bài 3-tt dsp

Name : Vo Phong Phu - 10117050

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:

% 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})')



pause

subplot(2,1,1)

plot(w/pi,abs(h));grid

title('Magnitude Spectrum |H(e^{j\omega})|')



subplot(2,1,2)

plot(w/pi,angle(h));grid

title('Phase Spectrum arg[H(e^{j\omega})]')


ylabel('Phase in radians');

Answers:

Q3.1 The expression of the DTFT being evaluated in Program P3_1 is

H(jw)=

2+e− jw

1−0 . 6e− jw

The function of the pause command is used to wait for user response by press any keys.

Q3.2 The plots generated by running Program P3_1 are shown below:

-4 -3 -2 -1 0 1 2 3 40

2

4

6

8Real part of H(ej)

/

Am

plitu

de

-4 -3 -2 -1 0 1 2 3 4-4

-2

0

2

4Imaginary part of H(ej)

/

Am

plitu

de

-4 -3 -2 -1 0 1 2 3 40

2

4

6

8Magnitude Spectrum |H(ej)|

/

Am

plitu

de

-4 -3 -2 -1 0 1 2 3 4-2

-1

0

1

2Phase Spectrum arg[H(ej)]

/

Phase in r

adia

ns

The DTFT is a continuous function of .

Its period is 2π.

The types of symmetries exhibited by the four plots are as follows:

• The real part is 2 π periodic and EVEN SYMMETRIC.

• The imaginary part is 2π periodic and ODD SYMMETRIC.

• The magnitude is 2 π periodic and EVEN SYMMETRIC.

• The phase is 2 π periodic and ODD SYMMETRIC.

Q3.3 The required modifications to Program P3_1 to evaluate the given DTFT of Q3.3 are given below:

clf;

clc;


w = 0:pi/511:pi;

num = [0.7 -0.5 0.3 1];

den = [1 0.3 -0.5 0.7];


% Plot the DTFT

subplot(2,1,1)





subplot(2,1,2)





pause

subplot(2,1,1)





subplot(2,1,2)





The plots generated by running the modified Program P3_1 are shown below:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1Real part of H(ej)

/A

mplitu

de

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5


/

Am

plitu

de

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1

1


/

Am

plitu

de

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2


/

Phase in r

adia

ns


Its period is 2π.

The jump in the phase spectrum is caused by the angle command. It give the angle lie between ±π.

The phase spectrum evaluated with the jump removed by the command unwrap is as given below:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1

1


/

Am

plitu

de

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2


/

Phase in r

adia

ns

Q3.4 The required modifications to Program P3_1 to evaluate the given DTFT of Q3.4 are given below:

clc;

clf;


w = -4*pi:8*pi/511:4*pi;

num=[1 3 5 7 9 11 13 15 17];

h = freqz(num, 1, w);

% Plot the DTFT

subplot(2,1,1)





subplot(2,1,2)





pause

subplot(2,1,1)





subplot(2,1,2)

plot(w/pi,unwrap(angle(h)));grid




The plots generated by running the modified Program P3_1 are shown below:

-4 -3 -2 -1 0 1 2 3 4-50

0

50

100Real part of H(ej)

/

Am

plitu

de

-4 -3 -2 -1 0 1 2 3 4-100

-50

0

50


/

Am

plitu

de

-4 -3 -2 -1 0 1 2 3 40

50


/

Am

plitude

-4 -3 -2 -1 0 1 2 3 4-4

-2

0

2


/

Phase in radia

ns


Its period is 2π.

The jump in the phase spectrum is caused by the same reason in Q3.3.

Q3.5 The required modifications to Program P3_1 to plot the phase in degrees are indicated below:

subplot(2,1,2)

plot(w/pi*180,angle(h)*180/pi);grid


xlabel('Degrees');

ylabel('Phase in degrees');

We get observation :

-4 -3 -2 -1 0 1 2 3 40

2

4

6


/

Am

plitu

de

-800 -600 -400 -200 0 200 400 600 800-100

-50

0

50


Degrees

Phase in d

egre

es

Project 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:

% Program P3_2B % Time-Shifting Properties of DTFT clf; w = -pi:2*pi/255:pi; % freqency vector for evaluating DTFT D = 10; % Amount of time shift in samples num = [1 2 3 4 5 6 7 8 9]; % h1 is the DTFT of original sequence % h2 is the DTFT of the time shifted sequence h1 = freqz(num, 1, w); h2 = freqz([zeros(1,D) num], 1, w); subplot(2,2,1) % plot the DTFT magnitude of the original sequence plot(w/pi,abs(h1));grid title('Magnitude Spectrum of Original Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the DTFT magnitude of the shifted sequence subplot(2,2,2) plot(w/pi,abs(h2));grid title('Magnitude Spectrum of Time-Shifted Sequence' ,'FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the DTFT phase of the original sequence subplot(2,2,3) plot(w/pi,angle(h1));grid title('Phase Spectrum of Original Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' ); % plot the DTFT phase of the shifted sequence subplot(2,2,4) plot(w/pi,angle(h2));grid title('Phase Spectrum of Time-Shifted Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' );

The parameter controlling the amount of time-shift is D

Q3.7 The plots generated by running the modified program are given below:

-1 -0.5 0 0.5 10

20

40

60Magnitude Spectrum of Original Sequence

-1 -0.5 0 0.5 10

20

40

60Magnitude Spectrum of Time-Shifted Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Original Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Time-Shifted Sequence

From these plots we make the following observations:

The magnitude spectrum is not changed.

