A Sequential Subspace Method for Millimeter Wave MIMO Channel Estimation

The mmWave MIMO channel sounding model is shown in Fig. 1, where the transmitter and receiver are equipped with $N_{t}$ and $N_{r}$ antennas, respectively. There are $N_{RF}\geq 2$ and $M_{RF}\geq 2$ RF chains at the transmitter and receiver, respectively. Without loss of generality, we assume $N_{t}$ is an integer multiple of $N_{RF}$, and $N_{r}$ is also an integer multiple of $M_{RF}$. In the considered mmWave channel sounding framework, one sounding symbol is transmitted over a unit time interval from the transmitter, which is defined as one channel use. It is assumed that the system employs $K$ channel uses for channel sounding. The received signal ${\mathbf{y}}_{(k)}\in{\mathbb{C}}^{M_{RF}\times 1}$ at the $k$th channel use is given by

where ${\mathbf{W}}_{(k)}\!\!=\!\!{\mathbf{W}}_{A,k}{\mathbf{W}}_{D,k}\in{\mathbb{C}}^{N_{r}\times M_{RF}}$ is the receive sounder composed of receive analog sounder ${\mathbf{W}}_{A,k}\in{\mathbb{C}}^{N_{r}\times M_{RF}}$ and receive digital sounder ${\mathbf{W}}_{D,k}\in{\mathbb{C}}^{M_{RF}\times M_{RF}}$ in series, ${\mathbf{f}}_{(k)}\!\!=\!\!{\mathbf{F}}_{A,k}{\mathbf{F}}_{D,k}{\mathbf{s}}_{k}\!\!\in\!\!{\mathbb{C}}^{N_{t}\times 1}$ is the transmit sounder composed of transmit analog sounder ${\mathbf{F}}_{A,k}\in{\mathbb{C}}^{N_{t}\times N_{RF}}$ and transmit digital sounder ${\mathbf{F}}_{D,k}\in{\mathbb{C}}^{N_{RF}\times N_{RF}}$ in series with transmitted sounding signal ${\mathbf{s}}_{k}$, and ${\mathbf{n}}_{(k)}\!\in\!{\mathbb{C}}^{{N_{r}}\times 1}$ is the noise.

Considering that the transmitted sounding signal ${\mathbf{s}}_{k}$ is included in ${\mathbf{f}}_{(k)}$, for convenience, we let ${\mathbf{s}}_{k}\!\!=\!\!\frac{1}{\sqrt{N_{RF}}}[1,\ldots,1]^{T}\in{\mathbb{R}}^{N_{RF}\times 1}$, which enables us to focus on the design of ${\mathbf{F}}_{A,k}$ and ${\mathbf{F}}_{D,k}$. It is worth noting that the analog sounders are constrained to be constant modulus, that is, $|[{\mathbf{W}}_{A,k}]_{i,j}|={1}/{\sqrt{N_{r}}}$, and $|[{\mathbf{F}}_{A,k}]_{i,j}|={1}/{\sqrt{N_{t}}},\forall i,j$. Without loss of generality, we assume the power of the transmit sounder is one, that is, $\|{\mathbf{f}}_{(k)}\|_{2}^{2}=1$. The noise ${\mathbf{n}}_{(k)}$ is an independent zero mean complex Gaussian vector with covariance matrix $\sigma^{2}{\mathbf{I}}_{N_{r}}$. Due to the unit power of transmit sounder, we define the signal-to-noise-ratio (SNR) as $1/{\sigma^{2}}$.¹¹1Here, the SNR is the ratio of transmitted sounder’s power to the noise’s power, which is a common practice in the channel estimation literature [4, 8, 9, 10, 16, 17]. The details of designing the receive and transmit sounders for facilitating the channel estimation will be discussed in Section III.

