Signals and systems 5

Text Books:
1. A.V. Oppenheim, A.S. Willsky and S.H. Nawab, “Signals and Systems”, 2nd Edition, PHI, 2009.

2. Simon Haykin and Van Veen, “Signals & Systems”, 2nd Edition, Wiley, 2005.

References:

1. BP Lathi, “Principles of Linear Systems and Signals”, 2nd Edition, Oxford University Press, 015.

2. Matthew Sadiku and Warsame H. Ali, “Signals and Systems A primer with MATLAB”, C R C Press,
2016.

3. Hwei Hsu, “Schaum's Outline of Signals and Systems”, 4th Edition, TMH, 2019.
 Signals and systems overview 6

What is Signal?
It is representation of physical quantity (Sound, temperature, intensity, Pressure,
etc..,) which varies with respect to time or space or independent or dependent
variable.
or
It is single valued func tion whic h c a rries of Amplitude,
information by me ans
Frequency and Phase.
Example: voice signal, video signal, signals on telephone wires
etc.
Signals and systems overview
7

Signal with different Phases, Amplitudes and Frequencies

 Classification of Signals
8

Types of Signals with respect to no. of variables or dimensions

▶ One Dimensional or 1-D Signal: If the signal is function of only one variable
or If Signal value varies with respect to only one variable then it is called “
One Dimensional or 1-D Signal”
Examples: Audio Signal, Biomedical Signals, temperature Signal etc.., in
which
signal is function “ time”
 Classification of Signals
9

▶ Two Dimensional or 2-D Signal: If the signal is function of two variable or If

Signal value varies with respect to two variable then it is called “Two
Dimensional or 2-D Signal”
Examples: Image Signal in which intensity is function of two spatial co-ordinates
“X”
& “ Y” i,.e I(X,Y)

▶ Three Dimensional or 3-D Signal: If the signal is

function of three variable or If Signal value varies
with respect to three varia b le then it is c a lled “
Three Dimensional or 3-D Signal”
Examples: Video Signal in which intensity is function of two
spatial co-ordinates “X” & “Y” and also time “t” i.e v(x,y,t)
 Classification of Signals
10

Types of Signal with respect to nature of the signal

Continuous Time Signal (CTS) or Analog Signal :
If the signal values continuously varies with respect to time then it is called
“Continuous Time Signal (CTS) or Analog Signal “. It contains infinite set of values
and it is represented as shown below.
Digital Signal: If the signal contains only two values then it is called
“Digital Signal”. Discrete Time Signal (DTS):
If signal contain discrete set of values with respect to time then it is called “Discrete
Time Signal (DTS)”. It contains finite set of values. Sampling process converts
Continuous time signal in to Discrete time signal.
Representation of Discrete Time Signal (DTS)
11
 Basic Types of Signals
12

▶ Unit Step Function

Unit step function is denoted by u(t). It is defined as u(t) = 1 when t
≥0
and 0 when t < 0

▶ It is used as best test signal.

▶ Area under unit step function is unity.
 Basic Types of Signals
13

 Unit Impulse Function

𝑡 ≠0
𝑡=0}
Impulse function is denoted by δ(t). and it is defined as δ(t) ={
0;
∞;
 Basic Types of Signals
14

▶ Ramp Signal

signal is denoted by r(t), and it is defined as r(t) =

Ramp

Area und er unit ramp is unity.

 Basic Types of Signals
15

▶ Parabolic Signal
Parabolic signal c an be defined as x(t) =
 Basic Types of Signals
16

▶ Signum Function
Signum function is denoted as sgn(t). It is defined as sgn(t) =
 Basic Types of Signals
17

▶ Exponential Signal
Exponential signal is in the form of x(t) = eαt
The shape of exponential c an be defined by α.
Case i: if α = 0 → x(t) = e0= 1

Case ii: if α< 0 i.e. -ve then x(t) = e−αt,

The shap e is c a lled d e c a ying exponentia l.

Case iii: if α> 0 i.e. +ve then x(t) = eαt,

The shap e is c a lled raising exp onentia l.
 Basic Types of Signals
18

Rectangular Signal
Let it b e d enoted a s x(t) a nd it is defined a s
 Basic Types of Signals
19

Triangular Signal
Let it be denoted as x(t),

Sinusoidal Signal
Sinusoidal signal is in the form of x(t) = A cos(w0±ϕ) or A sin(w0±ϕ)

Where T0 = 2π/w0
 Classification of Signals
20

Signals are classified into the following categories:

▶ Continuous Time and Discrete Time Signals

▶ Deterministic and Non-deterministic Signals

▶ Even and O d d Signals

▶ Periodic and Aperiodic Signals

▶ Energy and Power Signals

▶ Real and Imaginary Signals

 Classification of Signals
21

▶ Continuous Time and Discrete Time Signals

A signal is said to be continuous when it is defined for all instants of time.

A signal is said to be discrete when it is defined at only discrete instants of time.

 Classification of Signals
22

Deterministic and Non-deterministic Signals

A signal is said to be deterministic if there is no uncertainty with respect to its
value at any instant of time. Or, signals which c an be defined exactly by a
mathematical formula are known as deterministic signals.

