A signal is any time-varying or spatial-varying quantity.
Lecture: 11
Examples of signals:
Motion. The motion of a particle through some space can be considered to be a signal, or can be
represented by a signal. The domain of a motion signal is one-dimensional (time), and the range is
generally three-dimensional. Position is thus a 3-vector signal; position and orientation is a 6-vector signal.
Sound. Since a sound is a vibration of a medium (such as air), a sound signal associates a pressure
value to every value of time and three space coordinates. A microphone converts sound pressure at some
place to just a function of time, generating a voltage signal as an analog of the sound signal. Sound
signals can be sampled to on a discrete set of time points; for example, compact discs (CDs) contain
discrete signals representing sound, recorded at 44,100 samples per second; each sample contains data
for a left and right channel, which may be considered to be a 2-vector signal (since CDs are recorded in
stereo).
Images. A picture or image consists of a brightness or color signal, a function of a two-dimensional
location. A 2D image can have a continuous spatial domain, as in a traditional photograph or painting; or
the image can be discretized in space, as in a raster scanned digital image. Color images are typically
represented as a combination of images in three primary colors, so that the signal is vector-valued with
dimension three.
Videos. A video signal is a sequence of images. A point in a video is identified by its two-dimensional
position and by the time at which it occurs, so a video signal has a three-dimensional domain. Analog
video has one continuous domain dimension (across a scan line) and two discrete dimensions (frame and
line).
Biological membrane potentials. The value of the signal is a straightforward electric potential ("voltage").
The domain is more difficult to establish. Some cells or organelles have the same membrane potential
throughout; neurons generally have different potentials at different points. These signals have very low
energies, but are enough to make nervous systems work; they can be measured in aggregate by the
techniques of electrophysiology.
dr.laith@uotechnology.edu.iq
FA-18 Hornet breaking sound barrier
The white halo is formed by condensed water droplets thought to result from a drop in air pressure around the aircraft
View from the Window at Le Gras (1826), Nicéphore Niépce. Generally considered the first
surviving stabilized photograph of a scene from nature taken with a camera
The three main components that together form a
composite video signal are as follows:
1. The luma signal (or luminance) — contains the
intensity (brightness or darkness) information of the
video image
2. The chroma signal — contains the color information
of the video image
3. The synchronization signal — controls the scanning
of the signal on a display such as the TV screen
Composite video waveform: color bars.
Parts of the Video Signal
Complete NTSC Frame Scan
Cell membrane detailed diagram
An Introduction to Signals and Sequences
Lecture: 11
Understanding the basic concepts of signals and sequences is a requirement in the development of any
system that involves the processing, manipulation, and/or transmission of signals and/or sequences.
Signals and sequences are basically the same. A signal is considered analog and is operated in a
continuous-time system. A continuous-time system is a system that operates on and generate signals that
may vary over the entire time interval, usually t ∈[0,∞) . An example of a signal is best described in Fig. 1.
The figure shown in Fig. 1 is an example of a sine wave signal of the general equation y(t) = Asin ωt
in which, the amplitude A is equal to 1.0, and the angular frequency ω is also equal to 0.5, making the
equation:
1
y (t ) = sin( t )
2
y (t ) = sin( wt )
y (t ) = sin( 2πft )
yi = sin( 2π .i. f / f s )
y(t): any physical quantity (Analog signal)
w: angular frequency (measured in radians/sec)
f: frequency (measured Hertz, cycles/sec)
i: discrete sampling-points.
fs: sampled frequency (samples/sec)
Fig. 1. An example of a sinusoidal signal
(one of very commen used signals in
Physics & Electronics) in continuous-time.
A sequence, on the other hand, is considered discrete in nature, and is operated in a discrete-time system.
A discretetime system is a system that generally receives inputs at equally-spaced (uniform) time intervals.
Fig. 2 shows an example of a sine wave sequence of an angular frequency of 0.5, making the equation
1 
[
]
sin
y n =  n
for
2 
n ∈ [0,19]
Fig. 2. An example of a sinusoidal signal in discrete-time.
