ECE 2713 Homework Matlab code:

Transcription

1 ECE 7 Homework 7 Spring 8 Dr. Havlicek. Matlab code: P - Create and plot the signal x_[n] as a function of n. - Compute the DFT X_[k]. Plot the magnitude and phase as functions of k. - Plot the DFT magnitude as a function of the matlab array index. - Plot the DFT magnitude as a function of the discrete radian frequency w. - Compute and plot the IDFT. n = :7; time variable xn = [ ]; our 8-point signal Xk = fft(xn); compute the DFT Xkmag = abs(xk); magnitude of the DFT Xkarg = angle(xk); phase of the DFT plot the signal figure(); stem(n,xn); axis([ 7.]); title( Original Signal ); xlabel( n ); ylabel( x_[n] ); plot DFT magnitude and phase as functions of k k = :7; frequency index figure(); stem(k,xkmag); ylim([ 6]); title( DFT Magnitude ); xlabel( k ); ylabel( X_[k] ); figure(); stem(k,xkarg); title( DFT Phase ); xlabel( k ); ylabel( arg(x_[k]) );

2 plot DFT magnitude as a function of Matlab index Matlab_idx = [:8]; Matlab index figure(); stem(matlab_idx,xkmag); ylim([ 6]); title( DFT Magnitude ); xlabel( Matlab index ); ylabel( X_[index] ); plot DFT magnitude as a function of discrete frequency (radians per sample) w = [:*pi/8:7**pi/8]; discrete frequency figure(); stem(w,xkmag); ylim([ 6]); title( DFT Magnitude ); ylim([ 6]); xlabel( discrete radian frequency \omega ); ylabel( X_[\omega] ); Compute and plot the IDFT xn = ifft(xk); figure(6); stem(n,xn); axis([ 7.]); title( IDFT ); xlabel( n ); ylabel( IDFT );. Original Signal x [n]. 6 7 n

3 6 DFT Magnitude X [k] 6 7 k DFT Phase arg(x [k]) 6 7 k

4 6 DFT Magnitude X [index] Matlab index 6 DFT Magnitude X [ω] 6 discrete radian frequency ω

5 . IDFT IDFT. 6 7 n. Matlab code: P - Compute the centered DFT of x_[n]. - Plot the centered magnitude and phase as function of - the discrete radian frequency w, - the discrete hertzian frequency f. - Invert the DFT and plot. n = :7; time variable xn = [ ]; our 8-point signal Xk = fftshift(fft(xn)); compute the centered DFT Xkmag = abs(xk); magnitude of the centered DFT Xkarg = angle(xk); phase of the centered DFT plot centered DFT magnitude & phase as functions of radian freq w = [-**pi/8:*pi/8:**pi/8]; radian discrete freq figure(); stem(w,xkmag); ylim([ 6]); title( Centered DFT Magnitude ); xlabel( discrete radian frequency \omega ); ylabel( X_[\omega] ); figure(); stem(w,xkarg);

6 title( Centered DFT Phase ); xlabel( discrete radian frequency \omega ); ylabel( arg(x_[\omega]) ); plot centered DFT magnitude & phase as functions of Hertzian freq f = [-.:/8:/8]; Hertzian discrete freq figure(); stem(f,xkmag); ylim([ 6]); title( Centered DFT Magnitude ); xlabel( discrete Hertzian frequency f ); ylabel( X_[f] ); figure(); stem(f,xkarg); title( Centered DFT Phase ); xlabel( discrete Hertzian frequency f ); ylabel( arg(x_[f]) ); Compute and plot the IDFT xn = ifft(ifftshift(xk)); figure(); stem(n,xn); axis([ 7.]); title( IDFT ); xlabel( n ); ylabel( IDFT ); 6 Centered DFT Magnitude X [ω] discrete radian frequency ω 6

7 Centered DFT Phase arg(x [ω]) discrete radian frequency ω 6 Centered DFT Magnitude X [f] discrete Hertzian frequency f 7

8 Centered DFT Phase arg(x [f]) discrete Hertzian frequency f. IDFT IDFT. 6 7 n 8

9 . Matlab code: P Plot the centered DFT magnitude of x_[n] using normalized radian frequency. xn = [ ]; our 8-point signal Xk = fftshift(fft(xn)); compute the centered DFT Xkmag = abs(xk); magnitude of the centered DFT Xkarg = angle(xk); phase of the centered DFT w = [-**pi/8:*pi/8:**pi/8]; figure(); stem(w/pi,xkmag); axis([- 6]); title( Centered DFT Magnitude ); xlabel( \omega/\pi ); ylabel( X_[\omega/\pi] ); radian discrete freq 6 Centered DFT Magnitude X [ω/π] ω/π 9

