Therefore, the quality of DL channel estimation
is of crucial importance.

In massive MIMO systems, where the BS is typically equipped with a high number
of antennas, as many pilots as transmit antennas are required to be sent from
the BS to the MT to fully illuminate the channel, i.e., avoiding a systematic
error when relying on least squares (LS) DL CSI estimation at the MT. However,
the associated pilot overhead for complete channel illumination can be
prohibitive [2]. In scenarios with spatial correlation at the BS and the MT,
the DL training overhead can be significantly reduced by leveraging
statistical knowledge of the channel and the noise [3, 4, 5, 6, 7, 8], e.g.,
by using Bayesian estimation approaches.

Hence, for DL channel estimation, given a budget of $n_{\mathrm{p}}$ pilots, a
common approach to use for channel illumination is transmitting
$n_{\mathrm{p}}$ pilots equivalent to the $n_{\mathrm{p}}$ dominant
eigenvectors of the transmit-side spatial correlation matrix. However, the
aforementioned works rely on either perfect or estimated statistical knowledge
at the BS and/or at the MT side, which may be difficult to acquire.

_Contributions:_ In this work, we propose to utilize GMMs for pilot design.
The proposed scheme neither requires _a priori_ knowledge of the channel’s
statistics at the BS nor the MT. The statistical prior information captured
with the GMM in the offline phase is exploited to determine a feedback index
at the MT in the online phase utilizing the GMM, which is shared between the
BS and the MT. This feedback index is sufficient to establish common knowledge
of the pilot matrix, which is selected from a codebook of pilot matrices.
Thus, no online pilot optimization is required. The inference of the feedback
information involves computing the responsibilities via the GMM and conducting
a maximum a posteriori (MAP) estimation, which selects the index of the GMM
component with the largest responsibility as the feedback index. The
responsibilities determine how well each component of the GMM describes the
underlying channel, measured in terms of posterior probabilities. The
responsibilities are additionally used to obtain an estimated channel at the
MT by performing a convex combination of observation-dependent LMMSE filters,
which are parametrized by the GMM and applied to the observation. Moreover,
the analytic representation of the GMM generally allows the adaption to any
SNR level and pilot configuration without re-training. Based on extensive
simulations, we highlight the superior performance of the proposed GMM-based
pilot scheme compared to state-of-the-art approaches.

##  II System and Channel Model

We consider a BS equipped with $N_{\mathrm{tx}}$ antennas and a MT equipped
with $N_{\mathrm{rx}}$ antennas. We assume a block-fading model, cf., e.g.,
[9], where the DL signal for block $t$ received at the MT is expressed as

$\bm{Y}_{t}=\bm{H}_{t}\bm{P}^{\operatorname{T}}_{t}+\bm{N}_{t}\in\mathbb{C}^{N_{\mathrm{rx}}\times
n_{\mathrm{p}}}$ (1)

where $t=0,\dots,T$, with the MIMO channel
$\bm{H}_{t}\in\mathbb{C}^{N_{\mathrm{rx}}\times N_{\mathrm{tx}}}$, the pilot
matrix $\bm{P}_{t}\in\mathbb{C}^{n_{\mathrm{p}}\times N_{\mathrm{tx}}}$, and
the additive white Gaussian noise (AWGN)
$\bm{N}_{t}=[\bm{n}^{\prime}_{t,1},\dots,\bm{n}^{\prime}_{t,n_{\mathrm{p}}}]\in\mathbb{C}^{N_{\mathrm{rx}}\times
n_{\mathrm{p}}}$ with
$\bm{n}^{\prime}_{t,p}\sim\mathcal{N}_{\mathbb{C}}(\bm{0},\sigma_{n}^{2}\mathbf{I}_{N_{\mathrm{rx}}})$
for $p\in\\{1,2,\dots,n_{\mathrm{p}}\\}$ and $n_{\mathrm{p}}$ is the number of
pilots. We consider systems with reduced pilot overhead, i.e.,
$n_{\mathrm{p}}<N_{\mathrm{tx}}$. For the subsequent analysis, it is
advantageous to vectorize (1):

