--- title: MVDR 算法仿真 mathjax: true comments: true copyright: true toc: enable: true number: true wrap: true expand_all: true max_depth: 4 abbrlink: 86ade7bc date: 2024-02-01 10:09:39 tags: - Beamforming - 波束成形 - MVDR - 误码率分析 categories: - - Josh 的学习笔记 - 算法仿真 ---   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。 # 阵列模型   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 $$\begin{equation} \begin{cases} \mathbf{x}(t) = \mathbf{A}(\boldsymbol{\theta})\mathbf{s}(t) + \mathbf{n}(t) \\ y(t) = \mathbf{w}^\mathrm{H}\mathbf{x}(t) \end{cases},\quad t = 1,\cdots,T \end{equation}$$ 其中 - 阵列接收信号 $\mathbf{x}(t) = [x_1(t),\cdots,x_N(t)]^\mathrm{T}\in\mathbb{C}^{N\times 1}$ - 噪声 $\mathbf{n}(t) = [n_1(t),\cdots,n_N(t)]^\mathrm{T}\in\mathbb{C}^{N\times 1}$ - 入射信号 $\mathbf{s}(t) = [s_1(t),\cdots,s_K(t)]^\mathrm{T}\in\mathbb{C}^{K\times 1}$ - 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 $\boldsymbol{\theta} = \{\theta_i\}_{i=1}^K$ - 波束形成复加权 $\mathbf{w} = [w_1,\cdots,w_N]^\mathrm{T}$ # MVDR 算法简述   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 $$\begin{equation} \mathbf{x}(t) = \underset{\mathbf{x}_\mathrm{d}(t)}{\underbrace{\mathbf{a}_\mathrm{d}s_1(t)}} + \underset{\mathbf{x}_\mathrm{i+n}(t)}{\underbrace{\sum_{k=2}^{K}\mathbf{a}(\theta_k)s_k(t) + \mathbf{n}(t)}} \end{equation}$$ 至此，可以给出 MVDR 问题的基本表述 $$\begin{equation} \begin{aligned} &\underset{\mathbf{w}}{\text{minimize}} &&\mathbf{w}^\mathrm{H}\mathbf{R}_{\mathrm{i+n}}\mathbf{w} \\ &\text{subject to} && \mathbf{w}^\mathrm{H}\mathbf{a}_\mathrm{d} = 1 \end{aligned} \end{equation}$$ 其闭式解可以由 Lagrange 乘子法得到 $$\begin{equation} \mathbf{w} = \alpha\mathbf{R}_{\mathrm{i+n}}^{-1}\mathbf{a}_\mathrm{d} \end{equation}$$ 其中 - 噪声加干扰协方差矩阵 $\mathbf{R}_{\mathrm{i+n}} = \displaystyle\frac{1}{T} \sum_{t=1}^{T}\mathbf{x}_\mathrm{i+n}(t)\mathbf{x}_\mathrm{i+n}^\mathrm{H}(t)$ - 归一化因子（使得复加权对于期望信号的增益为 1） $\alpha = \displaystyle\frac{1}{\mathbf{a}_\mathrm{d}^\mathrm{H}\mathbf{R}_{\mathrm{i+n}}^{-1}\mathbf{a}_\mathrm{d}}$ # MVDR 算法仿真 ## 理想空域谱 ![理想 Capon 谱](../images/post/2024-02-01-algo-MVDR/2024-02-01-algo-MVDR-010-IdealCaponSpectrum.svg){width=800px} {% note primary MATLAB Code %} ``` matlab mvdr_idea.m %% High-Resolution Frequency-Wavenumber Spectrum Analysis % J. Capon, “High-Resolution Frequency-Wavenumber Spectrum Analysis,” _Proceedings % of the IEEE_, vol. 57, no. 8, pp. 1408–1418, Aug. 1969, doi: . % % H. L. Van Trees, _Optimum Array Processing: Part IV of Detection, Estimation, % and Modulation Theory_. New York, NY: Wiley, 2002. clear close all %% Parameters rng(253) % Array Parameters % elements in array elemNum = 12; % spacing wrt wavelength elemSpac = 0.5; % element locations elemLoc = 0:elemSpac:(elemNum-1)*elemSpac; % Signal Parameters % signal number sigNum = 6; % direction of arrival sigAng = rand(1,sigNum)*180; % signal to noise ratio snr = [30,70]; % sample number sampleNum = 9000; % parameter match check if (length(snr) ~= sigNum) | (length(sigAng) ~= sigNum) disp('WARN: Signal number and parameter number NOT match.') end %% Impinging Waves/Signals % Transmitted Signals % wave generation, here are some sets of uncorrelated white noise txWave = randn(sigNum,sampleNum); % transmit power control based on given snr if length(snr) == 1 txWave = db2mag(snr)*txWave; elseif