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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
EEEG313-Signals and Systems
In the signals and systems course, we basically intend to acquire knowledge (know about tools) and
skill (know about techniques of using them) for signal and system representation and analysis. We
discuss the tools and techniques to
 interpret various kinds of signals according to their properties,
 interpret various kinds of systems according to their properties, and
 manipulate signals using various systems
Lecture Objectives:
 Define signal and systems
 Identify the signal and systems of interest.
 Know what signals are for and systems are for.
 Distinguish between CT and DT signals.
 Distinguish between power and energy signals
 Distinguish between periodic and aperiodic signals[Extended during revision]
 Distinguish between odd and even signal[Extended during revision]
 Acquaint with deterministic and random signal [Extended during revision]
TABLE OF CONTENTS
1.
General Background ......................................................................................................................... 1
1.1 Defining signal and identifying signals of interest: .................................................................. 1
1.2 Defining systems and identifying systems of interest: ............................................................. 2
1.3 What signals are for? .................................................................................................................. 2
1.4 What systems are for? ................................................................................................................ 2
1.5 Signals and systems for electrical and electronics engineers? ................................................ 2
2. Signal Properties and Classification ............................................................................................... 4
1.1 Continuous time or analog, discrete time, and digital signal ...................................................... 4
1.1.1 Continuous time and analog signal .................................................................................... 4
1.1.2 Discrete time signals ............................................................................................................ 5
1.1.3 Digital signals ....................................................................................................................... 5
1.2 Power and Energy Signal ............................................................................................................... 6
1.3 Periodic and Non-Periodic Signal.................................................................................................. 7
1.4 Even and Odd Signal ...................................................................................................................... 7
1.5 Deterministic and Random/Stochastic Signal .............................................................................. 7
3. Signal Representation ....................................................................................................................... 8
1. General Background
1.1 Defining signal and identifying signals of interest:
Signal is a dependent variable, a physical quantity, which can be represented mathematically as a
function of one or more independent variables. Signals usually describe wide varieties of physical
phenomena. Any physical phenomenon indicator (which indicates what is happening) can be utilized,
interpreted, and processed/manipulated for extended utilization and interpretation. Our signals of interest
can be speech, audio, video, power supply voltage variation, temperature of a room, wind velocity
outside, intensity of light outside, and so on.
Hence, a signal is a phenomenon indicator. Moreover, a phenomenon indicator can be a signal only if
you are interested in it. If you are interested, you can find signals everywhere.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
1.2 Defining systems and identifying systems of interest:
Systems in a broad sense are components or an interconnection of components, devices or subsystems.
Systems normally respond to the input by producing certain output. The relationship between input and
output of a system is definite, that is, if an input is given to a system, we will be able to find what output
it produces if we have an adequate knowledge about the system. In a physical system, the phenomenon
occurring in the system components get influenced by the input. The components respond and produce
outputs. As with signal, what we take as the system’s output is a kind of phenomenon indicator. A
system may have several components whose behaviour can be taken as the output. In other words,
system may have many nodes and one or many node variables can be considered as the output/outputs.
It is normal that you tend to take the variable as output in which you are interested. Our system of
interest can be a heater, an electronic wave filter, a temperature controlled oven, a motor, a generator,
and so on.
Hence, a system is a component or an interconnection of components that responds to the input to it in a
definite way producing corresponding output. In some way, system performs transformation of input
signal to the output signal. If we are interested in the output from a particular system, then the system is
the system of our interest.
1.3 What signals are for?
From the utilization point of view, signals are basically for
 delivering information, for example, speech, audio, video, etc. In this case, the information
that the signal delivers may be qualitative or quantitative.
 delivering power or energy, for example, mains supply voltage.
 or both.
Furthermore, as signals are physical phenomenon indicators, signals are for analysing, interpreting, and
utilizing any physical phenomena. In the process of doing that we may utilize various tools for analysis,
various systems for signal processing or manipulation, various systems for signal conversion from one
form to another, and various systems for utilizing the signals.
1.4 What systems are for?
Systems are for producing outputs of our interest. Systems facilitates
 processing or manipulation of signal, for example, amplifier, filter, etc.
 utilization of signal, for example, speaker, heater, lamp, motor, etc.
 or both.
Our interest in a system may be in analyzing, utilizing, and/or controlling the output of the system. We
may wish to design and/or produce our own system of interest after having analysed various systems and
system components. Systems utilize the signals for producing meaningful output.
1.5 Signals and systems for electrical and electronics engineers?
Electrical and electronics engineers tend to convert every signal into the variation of voltage/current
with respect to time, so that it can be easily stored and manipulated/processed electrically. They use
various input transducers and mechanisms to convert the signal from its other physical form to electrical
form, for example, microphone, video-camera, thermocouple, phototransistor, anemometer, etc. If
needed for utilization and interpretation, they use various output transducers and mechanisms to
reproduce the signal to its original form or convert to an utilizable form, for example, speaker, monitor,
heater, light emitting diode, motor, etc. For facilitating better and appropriate utilization/interpretation
signal, they process signal using various systems, for example, amplifier, signal conditioner, filters,
wave-shapers, transformers, and so on.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
Hence, for an electrical-and-electronics engineer, signal is mostly a voltage or a current varying in time.
The dependent variable is voltage or current and the independent variable is time. In general, we use
functions like x(t ) , y (t ) , v (t ) , i (t ) , f (t ) , etc. to denote signals.
In similar way, the system mostly performs a transformation of input voltage/current to output
voltage/current. Both input and output variables can be actually taken as signals. For example, if a
system input is x(t ) and it produces output y (t ) , with transformation relation indicated by T {} , then we
may indicate that as follows.
Input
x(t )
System T {}
Output
y (t )
y (t )  T {x(t )}
T {}
x(t )  y (t )
x(t )  y (t )
The transformation relation however can be made clearer by utilizing impulse response, system function,
frequency response, convolution, I/O mapping graphs, etc., which will be discussed later.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
2. Signal Properties and Classification
According to the purpose or what the signal is intended for, signal may be classified as
 Information signal
 Power signal
 Energy signal
According to number of independent variables, signals may be classified as
 One dimensional with one independent variable
 Multi-dimensional with multiple independent variable
According to nature of variation, signals may be classified as
 Deterministic
 Random
 Stochastic
According to the nature of the independent and dependent variable
 Continuous time
 Discrete time
 Digital
According to periodicity,
 Periodic
 Non-periodic or aperiodic
According to symmetry or anti-symmetry of the dependent variable about the independent variable
axis
 Even
 Odd
According to the nature of value of independent variable
 Real
 Complex
We will hereby see properties of some of the important types of signals in a slight detail.
2.1 Continuous time or analog, discrete time, and digital signal
2.1.1 Continuous time and analog signal
In a continuous time signal, the independent variable can take any real value from   to   .
The independent variable which is normally the time is denoted by t and the signal is denoted by
an analytical function of time, e.g. x(t ) . There is also no restriction on the value of x(t ) and it may
contain discrete jumps. A continuous time signal is also often called analog signal; moreover the
analog signals have their value continuous. Most of physical processes are analog and they give
rise to analog signal. The signal is graphically illustrated below.
x(t )
t
The signals such as speech, ambient temperature, etc. are examples of continuous time signal.
They originate from natural processes and indicate natural phenomenon.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
2.1.2 Discrete time signals
In the discrete time signal, the independent variable or time is allowed only certain discrete values
at regular interval, for example, …-3, -2, -1, 0, 1, 2, … seconds. In such a case, if we know the
value of the interval T , then it is not needed to express it explicitly. We rather can use an index
variable n for denoting the independent variable. The independent variable then is restricted to
possess only integer values from   to   . The actual instances it denotes is given by nT . The
dependent variable is normally denoted by x[n ] .The independent variable however can take any
value. In graphical representation, the discrete time signal is indicated as shown below.
x[n ]