Shift in time-domain is shrink in frequency-domain. The phase changes rapidly.

Q3.8 Program P3_2 was run for the following value of the time-shift

The plots generated by running the modified program are given below:

D=15 :

-1 -0.5 0 0.5 10

20

40


-1 -0.5 0 0.5 10

20

40


-1 -0.5 0 0.5 1-4

-2

0

2


-1 -0.5 0 0.5 1-4

-2

0

2


D=25 :

-1 -0.5 0 0.5 10

20

40


-1 -0.5 0 0.5 10

20

40


-1 -0.5 0 0.5 1-4

-2

0

2


-1 -0.5 0 0.5 1-4

-2

0

2


From these plots we make the following observations: the time shift has no effect on the magnitude spectrum. However, there is a very significant change to the phase.

The higher delay-time is, the higher shrink is. The phase changes more rapidly.

Q3.9 Program P3_2 was run for the following values of the time-shift: 15 and for the following values of length for the sequence: 12


-1 -0.5 0 0.5 10

20

40

60


-1 -0.5 0 0.5 10

20

40

60


-1 -0.5 0 0.5 1-4

-2

0

2


-1 -0.5 0 0.5 1-4

-2

0

2



Magnitude is increased when the length of sequence is increased.


% Program P3_3B % Frequency-Shifting Properties of DTFT clf; w = -pi:2*pi/255:pi; % frequency vector for evaluating DTFT wo = 0.4*pi; % Amount of frequency shift in radians % h1 is the DTFT of the original sequence % h2 is the DTFT of the frequency shifted sequence 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); % plot the DTFT magnitude of the original sequence subplot(2,2,1) plot(w/pi,abs(h1));grid title('Magnitude Spectrum of Original Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the DTFT magnitude of the freq shifted sequence subplot(2,2,2) plot(w/pi,abs(h2));grid title('Magnitude Spectrum of Frequency-Shifted Sequence' ,'FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the DTFT phase of the original sequence subplot(2,2,3) plot(w/pi,angle(h1));grid title('Phase Spectrum of Original Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' ); % plot the DTFT phase of the shifted sequence subplot(2,2,4) plot(w/pi,angle(h2));grid title('Phase Spectrum of Frequency-Shifted Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' );

The parameter controlling the amount of frequency-shift is “wo”.


-1 -0.5 0 0.5 10

50


-1 -0.5 0 0.5 10

50

100Magnitude Spectrum of Frequency-Shifted Sequence

-1 -0.5 0 0.5 1-4

-2

0

2


-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Frequency-Shifted Sequence


Magnitude spectrum of frequency shifted sequence and phase spectrum are shifted to the right, but the altitude is no changed. The shift range is 0.4π.

Due to the property

ejw0 n x [n ]↔X (e

j (w−w0 ))

Q3.12 Program P3_3 was run for the following value of the frequency-shift : -0.6*pi.


-1 -0.5 0 0.5 10

50


/

Am

plitu

de

-1 -0.5 0 0.5 10

50


/

Am

plitu

de

-1 -0.5 0 0.5 1-4

-2

0

2


/

Phase in r

adia

ns

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Frequency-Shif ted Sequence

/

Phase in r

adia

ns

From these plots we make the following observations: when wo is increased the magnitude spectrum is shifted more and more to the left.

Q3.13 Program P3_3 was run for the following values of the frequency-shift: 0.8*pi and for the following values of length for the sequence: 12

-1 -0.5 0 0.5 10

50

100


-1 -0.5 0 0.5 10

50

100


-1 -0.5 0 0.5 1-4

-2

0

2


-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Frequency-Shifted Sequence



When the length of sequence is increased, the altitude of magnitude spectrum is increased too.