A sign a l is said to b e non-deterministic

if there is uncertainty with respect to its
value
at some instant of time. Non-deterministic signals
are random in nature hence they are
called random signals. Ra ndom signa
ls c a nnot be described by a
mathematical equation.
They are modelled in probabilistic terms.
 Classification of Signals
23

Even and Odd signals

A signal is said to be even when it satisfies the condition x(t) = x(-t)
Example 1: t2, t4… c ost etc .
Let x(t) = t2

x(-t) = (-t)2 = t2 = x(t)

∴ t2 is even func tion
Example 2: As shown in the following diagram, rectangle function x(t) = x(-t) so it is also even function.

A signal is said to be odd when it satisfies the condition x(t) = -x(-t)

 Classification of Signals
24

Periodic and Aperiodic Signals

A signal is said to be periodic if it satisfies the condition x(t) = x(t + T) or x(n) =
x(n + N). Where, T= fund a menta l time p eriod,
1/T = f = fund a menta l freq uenc y.

The above signal will repeat for every time interval T0 hence it is periodic with
period T0.
 Classification of Signals
25

Energy and Power Signals

A signal is said to be energy signal when it has finite energy.

A signal is said to be power signal when it has finite power.

NOTE:A signal cannot be both, energy and power simultaneously. Also, a signal
may b e neither energy nor power signal.

Power of energy signal = 0 and Energy of pow er signal = ∞

 Classification of Signals
26

Real and Imaginary Signals

A signal is said to be real when it satisfies the condition x(t)

= x*(t) A signal is said to be odd when it satisfies the

condition x(t) = -x*(t)

Example:
If x(t)= 3 then x*(t)=3*=3, here x(t) is a real signa l.
If x(t)= 3j then x*(t)=3j* = -3j = -x(t), henc e x(t) is a od d signa l.

Note: For a real signal, imaginary part should be zero. Similarly for an imaginary
signal, real part should be zero.
 Basic Operations on Signals
27

There are two variable parameters in general:

▶ Amplitude
▶ Time
The following operation can be performed with amplitude:
Amplitude Scaling
C x(t) is a amplitude scaled version of x(t) whose amplitude is scaled by a
factor C.
 Basic Operations on Signals
28

Addition
Addition of two signals is nothing but addition of their corresponding amplitudes.
This
c an be best explained by using the following example:

As seen from the previous diagram,

-10 < t < -3 am p litud e of z(t) = x1(t) + x2(t) = 0 + 2 = 2
-3 < t < 3 amplitude of z(t) = x1(t) + x2(t) = 1 + 2 = 3
3 < t < 10 am p litud e of z(t) = x1(t) + x2(t) = 0 + 2 = 2
 Basic Operations on Signals
29

Subtraction
subtraction of two signals is nothing but subtraction of their corresponding
amplitudes.
This c an be best explained by the following example:

As seen from the diagram above,

-10 < t < -3 amplitude of z (t) = x1(t) - x2(t) = 0 - 2 = -2
-3 < t < 3 amplitude of z (t) = x1(t) - x2(t) = 1 - 2 = -1
3 < t < 10 amplitude of z (t) = x1(t) - x2(t) = 0 - 2 = -2
 Basic Operations on Signals
30

Multiplication
M ultip lic a tio n two signals is nothing but multiplication of their corresponding
of
amplitudes.
This c a n b e b est expla ined b y the follo wing exa m ple:

As seen from the diagram above,

-10 < t < -3 am p litud e of z (t) = x1(t) ×x2(t) = 0 ×2 = 0
-3 < t < 3 amplitude of z (t) = x1(t) - x2(t) = 1 ×2 = 2
3 < t < 10 am p litud e of z (t) = x1(t) - x2(t) = 0 × 2 = 0
 Basic Operations on Signals
31

The following operations can be performed with time:

Time Shifting
x(t ±t0) is time shifted version of the signal x(t).
x (t + t0) →ne g a tive shift
x (t - t0) →p o sitive shift
 Basic Operations on Signals
32

Time Scaling
x(At) is time scaled version of the signal x(t). where A is always positive.
| A | > 1 → C om pression of the signa l
| A | < 1 → Expansion of the signal

Note: u(at) = u(t) time scaling is not applicable for unit step function.
 Basic Operations on Signals
33

Time Reversal
x(-t) is the time reversal of the sign a l x(t).
 Basic Operations on Signals
34

Convolution: Convolution between two continuous time signals c an be written as

The following operations are required to compute convolution

1. Time reversal
2. Time Shifting ( Delay & Advance)
3. Signal Multiplication
4. Integration
Note: If two signals are finite duration then Graphical Method is used and Else
Function
Method is employed to compute Convolution
 System Definition 35

What is System?
System is a device or combination of devices, which c an operate on signals
and produces corresponding response. Input to a system is called as
excitation and outp ut from it is c a lled as response.
For one or more inputs, the system c an have one or more outputs.
Example: Communication System
 Classification of Systems
36

Systems are classified into the following categories:

