Low-Complexity Hybrid Precoding in Massive Multiuser MIMO Systems Reading Notes 1

basic knowledge

Why can a diagonal matrix be introduced for column power normalization?

For a complex matrix, the square of the modulus of each element represents the power of that element. For a matrix $A$ , the power of each column can be obtained by calculating the sum of the squares of the modulus of the column. Therefore, if you want to normalize the column power, you need to adjust the power of each column to be equal, that is, make the sum of the squares of the modulus lengths of each column equal.

In order to achieve this purpose, the sum of the squares of the modulus lengths of each column can be used as the diagonal elements of the diagonal matrix, and the diagonal matrix can be multiplied by the original matrix to achieve column power normalization. This diagonal matrix is ​​called the scaling matrix ( Scaling Matrix ) or the magnitude matrix ( Amplitude Matrix ), which is a matrix whose diagonal elements are the sum of the squares of the modulus of each column.

Specifically, for a matrix $A$ , assuming that the sum of the squares of the modulus of each column is$c_1,c_2,\cdots ,c_N$, then the scaling matrix can be expressed as:

$$L=\left[\begin{array}{cccc}\frac{1}{\sqrt{c_1}}& 0& \cdots & 0\\ 0& \frac{1}{\sqrt{c_2}}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & \frac{1}{\sqrt{c_N}}\end{array}\right]$$

Then for $A\text{'}s$ operation of column power normalization can be expressed as:

where ${\mathbf{A}}_{\mathrm{norm}}$Represents the normalized matrix, $\Lambda$ denotes the scaling matrix. Therefore, the scaling matrix can be introduced into the column power normalization process to realize the operation of normalizing the power of each column to be equal.

What does beamforming mean?

Beamforming is a signal processing technique that uses the combination of multiple antennas to control the direction and shape of signal transmission. Beamforming can be achieved by changing the distribution of signal phase and amplitude in order to focus the signal in a specific direction or area.

Reference RS signal?

A reference signal (Reference Signal, RS) is a basic signal used in a wireless communication system for operations such as channel estimation and modulation and demodulation.

In wireless communication systems such as LTE and 5G, reference signals are usually sent by the base station to estimate and compensate the channel at the receiving end. These reference signals generally include two types of uplink reference signals and downlink reference signals .

What is the Frobenius norm of a matrix?

Frobenius norm, referred to as F-norm, is a matrix norm, denoted as $\Vert \cdot {\Vert }_F$。

定义：设 $A={\left[a_{ij}\right]}_{m\times n}$, is a $m\times n$ 的矩阵，称 $\Vert A{\Vert }_F=\sqrt{\mathrm{tr}\left(A^{T}A\right)}=\sqrt{{\sum }_{i,j}{a}_{ij}^2}$is the matrix $A$ 's Frobenius norm.

with matrix $B$ approximation matrix$A$，即 $B=\underset{B}{\mathrm{argmin}}\Vert A-B{\Vert }_F$。

Zero Forcing Precoding

Depending on the characteristics of the wireless channel, each receive antenna receives data from all transmit antennas. The channel through which each antenna transmits a signal may be represented by channel state information ( channel state information, CSI ).

Therefore, the channel matrix of the wireless channel can be expressed as:

$$\mathbf{H}=\left\{\begin{array}{cccc}{h}_{11}& h_{12}& \cdots & h_{1N}\\ h_{21}& h_{22}& \ddots & h_2\\ ⋮& \ddots & \ddots & ⋮\\ h_K& h_{K2}& \cdots & h_{KN}\end{array}\right\}$$

$$\mathbf{y}=\sqrt{r_f}\mathbf{H}\mathbf{x}+n$$

where:$r_f$is the sending power; $n$ represents additive white Gaussian noise. Since users are independent from each other, it is difficult to obtain CSI of other users, which means it is difficult to collaborate with other users. But the CSI of all communication users can be obtained at the base station (base station, BS) side. In Time Division Duplex (TDD) systems, CSI can be obtained through BS channel estimation. Therefore, precoding technology can be used to suppress MUI and improve the performance of the communication system.

