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

{width=800px}

{% note primary MATLAB Code %}

%% 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: <https://doi.org/10.1109/PROC.1969.7278
% 10.1109/PROC.1969.7278>.
%
% 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

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.
H. L. Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. New York, NY: Wiley, 2002.

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