Both magnitude and phase spectrum is shifted more and more to the right .wo=0.8*pi


% Program P3_4B % Convolution Property of DTFT clf; w = -pi:2*pi/255:pi; % frequency vector for evaluating DTFT x1 = [1 3 5 7 9 11 13 15 17]; % first sequence x2 = [1 -2 3 -2 1]; % second sequence y = conv(x1,x2); % time domain convolution of x1 and x2 h1 = freqz(x1, 1, w); % DTFT of sequence x1 h2 = freqz(x2, 1, w); % DTFT of sequence x2 % hp is the pointwise product of the two DTFT's hp = h1.*h2; % h3 is the DTFT of the time domain convolution; % it should be the same as hp h3 = freqz(y,1,w); % plot the magnitude of the product of the two original spectra subplot(2,2,1) plot(w/pi,abs(hp));grid title('Product of Magnitude Spectra' ,'FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the magnitude spectrum of the time domain convolution subplot(2,2,2) plot(w/pi,abs(h3));grid title('Magnitude Spectrum of Convolved Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude');

% plot the phase of the product of the two original spectrasubplot(2,2,3) plot(w/pi,angle(hp));grid title('Sum of Phase Spectra','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' ); % plot the phase spectrum of the time domain convolution subplot(2,2,4) plot(w/pi,angle(h3));grid title('Phase Spectrum of Convolved Sequence' ,'FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' ); Q3.15 The plots generated by running the modified program are given below:

-1 -0.5 0 0.5 10

50

100Product of Magnitude Spectra

-1 -0.5 0 0.5 10

50

100Magnitude Spectrum of Convolved Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Sum of Phase Spectra

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Convolved Sequence


Both phase are the same. Both magnitude are the same.

We see that the convolution in time domain is the manipulation in frequency domain.

Q3.16 Program P3_4 was run for the following two different sets of sequences of varying lengths. Length of x1 = 12, x2 = 7.

x1 = [1 3 5 7 9 11 13 15 17 19 21 23 ];

x2 = [1 -2 3 -2 1 3 4];

-1 -0.5 0 0.5 10

500

1000

1500Product of Magnitude Spectra

-1 -0.5 0 0.5 10

500

1000

1500Magnitude Spectrum of Convolved Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Sum of Phase Spectra

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Convolved Sequence


When the length changes, the properties discovered in Q3.16 doesn’t change. Both phase are the same. Both magnitude are the same.


% Program P3_5B % Modulation Property of DTFT clf; w = -pi:2*pi/255:pi; % freqency vector for evaluating DTFT x1 = [1 3 5 7 9 11 13 15 17]; % first sequence x2 = [1 -1 1 -1 1 -1 1 -1 1]; % second sequence % y is the time domain pointwise product of x1 and x2 y = x1.*x2; h1 = freqz(x1, 1, w); % DTFT of sequence x1 h2 = freqz(x2, 1, w); % DTFT of sequence x2 h3 = freqz(y,1,w); % DTFT of sequence y % plot the magnitude spectrum of x1 subplot(3,1,1) plot(w/pi,abs(h1));grid title('Magnitude Spectrum of First Sequence' ) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the magnitude spectrum of x2 subplot(3,1,2) plot(w/pi,abs(h2));grid title('Magnitude Spectrum of Second Sequence') xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the magnitude spectrum of y % it should be 1/2pi times the convolution of the DTFT's % of the two original sequences. subplot(3,1,3) plot(w/pi,abs(h3));grid title('Magnitude Spectrum of Product Sequence' )

xlabel('\omega /\pi' ); ylabel('Amplitude');

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

50

100Magnitude Spectrum of First Sequence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10Magnitude Spectrum of Second Sequence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

50

100Magnitude Spectrum of Product Sequence



x1(n) H1(w)

x2(n) H2(w)

x1(n).x2(n) 1/(2.pi) H1(w)*H2(w)

thus, we have:

H3(w) = 1/(2.pi) H1(w)*H2(w)

Q3.19 Program P3_5 was run for the following two different sets of sequences of varying lengths:

x1 = [1 3 5 7 9 11 13 15 17 19 21 23 25]

x2 = [1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1]


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

100

200Magnitude Spectrum of First Sequence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20Magnitude Spectrum of Second Sequence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

100

200Magnitude Spectrum of Product Sequence

From these plots we make the following observations: When the length changes, modulation property dosen’t change.


% Program P3_6

% Program P3_6B % Time Reversal Property of DTFT clf; w = -pi:2*pi/255:pi; % freqency vector for evaluating DTFT % original ramp sequence % note: num is nonzero for 0 <= n <= 3. num = [1 2 3 4]; L = length(num)-1; h1 = freqz(num, 1, w); % DTFT of original ramp sequence h2 = freqz(fliplr(num), 1, w); h3 = exp(w*L*i).*h2; % plot the magnitude spectrum of the original ramp sequence subplot(2,2,1) plot(w/pi,abs(h1));grid title('Magnitude Spectrum of Original Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the magnitude spectrum of the time reversed ramp sequence subplot(2,2,2) plot(w/pi,abs(h3));grid title('Magnitude Spectrum of Time-Reversed Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Amplitude'); % plot the phase spectrum of the original ramp sequence subplot(2,2,3) plot(w/pi,angle(h1));grid title('Phase Spectrum of Original Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' );

% plot the phase spectrum of the time reversed ramp sequence subplot(2,2,4) plot(w/pi,angle(h3));grid title('Phase Spectrum of Time-Reversed Sequence','FontSize' ,8) xlabel('\omega /\pi' ); ylabel('Phase in radians' );

