Lec2

Principles of Signals and Systems
Prof. Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology, Kanpur
Lecture – 02
Signal Classification – Deterministic/Random, Even/ Odd, Periodic Signals
Keywords: Deterministic and Random Signals, Even and Odd Signals
Hello welcome to another module in this massive open online course. So we are looking
at the classification of signals. And let us look at a different classification of signals that
is analog and digital signals. Analog signals are basically continuous time signals which
can take all possible values over a continuous interval.
(Refer Slide Time: 00:35)

So an analog signal x(t) is a continuous time signal which can take values belonging to
an interval  ,a b . We have several examples of analog signals, for instance,sin(2 )ft ,
which is a sinusoidal signal. It takes all values that belong to the interval 1,1 . Another
such example is t
e which can take all positive values belonging to the interval 0, . So
t
e is a continuous time signal.

On the other hand, a digital signal is a discrete time signal, which can take only values
that belong to a discrete or finite set.
For instance, x(n) can either be -1 or 1 and this is not an interval that is x(n) can take
only values from a discrete set of two possible values.
In communication, this is termed as a binary phase shift keying system, where
transmitted digital symbol can either be -1 or 1 or voltage level can be -1 volt or 1 volt to
indicate the information symbol 0 or 1.

Now another classification of signals can be real and complex signals. So a signal is a
real signal, either it be continuous time or discrete time signal, if it belongs to the set of
real numbers that is you can take only real values.
For instance, sin(2 )ft or t
e these take only real values, so these are real signals. On the
other hand if you look at other examples such as for instance, on the other hand if you
take other signals such as x(t) or discrete time signal x(n), which can take values
belonging to the set of complex numbers, these are termed as complex signals.

For instance, the classic example of a complex signal is 2
( ) j ft
x t e 
 which can also be
written as cos(2 ) sin(2 )ft j ft  where j equals the imaginary number which is 1 .
This is a complex number also termed as a complex sinusoid which can take values
belonging to the set of complex numbers. Complex signals are very useful in the
representation of signals in fact, if you go back to the analysis of communication
systems, all communication signals can be analyzed or represented as complex signals
assuming inphase and quadrature components.

So complex signals have a great utility in the study of signals and carrying out analysis
in various areas to understand different concept and carry out the analysis in various
areas of electronics and communication engineering. So these are also an important class
of signals.
Another important classification of signals is deterministic and random signals. These
can also be either continuous time or discrete time signals. A deterministic signal is
completely specified that is deterministically at any given time instant.

For instance, sin(2 )ft , t
e these are deterministic signals in the sense that at a given
time instant t there is no ambiguity, they are completely specified that is, given a time
instant one can exactly determine what is the signal. However this is not the case of a
random signal, as the name implies this is random in nature that is it takes random values
at various time instants.
So this takes random values at different time instants and hence it is not completely
determined ahead. So let us say your signal represents the outcome of a coin toss
experiment. So if its outcome is a head it signaled by plus 1 if the outcome is tail it is
signaled by minus 1. So signal x(n) = 1 if outcome equals heads or = - 1, if outcome
equals tails. So this is basically representing a coin toss experiment and what this means
is that if at every instant of time you are tossing a coin, if the outcome is a head, you are
representing it by 1 and if the outcome is a tail you, you are representing it by 1 and
therefore, since the outcome of the coin toss experiment is random the signal itself is
random in nature and this is a discrete time random signal.

A classic example of a continuous time random signal is noise and it is some kind of a
signal which looks like as shown in slide. The noise limits the performance of a system
and it is important to understand the properties and behavior of noise to completely
characterize the performance and behavior of a system.
Noise is an inevitable component and whenever we analyze a signal, there is also an
underlying noise component that is present, its power may be less or the power level
relative to the signal might be varying.

Once again going back to our example of communication systems, to understand the
accuracy with which information can be transferred for instance from a base station to
the mobile, it is very important to understand and characterize the noise properties of the
system.
Another classification is even and odd signals. An even function is ( ) ( )x t x t  or for a
discrete time signal ( ) ( )x n x n  . For instance, you have a classic example that
iscos(2 )ft . Socos(2 ) cos( 2 )ft ft   . This is an example of an even signal.
(Refer Slide Time: 25:37).

On the other hand, an odd signal satisfies ( ) ( )x t x t  or ( ) ( )x n x n  . Another classic
example of an odd signal is ( ) sin(2 )x t ft .
Here the value for
2

, that is at
1
4
t
f
 is 1 and this is the value correspondingly at
1
4
t
f
  and you can see this is basically 1 and at
1
4
t
f
  it is -1. So this satisfies
basicallysin(2 ) sin( 2 )ft ft    . So this is a classic example of an odd signal. The
concept of even and odd signals comes in handy when analyzing the properties of the
behavior of signals. So even signal has even symmetry that is symmetric about 0 and odd
signal has odd symmetry.

Another very important classification of signals is periodic versus aperiodic signals. So
x(t) is periodic if there a time period T such that ( ) ( )x t T x t  for all t.
Let us consider a periodic triangular wave. So this is 0, this is T this is 2T, 3T, -T and so
on. And you can see that for any T, that is if you look at values T apart, they are all the
same.

Consider the classic example again that is the sine which is the periodic signal. So you
can see that sin(2 )Ft this is equal to sin(2 )Ft plus the period, the period here is going
to be
1
F
which is equal to sin(2 )t . So this is sin(2 ( 1)) sin(2 2 ) sin(2 )t t t       .
So this is a period of T = 1.
Now if T is a period of the periodic signal, then mT is also a period, where m is any
integer. We have ( ) ( )x t mT x t  and this also holds for all t.

So therefore the fundamental period is the smallest positive number or it is the smallest
time period such that ( ) ( )x t T x t  holds for all t. All other periods are basically
multiples of this fundamental period. So let us go back to our example, sin(2 )t and
here T=1is the fundamental period.
Any multiple of the fundamental period is also a period and the fundamental period is the
smallest possible duration, such that ( ) ( )x t T x t  for all time instance t. Now the same
can be defined for a discrete time signal again.

For a discrete periodic signal, we must have ( ) ( )x n N x n  and this must hold for all n.
And the smallest N for which this holds is known as the fundamental period N0. So that
is basically continuous time periodic signals and discrete time periodic signals.
So we have seen various classes of signals such as deterministic and random signals,
even and odd signals and periodic signals. So you can go over these different classes and
try to understand that better alright. So we will stop here and continue with other aspects
in the subsequent modules. Thank you very much.

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