▶ line a r a nd Non-lin ea r System s

▶ Time Variant and Time Invariant Systems

▶ linear Time variant and linear Time invariant systems

▶ Sta tic a nd Dynamic System s

▶ Causal and Non-causal Systems

▶ Invertib le a nd Non-Invertib le System s

▶ Sta ble a nd Unstable System s

 Classification of Systems
37

Linear and Non-linear Systems

A system is said to be linear when it satisfies superposition and homogenate principles.
Consider two systems with inputs as x1(t), x2(t), and outputs as y1(t), y2(t) respectively.
Then, according to the superposition and homogenate principles,
T[a 1 x1(t) + a 2 x2(t)] = a 1 T[x1(t)] + a 2 T[x2(t)]
∴ T [a 1 x1(t) + a 2 x2(t)] = a 1 y1(t) + a 2 y2(t)
From the above expression, is clear that response of overall system is equal to response of
individual
system.
Exampl
y(t) = x2(t)
e:
Solution:
y1 (t) = T[x1(t)] = x12(t)
y2 (t) = T[x2(t)] = x22(t)
T[a 1 x1(t) + a 2 x2(t)] = [ a 1 x1(t) + a 2 x2(t)]2
Which is not equal to a1 y1(t) + a2 y2(t). Hence the system is said to be non linear.
 Classification of Systems
38

Time Variant and Time Invariant Systems

A system is said to be time variant if its input and output characteristics vary with time.
Otherwise, the system is considered as time invariant. The condition for time invariant system is:
y (n , t) = y(n-t)
The c o ndition for time va ria nt system is:
y (n , t) ≠y(n-t)
Where y (n , t) = T[x(n-t)] = input cha nge
y (n-t) = output cha nge
Example:
y(n) = x(-n)
y(n, t) = T[x(n-t)] = x(-n-t)
y(n-t) = x(-(n-t)) = x(-n + t)
∴ y(n, t) ≠y(n-t). Hence, the system is time variant.
 Classification of Systems
39

Liner Time variant (LTV) and Liner Time Invariant (LTI) Systems
If a system is both liner and time variant, then it is called liner time variant (LTV)
system.
If a system is both liner and time Invariant then that system is called liner time
invariant (LTI)
system.
Static and Dynamic Systems
Static system is memory-less whereas dynamic system is a memory system.
Example 1: y(t) = 2 x(t)
For present value t=0, the system output is y(0) = 2x(0). Here, the output is only
dependent
upon present input. Hence the system is memory less or static.
Example 2: y(t) = 2 x(t) + 3 x(t-3)
For present value t=0, the system output is y(0) = 2x(0) + 3x(-3).
Here x(-3) is past value for the present input for which the system requires memory to
get
this output. Hence, the system is a dynamic system.
 Classification of Systems
40

Causal and Non-Causal Systems

A system is said to be causal if its output depends upon present and past inputs, and
does
not depend upon future input.
For non causal system, the output depends upon future inputs also.
Example 1: y(n) = 2 x(t) + 3 x(t-3)
For present value t=1, the system output is y(1) = 2x(1) + 3x(-2).
Here, the system output only depends upon present and past inputs. Hence, the
system is
causal.
Example 2: y(n) = 2 x(t) + 3 x(t-3) + 6x(t + 3)
For present value t=1, the system output is y(1) = 2x(1) + 3x(-2) + 6x(4) Here, the
system output depends upon future input. Hence the system is non-causal
system.
 Classification of Systems
41

Invertible and Non-Invertible systems

A system is said to invertible if the input of the system appears at the output.

Y(S) = X(S) H1(S) H2(S)

= X(S) H1(S) · 1(H1(S))
Sinc e H2(S) =1/( H1(S) )
∴ Y(S) = X(S)
→ y(t) = x(t)
Henc e, the system is invertib le.
If y(t) ≠x(t), then the system is said to be non-invertible.
 Classification of Systems
42

Stable and Unstable Systems

The system is said to be stable only when the output is bounded for bounded
input. For a
bounded input, if the output is unbounded in the system then it is said to be
unstable.
Note: Fo r a b ounded signal, am p litud e is finite.
Example 1: y (t) = x2(t)
Let the input is u(t) (unit step bounded input) then the output y(t) = u2(t) = u(t) =
bounded
output.
Henc e, the system is sta b le.
Example 2: y (t) = ∫x(t)dt
Let the input is u (t) (unit step bounded input) then the output y(t) = ∫u(t)dt =
ramp signal (unbounded because amplitude of ramp is not finite it goes to
 infinite when t → infinite).
Henc e, the system is unsta b le.
Convolution and correlation of signals
43
Convolution and correlation of signals
44
Convolution and correlation of signals
45
Convolution and correlation of signals
46
Convolution and correlation of signals
47
Convolution and correlation of signals
48
Convolution and correlation of signals
49
Convolution and correlation of signals
50
Convolution and correlation of signals
51
Convolution and correlation of signals
52
Convolution and correlation of signals
53
Convolution and correlation of signals
54
Convolution and correlation of signals
55
Convolution and correlation of signals
56
Convolution and correlation of signals
57
Convolution and correlation of signals
58
Convolution and correlation of signals
59
 Analogy between vectors and signals
60

There is a p erfect analogy b etween vecto rs and signals.

