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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_ % Generation of a Unit Sample Sequence clf; % Generate a vector from -10 to 20 n = -10:20; % Generate the unit sample sequence u = [zeros(1,10) 1 zeros(1,20)]; % Plot the unit sample sequence stem(n,u); xlabel('Time index n');ylabel('Amplitude'); title('Unit Sample Sequence'); axis([-10 20 0 1.2]);
Answers :
Q1.1 The unit sample sequence u[n] generated by running Program P1_1 is shown below:
-10 -5 0 5 10 15 20
0
1
Time index n
Amplitude
Unit Sample Sequence
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 by running this program.
% Program P1_1, MODIFIED for Q1. % Generation of a DELAYED Unit Sample Sequence clf; % Generate a vector from -10 to 20 n = -10:20; % Generate the DELAYED unit sample sequence u = [zeros(1,21) 1 zeros(1,9)]; % Plot the DELAYED unit sample sequence stem(n,u); xlabel('Time index n');ylabel('Amplitude'); title('DELAYED Unit Sample Sequence'); axis([-10 20 0 1.2]);
-10 -5 0 5 10 15 20
0
1
Time index n
Amplitude
DELAYED Unit Sample Sequence
-10 -5 0 5 10 15 20
0
1
Time index n
Amplitude
ADVANCED Unit Step Sequence
Project 1.2 Exponential signals
A copy of Programs P1_2 and P1_3 are given below.
% Program P1_ % Generation of a complex exponential sequence clf; c = -(1/12)+(pi/6)i; K = 2; n = 0:40; x = Kexp(c*n); subplot(2,1,1); stem(n,real(x)); xlabel('Time index n');ylabel('Amplitude'); title('Real part'); subplot(2,1,2); stem(n,imag(x)); xlabel('Time index n');ylabel('Amplitude'); title('Imaginary part');
% Program P1_ % Generation of a real exponential sequence clf; n = 0:35; a = 1.2; K = 0.2; x = K*a.^n; stem(n,x); xlabel('Time index n');ylabel('Amplitude');
Answers:
Q1.6 The complex-valued exponential sequence generated by running Program P1_2 is shown
below:
Time index n
Amplitude
Real part
Time index n
Amplitude
Imaginary part
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 the envelope of
the signal will grow with n instead of decay with n.
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.
Q1.14 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:
Time index n
Amplitude
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 an example
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.
and computed using the command sum are - 4.5673e+004 and 2.1042e+003.
Project 1.3 Sinusoidal sequences
A copy of Program P1_4 is given below.
% Program P1_ % Generation of a sinusoidal sequence n = 0:40; f = 0.1; phase = 0; A = 1.5; arg = 2pifn - phase; x = Acos(arg); clf; % Clear old graph stem(n,x); % Plot the generated sequence axis([0 40 -2 2]); grid; title('Sinusoidal Sequence'); xlabel('Time index n'); ylabel('Amplitude'); axis;
Answers:
Q1.17 The sinusoidal sequence generated by running Program P1_4 is displayed below.
Sinusoidal Sequence
Time index n
Amplitude
Sinusoidal Sequence
Time index n
Amplitude
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.
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
is identical to the one we got in Q1.17 with f = +0.1 Hz/sample.
A sinusoidal sequence of frequency 1.1 generated by modifying Program P1_4 is shown below.
Sinusoidal Sequence
Time index n
Amplitude
A comparison of this new sequence with the one generated in Question Q1.17 shows - the
graph here is again identical to the one in Q1.17. This is because a cosine of frequency
f = 1.1 Hz/sample is identical to one with frequency f = 1.1 – 1 = 0.1 Hz/sample, which
was the frequency used in Q1.17.
Q1.24 By replacing the stem command in Program P1_4 with the plot command, the plot
obtained is as shown below:
0 5 10 15 20 25 30 35 40
-1.
-0.
0
1
2
Sinusoidal Sequence
Time index n
Amplitude
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 straight line segments, which approximates the graph of a
continuous-time cosine signal.
Q1.25 By replacing the stem command in Program P1_4 with the stairs command the plot
obtained is as shown below:
0 5 10 15 20 25 30 35 40
-1.
-0.
0
1
2
Sinusoidal Sequence
Time index n
Amplitude
The difference between the new plot and those generated in Questions Q1.17 and Q1.24 is –
the “stairs” command produces a stairstep plot as opposed to the stem graph that
was generated in Q1.17 and the straight-line interpolation plot that was generated in
Q1.24.
