Xử lý tín hiệu số bai3A

ylabel'Phase in radians';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

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Laboratory Exercise 3 DISCRETE-TIME SIGNALS: FREQUENCY-DOMAIN REPRESENTATIONS

ylabel('Amplitude');

pausesubplot(2,1,1)plot(w/pi,abs(h));gridtitle('Magnitude Spectrum |H(e^{j\omega})|')

xlabel('\omega /\pi');

subplot(2,1,2)plot(w/pi,angle(h));gridtitle('Phase Spectrum arg[H(e^{j\omega})]')xlabel('\omega /\pi');

The function of the pause command is – Dùng để dừng chương trình

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

The DTFT is a 2+e−jw

function of ω

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Its period is - 2

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

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

are given below:

xlabel('\omega /\pi');

subplot(2,1,2)plot(w/pi,angle(h));gridtitle('Phase Spectrum arg[H(e^{j\omega})]')xlabel('\omega /\pi');

ylabel('Phase in radians');

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

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The DTFT is a _ function of ω

Its period is -

The jump in the phase spectrum is caused by -

The phase spectrum evaluated with the jump removed by the command unwrap is

subplot(2,1,2)plot(w/pi,imag(h));gridtitle('Imaginary part of

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e e

2

56

.04

23

Its period is -

The jump in the phase spectrum is caused by -

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

subplot(2,1,2)plot(w,imag(h));grid

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ylabel('Phase in radians');

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 beinggenerated by the program is given below:

xlabel('time');

subplot(2,2,4)plot(w/pi,angle(h2));gridtitle('Phase Spectrum of Time-ShiftedSequence')

xlabel('time');

The parameter controlling the amount

of time-shift is – D=10

Q3.7 The plots generated by running the

modified program are given below:

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observations:Phổ pha dịch sát lại nhau

Q3.8 Program P3_2 was run for the

following

value of the timeshift

-The plots generated by running the modified

program are given below:

Q3.9 Program P3_2 was run for the following

values of the time-shift and for thefollowing values of length for thesequence - The plots generated byrunning the modified program are givenbelow:

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From these plots we make the following observations:Tần số càng lớn,D càng lớn phổ

pha càng dịch ra xa

subplot(2,2,3)plot(w/pi,angle(h1));gridtitle('Phase Spectrum of Original Sequence')

xlabel('Frequency');

subplot(2,2,4)plot(w/pi,angle(h2));gridtitle('Phase Spectrum of Frequency-Shifted Sequence')

xlabel('Frequency-Shifted');

The parameter controlling the amount of frequency-shift is - wo = 0.4*pi;

Q3.11 The plots generated by running the modified program are given below:

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From these plots we make the following observations:Phổ biên độ bị dịch

Q3.12 Program P3_3 was run for the following value of the frequency-shift - wo

= 0.9*pi

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

From these plots we make the following observations:Phổ biên độ bị dịch theo w0

Q3.13 Program P3_3 was run for the following values of the frequency-shift and

for the following values of length for the sequence –

wo = 0.9*pi;

num1 = [1 3 5 7 9 11 13 15 17 8 9 1 7];

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From these plots we make the following observations: Phổ biên độ bị dịch theo w0

xlabel('Time index');

subplot(2,2,3)plot(w/pi,angle(hp));gridtitle('Sum of Phase Spectra')xlabel('Time index');

subplot(2,2,4)plot(w/pi,angle(h3));gridtitle('Phase Spectrum of Convolved Sequence')

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From these plots we make the following observations:

Q3.16 Program P3_4 was run for the following two different sets of

sequences of varying lengths –

x1 = [1 3 5 7 9 11 8 -7 5 3];

x2 = [1 -2 3 -2 1 5 9 2];

From these plots we make the following observations:Phổ biên độ và pha không khác

biệt với tín hiệu gốc

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subplot(3,1,3)plot(w/pi,abs(h3));gridtitle('Magnitude Spectrum of ProductSequence')

Q3.18 The plots generated by running the modified program are given below:

From these plots we make the following observations: Phổ biên độ giảm khi nhân 2 tín hiệu

