UNIT 11 SIGNALS AND SYSTEMS

Structure Page No.

11.1 Introduction
Objectives
11.2 Basics of Signals and Systems 76
Preliminaries
Some Special Signals and System Classifications
11.3 Characterization of Linear Time Invariant System 85
11.4 Summary 88
1 1.5 Hints/Solutions 89

1 . INTRODUCTION
In most engineering applications we come across systems which are either
time-invariant or can be approximated in a given interval by time-invariant
systems with reasonable accuracy. So appropriate characterizations are required
for the manipulation, design and analysis of such systems. There are various
ways in which these systems can be classified. Different classifications are
based on different physical attributes of the system of various systems. The
exponential functions can be easily shown to be eigen functions of such
systems. Hence one of the important characterization of such systems is in
terms of their eigen functions namely exponentials. As will become clear later
in the sequel that in order to obtain the characterization, it is required to express
a signal a complex valued function on real line in terms of the system eigen
functions, i.e. exponentials. Clearly the Fourier Series and Fourier Transform
representation precisely serve this purpose. Whereas the Fourier Series
representation caters to the class of periodic signals only the Fourier Transform
representation are applicable for signals which are absolutely integrable. We
will also see that under the category of special functions, it would be possible to
define Fourier Transform for periodic signals. So with the Fourier
Transformation at our disposal we can easily obtain a characterization for
Linear Time Invariant(LT1) systems which plays a fimdamental role in the study
of such systems. We begin this unit with a brief discussion of some definitions
and preliminaries which are required to appreciate the results discussed later.
Objectives
After studying this unit, you should be able to
give the definition of a signal and a system;
explain the following system
1. Translation system
2. Scaling system
3. Reflection system
4. Illustrate the following properties based on which the system can be
classified.
i) System with memory
ii) Stable system
 iii) Causal system -
iv) Time-invariant system
v) Invertible system
vi) Linear system
t

5. Give a characterisation of Linear Time invariant systems.

1 . 2 BASICS OF SIGNALS AND SYSTEMS

In this section we shall define signals and systems in mathematical terms. We
shall consider some special signals and give some classifications of.systems
based on different attributes which will be introduced in this unit.
We start with some preliminaries.

1 1.2.1 Preliminaries
Definition 1: Any map f : R +C will be termed a signal.

In practice, we are interested in analysing, understanding, manipulating or

interpreting a physical quantity which varies with either time or space. For
instance, suppose somebody wants to communicate through telephone, then her
speech gives rise to air-pressure variations which can be considered as a
function of time at a particular point. This physical quantity, i.e. air-pressure at
a particular point is converted into time-variation of another quantity, called
voltage, via a device called microphone. So a mapping is a natural way to
represent any time/opatialy varying physical quantity. Depending upon the
situation, the underlying physical quantity may require some additional
constraints to be satisfied, say, for instance, finite energy condition. This can be
mathematically stated in the following manner.
A physical quantity can be expressed as a measurable function f : R + C and
the finite energy can then be written as

This means that the function is an element of L2(R).

Usually we assume that the signals are elements of the Hilbert space (discussed
in Unit 12) L ~ ( R ) .
In some other context, we may be dealing with signals which are periodic in
nature and have finite power which can be mathematically put as:
3 T > 0 3 f(t + T ) = f(t)Vt E Rand -
:IT If(t)12dt< m. Again in this context
the signals are in L2( [O, TI). Thus, our signals are the elements of the Hilbert
Space L2[O, TI.
Here onwards we will use the notation 'S' to denote the set of signal of our
interest.
Having understood the concept of signal in different contexts we now define the
concept of a system.
Definition 2: Any map R : S -+ S , will be called a system. Signals and Systems

So a system is a transformation which takes signal and maps it to another

signal. Since this definition is too general and we are not interested in arbitrary
mappings from S to S?we define some properties which can be used to classify
systems and then focus on the analysis of systems of our interest.
Here we shall recall some general properties of Transformaticns (mapsj from
one vector space to another vector space which you might have studied in your
undergraduate Linear Algebra course (You can refer to IGNOU course MTE-02
Linear Algebra, Block 2 which will be available at your programme centre.)
You might be already familiar with the property of linearity of a map.
If a system R is linear as a map from S to S , i.e. for a,. . . . , (1, E C and
f i , . . . ,f, E S, we have
R(alfi + . . . t-rr,f,,) = o l R ( f l ) + . . . + a,R(f,,) (1)
then wc say that the system is linear. Most of the examples we will be
considering in this unit will be linear only. Other properties which you are
familiar with maps are "injective", c'suqective" and "invertible" maps.
If a system 72 is injective as a map from S to S, i.e. for fi, fi E S with fi # fi,
we have
Rjf~#
) ~ ( f , ) ,

