# Filter design by convex iteration

(Difference between revisions)
 Revision as of 00:58, 26 August 2010 (edit) (→demonstration in Matlab)← Previous diff Revision as of 03:30, 17 October 2010 (edit) (undo) (→demonstration in Matlab)Next diff → Line 55: Line 55: ==demonstration in Matlab== ==demonstration in Matlab==

-                                                                                      % CVX by Almir Mutapcic in 2006.                                               +                                                                                          % CVX code by Almir Mutapcic in 2006.
% Adapted in 2010 for impulse response peak-minimization by convex iteration.                                                                                             % Adapted in 2010 for impulse response peak-minimization by convex iteration.
%                                                                                                                                                                         %

## Revision as of 03:30, 17 October 2010 $LaTeX: H(\omega) = h(0) + h(1)e^{-j\omega} + \cdots + h(N-1)e^{-j(N-1)\omega}$  where $LaTeX: h(n)\in\mathbb{C}^N$ denotes impulse response.

For a low pass filter, frequency domain specifications are: $LaTeX: \begin{array}{ll} \frac{1}{\delta_1}\leq|H(\omega)|\leq\delta_1\,, &\omega\in[0,\omega_p]\\ |H(\omega)|\leq\delta_2\,, &\omega\in[\omega_s\,,\pi] \end{array}$

To minimize peak magnitude of $LaTeX: h\,$ , the problem becomes $LaTeX: \begin{array}{cll} \textrm{minimize} &\|h\|_\infty\\ \textrm{subject~to} &\frac{1}{\delta_1}\leq|H(\omega)|\leq\delta_1\,, &\omega\in[0,\omega_p]\\ &|H(\omega)|\leq\delta_2\,, &\omega\in[\omega_s\,,\pi] \end{array}$

But this problem statement is nonconvex. $LaTeX: G\triangleq h(n)h(n)^{\rm H} \in\mathbb{C}^{N\times N}$

is a positive semidefinite rank 1 matrix.

Summing along each of $LaTeX: 2N-1\,$ subdiagonals gives entries of the autocorrelation function $LaTeX: r\,$ of $LaTeX: h\,$ .

In particular, the main diagonal of $LaTeX: G\,$ holds squared entries of $LaTeX: h\,$ .

Minimizing $LaTeX: \|h\|_\infty$ is therefore equivalent to minimizing $LaTeX: \|\textrm{diag}(G)\|_\infty$ .

Define $LaTeX: I_0\triangleq I\,$ and define $LaTeX: I_n\,$ as a zero matrix having vector $LaTeX: \mathbf{1}$ along the $LaTeX: n^{\rm{th}}\,$ superdiagonal when $LaTeX: n\,$ is positive or $LaTeX: \mathbf{1}$ along the $LaTeX: n^{\rm{th}}\,$ subdiagonal when $LaTeX: n\,$ is negative.

By spectral factorization, $LaTeX: R(\omega)=|H(\omega)|^2\,$ , an equivalent problem is: $LaTeX: \begin{array}{cll} \textrm{minimize}_{G\,,\,r}&\|\textrm{diag}(G)\|_\infty\\ \textrm{subject~to} & R(\omega) = r(0)+\!\sum\limits_{n=1}^{N-1}2r(n)\cos(\omega n)\\ &\frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2 &\omega\in[0,\omega_p]\\ & R(\omega)\leq\delta_2^2 & \omega\in[\omega_s\,,\pi]\\ & R(\omega)\geq0 & \omega\in[0,\pi]\\ & r(n)=\textrm{trace}(I_n^{\rm T}G) &n=0\ldots N-1\\ & G\succeq0\\ & \textrm{rank}(G) = 1 \end{array}$

Excepting the rank constraint, this problem is convex.

## demonstration in Matlab

% CVX code by Almir Mutapcic in 2006.
% Adapted in 2010 for impulse response peak-minimization by convex iteration.
%
% S.-P. Wu, S. Boyd, and L. Vandenberghe,
% "FIR Filter Design via Spectral Factorization and Convex Optimization":
%
%   minimize   max |H(w)|                      for w in stopband
%       s.t.   1/delta <= |H(w)| <= delta      for w in passband
%
% Change variables via spectral factorization method and get:
%
%   minimize   max R(w)                        for w in stopband
%       s.t.   (1/delta)^2 <= R(w) <= delta^2  for w in passband
%              R(w) >= 0                       for all w
%
% where R(w) is squared magnitude of frequency response
% (and Fourier transform of autocorrelation coefficients r).
% Variables are coeffients r and G := hh' where h is impulse response.
% delta is allowed passband ripple.
% delta2 is specified stopband attenuation.