To model the point-to-point sparse mmWave MIMO channel, we assume there are $L$ clusters with $L\ll\min\{N_{r},N_{t}\}$, and each constitutes a propagation path. The channel model can be expressed as [18, 19],

where ${\mathbf{a}}_{r}(\theta_{r,l})\!\in\!{\mathbb{C}}^{N_{r}\!\times 1}\!$ and ${\mathbf{a}}_{t}(\theta_{t,l})\!\in\!{\mathbb{C}}^{N_{t}\!\times 1}\!$ are array response vectors of the uniform linear arrays (ULAs) at the receiver and transmitter, respectively. We extend it to the channel model with 2D uniform planar arrays (UPAs) in Section IV-C. In particular, ${\mathbf{a}}_{r}(\theta_{r,l})$ and ${\mathbf{a}}_{t}(\theta_{t,l})$ are expressed as

where $\lambda$ is the wavelength, $d=0.5\lambda$ is the antenna spacing, $\theta_{r,l}$ and $\theta_{t,l}$ are the angle of arrival (AoA) and angle of departure (AoD) of the $l$th path uniformly distributed in $[-\pi/2,\pi/2)$, respectively, and $h_{l}\sim{\mathcal{C}}{\mathcal{N}}(0,\sigma_{h,l}^{2})$ is the complex gain of the $l$th path.

The channel model in (2) can be rewritten as

where $\!\!{\mathbf{A}}_{r}\!\!=\!\![{\mathbf{a}}_{r}(\theta_{r,1}),\ldots,{\mathbf{a}}_{r}(\theta_{r,L})]\!\!\in\!\!{\mathbb{C}}^{N_{r}\times L}$, ${\mathbf{A}}_{t}\!\!=\!\![{\mathbf{a}}_{t}(\theta_{t,1}),\ldots,{\mathbf{a}}_{t}(\theta_{t,L})]\!\in\!{\mathbb{C}}^{N_{t}\times L}$, and $\mathbf{h}\!\!=\!\![h_{1},\cdots,h_{L}]^{T}\!\!\in\!\!{\mathbb{C}}^{L\times 1}$. The channel estimation task is to obtain an estimate of ${\mathbf{H}}$, i.e., $\widehat{{\mathbf{H}}}$, from ${\mathbf{y}}_{(k)}$, ${\mathbf{W}}_{(k)}$, and ${\mathbf{f}}_{(k)}$, $k\!=\!\!1,\cdots,K$ in (1).

To evaluate the channel estimation performance, the achieved spectrum efficiency by utilizing the channel estimate $\widehat{{\mathbf{H}}}$ is discussed in the following. Conventionally, the precoder $\widehat{{\mathbf{F}}}\!\in\!{\mathbb{C}}^{N_{t}\times N_{d}}$ and combiner $\widehat{{\mathbf{W}}}\!\in\!{\mathbb{C}}^{N_{r}\times N_{d}}$ are designed, based on the estimated $\widehat{{\mathbf{H}}}$, where $N_{d}$ is the number of transmitted data streams with $N_{d}\!\leq\!\min\{N_{RF},M_{RF}\}$. Here, when evaluating the channel estimation performance, it is assumed the number of transmitted data streams is equal to the number of dominant paths, i.e., $N_{d}=L$. After the design of precoder and combiner, the received signal for the data transfer is given by

where the signal follows ${\mathbf{s}}\sim{\mathcal{C}}{\mathcal{N}}(\mathbf{0}_{L},\frac{1}{L}{\mathbf{I}}_{L})$ and ${\mathbf{n}}\sim{\mathcal{C}}{\mathcal{N}}(\mathbf{0}_{N_{r}},\sigma^{2}{\mathbf{I}}_{N_{r}})$. It is worth noting that (4) is for data transmission, while (1) is for channel sounding. The spectrum efficiency achieved by $\widehat{{\mathbf{W}}}$ and $\widehat{{\mathbf{F}}}$ in (4) is defined in [20] as,

where ${\mathbf{H}}_{e}\!=\!\widehat{{\mathbf{W}}}^{H}{\mathbf{H}}\widehat{{\mathbf{F}}}\in{\mathbb{C}}^{{L}\times{L}}$ and ${\mathbf{R}}_{n}\!=\!\widehat{{\mathbf{W}}}^{H}\widehat{{\mathbf{W}}}\in{\mathbb{C}}^{{L}\times{L}}$. In this work, we assume that the precoder and combiner are unitary, such that $\widehat{{\mathbf{W}}}^{H}\widehat{{\mathbf{W}}}={\mathbf{I}}_{L}$ and $\widehat{{\mathbf{F}}}^{H}\widehat{{\mathbf{F}}}={\mathbf{I}}_{L}$. Under this assumption, we have ${\mathbf{R}}_{n}={\mathbf{I}}_{L}$ in (5).