The program implements the time-reversal operation as follows :

A new sequence is formed by using fliplr this new sequence contains the samples of the original ramp sequence in time reversed order

h2 = freqz(fliplr(num), 1, w);

h3 = exp(w*L*i).*h2;


-1 -0.5 0 0.5 12

4

6

8


w

A

-1 -0.5 0 0.5 12

4

6

8

10Magnitude Spectrum of Time-Reversed Sequence

w

A

-1 -0.5 0 0.5 1-4

-2

0

2


w

Phase

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Time-Reversed Sequence

w

Phase

From these plots we make the following observations: x(-n) X(-w)

Both the original and time reversed ramp sequences are real-valued.

Q3.22 Program P3_6 was run for the following two different sets of sequences of varying lengths : num = [1 2 3 4 5 6];


-1 -0.5 0 0.5 10

10

20


-1 -0.5 0 0.5 10

10

20

30Magnitude Spectrum of Time-Reversed Sequence

-1 -0.5 0 0.5 1-4

-2

0

2


-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Time-Reversed Sequence


Both the original and time reversed ramp sequences are real-valued.

3.2 DISCRETE FOURIER TRANSFORM

Project 3.3 DFT and IDFT Computations

Answers:

Q3.23 The MATLAB program to compute and plot the L-point DFT X[k] of a length-N sequence x[n] with L N and then to compute and plot the IDFT of X[k] is given below:

clf;

clc;

N=20;

x=randint(1,N);

subplot(3,1,1)

plot(x);

title('original signal : x');

y=fft(x,32);

subplot(3,1,2)

plot(abs(y));

title(' magnitude spectrum of x');

ix=ifft(y,32);

subplot(3,1,3)

plot(abs(ix));

title('signal after transforming')

Figure

0 2 4 6 8 10 12 14 16 18 200

0.5

1original signal : x

0 5 10 15 20 25 30 350

5

10 magnitude spectrum of x

0 5 10 15 20 25 30 350

0.5

1signal after transforming

The DFT and the IDFT pairs generated by running the program for sequences of different lengths N and for different values of the DFT length L are shown below:

N = 30, quantity of point in DFT : 64, this value in IDFT is 32.

0 5 10 15 20 25 300

0.5

1original signal : x

0 10 20 30 40 50 60 700

10

20 magnitude spectrum of x

0 5 10 15 20 25 30 350

1

2signal after transforming


DFT and IDFT must have the same length to give exact signal transformed.

Q3.24 The MATLAB program to compute the N-point DFT of two length-N real sequences using a single N-point DFT and compare the result by computing directly the two N-point DFTs is given below:

clf;

clc;

N=30;

x1=randint(1,N);

x2=randint(1,N);

x=x1+i*x2;

X=fft(x,64);

dX1=fft(x1,64);

dX2=fft(x2,64);

Xcon=conj(X);

Xconf=[Xcon(1) fliplr(X(2:length(Xcon)))];

X1=0.5*(X + Xconf);

X2=(0.5/i)*(X - Xconf);

subplot(4,1,1)

plot(abs(dX1));

title('fft of g(n)( direct transform )');

subplot(4,1,2)

plot(abs(X1));

title(' fft of g(n) (indirect)');

subplot(4,1,3)

plot(abs(dX2));

title('fft of h(n)( direct transform )');

subplot(4,1,4)

plot(abs(X2));

title(' fft of h(n) (indirect)');

%Q3.24clc;clear all;g=1:10;h=10:-1:1;x=g+i*h;Xk=fft(x);Gk=fft(g);Hk=fft(h);Xk1=fliplr(Xk);Xkk=[Xk1(length(Xk1)) Xk1(1:length(Xk1)-1)];Gk1=0.5*(Xk+conj(Xkk));Hk1=0.5*(-i)*(Xk-conj(Xkk));dg=Gk-Gk1;dh=Hk-Hk1;subplot(2,1,1)stem(dg)title('deviation of Gk')subplot(2,1,2)stem(dh)title('deviation of Hk')

%Q3.25clc;clear all;v=1:20;g=v(1:2:length(v));h=v(2:2:length(v));Vk=fft(v);N=length(v)/2;Gk=fft(g,N);Hk=fft(h,N);w=exp(-i*2*pi/(2*N));k=0:2*N-1;wk=w.^k;Vkk=[Gk Gk]+wk.*[Hk Hk];d=Vk-Vkksubplot(3,1,1)stem(Vkk)title('2N-point DFT')subplot(3,1,2)stem(Vk)title('N-point DFT')subplot(3,1,3)stem(d)title('deviation')

The DFTs generated by running the program for sequences of different lengths N are shown below:

0 10 20 30 40 50 60 700

10

20fft of g(n)( direct transform )

0 10 20 30 40 50 60 700

10

20 fft of g(n) (indirect)

0 10 20 30 40 50 60 700

10

20fft of h(n)( direct transform )

0 10 20 30 40 50 60 700

10

20 fft of h(n) (indirect)

From these plots we make the following observations: between direct and indirect the transform is nearly the same.

