Bayesian Channel Estimation for Intelligent Reflecting Surface-Aided mmWave Massive MIMO Systems With Semi-Passive Elements

The channel is composed of the UE-IRS link $\bar{\mathbf{f}}_{k}\in\mathbb{C}^{N}$ for $k\in\llbracket K\rrbracket$, IRS-BS link $\bar{\mathbf{G}}\in\mathbb{C}^{M\times N}$, and UE-BS link $\bar{\mathbf{h}}_{k}\in\mathbb{C}^{M}$ for $k\in\llbracket K\rrbracket$. Since the scatterers are limited in the mmWave band as the path loss is severe, the links are typically modeled as [23]

where a uniform linear array (ULA) geometry at the base station and uniform planar array (UPA) geometry at the IRS with half-wavelength spacings are assumed without loss of generality. For each link, $\alpha_{\ell}\sim\mathcal{CN}(0,1/\mathrm{PL})$ is the $\ell$-th path gain where $\mathrm{PL}$ is the distance- and frequency-dependent path loss, $\theta_{\ell}$ is the $\ell$-th azimuth/zenith angle of arrival/departure, $\kappa$ is the Rician K-factor, and $L$ is the number of non-LoS (NLoS) paths. In addition, $\mathbf{a}_{\mathrm{B}}(\cdot)\in\mathbb{C}^{M}$ and $\mathbf{a}_{\mathrm{I}}(\cdot,\cdot)\in\mathbb{C}^{N}$ are the array response vectors of the base station and IRS.

Let $\bm{\Omega}[t]\in\{0,1\}^{N}$ and $\bm{\Omega}^{\mathrm{c}}[t]=\mathbf{1}_{N}-\bm{\Omega}[t]$ denote the index vectors of the active sensors and passive reflecting elements that constitute the IRS at time $t$. In addition, define $\bar{\mathbf{F}}=[\bar{\mathbf{f}}_{1},\dots,\bar{\mathbf{f}}_{K}]$ and $\bar{\mathbf{H}}=[\bar{\mathbf{h}}_{1},\dots,\bar{\mathbf{h}}_{K}]$ for notational simplicity.

Then, the received signal $\mathbf{y}[t]\in\mathbb{C}^{M}$ at the base station is

where $\mathbf{v}[t]\in\mathbb{C}^{N}$ is the passive reflection vector with the reflection amplitude $|v_{n}[t]|\leq 1$ and phase shift $\angle v_{n}[t]\in[0,2\pi)$, $\mathbf{x}[t]\in\mathbb{C}^{K}$ is the transmit signal of the users under the transmit power constraint $\mathbb{E}\{|x_{k}[t]|^{2}\}\leq P_{k}[t]$, and $\mathbf{n}_{\mathrm{B}}[t]\sim\mathcal{CN}(\mathbf{0}_{M},\sigma_{\mathrm{B}}^{2}\mathbf{I}_{M})$ is the additive white Gaussian noise (AWGN) at the base station. Likewise, the quantized received signal $\mathbf{z}[t]\in\mathbb{C}^{N}$ at the active sensors forwarded to the base station is

where $\mathbf{n}_{\mathrm{I}}[t]\sim\mathcal{CN}(\mathbf{0}_{N},\sigma_{\mathrm{I}}^{2}\mathbf{I}_{N})$ is the AWGN at the active sensors. The $B$-bit quantizer $\mathrm{Q}(\cdot)$ is applied to the real and imaginary parts elementwise as

where $z^{\mathrm{lo}}\in\mathbb{C}$ and $z^{\mathrm{up}}\in\mathbb{C}$ are the lower and upper thresholds associated with $z\in\mathbb{C}$. In other words, the real and imaginary parts of $z$, $z^{\mathrm{lo}}$, and $z^{\mathrm{up}}$ correspond to one of the $2^{B}$ quantization intervals.

