Qassim University
College of Engineering
Electrical Engineering Department
Course: EE301: Signals and Systems Analysis
Prerequisite: EE202
Text Books:
* B.P. Lathi, “Modern Digital and Analog Communication Systems”, 3th edition,
Oxford University Press, Inc., 1998
* Sanjay Sharma; “Signals and Systems”, S.K. Kataria & Sons, 6th edition,
2008.
Ref Books:
* Oppenheim, A. Willsky and W. Nawab; “Signals and Systems”, Prentice- Hall,
1997
Instructor: Associate Prof. Dr. Ahmed Abdelwahab
INTRODUCTION
Nowadays, electrical systems can play very vital roles with
diverse applications such as circuit design, communications,
biomedical engineering, machines, energy generation and
distribution, etc. For example, wireless communications can
transmit information signals over much longer distances (even
to distant planets and galaxies) and at the speed of light.
Moreover, important discussions now mostly communicated
face to face in meetings or conferences, often requiring travel,
are increasingly using teleconferencing.
Similarly, teleshopping and telebanking provide reliable and
economic services by electronic communication, and
newspapers are now replaced by electronic news services.
Signals and Systems
Signals are processed by systems
• A signal, as the term implies, is a set of information or data.
Examples include a telephone or a television signal, monthly sales of
a corporation, or the daily closing prices of a stock market (e.g., the
Dow Jones averages). However, in electrical sense, the signal
(voltage or current) is defined as a function of one or more
independent variables such as time and/or space.
• A system may be made up of physical components connected
together to produce an output signal in response to an input signal, as
in electrical, mechanical, or hydraulic systems (hardware realization),
or it may be an algorithm that computes an output from an input
signal (software realization). The response (output) of the system
depends upon the transfer function of the system.
CLASSIFICATION OF SIGNALS
A signal that is specified for every value of time t is a continuous-time signal, g(t).
A signal that is specified only at discrete values of t is a discrete-time signal, g(nT).
Where T is the time duration between two successive samples (data values). Simply it can be
written as g(n), where n is any integer number to represent the time independent discrete variable.
A signal whose amplitude can take on any value in a continuous finite range is an analog signal.
This means that an analog signal amplitude can take on an infinite number of values within a
finite range. A digital signal, on the other hand, is one whose amplitude can take on only a finite
number of values. A digital signal whose amplitudes can take on M values is called an M-ary
signal of which binary (M = 2) is a special case. The terms continuous time and discrete time
qualify the nature of a signal along the time (horizontal) axis. The terms analog and digital, on the
other hand, qualify the nature of the signal amplitude (vertical axis).
A signal is said to be periodic when g(t) = g (t ± To) for all t, where To is the signal period.
Similarly, g(n) = g (n ± N), where N is the period of the signal.
A signal is said to be aperiodic if it is not periodic. The second important property of a periodic
signal g(t) is that g(t) can be generated by periodic extension of any segment of g(t) of duration
To (the period).
The sum of two or more sinusoids may be periodic if the ratio
of their periods is a rational number.
A signal whose physical description is known completely, in
either a mathematical form or a graphical form, is
a deterministic signal. If a signal is known only in terms of
its probabilistic description, such as mean value, mean squared
value, and so on, rather than its complete mathematical
or graphical description, is a random signal. Most of the
noise signals encountered in practice are random signals. All
message signals are random signals because, as will be shown
later, a signal, to convey information, must have some
uncertainty (randomness) about it.
Multichannel and Multidimensional Signals
• Multichannel signal is generated by multiple
sources (or sensors). Thee resultant signal is the
vector sum of signals from all channels such as
ECG signal where different leads are connected
to the body of the patient and each lead is acting
as an individual channel.
• Multidimensional signal is a function of more
than one independent variable. For example,
speech signal is one-dimensional signal while,
an image is two-dimensional signal and the
moving picture is three-dimensional signal.
Size of A Signal
The area under a signal g(t) may be considered as a
possible measure of its size, because it takes account
of not only the amplitude, but also the duration.