In a massive MIMO system, the existence of a large number of antennas makes MUI affect the communication performance. ZF precoding improves the anti-interference performance, thereby improving the reliability of wireless communication.

The precoding model is shown in Figure 2, and the signal to be transmitted is processed by the precoding matrix. The receiver signal is as follows:
$$\mathbf{y}=\sqrt{r_f}\mathbf{H}{\mathbf{W}}_{ZF}\mathbf{s}+\mathbf{n}$$

The scheme in this paper uses ZF precoding to reduce the impact of MUI, and the ZF precoding algorithm uses channel matrix inversion to eliminate MUI. The traditional ZF precoding matrix ${\mathbf{W}}_{ZF}$It can be expressed as:

$${\mathbf{W}}_{ZF}={\mathbf{H}}^H{\left(\mathbf{H}{\mathbf{H}}^{\mathrm{H}}\right)}^{-1}$$

Therefore, the received signal can be expressed as:

$$\mathbf{y}=\mathbf{s}+\mathbf{n}$$

        %   =============  ZF preocidng, numerical  ================ 
        WtZF = H'*inv(H*H');%H'是Hermitian转置符
        WZF = WtZF*inv(sqrt(diag(diag(WtZF'*WtZF)))); % normalized columns

Signal to be transmitted $s$ The impact of multi-user interference in MIMO systems can be reduced by ZF precoding. The principle is to use the knownCSIthe basic distortion of the transmitted data caused bythe fading of the wireless communication channel and MUI (multi-user interference, MUI

Since the ZF precoding process requires matrix inversion operation, in the traditional MIMO communication system, its computational complexity is limited. However, in the massive MIMO system, due to the rapid increase in the number of base station antennas and users in the system, the ZF precoding process The complexity of matrix inversion rises sharply, making the computational requirements of massive MIMO systems extremely demanding. Therefore, a low-complexity precoding scheme is necessary for ZF precoding in massive MIMO systems.

complex normal distribution

$$h\sim \mathcal{C}\mathcal{N}(0,1)$$

This mathematical expression represents the random variable $h$ follows a complex normal distribution with a mean of 0 and a variance of 1. Let's parse this expression step by step:

• $h$ is a random variable representing a random phenomenon or quantity.

• $The\; \sim$ sign means "$h$ obeys" or "follows", that is, the random variable$h$ follows a certain distribution.

• $\mathcal{C}\mathcal{N}(0,1)$ represents a complex normal distribution, where$C$ and$N$ stands for "complex" and "normal", respectively. This is a probability distribution that describes the probability density of a complex random variable. The two parameters in parentheses represent the mean and variance of the distribution, respectively:

• $0$ is the mean and represents the center of the distribution. Here, it represents the random variablehhBoth the real and imaginary parts of$h\; have\; an\; expected\; value\; of\; 0.$

• $1$ is the variance, indicating how discrete the distribution is. Here, it represents the random variablehhBoth the real and imaginary parts of$h\; have\; a\; variance\; of\; 0.5.$Thus, the entire expression represents the random variable$h$ follows a complex normal distribution with mean 0 and variance 1.

1. Introduction

It is well known that massive multiple-input multiple-output (MIMO) can achieve high-capacity performance through simplified transmit precoding/receive joint design. Most notably, simple linear precoding schemes such as zero-forcing (ZF) are nearly optimal and comparable to nonlinear precoding such as dirty-paper coding (DPC) in massive MIMO systems. However, to take advantage of multiple antennas, the convention is to modify the amplitude and phase of the complex signal at baseband, then after passing through a digital-to-analog (D/A) converter, mixer, and power amplifier (often referred to as a radio frequency RF link), the The processed signal is up-converted to near the carrier frequency. The output of the RF chain needs to be coupled to the antenna elements, that is, each antenna element needs to be supported by a dedicated RF chain. This is actually too expensive to implement in massive MIMO systems due to the large number of antenna elements.