Vector
A vector contains magnitude and direction. The name of the vector is denoted
by
bold face type and their magnitude is denoted by light face type.
Example: V is a vector with magnitude V. Consider two vectors V1 and V2 as
shown in the following diagram. Let the component of V1 along with V2 is
given by C12V2. The component of a vector V1 along with the vector V2 c
an obtained by taking a perpendicular from the end of V1 to the
vector V2 as shown in diagram:

The vector V1 c an be expressed in terms of

vector V2 V 1=C 12V 2 + Ve
▶
▶ Where Ve is the error vecto r.
 Analogy between vectors and signals
61

But this is not the only way of expressing vector V1 in terms of V2. The
alternate
possibilities are:
V1=C1V2+Ve1

V 2=C2V 2+Ve 2

The error signal is minimum for large component value. If C12=0, then two signals are said to
be
orthogonal.
Dot Prod uct of Tw o Vecto rs V 1 . V 2 = V 1.V2 c o sθ
SUB:ES UNIT:2
θ = Angle betwe en V 1 a nd V 2 V 1. V 2
=V2.V1
 Analogy between vectors and signals
62

The error signal is minimum for large component value. If C12=0, then two signals
are said to be orthogonal.
Dot Product of Two
Vectors V 1 . V 2 = V 1.V2
c osθ
θ = Angle b etween V 1 and V 2 V 1. V 2 =V2.V1
From the diagram, components of V1 a long V2 = C 12 V2
 Analogy between vectors and signals
63

Signal
The concept of orthogonality c a n be applied to signals. Let us consider two signals f1(t) and f2(t).
Similar to vectors, you c a n approximate f1(t) in terms of
f2(t) as f1(t) = C 12 f2(t) + fe(t) for (t1 < t < t2)
⇒ fe(t) = f1(t) – C 12 f2(t)
One possible way of minimizing the error is integrating over the interval t1 to t2.

However, this step also does not reduce the error to appreciable extent. This c a n be
corrected by taking the square of error function.
 Analogy between vectors and signals
64

Where ε is the mean square value of error signal. The value of C12 which
minimizes the error, you need to calculate dε/dC12=0

Derivative of the terms which do not have C12 term are zero.

Put C 12 = 0 to get condition for orthogonality.

 Analogy between vectors and signals
65

Orthogonal Vector Space

A complete set of orthogonal vectors is referred to as orthogonal vector space.
Consider
a three dimensional vector space as shown below:

Consider a vector A at a point (X1, Y1, Z1). Consider three unit vectors (VX, VY,
VZ) in the direction of X, Y, Z axis respectively. Since these unit vectors are
mutually orthogonal, it satisfies that
 Analogy between vectors and signals
66

The vector A c an be represented in terms of its components and unit vectors as

Any vectors in this three dimensional space c an be represented in terms of these

three unit
vectors only.
If you consider n dimensional space, then any vector A in that space c an be
represented
as

As the magnitude of unit vectors is unity for any vector A a long Z

Th e c omp onent of A a long x a xis a xis =
= A.VX Th e c omponent of A along A.VZ
Y axis = A.VY Th e c omp onent of A Similarly, for n dimensional space, the
component of A along some G axis
=A.VG (3)

SUB:ES UNIT:2
 Analogy between vectors and signals
67

Substitute equation 2 in equation 3.

 Analogy between vectors and signals
68

Orthogonal Signal Space

Let us consider a set of n mutually orthogonal functions x1(t), x2(t)... xn(t) over
the interval t1 to t2. As these functions are orthogonal to e ac h other, any two
signals xj(t), xk(t) have to satisfy the orthogonality condition. i.e.

Let a function f(t), it c an be approximated with this orthogonal signal space by

adding the
components along mutually orthogonal signals i.e.
 Analogy between vectors and signals
69

The component which minimizes the mean square error c an be found by

All terms that do not contain Ck is zero. i.e. in summation, r=k term remains and all other terms are zero.
 Analogy between vectors and signals
70

Mean Square Error

The average of square of error function fe(t) is called as mean square error. It is
denoted by ε (epsilon).
 Fourier Series
71

To represent any periodic signal x(t), Fourier developed an expression called

Fourier series. This is in terms of an infinite sum of sines and cosines or
exponentials. Fourier series uses orthoganality condition.
Fourier Series Representation of Continuous Time Periodic Signals
A signal is said to be periodic if it satisfies the condition x (t) = x (t + T) or x (n) = x (n
+ N).
Where T= fund a menta l time p eriod,
ω 0= fund a menta l frequenc y = 2π/T
There are two basic periodic signals: x(t)=cosω0t(sinusoidal) & x(t)=ejω0t(complex
exponential)
These two signals are periodic with period T=2π/ω0
A set of harmonically related complex exponentials c an be represented as {ϕk(t)}

All these signals are periodic with period T

 Fourier Series
72

According to orthogonal signal space approximation of a function x (t) with n,

mutually
orthogonal functions is given by

Where ak = Fourier coefficient = coefficient of

approximation. This signal x(t) is also periodic with period
T.
Equation 2 represents Fourier series representation of periodic
signal x(t). The term k = 0 is constant.
▶ The term k=±1 having fundamental frequency ω0 , is called as 1st harmonics.
▶ The term k=±2 having fundamental frequency 2ω0 , is called as 2nd
harmonics, and so on...
▶ The term k=±n having fundamental frequency nω0, is called as nth harmonics.
 Fourier Series
73

Deriving Fourier Coefficient

We know that

Multiply e−jnω0t on both sides. Then

Consider integral on both sides.

