实验三离散时间信号的频域分析第一部分.doc

Name: Yifan Chen 20112121006
Section:
Laboratory Exercise 3
DISCRETE-TIME SIGNALS: FREQUENCY-DOMAIN REPRESENTATIONS
3.1 DISCRETE-TIME FOURIER TRANSFORM
Project 3.1 DTFT Computation
A copy of Program P3_1 is given below:
< Insert program code here. Copy from m-file(s) and paste. >
% Program P3_1
% Evaluation of the DTFT
clf;
% Compute the frequency samples of the DTFT
w = -4*pi:8*pi/511:4*pi;
num = [2 1];den = [1 -0.6];
h = freqz(num, den, w);
% Plot the DTFT
subplot(2,1,1)
plot(w/pi,real(h));grid
title('Real part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
plot(w/pi,imag(h));grid
title('Imaginary part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
pause
subplot(2,1,1)
plot(w/pi,abs(h));grid
title('Magnitude Spectrum |H(e^{j\omega})|')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
plot(w/pi,angle(h));grid
title('Phase Spectrum arg[H(e^{j\omega})]')
xlabel('\omega /\pi');
Answers:
Q3.1 The expression of the DTFT being evaluated in Program P3_1 is -
The function of the pause command is - causes M-files to stop and wait for you to press any key before continuing.

Q3.2 The plots generated by running Program P3_1 are shown below:
< Insert MATLAB figure(s) here. Copy from figure window(s) and paste. >

The DTFT is a period function of w.
Its period is - 2 pi
The types of symmetries exhibited by the four plots are as follows:
Real part of H(e^{j\omega}) is eve symmetries;
Imaginary part of H(e^{j\omega}) is ode symmetries;
Magnitude Spectrum |H(e^{j\omega})| is eve symmetries;
Phase Spectrum arg[H(e^{j\omega})] is ode symmetries;

Q3.3 The required modifications to Program P3_1 to evaluate the given DTFT of Q3.3 are given below:
< Insert program code here. Copy from m-file(s) and paste. >

% Program Q3_3
% Evaluation of the DTFT
clf;
% Compute the frequency samples of the DTFT
w = 0:pi/511:pi;
num = [0.7 -0.5 0.3 1];den = [1 0.3 -0.5 0.7];
h = freqz(num, den, w);
% Plot the DTFT
subplot(2,1,1)
plot(w/pi,real(h));grid
title('Real part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
plot(w/pi,imag(h));grid
title('Imaginary part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
pause
subplot(2,1,1)
plot(w/pi,abs(h));grid
title('Magnitude Spectrum |H(e^{j\omega})|')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
plot(w/pi,angle(h));grid
title('Phase Spectrum arg[H(e^{j\omega})]')
xlabel('\omega /\pi');
ylabel('Phase, radians');

The plots generated by running the modified Program P3_1 are shown below:
< Insert MATLAB figure(s) here. Copy from figure window(s) and paste. >

The DTFT is a perio function of w.
Its period is – 2pi
The jump in the phase spectrum is caused by - the real parts saltus step,the phase saltus set up.
The phase spectrum evaluated with the jump removed by the command unwrap is as given below:
< Insert MATLAB figure(s) here. Copy from figure window(s) and paste. >

% Program Q3_3
% Evaluation of the DTFT
clf;
% Compute the frequency samples of the DTFT
w = 0:pi/511:pi;
num = [0.7 -0.5 0.3 1];den = [1 0.3 -0.5 0.7];
h = freqz(num, den, w);
subplot(2,1,1)
plot(w/pi,abs(h));grid
title('Magnitude Spectrum |H(e^{j\omega})|')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
p = angle(h(:));
plot(w/pi,unwrap(p));grid
title('Phase Spectrum arg[H(e^{j\omega})]')
xlabel('\omega /\pi');
ylabel('Phase, radians');