A coherence block of length $T_{\mathrm{c}}$ in the uplink consists of the channel estimation phase of length $T$ and data transmission phase of length $T_{\mathrm{c}}-T$. To proceed, let $\mathcal{T}_{\mathrm{c}}=\llbracket T\rrbracket$ and $\mathcal{T}_{\mathrm{d}}=\llbracket T_{\mathrm{c}}\rrbracket\setminus\llbracket T\rrbracket$ be the time slots for the channel estimation phase and data transmission phase. Then, the received signals $\mathbf{Y}\in\mathbb{C}^{M\times T}$ and $\mathbf{Z}\in\mathbb{C}^{N\times T}$ at the base station and active sensors over the channel estimation phase of length $T$ are expressed as

using the notations $\mathbf{S}=[\mathbf{s}[1],\dots,\mathbf{s}[T]]$, $\bm{\Omega}=[\bm{\Omega}[1],\dots,\bm{\Omega}[T]]$, $\mathbf{X}=[\mathbf{x}[1],\dots,\mathbf{x}[T]]$, $\mathbf{N}_{\mathrm{B}}=[\mathbf{n}_{\mathrm{B}}[1],\dots,\mathbf{n}_{\mathrm{B}}[T]]$, and $\mathbf{N}_{\mathrm{I}}=[\mathbf{n}_{\mathrm{I}}[1],\dots,\mathbf{n}_{\mathrm{I}}[T]]$.

In addition, the semi-passive elements are configured as

which means that $N_{\mathrm{a}}$ semi-passive elements are activated to the active sensing mode in the channel estimation phase and deactivated to the passive reflecting mode in the data transmission phase. The switching performance of switching networks that connect the IRS elements to RF chains is determined by the switching period $T_{\mathrm{sn}}\geq 1$ defined as the time required for the 0-1 pattern of $\bm{\Omega}[t]$ to change such that

and we define the switching frequency as $f_{\mathrm{sn}}=1/T_{\mathrm{sn}}\leq 1$. Therefore, the 0-1 pattern of $\bm{\Omega}[t]$ can change according to (7) and (8).

We formulate the channel estimation problem using the virtual channel representation in conjunction with the channel model and signal model introduced in the previous subsections. To proceed, define the overcomplete dictionaries

over the predefined grids $\{\hat{\theta}_{m}\}_{m\in\llbracket M_{\mathrm{g}}\rrbracket}$ and $\{(\hat{\theta}_{n}^{\mathrm{A}},\hat{\theta}_{n}^{\mathrm{Z}})\}_{n\in\llbracket N_{\mathrm{g}}\rrbracket}$ where $M_{\mathrm{g}}\geq M$ and $N_{\mathrm{g}}\geq N$ are the grid resolutions. Then, the virtual channel representation admits the transformations [24]

where $\mathbf{F}\in\mathbb{C}^{N_{\mathrm{g}}\times K}$, $\mathbf{G}\in\mathbb{C}^{M_{\mathrm{g}}\times N_{\mathrm{g}}}$, and $\mathbf{H}\in\mathbb{C}^{M_{\mathrm{g}}\times K}$ are the equivalent angular-domain channels. In practice, the angular-domain channels are sparse because there are limited scatterers that constitute the channels in the mmWave band [25, 26].

Now, we can reexpress (5) and (6) using the virtual channel representation as

and our goal is to estimate $\{\mathbf{F},\mathbf{G},\mathbf{H}\}$ from $\{\mathbf{Y},\mathbf{Z}\}$. Since all the links are sparse, (14) and (15) is a combination of low-rank matrix factorization [27] and low-rank matrix completion [28] problems, which are NP-hard in general. To the best of our knowledge, our work is the first attempt to jointly exploit the received signal at the base station and additional information acquired from the active sensors at the IRS to estimate the UE-IRS and IRS-BS links. In contrast, recent works [12, 13] on channel estimation with semi-passive elements cannot be considered as semi-passive element-aided but rather driven because only the received signal at the semi-passive elements is leveraged. As a result, [12, 13] require both uplink and downlink training signals to estimate the UE-IRS and BS-IRS links, whereas we only use uplink training signals to estimate all the links.

To facilitate the analysis, we propose a pseudo-measurement model

where the elements of $\hat{\mathbf{Z}}\in\mathbb{C}^{N\times T}$ corresponding to the nonzero pattern of $\bm{\Omega}$ are equal to the nonzero elements of $\mathbf{Z}$. In addition, the lower and upper pseudo-thresholds $\hat{\mathbf{Z}}^{\mathrm{lo}}\in\mathbb{C}^{N\times T}$ and $\hat{\mathbf{Z}}^{\mathrm{up}}\in\mathbb{C}^{N\times T}$ associated with $\hat{\mathbf{Z}}$ are defined as in (4). The elements of $\mathbf{Z}$ and $\hat{\mathbf{Z}}$ corresponding to the zeros of $\bm{\Omega}$, however, are not the same. Nevertheless, (15) and (16) are statistically equivalent because the elements of $\mathbf{Z}$ and $\hat{\mathbf{Z}}$ corresponding to the nonzero pattern of $\bm{\Omega}$ bear the same information about $\mathbf{F}$, while those corresponding to the zeros of $\bm{\Omega}$ have no meaningful information about $\mathbf{F}$ as evident from (15) and (16). Therefore, the elements of $\hat{\mathbf{Z}}$, $\hat{\mathbf{Z}}^{\mathrm{lo}}$, and $\hat{\mathbf{Z}}^{\mathrm{up}}$ corresponding to the zeros of $\bm{\Omega}$—the elements that cannot be determined from $\mathbf{Z}$—can be assigned arbitrarily from the $2^{B}$ quantization intervals of the $B$-bit quantizer without loss of generality. Since the pseudo-measurement model is more convenient to deal with, we estimate $\{\mathbf{F},\mathbf{G},\mathbf{H}\}$ from $\{\mathbf{Y},\hat{\mathbf{Z}}\}$ in (14) and (16) in the sequel.