Project 1.4 Random signals
Answers:
Q1.26 The MATLAB program to generate and display a random signal of length 100 with elements uniformly distributed in the interval [–2, 2] is given below along with the plot of the random
sequence generated by running the program:
% Program Q1_ % Generation of a uniform random sequence n = 0:99; A = 2; rand('state',sum(100clock)); % "seed" the generator % rand(1,100) is uniform in [0,1] % rand(1,100)-0.5 is uniform in [-0.5,0.5] % 4(rand(1,100)-0.5) is uniform in [-2,2] x = 2A(rand(1,length(n))-0.5); clf; % Clear old graph stem(n,x); % Plot the generated sequence axis([0 length(n) -round(2(A+0.5))/2 round(2(A+0.5))/2]); grid; title('Uniform Random Sequence'); xlabel('Time index n'); ylabel('Amplitude'); axis;
Gaussian Random Sequence
Time index n
Amplitude
Q1.28 The MATLAB program to generate and display five sample sequences of a random sinusoidal signal of length 31
{X[n]} = {A·cos(ωon + φ)}
where the amplitude A and the phase φ are statistically independent random variables with
uniform probability distribution in the range 0 ≤ A ≤ 4 for the amplitude and in the range
0 ≤ φ ≤ 2π for the phase is given below. Also shown are five sample sequences generated by
running this program five different times.
% Generates the "deterministic stochastic process" % called for in Q1.28. n = 0:30; f = 0.1; Amax = 4; phimax = 2pi; rand('state',sum(100clock)); % "seed" the generator A = Amaxrand; % NOTE: successive calls to rand without arguments % return a random sequence of scalars. Since this % random sequence is "white" (uncorrelated), it is % not necessary to re-seed the generator for phi. phi = phimaxrand; % generate the sequence arg = 2pifn + phi; x = Acos(arg); % plot clf; % Clear old graph
stem(n,x); % Plot the generated sequence Ylim = round(2*(Amax+0.5))/2; axis([0 length(n) -Ylim Ylim]); grid; title('Sinusoidal Sequence with Random Amplitude and Phase'); xlabel('Time index n'); ylabel('Amplitude'); axis;
0 5 10 15 20 25 30
0
1
2
3
4
Sinusoidal Sequence with Random Amplitude and Phase
Time index n
Amplitude
0 5 10 15 20 25 30
0
1
2
3
4
Sinusoidal Sequence with Random Amplitude and Phase
Time index n
Amplitude
0 5 10 15 20 25 30
0
1
2
3
4
Sinusoidal Sequence with Random Amplitude and Phase
Time index n
Amplitude
1.2 SIMPLE OPERATIONS ON SEQUENCES
Project 1.5 Signal Smoothing
A copy of Program P1_5 is given below.
% Program P1_ % Signal Smoothing by Averaging clf; R = 51; d = 0.8(rand(R,1) - 0.5); % Generate random noise m = 0:R-1; s = 2m.*(0.9.^m); % Generate uncorrupted signal x = s + d'; % Generate noise corrupted signal subplot(2,1,1); plot(m,d','r-',m,s,'g--',m,x,'b-.'); xlabel('Time index n');ylabel('Amplitude'); legend('d[n] ','s[n] ','x[n] '); x1 = [0 0 x];x2 = [0 x 0];x3 = [x 0 0]; y = (x1 + x2 + x3)/3; subplot(2,1,2); plot(m,y(2:R+1),'r-',m,s,'g--'); legend( 'y[n] ','s[n] ');
xlabel('Time index n');ylabel('Amplitude');
Answers:
Q1.29 The signals generated by running Program P1_5 are displayed below:
0 5 10 15 20 25 30 35 40 45 50
0
5
10
Time index n
Amplitude
d[n] s[n] x[n]
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
Time index n
Amplitude
y[n] s[n]
Q1.30 The uncorrupted signal s[n]is - the product of a linear growth with a slowly decaying
real exponential.
The additive noise d[n]is – a random sequence uniformly distributed between -0.4 and
Q1.31 The statement x = s + d CANNOT be used to generate the noise corrupted signal
because – d is a column vector, whereas s is a row vector; it is necessary to transpose
one of these vectors before adding them.
Q1.32 The relations between the signals x1, x2, and x3, and the signal x are – all three signals
x1, x2, and x3 are extended versions of x, with one additional sample appended at the
left and one additional sample appended to the right. The signal x1 is a delayed
version of x, shifted one sample to the right with zero padding on the left. The signal
x2 is equal to x, with equal zero padding on both the left and right to account for the
extended length. Finally, x3 is a time advanced version of x, shifted one sample to the
left with zero padding on the right.