then we say that the system is injective. We also say that the system is one-one.
In the same way a system R is called surjective (onto), i.e. for each
g 6 S. 3 f E S such that R(f) = g. You recall here that a injective and
surjective map is called an "invertible map". Correspondingly, if system is
both injective and surjective, then it is called an invertible system.
In the theory of systems, it is coustomory to call a system invertible as soon as it
is injective, even though it is invertible as a map from S to R ( S ) ,i.e. onto on its
range space.
In the next section, we shall discuss some special signal classes and also some
basic systems which are of great practical value.

11.2.2 Some Special Signals and System Classifications

We first consider some special signals and some basic signal transformations.
They play important role in general system theory.
1. Damped Exponentials

+
Let z = cr iw be a complex number and A be a positive real number. Consider
a complex valued signal f given by

For t E: R, we can rewrite f(t) as

f(t) = Ae(0+iw)t= Aeateiwt
= AeUt(coswt + i sir1wt)
f(t) = Ae"' cos wt + ieat sin wt
Measure and Integral Clearly the real and imaginary parts of the signal fare damped sinusoidal
functions if o < 0 and would be growing sinusoides if cv > 0.
2. The Unit Step Function
We define a function as u as given by

Then u is called Unit step function. This is not a "step function" because it
does not vanish outside a bounded interval. The term "unit step function" is
used because this is what is a coustomarily called.
3. Delta function which we denote by S(t) is defined in terms of its properties.
Here we provide two equivalent definitions of the same.
Let 6 be a function from R -+ C satisfying.

for any function f which is continuous in some neighbourhood of '0'. Then 6 is

called the Dirac delta function. Equivalently we can define it also as the one
which satisfies the following.

which means that the entire area is concentrated in the neighbourhood of '0'.
The Dirac delta function is, in fact, a distribution but the misnoma is highly
prevalent particularly in engineering literature.
Next we shall discuss some classification of basic signal transformations, i.e.
systems based on different physical attributes and which are of great practical
value.
Basic Signal Transformation
(1) Translation: Let t+ E R. Let a signal f be input to a system and let g be the
output defined as follows:

We now impose the condition that for any input f i n S , the output Rt0fis also in
S . The map f -, Rt0fto itself then gives rise to a system which will be denoted
by Go.Such a system R,,is called a translation system.
The function g is denoted by R,,f. The above system is called translation
system. When to > 0, we say that it is a delay system which has delayed the
input by to and if to < 0 then we say that it is advance system, i.e. the system
has advanced the input by j to1.
(2) Scaling: The second system of interest is the scaling system, denoted here
as: Ra and is defined as follows:
For any input f E S , output g be defined as Signals and Systems

here a # 0. As in the earlier case, we now impose the condition that for any
input f i n S, the output Raf also is in S . Then the map f --, Raf gives rise to a
system which is called scaling system and will be denoted by Ra.
This system either compresses the signal (if a > 1) or expands the signal if
la( < 1. That is the reason for calling it by the scaling system.
(3) Reflection System: This system, although a special case of scaling system
with a = -1, is treated separately because of its practical utility. The system is
denoted as R- and is defined similarly as in (2). Under this system an input f
is mapped to an output g defined as

Next we shall discuss the following result.

Proposition 1: Show that scaling the translation systems do not commute.
Infact for a # 0 # to

if the signal set S separates points of R in the sense that for s, t E R with
s # t, 3 f E S satisfying f(s) # f(t).
Proof: To show this, we note that
(RaRt,f)(t) = f(at - to)
(R,,,Rafj(t) = Raf(t - to)

= f(at - to)

Now we put s = at to find f, s, to, a such that

Suppose that f is a non constant. Then

Let tl = s - to and tP = s - a b
Then

t2 - tl = -ato + t o = k ( l - a).

Then

1
Take to = - and a = -1
2
Hence the result.
 Measure and Integral You can try this exercise now.

E l ) Show that R-'R,, # R,R-l(to # 0) if the signal set contains atleast one
hnction which is not periodic with 2to as a period.