0
1 2
3

n
The discrete time signal either originates from discrete time processes where an event occurs only
at discrete time intervals (e.g., monthly salary deposits in a bank account) or originates due to
sampling of a continuous time process. The obtaining of discrete time signal by sampling a
continuous time signal with sampling period T is illustrated below.
.
discrete time signal x[n]  x(nT )
continuous time signal x(t )


t
n
T 2T 3T
0
1 2
3
2.1.3 Digital signals
Digital signals are discrete time signals with the value of signal also limited to quantized values.
Both the time index n and dependent variable x[n ] are restricted to discrete values. Hence, if a
signal is digital then the value is often said quantized or digitized. If  is the quantization step size,
then x[n ] , at any value of n , can only take a value that is equal to m , where m is an integer
variable. The figure below illustrates the digital version of the discrete time signal of previous
example.
quantized signal x[n]
x[n ]


0
1 2 3

allowable values
n
The digital signal thus can be represented by integer values as the function of the independent time
index integer variable.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
2.2 Power and Energy Signal
Apart from information, many signals are for delivering power or energy. For example, the mains
supply voltage is meant for delivering power and a battery charging signal may be taken as the
signal that is for delivering energy to the battery.
In signal analysis, we define the instantaneous power associated with a CT signal x(t ) as follows.
p (t )  x(t )  x 2 (t )
2
(2.1)
The energy associated with this signal in the interval of t1  t  t 2 can be obtained by
t2
t2
 p(t )dt   x(t ) dt
t1
2
(2.2)
t1
The average power associated with this signal in the interval of t1  t  t 2 can be obtained by
1
t 2  t1
t2
1
t 2  t1
 p(t )dt 
t1
t2
 x(t )
2
dt
(2.3)
t1
Similarly, in case of a DT signal x[n ] , the instantaneous power is defined as
p[n]  x[n]  x 2 [n]
2
(2.4)
The energy associated with this signal in the interval of n1  n  n2 can be obtained by
n2
 x[n]
2
(2.5)
n  n1
The average power associated with this signal in the interval of n1  n  n2 can be obtained by
n2
1
2
x[n]
(2.6)

n2  n1  1 n  n1
In order to have a distinctive understanding of the purpose of a signal, whether it is intended to
deliver power or deliver energy, we normally tend to define the energy or power in a duration that
spans from   to   as E  or P as follows.
(A) In case of CT signal
T
E  lim
T 
 p(t )dt
and
T
1
T  2T
P  lim
T
 p(t )dt
(2.7)
T
(B) In case of DT signal
N
1
2
x[n]
(2.8)