$\bm{y}_{t}=(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})\bm{h}_{t}+\bm{n}_{t}$
(2)

where $\bm{h}_{t}=\operatorname{vec}(\bm{H}_{t})$,
$\bm{y}_{t}=\operatorname{vec}(\bm{Y}_{t})$,
$\bm{n}_{t}=\operatorname{vec}(\bm{N}_{t})$, and
$\bm{n}_{t}\sim\mathcal{N}_{\mathbb{C}}(\mathbf{0},{\bm{\Sigma}})$ with
${\bm{\Sigma}}=\sigma_{n}^{2}\mathbf{I}_{N_{\mathrm{rx}}n_{\mathrm{p}}}$.

We adopt the 3rd Generation Partnership Project (3GPP) spatial channel model
(see [10, 9]) where channels are modeled conditionally Gaussian, i.e.,
$\bm{h}_{t}|{\bm{\delta}}\sim\mathcal{N}_{\mathbb{C}}({\bm{0}},\bm{C}_{{\bm{\delta}}})$.
The covariance matrix $\bm{C}_{{\bm{\delta}}}$ is assumed to remain constant
over $T+1$ blocks. The random vector ${\bm{\delta}}$ comprises the main angles
of arrival/departure of the multi-path propagation cluster between the BS and
the MT. The main angles of arrival/departure are drawn independently and are
uniformly distributed over $[\frac{-\pi}{2},\frac{\pi}{2}]$. The BS as well as
the MT employ a uniform linear array (ULA) such that the transmit- and
receive-side spatial channel covariance matrices are given by

$\bm{C}_{{\bm{\delta}}}^{\mathrm{\\{rx,tx\\}}}=\int_{-\pi}^{\pi}g^{\mathrm{\\{rx,tx\\}}}(\theta;{\bm{\delta}})\bm{a}^{\mathrm{\\{rx,tx\\}}}(\theta)\bm{a}^{\mathrm{\\{rx,tx\\}}}(\theta)^{\operatorname{H}}d\theta,$
(3)

where
$\bm{a}^{\mathrm{\\{rx,tx\\}}}(\theta)=[1,e^{j\pi\sin(\theta)},\dots,e^{j\pi(N_{\mathrm{\\{rx,tx\\}}}-1)\sin(\theta)}]^{\operatorname{T}}$
is the array steering vector for an angle of arrival/departure $\theta$ and
$g^{\mathrm{\\{rx,tx\\}}}$ is a Laplacian power density whose standard
deviation describes the angular spread
$\sigma_{\text{AS}}^{\mathrm{\\{rx,tx\\}}}$ of the propagation cluster at the
BS
($\sigma_{\text{AS}}^{\mathrm{tx}}=$2\text{\,}\mathrm{\SIUnitSymbolDegree}$$)
and MT
($\sigma_{\text{AS}}^{\mathrm{rx}}=$35\text{\,}\mathrm{\SIUnitSymbolDegree}$$)
side [10]. The overall channel covariance matrix is constructed as
$\bm{C}_{{\bm{\delta}}}=\bm{C}_{{\bm{\delta}}}^{\mathrm{tx}}\otimes\bm{C}_{{\bm{\delta}}}^{\mathrm{rx}}$
due to the assumption of independent scattering in the vicinity of transmitter
and receiver, see, e.g., [11]. In the case of a MT equipped with a single
antenna, $\bm{C}_{{\bm{\delta}}}$ degenerates to the transmit-side covariance
matrix $\bm{C}_{{\bm{\delta}}}^{\mathrm{tx}}$.

With

$\mathcal{H}=\\{\bm{h}^{\ell}\\}_{\ell=1}^{L},$ (4)

we denote the training data set consisting of $L$ channel samples. For every
channel sample $\bm{h}^{\ell}$, we generate random angles, collected in
${\bm{\delta}}^{\ell}$, and then draw the sample as
$\bm{h}^{\ell}\sim\mathcal{N}_{\mathbb{C}}({\bm{0}},\bm{C}_{{\bm{\delta}}^{\ell}})$.
These channels represent a communications environment with unknown probability
density function (PDF) $f_{\bm{h}}$.

