278 lines
7.8 KiB
Markdown
278 lines
7.8 KiB
Markdown
---
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title: MVDR 算法仿真
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mathjax: true
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comments: true
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copyright: true
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toc:
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enable: true
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number: true
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wrap: true
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expand_all: true
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max_depth: 4
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abbrlink: 86ade7bc
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date: 2024-02-01 10:09:39
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tags:
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- Beamforming
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- 波束成形
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- MVDR
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- 误码率分析
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categories:
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- - Josh 的学习笔记
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- 算法仿真
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---
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  MVDR （Minimum Variance Distortionless Response，最小方差无失真响应）算法是 Capon 于 1969 年提出的经典波束形成算法，其解决的是无失真约束下输出噪声加干扰功率最小化问题。
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# 阵列模型
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  首先对阵列接收信号进行建模。假设 $N$ 元阵列天线在 $t$ 时刻的阵列接收信号和阵列输出信号分别为 $\mathbf{x}(t)$ 和 $y(t)$，则对于 $K$ 个入射信号，有
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$$\begin{equation}
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\begin{cases}
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\mathbf{x}(t) = \mathbf{A}(\boldsymbol{\theta})\mathbf{s}(t) + \mathbf{n}(t) \\
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y(t) = \mathbf{w}^\mathrm{H}\mathbf{x}(t)
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\end{cases},\quad t = 1,\cdots,T
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\end{equation}$$
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其中
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- 阵列接收信号 $\mathbf{x}(t) = [x_1(t),\cdots,x_N(t)]^\mathrm{T}\in\mathbb{C}^{N\times 1}$
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- 噪声 $\mathbf{n}(t) = [n_1(t),\cdots,n_N(t)]^\mathrm{T}\in\mathbb{C}^{N\times 1}$
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- 入射信号 $\mathbf{s}(t) = [s_1(t),\cdots,s_K(t)]^\mathrm{T}\in\mathbb{C}^{K\times 1}$
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- 阵列流形矩阵 $\mathbf{A}(\boldsymbol{\theta}) = [\mathbf{a}(\theta_1),\cdots,\mathbf{a}(\theta_K)]\in \mathbb{C}^{N\times K}$，导向矢量 $\mathbf{a}(\theta_i)\in\mathbb{C}^{N\times 1}$，$K$ 个信号来向集合 $\boldsymbol{\theta} = \{\theta_i\}_{i=1}^K$
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- 波束形成复加权 $\mathbf{w} = [w_1,\cdots,w_N]^\mathrm{T}$
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<!-- more -->
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# MVDR 算法简述
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  假设第 1 个信号是期望信号（desired signal），令其导向矢量 $\mathbf{a}_\mathrm{d} = \mathbf{a}(\theta_1)$，则阵列接收信号可以进一步写为
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$$\begin{equation}
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\mathbf{x}(t) = \underset{\mathbf{x}_\mathrm{d}(t)}{\underbrace{\mathbf{a}_\mathrm{d}s_1(t)}}
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+
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\underset{\mathbf{x}_\mathrm{i+n}(t)}{\underbrace{\sum_{k=2}^{K}\mathbf{a}(\theta_k)s_k(t) + \mathbf{n}(t)}}
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\end{equation}$$
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至此，可以给出 MVDR 问题的基本表述
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$$\begin{equation}
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\begin{aligned}
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&\underset{\mathbf{w}}{\text{minimize}} &&\mathbf{w}^\mathrm{H}\mathbf{R}_{\mathrm{i+n}}\mathbf{w}
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\\
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&\text{subject to} && \mathbf{w}^\mathrm{H}\mathbf{a}_\mathrm{d} = 1
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\end{aligned}
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\end{equation}$$
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其闭式解可以由 Lagrange 乘子法得到
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$$\begin{equation}
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\mathbf{w} = \alpha\mathbf{R}_{\mathrm{i+n}}^{-1}\mathbf{a}_\mathrm{d}
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\end{equation}$$
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其中
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- 噪声加干扰协方差矩阵 $\mathbf{R}_{\mathrm{i+n}} = \displaystyle\frac{1}{T} \sum_{t=1}^{T}\mathbf{x}_\mathrm{i+n}(t)\mathbf{x}_\mathrm{i+n}^\mathrm{H}(t)$
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- 归一化因子（使得复加权对于期望信号的增益为 1） $\alpha = \displaystyle\frac{1}{\mathbf{a}_\mathrm{d}^\mathrm{H}\mathbf{R}_{\mathrm{i+n}}^{-1}\mathbf{a}_\mathrm{d}}$
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# MVDR 算法仿真
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## 理想空域谱
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{width=800px}
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{% note primary MATLAB Code %}
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``` matlab mvdr_idea.m
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%% High-Resolution Frequency-Wavenumber Spectrum Analysis
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% J. Capon, “High-Resolution Frequency-Wavenumber Spectrum Analysis,” _Proceedings
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% of the IEEE_, vol. 57, no. 8, pp. 1408–1418, Aug. 1969, doi: <https://doi.org/10.1109/PROC.1969.7278
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% 10.1109/PROC.1969.7278>.
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%
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% H. L. Van Trees, _Optimum Array Processing: Part IV of Detection, Estimation,
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% and Modulation Theory_. New York, NY: Wiley, 2002.
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clear
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close all
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%% Parameters
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rng(253)
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% Array Parameters
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% elements in array
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elemNum = 12;
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% spacing wrt wavelength
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elemSpac = 0.5;
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% element locations
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elemLoc = 0:elemSpac:(elemNum-1)*elemSpac;
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% Signal Parameters
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% signal number
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sigNum = 6;
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% direction of arrival
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sigAng = rand(1,sigNum)*180;
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% signal to noise ratio
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snr = [30,70];
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% sample number
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sampleNum = 9000;
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% parameter match check
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if (length(snr) ~= sigNum) | (length(sigAng) ~= sigNum)
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disp('WARN: Signal number and parameter number NOT match.')