As mentioned earlier we now discuss various attributes which can be used to

classify systems. So while dealing with a particular physical system first thing
we will have to do is to find an appropriate mathematical model for the system.
The system properties to be mentioned below help in finding mathematical
characterization of the system. Once we have a suitable mathematical model,
the rich mathematical theories can be used to understand, analyse and also
design systems for a similar class.
Recall that a system is any map from a signal space to a signal space.
We shall briefly describe system properties which are important for
understanding its behaviour. This understanding is required as it helps a
designer in predicting its behaviour under arbitrary input conditions and also
provide a basis for mathematical modeling of the system. The primary reason
why computers have revolutionized all Engineering and Science disciplines is
the advances in the theory for mathematical modeling of systems. Since once a
mathematical model for a system becomes available it is possible to carry out a
study by computer simulation under all input conditions. This is a great help to
any system design since it does not require any physical proto type of the
system to be made before it is fully tested. So we begin by definitions. .

Definition 3: A system R : S -+S is called memory-less if the value of the

output at instant 't' depends on the value of the input at that instant only.

-
Mathematically translated it means if we denote input signal as f(t), output
signal as g(t), i.e. g(t) (Rx)(t), then g(t) = f(t, x(t)).Vt.

So any system which is not memory-less will be called system with memory.
Example 1: Let y(t) = (Rx)(t) tx(t). Clearly y(t) = f(t, f(t)) and hence the
system is memory less.

I You can try this exercise now.

E2) Check whether the system R : f i

nt
g given by
g(t) = (Rf)(t) = / f(r)e-"-'ld~ is memory less or not.

Definition 4: A system R : S -t S is called a stable system if bounded input

produces bounded output. Recall that a function f is called bounded if 3 a
constant m < m such that Jf(t)I < m t,t. So the definition can be alternately
stated as : Given If(t)1 < m t, t, 3 mo < co which is independent o f t such that
Is(t>l= I(Rf)(t)l < m, < 00 t.

Example 2: Consider the system given in E2, i.e.

 Signals and Systems

then assuming If(t) < M 'd t, we have

where f is any arbitrary bounded input.

SO we have shown that any arbitrary bounded input produces a bounded output
and hence the system is a stable system.
***
Example 3: Let us consider the system 72 : f 4 g, given by
g(t) = (Rf/(t) = J"
-30
f(.r)dr, where f E S & L1(R)that is intuitively we may
say that i f f takes only positive values and g will keep growing and then it may
not be possible to find any Mo < oo such that it is bounded. So in order to show
that this system indeed is unstable just one counter example would serve the
purpose. For example let us consider the input f given by,

+
Clearly f is bounded by any M = 1 E, E > 0. But the output for this particular
input could be g(t) = t t 2 0, which is increasing with time and one cannot
find any constant Mo < cc such that Ig(t)1 < mo < co V t.
Hence the system is unstable.
***
Definition 5: A system 'R : S -+ S, is called causal if for any pair of input
signals fl and f2 the following condition is satisfied:
fi (t) = f2(t) V t < to * (Rfi)(t) = (Rf2)(t) Vt < to
This condition implies that the system is non-anticipatory, i.e. present output
depends on present and past values of the input only. (Note the any time 'to' the
time instant t 5 to are considered past and t > to corresponds to future (to to)

Example 4: Let us consider the system discussed in Example 3. Clearly the

system is causal as g(t) depends on the value o f f which are past at time t. The
condition actually means that if one delays the input x(t) by an amount then the
response (output) also gets delayed by the same amount 'T'.
***
Example 5: We modify the above example slightly and let
Measure and Integral if we consider t = -2, the evaluation of g(-2) would require values o f f upto
time t = - 1, which is located to the future o f t = -2. Therefore the system is
non-causal. I
Now we want you to try an exercise. 1
E3) Let R : f -+ g be given by

g(t) = R(f(t))= / -x
2t
f(~)dr.

Show that the system R is causal.

Definition 6: A system R : S + S is called time-invariant if the system

operator commutes with the delay operator defined as (R,,) (t) = x(t - to). So
if R is any system under test, then

where to E R.

Example 6: Let R as given in Example 1, i.e. g(t) = (Rf)(tj = t f(t). Then we

first compute the response under both the systems given. Then

('learly R,,,'R # R'R, and hence the system R is time-varying system, i.e. not
tlme-lncariant.

Example 7: Let us check whether the integrater system given in Example 3

I

/
,I

p(tl = ( R f , ( t ) = f ( r j d i is time-invariant system.

X

We have
(,R,,lfl(tj = f(t - ti,)
Then

and

Clearly R(R,,,f) = R , , R ( f ) .
Hence the system is time-invariant. Signals and Systems

Definition 7: A system R : S -t S is called an invertible system if R

one-one, i.e. if distinct inputs go to distinct outputs.