Inspired by [6], this paper proposes to design an RF precoder by extracting the conjugate transpose phase of the downlink aggregation channel to obtain huge array gain in massive MIMO systems, considering the practical constraints of the RF chain. Then, the low-dimensional baseband ZF precoding is performed based on the equivalent channel obtained by multiplying the RF precoder and the actual channel matrix. This hybrid precoding scheme, called PZF, is shown to approach the performance of nearly optimal but practically unachievable full-complexity ZF precoding in massive multiuser MIMO scenarios.

2. System model

We consider arranging $K$ single-antenna users, each of which supports single-stream transmission. $W$ is$K\times K$ , is the precoding of the baseband;$F$ is$N_t\times K$ , is the precoding of RF. Baseband precoder$W$ can adjust the amplitude and phase, and RF precoder$F$ can only adjust the phase through variable phase shifters (variable phase shifters) and synthesizers (combiners). $Each\; element\; in\; F$ is normalized to satisfy the condition$\left|{\mathbf{F}}_{i,j}\right|=\frac{1}{\sqrt{N_t}}$，其中 $|F_{i,j}|$ means$F$中$(i,j)$ Magnitude of the th element.

We adopt a narrowband flat fading channel and getSampled baseband signals received at$k\; users:$

$$y_k={\mathbf{h}}_k^H\mathbf{FW}\mathbf{s}+n_k\text{\hspace{1em}(1)}$$

$h_k^H$is from BS to $The\; downlink\; channel\; of\; k$ users. $\mathbf{s}\in {\mathbb{C}}^{K\times 1}$ means a total of$The\; signal\; vectors\; of\; k$ users satisfy the condition$\mathbf{E}\left[{\mathbf{ss}}^H\right]=\frac{P}{K}{\mathbf{I}}_K$, where $P$ is the total transmission power at the base station,$E[\cdot ]$ represents the mathematical expectation operator. In order to further satisfy the transmission power limit, we will$W$ is normalized to satisfy$\Vert \mathbf{FW}{\Vert }_F^2=K$。$n_k$Represents additional circular symmetric Gaussian distribution white noise with unit variance, ie $n_k\sim \mathcal{C}\mathcal{N}(0,1)$。

The kkth at the receiving end$The\; signal-to-interference\; ratio\; of\; k$ users can be expressed by the following formula:

$${\mathrm{SINR}}_k=\frac{\frac{P}{K}{\left|h_k^{H}F{w}_k\right|}^2}{1+{\sum }_{j\ne k}\frac{P}{K}{\left|{\mathbf{h}}_k^H\mathbf{F}{\mathbf{w}}_j\right|}^2}\text{\hspace{1em}(2)}$$

where ${\mathbf{w}}_j$means W \mathbf{W}The jjthof$the\; W\; matrix$$column\; j$ . If a Gaussian input signal is used, the long-term average spectral efficiency that the system can achieve is:

$$R=\sum _{k=1}^K\mathbb{E}\left[{\log }_2\left(1+{SINR}_k\right)\right]\text{\hspace{1em}(3)}$$

3. Hybrid precoding in Massive MIMO system

In massive MIMO systems, due to the asymptotic orthogonality of user channels in rich scattering environments [2], ZF precoding is considered to be an important linear precoding scheme that can achieve almost optimal capacity performance. It is usually implemented by baseband processing and requires $N_t$An RF chain performs RF-to-baseband frequency conversion and A/D conversion. However, this huge hardware requirement limits the expansion of the array size.

In order to alleviate the hardware limitation while fully exploiting the potential of massive multi-user MIMO systems, we plan to use only phase modulation to convert $K$ RF link results and$N_t$Coupled with two transmit antennas, use cheap RF phase shifters. Low-dimensional multi-user stream processing is then performed at baseband to manage inter-user interference. The proposed low-complexity hybrid precoding scheme, called phased-ZF (PZF), can approach the performance of full-complexity ZF precoding, which, as mentioned earlier, is practically unrealizable because Each antenna needs to be supported with a dedicated RF chain. We next analyze the spectral efficiency that the proposed PZF scheme can achieve.

