tai lieu hoc DSP ( xử lý số tín hiệu 7 labex )

BÁO CÁO CUỐI KÌ THÍ NGHIỆM DSP Laboratory Exercise 1 DISCRETE TIME SIGNALS TIME DOMAIN REPRESENTATION 1 1 GENERATION OF SEQUENCES Project 1 1 Unit sample and unit step sequences A copy of Program P1 1 is given below % Program P1 1 clf; n = 10 20; u = zeros(1,10) 1 zeros(1,20); stem(n,u); xlabel(Time index n);ylabel(Amplitude); title(Unit Sample Sequence); axis( 10 20 0 1 2); Q1 1 The unit sample sequence un generated by running Program P1 1 is shown below Q1 2 The purpose of clf comm.

BÁO CÁO CUỐI KÌ THÍ NGHIỆM DSP

Laboratory Exercise 1 DISCRETE-TIME SIGNALS: TIME-DOMAIN REPRESENTATION

1.1 GENERATION OF SEQUENCES

Project 1.1 Unit sample and unit step sequences

A copy of Program P1_1 is given below

Q1.2 The purpose of clf command is – clear the current figure

The purpose of axis command is – control axis scaling and appearance

The purpose of title command is – add a title to a graph or an axis and specify text properties

The purpose of xlabel command is – add a label to the x-axis and

specify text properties

The purpose of ylabel command is – add a label to the y-axis and specify the text properties

Q1.3 The modified Program P1_1 to generate a delayed unit sample sequence ud[n]

with a delay of 11 samples is given below along with the sequence generated byrunning this program

Q1.4 The modified Program P1_1 to generate a unit step sequence s[n] is

given below along with the sequence generated by running thisprogram

Q1.5 The modified Program P1_1 to generate a unit step sequence sd[n]

with an advance of 7 samples is given below along with thesequence generated by running this program

title('ADVANCED Unit Step

Sequence'); axis([-10 20 0 1.2]);

Project 1.2 Exponential signals

A copy of Programs P1_2 and P1_3 are given below

% Program P1_2

4 4

Q1.6 The complex-valued exponential sequence generated by running

Program P1_2 is shown below:

Q1.7 The parameter controlling the rate of growth or decay of this sequence is – the real part of c

The parameter controlling the amplitude of this sequence is - K

Q1.8 The result of changing the parameter c to (1/12)+(pi/6)*i is – since exp(-1/12) is

less than one and exp(1/12) is greater than one, this change means that theenvelope of the signal will grow with n instead of decay with n

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Q1.9 The purpose of the operator real is – to extract the real part of a Matlab vector

The purpose of the operator imag is – to extract the imaginary part of a Matlab vector

Q1.10 The purpose of the command subplot is – to plot more than one graph in the same

Matlab figure

Q1.11 The real-valued exponential sequence generated by running Program

P1_3 is shown below:

Q1.12 The parameter controlling the rate of growth or decay of this sequence is - a

The parameter controlling the amplitude of this sequence is - K

Q1.13 The difference between the arithmetic operators ^ and ^ is – “^” raises a squarematrix to

a power using matrix multiplication “.^” raises the elements of a

matrix or vector to a power; this is a “pointwise” operation

Q1.14 The sequence generated by running Program P1_3 with the

parameter a changed to 0.9 and the parameter K changed to 20 is

shown below:

Q1.15 The length of this sequence is - 36

It is controlled by the following MATLAB command line: n = 0:35;

It can be changed to generate sequences with different lengths as follows (give anexample command line and the corresponding length): n = 0:99; makes the length

100

Q1.16 The energies of the real-valued exponential sequences x[n]generated in Q1.11

and Q1.14 and computed using the command sum are - 4.5673*e+004 and2.1042*e+003

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Project 1.3 Sinusoidal sequences

A copy of Program P1_4 is given below

Q1.18 The frequency of this sequence is - f = 0.1 cycles/sample

It is controlled by the following MATLAB command line: f = 0.1;

A sequence with new frequency 0.2 can be generated by the following command line:

f = 0.2;

The parameter controlling the phase of this sequence is - phase

10 10

The parameter controlling the amplitude of this sequence is - A

The period of this sequence is - 2π/ω = 1/f = 10

Q1.1

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The length of this sequence is - 41

It is controlled by the following MATLAB command line: n = 0:40;

A sequence with new length 81_ can be generated by the following command line:

2 The modified Program P1_4 to generate a sinusoidal sequence of frequency 0.9 is

given below along with the sequence generated by running it

A comparison of this new sequence with the one generated in Question Q1.17 shows - the

two graphs are identical This is because, in the modified program,

we have ω = 0.9*2π This generates the same graph as a cosine

with angular frequency ω - 2π = −0.1*2π Because cosine is an

even function, this is the same as a cosine with angular frequency

+0.1*2π, which was the value used in P1_4.m in Q1.17

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In terms of Hertzian frequency, we have for P1_4.m in Q1.17 that f