Mathematically, R ( f l ) = R ( f 2 ) 3 fl = f2

or equivalently f l + f2 + R f l # Rf2
Example 8: Let R : f + g be such that
g(t) = (Rf? (t) = 5f(t),
Suppose that ( R f l ) ( t ) = (Rf2)(t) 'd fl, f2 E S,t E R

So the system is one-one.

***
Example 9: Let 'R, : f -t g be such that g(t) = (RF)(t) = a sin(f(t)). C
we consider two signals
f, (t) and f2(t) = f~(t) + 27~
Then fl (t) # f2(t)

But (Rfl)(t) = ( R f 2 )(t) 'd t

+ 'R is not one-one and hence the system is not invertible.
. A *

Definition 8: A system R : S -+ S is called a l~nearsystem ~f

i) R(fl+f2)=R(fi)+R(fi2) v f 1 , f 2 ~ S
ii) R(af) = a R f V a E K 'df E S

where K is underlying field which in our case is C the field of complex

numbers. The condition (i) is called super position and (ii) is called
Homogeneity. These two conditions can be combined into one and the
condition of linearity can be equivalently stated as follows:
R is linear w R(alfl + a&) = a l R ( f l ) + a2R(f2) 'd al, a2 E K
'd fi, f, E S
Example 10: Let h be a scalar-valued function defined on R function. Define
R : S + S which takes f + g given by g(t) = (Rf)(t) = J'=' h(r)x(t - T ) ~ T
Note that for t E R, g(t) can also be written as
Measure and Integral Clearly

So the system is a linear system.

***
Try these exercises now.
i
E4) Let R : f + g be such that R(fl = g where g is such that
g(t) = (Rf)(t) = sin(f(t)). Check whether R is linear or not.

E5) Consider the following example of the system R : f + g given by

g(t) = (Rf)(t) = a + f(t)

Show that R is not linear.

Though the system R : f -+ g is not a linear system but this system is of great
interest. So we are motivated by such examples to define another class of
systems which are nearly linear in the sense of this example.

Definition 9: Let R : S + S be a system. Then R is called an incrementally

linear system if 3 a linear system 2 : S + S that satisfies.

The first condition states that the difference Rfi - Rf2 of output corresponding
to fi and G is the difference of the two inputs. We can verify that the Example
10 discussed above is an incrementally linear system.
With this we conclude our discussion on classification of systems. Now onward
we focus on characterization of only specific class of system namely
Linear-Time Invariant Systems.

Linear-Time Invariant Systems

A system which is linear as well as time invariant is called an Linear-Time
Invariant (or LTI) system. In the next section, we shall discuss this system.
 Signals and Systems
11.3 CHARACTERIZATION OF LINEAR TIME
INVARIANT SYSTEM

In this section, we shall obtain a characterisation for LTI system.

Let us consider a signal space S. We assume that S L'(R) n L1(R). YOU
recall that a system of S is a transformation on S. Here we shall consider a
system R which is defined as follows:

Definition 10: For every s E R , let k(t. T ) be a real valued function defined on
R such that for every f E S the integral given by

1: k(t. T ) ~ ( T ) ~ T

exists. The function k is called a Kernal.

We have already noticed that the exponential functions act as a Kemal.

An example of a Kemal is given by

Now we use such Kei-nals to define an operator on S. For any f E S and k(t, T),
a kernal define an operator R on S, given by

Then R is called an integral operator with Kemal k(t, T).

Suppose that R is an LTI system defined as an integral operator with kemal

k(t, 7). Then by definition R commutes with the delay operator Rt, for some
to E R . This means that

Now, for any t E R and f E S

Also, for any t E R and f E S, we have

R,, (Rf) (t) = Rf(t - to)

=I I ( k ( t - to). r ) f ( ~ ) d ~
Measure and Integral Since (2) and (3) are true for any f E S, we get that

Since this is true for all to E R , in particular if to = -7, we get

This indicates that kernal is a function of difference of two indices, rather than
depending on the variables independently. Now you recall the definition of the
L ' ~ ~ n v ~ l u t ioperator
on" "*" defined in Unit 9 (Refer Unit 9, ~ e c . 9 . 5 then
) ~ we
get that

Rf(t) =

= k'
S-m_* k(t - T ) ~ ( T ) ~ T

f(t).
where kt is a real-valued function defined on R. This means that the
corresponding to R, we get a scalar valued function h defined on R such that

Thus, we have the following theorem.