3.1 Hybrid precoding design

The proposed hybrid baseband and RF joint processing is performed with the structure shown in Fig. 1, where the baseband precoder W modifies the amplitude and phase of the incoming complex symbols and the RF precoder F controls the phase of the upconverted RF signal. We propose pure phase control in the RF domain by extracting the phase of the conjugate transpose of the aggregated downlink channel from the BS to multiple users. This is to align the phases of the channel elements so that the enormous array gain provided by the excess antennas in massive MIMO systems can be obtained.

For clarity, ${\mathbf{F}}_{i,j}$Expressed as $(i,j)$th element of $\mathbf{F}$。

$$F_{i,j}=\frac{1}{\sqrt{N_t}}{e}^{j{\phi }_{i,j}}\text{\hspace{1em}(4)}$$

${Phi}_{i,j}$is the conjugate transpose matrix $H^H$ 's$(i,j)$ th element$h_{i,j}$phase. Here we implicitly assume that perfect channels are only available at the BS.

At baseband, we observe an equivalent channel ${\mathbf{H}}_{eq}=HF$ , which is low-dimensional$K\times K$ 的。$H=[{\mathbf{h}}_1,\cdots ,{\mathbf{h}}_K{]}^H$ is a downlink composite link. Therefore, for$H_{eq}$Multi-stream baseband precoding can be performed, where simple low-dimensional ZF precoding can be expressed as:
$$\mathbf{W}=H_{eq}^H{\left({\mathbf{H}}_{eq}{H}_{eq}^H\right)}^{-1}L\text{\hspace{1em}(5)}$$

For example, $\Lambda$ is a diagonal matrix introduced for column power normalization. Using this PZF scheme, in order to support$Simultaneous\; transmission\; of\; K$ streams greatly reduces hardware complexity, of which only$K$ RF chains, and N t N_trequired for full-complexity ZF precoding$N_t$。

Quantized RF phase control: According to (4), the RF precoder $Each\; element\; of\; F$ is assumed to be continuously variable in phase. However, in practical implementations, the phase of each entry tends to be heavily quantized due to the practical constraints of variable phase shifters. Therefore, we need to study the performance of our proposed PZF precoding scheme in this real-world scenario, namely$\mathbf{F}$ 的 $K{N}_t$The phase of the element is quantized to $With\; B$ -bit precision, each element is quantized to its nearest neighbor position according to the smallest Euclidean distance. Therefore,F \mathbf{F}The phase of each element of$F$$\widehat{Phi}=(2p\hat{n})/\left(2^B\right)$, where$\hat{n}$ is chosen according to:
n ^ = arg ⁡ min ⁡ n ∈ { 0 , ⋯ , 2 B − 1 } ∣ φ − 2 π n 2 B ∣ \hat{n}=\arg \min _{n \ in\left\{0, \cdots, 2^{B}-1\right\}}\left|\varphi-\frac{2 \pi n}{2^{B}}\right|n^=argn∈{ 0,⋯,2B−1}min​ ​Phi−2B2πn​ ​

Among them, $\phi$ is the unquantized phase obtained from equation (4). Next, baseband precoding needs to pass through (5) and quantized$F$ to calculate.

In the MATLAB code is Quantthe quantization process of function calculation.

function r = Quant(B, W)
delta = 2*pi/2^B; % quantization interval
r = zeros(size(W, 1), size(W, 2));% ininitialize quantized matrix

for i1 = 1 : size(W, 1)
    for i2 = 1 : size(W, 2)
        ph = angle(W(i1, i2)); % ph in [-pi, pi]
        phq = floor(ph/delta)*delta +(mod(ph, delta) > delta/2)*delta ;% quantized phase
        r(i1, i2) = exp(1j*phq);
    end
end
r = 1/sqrt(size(W, 1)) * r;
end

3.2 Spectral Efficiency Analysis in Sharp Fading Channels

In this section, we analyze our proposed PZF precoding and full-complexity ZF precoding in a large-scale number of transmit antennas $N_t$, assuming that the channel is Rayleigh fading spectral efficiency. Closed-form expressions are derived, revealing the effect of different parameters on the system capacity.

