Exp 1 Introduction to Signal and System

ENT 281 SIGNALS AND SYSTEMS

EXPERIMENT 1

INTRODUCTION ON SIGNALS AND SYSTEMS

1.0 OBJECTIVES:

After completing this section you will be able to :

1.1 Generate basic signals.

1.2 Differentiate type of signaling generated.

2.0 TOOLS

The MATLAB Signal Processing Toolbox.

3.0 M-FILES

It is useful for a large number of commands. We can write and save our commands in

script files called M-Files. When an M-File is run, MATLAB sequentially executes the

commands found in the file. The advantage of having M-File is that commands are saved

and can be easily modified without retyping the entire list of commands.

Example 3.0.1

Start by creating an M-File file:

Type this text in new M-File

t = linspace(1,3,200) %---- to create 200 points between 1 to 3

x1 = t. ^ 2; %---- signal x1,

x2 = sin ( 10.*t); %---- signal x2,

x = x1.*x2; %---- Product of signals.

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Then, Save As ( to (C:) Directory ) as Revp1.m (as File name)

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Example 3.0.2

Consider the signal,

x(t) = t2 sin (10t), 0 ≤ t ≤ 3 ___________ (i)

We would like to plot this signal. To generate the signal in (i), two intermediate signals are

defined.

x1(t) = t2 , and x2(t) = sin (10t) ___________ (ii)

Then the signal in (i) is the product of the signals in (ii).

Solution to the above example;

clear % generate signal;type Revp1 .m % recall from m.filesplot (t,x) % Plot Signalxlabel (‘time’) % Label x-axisylabel (‘x’) % Label y-axistitle (‘Product of Function’) % Puts a title on the plot

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3.1 OVERLAY PLOTS

There are three different ways of generating overlay plots in MATLAB:

the plot, hold, and line commands

Plot command

For example, If we have three sets of data, and ,

the command plot ;

Plots , with a solid line, , with a dotted line, and as

uncorrected points marked by small circles ‘o’, all on the same graph.

Consider the three signals below and using plot command to generate overlay plots.

, and

Plot 100 points vector, sets the x-axis from 0 to 5 and y-axis from -1 to 5.

Example 3.1.1

>> clear

>> t = linspace (0, 2.*pi, 100); % generate vector t

>> y1 = sin (t); y2 = t;

>> y3 = t-(t.^3)./6 + (t.^5)/120;

>> plot (t, y1, t, y2, ‘--‘, t, y3, ‘o’)

>> axis ( [0 5 -1 5] ) % Zoom in new axis limits

>> xlabel (‘t’)

>> ylabel (‘Approximations of sin (t)’)

>> title (‘Fun with sin (t)’)

>> text (3.5,0, ‘sin (t)’)

>> gtext (‘Linear approximation’)

>> gtext (‘First 3 terms’)

>> gtext (‘in Taylor Series’)

“gtext” writes the specified string at a location clicked with the mouse in the graphics

window. So, after hitting return at the end of gtext command, go to the graphic window

and click a location.

Example 3.1.2

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, 0 ≤ x ≤ 20 , using plot command

x = 0:0.1:20; % create vector x

y = exp(0.1.*x).*sin(x); % calculate y

plot (x,y) % plot x vs. y

xlabel (‘Time (t) in seconds’) % label x-axis

ylabel (‘The Response Amplitude in mm’) % label y-axis

title (‘A Simple 2-D Plot’) % put a title

Example 3.1.3

, 0 ≤ x ≤ 20 , using fplot

fplot (‘exp(0.1.*x).*sin(x)’,[0,20])

xlabel (‘x’),ylabel(‘f(x)=e^{x/10}sin(x)’)

title (‘A function plotted with fplot’)

Example 3.1.4

In this example, the magnitude of “F(Z)” is manipulated by MATLAB program.

F (Z) =

along the path Z = x + j2 ; -6 ≤ x ≤ 4. Create 100 points between -6 to 4.