It is worth noting that the spectrum efficiency in (5) is invariant to the right rotations of the precoder and combiner, i.e., substituting $\widetilde{{\mathbf{F}}}=\widehat{{\mathbf{F}}}{\mathbf{R}}_{\mathbf{F}}$ and $\widetilde{{\mathbf{W}}}=\widehat{{\mathbf{W}}}{\mathbf{R}}_{\mathbf{W}}$ into (5), where ${\mathbf{R}}_{\mathbf{F}}\in{\mathbb{C}}^{L\times L}$ and ${\mathbf{R}}_{\mathbf{W}}\in{\mathbb{C}}^{L\times L}$ are unitary matrices, does not change the spectrum efficiency. Thus, the $R$ in (5) is a function of subspaces spanned by the precoder and combiner, i.e., $\mathop{\mathrm{col}}(\widehat{{\mathbf{F}}})$ and $\mathop{\mathrm{col}}(\widehat{{\mathbf{W}}})$. Moreover, the highest spectrum efficiency can be achieved when $\mathop{\mathrm{col}}(\widehat{{\mathbf{F}}})$ and $\mathop{\mathrm{col}}(\widehat{{\mathbf{W}}})$ respectively equal to the row and column subspaces of ${\mathbf{H}}$.

Apart from the spectrum efficiency achieved by the signal model in (4), we consider the effective SNR at the receiver,

The received SNR $\gamma$ in (6) has the same rotation invariance property as the spectrum efficiency. In other words, the $\gamma$ in (6) is a function of the estimated column and row subspaces. The maximum of the $\gamma$ is also achieved when $\widehat{{\mathbf{W}}}$ and $\widehat{{\mathbf{F}}}$ span the column and row subspaces of ${\mathbf{H}}$, respectively.

Inspired by the definition in (6), in this paper, the accuracy of subspace estimation is defined as the ratio of the power captured by the transceiver matrices [21] $\widehat{{\mathbf{W}}}$ and $\widehat{{\mathbf{F}}}$ to the power of the channel,

Similarly, the measures for the accuracy of column subspace and row subspace estimation, i.e., $\eta_{c}(\widehat{{\mathbf{W}}})$ and $\eta_{r}(\widehat{{\mathbf{F}}})$, are respectively defined as the ratio of the power captured by $\widehat{{\mathbf{W}}}$ and $\widehat{{\mathbf{F}}}$ to the power of the channel in the following,

Moreover, $\eta_{c}$ and $\eta_{r}$ are also rotation invariant. When the values of $\eta_{c}$ or $\eta_{r}$ are closed to one, the corresponding $\widehat{{\mathbf{W}}}$ or $\widehat{{\mathbf{F}}}$ can be treated accurate subspace estimates.

The illustration of the proposed SASE algorithm is shown in Fig. 2. It consists of two stages: one is column subspace estimation and the other is row subspace estimation. In particular, the training sounders of the second stage can be optimized by fully adapting them to the estimated column subspace, which would reduce the number of channel uses and improve the estimation accuracy.

In this subsection, we present the design of transmit and receive sounders along with the method for obtaining the column subspace of the mmWave channel. To begin with, the following lemma shows that under the mmWave channel model in (3), the column subspaces of ${\mathbf{H}}$ and sub-matrix ${\mathbf{H}}_{S}$ are equivalent.

Lemma 1 reveals that when $\mathop{\mathrm{col}}({\mathbf{H}}_{S})=\mathop{\mathrm{col}}({\mathbf{H}})$, to obtain the column subspace of ${\mathbf{H}}$, it suffices to sample the first $m$ columns of ${\mathbf{H}}$, i.e., ${\mathbf{H}}_{S}$, which reduces the number of channel uses. However, the mmWave hybrid MIMO architecture can not directly access the entries of ${\mathbf{H}}$ due to the analog array constraints. This can be overcome by adopting the technique proposed in [16]. Specifically, to sample the $i$th column of ${\mathbf{H}}$, i.e., $[{\mathbf{H}}]_{:,i}$, the transmitter needs to construct the transmit sounder ${\mathbf{f}}_{(i)}={\mathbf{e}}_{i}\in{\mathbb{C}}^{N_{t}\times 1}$, where ${\mathbf{e}}_{i}$ is the $i$th column of ${\mathbf{I}}_{N_{t}}$. This is possible due to the fact that any precoder vector can be generated by $N_{RF}\geq 2$ RF chains [22]. To be more specific, there exists ${\mathbf{F}}_{A,i}$, ${\mathbf{F}}_{D,i}$, and ${\mathbf{s}}_{i}$ such that ${\mathbf{e}}_{i}={\mathbf{F}}_{A,i}{\mathbf{F}}_{D,i}{\mathbf{s}}_{i}$,

