Mehran University of Engineering and Technology, Jamshoro
Department of Computer Systems Engineering
Analogue and Digital Signal Processing
Lab 2:
Generation of Commonly Used Signals
1.
•
•
Objectives:
After this lab, students should be able to:
Generate and plot commonly used
continuous time signals.
Generate and plot commonly used discrete
time signals
Software Required: MATLAB with Signal
Processing Toolbox
2.0
Generation of
Periodic Signals:
Continuous
time
2.1
Continuous time Periodic Triangular
Wave:
MATLAB has a built-in command sawtooth to
generate a periodic triangular waveform.
Following example help you draw such a
waveform:
Example: Draw a triangular wave of amplitude
= 2 units, a frequency of π radians per second
and a width of 0.5 units.
Solution: Use the following code
A=2;
% amplitude
w =pi;
% frequency
W=0.5;
% width
t=0:0.001:10;
% time vector
y=A*sawtooth(w*t+W);
plot(t,y)
xlabel(‘Time in seconds’)
ylabel(‘Triangular Wave’)
2.2 Continuous time Periodic Square Wave
A periodic square wave can be generated by
using the following command:
A*square(w0*t);
Example: Generate a square wave of amplitude
1, fundamental frequency 10π radian per second.
A=1;
w0=10*pi;
t=0:0.001:1; % you may also use linspace here
q=A*square(w0*t);
plot(t, q)
axis([0 1 -2 2])
% optional
2.3 Discrete time signals
To visualize a discrete time signal, we may use
the stem command. Specifically, stem(n,x)
depicts the data contained in vector x as a
discrete time signal at the time values defined by
n. The vectors n and x must, of course, have
compatible dimensions.
Consider, for example, the discrete time square
wave, with frequency π/4 radians per second and
amplitude = 1 unit. This signal is generated
using the following commands:
A=1;
w=pi/4;
n=-10:1:10;
x=A*square(w*n);
stem(n,x)
Exercise 1:
(a) Generate a discrete time triangular wave of
unity amplitude with width 0 and frequency π/8
radians per second. Plot for 100 seconds (0 to
100 sec).
(b) Draw the following sinusoidal signals for 100
seconds:
(i) Acos(wn + φ) (ii) Asin(wn + φ)
where A = 4; w = π/8 and φ = 30 degrees.
Note: Convert degrees into radians.
2.4 Exponential Signals:
There are two types of exponential signals:
decaying exponentials and growing exponentials.
The MATLAB command for decaying
exponential Be-at is
B*exp(-a*t);
To generate a growing exponential Beat ,we use
command
B*exp(a*t);
In both cases, the exponential parameter a is
positive.
Exercise 2: Draw the following signals:
(a) x(t) = 5e-6t (b) y(t) = 3e5t
(b) x(n) = 2(0.85)n (c) z(t) = 60sin(20πt)e-6t
(d) y(n) = 60sin(20πn)e-6n.
2.5 Other signals:
A discrete time unit step function u(n) may be
created as follows:
n = 0:1:20;
x = ones(1, length(n));
stem(n,x)
axis([-1 25 0 2])
% optional
Exercise 3:
(i) Draw the signal x[n] = n (ramp function)
(ii) Plot continuous time and discrete time sinc
function (i.e. sin(x)/x) for x between -5 and 5.
Use the built-in function sinc.
(iii) Plot a rectangular function of width 3 units.
(use the built-in function “rectpuls”.
(iv) Draw a discrete time triangular pulse using
the built-in function “tripuls”. Plot the function
from -3 to 3 seconds.
(v) Find and plot u(n) – u(n-5), where u(n) is a
discrete time unit step signal.
2.6 Even and Odd Signals
The even and odd parts of a signal x(t) are
computed by using the following formula:
1
[x(t ) + x(− t )]
2
1
x o (t ) = [x(t ) − x( − t )]
2
xe (t ) =
Example: Plot the signal given below. Also plot
even and odd parts of the signal.
1
-1
1
Solution: Mathematically, the above signal is
represented as follows:
 12 (t + 1) − 1 ≤ t ≤ 1
x(t ) = 
elsewhere
 0
This signal can be plotted by using the following
code
t = -1:0.0001:1;
x = ½*(t + 1);
plot(t, x), grid
Now even and odd parts of the signal may be
plotted as follows:
x1 = fliplr(x);
% folded version of the signal
xe = ½*(x + x1); % Even part
xo = ½*(x – xe); % Odd part
subplot(2,1,1)
plot(t, xe)
xlabel (‘Time’)
ylabel(‘Even part of x(t)’)
subplot(2,1,2)
plot (t, xo)
xlabel(‘Time’)
ylabel(‘Odd part of x(t)’)
Exercise 4: (a) Plot the following signals for
-5≤t≤5 seconds. Also plot even and odd parts of
each.
(i)
x(t) = cos(t )+ sin(t) + cos(t)sin(t)
(ii)
x(t) = (1 + t3)cos3(10t)
(b) Repeat exercise 4(a) for the following
discrete time signals.
(i) x(n) = [1 1 1 1 0 0 0];
(ii) y(n) = [1 0 0 1 0 1 1 0 1 1]
(c) Compute the energy of x(n) and y(n) in part
(b). Note: Energy of a signal x(n) may be
computed as sum(abs(x.^2)).