To account for the interaction among $\{\mathbf{F},\mathbf{G},\mathbf{H},\mathbf{U},\mathbf{Y},\hat{\mathbf{Z}}\}$, we treat all the variables as random variables that constitute a hierarchical Bayesian model as shown in Fig. 2. In addition, we introduce the random variables $\bm{\Gamma}_{\mathbf{F}}\in\mathbb{C}^{N_{\mathrm{g}}\times K}$, $\bm{\Gamma}_{\mathbf{G}}\in\mathbb{C}^{M_{\mathrm{g}}\times N_{\mathrm{g}}}$, and $\bm{\Gamma}_{\mathbf{H}}\in\mathbb{C}^{M_{\mathrm{g}}\times K}$ to capture the sparse nature of $\{\mathbf{F},\mathbf{G},\mathbf{H}\}$, which is the well-known SBL framework [14, 15]. Before moving on, we introduce the equivalent vector forms of the measurement $\mathcal{Y}=\{\mathbf{y},\hat{\mathbf{z}}\}$, hidden variable $\mathcal{X}=\{\mathbf{f},\mathbf{g},\mathbf{h},\bm{\gamma}_{\mathbf{f}},\bm{\gamma}_{\mathbf{g}},\bm{\gamma}_{\mathbf{h}},\mathbf{u}\}$, and pseudo-threshold $\{\hat{\mathbf{z}}^{\mathrm{lo}},\hat{\mathbf{z}}^{\mathrm{up}}\}$ to facilitate the analysis. For example, $\mathbf{f}=\mathrm{vec}(\mathbf{F})$, $\hat{\mathbf{z}}^{\mathrm{lo}}=\mathrm{vec}(\hat{\mathbf{Z}}^{\mathrm{lo}})$, and $\hat{\mathbf{z}}^{\mathrm{up}}=\mathrm{vec}(\hat{\mathbf{Z}}^{\mathrm{up}})$.

In essence, the goal of SBL is to perform posterior inference on $\mathcal{X}$ from $\mathcal{Y}$ where the interaction among $\{\mathcal{X},\mathcal{Y}\}$ is captured by the conditional distributions listed as follows. First, $\{\mathbf{f},\mathbf{g},\mathbf{h}\}$ conditioned on $\{\bm{\gamma}_{\mathbf{f}},\bm{\gamma}_{\mathbf{g}},\bm{\gamma}_{\mathbf{h}}\}$ are assumed to be Gaussian distributed as

where $\bm{\Gamma}_{\mathbf{f}}=\mathrm{diag}(\bm{\gamma}_{\mathbf{f}})$, $\bm{\Gamma}_{\mathbf{g}}=\mathrm{diag}(\bm{\gamma}_{\mathbf{g}})$, and $\bm{\Gamma}_{\mathbf{h}}=\mathrm{diag}(\bm{\gamma}_{\mathbf{h}})$ are the precision matrices of the Gaussian distributions above. Meanwhile, the hyperpriors of the hyperparameters $\{\bm{\gamma}_{\mathbf{f}},\bm{\gamma}_{\mathbf{g}},\bm{\gamma}_{\mathbf{h}}\}$ are modeled as independent and identically distributed (i.i.d.) Gamma distributions