clear all, clc, close all, fclose('all');
rand('twister',sum(100*clock));
randn('state',sum(100*clock));
%*********************************************************************
% filter specs (for a low-pass filter)
%*********************************************************************
% number of FIR coefficients (including zeroth)
N = 32;

wpass = 0.12*pi;   % end of the passband
wstop = 0.20*pi;   % start of the stopband
delta = 1;         % maximum passband ripple in dB (+/- around 0 dB)
delta2 = -20;      % stopband attenuation desired in dB

%*********************************************************************
% optimization parameters
%*********************************************************************
% rule-of-thumb discretization (from Cheney's Approximation Theory)
m = 15*N;
w = linspace(0,pi,m)';  % omega

% A is the matrix used to compute the power spectrum
% A(w,:) = [1 2*cos(w) 2*cos(2*w) ... 2*cos(N*w)]
A = [ones(m,1) 2*cos(kron(w,[1:N-1]))];

% passband 0 <= w <= w_pass
ind = find((0 <= w) & (w <= wpass));    % passband
Lp  = 10^(-delta/20)*ones(length(ind),1);
Up  = 10^(+delta/20)*ones(length(ind),1);
Ap  = A(ind,:);

% stopband (w_stop <= w)
ind = find((wstop <= w) & (w <= pi));   % stopband
Sp  = 10^(delta2/20)*ones(length(ind),1);
As  = A(ind,:);

% make I matrices
B = zeros(N,N^2);
for i=0:N-1
C = zeros(N,N);
C = spdiags(ones(N,1),i,C);
B(i+1,:) = vect(C)';
end

%initial direction vector
W = eye(N);

% peak impulse response
peak = 0.125;

%********************************************************************
% optimization
%********************************************************************
% formulate and solve the magnitude design problem
iteration = 1;
while 1
tic
fprintf('\nMinimizing impulse response peak: iteration %d\n', iteration);
cvx_quiet(true);
%   cvx_solver('sdpt3');
cvx_begin
variable r(N,1);
variable G(N,N) symmetric;
minimize(trace(W'*G));
% passband constraints
Ap*r >= Lp.^2;
Ap*r <= Up.^2;
%stopband constraint
As*r <= Sp.^2;
% nonnegative-real constraint
A*r >= 0;
% relate r to h
r == B*vect(G);
G == semidefinite(N);
%minimize peak of h
diag(G) <= peak^2;
cvx_end
toc

%computer new direction vector
[v,d,q] = svd(G);
f = diag(d); fprintf('first few eigenvalues of G:\n%f\n%f\n%f\n%f\n%f\n%f\n%f\n', f(1:7));
W = v(:,2:N)*v(:,2:N)';
%   W = v(:,2:N)*diag(rand(N-1,1))*v(:,2:N)';
rankG = sum(diag(d) > max(diag(d))*1e-5);
fprintf('rank(G)=%d,  trace(W*G)=%f\n', rankG, trace(G*W));
if (rankG == 1)
break
end

figure(1)
% FIR impulse response
h = G(:,1)/sqrt(G(1,1));
plot([0:N-1],h','o',[0:N-1],h','b:')
xlabel('t'), ylabel('h(t)')
pause(1)

% check if problem was successfully solved
disp(['Problem is ' cvx_status])
if ~strfind(cvx_status,'Solved')
return
end

iteration = iteration + 1;
end

% compute the min attenuation in the stopband (convert to original vars)
Ustop = delta2;
fprintf(1,'Min attenuation in the stopband is %3.2f dB.\n',Ustop);

%*********************************************************************
% plotting routines
%*********************************************************************
% frequency response of the designed filter, where j = sqrt(-1)
h = G(:,1)/sqrt(G(1,1));
H = [exp(-j*kron(w,[0:N-1]))]*h;

figure(2)
% magnitude
plot(w,20*log10(abs(H)), ...
[0 wpass],[delta delta],'r--', ...
[0 wpass],[-delta -delta],'r--', ...
[wstop pi],[Ustop Ustop],'r--')
xlabel('w')
ylabel('mag H(w) in dB')
axis([0 pi -50 5])

%compare impulse response designed by conventional method
figure(3)
h = spectral_fact(r);  %from CVX distribution, Examples subdirectory
plot([0:N-1],h','o',[0:N-1],h','b:')
xlabel('t'), ylabel('h(t)')

figure(1)
% FIR impulse response
h = G(:,1)/sqrt(G(1,1));
plot([0:N-1],h','o',[0:N-1],h','b:')
xlabel('t'), ylabel('h(t)')