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Properties of discrete-time Fourier transform. (1) Linearity. (2) Time shifting. (3) Time reversal. (4) Time scaling. if n is a multiple of k. otherwise. (4) Time scaling. clear; clf ; N = 5; x(1:N) = 1; r = 3; Gn = floor(N/r); for k = 1:N*r gn = floor(k/r); if k/r == gn - PowerPoint PPT Presentation
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16.362 Signal and System I • Properties of discrete-time Fourier transform
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16.362 Signal and System I
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clear;clf;N = 5;x(1:N) = 1;r = 3;Gn = floor(N/r);for k = 1:N*r gn = floor(k/r); if k/r == gn g(k) = x(gn); else g(k) = 0; endendod = 0.001*2*pi;omega = -2*2*pi*r:od*r:2*2*pi*r;for m = 1:length(omega) X(m) = 0; G(m) = 0; for n=1:length(x) X(m) = X(m)+x(n)*exp(-j*omega(m)*n); end for nn=1:length(g) G(m) = G(m)+g(nn)*exp(-j*omega(m)*nn); endendomegap = -2*2*pi:od:2*2*pi;for m = 1:length(omegap) Gp(m) = X(m);end
plot(omega/(pi), abs(X));zoom on;hold on;plot(omega/(pi), abs(G),'r');hold off;figure(2)stem(x);hold on;stem(g, 'r');hold off;zoom on;figure(3)plot(omega/(pi), abs(X));zoom on;hold on;plot(omegap/(pi), abs(Gp),'g');hold off;
16.362 Signal and System I
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16.362 Signal and System I clear;clf;N = 10;x(1:N) = 1;r = 3;Gn = floor(N/r);for k = 1:Gn g(k) = x(r*k);endod = 0.001*2*pi;omega = -2*2*pi:od:2*2*pi;for m = 1:length(omega) X(m) = 0; G(m) = 0; for n=1:length(x) X(m) = X(m)+x(n)*exp(-j*omega(m)*n); end for nn=1:length(g) G(m) = G(m)+g(nn)*exp(-j*omega(m)*nn); endendodp = od*2;omegap = -2*2*pi:odp:2*2*pi;for m = 1:length(omegap) Gp(m) = 0; for k = 0:r-1 sm =floor((omegap(m)/r+(2*pi/r)*k+2*2*pi)/od+1); Gp(m) = Gp(m)+X(sm); end Gp(m) = Gp(m)/r;end
plot(omega/(pi), abs(X));zoom on;hold on;plot(omega/(pi), abs(G),'r');hold off;figure(2)stem(x);hold on;stem(g, 'r');hold off;zoom on;figure(3)plot(omega/(pi), abs(X));zoom on;hold on;plot(omegap/(pi), abs(Gp),'g');hold off;
16.362 Signal and System I
(5) Conjugation and conjugate summary
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16.362 Signal and System I Differentiation in Frequency
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16.362 Signal and System I Parseval’s Relation
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16.362 Signal and System I The convolution property
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