In this section, we present the beam-domain signal processing in hybrid architecture massive MIMO systems. First, we discuss the DOA estimation based on the hybrid architecture with limited RF chains. Then, low-complexity channel estimation is proposed based on the estimated beam information. Finally, we study the power allocation to improve the system performance.
Due to the angle reciprocity, the direction-of-departure (DOD) of the forward-link channel is the same as the DOA of the reverse-link channel, even for the frequency division duplex (FDD) systems [
8].
The DOA information
\(\theta _{l}^{t}\) and
\(\phi _{l}^{t}\) can be estimated using the limited RF chains in two time slots, respectively. For the reverse link,
Kt contiguous antennas in a row at the receiver (which is the transmitter in the forward link) are first chosen in the horizontal direction as in Fig.
1, and then linked with
Kt RF chains to estimate the azimuth angle
\(\theta _{l}^{t}\) in the first time slot. Similarly, the elevation angle
\(\phi _{l}^{t}\) can be estimated by employing
Kt antennas in a column in the second time slot. Here, the estimated angles are defined as (
\(\theta _{l}^{t}, \phi _{l}^{t}\)).
By using only a small number of antennas, this estimation strategy has a low complexity and a high accuracy, which have been analyzed in our previous work [
18]. For example, while
Kt=16, the root mean square error (RMSE) of the estimated DOA has been less than 1
∘ by costing only 32 snapshots [
18]. Many DOA estimation algorithms can be adopted in this proposed estimation strategy, such as the well-known MUltiple SIgnal Classification (MUSIC) and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) [
19‐
21]. In each time slot, there are a number of snapshots to obtain approximate covariance information for DOA estimation with MUSIC or ESPRIT algorithm.
Practically, considering the low-cost, finite-resolution phase shifters are implemented at the transmitter and receiver for RF processing. For the Mt×Nt URA, there are Mt and Nt phase shifters with finite-resolution in the horizontal and vertical directions, respectively. When all the phase shifters have the same resolution \(\frac {1}{P}\pi \), for the azimuth angle θ∈[0,π) in the horizontal direction, the beam codebook consists of P basic beams by sampling the angle space [0,π) as \(G_{M}=\left \{0,\frac {1}{P}\pi,\cdots,\frac {P-1}{P}\pi \right \}\); for the elevation angle \(\phi \in [0,\frac {\pi }{2})\) in the vertical direction, the beam codebook consists of \(\left (\left \lfloor \frac {P}{2}\right \rfloor +1\right)\) basic beams by sampling the angle space \([0,\frac {\pi }{2})\) as \(G_{N}=\left \{0,\frac {1}{P}\pi,\cdots,\frac {\left \lfloor \frac {P}{2}\right \rfloor }{P}\pi \right \}\). Thus, for an Mt×Nt URA, the beam codebook consists of \(P\left (\left \lfloor \frac {P}{2}\right \rfloor +1\right)\) basic 2-D beams (θ,ϕ).
Finally, the estimated angles (\(\theta _{l}^{t}, \phi _{l}^{t}\)) should be mapped to the beam codebooks for practical transmission. The DOA information can be easily estimated as \((\hat {\theta }_{l}^{t},\hat {\phi }_{l}^{t})=\text {arg} \mathop {\text {min}}\limits _{\theta \in G_{M},\\ \phi \in G_{N}}\left \{|\theta \,-\,\theta _{l}^{t}|\,+\,|\phi \,-\,\phi _{l}^{t}|\right \}\) for practical RF processing. Notably, the DOA estimation strategy can also be directly applied at the receiver in forward link to obtain \((\hat {\theta }_{l}^{r}, \hat {\phi }_{l}^{r})\).
With the estimated beams, RF precoder at the transmitter and RF combiner at the receiver are given as
$$\begin{array}{@{}rcl@{}} \hat{\mathbf{F}}_{t}=\frac{1}{\sqrt{M_{t}N_{t}}}\left[\mathbf a_{t}\left(\hat{\theta}_{1}^{t}, \hat{\phi}_{1}^{t}\right) \cdots \mathbf a_{t}(\hat{\theta}_{L}^{t}, \hat{\phi}_{L}^{t})\right]\textrm{,} \end{array} $$
(4)
$$\begin{array}{@{}rcl@{}} \hat{\mathbf{F}}_{r}=\frac{1}{\sqrt{M_{r}N_{r}}}\left[\mathbf a_{r}\left(\hat{\theta}_{1}^{r}, \hat{\phi}_{1}^{r}\right) \cdots \mathbf a_{r}(\hat{\theta}_{L}^{r}, \hat{\phi}_{L}^{r})\right]\textrm{.} \end{array} $$
(5)
3.2 Channel estimation
Channel estimation is crucial but intractable in massive MIMO systems [
22,
23]. In this subsection, we propose a low-complexity channel estimation method in the beam domain. The proposed method includes the following three steps.