Project 3.4 DFT Properties

Answers:

Q3.26 The purpose of the command rem in the function circshift is used to give us the remainder of division.

Q3.27 The function circshift operates as follows :

Array will be shifted in circular by M samples.

Q3.28 The purpose of the operator ~= in the function circonv is used to compare two objects. This is the binary relational NOT EQUAL operator.

Q3.29 The function circonv operates as follows:

This function computate convolution in circle.

Q3.30 The modified Program P3_7 created by adding appropriate comment statements, and adding program statements for labeling each plot being generated by the program is given below:

% Program P3_7B % Illustration of Circular Shift of a Sequence clf; % initialize shift amount M M = 6; % initialize sequence a to be shifted a = [0 1 2 3 4 5 6 7 8 9]; b = circshift(a,M); % perform the circular shift L = length(a)-1; % plot the original sequence a and the circularly shifted sequence b n = 0:L; subplot(2,1,1); stem(n,a);axis([0,L,min(a),max(a)]); title('Original Sequence'); xlabel('time index n'); ylabel('a[n]'); subplot(2,1,2); stem(n,b);axis([0,L,min(a),max(a)]); title(['Sequence Obtained by Circularly Shifting by ',num2str(M),' Samples' ]); xlabel('time index n'); ylabel('b[n]');

The parameter determining the amount of time-shifting is M.

If the amount of time-shift is greater than the sequence length then M = rem(M,length(x));


0 1 2 3 4 5 6 7 8 90

2

4

6

8

Original Sequence

n

altitude

0 1 2 3 4 5 6 7 8 90

2

4

6

8

Sequence Obtained by Circularly Shifting by 6 Samples

n

altitude

From these plots we make the following observations: The sequence is shift to the right by 6 samples. Because it’s a circular shift, the samples will replace other samples shifted.

Q3.32 The modified Program P3_8 created by adding appropriate comment statements, and adding program statements for labeling each plot being generated by the program is given below:

% Program P3_8B % Circular Time-Shifting Property of DFT clf; x = [0 2 4 6 8 10 12 14 16]; % original sequence x N = length(x)-1; n = 0:N; % time index vector % set y equal to the circular shift left of x y = circshift(x,5); XF = fft(x); % DFT of x YF = fft(y); % DFT of y subplot(2,2,1); % plot the spectral magnitudes of the original and shifted sequences stem(n,abs(XF));grid; title('Magnitude of DFT of Original Sequence'); xlabel('Frequency index k'); ylabel('|X[k]|' ); subplot(2,2,2); stem(n,abs(YF));grid; title('Magnitude of DFT of Circularly Shifted Sequence'); xlabel('Frequency index k'); ylabel('|Y[k]|' ); % plot the spectral phases of the original and shifted sequences subplot(2,2,3); stem(n,angle(XF));grid; title('Phase of DFT of Original Sequence' ); xlabel('Frequency index k'); ylabel('arg(X[k])'); subplot(2,2,4); stem(n,angle(YF));grid; title('Phase of DFT of Circularly Shifted Sequence' ); xlabel('Frequency index k'); ylabel('arg(Y[k])');

The amount of time-shift is 5


0 2 4 6 80

20

40

60

80Magnitude of DFT of Original Sequence

0 2 4 6 80

20

40

60

80Magnitude of DFT of Circularly Shifted Sequence

0 2 4 6 8-4

-2

0

2

4Phase of DFT of Original Sequence

0 2 4 6 8-4

-2

0

2

4Phase of DFT of Circularly Shifted Sequence


The altitude is not changed when sequence is shifted.

The phase is changed with rule:

x(n) X(k)

x(n-n0) X(k).exp (-j2πkn0/N)

Q3.34 The plots generated by running the modified program for the following two different amounts of time-shifts, with the amount of shift indicated, are shown below:

When time-shift = 7, we have

0 2 4 6 80

20

40

60


Altitude (

V)

w 0 2 4 6 8

0

20

40

60


Altitude (

V)

w

0 2 4 6 8-4

-2

0

2


Phase

w 0 2 4 6 8

-4

-2

0

2


Phase

w

when time-shift = 10 , we have

0 2 4 6 80

20

40

60


Altitude (

V)

w 0 2 4 6 8

0

20

40

60


Altitude (

V)

w

0 2 4 6 8-4

-2

0

2


Phase

w 0 2 4 6 8

-4

-2

0

2


Phase

w

From these plots we make the following observations: When time-shift is changed, we observe that only the phase of shifted sequence is changed.

