% Minimize stopband ripple of a linear phase lowpass FIR filter % "Filter design" lecture notes (EE364) by S. Boyd % (figures are generated) % % Designs a linear phase FIR lowpass filter such that it: % - minimizes the maximum passband ripple % - has a constraint on the maximum stopband attenuation % % This is a convex problem. % % minimize delta % s.t. 1/delta <= H(w) <= delta for w in the passband % |H(w)| <= atten_level for w in the stopband % % where H is the frequency response function and variables are % delta and h (the filter impulse response). % % Written for CVX by Almir Mutapcic 02/02/06 %******************************************************************** % user's filter specifications %******************************************************************** % filter order is 2n+1 (symmetric around the half-point) n = 10; wpass = 0.12*pi; % passband cutoff freq (in radians) wstop = 0.24*pi; % stopband start freq (in radians) atten_level = -30; % stopband attenuation level in dB %******************************************************************** % create optimization parameters %******************************************************************** N = 30*n+1; % freq samples (rule-of-thumb) w = linspace(0,pi,N); A = [ones(N,1) 2*cos(kron(w',[1:n]))]; % matrix of cosines % passband 0 <= w <= w_pass ind = find((0 <= w) & (w <= wpass)); % passband Ap = A(ind,:); % transition band is not constrained (w_pass <= w <= w_stop) % stopband (w_stop <= w) ind = find((wstop <= w) & (w <= pi)); % stopband Us = 10^(atten_level/20)*ones(length(ind),1); As = A(ind,:); %******************************************************************** % optimization %******************************************************************** % formulate and solve the linear-phase lowpass filter design cvx_begin variable delta variable h(n+1,1); minimize( delta ) subject to % passband bounds Ap*h <= delta; inv_pos(Ap*h) <= delta; % stopband bounds abs( As*h ) <= Us; cvx_end % check if problem was successfully solved disp(['Problem is ' cvx_status]) if ~strfind(cvx_status,'Solved') return else % construct the full impulse response h = [flipud(h(2:end)); h]; fprintf(1,'The optimal minimum passband ripple is %4.3f dB.\n\n',... 20*log10(delta)); end %******************************************************************** % plots %******************************************************************** figure(1) % FIR impulse response plot([0:2*n],h','o',[0:2*n],h','b:') xlabel('t'), ylabel('h(t)') figure(2) % frequency response H = exp(-j*kron(w',[0:2*n]))*h; % magnitude subplot(2,1,1) plot(w,20*log10(abs(H)),[wstop pi],[atten_level atten_level],'r--'); axis([0,pi,-40,10]) xlabel('w'), ylabel('mag H(w) in dB') % phase subplot(2,1,2) plot(w,angle(H)) axis([0,pi,-pi,pi]) xlabel('w'), ylabel('phase H(w)')