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end
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%% Impinging Waves/Signals
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% Transmitted Signals
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% wave generation, here are some sets of uncorrelated white noise
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txWave = randn(sigNum,sampleNum);
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% transmit power control based on given snr
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if length(snr) == 1
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txWave = db2mag(snr)*txWave;
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elseif length(snr) >= sigNum
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if length(snr) > sigNum
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disp('WARN: Extra SNRs are ignored.')
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end
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for i = 1:sigNum
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txWave(i,:) = db2mag(snr(i))*txWave(i,:);
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end
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elseif length(snr) < sigNum
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disp('WARN: The last SNR is used for the rest of the signals.')
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for i = 1:sigNum
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if i <= length(snr)
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txWave(i,:) = db2mag(snr(i))*txWave(i,:);
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else
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txWave(i,:) = db2mag(snr(end))*txWave(i,:);
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end
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end
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end
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% Array Manifold Matrix
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% For each signal, the array manifold vector is
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%
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% $$\mathbf{a}\left(\theta \right)=\exp \left(-j\frac{2\pi \mathbf{d}}{\lambda
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% \;}\cos \theta \;\right)$$
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%
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% in which the element locations are
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%
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% $$\mathbf{d}={\left\lbrack 0,\frac{\lambda }{2},\cdots ,\left(N-1\right)\frac{\lambda
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% }{2}\right\rbrack }^{\mathrm{T}}$$
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if length(sigAng) > sigNum
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disp('WARN: Extra angles are ignored.')
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elseif length(sigAng) < sigNum
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disp('ERROR: Angle number is less than signal number.')
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return
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end
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arrayMF = exp(-1i*pi*elemLoc'*cos(sigAng(1:sigNum)/180*pi));
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if sigNum > 1
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% signal & interference manifold vector
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sigMF = arrayMF(:,1);
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intMF = arrayMF(:,2:end);
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% impinging signal & interference
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rxSig = sigMF*txWave(1,:);
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rxIpn = intMF*txWave(2:end,:) + randn(elemNum,sampleNum);
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else
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% signal manifold vector
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sigMF = arrayMF;
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% impinging signal
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rxSig = sigMF*txWave;
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rxIpn = randn(elemNum,sampleNum);
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end
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% Received Wave
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rxWave = rxSig + randn(elemNum,sampleNum);
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%% Array Processing
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% The MVDR problem is to minimize output interference-plus-noise power under
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% distortionless contraint of target signal
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%
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% $$\begin{array}{l}\underset{\mathbf{w}}{\mathrm{minimize}\;\;} \;{\mathbf{w}}^{\mathrm{H}}
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% {\mathbf{R}}_{\mathrm{i}+\mathrm{n}} \mathbf{w}\\\mathrm{subecto}\;\mathrm{to}\;\;{\mathbf{w}}^{\mathrm{H}}
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% \mathbf{a}=1\end{array}$$
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%
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% The problem can be solve using the Lagrange multiplier method, and the solution
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% is
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%
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% $$\mathbf{w}=\alpha {\mathbf{R}}_{\mathrm{i}+\mathrm{n}}^{-1} \mathbf{a}$$
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%
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% where $\alpha \;$is the normalization factor that ensures the power of array
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% input and output desired signal remains unchanged
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%
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% $$\alpha =\frac{1}{{\mathbf{a}}^{\mathrm{H}} {\mathbf{R}}_{\mathrm{i}+\mathrm{n}}
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% \mathbf{a}}$$
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% interference-plus-noise covariance matrix
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ipnCovar = rxIpn*rxIpn';
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% MVDR weight vector
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weight = ipnCovar\sigMF/(sigMF'*(ipnCovar\sigMF));
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% = inv(ipnCovar)*sigMF/(sigMF'*inv(ipnCovar)*sigMF);
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% output SINR
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sinrOut = pow2db(var(weight'*rxSig)/var(weight'*rxIpn));
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%% Spectrum & Figure
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% scan grid
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scanGrid = 0:0.1:180;
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% spectrum
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spec = zeros(size(scanGrid));
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for i = 1:length(scanGrid)
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% steering vector
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scanMF = exp(-1i*pi*elemLoc'*cos(scanGrid(i)/180*pi));
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% MVDR spectrum
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spec(i) = weight'*scanMF;
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end
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% normalized spectrum
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specNorm = pow2db(abs(spec));
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% spectrum & signal location indicator
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plot(scanGrid,specNorm)
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hold on
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for i = 1:sigNum
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if i == 1
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plot([sigAng(i),sigAng(i)],[max(specNorm),min(specNorm)],'-.r');
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else
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plot([sigAng(i),sigAng(i)],[max(specNorm),min(specNorm)],'-.g');
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end
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end
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hold off
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axis tight
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title('Ideal Capon Spectrum')
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xlabel('Degree/°')
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ylabel('Magnitude/dB')
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grid on
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grid minor
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```
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{% endnote %}
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## 理想误码率
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## 失配输出 SINR
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## 失配误码率
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# Reference
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1. J. Capon, “High-Resolution Frequency-Wavenumber Spectrum Analysis,” _Proceedings of the IEEE_, vol. 57, no. 8, pp. 1408–1418, Aug. 1969, doi: [10.1109/PROC.1969.7278](https://doi.org/10.1109/PROC.1969.7278).
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2. H. L. Van Trees, _Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory_. New York, NY: Wiley, 2002.
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