Signal & Systems _EDUSAT

8/12/2019 Signal & Systems _EDUSAT

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Introduction to Signals &

Systems

By:

Dr. AJAY KUMAR

Associate Prof. ECE

BCET Gurdaspur


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Introduction to Signals

A Signal is the function of one or more independent

variables that carries some information to represent a

physical phenomenon.

A continuous-time signal, also called an analog signal, is

defined along a continuum of time.


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A discrete-time signal is defined at discrete

times.


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Elementary Signals

Sinusoidal & Exponential Signals Sinusoids and exponentials are important in signal and

system analysis because they arise naturally in the

solutions of the differential equations.

Sinusoidal Signals can expressed in either of two ways :

cyclic frequencyform- A sin 2fot =A sin(2/To)t

radian frequency form- A sin ot

o = 2fo = 2/To

To= Time Period of the Sinusoidal Wave


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Sinusoidal & Exponential Signals Contd.

x(t) = A sin (2fot+ )= A sin (ot+ )

x(t) = Aeat Real Exponential

= Aejt = A[cos (ot) +j sin (ot)]Complex Exponential

= Phase of sinusoidal wave

A = amplitude of a sinusoidal or exponential signal

fo= fundamental cyclic frequency of sinusoidal signal

o = radian frequency

Sinusoidal signal


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Unit Step Function

1 , 0

u 1/ 2 , 0

0 , 0

tt t

t

Precise Graph Commonly-Used Graph


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Signum Function

1 , 0

sgn 0 , 0 2u 1

1 , 0

tt t t

t

Precise Graph Commonly-Used Graph

The signum function, is closely related to the unit-step

function.


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Unit Ramp Function

, 0

ramp u u0 , 0

tt tt d t t

t

The unit ramp function is the integral of the unit step function.

It is called the unit ramp function because for positive t, its

slope is one amplitude unit per time.


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Rectangular Pulse or Gate Function

Rectangular pulse, 1/ , / 2

0 , / 2a

a t at

t a


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Unit Impulse Function

As approaches zero, g approaches a unit

step andg approaches a unit impulse

a t

t

So unit impulse function is the derivativeof the unit step

function or unit step is the integral of the unit impulse

function

Functions that approach unit step and unit impulse


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Representation of Impulse Function

The area under an impulse is called its strength or weight. It is

represented graphically by a vertical arrow. An impulse with a

strength of one is called a unit impulse.


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Properties of the Impulse Function

0 0g gt t t dt t

The Sampling Property

0 01

a t t t t

a

The Scaling Property

The Replication Property

g(t)(t) = g (t)


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Unit Impulse Train

The unit impulse train is a sum of infinitely uniformly-spaced impulses and is given by

, an integer Tn

t t nT n


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The Unit Rectangle Function

The unit rectangle or gate signal can be represented as combinationof two shifted unit step signals as shown


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The Unit Triangle Function

A triangular pulse whose height and area are both one but its base

width is not, is called unit triangle function. The unit triangle is

related to the unit rectangle through an operation called

convolution.


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Sinc Function

sinsinc ttt


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Discrete-Time Signals

Samplingis the acquisition of the values of a

continuous-time signal at discrete points in time

x(t) is a continuous-time signal, x[n] is a discrete-time signal

x x where is the time between sampless sn nT T


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Discrete Time Exponential and

Sinusoidal Signals

DT signals can be defined in a manner analogous to theircontinuous-time counter part

x[n] = A sin (2n/No+)

= A sin (2Fon+ )

x[n] = an

n = the discrete time

A = amplitude

= phase shifting radians,No= Discrete Period of the wave

1/N0 = Fo = o/2 = Discrete Frequency

Discrete Time Sinusoidal Signal

Discrete Time Exponential Signal


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Discrete Time Sinusoidal Signals


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Discrete Time Unit Step Function or

Unit Sequence Function

1 , 0

u0 , 0

nn

n


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Discrete Time Unit Ramp Function

, 0

ramp u 10 , 0

n

m

n nn m

n


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Discrete Time Unit Impulse Function or

Unit Pulse Sequence

1 , 0

0 , 0

nn

n

for any non-zero, finite integer .n an a


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Unit Pulse Sequence Contd.

