Inverse DTFT using the DFT Find the approximate inverse DTFT of [latex]X(F) = [rect(50(F − 1/4)) + rect(50(F + 1/4))] ∗ δ_1(F)[/latex] using the DFT.

Question 7.8: Inverse DTFT using the DFT Find the approximate inverse DTFT......

Signals and Systems Analysis Using Transform Methods and MATLAB [1050783]

Inverse DTFT using the DFT Find the approximate inverse

DTFT of

X(F) = [rect(50(F - 1/4)) + rect(50(F + 1/4))] * δ_1(F)

using the DFT.

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Script File

N = 512 ; % Number of pts to approximate X(F)
k = [0:N-1]’ ; % Harmonic numbers
% Compute samples from X(F) between 0 and 1 assuming
% periodic repetition with period 1
X = rect(50*(k/N - 1/4)) + rect(50*(k/N - 3/4)) ;
% Compute the approximate inverse DTFT and
% center the function on n = 0
xa = real(fftshift(ifft(X))) ;
n = [-N/2:N/2-1]’ ; % Vector of discrete times for plotting

% Compute exact x[n] from exact inverse DTFT
xe = sinc(n/50).*cos(pi*n/2)/25 ;
% Graph the exact inverse DTFT
subplot(2,1,1) ; p = stem(n,xe,’k’,’fi lled’) ; set(p,’LineWidth’,1,
’MarkerSize’,2) ;
axis([-N/2,N/2,-0.05,0.05]) ; grid on ;
xlabel(‘\itn’,’FontName’,’Times’,’FontSize’,18) ;
ylabel(‘x[{\itn}]’,’FontName’,’Times’,’FontSize’,18) ;
title(‘Exact’,’FontName’,’Times’,’FontSize’,24) ;
% Graph the approximate inverse DTFT
subplot(2,1,2) ; p = stem(n,xa,’k’,’fi lled’) ; set(p,’LineWidth’,1,
’MarkerSize’,2) ;
axis([-N/2,N/2,-0.05,0.05]) ; grid on ;
xlabel(‘\itn’,’FontName’,’Times’,’FontSize’,18) ;
ylabel(‘x[{\itn}]’,’FontName’,’Times’,’FontSize’,18) ;
title(‘Approximation Using the DFT’,’FontName’,’Times’,’FontSize’,
24) ;

The exact and approximate inverse DTFT results are illustrated in Figure 7.19. Notice that the exact and approximate x[n] are practically the same near n = 0 but are noticeably different near n = ±256. This occurs because the approximate result is periodic and the overlap of the periodically repeated sinc functions causes these errors near plus or minus half a period.
Example 7.9 illustrates a common analysis problem and a different kind of solution.