##  III Pilot Optimization with Perfect Statistical Knowledge

Given the knowledge of ${\bm{\delta}}$, the observation $\bm{y}_{t}$ is
jointly Gaussian with the channel $\bm{h}_{t}$ [see (2)], and we can compute a
genie LMMSE channel estimate with [9]

$\displaystyle\hat{\bm{h}}_{t,\text{gLMMSE}}=\operatorname{E}[\bm{h}_{t}\mid\bm{y}_{t},{\bm{\delta}}]$
(5)
$\displaystyle=\bm{C}_{{\bm{\delta}}}(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})^{\operatorname{H}}((\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})\bm{C}_{{\bm{\delta}}}(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})^{\operatorname{H}}+{\bm{\Sigma}})^{-1}\bm{y}_{t}.$
(6)

The goal of pilot optimization is to design the pilot matrix $\bm{P}_{t}$ such
that the mean squared error (MSE) between $\hat{\bm{h}}_{t,\text{gLMMSE}}$ and
the actual channel $\bm{h}_{t}$ is minimized [7, 6, 3]:

$\bm{P}_{t}^{\star}=\operatorname*{arg\,min}_{\bm{P}_{t}}\
\operatorname{E}[\|\hat{\bm{h}}_{t,\text{gLMMSE}}-\bm{h}_{t}\|_{2}^{2}]$ (7)

where the pilot matrix $\bm{P}_{t}$ typically satisfies either a total power
constraint as in [7, 6] or an equal power per pilot vector constraint as in
[3]. In this work, we will consider the latter case. For a given
${\bm{\delta}}$, the optimal pilot matrix $\bm{P}_{t}^{\star}$ for every block
is the same, i.e., $\bm{P}^{\star}_{t}=\bm{P}_{\text{genie}}$ for all
$t=0,\dots,T$. In particular, $\bm{P}_{\text{genie}}$ is a sub-unitary matrix
[3]

$\bm{P}_{\text{genie}}=\sqrt{\rho}\bm{U}_{\bm{\delta}}^{\operatorname{H}}[:n_{\mathrm{p}},:],$
(8)

which is composed of the $n_{\mathrm{p}}$ dominant eigenvectors of the
transmit-side covariance matrix
$\bm{C}_{{\bm{\delta}}}^{\mathrm{tx}}=\bm{U}_{\bm{\delta}}\bm{\Lambda}_{\bm{\delta}}\bm{U}_{\bm{\delta}}^{\operatorname{H}}$
corresponding to the $n_{\mathrm{p}}$ largest eigenvalues, where $\rho$
denotes the transmit power per pilot vector. Note that power loading across
pilot vectors generally achieves better performance but requires additional
processing. Additionally, with a sub-unitary pilot design, our proposed scheme
yields a codebook consisting of pilot matrices that do not depend on the SNR,
resolving the burden of saving SNR level-specific pilot codebooks, see
Subsection IV-D.

##  IV GMM-based Pilot Design and Downlink Channel Estimation

Any channel $\bm{h}_{t}$ of a MT located anywhere within the BS’s coverage
area can be interpreted as a realization of a random variable with PDF
$f_{\bm{h}}$ for which however, no analytical expression is available.
Therefore, we utilize a GMM to approximate the PDF $f_{\bm{h}}$, similar to
[12, 13]. This learned model is then shared between the BS and the MT to
establish common awareness of the channel characteristics. The GMM is then
utilized to infer feedback information for pilot matrix design and for DL
channel estimation at the MT in the online phase. Thereby, the feedback
information of the MT of a preceding fading block $t-1$ is leveraged at the BS
to select the pilot matrix for the subsequent fading block $t>0$.