where ${\mathbf{F}}_{A,i}\!\!=\!\!\frac{1}{\sqrt{N_{t}}}\mathbf{1}_{N_{t},N_{RF}}$ except for $[{\mathbf{F}}_{A,i}]_{i,2}\!\!=\!\!-\frac{1}{\sqrt{N_{t}}}$, the ${\mathbf{F}}_{D,i}\!\!=\!\!\mathbf{0}_{N_{RF},N_{RF}}$ except for $[{\mathbf{F}}_{D,i}]_{1,1}\!\!=\!\!\frac{\sqrt{N_{RF}N_{t}}}{2},[{\mathbf{F}}_{D,i}]_{2,1}\!\!=\!\!-\frac{\sqrt{N_{RF}N_{t}}}{2}$, and ${\mathbf{s}}_{i}\!\!=\!\!\frac{1}{\sqrt{N_{RF}}}[1,\ldots,1]^{T}\in{\mathbb{R}}^{N_{RF}\times 1}$.

At the receiver side, we collect the receive sounders of $N_{r}/M_{RF}$ channel uses to form the full-rank matrix,

where ${\mathbf{W}}_{(i,j)}\in{\mathbb{C}}^{N_{r}\times M_{RF}},~{}j=1,\ldots,N_{r}/M_{RF}$, denotes the $j$th receive sounder corresponding to transmit sounder ${\mathbf{e}}_{i}$. In order to satisfy the analog constraint where the entries in analog sounders should be constant modulus, we let the matrix ${\mathbf{M}}$ in (10) be the discrete Fourier transform (DFT) matrix. Specifically, the analog and digital receive sounders associated with ${\mathbf{W}}_{(i,j)}$ in (10) are expressed as follows

Thus, the received signal ${\mathbf{y}}_{(i,j)}\!\in\!{\mathbb{C}}^{M_{RF}\times 1}$ under the transmit sounder ${\mathbf{e}}_{i}$ and receive sounder ${\mathbf{W}}_{\!(i,j)}$ is expressed as follows

where ${\mathbf{n}}_{(i,j)}\in{\mathbb{C}}^{N_{r}\times 1}$ is the noise vector with ${\mathbf{n}}_{(i,j)}\sim{\mathcal{C}}{\mathcal{N}}(\bm{0}_{N_{r}},\sigma^{2}{\mathbf{I}}_{N_{r}})$. Then we stack the observations of $N_{r}/M_{RF}$ channel uses as ${\mathbf{y}}_{i}=[{\mathbf{y}}_{(i,1)}^{T},\cdots,{\mathbf{y}}^{T}_{(i,N_{r}/M_{RF})}]^{T}\in{\mathbb{C}}^{N_{r}\times 1}$,

where $\tilde{{\mathbf{n}}}_{i}\in{\mathbb{C}}^{N_{r}\times 1}$ is the effective noise vector after stacking, whose covariance matrix is expressed as,

Because the DFT matrix ${\mathbf{M}}$ in (10) satisfies ${\mathbf{M}}^{H}{\mathbf{M}}={\mathbf{M}}{\mathbf{M}}^{H}={\mathbf{I}}_{N_{r}}$, the following holds

Substituting (15) into (12), we can verify that ${\mathbb{E}}[\tilde{{\mathbf{n}}}_{i}\tilde{{\mathbf{n}}}_{i}^{H}]=\sigma^{2}{\mathbf{I}}_{N_{r}}$, and precisely, $\tilde{{\mathbf{n}}}_{i}\sim{\mathcal{C}}{\mathcal{N}}(\mathbf{0}_{N_{r}},\sigma^{2}{\mathbf{I}}_{N_{r}})$. Moreover, by denoting $\widetilde{{\mathbf{N}}}=[\tilde{{\mathbf{n}}}_{1},\cdots,\tilde{{\mathbf{n}}}_{m}]\in{\mathbb{C}}^{N_{r}\times m}$, it is straightforward that the entries in $\widetilde{{\mathbf{N}}}$ are independent, identically distributed (i.i.d.) as ${\mathcal{C}}{\mathcal{N}}(0,\sigma^{2})$. Here, for convenience, we denote $\widetilde{{\mathbf{Y}}}_{S}=[{\mathbf{y}}_{1},\cdots,{\mathbf{y}}_{m}]\in{\mathbb{C}}^{N_{r}\times m}$ where ${\mathbf{y}}_{i}$ is defined in (11). Then, we apply DFT to the collected observation $\widetilde{{\mathbf{Y}}}_{S}$, and obtain ${\mathbf{Y}}_{S}={\mathbf{M}}\widetilde{{\mathbf{Y}}}_{S}\in{\mathbb{C}}^{N_{r}\times m}$ as