Q3.5 The required modifications to Program P3_1 to plot the phase in degrees are indicated below:
< Insert program code here. Copy from m-file(s) and paste. >
clf
w=-4*pi:8*pi/511:4*pi;
num=[2 1];den=[1 -0.6];
h=freqz(num,den,w);
subplot(2,1,1)
plot(w,real(h));grid;
title('H(e^{j\angle}的实部')
xlabel('\angle');
ylabel('振幅');
subplot(2,1,2)
plot(w/pi,imag(h));
grid
title('H(e^{j\angle}的虚部')
xlabel('\angle/');
ylabel('振幅');
figure(2);
subplot(2,1,1)
plot(w,abs(h));grid
title('|H(e^{j\angle})|幅度谱')
xlabel('\angle/');
ylabel('振幅');
subplot(2,1,2)
plot(w,angle(h));grid
title('相位谱arg[H(e^{j\angle})]')
xlabel('angle');
ylabel('以角度为单位的相位');

ct 3.2 DTFT Properties
Answers:
Q3.6 The modified Program P3_2 created by adding appropriate comment statements, and adding program statements for labeling the two axes of each plot being generated by the program is given below:
< Insert program code here. Copy from m-file(s) and paste. >

% Program P3_2
% Time-Shifting Properties of DTFT
clf;
w = -pi:2*pi/255:pi; wo = 0.4*pi;D=10;
num=[1 2 3 4 5 6 7 8 9];
h1 = freqz(num, 1, w);
h2 = freqz([zeros(1,D) num], 1, w);
subplot(2,2,1)
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,2)
plot(w/pi,abs(h2));grid
title('Magnitude Spectrum of Time-Shifted Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,3)
plot(w/pi,angle(h1));grid
title('Phase Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');
subplot(2,2,4)
plot(w/pi,angle(h2));grid
title('Phase Spectrum of Time-Shifted Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');

The parameter controlling the amount of time-shift is - D=10，[zeros(1,D) num]
Q3.7 The plots generated by running the modified program are given below:
< Insert MATLABfigure(s) here. Copy from figure window(s) and paste. > figure(s) here. Copy from figure window(s) and paste. >

From these plots we make the following observations:In the limited length of vector, in the original sequence for discrete time Fourier transform, amplitude and phase is changed. From the continuous time, continuous frequency, is a continuous time, discrete frequency.

Q3.10 The modified Program P3_3 created by adding appropriate comment statements, and adding program statements for labeling the two axes of each plot being generated by the program is given below:
< Insert program code here. Copy from m-file(s) and paste. >
% Program P3_3
% Time-Shifting Properties of DTFT
clf;
w = -pi:2*pi/255:pi; wo = 0.4*pi;
num1=[1 3 5 7 9 11 13 15 17];
L = length(num1);
h1 = freqz(num1, 1, w);
n = 0:L-1;
num2 = exp(wo*i*n).*num1;
h2 = freqz(num2, 1, w);
subplot(2,2,1)
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,2)
plot(w/pi,abs(h2));grid
title('Magnitude Spectrum of Time-Shifted Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,3)
plot(w/pi,angle(h1));grid
title('Phase Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');
subplot(2,2,4)
plot(w/pi,angle(h2));grid
title('Phase Spectrum of Time-Shifted Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');

The parameter controlling the amount of frequency-shift is -wo
Q3.12 Program P3_3 was run for the following value of the frequency-shift -
The plots generated by running the modified program are given below:
< Insert MATLAB figure(s) here. Copy from figure window(s) and paste. >
clf;
w = -pi:2*pi/255:pi; wo = 3*pi;
num1=[1 3 5 7 9 11 13 15 17];
L = length(num1);
h1 = freqz(num1, 1, w);
n = 0:L-1;
num2 = exp(wo*i*n).*num1;
h2 = freqz(num2, 1, w);
subplot(2,2,1)
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,2)
plot(w/pi,abs(h2));grid
title('Magnitude Spectrum of Time-Shifted Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,3)
plot(w/pi,angle(h1));grid
title('Phase Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');
subplot(2,2,4)
plot(w/pi,angle(h2));grid
title('Phase Spectrum of Time-Shifted Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');

From these plots we make the following observations:Select different values of frequency shift, can get different results.