###  IV-A Modeling the Channel Characteristics at the BS – Offline

The channel characteristics are captured offline using a GMM comprised of
$K=2^{B}$ components,

$f^{(K)}_{\bm{h}}(\bm{h}_{t})=\sum\nolimits_{k=1}^{K}\pi_{k}\mathcal{N}_{\mathbb{C}}(\bm{h}_{t};{\bm{\mu}}_{k},\bm{C}_{k})$
(9)

where each component of the GMM is defined by the mixing coefficient
$\pi_{k}$, the mean ${\bm{\mu}}_{k}$, and the covariance matrix $\bm{C}_{k}$.

Motivated by the observation that the channel exhibits an unconditioned zero
mean and similar to [14], we enforce the means of the GMM-components to zero,
i.e., ${\bm{\mu}}_{k}=\bm{0}$ for all $k\in\\{1,\dots,K\\}$. This also reduces
the number of learnable parameters and, thus, prevents overfitting. Note that
the parameters of the GMM, i.e., $\\{\pi_{k},\bm{C}_{k}\\}_{k=1}^{K}$, remain
constant across all blocks. To obtain maximum likelihood estimates of the GMM
parameters, we utilize the training dataset $\mathcal{H}$ [see (4)] and employ
an expectation maximization (EM) algorithm, as described in [15, Subsec.
9.2.2], where we enforce the means to zero in every M-step of the EM
algorithm.

For MIMO channels, we further impose a Kronecker factorization on the
covariances of the GMM, i.e.,
$\bm{C}_{k}=\bm{C}^{\mathrm{tx}}_{k}\otimes\bm{C}^{\mathrm{rx}}_{k}$. Thus,
instead of fitting an unconstrained GMM with $N\times N$-dimensional
covariances (where $N=N_{\mathrm{tx}}N_{\mathrm{rx}}$), we fit separate GMMs
for the transmit and receive sides. These transmit-side and receive-side GMMs
possess $N_{\mathrm{tx}}\times N_{\mathrm{tx}}$ and $N_{\mathrm{rx}}\times
N_{\mathrm{rx}}$-dimensional covariances, respectively, with $K_{\mathrm{tx}}$
and $K_{\mathrm{rx}}$ components. Then, by computing the Kronecker products of
the corresponding transmit-side and receive-side covariance matrices, we
obtain a GMM with $K=K_{\mathrm{tx}}K_{\mathrm{rx}}$ components and $N\times
N$-dimensional covariances. Imposing this constraint on the GMM covariances
not only significantly decreases the duration of offline training, facilitates
parallelization of the fitting process, and demands fewer training samples due
to the reduced number of parameters to be learned, cf. [13, 12], but also
ensures access to a transmit-side covariance during pilot design in the online
phase, as discussed in Subsection IV-D.

Using a GMM, we can calculate the posterior probability that the channel
$\bm{h}_{t}$ originates from component $k$ as [15, Sec. 9.2],

$p(k\mid\bm{h}_{t})=\frac{\pi_{k}\mathcal{N}_{\mathbb{C}}(\bm{h}_{t};\bm{0},\bm{C}_{k})}{\sum_{i=1}^{K}\pi_{i}\mathcal{N}_{\mathbb{C}}(\bm{h}_{t};\bm{0},\bm{C}_{i})}.$
(10)

These posterior probabilities are commonly referred to as responsibilities.

###  IV-B Sharing the Model with the MTs – Offline

For a MT to infer the feedback information, it must have access to the
parameters of the GMM. Conceptually, this involves sharing the model
parameters, i.e., $\\{\pi_{k},\bm{C}_{k}\\}_{k=1}^{K}$, with the MTs upon
entering the coverage area of the BS. This transfer is required only once
since the GMM remains unchanged for a specific BS environment.

Incorporating model-based insights to restrict the GMM covariances, as
discussed in Subsection IV-A, additionally significantly reduces the model
transfer overhead. Due to specific antenna array geometries, the GMM
covariances can be further constrained to a Toeplitz or block-Toeplitz matrix
with Toeplitz blocks, in case of a ULA or uniform rectangular array (URA),
respectively, cf. [16], with even fewer parameters. In [16], it is also
further discussed how GMMs with variable bit lengths can be obtained. These
further structural constraints, as well as the analysis with variable bit
lengths, are out of the scope of this work.