用${\mathbf{f}}_k$Represents the matrix F \mathbf{F}The kth column of$F.$

$$y_k=\left[{\mathbf{h}}_k^H{f}_1,\cdots ,{\mathbf{h}}_k^H{f}_k,\cdots ,h_K^H{f}_K\right]\mathbf{Ws}+n_k\text{\hspace{1em}(7)}$$

Diagonal item

As mentioned in Section III, $f_k$is obtained by extracting $h_k$The phase obtained. For the diagonal terms, we have

$${\mathbf{h}}_k^H{\mathbf{f}}_k=\frac{1}{\sqrt{N_t}}\sum _{i=1}^{N_t}\left|h_{i,k}\right|\text{\hspace{1em}(8)}$$

$h_{i,k}$Is the channel matrix conjugate transpose matrix ${\mathbf{H}}^H=[{\mathbf{h}}_1,\cdots ,h_K]$ of$(i,k)$ the th element. Also the vector$h_k$target $i$ elements. Assume that each$h_k$The elements in are all independent and identically distributed (iid ) Gaussian random variables with unit variance and zero mean, ie, $h\sim \mathcal{C}\mathcal{N}(0,1)$ . We can know$|h|$ is a Rayleigh distribution with mean$\frac{\sqrt{Pi}}{2}$, the variance is $1-\pi /4$ . When$N_t$close to infinity, according to the central limit theorem we know

$$h_k^H{\mathbf{f}}_k\sim N\left(\frac{\sqrt{\pi N_t}}{2},1-\frac{}{4}\right)\text{\hspace{1em}(9)}$$

off-diagonal items

For off-diagonal terms, ie, $j\ne k$，我们有${\mathbf{h}}_k^H{\mathbf{f}}_j=\frac{1}{\sqrt{N_t}}{\sum }_{i=1}^{N_t}{h}_{i,k}^{\ast }{e}^{j{\phi }_{i,j}}$. Its distribution is given in the following theorem.

Lemma 1 : In sharp fading channels, the off-diagonal term
$h_k^H{f}_j\sim \mathcal{C}\mathcal{N}(0,1)$

Through Lemma 1, we can analyze that the off-diagonal term $\left|{\mathbf{h}}_k^H{\mathbf{f}}_j\right|$The amplitude obeys the mean value of $\frac{\sqrt{Pi}}{2}$, with a variance of $1-\frac{}{4}$sharp distribution. According to the same formula (9), the diagonal term $h_k^H{f}_k$For comparison, it can be proved that even without baseband processing, in the large number of antennas $N_t$In the case of , the inter-user interference is essentially negligible! Because at this time the equivalent channel matrix an equivalent channel ${\mathbf{H}}_{eq}=HF$ is close to the identity matrix. The values ​​on the diagonal are very large.

However, we note that when $N_t$When taking a moderately large value, the crosstalk between users may still deteriorate the system performance. Therefore we need to use our proposed baseband precoding scheme.

Remark 1: Considering that when $N_t$When it is very large, the off-diagonal ${\mathbf{h}}_k^H{\mathbf{f}}_j$is basically negligible, and we expect that the upper bound of the derived closed-form expression will be very tight in systems with large antennas. This further confirms the simulation results in Figure 2. Therefore, the upper bound of the closed-form expression is at a larger value $N_t$In the case of , a very good estimate of the spectral efficiency of the PZF precoding scheme is used.