 Fourier Series
74

by Euler's formula,

Hence in equation 2, the integral is zero for all values of k except at k = n. Put k =
n in
equation 2.

Replace n by k
 Fourier Series Properties
75

Properties of Fourier
series: Linearity
Property

Time Shifting Property

 Fourier Series Properties
76

Frequency Shifting Property

Time Reversal Property

Time Scaling Property

 Fourier Series Properties
77

Differentiation and Integration Properties

Multiplication and Convolution Properties

 Fourier Series Properties
78

Conjugate and Conjugate Symmetry Properties

 Trigonometric Fourier Series
79

Trigonometric Fourier Series (TFS)

sinnω0t and sinmω0t are orthogonal over the interval (t0,t0+2πω0). So
sinω0t,sin2ω0t forms an orthogonal set. This set is not complete without
{cosnω0t }because this cosine set is also orthogonal to sine set. So to complete
this set we must include both cosine and sine terms. Now the complete
orthogonal set contains all cosine and sine terms i.e.
{sinnω0t,cosnω0t } where n=0, 1, 2...

The above equation represents trigonometric Fourier series representation of x(t).

Trigonometric Fourier Series
80
 Exponential Fourier Series
81

Exponential Fourier Series (EFS):

a set of complex exponential functions

Consider
which is orthogonal over the interval (t0,t0+T). Where T=2π/ω0 . This is a complete set
so it is
possible to represent any function f(t) as shown below

Equation 1 represents exponential Fourier series representation of a signal f(t)

over the interval (t0, t0+T).
Exponential Fourier Series
82
 Exponential Fourier Series
83

Relation Between Trigonometric and Exponential Fourier Series:

Consider a periodic signal x(t), the TFS & EFS representations are given below
respectively
 Continuous Time Fourier Transform
84

INTRODUCTION:
The main drawback of Fourier series is, it is only applicable to periodic signals.
There are some naturally produced signals such as nonperiodic or aperiodic,
which we cannot represent using Fourier series. To overcome this
shortcoming, Fourier developed a mathematical model to transform signals
between time (or spatial) domain to frequency domain & vice versa, which
is called 'Fourier transform'.
Fourier transform has many applications in physics and engineering such as
analysis of LTI systems, RADAR, astronomy, signal processing etc.
Deriving Fourier transform from Fourier series:
Consider a periodic signal f(t) with period T. The complex Fourier series representation
of
f(t) is given a s
Continuous Time Fourier Transform
85
 Continuous Time Fourier Transform
86

In the limit as T→∞,Δf approaches differential df, kΔf becomes a continuous variable
f,
and summation becomes integration

Fourier transform of a signal

Inverse Fourier Transform is

 Fourier Transform of Basic
functions 87

FT of GATE
Function

FT of Impulse Function:
 Fourier Transform of Basic functions
88

FT of Unit Step Function:

FT of Exponentials:

FT of Signum Function :
 Continuous Time Fourier Transform
89

Conditions for Existence of Fourier Transform:

Any function f(t) c an be represented by using Fourier transform only when
the function
satisfies Dirichlet’s conditions. i.e.
▶ The function f(t) has finite number of maxima and minima.
▶ There must be finite number of discontinuities in the signal
f(t),in the given interval of time.
▶ It must be absolutely integrable in the given interval of time i.e.
 Fourier Transform Properties
90

Linearity Property:

Then linearity property states that

Time Shifting Property:

Then Time shifting property states that

 Fourier Transform Properties
91

Frequency Shifting Property:

Then frequency shifting property states that

Time Reversal Property:

Then Time reversal property states that

 Fourier Transform Properties
92

Time Scaling Property:

Time scaling property states that

Then
Differentiation and Integration Properties:

Then Differentiation property states that

and integration property states that

 Fourier Transform Properties
93

Multiplication and Convolution Properties:

Then multiplication property states that

and convolution property states that

 Sampling theorem of low pass signals
94

Statement of Sampling Theorem:

A band limited signal c an be reconstructed exactly if it is sampled at a rate
atleast twice
the maximum frequency component in it.“
The following figure shows a signal g(t) that is band limited.

Figure1: Spectrum of band limited signal g(t)

The maximum frequency component of g(t) is fm. To recover the signal g(t)
exactly from its samples it has to be sampled at a rate fs ≥2fm.
The minimum required sampling rate fs = 2fm is called “Nyquist rate”.
Sampling theorem of low pass signals
95

Figure 2: (a) Original signal g(t) (b) Spectrum G(ω)

 Sampling theorem of low pass signals
96

Let g s(t) be the sampled signal. Its Fourier Transform Gs(ω) is given by
Sampling theorem of low pass signals
97
 Sampling theorem of low pass signals
98

Aliasing:
Aliasing is a phenomenon where the high frequency components of the sampled
signal
interfere with e ac h other because of inadequate sampling ωs < ωm

Aliasing leads to distortion in recovered signal. This is the reason why sampling
frequency
should be atleast twice the bandwidth of the signal.
 Sampling theorem of low pass signals
99

Oversampling:
In practice signal are oversampled, where fs is significantly higher than Nyquist
rate to avoid aliasing.
 Discrete Time Fourier Transform
100