###  IV-C Inferring the Feedback Information and Estimating the Channel at the
MTs – Online

In the online phase, the MT infers feedback information given the observation
$\bm{y}_{t}$ utilizing the GMM. The joint Gaussian nature of each GMM
component [see (9)] combined with the AWGN, allows for simple computation of
the GMM of the observations with the GMM from (9) as

$\small
f_{\bm{y}}^{(K)}(\bm{y}_{t})=\sum\nolimits_{k=1}^{K}\pi_{k}\mathcal{N}_{\mathbb{C}}(\bm{y}_{t};\bm{0},(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})\bm{C}_{k}(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})^{\operatorname{H}}+{\bm{\Sigma}}).$
(11)

Thus, the MT can compute the responsibilities based on the observations
$\bm{y}_{t}$ as

$\small
p(k\mid\bm{y}_{t})=\frac{\pi_{k}\mathcal{N}_{\mathbb{C}}(\bm{y}_{t};\bm{0},(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})\bm{C}_{k}(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})^{\operatorname{H}}+{\bm{\Sigma}})}{\sum_{i=1}^{K}\pi_{i}\mathcal{N}_{\mathbb{C}}(\bm{y}_{t};\bm{0},(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})\bm{C}_{i}(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})^{\operatorname{H}}+{\bm{\Sigma}})}.$
(12)

The feedback information $k_{t}^{\star}$ is then determined through a MAP
estimation as

$k^{\star}_{t}=\operatorname*{arg\,max}_{k}~{}{p(k\mid\bm{y}_{t})}$ (13)

where the index of the component with the highest responsibility for the
observation $\bm{y}_{t}$ serves as the corresponding feedback information. The
responsibilities measure how well each component of the GMM explains the
underlying channel $\bm{h}_{t}$ of the observed pilot signal $\bm{y}_{t}$.
Hence, the feedback information is simply the index of the GMM component that
best explains the underlying channel. Subsequently, the responsibilities are
utilized to obtain a channel estimate via the GMM by calculating a convex
combination of per-component LMMSE estimates, as discussed in [13, 12]. In
particular, the MT estimates the channel by computing

$\hat{\bm{h}}_{t,\text{GMM}}(\bm{y})=\sum_{k=1}^{K}p(k\mid\bm{y}_{t})\hat{\bm{h}}_{t,\text{LMMSE},k}(\bm{y}_{t}),$
(14)

using the responsibilities $p(k\mid\bm{y}_{t})$ from (12) and

$\displaystyle\hat{\bm{h}}_{t,\text{LMMSE},k}(\bm{y}_{t})$
$\displaystyle=\bm{C}_{k}(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})^{\operatorname{H}}((\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})\bm{C}_{k}(\bm{P}_{t}\otimes\mathbf{I}_{N_{\mathrm{rx}}})^{\operatorname{H}}+{\bm{\Sigma}})^{-1}\bm{y}_{t}.$
(15)

###  IV-D Designing the Pilots at the BS– Online

Consider the eigenvalue decomposition of each of the GMM’s transmit-side
covariances, i.e.,
$\bm{C}_{k}^{\mathrm{tx}}=\bm{U}_{k}\bm{\Lambda}_{k}\bm{U}^{\operatorname{H}}_{k}$.
For $t>0$, given the feedback information $k_{t-1}^{\star}$ of the MT from the
preceding block $t-1$, we propose employing the pilot matrix $\bm{P}_{t}$ at
the BS for the subsequent block $t$ as [cf. (8)]

$\bm{P}_{t}=\sqrt{\rho}\bm{U}^{\operatorname{H}}_{k^{\star}_{t-1}}[:n_{\mathrm{p}},:],$
(16)

i.e., the $n_{\mathrm{p}}$ dominant eigenvectors of the $k^{\star}_{t-1}$-th
transmit-side covariance matrix $\bm{C}_{k^{\star}_{t-1}}^{\mathrm{tx}}$ are
selected as the pilot matrix. Since the GMM-covariances remain fixed, we can
store a set of pilot matrices
$\mathcal{P}=\\{\bm{U}^{\operatorname{H}}_{k}[:n_{\mathrm{p}},:]\\}_{k=1}^{K}$,
and the online pilot design utilizing the proposed GMM-based scheme simplifies
to a simple selection task based on the feedback information $k^{\star}_{t-1}$
from the previous block. For the initial block $t=0$, we employ a discrete
Fourier transform (DFT)-based pilot matrix.