We can infer that even with ZF precoding on the channel to eliminate inter-user interference, we still get more spectral efficiency than the off-diagonal term $h_k^H{f}_j$All are 0 to be low. In other words, $h_k^H{f}_j$The case of all 0 is the upper bound of the spectral efficiency of the PZF precoding scheme, that is, $K\mathcal{R}$，其中 $\mathcal{R}=\mathbb{E}\left[{\log }_2\left(1+\frac{P}{K}{\left|{\mathbf{h}}_k^H{\mathbf{f}}_k\right|}^2\right)\right]$。

$$\begin{array}{rl}\mathcal{R}& =\mathbb{E}\left[{\log }_2\left(1+\frac{P}{K}{\left(y+\frac{\sqrt{\pi N_t}}{2}\right)}^2\right)\right]\\ & ={\log }_2\left(1+\frac{}{4}\frac{P{N}_t}{K}\right)+\underset{D}{\underbrace{\mathbb{E}\left[{\log }_2\left(\frac{1+\frac{P}{K}{\left(y+\frac{\sqrt{\pi N_t}}{2}\right)}^2}{1+\frac{\pi N_t}{4}\frac{P}{K}}\right)\right]}}\end{array}$$

∼ $y\sim \mathcal{N}\left(0,p^2\right)$，$p=\sqrt{1-\frac{}{4}}$. We can prove that $The\; upper\; and\; lower\; bounds\; of\; \Delta$ are both 0 to prove that${\lim }_{N_t\to \infty }D=0$。

for the $For\; k$ user streams, the full complexity ZF precoding vector (with unit norm) will be${\mathbf{h}}_k$ 投影到 ${\tilde{\mathbf{H}}}_k={\left[{\mathbf{h}}_1,\cdots ,{\mathbf{h}}_{k-1},h_{k+1},\cdots ,h_K\right]}^{}$in the null space of${}^{H.}$In the spectral efficiency analysis, we exploit the asymptotic orthogonality of user channels in massive user MIMO systems. It shows that: full-complexity ZF precoding converges to conjugate beamforming, the interference between users is eliminated to 0, and${\mathrm{SINR}}_k\to \frac{P}{K}{\left|{\mathbf{h}}_k\right|}^2$，当 $N_t\to \infty$ . Therefore, according to formula (3), we get the full complexity ZF precoding in large$N_t$The spectral efficiency under the limit is

$$\begin{array}{rl}{R}_{\mathrm{FC}-\mathrm{ZF}}& \to KE\left[{\log }_2\left(1+\frac{P}{K}{\left|{\mathbf{h}}_k\right|}^2\right)\right]\\ & =K{e}^{\frac{K}{P}}{\log }_{2}e\sum _{n=1}^{N_t}{E}_n\left(\frac{K}{P}\right)\end{array}\text{\hspace{1em}(11)}$$

where $|{\mathbf{h}}_k{|}^2$ subject to degrees of freedom is$2N{\_}_t$The chi-square distribution of $E_n\left(x\right)$ isnExponential integral of order$n\; .$

$$E_n(x)={\int }_1^{\infty }\frac{e^{-xt}}{t^n}dt$$

$$E_n(z)=\frac{(-z{)}^{n-1}}{(n-1)!}{E}_1\left(z\right)+\frac{{\mathrm{e}}^{-z}}{(n-1)!}\sum _{k=0}^{n-2}(n-k-2)!(-z{)}^k$$

function r = expintn_noMaple(x, n)

if n >= 2% Calculation using series expansion
    sum = 0;
    for m = 0 : (n-2)
        sum = sum + (-1)^m*factorial(m)/x^(m+1);
    end

    minuend = x^(n-1) * (-1)^(n+1)/factorial(n-1) * expint(x);
    subtractor = x^(n-1) * (-1)^(n+1)/factorial(n-1) * exp(-x) * sum;
    r = minuend - subtractor;
else % Calculation using built-in expint function
    r = expint(x);
end