Q3.35 The plots generated by running the modified program for the following two different sequences of different lengths, with the lengths indicated, are shown below:

When :

x = [0 2 4 6 8 10 12 14 16 18 20 22];

0 5 10 150

50

100


Altitude (

V)

w 0 5 10 15

0

50

100


Altitude (

V)

w

0 5 10 15-4

-2

0

2


Phase

w 0 5 10 15

-4

-2

0

2


Phase

w

When:

x = [0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 16];


0 5 10 150

100

200


Altitude (

V)

w 0 5 10 15

0

100

200


Altitude (

V)

w

0 5 10 15-4

-2

0

2


Phase

w 0 5 10 15

-4

-2

0

2


Phase

w

When the length of sequence changes, the magnitude changes but the original sequence and the shifted signal are the same.

The phase changes.

Q3.36 A copy of Program P3_9 is given below along with the plots generated by running this program:

% Program P3_9

% Circular Convolution Property of DFT

g1 = [1 2 3 4 5 6]; g2 = [1 -2 3 3 -2 1];

ycir = Circonv(g1,g2);

disp('Result of circular convolution = ');disp(ycir)

G1 = fft(g1); G2 = fft(g2);

yc = real(ifft(G1.*G2));

disp('Result of IDFT of the DFT products = ');disp(yc)

Result of circular convolution =

12 28 14 0 16 14

Result of IDFT of the DFT products =

12 28 14 0 16 14

the modified P3_9 to show figure is:

% q3.36

% Circular Convolution Property of DFT

g1 = [1 2 3 4 5 6]; g2 = [1 -2 3 3 -2 1];

ycir = Circonv(g1,g2);

disp('Result of circular convolution = ');disp(ycir)

G1 = fft(g1); G2 = fft(g2);


disp('Result of IDFT of the DFT products = ');disp(yc)

subplot(2,1,1)

stem(ycir)

title('Result of circular convolution ')

subplot(2,1,2)

stem(yc)

title('Result of IDFT of the DFT products ')

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

10

20

30Result of circular convolution

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

10

20

30Result of IDFT of the DFT products


The result when we use direct convolution and when we use DFT are the same.

Q3.37 Program P3_9 was run again for the following two different sets of equal-length sequences:

The plots generated are shown below:

When g1 = [1 2 -4 -3 5 6]; g2 = [1 -2 3 3 -2 1];

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-40

-20

0

20


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-40

-20

0

20


When g1 = [1 2 -4 -3 5 6 3]; g2 = [1 -2 3 3 -2 1 4];

1 2 3 4 5 6 7-20

0

20


1 2 3 4 5 6 7-20

0

20



Although the elements of sequence get different value or different length, the result when we change method are the same.

Q3.38 A copy of Program P3_10 is given below along with the plots generated by running this program:

% Program P3_10

% Linear Convolution via Circular Convolution

g1 = [1 2 3 4 5];g2 = [2 2 0 1 1];

g1e = [g1 zeros(1,length(g2)-1)];


ylin = circonv(g1e,g2e);

disp('Linear convolution via circular convolution = ');disp(ylin);

y = conv(g1, g2);

disp('Direct linear convolution = ');disp(y)

The modified program

% Modified Program P3_10 to show figure

% Linear Convolution via Circular Convolution

g1 = [1 2 3 4 5];g2 = [2 2 0 1 1];



ylin = circonv(g1e,g2e);

disp('Linear convolution via circular convolution = ');disp(ylin);

y = conv(g1, g2);

disp('Direct linear convolution = ');disp(y)

subplot(2,1,1)

stem(ylin)

title('Circular convolution')

subplot(2,1,2)

stem(y)

title('Linear convolution')

Figure :

1 2 3 4 5 6 7 8 90

10

20

30Circular convolution

1 2 3 4 5 6 7 8 90

10

20

30Linear convolution

From these plots we make the following observations: When padding zero to make the same length sequences (Because linear convolution of 2 sequences give the length of (L1+L2-1)).Linear convolution and circular convolution get the same result.

Q3.39 Program P3_10 was run again for the following two different sets of sequences of unequal lengths:

g1 = [1 2 3 4 5 6];g2 = [2 2 0 1 1];

1 2 3 4 5 6 7 8 9 100

10

20


1 2 3 4 5 6 7 8 9 100

10

20


g1 = [1 2 3 4 5 6];g2 = [2 2 0 1 1 7 8];

0 2 4 6 8 10 120

50


0 2 4 6 8 10 120

50


From these plots we make the following observations: When length of sequences are different, the convolution is the same.

Q3.40 The MATLAB program to develop the linear convolution of two sequences via the DFT of each is given below:

% Linear Convolution via the DFT

g1 = [1 2 3 4 5];g2 = [2 2 0 1 1];

y = conv(g1, g2);



G1 = fft(g1e); G2 = fft(g2e);


subplot(2,1,1)

stem(y)

title('Sequence (Direct linear convolution)')

subplot(2,1,2)

stem(yc)

title(' Sequence (Using DFT)')

The plots generated by running this program for the sequences of Q3.38 are shown below:

1 2 3 4 5 6 7 8 90

10

20

30Sequence (Direct linear convolution)

1 2 3 4 5 6 7 8 90

10

20

30 Sequence (Using DFT)

From these plots we make the following observations: The result in 2 methods is the same.