###  IV-E Complexity Analysis

The online computational complexity of the proposed GMM-based scheme can be
divided into three parts: the inference of the feedback information, the
channel estimation, and the pilot design.

Matrix-vector multiplications dominate the computational complexity for
inferring the feedback information. This is because the computation of the
responsibilities in (12) entails evaluating Gaussian densities, and the
calculations involving determinants and inverses can be pre-computed for a
specific SNR level due to the fixed GMM parameters. Thus, the inference of the
feedback information using (13) in the online phase at the MT has a complexity
of $\mathcal{O}(KN_{\mathrm{rx}}^{2}n_{\mathrm{p}}^{2})$.

The additional processing to calculate an estimated channel with the GMM via
(14) further involves the application of the per-component LMMSE filters [see
(IV-C)] which exhibits a computational complexity of
$\mathcal{O}(KN_{\mathrm{rx}}^{2}N_{\mathrm{tx}}n_{\mathrm{p}})$, since also
the LMMSE filters for a given SNR level can be pre-computed. For the feedback
inference and application of the LMMSE filters, parallelization concerning the
number of components $K$ is possible.

The computational complexity of the online pilot design is $\mathcal{O}(K)$,
as it only involves traversing the pre-computed set of pilot matrices
$\mathcal{P}$. Thus, in the online phase, our scheme avoids computing an
eigenvalue decomposition, which is required for solving the optimization
problem from (8).

##  V Baseline Channel Estimators and Pilot Schemes

Figure 1: The NMSE over the SNR for a MIMO system ($N_{\mathrm{tx}}=16$,
$N_{\mathrm{rx}}=4$) with $B=7$ feedback bits and $n_{\mathrm{p}}=4$ pilots.

In addition to the utopian genie LMMSE approach (5), where we assume perfect
knowledge of ${\bm{\delta}}$ (at the BS to design the optimal pilots and at
the MT to apply the genie LMMSE estimator), we consider following channel
estimators and pilot matrices.

Firstly, we consider the LMMSE estimator $\hat{\bm{h}}_{\text{LMMSE}}$, where
the sample covariance matrix is formed using the set $\mathcal{H}$ [see (4)],
as discussed in, e.g., [13, 12].

As another baseline, we consider a compressive sensing estimation method
$\hat{\bm{h}}_{\text{OMP}}$ employing the orthogonal matching pursuit (OMP)
algorithm, cf. [13, 17].

Additionally, we compare to an end-to-end deep neural network (DNN) approach
for DL channel estimation with a jointly learned pilot matrix
$\bm{P}_{\text{DNN}}$, similar to [18]. To determine the hyperparameters of
the DNN, we utilize random search [19], with the MSE serving as the loss
function. The DNN architecture comprises $D_{\text{CM}}$ convolutional
modules, each consisting of a convolutional layer, batch normalization, and an
activation function, where $D_{\text{CM}}$ is randomly selected within
$[3,9]$. Each convolutional layer contains $D_{\text{K}}$ kernels, where
$D_{\text{K}}$ is randomly selected within $[32,64]$. The activation function
in each convolutional module is the same and is randomly selected from
$\\{\text{ReLu, sigmoid, PReLU, Leaky ReLU, tanh, swish}\\}$. Following a
subsequent two-dimensional max-pooling, the features are flattened, and a
fully connected layer is employed with an output dimension of
$2N_{\mathrm{tx}}N_{\mathrm{rx}}$ (concatenated real and imaginary parts of
the estimated channel). We train a distinct DNN for each pilot configuration
and SNR level, running $50$ random searches per pilot configuration and SNR
level and selecting the best-performing DNN for each setup.