where $\Gamma(\cdot)$ is the Gamma function, and $a$ and $b$ are the shape and rate to be chosen. The reason for choosing the Gaussian-Gamma distribution is because $\{p(\bm{\gamma_{\mathbf{f}}}),p(\bm{\gamma_{\mathbf{g}}}),p(\bm{\gamma_{\mathbf{h}}})\}$ are the conjugate priors for the likelihood functions $\{p(\mathbf{f}|\bm{\gamma}_{\mathbf{f}}),p(\mathbf{g}|\bm{\gamma}_{\mathbf{g}}),p(\mathbf{h}|\bm{\gamma}_{\mathbf{h}})\}$ that make posterior inference tractable [29]. Furthermore, the Gaussian-Gamma distribution captures the sparse nature of $\{\mathbf{f},\mathbf{g},\mathbf{h}\}$ by making the marginal priors $\{p(\mathbf{f}),p(\mathbf{g}),p(\mathbf{h})\}$ Student’s-$t$ distributions, which are sparsity-promoting priors under the appropriate choice of $a$ and $b$ [14, 15]. In the simulation, we adhere to the convention that assumes uninformative priors by setting $a=b=10^{-6}$.

Moving on to the measurement model, the conditional distribution of $\mathbf{y}$ is

which follows from (14). Likewise, the conditional distributions of $\mathbf{u}$ and $\hat{\mathbf{z}}$ in the pseudo-measurement model are

where (24) and (25) come from (16) and (4). The indicator function $\mathbb{I}(\cdot)$ is equal to one if the argument is true and zero otherwise.

Now, recall that the goal of SBL is to perform exact posterior inference on $\mathcal{X}$ from $\mathcal{Y}$ using (17)-(25). The problem, however, is that computing $p(\mathcal{X}|\mathcal{Y})=p(\mathcal{X},\mathcal{Y})/\int p(\mathcal{X},\mathcal{Y})d\mathcal{X}$ is intractable in general. Therefore, we focus on finding the approximate posterior distribution using the VI approach.

First, we explain the idea behind VI with the mean-field approximation [16, 17]. Then, we derive the approximate posterior distributions, which enable us to compute the posterior means. To proceed, consider the decomposition of the log-evidence

where $q(\mathcal{X})$ is any probability distribution that approximates $p(\mathcal{X}|\mathcal{Y})$, $D_{\mathrm{KL}}(q\|p)$ is the Kullback–Leibler (KL) divergence that measures the distance between $q(\mathcal{X})$ and $p(\mathcal{X}|\mathcal{Y})$, and $\mathcal{L}(q)$ is known as the negative variational free energy. Then, we can minimize $D_{\mathrm{KL}}(q\|p)$ with respect to $q(\mathcal{X})$ by maximizing $\mathcal{L}(q)$ because $\log p(\mathcal{Y})$ is constant.

In general, the variational free energy minimization problem is intractable for a general class of $q(\mathcal{X})$. Therefore, we adopt the mean-field approximation by considering the class of $q(\mathcal{X})$ that assumes independence among the partition of $\mathcal{X}$ such that $q(\mathcal{X})=\prod_{i}q(\mathcal{X}_{i})$. Then, $\{q(\mathcal{X}_{i})\}_{\forall i}$ is the global minimum of the variational free energy minimization problem if and only if [16, 17]

where $Z_{i}$ is the normalization constant that makes $q(\mathcal{X}_{i})$ a valid probability distribution. Here, $\langle\cdot\rangle_{\mathcal{X}_{i}}$ indicates the expectation with respect to $\prod_{j\neq i}q(\mathcal{X}_{j})$. In addition, we use the notation $\langle\cdot\rangle$ to indicate the expectation with respect to $q(\mathcal{X})$ in the sequel.

Then, our goal is to derive the functional forms of $\{q(\mathcal{X}_{i})\}_{\forall i}$ from (27) that enable us to deduce the distributions of $\{q(\mathcal{X}_{i})\}_{\forall i}$ by inspection. Since the distributions of $\{q(\mathcal{X}_{i})\}_{\forall i}$ are coupled in an intertwined manner as evident from (27), however, $q(\mathcal{X}_{i})$ is updated for fixed $\{q(\mathcal{X}_{j})\}_{j\neq i}$ by cycling through $i$, which leads us to a local minimum. Now, we derive the functional forms of $\{q(\mathcal{X}_{i})\}_{\forall i}$ to determine the approximate posterior distributions for the partition $\mathcal{X}=\{\mathbf{f},\mathbf{g},\mathbf{h},\bm{\gamma}_{\mathbf{f}},\bm{\gamma}_{\mathbf{g}},\bm{\gamma}_{\mathbf{h}},\mathbf{u}\}$.