Solution Example 3.1.4

>> clear

>> x = linspace (-6 , 4,100) ; % Create a 100 point independent variable

>> Z = x + 2i ; % now the range of Z has been defined

>> F = 2*(Z+4)./(Z-1); % Calculate the function for each value of Z

>> plot (x, 180*angle (F)/pi) % plot phase in degrees

>> grid

>> xlabel (‘x’)

>> ylabel (‘phase in degrees’)

3.2 INTRODUCTION TO SIGNALS

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fplot (‘function’,[Xmin Xmax])


A signal is formally defined as a function of one or more variables that conveys

information on the nature of a physical phenomenon.

The function which is chosen to describe a signal is called the representation of

the signal.

A signal that depends on the discrete variable n and that model a physical

variable that evolves in discrete time is called a discrete-time signal.

A signal that depends on the continuous variable t and that model a physical

variable that evolves in continuous time is called a continuous-time signal.

A system is formally defined as an entity that manipulates one or more signals to

accomplish a function, thereby yielding new signals.

The interaction between a system and its associated signal is illustrated

schematically in figure below;

Input signal Output Signal

If the input signal and output signals of a system are continuous time signal, then

the system is called a continuous-time system.

If the input signal and output signals of a system are discrete-time signals, then

the system is called a discrete-time system.

3.2.1 EVEN AND ODD SIGNALS

Even and odd signals bear some important symmetry properties. Under

reversal of independent variable, these signals either remain the same (even

signal) or get reflected or flipped (odd signal) about the horizontal axis.

Equations or definitions (3.1) and (3.2) mathematically express these properties for

both continuous and discrete time cases.

Even Signals: x(t) = x(-t), x[n] = x[-n] (3.1)

Odd Signals: x(t) = -x(-t), x[n] = -x[-n] (3.2)

To explore even and odd signals in continuous time, let us create one

even and one odd signal. The even one will be a cosine and the odd will be a

sine, both will have frequency of 1 Hz, defined over a time axis extending from

-1 to 1 with a sampling interval of 0.01s, each totaling 201 points or samples.

» f=1;

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H y (t)x (t)


» t=-1:0.01:1;

» xe=cos(2*pi*f*t);

» xo=sin(2*pi*f*t);

We can now see what happens to our functions under time reversal. We

first examine the even one, and then the odd one.

» subplot(2,2,1);plot(t,xe);

» subplot(2,2,2);plot(-t,xe);

» subplot(2,2,3);plot(t,xo);

» subplot(2,2,4);plot(-t,xo);

What do you observe? How do you interpret the pictures you got?

3.2.2 PERIODIC SIGNALS

Periodic signals can be generated by square waves and triangular waves. To

generate a square waves signals, use the basic command:

A*square(w0*t + rho);

A is an amplitude, w0 is a fundamental frequency measured rad/sec and rho is a duty

cycle for which the signal is positive.

Example

Use the following set of commands for generating:

(i) Square waves

A = 1 and Period, T = 0.2s * f = 1/T ; = 2πf = 2/T

>> A = 1; % Amplitude

>> w0 = 10*pi; % Fundamental frequency

>> rho = 0.5; % Duty cycle

>> t = 0: .001 : 1; % To create 1000 time vector points

>> sq = A*square(w0*t + rho);

>> plot (t,sq)

>> axis ( [ 0 1 -1.1 1.1 ] ) % Sets graph axes: axis([xstart xend ystart yend])

Explanation: The plot command is used to view the square waves : the command

plot draws lines connecting the successive values of the signal and thus gives

the appearance of a continuous-time signal.

(ii) Triangular waves

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A*sawtooth (w0*t,w);

>> A = 1;

>> w0 = 10*pi;

>> w = 0.5;

>> t = 0: .001 : 1;

>> tri = A*sawtooth(w0*t,w);

>> plot (t,tri)

Explanation: The plot command is used to view the triangular waves : the

command plot draws lines connecting the successive values of the signal and

thus gives the appearance of a continuous-time signal.

To visualize a discrete-time signal, used 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 vector n and x must have compatible dimensions.

Example

The discrete-time square wave is generated by using the following commands;

>> A = 1;

>> omega = pi/4;

>> n = -10: 10; % Starting point from -10 to 10

>> x = A*square(omega*n);

>> stem (n,x)

3.2.3 EXPONENTIAL SIGNALS

Exponential signal is written as x(t) = Beat , where both B and a are parameters.

The parameter B is the amplitude of the exponential signal measured at time t = 0.

Depending on whether the other parameter a is positive or negative, its identify

two special cases:

Decaying exponential, for it which a < 0, B*exp (-a*t);

Growing exponential, for it which a > 0, B*exp (a*t);

Example

>> B = 5;

>> a = 6;

>> t = 0: .001: 1;

>> x = B*exp (-a*t); % Decaying exponential

>> plot (t,x)

Explanation: The plot command is used to view the decaying exponential signal.

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3.2.4 SINUSOIDAL SIGNALS

A*cos (w0*t + phi) ; for continuous

A*sin (w0*t + phi); (replace ‘t’ with ‘n’ for discrete)

These two commands are basically to generate the sinusoidal signals. A cosine

signal of amplitude A, frequency w0 (measured in radians per second), and phase

angle phi (in radians) is obtained by using the command.

Example

% generate of a sinusoidal sequence for discrete signal

>> n = 0:40;>> f = 0.1;>> phase = 0;>> A=1.5;>> arg = (2*pi*f*n) – phase;>> x = A*cos(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;

3.2.5 STEP, IMPULSE, AND RAMP FUNCTIONS

Step Function

A unit-amplitude step function is generated by unity

u = [zeros (1, 50), ones (1, 50)];

Discrete-time Impulse Function

Delta = [Zeros (1, 49), 1, zeros (1, 49)];

Ramp Function

To generate a ramp sequence, we simply write

ramp = 0: .1: 10

Example

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Try these commands and observe the output

x(t) = A , 0≤ ׀t 0.5 ≥ ׀ t׀ 0 0.5 > ׀

>> t = -1: 1/500 : 1 % time running from 1 second to 1 second

>> u1 = [ zeros (1,250) , ones ( 1, 751)]; % beginning at t = -0.5 second

>> u2 = [ zeros (1,751) , ones ( 1, 250)]; % begin at t = 0.5

>> u = u1 – u2; % produce a rectangular pulse of unit amplitude and unit

duration centered on the origin.

>> plot(t,u)

Example

Try this command and generate the signal as shown in Figure 1 below;

>> dt = 0.05; % time increment for plotting>> t1 = [-1 : dt : -dt]; % first interval>> x1 = ones (size (t1)); % signal on first interval>> t2 = [0: dt : 1]; % second interval>> x2 = 1-t2; % signal on 2nd interval>> t3 = [1+dt : dt : 2]; % third signal>> x3 = 2*ones(size(t3)); % signal on 3rd interval>> t4 = [ 2+dt: dt: 4]; % fourth interval>> x4 = (t4-3); % signal on 4th interval>> t = [ t1, t2, t3, t4]; % total time interval>> x = [ x1, x2, x3, x4]; % total signal>> plot (t,x)

Figure 1

3.3 COMMANDS RELATED SIGNALS TO SYSTEMS

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The following commands are organized by topics in signals and systems. Each of

these commands has a number of options that extend its usefulness.

3.4 PLOTTING

Plotting is a basic skill in MATLAB that will be used frequently. Basically, a signal

can be represented by a vector that represents values of the signal at specified

point in time.

3.4.1 Simple plotting commands

The simple 2D plotting commands include

plot Plot in linear coordinates as a continuous function

stem Plot in linear coordinates as discrete samples

loglog Logarithmic x and y axes

semilogx Linear y and logarithmic x axes

semilogy Linear x and logarithmic y axes

bar Bar graph

errorbar Error bar graph

hist Histogram

3.4.2 Customization of plots

There are many commands used to customize plots by annotations, titles,

axes labels, etc.

A few of the most frequently used commands are

xlabels Labels x-axis

ylabel Labels y-axis

title Puts a title on the plot

grid Adds a grid to the plot

gtext Allows positioning of text with the mouse

text Allows placing text at specified coordinates of the plot

axis Allows changing the x and y axes

figure Create a figure for plotting

figure (n) Make figure number n the current figure

hold on Allows multiple plots to be superimposed on the same axes

hold off Release hold on current plot

close (n) Close figure number n

subplot (a,b,c) Create an a x b matrix of plots with c the current figure

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orient Specify orientation of a figure

String modifiers can be used to change color, data point, and line styles.

Colors Point style Line style

b blue . point - solid

g green o circle : dotted

r red x x-mark -. dashdot

c cyan + plus -- dashed

m magenta * star (none) no line

y yellow s square

k black d diamond

v triangle (down)

^ triangle (up)

< triangle (left)

> triangle (right)

p pentagram

h hexagram

3.5 Polynomials

Polynomials arise frequently in systems theory. MATLAB represents polynomials

as row vectors of polynomial coefficients. For example, the polynomial s2 + 4s – 5

is represented in MATLAB by the polynomial >> p = [1 4 -5]. The following is a

list of the more important commands for manipulating polynomials.

roots(p) Express the roots of polynomial p as a column vector.

polyval(p,x) Evaluate the polynomial p at the values contained in the vector x.

conv(p1,p2) Compute the product of the polynomials p1 and p2

deconv(p1,p2) Compute the quotient of p1 divided by p2

poly2str(p,’s’) Display the polynomial as an equation in s

( asn+bsn-1+csn2+……+……)

poly(r) Compute the polynomial given a column vector of roots r

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TASKS OF THE DAY

1. Plot the following signal on the same graph and determine whether it’s odd or

even signal. -2 ≤ t ≤ 2

* use subplot command

(i) x(t)=4t

(ii) x(t)=5cos(3t)

(iii) x(t)=5sin(3t)

2. Plot the following signals on the same graph and identify whether it’s periodic or

non periodic signal.

(i) x(t) = cos (10 πt + 8 sin (πt)), 0 ≤ t ≤ 2 for 200 points vectors

(ii) x(t) = cos (10 πt + 8 ( t/4) ) ,

3. Consider the signal

x(t) = A sin (t) + B sin (ωt + θ )

A = 0.9 ; B = 2.5 ; 0 ≤ t ≤ 1

frequency = 1 Hz .

Write a M-File to plot this signal for phase θ = 0, π/2, π, 3π/2, and 2π in one

graph. Investigate the waveforms obtained.

Use string argument to differentiate the signal and “gtext” function

4. This exercise examines the properties of the exponential and sinusoidal signals in

both continuous and discrete forms.

(a) The Continuous Signal, x(t) = 2eat

(i) For values of ‘a’ equal to 0.5,1.0 and 1.5, plot the corresponding

signals all on the same graph using a timescale -3 to +3.

(ii) Repeat (i) for values of ‘a’ equal to -0.5,-1.0 and -1.5, again using

the time scale -3 to +3.

Use ‘subplot’ to show both two plots on one screen

(b) The Discrete Signal, x(n) = 2eanT with T = 0.25

(i) For values of ‘a’ equal to 0.5, 1.0 and 1.5, plot the corresponding

signals using the command ‘stem’ over a range of ‘n’ from n = -12

to n = +12. Plot the three signal in one graph.

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5.

Figure 2

Consider the signal in Figure 2:

(a) Find representation for this signal.

(b) Plot this signal using MATLAB.

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Exp 1 Introduction to Signal and System

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