= 0.1 Hz/sample For the modified program in Q1.22, we have f =0.9 Hz/sample, which generates the same graph as f = 0.9 – 1 =

−0.1 Again because cosine is even, this makes a graph that isidentical to the one we got in Q1.17 with f = +0.1 Hz/sample

A sinusoidal sequence of frequency 1.1 generated by modifying ProgramP1_4 is shown below

A comparison of this new sequence with the one generated in Question

Q1.17 shows - the graph here is again identical to the one inQ1.17 This is because a cosine of frequency f = 1.1 Hz/sample isidentical to one with frequency f = 1.1 – 1 = 0.1 Hz/sample, whichwas the frequency used in Q1.17

Q1.23 The sinusoidal sequence of length 50, frequency 0.08, amplitude 2.5,

and phase shift of 90 degrees generated by modifying ProgramP1_4 is displayed below

NOTE: for this program, it is necessary to convert the phase of 90 deg to

radians and account for the minus sign that appears in thestatement “arg = 2*pi*f*n - phase;” as opposed to the plus signshown in eq (1.12) of the lab manual The correct statement to

generate the phase is “phase = -90/(2*pi);” It is also necessary tomodify the axis command to account for the new length andamplitude of the signal The correct axis statement is “axis([0 50-3 3]);”

The period of this sequence is - 2π/ω = 1/f = 1/0.08 = 1/(8/100) = 100/8 =

25/2 Therefore, the fundamental period is 25 and the graph hasthe “appearance” of going through 2 cycles of a cosine waveformduring each period

Q1.24 By replacing the stem command in Program P1_4 with the plot

command, the plot obtained is as shown below:

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The difference between the new plot and the one generated in Question

Q1.17 is – instead of drawing stems from the x-axis to the points

on the curve, the “plot” command connects the points with straightline segments, which approximates the graph of a continuous-timecosine signal

Q1.25 By replacing the stem command in Program P1_4 with the stairs

command the plot obtained is as shown below:

The difference between the new plot and those generated in Questions

Q1.17 and Q1.24 is – the “stairs” command produces a stairstepplot as opposed to the stem graph that was generated in Q1.17 andthe straight-line interpolation plot that was generated in Q1.24

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Laboratory Exercise 2 DISCRETE-TIME SYSTEMS: TIME-DOMAIN REPRESENTATION 2.1 SIMULATION OF DISCRETE-TIME SYSTEMS

Project 2.1 The Moving Average System

A copy of Program P2_1 is given below:

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

title('Signal #1'); subplot(2,2,2); plot(n, s2);

axis([0, 100, -2, 2]);

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

title('Signal #2'); subplot(2,2,3);

plot(n, x); axis([0, 100, -2, 2]);

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

title('Input Signal'); subplot(2,2,4); plot(n, y);

axis([0, 100, -2, 2]);

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

title('Output Signal'); axis;

Answers:

Q2.1 The output sequence generated by running the above program for M = 2 with x[n]

= s1[n]+s2[n] as the input is shown below

Q2.2 Program P2_1 is modified to simulate the LTI system y[n] = 0.5(x[n]–x[n–1])

and process the input x[n] = s1[n]+s2[n] resulting in the output sequence shownbelow:

Note: the code is not required; however, it is included here to demonstrate a tricky way

of making the modification to P2_1

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

title('Signal #1'); subplot(2,2,2); plot(n, s2);

axis([0, 100, -2, 2]);

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

title('Signal #2'); subplot(2,2,3); plot(n, x);

axis([0, 100, -2, 2]);

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

title('Input Signal'); subplot(2,2,4); plot(n, y);

axis([0, 100, -2, 2]);

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

title('Output Signal'); axis;

Q2.3 Program P2_1 is run for the following values of filter length M and following values of the frequencies of the sinusoidal signals s1[n] and s2[n] The output generated for these different values of M and the frequencies are shown below

From these plots we make the following observations – with M=4, this filter performs more smoothing than in the case M=2 Both s1 and s2 are high frequency in this case, and they are both substantially attenuated in the output.

Q2.4 The required modifications to Program P2_1 by changing the input sequence to a

sweptfrequency sinusoidal signal (length 101, minimum frequency 0, and amaximum frequency 0.5) as the input signal (see Program P1_7) are listedbelow:

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

title('Input Signal'); subplot(2,1,2); plot(n, y);

axis([0, 100, -1.5, 1.5]);

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

title('Output Signal'); axis;

The output signal generated by running this program is plotted below

Project 2.2 (Optional) A Simple Nonlinear Discrete-Time System

A copy of Program P2_2 is given below:

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0 20 40 60 80 100 120 140 160 180 200 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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

title('Input Signal') subplot(2,1,2) plot(n,y)

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

title('Output signal');

Answers:

Q2.5 The sinusoidal signals with the following frequencies as the input signals were used

to generate the output signals: f=0.05, f=0.1, f=0.25

The output signals generated for each of the above input signals are displayed below:

Q2.6 The output signal generated by using sinusoidal signals of the form x[n] = cos(won)