CONTINUOUS-TIME SIGNALS
A. Unit Step Function
A unit step function, denoted by u (t ) is a signal (or function) that has an amplitude of 1 for time equal or
greater than 0 (t ≥ 0 ), and 0.0 for time less than 0 (t < 0 ). It mathematical definition is
1,
u (t ) = 
0,
t≥0
t<0
The unit step function (see Fig. 3) is one of the most important functions used by mathematicians and
engineers in the analysis and design of continuous-time systems. In addition to being useful, step functions
also provide a way to “turn on” and “turn off” other functions as well, due to its “on-off” characteristics. Thus,
we can think of step functions as mathematical switches.
Fig. 3. The unit step function.
B. Unit Impulse Function
A unit impulse function, denoted by δ (t ) is not really a function in the normal sense, because it is generally
defined as the limit of a function or through its properties. The general definition of a unit impulse function is
δ (σ ) = lim f (∆, σ )
∆ →0
where, f (Δ, σ ) is any function such that when Δ → 0 , the height of the function becomes infinitely large,
making its area equal to 1.0. A convenient mathematical representation of unit impulse function is
1,
δ (t ) = 
0,
t =0
t≠0
which
∞
∫ δ (t ) = 1
−∞
Fig. 4. The unit impulse function.
The unit impulse (Fig. 4) is also an important signal in the study of continuous-time systems because this
signal can represent any arbitrary analog signal. This is because an arbitrary analog signal can be
represented (or approximated) by series of impulses in the time domain [ t = τ1, τ2] .
C. Ramp Function
The ramp function, denoted by r(t) is a signal whose amplitude increases proportionally as time increases. The
mathematical definition of a ramp signal is
kt ,
r (t ) = 
 0,
t≥0
t<0
Fig. 5 shows a ramp signal of the function
2t ,
r (t ) = 
 0,
t≥0
t<0
A discrete-time form a ramp signal is called a ramp
sequence and is shown in Fig. 6.
Fig. 5. The ramp function.
Fig. 6. The ramp sequence.
D. Exponential Function
The exponential function is a signal whose amplitude exponentially increases or exponentially decreases,
depending on the value of a , as time approaches infinity. The exponential function is defined by the equation
x(t ) = a t
To get an exponential function whose amplitude increases to infinity, the value of a should be a positive whole
number. Fig. 7 shows an increasing exponential signal with a = 2 . To get an exponential function whose
amplitude decreases to zero, the value of a should be a positive fraction which has a value less than one. Fig.
8 shows a decreasing exponential signal with a= ½. Figs. 9 and 10 presents the discrete-time equivalents of
Fig. 7 and 8, respectively.
Fig. 7. Example of
an increasing
exponential
function.
Fig. 9. Example
of an increasing
exponential
sequence.
Fig. 8. Example
of a decreasing
exponential
function.
Fig. 10. Example
of a decreasing
exponential
sequence.
Exponential signals (complex representation)
MATLAB example: Discrete-time Signal Representation
Using the MATLAB functions to compute and plot the following functions
t = linspace(-2*pi,2*pi,10);
h = stem(t,cos(t),'fill','--');
%********************************************
t = linspace(0,15,100);
h = stem(t,sin((pi/5)*t),'fill','--');
%********************************************
t = linspace(0,15,10);
h = stem(t,sin(((3*pi)/5)*t),'fill','--');
%********************************************
t = linspace(0,32,200);
h = stem(t,(sin((pi/4)*t).*cos ((pi/4)*t)),'fill','--');
Dr. Laith Abdullah Mohammed
Email: dr.laith@uotechnology.edu.iq
t = (0:0.001:1)';
imp= [1; zeros(99,1)]; % Impulse
unit_step = ones(100,1); % Step (with 0 initial cond.)
ramp_sig= t; % Ramp
quad_sig=t.^2; % Quadratic
sq_wave = square(4*pi*t); % Square wave with period 0.5
plot(t,quad_sig);
%Impulse Function
%n=[-3:3];
%x = [(n-0)==0];
%stem(n,x,'ro');
%Unit step Function
%n=[-3:3];
%x=[(n-0)>=0];
%stem(n,x,'ro');
%Exponential Function
n=[0:10];
x=(0.9).^n;
stem(n,x,'ro')
Dr. Laith Abdullah Mohammed
Email: dr.laith@uotechnology.edu.iq