Theorem 1: If system defined by a kernal operator is LTI, then the system can
be expressed as a convolution with a scalar-valued function.
We have already observed in previous section for instance in Example 12, that
the converse of Theorem 2 is true.
Then we have the following result.
Theorem 2: A system defined by a kernal operator is LTI if and only if it is
the convolution product of a real-valued function h.
Next we shall illustrate the importance of this theorem.
Suppose we take f(t) = eat as the input to an LTI system '72, with Kernal
e-a(t-7)

Then
g(t) = (Rf) (t) = eat 1 00

-00
f(r)e-"'dr

Denoting H(o) = Lm00

f ( ~ ) e - ~ ~we
d rget
, g(t) = H(o)eat. This means that the
response of LTI system to an exponential input is same exponential but for a
scaling factor which depends on 'a' called frequency of the exponential. From
the operator theoretic point of view we can say that complex exponentials are
eigen functions of LTI systems. This motivates us to look for representation of
an arbitrary signal in terms of system eigen functions or exponentials. Since if
we can express a signal in terms of exponentials, the response of an LTI system
to thissignal would be in terms of these exponentials only, i.e. let

g(t) = ( R f (t)
) = x akl + (a&"''
 So if we know the function h, of an LTI system, which is known as impulse Signals and Systems
response of the system, then we can compute H(ctk), which in turn can be used
to give response of the systems to f(t). If we take cr = i ~ uin
' the expression for
U3

exponential 'eat' then H(iw) = [

J-CG
h(r)e-"'dr is known as frequency
response of the LTI system (which has impulse response h(t)). Notice that
H(iw) is a complex quantity so it has; B(w) magnitude as well as phase part.
That is H(iw) = (H(iw)lewhere (H(iw)lis the magnitude part and H(w) is this
phase part. The magnitude part, JH(iw)I,is called magnitude response and the
phase part, i.e. t j ( d ) ,is called phase response of the system.
~ eust see some examples.
C
1 -w<LJ<w
Example 11: Let H(iw) =
0 otherwise
b' W
i-t
i-t
Let us find the system response to the signal f(t) = e 4 + 5e 8 + 7ei2"'
We first notice that H (7) = I =H ):( & H(2W) = 0. Then the system
response g(t) is given by
W W
i-t i-t
g(t) = e 4 + 5e 8 .

Example 12: Let h(t) = e-2'u(t). Let us find system response to the input

In order to compute H(i3k), we find the Fourier Transform of h(t)

=-=+ H(i3k) = - Substituting this, we can obtain y(t),

1
i3k 2' +
***
Bleasure and.Integral We summarise our discussion now. Given a signal f(t), its Fourier Transform
i(iw) is given as f(iu) = f(t)e-iwtdt.1f E L1(R) then by Fourier Inversion
1 rx.'
Theorem, we get that for almost all w E R, f(t) = I
this we can easily see that if
277 I---
-
~(iw)e'"'dw.Using

f(t) t-+ F(iw), i.e. they are Fourier Transform pair

Then

Thus, we observed that The Fourier transform is a tool that can resolve a given
signal into its exponential components. The function F(iw) is the direct Fourier
transform of f(t) and represents relative amplitudes of various frequency
components. Therefore F(iw) is the frequency-domain representation of f(t).
Time-domain representation specifies a function at each'instant of time,
whereas frequency-domain representation specifies the relative amplitudes of
the frequency components of the function. Either representation uniquely
specifies the function.
1
With this we come to an end of this unit. t

1 . 4 SUMMARY

In this unit, we have covered the following points:

1. We have defined signals and system and illustrated with examples.
2. We have considered some special signals.
i) Damped Exponentials
ii) The unit step and Dirac Delta functions
3. We have described the following basic signal transformation
i) Translation
ii) Scaling
iii) Reflection system
4. We have explained that the system can be classified into the following
types
i) System which is memory-less
ii) Stable system
iii) Causal System
iv) Time-invariant system
v) Linear system
vi) Invertible system
5. We have shown that LTI system can be characterised as convolution with a
scalar-valued function.
 "
1
6l
..
.
10 .
1
" U J I.C...U

1 . 5 HINTSISOLUTIONS

E l ) Hint: Apply
- a- the definitions and observe the importance of the condition. .
E2) Hint: Note that g(2) = /
J-m
1
f(r)dr.

E3) Clearly g(2) depends on the value o f f at points other than '2'. Therefore
the systems is with memory.
E4) Clearly (R(af))(t) = sin(af(t)) # a(Rf) (t) = a sin f(t) (for instance you
can try with a = 2).
So this is not a linear system.
E5) Hint: Apply the definition.
NOTES