信息处理 - 如何在 MATLAB 中执行这种谱分解？ - 吾爱随笔录

给定一个过滤器 $X(z)$ 我想找到 $G(z)$ 使得它是稳定的、因果的和最小的相位，并且它实现了

X (z) = K_{0} G (z) G^{*} (1 / z^{*})

$X(z)=K_0G(z)G^*(1/z^*)$

在哪里 $K_0\in\mathbb{R}$ . 当然， $G^*(1/z^*)$ 然后将是反因果和最大相位。过滤器 $x[n]$ 有复系数。

我试图在 MATLAB 中实现这一点，但我发现它比我想象的要困难。

% This is my filter x[n]. You can try with any coefficients, it doesn't matter
x = dfilt.dffir(q_k + 1/(10^(SNR_MFB/10)));
% Here I find its zeros
zeros_x = zpk(x);
% And now I identify those who are inside and outside the unit circle
zeros_min = zeros_x(abs(zeros_x) < 1);
zeros_max = zeros_x(abs(zeros_x) > 1);
% Now I wanted to build the filter g[n] using this information but don't know how

我尝试使用spectralfact()，但它不支持具有复杂系数的输入。

有谁知道如何获得系数 $g[n]$ ?

N = 10; % signal length is 2*N-1 x = randn(N,1) + 1i*randn(N,1); x = conv(x,conj(x(end:-1:1))); % x has conjugate symmetry and its spectrum is non-negative r = roots(x); I = find( abs(r) < 1 ); ri = r(I); g = poly(ri); K = real(x(N))/real(g(:)'*g(:)); x2 = K * conv(g,conj(g(end:-1:1))); % same as x up to numerical errors