$$\begin{array}{l}{\lim }_{N_{t\to \infty }}D\ge {\lim }_{a\to \infty }\mathbb{E}\left[{\log }_2{\left(1+\frac{y}{a}\right)}^2\right]+{\lim }_{a\to \infty }{\log }_2\frac{a^2}{\frac{1}{r}+a^2}\\ \stackrel{(a)}{=}{\lim }_{a\to \infty }\frac{2a\_{\log }_{2}e}{\sqrt{2p.m}p}{\int }_0^{+\infty }(\ln x){and}^{-\frac{a^2(x-1{)}^2}{2p^2}}\left(1+e^{-\frac{2a{\_}^2}{p^2}}\right)dx\\ \stackrel{(b)}{\ge }{\lim }_{a\to \infty }\frac{2a\_{\log }_{2}e}{\sqrt{2p.m}p}\left(1+e^{-\frac{2a{\_}^2}{p^2}}\right){\int }_0^1\left(\ln x\right){and}^{-\frac{a^2(x-1{)}^2}{2p^2}}dx\\ \stackrel{(c)}{=}{\lim }_{a\to \infty }\frac{2a{e}^{-\frac{a^2}{2p^2}}}{\sqrt{2p.m}p\ln 2}\left(1+e^{-\frac{2a{\_}^2}{p^2}}\right){\int }_0^1(\ln x){and}^{\frac{a^2}{2p^2}}dx\\ \stackrel{(d)}{\ge }{\lim }_{a\to \infty }\frac{2a{e}^{-\frac{a^2}{2p^2}}}{\sqrt{2p.m}p\ln 2}\left(1+e^{\frac{2a{\_}^2}{p^2}}\right)\frac{2p^2\left(1-e^{\frac{a^2}{2p^2}}\right)}{2}=0\end{array}$$

$$\begin{array}{l}{\lim }_{N_{t\to \infty }}D\stackrel{(d)}{\ge }{\lim }_{a\to \infty }\frac{2}{\sqrt{2p.m}p\ln 2}\left(1+e^{-\frac{2a{\_}^2}{p^2}}\right)e^{-\frac{a^2(x^2-2\xi }{2p^2}}{\int }_0^1(\ln x)dx\\ ={\lim }_{a\to \infty }\frac{2}{\sqrt{2p.m}p\ln 2}{e}^{-\frac{a^2(\xi -1{)}^2}{2p^2}}{\int }_0^1\left(\ln x\right)dx\\ ={\lim }_{a\to \infty }-\frac{2}{\sqrt{2p.m}p\ln 2}{e}^{-\frac{a^2(x^2-2\xi }{2p^2}}\\ =0\end{array}$$

$$\begin{array}{rl}\underset{N_t\to \infty }{\lim }D& \ge \underset{a\to \infty }{\lim }\mathbb{E}\left[{\log }_2{\left(1+\frac{y}{a}\right)}^2\right]+\underset{a\to \infty }{\lim }{\log }_2\frac{a^2}{\frac{1}{r}+a^2}\\ & =\underset{a\to \infty }{\lim }E\left[{\log }_2{\left(1+\frac{y}{a}\right)}^2\right]=\underset{a\to \infty }{\lim }\frac{1}{\sqrt{2p.m}p}{\int }_{-\infty }^{\infty }{\log }_2{\left(1+\frac{y}{a}\right)}^2{e}^{-\frac{y^2}{2p^2}}\mathrm{d}y\\ & =\underset{a\to \infty }{\lim }\frac{1}{\sqrt{2p.m}p}{\int }_{-\infty }^{\infty }{\log }_2{x}^{2e}{\_}^{-\frac{(a(x-1){)}^2}{2p^2}}a\mathrm{d}x,x=1+\frac{y}{a}\\ & =\underset{a\to \infty }{\lim }\frac{2}{\ln 2\cdot \sqrt{2p.m}p}\left({\int }_0^{\infty }\ln x{e}^{-\frac{a^2(x-1{)}^2}{2p^2}}\mathrm{d}x+{\int }_{-\infty }^0\ln (-x)e^{-\frac{a^2(x-1{)}^2}{2p^2}}\mathrm{d}x\right)\\ & =\underset{a\to \infty }{\lim }\frac{2}{\ln 2\cdot \sqrt{2p.m}p}\left({\int }_0^{\infty }\ln x{e}^{-\frac{a^2(x-1{)}^2}{2p^2}}\mathrm{d}x+{\int }_0^{\infty }\ln x{e}^{-\frac{a^2(x+1{)}^2}{2p^2}}dx\right)\\ & =\underset{a\to \infty }{\lim }\frac{2a\_{\log }_{2}e}{\sqrt{2p.m}p}\left({\int }_0^{\infty }\ln x{e}^{-\frac{a^2(x-1{)}^2}{2p^2}}\left(1+e^{-\frac{a^{2}4}{2p^2}}\right)dx\right)\\ & =\underset{a\to \infty }{\lim }\frac{2a\_{\log }_{2}e}{\sqrt{2p.m}p}\left({\int }_0^{\infty }\ln x{e}^{-\frac{a^2(x-1{)}^2}{2p^2}}\left(1+e^{-\frac{2a{\_}^2}{p^2}}\right)\mathrm{d}x\right)\end{array}$$