The plots generated by running this program for the sequences of Q3.39 are shown below:

1 2 3 4 5 6 7 8 90

10

20

30

40Sequence (Direct linear convolution)

1 2 3 4 5 6 7 8 90

10

20

30

40 Sequence (Using DFT)

From these plots we make the following observations: the result is not changed.

Q3.41 A copy of Program P3_11 is given below:

% Program P3_11

% Relations between the DFTs of the Periodic Even

% and Odd Parts of a Real Sequence

x = [1 2 4 2 6 32 6 4 2 zeros(1,247)];

x1 = [x(1) x(256:-1:2)];

xe = 0.5 *(x + x1);

XF = fft(x);

XEF = fft(xe);

clf;

k = 0:255;

subplot(2,2,1);

plot(k/128,real(XF)); grid;


title('Re(DFT\{x[n]\})');

subplot(2,2,2);

plot(k/128,imag(XF)); grid;


title('Im(DFT\{x[n]\})');

subplot(2,2,3);

plot(k/128,real(XEF)); grid;

xlabel('Time index n');ylabel('Amplitude');

title('Re(DFT\{x_{e}[n]\})');

subplot(2,2,4);

plot(k/128,imag(XEF)); grid;


title('Im(DFT\{x_{e}[n]\})');

The relation between the sequence x1[n] and x[n] is x1[n] = x[-n].

Q3.42 The plots generated by running Program P3_11 are given below:

0 0.5 1 1.5 2-50

0

50

100

Am

plitu

de

Re(DFT{x[n]})

0 0.5 1 1.5 2-100

-50

0

50

100

Am

plitu

de

Im(DFT{x[n]})

0 0.5 1 1.5 2-50

0

50

100

Time index n

Am

plitu

de

Re(DFT{xe[n]})

0 0.5 1 1.5 2

-0.5

0

0.5

x 10-14

Time index n

Am

plitu

de

Im(DFT{xe[n]})

The imaginary part of XEF is equal to zero. This result can be explained as follows:

xe(n) = ½(x(n) + x(-n))

If X(k) is the N-point DFT of x(n), the N-point DFT of xe(n) is the real part of X(k), so Re(DFT(xe(n))) is the same of Re(DFT(x(n))).

Q3.43 The required modifications to Program P3_11 to verify the relation between the DFT of the periodic odd part and the imaginary part of XEF are given below along with the plots generated by running this program:

% Relations between the DFTs of the Periodic Odd part

% and the imaginary part of XEF

x = [1 2 4 2 6 32 6 4 2 zeros(1,247)];

x1 = [x(1) x(256:-1:2)];

xe = 0.5 *(x + x1);

xo = 0.5 *(x - x1);

XEF = fft(xe);

XOF = fft(xo);

clf;

k = 0:255;

subplot(2,2,1);

plot(k/128,real(XOF)); grid;


title('Re(DFT\{x_{o}[n]\})');

subplot(2,2,2);

plot(k/128,imag(XOF)); grid;


title('Im(DFT\{x_{o}[n]\})');

subplot(2,2,3);

plot(k/128,real(XEF)); grid;


title('Re(DFT\{x_{e}[n]\})');

subplot(2,2,4);

plot(k/128,imag(XEF)); grid;


title('Im(DFT\{x_{e}[n]\})');

0 0.5 1 1.5 2

-0.5

0

0.5

x 10-14

Am

plitu

de

Re(DFT{xo[n]})

0 0.5 1 1.5 2-100

-50

0

50

100

Am

plitu

de

Im(DFT{xo[n]})

0 0.5 1 1.5 2-50

0

50

100

Time index n

Am

plitu

de

Re(DFT{xe[n]})

0 0.5 1 1.5 2

-0.5

0

0.5

x 10-14

Time index n

Am

plitu

de

Im(DFT{xe[n]})


xo(n) = ½(x(n) - x(-n))

N-points DFT of xo(n) is the imaginary part of x(n). The imaginar part of XOF is unequal to zero, whereas the imaginary part of XEF is nearly zero.

Q3.44 A copy of Program P3_12 is given below:

% Program P3_12

% Parseval's Relation

x = [(1:128) (128:-1:1)];

XF = fft(x);

a = sum(x.*x)

b = round(sum(abs(XF).^2)/256)

The values for a and b we get by running this program are

a = 1414528

b = 1414528

Q3.45 The required modifications to Program P3_11 are given below:

% Modified Program P3_12

% Parseval's Relation

clc;

x = [(1:128) (128:-1:1)];

XF = fft(x);

a = sum(x.*x)

absXF=XF.*conj(XF);

b = round(sum(absXF)/256)

We have the same result as in Q3.44.