First, the magnitude of the channel gain
\(|\hat {\gamma }_{l}|\) and DOAs
\((\hat {\theta }_{l}^{t},\hat {\phi }_{l}^{t})\) mentioned above are simultaneously derived. In the reverse link, the received signal as the output of the URA at the receiver, which is the transmitter in the forward link, is expressed as [
20,
21]
$$\begin{array}{@{}rcl@{}} \mathbf{Y}_{1}=\mathbf{H}_{1}\mathbf{X}_{1}+\mathbf{W}_{1}=\mathbf A_{t}\mathbf{G}\mathbf{X}_{1}+\mathbf{W}_{1}\textrm{,} \end{array} $$
(6)
where
X1 is the transmitted signal matrix from one antenna for DOA estimation with
\(\mathbb E\left \{\mathbf {X}_{1}\mathbf {X}_{1}^{H}\right \}=\sigma _{X}^{2}\mathbf {I}_{L}, \mathbf {H}_{1}=\mathbf A_{t}\mathbf {G}\) is the reverse-link channel matrix,
\(\mathbf A_{t}=\left [\mathbf a_{t}\left (\hat {\theta }_{1}^{t}, \hat {\phi }_{1}^{t}\right) \cdots \mathbf a_{t}\left (\hat {\theta }_{L}^{t}, \hat {\phi }_{L}^{t}\right)\right ]\) is obtained from the estimated beam information,
G=diag(
γ1,⋯,
γL) is a
L×
L diagonal matrix with beam gains {
γ1,⋯,
γL} on the main diagonal, and
\(\mathbf {W}_{1}\sim \mathcal {CN}\left (\mathbf {0},\sigma _{W}^{2}\mathbf {I}_{M_{t}N_{t}}\right)\) denotes the AWGN matrix. From (
6), the covariance matrix of the received signal can be written as
$$\begin{array}{@{}rcl@{}} {\mathbf{R}_{YY}}&{=}&{\mathbb{E}\{\mathbf{Y}_{1}\mathbf{Y}_{1}^{H}\}}\\ &{=}&{\sigma_{X}^{2}\mathbf A_{t}\mathbf{G}\mathbf{G}^{H}\mathbf A_{t}^{H}+\sigma_{W}^{2}\mathbf{I}_{M_{t}N_{t}}} \\ &{=}&{\sigma_{X}^{2}\mathbf A_{t}} \left[\begin{aligned} |&\gamma_{1}|^{2}&\cdots & &0&\\ &\vdots & \ddots & &\vdots&\\ &0&\cdots & &|\gamma_{L}&|^{2} \end{aligned}\right] {\mathbf A_{t}^{H}+\sigma_{W}^{2}\mathbf{I}_{M_{t}N_{t}}\textrm{.}} \end{array} $$
(7)
Based on (
7), the magnitude of the beam gain is obtained as
$$\begin{array}{@{}rcl@{}} |\hat{\gamma}_{l}|=\sqrt{\left[\frac{1}{\sigma_{X}^{2}}\mathbf A_{t}^{\dag}(\mathbf R_{YY}-\sigma_{W}^{2}\mathbf{I}_{M_{t}N_{t}})(\mathbf A_{t}^{H})^{\dag}\right]_{l,l}}\textrm{.} \end{array} $$
(8)
where \(\mathbf A_{t}^{\dag }=\left (\mathbf A_{t}^{H}\mathbf A_{t}\right)^{-1}\mathbf A_{t}^{H}\) is the pseudo-inverse of At, and \(\left (\mathbf A_{t}^{H}\right)^{\dag }=\mathbf A_{t}\left (\mathbf A_{t}^{H}\mathbf A_{t}\right)^{-1}\) is the pseudo-inverse of \(\mathbf A_{t}^{H}\). By using the statistic characteristic, the estimated magnitude of the beam gain will have higher accuracy than the pilot-based estimation method.