+ k as the input signal is shown below for the following values of wo and k -

wo = 0.2pi (f=0.1); K = 0.5

Project 2.3 Linear and Nonlinear Systems

A copy of Program P2_3 is given below:

title('Weighted Output: a \cdot y_{1}[n] + b \cdot y_{2}[n]');

subplot(3,1,3) stem(n,d);

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

title('Difference Signal');

Q2.7 The outputs y[n], obtained with weighted input, and yt[n], obtained by combining

the two outputs y1[n] and y2[n] with the same weights, are shown below along with thedifference between the two signals:

The two sequences are – the same up to numerical roundoff

The system is – Linear

Q2.8 Program P2_3 was run for the following three different sets of values of the

weighting constants, a and b, and the following three different sets of inputfrequencies:

Based on these plots we can conclude that the system with different weights is – Linear

Q2.9 Program 2_3 was run with the following non-zero initial conditions – ic = [5 10];

The plots generated are shown below -

Based on these plots we can conclude that the system with nonzero initialconditions is – Nonlinear

Q2.10 Program P2_3 was run with nonzero initial conditions and for the following three

different sets of values of the weighting constants, a and b, and the followingthree different sets of input frequencies:

Based on these plots we can conclude that the system with nonzero initial conditions and

different weights is – Nonlinear

Q2.11 Program P2_3 was modified to simulate the system:

y2[n]

10

yt[n]

50

Project 2.4 Time-invariant and Time-varying Systems

A copy of Program P2_4 is given below:

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

title('Difference Signal'); grid;

Answers:

Q2.12 The output sequences y[n] and yd[n] generated by

running Program P2_4 are shown below -

These two sequences are related as follows – y[n-10] = yd[n]

The system is – Time Invariant

Q2.13 The output sequences y[n] and yd[n] generated by running Program P2_4 for the

following values of the delay variable D – 2; 6; 8

are shown below -

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In each case, these two sequences are related as follows – y[n-D] = yd[n]

The system is – Time Invariant

Q2.14 The output sequences y[n] and yd[n] generated by running Program P2_4 for the

following values of the input frequencies –

1 f1=0.05; f2=0.40;

2 f1=0.10; f2=0.25;

3 f1=0.15; f2=0.20; are shown below –

In each case, these two sequences are related as follows – y[n-10] = yd[n].

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The system is – Time Invariant

Q2.15 These two sequences are related as follows – yd[n] is NOT equal to the shift of

y[n]

The system is – Time Varying

Q2.16 The output sequences y[n] and yd[n] generated by running Program P2_4 for

non-zero initial conditions and following values of the input frequencies –

The system is – Time Varying

Q2.17 The modified Program 2_4 simulating the system

title(['Output due to Delayed Input x[n -',

num2str(D),']']); grid; subplot(3,1,3)

stem(n,d);

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

Signal'); grid;

The system is – Time Varying

Q2.18 (optional) The modified Program P2_3 to test the linearity of the system of (2.16)

title('Output Due to Weighted Input: a \cdot x_{1}[n] + b \cdot x_{2}

The two sequences are – The same up to numerical roundoff

The system is – Linear

2.5 LINEAR TIME-INVARIANT DISCRETE-TIME

SYSTEMS Project 2.5 Computation of Impulse

Responses of LTI Systems

A copy of Program P2_5 is shown below:

Q2.19 The first 41 samples of the impulse response of the discrete-time system of

Project 2.3 generated by running Program P2_5 is given below:

Q2.20 The required modifications to Program P2_5 to generate the impulse response of

the following causal LTI system:

y[n] + 0.71y[n-1] – 0.46y[n-2] – 0.62y[n-3]

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= 0.9x[n] – 0.45x[n-1] + 0.35x[n-2] + 0.002x[n-3] are given below:

Q2.21 The MATLAB program to generate the impulse response of a causal LTI system

of Q2.20 using the filter command is indicated below:

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

Q2.22 The MATLAB program to generate and plot the step response of a causal LTI

system is indicated below:

Project 2.6 Cascade of LTI Systems

A copy of Program P2_6 is given below:

% Program P2_6 %

clf;

title('Output of 4th order Realization'); grid;

subplot(3,1,2); stem(n,y2) ylabel('Amplitude');

title('Output of Cascade Realization'); grid;

subplot(3,1,3); stem(n,d)

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

Signal'); grid;

Q2.23 The output sequences y[n], y2[n], and the difference signal d[n] generated by running

Program P2_6 are indicated below:

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The relation between y[n]and y2[n] is – They are the SAME up to numerical roundoff

Q2.24 The sequences generated by running Program P2_6 with the input changed to a

sinusoidal sequence are as follows:

The relation between y[n] and y2[n] in this case is – The are the same up to numerical

roundoff

Q2.25 The sequences generated by running Program P2_6 with non-zero initial

condition vectors are now as given below:

The relation between y[n]and y2[n] in this case is – They are NOT the same

Q2.26 The modified Program P2_6 with the two 2nd-order systems in reverse order and

with zero initial conditions is displayed below: