Exercise#

Gaining insight using computers#

To help us answer these questions, let’s use our Mathematical tools to plot a signal like this and explore it. The example we will use is from Signals and Systems for Dummies (SS4D: page 12):

plot 3 cos(2 pi 2 t - 3 pi/4)

MATLAB#

In MATLAB we would need to tackle this by breaking down the steps.

% Make sure that we have a clean workspace
clear all
format compact
setappdata(0, "MKernel_plot_format", 'svg')

Define t

t = linspace(0, 1, 100);

Define x

x = 3 * cos(2*pi*2*t - 3*pi/4);

Plot result and label plot

plot(t,x)
title('A Sinusoidal Signal')
xlabel('Time t (s)')
ylabel('Amplitude')
grid

If you want to execute this in MATLAB, you can create a file by copying this text into an m-file called sinewave.m:

% SINEWAVE - plot function $x(t) = 3 \cos(2\pi t - 3 \pi/4)$ for $t = 0\ldots 1$

%% Set up the problem
% define t
t = linspace(0, 1, 100);
% define x
x = 3 * cos(2*pi*2*t - 3*pi/4);
%% Plot result and label plot
plot(t,x)
title('A Sinusoidal Signal')
xlabel('Time t (s)')
ylabel('Amplitude')
grid

To run this script, just type the filename without the .m extension.

sinewave

(Download sinewave.m)

To run this script, just type the filename without the .m extension. Try adjusting the values of the numerical constants and see what insights you gain.

Returning to the Question#

Sinusoidal signals (e.g. AC) are pretty fundamental in electrical engineering. The mathematical model of a sinusoidal signal is:

Supplementary question#

What is the period of the waveform in seconds?

Notes

• In communications and electronic signal processing, the frequency of sinusoidal signals us usually given in cycles per second or Hz.

• In mathematics, the frequency is always expressed in radians per second.

• In some courses, including later in this one and in EG-243 Control Systems, the frequency $2\pi f_0$ is often called the natural frequency and is usually written ${\omega }_n$.

Try This Yourself#

• Use any or all of computing tools that you have access to to explore other sinusoids. Change the values of the variables and explain what happens.

• Try adding sinusoids of different amplitudes and different frequencies together and see what happens.

• Change cos to sin and see what happens.

Example 1#

An example of a continuous-time system is an electronic amplifier with a gain of 5 and level shift of 2: $y(t)=H\{x(t)\}=5x(t)+2$.

In this course, we will model such systems as block diagram models in Simulink.

Example 2#

Consider the following simple signal, a pulse sequence:

We can plot this in MATLAB as a stem plot

Procedure#

Define function using this text:

% Define the function
function [ y ] = x( n )
  if n < 0 | n >= 10 
      y = 0;
  else 
      y = 5;
  end
end

Save as y.m

%% Define sample points
n = -15:18;
%% Make space for the signal
xn = zeros(size(n));

%% Define sample points
n = -15:18;
%% Make space for the signal
xn = zeros(size(n));

%% Compute the signal x[n]
for i = 1:length(xn)
    xn(i) = y(n(i));
end

%% Plot the result
stem(n,xn)
axis([-15, 18, 0, 6])
title('Stem Plot for a Discrete Signal')
xlabel('Sample n')
ylabel('Signal x[n]')
grid

Periodic#

Signals that repeat over and over are said to be periodic. In mathematical terms, a signal is periodic if:

• Continuous signal $x(t+T)=x(t)$

• Discrete signal $x[n+N]=x[n]$

The smallest $T$ or $N$ for which the equality holds is the signal period.

The sinusoidal signal we saw earlier is periodic because of the $\mathrm{mod}2\pi$ property of cosines. The signal of the sinusoid has a period 0.5 seconds (s), which turns out to be the reciprocal of the frequency $1/f_0$ Hz.

Square Wave#

This code generates a square wave.

%% A Periodic signal (square wave)
t = linspace(0, 1, 500);
x = square(2 * pi * 5 * t);

This Square wave is a 5 Hz waveform sampled at 500 Hz for 1 second

plot(t, x);
ylim([-2, 2]);
grid()
title('A Periodic Signal')
xlabel('Time t (s)')
ylabel('Amplitude')

What is the period $T$ in milliseconds?

Aperiodic#

Signals that are deterministic (completely determined functions of time) but not periodic are known as aperiodic. Point of view matters. If a signal occurs infrequently, you may view it as aperiodic.

This is how we generate an aperiodic rectangular pulse of duration $\tau$ in Matlab:

%% An aperiodic function
tau = 1;
x = linspace(-1,5,1000);
y = rectangularPulse(0,tau,x);

plot(x,y)
ylim([-0.2,1.2])
grid
title('An Aperiodic Signal')
xlabel('Time t (s)')
ylabel('Amplitude')

Random#

A signal is random if one or more signal attributes takes on unpredictable values in a probability sense.

Engineers working with communication receivers are concerned with random signals, especially noise.

%% Plot a Random Signal
plot(0.5 + 0.25 * rand(100,1))
ylim([0,1])
grid
title('Random Signal')
xlabel('Time t (s)')
ylabel('Amplitude')

Other Domains you will encounter#

The most commonly used domains used when analyzing continuous-time signals are the frequency domain ($f$ or $\omega$) and the Laplace $s$-domain ($s$).

Similarly, for discrete-time signals, you may need to transform from the discrete-time domain ($n$) to the frequency domain ($\hat{\omega }$) or the z-domain ($z$).

This section briefly introduces the world of signals and systems in the frequency, s-, and z-domains. More on these domains will follow.

Systems, continuous and discrete, can also be transformed to the frequency and s- and z-domains, respectively. Signals can, in fact, be passed through systems in these alternative domains. When a signal is passed through a system in the frequency domain, for example, the frequency domain output signal can later be returned to the time domain and appear just as if the time-domain version of the system operated on the signal in the time domain.

Consider the sum of a two-sinusoids signal

This can be coded as

% TWO_SINES - plot two sinusoids signal

A1 = 2; f1 = 1;
A2 = 1.5;f2 = 2.2;
t = linspace(0, 3*(1/f1), 1000);
s1 = A1*cos(2*pi*f1*t);
s2 = A2*cos(2*pi*f2*t);
subplot(4,1,1)
plot(t,s1),ylim([-5,5]),xlabel('Time t (s) '),ylabel('s1')
subplot(4,1,2)
plot(t,s2),ylim([-5,5]),xlabel('Time t (s) '),ylabel('s2')
subplot(4,1,3)
plot(t,s1+s2),ylim([-5,5]),xlabel('Time t (s) '),ylabel('x(t) = s2 + s2')
subplot(4,1,4)
axis([0,4,0,2.5])
arrow([f1,0],[f1,A1]),text(f1,A1+0.2,'A1'),text(f1+0.05,0.2,'f1')
arrow([f2,0],[f2,A2]),text(f2,A2+0.2,'A2'),text(f2+0.05,0.2,'f2')
ylabel('Frequency Spectrum X(f)'),xlabel('Frequency (Hz)')

Save this as two_sines.m

Run

two_sines

(Download two_sines.m)

Viewing Signals in the Frequency Domain#

The top waveform plot, denoted $s_1$, is a single sinusoid at frequency $f_1$ and peak amplitude $A_1$. The waveform repeats every period $T_1=1/f_1$. The second waveform plot, denoted $s_2$, is a single sinusoid at frequency $f_2>f_1$ and peak amplitude $A_2<A_1$. The sum signal, $s_1+s_2$, in the time domain is a squiggly line (third waveform plot), but the amplitudes and frequencies (periods) of the sinusoids aren’t as clear here as they are in the first two plots. The frequency spectrum (bottom plot) reveals that $x(t)$ is composed of just two sinusoids, with both the frequencies and amplitudes discernible.

Think about tuning in a radio station. Stations are located at different center frequencies. The stations don’t interfere with one another because they’re separated from each other in the frequency domain. In the frequency spectrum plot, imagine that $f_1$ and $f_2$ are the signals from two radio stations, viewed in the frequency domain. You can design a receiving system to filter $s_1$ from $s_1+s_2$. The filter is designed to pass $s_1$ and block $s_2$.

Fourier Transform#

We use the Fourier transform to move away from the time domain and into the frequency domain. To get back to the time domain, use the inverse Fourier transform. We will find out more about these transforms in this module.

Laplace and Z-Transform Domains#

From the time domain to the frequency domain, only one independent variable, $t\to f$, exists. When a signal is transformed to the s-domain, it becomes a function of a complex variable $s=\sigma +j\omega$. The two variables (real and imaginary parts) describe a location in the s-plane.

In addition to visualization properties, the s-domain reduces differential equation solving to algebraic manipulation. For discrete-time signals, the z-transform accomplishes the same thing, except differential equations are replaced by difference equations.