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Highlights from
MatLab Solutions: "Introduction to Digital Signal Processing: A Computer Laboratory Textbook".

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MatLab Solutions: "Introduction to Digital Signal Processing: A Computer Laboratory Textbook".

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29 Oct 2012 (Updated )

These files are the MatLab solutions of exercises contained in the above DSP lab textbook.

ex644.m
% Exercise 6.4.4. Overlap and Save Method.

clc; clear; close all;

%% a.  Generate the triangular sequence x[n]:
Period = 65;
n = 0:(Period-1)/2;
A = 1;

half_block = 2*A*n/(Period-1); 
second_half = fliplr(half_block(2:end));
block = [half_block second_half];
x = [block block block block block block block block block block block block block block block];
N = 975;   % 
x = x(1:N); % Truncate the sequence to create the desired length.
% x = randn(1,N);

% Create a low-pass filter with cut-off frequency pi/2 first and then convert it to it's high-pass conjugate.
M = 63;
h_low = [ sin(pi/2.*(-31:-1))./(pi.*(-31:-1)) 1/2 sin(pi/2.*(1:31))./(pi.*(1:31)) ];
h_high = firlp2hp(h_low);
%a_high = 1;
%NSamples = 1024;
%w = 0:pi/NSamples:pi;

% Plot its frequency response.
% figure(1);
% H = freqz(h_high,a_high,w);
% plot(w,20*log10(abs(H)));
% set(gca,'XTick',0:pi/6:pi);
% set(gca,'XTickLabel',{'0','pi/6','pi/3','pi/2','2pi/3','5pi/6','pi' })
% xlim([0 pi]);
% ylim([-80 5]);
% ylabel('(dB)');
% grid on;
% title('20log(|{\itH}(j\omega)|)');
% It is a high-pass filter.

y = conv(h_high,x);
figure('Name',' Exercise 6.4.4. Overlap and Save Method');
subplot(2,1,1);
stem(0:N-1,x);
title('x[n]');
grid on;
axis tight;

subplot(2,1,2);
stem(0:N+M-2,y,'r.');
title('High-Pass Filtered Output y[n]');
grid on;
axis tight;

%% b. Evaluate the convolution using the macro from Ex. 6.4.1.
N1 = 1037;
x1 = [x           zeros(1,N1-size(x,2))         ]; % Both sequences are of size N1.
h   = [h_high zeros(1,N1-size(h_high,2))];
y1 = fastconv(h,x1);

figure('Name',' Exercise 6.4.4. Overlap and Save Method');
stem(0:N1-1,y1,'b');
hold on;
grid on;
axis tight;

%% c. Overlap and Save (finally).
L1 = 115; % This is the selected block size for the long sequence x[n].
L2 = 63;  % This is the size of the short sequence h_high.

y2 = overlap_save(x,h_high,L1);
 
%                y2 = zeros(1,N+M-1); 
% y_previous = zeros(1,L1);
%    y_current = zeros(1,L1);
%        
%    x = [x zeros(1,25)];  % To create a version of x[n] with length(x)/L1 = integer. 
%    h_block = h_high;
%     
% for k=0:10
%     
%      if k ==0  % First x_block processing. We introduce some zeros in the x_block as necessary.
%          x_block = [zeros(1,L2) x(1:L1)];  
%          y_current = cconv(x_block,h_block,L1+L2);
%          y2(1:L1) = y_current(L2+1:end);
%      elseif k<10
%           x_block = x(k*L1- L2+1:(k+1)*L1);                % Successive x_blocks are of size L1+L2 overlapping by L2 samples.
%           y_current = cconv(x_block,h_block,L1+L2); % Compute the circular convolution of size L1 + L2.
%           y2(k*L1:(k+1)*L1)  = y_current(L2:end) ;      % Keep only the correct portion of y_current.
%       elseif k == 10  % Last x_block processing...
%          x1 = x(k*L1-L2+1:end);
%          x_block = [x1   zeros(1,L1+L2- size(x1,2))];  
%          y_current = cconv(x_block,h_block,L1+L2);   
%          y2(k*L1:(k+1)*L1-L2) = y_current(L2:end-L2) ;
%       end
%        
% end

stem(0:length(y2)-1,y2,'g.');
title([num2str(L1+L2),'-point block, Overlap and Save Convolution (dots) vs built-in Convolution (circles)']);
grid on;
axis tight;

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