$$\underset{N_t\to \infty }{\lim }\frac{\mathcal{R}}{{\log }_2\left(1+\frac{\pi }{4}\frac{P{N}_t}{K}\right)}=1$$

四、仿真结果

4.1 大型锐利衰减信道

在图2中，我们数值上比较了我们提出的 PZF 预编码方案和全复杂度ZF方案，包括我们方案的量化版本。后者在大阵列体制下被认为是几乎是完美的，但是由于需要 $N_t$ 个昂贵的 RF 射频链路，实际上是不可能的。可以观察到，我们提出的 PZF 预编码方案表现适度地接近全复杂度ZF预编码，只有不到 1dB 的损失，但是大大降低了复杂度。至于严重量化后的相位控制，我们发现 2 比特的量化精度，即相位控制在 $\left\{0,\pm \frac{\pi }{2},\pi \right\}$ 中变化，我们提出的方案只遭受到了忽略不计的退化，即小于 1dB.

The derived spectral efficiency expressions analytical solutions (10) and (11) are also plotted in Fig. 2. We observe that the derived closed-form expressions are quite accurate in characterizing the proposed PZF precoding and full-complexity ZF precoding schemes over the entire range of signal-to-noise ratio (SNR), which provides support for practical system design Effective guidance is provided.

4.2 Large mmWave Multiuser Channels

In addition to the ideal sharp fading channel, our proposed PZF scheme is also suitable for mmWave communication (known to have very limited multipath components). In order to capture this poor scattering property, in the simulation, we use the geometric channel model [8-10]

$${\mathbf{h}}_k^H=\sqrt{\frac{N_t}{N_p}}\sum _{l=1}^{N_p}{a}_l^k{\mathbf{a}}^H\left({\varphi }_l^k,i_l^k\right)$$

Among them, each user we assume can observe the same number of transmission paths, namely $N_p$, by the kkthThe llthobserved by$k\; users$$The\; strength\; of\; l$ paths is expressed as$a_l^k$(assuming $a_l^k\sim \mathcal{C}\mathcal{N}(0,1)$ ),${\varphi }_l^k\left(i_l^k\right)$ is a random azimuth and elevation angle, subject to$[0,2\pi ]$ is a random variable$\mathbf{a}\left({\varphi }_l^k,i_l^k\right)$ is the corresponding vector of the array related only to the antenna structure. Here we consider an array of equally spaced antennas, which has a simple array corresponding expression given by [9, eq(6)].

We compare our proposed PZF scheme with the Beam Spatial MIMO (B-MIMO) scheme proposed in [10] in Figure 3. This scheme essentially steers the flow to an approximate strongest path in the RF domain (using DFT matrix processing), and performs low-dimensional baseband ZF precoding based on the equivalent channel. For a fair comparison, it is also assumed that BS has a total of $K$ chains. The B-MIMO scheme achieves ideal performance in line-of-sight (LoS) channels, but fails to capture sparse multipath components in non-LoS channels.

V. Conclusion

In this paper, we study massive multiuser MIMO systems under the constraints of real radio frequency hardware. We propose a low-complexity hybrid PZF scheme to approach the ideal but infeasible full-complexity ZF precoding. RF processing is designed to achieve large power gains with reasonable complexity, and then a baseband precoder is introduced to facilitate multi-stream processing. Its performance has been characterized by closed-form expressions and further demonstrated by computer simulations in Rayleigh fading and low-scattering mmWave channels.