length(snr) >= sigNum if length(snr) > sigNum disp('WARN: Extra SNRs are ignored.') end for i = 1:sigNum txWave(i,:) = db2mag(snr(i))*txWave(i,:); end elseif length(snr) < sigNum disp('WARN: The last SNR is used for the rest of the signals.') for i = 1:sigNum if i <= length(snr) txWave(i,:) = db2mag(snr(i))*txWave(i,:); else txWave(i,:) = db2mag(snr(end))*txWave(i,:); end end end % Array Manifold Matrix % For each signal, the array manifold vector is % % $$\mathbf{a}\left(\theta \right)=\exp \left(-j\frac{2\pi \mathbf{d}}{\lambda % \;}\cos \theta \;\right)$$ % % in which the element locations are % % $$\mathbf{d}={\left\lbrack 0,\frac{\lambda }{2},\cdots ,\left(N-1\right)\frac{\lambda % }{2}\right\rbrack }^{\mathrm{T}}$$ if length(sigAng) > sigNum disp('WARN: Extra angles are ignored.') elseif length(sigAng) < sigNum disp('ERROR: Angle number is less than signal number.') return end arrayMF = exp(-1i*pi*elemLoc'*cos(sigAng(1:sigNum)/180*pi)); if sigNum > 1 % signal & interference manifold vector sigMF = arrayMF(:,1); intMF = arrayMF(:,2:end); % impinging signal & interference rxSig = sigMF*txWave(1,:); rxIpn = intMF*txWave(2:end,:) + randn(elemNum,sampleNum); else % signal manifold vector sigMF = arrayMF; % impinging signal rxSig = sigMF*txWave; rxIpn = randn(elemNum,sampleNum); end % Received Wave rxWave = rxSig + randn(elemNum,sampleNum); %% Array Processing % The MVDR problem is to minimize output interference-plus-noise power under % distortionless contraint of target signal % % $$\begin{array}{l}\underset{\mathbf{w}}{\mathrm{minimize}\;\;} \;{\mathbf{w}}^{\mathrm{H}} % {\mathbf{R}}_{\mathrm{i}+\mathrm{n}} \mathbf{w}\\\mathrm{subecto}\;\mathrm{to}\;\;{\mathbf{w}}^{\mathrm{H}} % \mathbf{a}=1\end{array}$$ % % The problem can be solve using the Lagrange multiplier method, and the solution % is % % $$\mathbf{w}=\alpha {\mathbf{R}}_{\mathrm{i}+\mathrm{n}}^{-1} \mathbf{a}$$ % % where $\alpha \;$is the normalization factor that ensures the power of array % input and output desired signal remains unchanged % % $$\alpha =\frac{1}{{\mathbf{a}}^{\mathrm{H}} {\mathbf{R}}_{\mathrm{i}+\mathrm{n}} % \mathbf{a}}$$ % interference-plus-noise covariance matrix ipnCovar = rxIpn*rxIpn'; % MVDR weight vector weight = ipnCovar\sigMF/(sigMF'*(ipnCovar\sigMF)); % = inv(ipnCovar)*sigMF/(sigMF'*inv(ipnCovar)*sigMF); % output SINR sinrOut = pow2db(var(weight'*rxSig)/var(weight'*rxIpn)); %% Spectrum & Figure % scan grid scanGrid = 0:0.1:180; % spectrum spec = zeros(size(scanGrid)); for i = 1:length(scanGrid) % steering vector scanMF = exp(-1i*pi*elemLoc'*cos(scanGrid(i)/180*pi)); % MVDR spectrum spec(i) = weight'*scanMF; end % normalized spectrum specNorm = pow2db(abs(spec)); % spectrum & signal location indicator plot(scanGrid,specNorm) hold on for i = 1:sigNum if i == 1 plot([sigAng(i),sigAng(i)],[max(specNorm),min(specNorm)],'-.r'); else plot([sigAng(i),sigAng(i)],[max(specNorm),min(specNorm)],'-.g'); end end hold off axis tight title('Ideal Capon Spectrum') xlabel('Degree/°') ylabel('Magnitude/dB') grid on grid minor ``` {% endnote %} ## 理想误码率 ## 失配输出 SINR ## 失配误码率 # Reference 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). 2. H. L. Van Trees, _Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory_. New York, NY: Wiley, 2002.