Second, the phase of beam gain is estimated through a channel estimation algorithm with low pilot overhead and complexity. While the training signal is transmitted in reverse link, the received signal is given by
$$\begin{array}{@{}rcl@{}} \mathbf{Y}_{2}=\mathbf A_{t} \left[\begin{aligned} |\hat{\gamma}_{1}|&e^{j\varphi_{1}} & \cdots& & &0\\ &\vdots & \ddots & &\vdots\\ &0 & \cdots & &|\hat{\gamma}_{L}&|e^{j\varphi_{L}} \end{aligned}\right] \mathbf{X}_{2}+\mathbf{W}_{2}\textrm{,} \end{array} $$
(9)
where
φl denotes the phase of
\(\hat {\gamma }_{l}, \mathbf {X}_{2}\in \mathbb {C}^{L \times n_{s}}\) is the training signal consisting of
ns bits pilots, and
W2 is the AWGN matrix. Using (
9), the phase of the beam gain is estimated as
$$\begin{array}{@{}rcl@{}} e^{j\hat{\varphi}_{l}}=\frac{\left\{\mathbf{A}_{t}^{\dag}\mathbf{Y}_{2}\mathbf{X}_{2}^{\dag}\right\}_{l,l}}{|\hat{\gamma}_{l}|}\textrm{.} \end{array} $$
(10)
where
\(\mathbf X_{2}^{\dag }=\mathbf X_{2}^{H}\left (\mathbf X_{2}\mathbf X_{2}^{H}\right)^{-1}\) is the pseudo-inverse of
X2. With
L beams, there only needs
ns= log2
L bits pilots to estimate the phase of the beam gain. Due to the pseudo-inverse in (
10), there is a significant complexity reduction by using
ns bits pilots.
Finally, with the estimated beam gain
\(\hat {\gamma }_{l}=|\hat {\gamma }_{l}|e^{j\hat {\varphi }_{l}}\), the forward-link channel matrix in (
1) can be estimated as
$$\begin{array}{@{}rcl@{}} \hat{\mathbf H}=\frac{1}{\sqrt{L}}\sum_{l=1}^{L}\hat{\gamma}_{l}\mathbf a_{r}\left(\hat{\theta}_{l}^{r}, \hat{\phi}_{l}^{r}\right)\mathbf a_{t}^{H}\left(\hat{\theta}_{l}^{t}, \hat{\phi}_{l}^{t}\right)\textrm{.} \end{array} $$
(11)
In summary, the proposed channel estimation strategy has a low complexity for three reasons: (1) It directly uses the results of DOA estimation to obtain the magnitude of the channel gain as an additional product; (2) the DOA estimation strategy has a low complexity; (3) the
L beam gains are estimated with short training pilots. Furthermore, the proposed channel estimation strategy has a high accuracy. The reason is that the magnitude of channel gain is estimated from the statistic characteristic as shown in (
8), which has a higher accuracy than that of the conventional pilot-based channel estimation.
The proposed channel estimation method can also be adopted for multi-user systems, resulting in a great degradation of pilot overhead. When multiple users have similar DOAs and bring in multi-user interference, the proposed channel estimation method is still available. The reason is that the channel estimation of all users is obtained from the DOA estimation which needs to be implemented in a time-division method. As a result, the interference caused by the similar DOAs does not affect the time-division channel estimation of multiple users.
3.3 Power allocation
In this subsection, the power allocation problem is studied to improve the spectral efficiency of the hybrid architecture massive MIMO system. Let us denote the power allocation matrix
\(\mathbf P=\text {diag}(\sqrt {p_{1}},\cdots,\sqrt {p_{L}})\). The power allocation problem can be formulated as
$$\begin{array}{*{20}l} &{}\mathop{\text{max}}\limits_{\{p_{l}\}} R=\text{log}\:\text{det}\left(\mathbf{I}_{L}\,+\,\frac{(\hat{\mathbf{F}}_{r}^{H}\hat{\mathbf{F}}_{r})^{-1}(\hat{\mathbf{F}}_{r}^{H}\hat{\mathbf H}\hat{\mathbf{F}}_{t}\mathbf{P})(\hat{\mathbf{F}}_{r}^{H}\hat{\mathbf H}\hat{\mathbf{F}}_{t}\mathbf{P})^{H}}{\sigma_{n}^{2}}\right) \end{array} $$
(12a)
$$\begin{array}{*{20}l} &\textrm{s.t.}\qquad \!\!\!\!\!\!\!\! \sum_{l=1}^{L} p_{l}= P_{t} \end{array} $$
(12b)
$$\begin{array}{*{20}l} &\,p_{l}\geq0 \end{array} $$
(12c)
where Pt is the total transmit power.
To sum up, the main steps of the low-complexity beam-domain signal processing are summarized as follows:
The low-complexity beam-domain signal processing
-
In forward link, the receiver estimates beams \((\hat {\theta }_{l}^{r},\hat {\phi }_{l}^{r})\).
-
Estimate the beams
\((\hat {\theta }_{l}^{t},\hat {\phi }_{l}^{t})\) and the magnitude of beam gain
\(|\hat {\gamma }_{l}|\) by (
8) in reverse link.
-
With the estimated beams, RF precoder at the transmitter and RF combiner at the receiver are derived from (
4) and (
5), respectively.
-
Perform the power allocation only with \(|\hat {\gamma }_{l}|\) according to Theorem 1.