Discrete Time Fourier Transforms (DTFT)

Here we take the exponential signals to be where ‘w’is a real number. The
representation is motivated by the Harmonic analysis, but instead of following
the historical development of the representation we give directly
the defining equation.
Let {x[n]} be discrete time signal such that , that is sequence is
absolutely summable.
The sequence {x[n]} c an be represented by a Fourier integral of the form,

Where,
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Equation (1) and (2) give the Fourier representation of the signal.
Equation (1) is referred as synthesis equation or the inverse discrete time Fourier
transform
(IDTFT) and equation (2)is Fourier transform in the analysis equation.
Fourier transform of a signal in general is a complex valued function, we c an write,

where is m agnitud e a nd is the p ha se.

We also use the term Fourier spectrum or simply, the spectrum to refer is called
to. Thus the magnitude spectrum and is called the phase
spectrum.

Interchanging the order of integration,

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Example: Let
Fourier transform of this sequence will exist if it is absolutely summable. We have
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Fourier transform of Periodic Signals

For a periodic discrete-time signal,

its Fourier transform of this signal is periodic in w with period 2∏ , and is given

Now consider a periodic sequence x[n] with period N and with the Fourier
series representation

The Fourier transform is,

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Discrete Time Fourier
Transform 107
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Properties of the Discrete Time Fourier Transform:

Let {x[n]}and {y[n]} be two signal, then their DTFT is denoted by and. The
notation

is used to say that left hand side is the signal x[n] whose DTFT is given a t righ t ha
nd side.

1. Periodicity of the DTFT:

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2. Linearity of the DTFT:

3. Time Shifting and Frequency Shifting:

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4. Conjugation and Conjugate Symmetry:

From this, it follows that Re{X(e jw )} is an even function of w and Im{X (e jw )}

is an odd function of w . Similarly, the magnitude of X(e jw ) is an even
function and the phase angle is an odd function. Furthermore,
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5. Differencing and Accumulation

The impulse train on the right-hand side reflects the dc or average value that c an
result
from summation.
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6. Time Reversal

7. Time Expansion
For continuous-time signal, we have

For discrete-time signals, however, a should be an integer. Let us define a signal

with k a
positive integer,
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For k > 1, the signal is spread out and slowed down in time, while its Fourier transform
is
compressed.
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8. Differentiation in Frequency

The right-hand side of the above equation is the Fourier transform of - jnx[n] .
Therefore,
multip lying b oth sides by j , w e see tha t
9. Parseval’s Relation
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SYSTEMS 116

Linear Systems:
A system is said to be linear when it satisfies superposition and homogenate principles.
Consider two systems with inputs as x1(t), x2(t), and outputs as y1(t), y2(t) respectively.
Then, according to the superposition and homogenate principles,
T[a 1 x1(t) + a 2 x2(t)] = a 1 T[x1(t)] + a 2 T[x2(t)]
∴ T [a 1 x1(t) + a 2 x2(t)] = a 1 y1(t) + a 2 y2(t)
From the above expression, is clear that response of overall system is equal to response
of individual system.
Example: y(t) = 2x(t)
Solution:
y1 (t) = T[x1(t)] = 2x1(t)
y2 (t) = T[x2(t)] = 2x2(t)
T[a 1 x1(t) + a 2 x2(t)] = 2[ a 1 x1(t) + a 2 x2(t)]
Which is equal to a1y1(t) + a2 y2(t). Hence the system is said to UNIT:2
be liSnUeB:aESr.
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SYSTEMS 117

Impulse Response:
The impulse response of a system is its response to the input δ(t) when the
system is initially at rest. The impulse response is usually denoted h(t). In other
words, if the input to an initially at rest system is δ(t) then the output is named
h(t).

Liner Time variant (LTV) and Liner Time Invariant (LTI) Systems
▶ If a system is both liner and time variant, then it is called liner time variant (LTV)
system.

▶ If a system is both liner and time Invariant then that system is called liner time
invariant
(LTI) system.
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Response of a continuous-time LTIsystem and the convolution

integral Impulse Response:

The impulse response h(t) of a continuous-time LTI system (represented by T) is
defined to
be the response of the system when the input is δ(t), that is,
h(t)= T{ δ(t)} (1)
Response to an Arbitrary Input:
▶ The inp ut x( t) c a n b e expressed as
(2)
Since the system is linear, the response y( t of the system to an arbitrary input x( t )
c an be
expressed as
(3)
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SYSTEMS 119

Since the system is time-invariant, we

have
(4)

Substituting Eq. (4) into Eq. (3), we

obtain (5)

Equation (5) indicates that a continuous-time LTI system is completely characterized

by its impulse response h( t).
Convolution Integral:
Equation (5) defines the convolution of two continuous-time signals x ( t ) and h(t)
denoted
By-----------------------------------------------------------------(6)
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SYSTEMS 120

Eq ua tion (6) is c ommonly c a lled the c onvolution integral.

Thus, we have the fundamental result that the output of any continuous-time LTI
system is
the c onvolution of the inp ut x ( t ) with the im pulse response h(t) of the
system .

The fo llo wingfigure illustrates the definitio n of the

im pulse response h(t) a nd the
relationship of Eq. (6).