3.3 z-TRANSFORM

Project 3.5 Analysis of z-Transforms

Answers:

Q3.46 The frequency response of the z-transform obtained using Program P3_1 is plotted below:

With the the z-transform:

G( z )=2+5 z−1+9 z−2+5 z−3+3 z−4

5+45 z−1+2 z−2+z−3+z−4

Code

% q346

% Evaluation of the z-transform

clf;

w = -4*pi:8*pi/511:4*pi;

num = [2 5 9 5 3];den = [5 45 2 1 1];


subplot(2,1,1)


title('Magnitude Spectrum')



subplot(2,1,2)


title('Phase Spectrum')


ylabel('Phase in radians')

-4 -3 -2 -1 0 1 2 3 40

0.2

0.4

0.6

0.8Magnitude Spectrum

/

Am

plitude

-4 -3 -2 -1 0 1 2 3 4-4

-2

0

2

4Phase Spectrum

/

Phase in

radia

ns

Q3.47 The MATLAB program to compute and display the poles and zeros, to compute and display the factored form, and to generate the pole-zero plot of a rational z-transform is given below:

clc;

w = -4*pi:8*pi/511:4*pi;

num = [2 5 9 5 3];den = [5 45 2 1 1];%q3.47

zplane(num,den)

[z,p,k]=tf2zp(num,den);

disp('zeros')

disp(z);

disp('poles')

disp(p);

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

-4

-3

-2

-1

0

1

2

3

4

Real Part

Imagin

ary

Part

zeros

-1.0000 + 1.4142i

-1.0000 - 1.4142i

-0.2500 + 0.6614i

-0.2500 - 0.6614i

poles

-8.9576

-0.2718

0.1147 + 0.2627i

0.1147 - 0.2627i

Q3.48 From the pole-zero plot generated in Question Q3.47, the number of regions of convergence (ROC) of G(z) are 4 regions. First, it’s |z| > 8.9576. Second, it’s |z| < 0.2718. Third, it ‘s 0.2718<|z|<0.2866 (magnitude of pole inside the unit circle). The last one is 0.2866<|z|<8.9576.

From the pole-zero plot it can be seen that the DTFT converge with 0.2866<|z|<8.9576.

Q3.49 The MATLAB program to compute and display the rational z-transform from its zeros, poles and gain constant is given below:

clc;

z=[0.3 ;2.5 ;-0.2+i*0.4 ;-0.2-i*0.4];

p=[0.5 ;-0.75 ;0.6+i*0.7 ;0.6-i*0.7];

k=3.9;

[num,den]=zp2tf(z,p,k);

disp('numerator:')

disp(num)

disp('denominator:')

disp(den)

disp('gain : ')

disp(k)

The rational form of a z-transform with the given poles, zeros, and gain:

With

numerator:3.9000 -9.3600 -16.8870 44.4132 -11.5830

denominator:1.0000 -0.9500 -12.5650 -2.5225 4.4587

gain : 3.9000

We have the rational form G(z) = 3.9.

3. 9−9 .36 z−1−16 . 887 z−2+44 .41 z−3−11.58 z−4

1−0 .95 z−1−12.57 z−2−2.52 z−3−4 . 4587 z−4

Project 3.6 Inverse z-Transform

Answers:

Q3.50 The MATLAB program to compute the first L samples of the inverse of a rational z-transform is given below:

clc;

num=[2,5,9,5,3];den=[5,45,2,1,1];

L=50

[g,t]=impz(num,den,L);

stem(t,g);

xlabel('Discrete Time');


grid;

The plot of the first 50 samples of the inverse of G(z) of Q3.46 obtained using this program is sketched below:

0 5 10 15 20 25 30 35 40 45 50-16

-14

-12

-10

-8

-6

-4

-2

0

2x 10

45

Discrete Time

Am

plitu

de

Q3.51 The MATLAB program to determine the partial-fraction expansion of a rational z-transform is given below:

clc;

num=[2,5,9,5,3];den=[5,45,2,1,1];

[r,p,k]=residuez(num,den);

disp('residue')

disp(r)

disp('pole')

disp(p)

disp('direct term')

disp(k)

The partial-fraction expansion of G(z) of Q3.46 obtained using this program is shown below:

residue

0.3109

-1.0254 - 0.3547i

-1.0254 + 0.3547i

-0.8601

pole

-8.9576

0.1147 + 0.2627i

0.1147 - 0.2627i

-0.2718

direct item

3

We have

G(z)= 3+

0 .31

1+8 .9576 z−1+

−1. 03−0 .35 j

1−(0 . 11+0 . 26 j )z−1+

−1 .03+0 .35 j

1−(0 . 11−0 .26 j)z−1+

−0. 86

1+0 .27 z−1

bài 3-tt dsp

Documents

xlabelomega piylabelphase

given dtft of q3

modified program p3

dtft clf

frequency samples

continuous function

required modifications

radiansthe plots