Lastly, as further baseline pilot matrices, we utilize a DFT sub-matrix
$\bm{P}_{\text{DFT}}$ as the pilot matrix, see, e.g., [20], and alternatively,
we consider random pilot matrices denoted by $\bm{P}_{\text{RND}}$, see, e.g.,
[7].

##  VI Simulation Results

Figure 2: The NMSE over the block index $t$ for a MIMO system
($N_{\mathrm{tx}}=16$, $N_{\mathrm{rx}}=4$) with $B=7$ feedback bits,
$n_{\mathrm{p}}=4$ pilots, and
$\text{SNR}\in\\{$0\text{\,}\mathrm{d}\mathrm{B}$,$10\text{\,}\mathrm{d}\mathrm{B}$,$20\text{\,}\mathrm{d}\mathrm{B}$\\}$.

We use the set $\mathcal{H}$ [see (4)] with $L=10^{5}$ samples for fitting the
GMM and all other data-based baselines. We use a different dataset of
$J=10^{4}$ channel samples per block $t$ for evaluation purposes, where we set
$T=10$. The data samples are normalized to satisfy
$\operatorname{E}[\|\bm{h}\|^{2}]=N=N_{\mathrm{tx}}N_{\mathrm{rx}}$.
Additionally, we fix $\rho=1$, enabling the definition of the SNR as
$\frac{1}{\sigma_{n}^{2}}$. We employ the normalized mean squared error (NMSE)
as the performance measure. Specifically, for every block $t$, we compute a
corresponding channel estimate $\hat{\bm{h}}^{j}$ for each test channel in the
set $\\{\bm{h}^{j}\\}_{j=1}^{J}$, and calculate
$\text{NMSE}=\frac{1}{NJ}\sum_{j=1}^{J}\|\bm{h}^{j}-\hat{\bm{h}}^{j}\|^{2}.$
If not mentioned otherwise, we consider the block with index $t=5$ in the
subsequent simulations.

In Fig. 1, we simulate a system with $N_{\mathrm{tx}}=16$ BS antennas,
$N_{\mathrm{rx}}=4$ MT antennas, and $n_{\mathrm{p}}=4$ pilots. Since we have
a MIMO setup, we fit a Kronecker structured GMM with in total $K=2^{7}=128$
components ($B=7$ feedback bits), where $K_{\mathrm{tx}}=32$ and
$K_{\mathrm{rx}}=4$. The proposed GMM-based pilot design scheme denoted by
“GMM, $\bm{P}_{\text{GMM}}$” outperforms all baselines “{GMM, OMP, LMMSE},
{$\bm{P}_{\text{DFT}}$, $\bm{P}_{\text{RND}}$}” by a large margin, where the
GMM estimator, the OMP-based estimator, or the LMMSE estimator, are used in
combination with either DFT-based pilot matrices or random pilot matrices. The
proposed scheme also outperforms the DNN based approach denoted by “DNN,
$\bm{P}_{\text{DNN}}$,” which jointly learns the estimator and a global pilot
matrix for the whole scenario; thus, it cannot provide an MT adaptive pilot
matrix. This highlights the advantage of the proposed model-based technique
over the end-to-end learning technique, which is even trained for each SNR
level and pilot configuration. Furthermore, we can observe that the GMM-based
pilot design scheme performs only slightly worse than the baseline with
perfect statistical information at the BS and MTsee (5) and (8), denoted by
“gLMMSE, $\bm{P}_{\text{genie}}$”, being a utopian estimation approach. We can
observe a larger gap in the low SNR regime, where the feedback information
obtained through the responsibilities given an observation (see (13)) is less
accurate due to high noise.