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

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From these plots we make the following observations: Phổ biên độ giảm khi nhân 2 tín hiệu

subplot(2,2,4)plot(w/pi,angle(h3));gridtitle('Phase Spectrum of Time-Reversed Sequence')

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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 7];

From these plots we make the following observations: Đảo phổ pha

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] isgiven below:

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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 computingdirectly the two N-point DFTs is given below:

< Insert program code here Copy from m-file(s) and paste >

The DFTs generated by running the program for sequences of different lengths

N are shown below:

< Insert MATLAB figure(s) here Copy from figure window(s) andpaste >

Q3.25 The MATLAB program to compute the 2N-point DFT of a length-2N real

sequence using a single N-point DFT and compare the result by computingdirectly the 2N-point DFT is shown below:

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The DFTs generated by running the program for sequences of different lengths2N are shown below:

Project 3.4 DFT Properties

Answers:

Q3.26 The purpose of the command rem in the function circshift is –Lấy phần dư sau

khi chia M cho length(x)

Q3.27 The function circshift operates as follows: Dịch vòng

Q3.28 The purpose of the operator ~= in the function circonv is –Không bằng,L1 khác

L2

Q3.29 The function circonv operates as follows: Tổng chập vòng

and adding program statements for labeling each plot being generated by theprogram is given below:

The parameter determining the amount of time-shifting is - M=6

If the amount of time-shift is greater than the sequence length then – Vẫn dịch

theo vòng

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and adding program statements for labeling each plot being generated by theprogram is given below:

subplot(2,2,3)stem(n,angle(XF));gridtitle('Phase of DFT of Original Sequence');

subplot(2,2,4)stem(n,angle(YF));gridtitle('Phase of DFT of Circularly Shifted Sequence');

The amount of time-shift is - 5

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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 shownbelow:

Q3.35 The plots generated by running the modified program for the following two

different sequences of different lengths, with the lengths indicated, are shownbelow:

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

running this program:

Result of circular convolution = 12 28 14 0 16 14

Result of IDFT of the DFT products = 12 28 14 0 16 14

From these plots we make the following observations: Hai kết quả giống nhau

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

sequences:

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The plots generated are shown below:

Result of circular convolution = 58 86 74 54 34 22 50 54

Result of IDFT of the DFT products = 58 86 74 54 34 22 50 54

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

running this program:

Linear convolution via circular convolution = 2 6 10 15 21 15 7 9 5Direct linear convolution = 2 6 10 15 21 15 7 9 5

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

unequal lengths:

Linear convolution via circular convolution = 3 8 15 22 30 39 48 57 39

27 13 15 8

Direct linear convolution = 3 8 15 22 30 39 48 57 39 27 13 15 8

From these plots we make the following observations: Hai kết quả khác nhau

Q3.40 The MATLAB program to develop the linear convolution of two sequences via

the DFT of each is given below:

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

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

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

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

subplot(2,2,4);

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

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

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

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

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

The imaginary part of XEF is/is not equal to zero This result can beexplained as follows:

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:

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

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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 transform is given below:

Q3.48 From the pole-zero plot generated in Question Q3.47, the number of regions of

convergence (ROC) of G(z) are –2

All possible ROCs of this z-transform are sketched below: 0.5≤ |z|≤0.7

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From the pole-zero plot it can be seen that the DTFT -

Q3.49 The MATLAB program to compute and display the rational z-transform from its

zeros, poles and gain constant is given below:

b = [2 1];

a = [1 -0.6 ];

[b,a] = eqtflength(b,a);% Make lengths equal

[z,p,k] = tf2zp(b,a) The rational form of a z-transform with the given poles, zeros, and gain is found

Q3.50 The MATLAB program to compute the first L samples of the inverse of a rational

z-transform is given below:

L=input('sample of the inverse','L');

num=[2 5 9 5 3];

den=[5 45 2 1 1];

impz(num,den,L)The plot of the first 50 samples of the inverse of G(z) of Q3.46 obtained using thisprogram is sketched below:

num=[2 5 9 5 3];

den=[5 45 2 1 1];

impz(num,den,50)

Q3.51 The MATLAB program to determine the partial-fraction expansion of a rational

z-transform is given below:

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From the above partial-fraction expansion we arrive at the inverse z-transformg[n] as shown below:

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