10 . (a) Matlab code: Pa Show that the DFT is given by samples of the DTFT. - plot the DTFT magnitude of xhat from -pi to pi. - plot the centered DFT magnitude of x_[n] on the same graph. -plot the DTFT phase of xhat from -pi to pi. - plot the centered DFT phase of x_[n] on the same graph. Frequency vector for plotting the DTFT. Use points. w = linspace(-pi,pi,); The DTFT was computed analytically Xhat = sin(.*w)./sin(w/).* exp(-*j*w); Xhatmag = abs(xhat); Xhatarg = angle(xhat); Now compute the 8-point DFT xn = [ ]; our 8-point signal k = -:; frequency index for the centered DFT Xk = fftshift(fft(xn)); Xkmag = abs(xk); Xkarg = angle(xk); figure(); plot(w,xhatmag, -b ); plot the DTFT magnitude axis([-pi pi 6]); hold on; makes the next plot come out on the same graph plot(k**pi/8,xkmag, ro ); plot the centered DFT magnitude hold off; using a symbol, but no line and no stem. title( Magnitude of DTFT and centered 8-pt DFT ); xlabel( \omega, FontSize,); ylabel( $ \widehat X_(e^{j\omega}) $, $ X_[\omega] $,... Interpreter, latex, FontSize,); legend( DTFT, DFT ); figure(); plot(w,xhatarg, -b ); plot the DTFT phase axis([-pi pi - ]); hold on; plot(k**pi/8,xkarg, ro ); plot the centered DFT phase

11 hold off; title( Phase of DTFT and centered 8-pt DFT ); xlabel( \omega, FontSize,); ylabel( $\arg\widehat X_(e^{j\omega})$, $\arg X_[\omega]$,... Interpreter, latex, FontSize,); legend( DTFT, DFT );

12 6 Magnitude of DTFT and centered 8 pt DFT DTFT DFT X(e jω ), X[ω] ω Phase of DTFT and centered 8 pt DFT DTFT DFT arg X(e jω ), argx[ω] ω

13 (b) Matlab code: Pb Change the length of x_[n] to N=6. - plot the DTFT magnitude of xhat from -pi to pi. - plot the centered DFT magnitude of x_[n] on the same graph. -plot the DTFT phase of xhat from -pi to pi. - plot the centered DFT phase of x_[n] on the same graph. Frequency vector for plotting the DTFT. Use points. w = linspace(-pi,pi,); The DTFT was computed analytically Xhat = sin(.*w)./sin(w/).* exp(-*j*w); Xhatmag = abs(xhat); Xhatarg = angle(xhat); Now compute the 6-point DFT xn = [ ]; 6-point signal k = -8:7; frequency index for the centered DFT Xk = fftshift(fft(xn)); Xkmag = abs(xk); Xkarg = angle(xk); figure(); plot(w,xhatmag, -b ); plot the DTFT magnitude axis([-pi pi 6]); hold on; makes the next plot come out on the same graph plot(k**pi/6,xkmag, ro ); plot the centered DFT magnitude hold off; using a symbol, but no line and no stem. title( Magnitude of DTFT and centered 6-pt DFT ); xlabel( \omega, FontSize,); ylabel( $ \widehat X_(e^{j\omega}) $, $ X_[\omega] $,... Interpreter, latex, FontSize,); legend( DTFT, DFT ); figure(); plot(w,xhatarg, -b ); axis([-pi pi - ]); hold on; plot the DTFT phase

14 plot(k**pi/6,xkarg, ro ); plot the centered DFT phase hold off; title( Phase of DTFT and centered 6-pt DFT ); xlabel( \omega, FontSize,); ylabel( $\arg\widehat X_(e^{j\omega})$, $\arg X_[\omega]$,... Interpreter, latex, FontSize,); legend( DTFT, DFT ); 6 Magnitude of DTFT and centered 6 pt DFT DTFT DFT X(e jω ), X[ω] ω

15 Phase of DTFT and centered 6 pt DFT DTFT DFT arg X(e jω ), argx[ω] ω