Fig. : C ontinuous-time LTl system.

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Properties of the Convolution Integral:

The convolution integral has the following properties.
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SYSTEMS 122

Step Response:
The step response s(t) of a continuous-time LTI system (represented by T) is defined
to
be the response of the system when the input is u(t); that is,
S(t)= T{u(t)}
In many applications, the step response s(t) is also a useful characterization
of the system.
The step response s(t) c a n be easily determined by,

Thus, the step response s(t) c an be obtained by integrating the impulse response
h(t).
Differentiating the above equation with respect to t, we get
Thus, the impulse response h(t) c an be determined by differentiating the step
response
s(t).
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SYSTEMS 123

Distortion less transmission through a system:

Transmission is said to be distortion-less if the input and output have identical
wave shapes. i.e., in distortion-less transmission, the input x(t) and output y(t)
satisfy the condition:
y (t) = Kx(t - td)
Where td = dela y time a nd
k = constant.

Take Fourier transform on both sides

FT[ y (t)] = FT[Kx(t - td)]
= K FT[x(t - td)]
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SYSTEMS 124

According to time shifting property,

Thus, distortion less transmission of a signal x(t) through a system with impulse response h(t) is achieved
when

and (a m p litud e response)

|H(ω)|=K

A physical transmission system may have amplitude and phase responses as shown below:
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SYSTEMS 127

FILTERING
One of the most basic operations in any signal processing system is filtering.
Filtering is the process by which the relative amplitudes of the frequency
components in a signal are changed or perhaps some frequency
components are suppressed.
For continuous-time LTI systems, the spectrum of the output is that of the input
multiplied
by the frequency response of the system.
Therefore, an LTI system acts as a filter on the input signal. Here the word "filter" is
used to
denote a system that exhibits some sort of frequency-selective behavior.
Ideal Frequency-Selective Filters:
An id e a l frequenc y-selec tive filter is one tha t exac tly p a sses signa ls at
one set of frequencies and completely rejects the rest.
The band of frequencies passed by the filter is referred to as the pass band,
and the band of frequencies rejected by the filter is called the stop band.
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SYSTEMS 128

The most c o mmon typ es of ide a l

frequenc y-selective filters a re the following.
Ideal Low-Pass Filter:
An ideal low-pass filter (LPF) is specified by

The frequency wc is called the cutoff frequency.

Ideal High-Pass Filter:
An ideal high-pass filter (HPF) is specified by
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Ideal Bandpass Filter:

An ideal bandpass filter (BPF) is specified by

Ideal Bandstop Filter:

An ideal bandstop filter (BSF) is specified by
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SYSTEMS 130

The following figures shows the magnitude responses of ideal filters

Fig: Magnitude responses of ideal filters (a) Ideal Low-Pass Filter (b)Ideal High-Pass
Filter
© Id e a l Ba nd p a ss Filter (d) Id e a l Ba ndstop Filter
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SYSTEMS 133
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SYSTEMS 134
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 LAPLACETRANSFORM
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THE LAPLACE TRANSFORM

We know that for a continuous-time LTI system with impulse response h(t), the output
y(t)of
the system to the complex exponential input of the form est is,

Definition:
The function H(s) is referred to as the Laplace transform of h(t). For a general
continuous-
time signal x(t), the Laplace transform X(s) is defined as,

The variable s is generally complex-valued and is expressed as,

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Relation between Laplace and Fourier transforms:

Laplace transform of x(t)
 LAPLACETRANSFORM
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Inverse Laplace Transform:

We know that
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Conditions for Existence of Laplace Transform:

Dirichlet's conditions are used to define the existence of Laplace transform. i.e.
▶ The function f has finite number of maxima and minima.
▶ There must be finite number of discontinuities in the signal f ,in the given interval
of
time.
▶ It must be absolutely integrable in the given interval of time. i.e.
Initial and Final Value Theorems
If the Laplace transform of an unknown function x(t) is known, then it is possible to
determine
the initial and the final values of that unknown signal i.e. x(t) at t=0+ and t=∞.
Initial Value Theorem
Statement: If x(t) and its 1st derivative is Laplace transformable, then the initial value
of x(t) is
given by
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140

Final Value Theorem

Statement: If x(t) and its 1st derivative is Laplace transformable, then the final
value of x(t) is given by,

Properties of Laplace transform:

The properties of Laplace transform are:
Linearity Property
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Time Shifting Property

Frequency Shifting Property

Time Reversal Property

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Time Scaling Property

Differentiation and Integration Properties

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Multiplication and Convolution Properties

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Region of convergence
The range variation of σ for which the Laplace transform converges is called
region of
convergence.
Properties of ROC of Laplace Transform
▶ ROC contains strip lines parallel to jω axis in s-plane.