1. Derivation of $q(\mathbf{u})$: To derive $q(\mathbf{u})$, we only need to plug in (24) and (25) to (27). The reason for not plugging in (17)-(23) to (27) is because the expectations of (17)-(23) with respect to $\langle\cdot\rangle_{\mathbf{u}}$ are constants that do not affect the functional form of $q(\mathbf{u})$. In the posterior inference jargon, $\{\mathbf{f},\hat{\mathbf{z}}\}$ is said to be the Markov blanket of $\mathbf{u}$ [30]. In the sequel, we identify the Markov blanket by inspection without further explanation.

To proceed, define

from (24). Then, plugging in (24) and (25) to (27) yields

or equivalently

for the real as well as the imaginary part, from which we recognize that $\mathbf{u}$ is truncated Gaussian distributed. The posterior mean of $\mathbf{u}$ has a well-known form [31, 32, 33]

where $\phi(\cdot)$ and $\Phi(\cdot)$ are the standard normal PDF and cumulative distribution function (CDF), $\alpha_{i}=(\hat{z}_{i}^{\mathrm{lo}}-\langle b_{\mathbf{uf},i}\rangle)/(\sigma_{\mathrm{I}}/\sqrt{2})$, and $\beta_{i}=(\hat{z}_{i}^{\mathrm{up}}-\langle b_{\mathbf{uf},i}\rangle)/(\sigma_{\mathrm{I}}/\sqrt{2})$. To compute the expressions above, we need $\langle\mathbf{b}_{\mathbf{uf}}\rangle$, which we can obtain from (28) by replacing $\mathbf{f}$ with $\langle\mathbf{f}\rangle$.

2. Derivation of $q(\mathbf{f})$: First, define $\mathbf{A}_{\mathbf{fg}}$, $\mathbf{b}_{\mathbf{fh}}$, and $\mathbf{A}_{\mathbf{f}}$ from (23) and (24) as (32) and (33) at the bottom of the next page. Then, plugging in (17), (23), and (24) to (27) with some straightforward but tedious algebra leads to

which is a Gaussian distribution with

The posterior mean and covariance above require $\langle\mathbf{A}_{\mathbf{fg}}\rangle$, $\langle\mathbf{b}_{\mathbf{fh}}\rangle$, and $\langle\mathbf{A}_{\mathbf{fg}}^{\mathrm{H}}\mathbf{A}_{\mathbf{fg}}\rangle$ to be computed, from which the first two can be obtained from (32) by substituting $\mathbf{G}$ and $\mathbf{h}$ with $\langle\mathbf{G}\rangle$ and $\langle\mathbf{h}\rangle$. The expression for $\langle\mathbf{A}_{\mathbf{fg}}^{\mathrm{H}}\mathbf{A}_{\mathbf{fg}}\rangle$ is provided in Appendix A as the derivation is more involved.

3. Derivation of $q(\mathbf{g})$: First, we define $\mathbf{A}_{\mathbf{gf}}$ and $\mathbf{b}_{\mathbf{gh}}$ from (23) as (37) at the bottom of the next page. Then, expanding (27) after plugging in (18) and (23) gives

which is recognized as a Gaussian distribution associated with the posterior mean and covariance

Again, computing the expressions above requires $\langle\mathbf{A}_{\mathbf{gf}}\rangle$ and $\langle\mathbf{b}_{\mathbf{gh}}\rangle$ as well as $\langle\mathbf{A}_{\mathbf{gf}}^{\mathrm{H}}\mathbf{A}_{\mathbf{gf}}\rangle$, from which the terms corresponding to the first moment are given by (37) after replacing $\mathbf{F}$ and $\mathbf{h}$ with $\langle\mathbf{F}\rangle$ and $\langle\mathbf{h}\rangle$. The explicit form of $\langle\mathbf{A}_{\mathbf{gf}}^{\mathrm{H}}\mathbf{A}_{\mathbf{gf}}\rangle$ with a detailed derivation is provided in Appendix B.

4. Derivation of $q(\mathbf{h})$: First, define $\mathbf{A}_{\mathbf{h}}$ and $\mathbf{b}_{\mathbf{hfg}}$ from (23) as (41) at the bottom of the next page. By plugging in (19) and (23) to (27), we arrive at

which implies that $\mathbf{h}$ is Gaussian distributed parameterized by

The posterior mean and covariance above require $\langle\mathbf{b}_{\mathbf{hfg}}\rangle$ to be computed, which can be obtained from (41) by substituting $\mathbf{F}$ and $\mathbf{g}$ with $\langle\mathbf{F}\rangle$ and $\langle\mathbf{g}\rangle$.

5. Derivation of $q(\bm{\gamma}_{\mathbf{f}})$: Let us plug in (17) and (20) to (27) to obtain