where ${\mathbf{N}}_{S}\!=\!{\mathbf{M}}\widetilde{{\mathbf{N}}}\in{\mathbb{C}}^{N_{r}\times m}$ and ${\mathbf{H}}_{S}\!=\![{\mathbf{H}}]_{:,1:m}\in{\mathbb{C}}^{N_{r}\times m}$. Before talking about the noise part ${\mathbf{N}}_{S}$ in (16), the following lemma is a preliminary which gives the distribution of entries in the product of matrices.

Therefore, considering the noise part in (16), i.e., ${\mathbf{N}}_{S}={\mathbf{M}}\widetilde{{\mathbf{N}}}$, where ${\mathbf{M}}$ is unitary and $\widetilde{{\mathbf{N}}}$ has i.i.d. ${\mathcal{C}}{\mathcal{N}}(0,\sigma^{2})$ entries, the conclusion of Lemma 2 can be applied, which verifies that the entries of ${\mathbf{N}}_{S}$ in (16) are i.i.d. as ${\mathcal{C}}{\mathcal{N}}(0,\sigma^{2})$.

Given the expression in (16), the column subspace estimation problem is formulated as,

where one of the optimal solutions of (17) can be obtained by taking the dominant $L$ left singular vectors of ${\mathbf{Y}}_{S}$. Here, the number of paths, $L$, is assumed to be known as a priori. In practice, it is possible to estimate $L$ by comparing the singular values of ${\mathbf{Y}}_{S}$ [23]. Because ${\mathbf{Y}}_{S}={\mathbf{H}}_{S}+{\mathbf{N}}_{S}$ and $\mathop{\mathrm{rank}}({\mathbf{H}}_{S})=L$, there will be $L$ singular values of ${\mathbf{Y}}_{S}$ whose magnitudes clearly dominate the other singular values. Alternatively, we can set it to $L_{\text{sup}}$, which is an upper bound on the number of dominant paths such that $L\leq L_{\text{sup}}$.²²2Due to the limited RF chains, the dimension of channel subspaces for data transmission is less than $\min\{M_{RF},N_{RF}\}$. Thus, if the path number estimate is larger than $\min\{M_{RF},N_{RF}\}$, we let it be $\min\{M_{RF},N_{RF}\}$.

Now, we design the receive combiner $\widehat{{\mathbf{W}}}$ in (4) for data transmission to approximate the estimated $\widehat{{\mathbf{U}}}\in{\mathbb{C}}^{N_{r}\times L}$ in (17). Specifically, we design the analog combiner $\widehat{{\mathbf{W}}}_{A}\in{\mathbb{C}}^{N_{r}\times M_{RF}}$ and digital combiner $\widehat{{\mathbf{W}}}_{D}\in{\mathbb{C}}^{M_{RF}\times L}$ at the receiver by solving the following problem

The problem above can be solved by using the OMP algorithm [5] or alternating minimization method [24]. The designed receive combiner is given by $\widehat{{\mathbf{W}}}=\widehat{{\mathbf{W}}}_{A}\widehat{{\mathbf{W}}}_{D}\in{\mathbb{C}}^{N_{r}\times L}$ with $\widehat{{\mathbf{W}}}^{H}\widehat{{\mathbf{W}}}={\mathbf{I}}_{L}$. The methods in [5, 24] have shown to guarantee the near optimal performance, such as $\mathop{\mathrm{col}}(\widehat{{\mathbf{W}}})\approx\mathop{\mathrm{col}}(\widehat{{\mathbf{U}}})$. The details of our column subspace estimation algorithm are summarized in Algorithm 1.