Q3.14 The modified Program P3_4 created by adding appropriate comment statements, and adding program statements for labeling the two axes of each plot being generated by the program is given below:
< Insert program code here. Copy from m-file(s) and paste. >
% Program P3_4
% Convolution Property of DTFT
clf;
w = -pi:2*pi/255:pi;
x1=[1 3 5 7 9 11 13 15 17];
x2=[1 -2 3 -2 1];
y = conv(x1,x2);
h1 = freqz(x1, 1, w);
h2 = freqz(x2, 1, w);
hp = h1.*h2;
h3 = freqz(y,1,w);
subplot(2,2,1)
plot(w/pi,abs(hp));grid
title('Product of Magnitude Spectra')
xlabel('\omega /\pi');
ylabel('Product of Amplitude');
subplot(2,2,2)
plot(w/pi,abs(h3));grid
title('Magnitude Spectrum of Convolved Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,3)
plot(w/pi,angle(hp));grid
title('Sum of Phase Spectra')
xlabel('\omega /\pi');
ylabel('Phase, radians');
subplot(2,2,4)
plot(w/pi,angle(h3));grid
title('Phase Spectrum of Convolved Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');

Q3.17 The modified Program P3_5 created by adding appropriate comment statements, and adding program statements for labeling the two axes of each plot being generated by the program is given below:
< Insert program code here. Copy from m-file(s) and paste. >
% Program P3_5
% Modulation Property of DTFT
clf;
w = -pi:2*pi/255:pi;
x1=[1 3 5 7 9 11 13 15 17];
x2=[1 -1 1 -1 1 -1 1 -1 1];
y=x1.*x2;
h1 = freqz(x1, 1, w);
h2 = freqz(x2, 1, w);
h3 = freqz(y,1,w);
subplot(3,1,1)
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of First Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(3,1,2)
plot(w/pi,abs(h2));grid
title('Magnitude Spectrum of Second Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(3,1,3)
plot(w/pi,abs(h3));grid
title('Magnitude Spectrum of Product Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');

Q3.20 The modified Program P3_6 created by adding appropriate comment statements, and adding program statements for labeling the two axes of each plot being generated by the program is given below:
< Insert program code here. Copy from m-file(s) and paste. >
% Program P3_6
% Time-Reversal Property of DTFT
clf;
w = -pi:2*pi/255:pi;
num=[1 2 3 4];
L = length(num)-1;
h1 = freqz(num, 1, w);
h2 = freqz(fliplr(num), 1, w);
h3 = exp(w*L*i).*h2;
subplot(2,2,1)
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,2)
plot(w/pi,abs(h3));grid
title('Magnitude Spectrum of Time-Reversed Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,3)
plot(w/pi,angle(h1));grid
title('Phase Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');
subplot(2,2,4)
plot(w/pi,angle(h3));grid
title('Phase Spectrum of Time-Reversed Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');

The program implements the time-reversal operation as follows -
From these plots we make the following observations: First of all, using the reverse function will row vector of molecular coefficient inversion, that is, from the original [1, 2, 3, 4] gone bad now [4 3 2 1], and then after the coefficient inversion sequence multiplied by the exp (L * w * I), make all the original molecules exp (- w * I) into the exp (w * I), so the x (n), thus to realize the flip operation time
Q3.22 Program P3_6 was run for the following two different sets of sequences of varying lengths -
The plots generated by running the modified program are given below:
< Insert MATLAB figure(s) here. Copy from figure window(s) and paste. >
clf;
w = -pi:2*pi/255:pi;
num=[1 3 6 9];
L = length(num)-1;
h1 = freqz(num, 1, w);
h2 = freqz(fliplr(num), 1, w);
h3 = exp(w*L*i).*h2;
subplot(2,2,1)
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,2)
plot(w/pi,abs(h3));grid
title('Magnitude Spectrum of Time-Reversed Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,2,3)
plot(w/pi,angle(h1));grid
title('Phase Spectrum of Original Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');
subplot(2,2,4)
plot(w/pi,angle(h3));grid
title('Phase Spectrum of Time-Reversed Sequence')
xlabel('\omega /\pi');
ylabel('Phase, radians');

From these plots we make the following observations: Choose the different sequence length, can get different results.