However, this will be a defective measure because g(t)
could be a large signal, yet its positive and negative
areas could cancel each other, indicating a signal of
small size. This difficulty can be corrected by defining
the signal size as the area under g2(t), which is always
positive. This measure is called the signal energy Eg
defined (for a general complex signals) as n
E   | x ( n) | 2
n  
The signal energy must be finite for it to be a meaningful measure of
the signal size.
A necessary condition for the energy to be finite is that
Otherwise the integral in the previous equation will not converge. If the
amplitude of g(t) does not go to zero as |t| goes to ∞, the signal energy
is infinite.
A more meaningful measure of the signal size in such a case would be
the time average of the energy (if it exists), which is the average power
Pg defined (for a general complex signal) by
1 n N / 2
2
P  Lim
| x ( n) |

N  N
n N / 2
A signal with finite and nonzero energy is an energy
signal with zero power, while a signal with finite and
nonzero power is a power signal with infinite energy.
Every signal that can be generated in a lab bas a finite
energy. In other words, every signal observed in real life
is an energy signal. A power signal, on the other hand,
must necessarily have an infinite duration with infinite
energy. Otherwise, its power which is its average energy
(averaged over an infinitely large interval) will not
approach a nonzero limit. However, practical periodic
signals are considered power signals.
Note that periodic signals for which the area under |g(t)|2
over one period is finite are power signals, however, not
all power signals are periodic. Aperiodic signals are
energy signals with zero power.
Transformation of the independent variable
in the analysis of signals and systems, we need the
transformation of the independent variable. In case
of continuous-time signal, the independent variable
is time t, whereas in case of discrete-time signals,
the independent variable is time n.
Some manipulations or transformations of the
independent variable t are
(1) Time shifting.
(2) Time-scaling
(3) Time-Inversion or folding.
Time shifting
Time-scaling
The expansion or compression of a signal in
time is called as time-scaling.
g(2t) is the signal g(t) compressed in time
by a factor 2. This means that whatever
happens in g(t) at some instant t, also
happens in g(2t) at the instant t/2 as shown
in (b)
Similarly, g(t/2) is the signal g(t) expanded
in time by a factor of 2 as shown in (c)
To summarize, we note that to timescale a signal by a factor a, we should
replace (t) with (at). if a > 1, the scaling
is compression and if a < I, the scaling
will be expansion.
Example of time compression
(Fast Forward)
(a)
(b)
The signal x(3t) shown in Figure (b) is the signal x(t) shown in figure (a)
compressed in time by a factor of 3
This indicates that the values of x(t) at t = 6,12,15 and 24 occur in x(3t) at
the instants t = 2,4,5 and 8 respectively.
Example of time expansion
(Slow Forward)
The signal x(t/2) is the signal x(t) expanded in time by a factor of 2.
This means that the values of x(t) at t = 1, - 1 and - 3 occur in x(t/2) at instants
2, - 2-ahd – 6 respectively.
Time-Inversion or folding
Time inversion is achieved by replacing the independent variable t by - t. This
results in folding of the signal about the origin, i.e., t = 0.
Time-inversion or folding may be considered as a special case of time-scaling with
a = - 1.
Even (Symmetric) and Odd (Asymmetric) Signals
• For Even (symmetrical) signals:
g(t) = g(-t) or g(n) = g(-n)
• For Odd (asymmetrical) signals:
g(t) = - g(-t) or g(n) = - g(-n)
Any continuous (or discrete) time signal g(t) can be
expressed as the summation of an even signal ge(t) and an
odd signal go(t) :
g(t) =ge(t) + go(t)
then, g(-t) =ge(-t) + go(-t) = ge(t) - go(-t)
Therefore, ge(t) = ½ [g(t) +g(-t)].
Similarly, go(t) = ½ [g(t) - g(-t)].
Some Standard Signals
A basic sine wave
Delta or Unit Impulse Function δ(t)
g(t) δ(t) = g(0) δ(t)
g(t) δ(t-T) = g(T) δ(t-T)
g(t) * δ(t) = g(t)
δ(n) = u(n) – u(n-1)
g(n) δ(n) = g(0)
g(n) δ(n-no) = g(no)
Real Exponential Signals and damped Sinusoidal
x(t) = e-at
x(t) = eαt
Sinc Function (Pulse)