▶ If x(t) is absolutely integral and it is of finite duration, then ROC is entire s-plane.
▶ If x(t) is a right sided sequence then ROC : Re{s} >σo.
▶ If x(t) is a left sided sequence then ROC : Re{s} <σo.
▶ If x(t) is a two sided sequence then ROC is the combination of two regions.
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Example 1: Find the Laplace transform and ROC of x(t)=e− at u(t) x(t)=e−atu(t)
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Example 2: Find the Laplace transform and ROC of x(t)=e at u(−t) x(t)=eatu(−t)
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Example 3: Find the Laplace transform and ROC of x(t)=e −at u(t)+e at u(−t)
x(t)=e−atu(t)+eatu(−t)

Referring to the above diagram, combination region lies from –a to a. Hence, ROC:
−a SUB:ES UNIT:2
<Res<a
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Causality and Stability

For a system to be causal, all poles of its transfer function must be right half of s-

plane.
A system is said to be stable when all poles of its transfer function lay on the left half
of s-
plane.
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A system is said to be unstable when at least one pole of its transfer function is shifted
to the
right half of s-plane.

A system is said to be marginally stable when at least one pole of its transfer
function lies on the jω axis of s-plane
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LAPLACE TRANSFORMS OF SOME COMMON SIGNALS

Unit Impulse Function δ( t ):

Unit Step Function u(t ):

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Some Laplace Transforms Pairs:

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Z-Transform
Analysis of continuous time LTI systems c an be done using z-transforms. It is a
powerful mathematical tool to convert differential equations into algebraic
equations.

The bilateral (two sided) z-transform of a discrete time signal x(n) is given as

The unilateral (one sided) z-transform of a discrete time signal x(n) is given as

Z-transform may exist for some signals for which Discrete Time Fourier Transform (DTFT)
does
not SUB:ES UNIT:2
exist.
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153

Concept of Z-Transform and Inverse Z-Transform

Z-transform of a discrete time signal x(n) c an be represented with X(Z), and it is
defined as

The above equation represents the relation between Fourier transform and Z-
transform
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Inverse Z-transform:
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155

Z-Transform Properties:
Z-Transform has following properties:
Linearity Property:
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156

Time Shifting Property:

Multiplication by Exponential Sequence Property:

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157

Time Reversal Property:

Differentiation in Z-Domain OR Multiplication by n Property:

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158

Convolution Property:

Correlation Property:
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159

Initial Value and Final Value Theorems

Initial value and final value theorems of z-transform are defined for causal signal.
Initial Value Theorem
For a causal signal x(n), the initial value theorem states that

This is used to find the initial value of the signal without taking inverse z-transform
Final Value Theorem
For a causal signal x(n), the final value theorem states that

This is used to find the final value of the signal without taking inverse z-transform
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Region of Convergence (ROC) of Z-Transform

The range of variation of z for which z-transform converges is called region of
convergence of z- transform.
Properties of ROC of Z-Transforms
▶ ROC of z-transform is indicated with circle in z-plane.
▶ ROC does not contain any poles.
▶ If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-plane
except at z
= 0.
▶ If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire z-
plane except
at z = ∞.
▶ If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a.
i.e . | z | > a .
▶ If x(n) is a infinite duration anti-causal sequence, ROC is interior of the circle with radius
a . i.e . | z | < a .
▶ f x(n) is a finite duration two sided sequence, then the ROC is entire z-pSUlaB:nESee xcept at z
=UN0IT&:2 z = ∞.
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Example 1: Find z-transform and ROC of a n u[n]+a −nu[−n−1] anu[n]+a−nu[−n−1]

The plot of ROC has two conditions as a > 1 and a < 1, as we do not know a.

In this case, there is no combination ROC.

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162

Here, the combination of ROC is from a<|z|<1/a

Hence for this problem, z-transform is possible when a < 1.

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163

Causality and Stability

Causality condition for discrete time LTIsystems is as follows:
A discrete time LTI system is c a usal when,
▶ RO C is outsid e the outermost p ole.
▶ In The transfer function H[Z], the order of numerator cannot be grater than the
order of denominator.

Stability Condition for Discrete Time LTI Systems:

A discrete time LTI system is sta b le when
▶ its system function H[Z] include unit circle |z|=1.
▶ all poles of the transfer function lay inside the unit circle |z|=1.
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164

Some Properties of the Z- Transform:

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165

Inverse Z transform:
Three different methods are:
▶ Partial fraction method
▶ Power series method
▶ Long division method
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Z-TRANSFORM
167
Z-
TRANSFORM 168
Z-
TRANSFORM 169
Z-TRANSFORM
170
Z-
TRANSFORM 171
Z-TRANSFORM
172
Z-TRANSFORM
173
Z-TRANSFORM
174
Z-
TRANSFORM 175
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176

Example: A finite sequence x [n ]is defined as

Find X(z) and its ROC.
Sol: We know that

For z not equal to zero or infinity, e ac h term in X(z) will be finite and consequently
X(z) will converge. Note that X ( z ) includes both positive powers of z and
negative powers of z. Thus, from the result w e c onc lud e tha t the RO C of X ( z )
is 0 < lzl < m.
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177

Example: Consider the sequence

Find X ( z ) and plot the poles and zeros of X(z).

Sol:

From the above equation we see that there is a pole of ( N - 1)th order at z = 0 and
a pole at z =a . Since x[n] is a finite sequence and is zero for n < 0, the ROC is
IzI > 0. The N roots of the numerator polynomial are at
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178

The root at k = 0 cancels the pole at z = a. The remaining zeros of X ( z ) are at

The p ole -zero p lot is shown in the following figure with N=8