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a_nonuniform_design.m
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a_nonuniform_design.m
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clear all; close all; clc
%% we first consider a new modulation sequence obtained by shifting DFT's by 0.5 bin
% So with K bins, the period will be T = 2*K due to 0.5 bin shift.
% Let's check its Gamma matrix with a simple example
K = 3;
T = 2*K;
W = zeros(K, T);
for k=0:K-1
for t=0:T-1
W(k+1,t+1) = exp(-1j*2*pi*(k+0.5)*t/K);
end
end
W1 = zeros(T, K);
for t=0:T-1
for k=0:K-1
W1(t+1,k+1) = exp(1j*2*pi*(k+0.5)*t/K);
end
end
Gamma = W1*W
%% We will use these modulation sequence to design a nonsubsampled filterbank
fb = FilterBankStruct( );
fb.T = 64; % 4 ms with 16 KHz sample rate
fb.Gamma = [speye(fb.T/2), -speye(fb.T/2); -speye(fb.T/2), speye(fb.T/2)];
fb.B = 1; % no decimation
Lh = 88;
Lg = Lh;
fb.tau0 = fb.T; % 4 ms latency
eta = 1e4;
fb.w_cut = 0.9*pi/(fb.T/4);
lambda = 0;
best_cost = inf;
best_fb = fb;
for num_trial = 1 : 10
fb.h = rand(Lh,1); fb.g = rand(Lg,1);
[fb, cost, recon_err, iter] = FilterBankDesign(fb, eta, lambda, 100);
fprintf('Trial: %g; cost: %g; reconstruction error: %g; iterations %g\n', num_trial, cost, recon_err, iter)
if cost < best_cost
best_cost = cost;
best_fb = fb;
end
end
[fb, cost, recon_err, iter] = FilterBankDesign(best_fb, eta, lambda, 1000);
fprintf('Refinement. Cost: %g; reconstruction error: %g; iterations %g\n', cost, recon_err, iter)
figure
freqz(conv(fb.h, fb.g), 1, 32768)
%% these are the analysis-synthesis impulse responses of the K=T/2 filters
hs_combined = zeros(Lh + Lg - 1, fb.T/2);
for k = 0 : fb.T/2 - 1
h = fb.h;
g = fb.g;
for t = 0 : Lh - 1
h(t+1) = exp(1j*2*pi*(k+0.5)*(t+fb.i)/(fb.T/2)) * h(t+1); % eq. (1) in my paper
g(t+1) = exp(1j*2*pi*(k+0.5)*(t+fb.j)/(fb.T/2)) * g(t+1); % eq. (6) in my paper
end
hs_combined(:, k+1) = conv(h, g)/(fb.T/2);
end
%% phases of all bands are aligned so that we can merge any bands
figure;
subplot(7,2,1);
plot(real(hs_combined(:,1)), 'r');
hold on; plot(imag(hs_combined(:,1)), 'r--');
xlim('tight')
ylim('tight')
legend('real', 'imag')
ylabel('$h_1(t)$', 'Interpreter','latex')
title('Time domain')
subplot(7,2,3);
plot(real(hs_combined(:,2)), 'g');
hold on; plot(real(hs_combined(:,2)), 'g--');
xlim('tight')
ylim('tight')
legend('real', 'imag')
ylabel('$h_2(t)$', 'Interpreter','latex')
subplot(7,2,5);
plot(real(hs_combined(:,3) + hs_combined(:,4)), 'b');
hold on; plot(imag(hs_combined(:,3) + hs_combined(:,4)), 'b--');
xlim('tight')
ylim('tight')
legend('real', 'imag')
ylabel('$\sum_{k=3}^{4} h_k(t)$', 'Interpreter','latex', 'FontSize', 7)
subplot(7,2,7);
plot(real(hs_combined(:,5) + hs_combined(:,6) + hs_combined(:,7)), 'c');
hold on; plot(imag(hs_combined(:,5) + hs_combined(:,6) + hs_combined(:,7)), 'c--');
xlim('tight')
ylim('tight')
legend('real', 'imag')
ylabel('$\sum_{k=5}^{7} h_k(t)$', 'Interpreter','latex', 'FontSize', 7)
subplot(7,2,9);
plot(real(hs_combined(:,8) + hs_combined(:,9) + hs_combined(:,10) + hs_combined(:,11)), 'm');
hold on; plot(imag(hs_combined(:,8) + hs_combined(:,9) + hs_combined(:,10) + hs_combined(:,11)), 'm--');
xlim('tight')
ylim('tight')
legend('real', 'imag')
ylabel('$\sum_{k=8}^{11} h_k(t)$', 'Interpreter','latex', 'FontSize', 7)
subplot(7,2,11);
plot(real(hs_combined(:,12) + hs_combined(:,13) + hs_combined(:,14) + hs_combined(:,15) + hs_combined(:,16)), 'k')
hold on; plot(imag(hs_combined(:,12) + hs_combined(:,13) + hs_combined(:,14) + hs_combined(:,15) + hs_combined(:,16)), 'k--')
xlim('tight')
ylim('tight')
legend('real', 'imag')
ylabel('$\sum_{k=12}^{16} h_k(t)$', 'Interpreter','latex', 'FontSize', 7)
% NPR check
subplot(7,2,13);
stem(real(sum(hs_combined,2)), 'k.')
hold on; stem(imag(sum(hs_combined,2)), 'b.')
xlim('tight')
legend('real', 'imag')
ylabel('$\sum_{k=1}^{32} h_k(t)$', 'Interpreter','latex', 'FontSize', 7)
xlabel('Time')
subplot(1,2,2);
[H, F] = freqz(hs_combined(:,1), 1, 32768, 16000);
plot(F, 20*log10(abs(H)), 'r');
[H, F] = freqz(hs_combined(:,2), 1, 32768, 16000);
hold on; plot(F, 20*log10(abs(H)), 'g');
[H, F] = freqz(hs_combined(:,3) + hs_combined(:,4), 1, 32768, 16000);
hold on; plot(F, 20*log10(abs(H)), 'b');
[H, F] = freqz(hs_combined(:,5) + hs_combined(:,6) + hs_combined(:,7), 1, 32768, 16000);
hold on; plot(F, 20*log10(abs(H)), 'c');
[H, F] = freqz(hs_combined(:,8) + hs_combined(:,9) + hs_combined(:,10) + hs_combined(:,11), 1, 32768, 16000);
hold on; plot(F, 20*log10(abs(H)), 'm');
[H, F] = freqz(hs_combined(:,12) + hs_combined(:,13) + hs_combined(:,14) + hs_combined(:,15) + hs_combined(:,16), 1, 32768, 16000);
hold on; plot(F, 20*log10(abs(H)), 'k');
xlabel('Frequency (Hz)', 'Interpreter','latex')
ylabel('Magnitude (dB)', 'Interpreter','latex')
legend('1st band', '2nd band', '3rd band', '4th band', '5th band', '6th band', 'Interpreter','latex')
xlim('tight')
ylim([-80, 1])
title('Frequency domain')