N 
N  2 N  1
n N
n N
A signal is taken as a power signal (intended for delivering power) if P is finite. Naturally, for
such signal, E  is infinite. A signal is taken as an energy signal (intended for delivering energy),
if E  is finite. Naturally, for such signal, P is zero. Therefore a signal can be power signal only
when it originates from an infinite energy source.
E   lim
N
x[n]
2
and
P  lim
An alternating current is a power signal. Since it is periodic, the value of P for it is simply the
value of Pavg for a period.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
2.3 Periodic and Non-Periodic Signal
Periodic signals are important in various purposes, such as in delivering power to an electrical
appliance, frequency transformation of communication signals, testing of various circuits and
devices, etc. A CT periodic signal satisfies the following relation.
x(t )  x(t  T ) .
(2.9)
Where, T is the period of the signal. For any signal to be periodic, the value of T shall be finite
and non-zero. The periodic signal repeats its value after each non-zero time T . Also, if a signal is
periodic at T , then it is naturally periodic at kT , where k  1,2,3, . Periodicity of any analytical
signal can be tested by testing the validity of Equation (2.9) for finite real T .
In DT case, a periodic signal satisfies the following relation.
x[n]  x[n  N ] .
(2.10)
Where, N is the period of the signal. For a signal to be period, the value of N shall be finite nonzero integer. If a signal is periodic at N , then it is periodic at kN , where k  1,2,3, .
If a signal does not satisfy the above property, then it is called aperiodic signal. Periodic signal are
x (t ) or ~
x [ n] .
often denoted by a tilde above the function name, for example, ~
The smallest non-zero value of T or N for which the periodicity condition by a CT or DT signal
is satisfied is called the fundamental period of the signal.
2.4 Even and Odd Signal
Evenness (symmetry about t  0 ) and oddness (anti-symmetry about t  0 ) of signal are important
properties that can be effectively utilized in signal representation and analysis. An even and an odd
signal CT signal have the following properties, respectively.
x(t )  x(t )
x(t )   x(t )
and
(2.11)
In graphical representation, the even and the odd signal have the following kind of appearances.
x(t )  x(t )
x(t )   x(t )
Even signal
Odd signal
Similarly, even and odd DT signals exhibit the following properties, respectively.
x[n]  x[  n]
x[n]   x[n]
and
Any signal can be decomposed into its odd and even part given by
Evenx(t )  12 x(t )  x(t ) and
Odd x(t )  12 x(t )  x(t )
(2.12)
(2.13)
2.5 Deterministic and Random/Stochastic Signal
A signal whose values are completely known for all the times, either mathematically or graphically,
for example x(t )  sin( t ) , is called a deterministic signal. A signal whose value at any instant of
time cannot be determined precisely but is known in terms of probabilistic description is known as
random or stochastic signal. A completely random signal has an equal probability for having any
value at any instant, with a flat probability density function (PDF). A stochastic signal has a
probability density function that is not flat but for example Gaussian. Most of the nature’s
information providers provide stochastic/probabilistic/random signals with different types of PDF
and correlation properties.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
Dept. of EEE, Kathmandu University
3. Signal Representation
Deterministic signals are generally represented or described in two ways:
 Graphical representation
 Mathematical representation
The graphical representation is the plot of values of the dependent variable against the independent
variable, generally time t or n. In general, we need to have the definition of the signal for entire
range of t or n, that is, from   to   . One single mathematical expression may not be possible
in most of the cases to define the signal completely. We may however define the signal considering
appropriate interval and appropriate expression for each interval. Few example signals are
presented here.
Example 1: [Full wave rectified sine-wave]
~
x (t )
1
~
x (t )  sin( t / T ) 0  t  T
Mathematical representation
T
T
2T
3T
~
x (t ) indicates that is the function is periodic.
The definition of it over a period is then
sufficient to describe it mathematically.
Graphical representation
Example 2: [A rectangular pulse of unit amplitude]
x(t )
1
T
t0
 0
1 / 2 t  0, t  T

x(t )  
0t T
 1

t T
 0
Mathematical representation
Graphical representation
Example 3: [Discrete time rectangular pulse of unit amplitude and length 4]
x[n ]
1
0
1 2 3
Graphical representation
n
0

x[n]  1
0

n0
0n3
n4
Mathematical representation
Note the interval definition limits of the independent variable. As signal description needs to be
correct in each interval and for entire range of independent variable, correct use of limits is
necessary. At the point of finite discontinuity, if we consider the definition, the mid-value is taken.
Moreover, it is not generally necessary to define such point of discontinuity, as limits with
inequality and the different value of the function on the two sides indicate the discontinuity and the
value at the discontinuity if needed.
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Signals and Systems Lecture Notes by Dr. B. B. Chhetri
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Dept. of EEE, Kathmandu University
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