In Fig. 2, we analyze the performance of the GMM-based pilot design scheme
depending on the number of blocks $t$ for the same setup as before at three
different SNR levels, i.e.,
$\text{SNR}\in\\{$0\text{\,}\mathrm{d}\mathrm{B}$,$10\text{\,}\mathrm{d}\mathrm{B}$,$20\text{\,}\mathrm{d}\mathrm{B}$\\}$.
As discussed in Subsection IV-D, at $t=0$, we utilize DFT-based pilots. After
only one block, we can see a significant gain in performance of the proposed
GMM-based pilot design scheme due to the feedback of the index. The results
further reveal that with an increasing SNR, fewer blocks are required to
achieve a performance close to the utopian baseline “gLMMSE,
$\bm{P}_{\text{genie}}$” which requires perfect statistical knowledge at the
BS and the MT.

Figure 3: The NMSE over the SNR for a MISO system ($N_{\mathrm{tx}}=64$,
$N_{\mathrm{rx}}=1$) with $B=6$ feedback bits and
$n_{\mathrm{p}}\in\\{16,32,48\\}$ pilots.

In the remainder, we consider a multiple-input single-output (MISO) system
with $N_{\mathrm{tx}}=64$ BS antennas and $N_{\mathrm{rx}}=1$ antenna at the
MT. In Fig. 3 we utilize a GMM with $K=2^{6}=64$ components ($B=6$ feedback
bits) and consider setups with $n_{\mathrm{p}}\in\\{16,32,48\\}$ pilots. In
this case, the proposed scheme “GMM, $\bm{P}_{\text{GMM}}$,
$n_{\mathrm{p}}=16$”, performs only slightly worse than the genie-aided
approach “gLMMSE, $\bm{P}_{\text{genie}}$”. Moreover, the GMM-based pilot
design scheme outperforms the baselines “GMM, {$\bm{P}_{\text{DFT}}$,
$\bm{P}_{\text{RND}}$}, $n_{\mathrm{p}}\in\\{16,32,48\\}$”. In particular, the
proposed scheme with only $n_{\mathrm{p}}=16$ pilots outperforms random pilot
matrices with twice as much pilots ($n_{\mathrm{p}}=32$) or in case of DFT-
based pilots even thrice as much pilots ($n_{\mathrm{p}}=48$).

Lastly, in Fig. 4, we analyze the effect of varying the number of GMM-
components $K$, on the scheme’s performance, where we set $n_{\mathrm{p}}=16$
and consider three different SNR levels, i.e.,
$\text{SNR}\in\\{$0\text{\,}\mathrm{d}\mathrm{B}$,$10\text{\,}\mathrm{d}\mathrm{B}$,$20\text{\,}\mathrm{d}\mathrm{B}$\\}$.
We can observe that the estimation error decreases with an increasing number
of components $K$. Moreover, with an increasing SNR, the gap to the genie-
aided approach “gLMMSE, $\bm{P}_{\text{genie}}$” decreases. Above $K=32$
components, a saturation can be observed. Overall, these results suggest that
by varying the number of GMM components $K$, a performance-to-complexity
trade-off can be realized without sacrificing too much in performance for
$K\geq 16$ components.

Figure 4: The NMSE over the number of components $K=2^{B}$ for a MISO system
($N_{\mathrm{tx}}=64$, $N_{\mathrm{rx}}=1$) with $n_{\mathrm{p}}=16$ pilots,
and
$\text{SNR}\in\\{$0\text{\,}\mathrm{d}\mathrm{B}$,$10\text{\,}\mathrm{d}\mathrm{B}$,$20\text{\,}\mathrm{d}\mathrm{B}$\\}$.

##  VII Conclusion

In this work, we proposed to utilize GMMs for pilot design in FDD-MIMO
systems. A significant advantage of the proposed scheme is that it does not
require _a priori_ knowledge of the channel’s statistics at the BS and the MT.
Instead, it relies on a feedback mechanism, establishing common knowledge of
the pilot matrix. The same GMM can be utilized for DL channel estimation and
can be generally adapted to any desired SNR level and pilot configuration
without requiring re-training. Simulation results show that the performance
gains achieved with the proposed scheme allow the deployment of system setups
with reduced pilot overhead while maintaining a similar estimation
performance. In our future work, we will investigate the extension of the
proposed GMM-based pilot scheme to multi-user systems, see, e.g., [7, 8, 21,
22].

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