In general, $\mathop{\mathrm{col}}(\widehat{{\mathbf{W}}})$ is not equal to the column subspace of ${\mathbf{H}}$, i.e., $\mathop{\mathrm{col}}({\mathbf{U}})$ with ${\mathbf{U}}\in{\mathbb{C}}^{N_{r}\times L}$, due to the noise ${\mathbf{N}}_{S}$ in (16). To analyze the column subspace accuracy $\eta_{c}(\widehat{{\mathbf{W}}})$ defined in (8), we introduce the theorem [25] below.

We have the following proposition for the accuracy of the column subspace estimation in Algorithm 1.

From (20), the larger the value of $m$ is, the more accurate the column subspace estimation. Thus, when more columns are used for the column subspace estimation, the estimated column subspace will be more reliable. In particular, when the noise level is low such that $\sigma_{L}^{2}({\mathbf{H}}_{S})\!\!\gg\!m\sigma^{2}$ in (20), we have

It means that the column subspace estimation accuracy is linearly related to the value of $\sigma^{2}\!\!/\!\sigma_{L}^{2}({\mathbf{H}}_{S})$, i.e., ${\mathcal{O}}(\text{SNR})$. On the other hand, when the noise level is high such that $\sigma_{L}^{2}({\mathbf{H}}_{S})\!\ll\!m\sigma^{2}$, the bound in (20) can be written as

At low SNR, the column subspace estimation accuracy is quadratically related to $\sigma^{4}/\sigma_{L}^{4}({\mathbf{H}}_{S})$, i.e., ${\mathcal{O}}(\text{SNR}^{2})$.

In this subsection, we present how to learn the row subspace by leveraging the estimated column subspace matrix $\widehat{{\mathbf{W}}}$. Because we have already sampled the first $m$ columns of ${\mathbf{H}}$ in the first stage, we only need to sample the remaining $N_{t}-m$ columns to estimate the row subspace as shown in Fig. 2.

At the $k$th channel use of the second stage, we observe the $(m+k)$th column of ${\mathbf{H}}$, $k=1,\ldots,(N_{t}-m)$. To achieve this, we employ the transmit sounder as

For the receive sounder, given the estimated column subspace matrix $\widehat{{\mathbf{W}}}$ in the first stage, we just let the receive sounder of the second stage be $\widehat{{\mathbf{W}}}\in{\mathbb{C}}^{N_{r}\times L}$.³³3It is worth noting that because the estimated column subspace of the first stage is $\widehat{{\mathbf{W}}}\in{\mathbb{C}}^{N_{r}\times L}$, thus the dimension for receive sounder of second stage is ${N_{r}\times L}$ rather than ${N_{r}\times M_{RF}}$ in (1). It is worth noting $\widehat{{\mathbf{W}}}$ is trivially applicable for hybrid precoding architecture since $\widehat{{\mathbf{W}}}$ is obtained from (18). Therefore, under the transmit sounder ${\mathbf{f}}_{(k)}$ in (21) and receive sounder $\widehat{{\mathbf{W}}}$ in (18), the observation ${\mathbf{y}}_{(k)}\in{\mathbb{C}}^{L\times 1}$ at the receiver can be given by

where ${\mathbf{n}}_{(k)}\in{\mathbb{C}}^{N_{r}\times 1}$ is the noise vector with ${\mathbf{n}}_{(k)}\sim{\mathcal{C}}{\mathcal{N}}(\bm{0}_{N_{r}},\sigma^{2}{\mathbf{I}}_{N_{r}})$. Then, the observations $k=1,\ldots,(N_{t}-m)$ in (22) are packed into a matrix $\widehat{{\mathbf{Q}}}_{C}\in{\mathbb{C}}^{L\times(N_{t}-m)}$ as

where ${\mathbf{H}}_{C}=\left[[{\mathbf{H}}]_{:,m+1},\ldots,[{\mathbf{H}}]_{:,N_{t}}\right]\in{\mathbb{C}}^{N_{r}\times(N_{t}-m)}$, and ${\mathbf{N}}_{C}=[{\mathbf{n}}_{(1)},\ldots,{\mathbf{n}}_{(N_{t}-m)}]\in{\mathbb{C}}^{N_{r}\times(N_{t}-m)}$.

In addition, given the receive sounder $\widehat{{\mathbf{W}}}$ and observations ${\mathbf{Y}}_{S}$ of the first stage in (16), we define $\widehat{{\mathbf{Q}}}_{S}\in{\mathbb{C}}^{L\times m}$ as,