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Properties of discrete-time Fourier transform

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

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Properties of discrete-time Fourier transform

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  1. Properties of discrete-time Fourier transform (1) Linearity

  2. (2) Time shifting (3) Time reversal

  3. (4) Time scaling if n is a multiple of k otherwise

  4. (4) Time scaling

  5. 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; end end od = 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); end end omegap = -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;

  6. (4) Time scaling r is an integer

  7. (4) Time scaling

  8. (4) Time scaling

  9. clear; clf; N = 10; x(1:N) = 1; r = 3; Gn = floor(N/r); for k = 1:Gn g(k) = x(r*k); end od = 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); end end odp = 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;

  10. (5) Conjugation and conjugate summary Real

  11. Differentiation in Frequency

  12. Differentiation in Frequency

  13. Differentiation in Frequency

  14. Parseval’s Relation

  15. The convolution property Example #1

  16. Example #2 When

  17. Example #2 When

  18. Example #2 When

  19. The multiplication property

  20. The multiplication property, discrete-time The multiplication property, continuous-time

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