The discrete-time unit impulse is a function in the

ordinary sense in contrast with the continuous-

time unit impulse. It has a sampling property.

It has no scaling property i.e.

[n]= [an] for any non-zero finite integer a


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Operations of Signals

Sometime a given mathematical function maycompletely describe a signal .

Different operations are required for different

purposes of arbitrary signals. The operations on signals can be

Time Shifting

Time Scaling

Time Inversion or Time Folding


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Time Shifting

The original signalx(t) is shifted by anamount t.

X(t)X(t-to) Signal DelayedShift to theright


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Time Shifting Contd.

X(t)X(t+to) Signal AdvancedShift

to the left


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Time Scaling

For the given function x(t), x(at) is the time

scaled version of x(t)

For a 1,period of function x(t) reduces andfunction speeds up. Graph of the function

shrinks.

For a 1, the period of the x(t) increasesand the function slows down. Graph of the

function expands.


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Time scaling Contd.

Example: Givenx(t) and we are to findy(t) =x(2t).

The period ofx(t) is 2 and the period ofy(t) is 1,


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Time Reversal

Time reversal is also called time folding

In Time reversal signal is reversed with

respect to time i.e.

y(t) = x(-t) is obtained for the given

function


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Time reversal Contd.

O ti f Di t Ti


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0 0 , an integern n n n Time shifting

Operations of Discrete Time

Functions


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Operations of Discrete Functions Contd.

Scaling; Signal Compression

n Kn Kan integer > 1


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Classification of Signals

Deterministic & Non Deterministic Signals

Periodic & A periodic Signals

Even & Odd Signals

Energy & Power Signals


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D t i i ti & N D t i i ti Si l


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Deterministic & Non Deterministic Signals

Contd.

Non Deterministic or Random signals

Behavior of these signals is randomi.e. not predictable

w.r.t time.

There is an uncertainty with respect to its value at anytime.

These signals cantbe expressed mathematically. For example Thermal Noise generated is non

deterministic signal.


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Periodic and Non-periodic Signals

Given x(t) is a continuous-time signal

x (t) is periodic iff x(t) = x(t+T) for any T and any integer n

Example

x(t) = A cos(wt) x(t+T) = A cos[wt+T)] = A cos(wt+wT)= A cos(wt+2)

= A cos(wt)

Note: T=1/fo ; w2fo


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Periodic and Non-periodic Signals

Contd. For non-periodic signals

x(t) x(t+T)

A non-periodic signal is assumed to have a

period T =

Example of non periodic signal is an

exponential signal


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Important Condition of Periodicity for

Discrete Time Signals A discrete time signal is periodic if

x(n) = x(n+N)

For satisfying the above condition the

frequency of the discrete time signal should

be ratio of two integers

i.e. fo = k/N


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Sum of periodic Signals

X(t) = x1(t) + X2(t)

X(t+T) = x1(t+m1T1) + X2(t+m2T2)

m1T1=m2T2= T

= Fundamental period Example: cos(t/3)+sin(t/4)

T1=(2)/(/3)=6; T2 =(2)/(/4)=8;

T1/T2=6/8 = = (rational number) = m2/m1m1T1=m2T2Find m1 and m2

6.4 = 3.8 = 24 = T


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Sum of periodic Signalsmay not

always be periodic!

T1=(2)/(1)= 2; T2 =(2)/(sqrt(2));

T1/T2= sqrt(2);Note: T1/T2 = sqrt(2) is an irrational number

X(t) is aperiodic

tttxtxtx 2sincos)()()( 21


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Even and Odd SignalsEven Functions Odd Functions

g t gt g t gt


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Even and Odd Parts of Functions

g g

The of a function is g 2et t

t

even part

g g

The of a function is g2

o

t tt

odd part

A function whose even part is zero, is oddand a functionwhose odd part is zero, is even.


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Various Combinations of even and odd

functionsFunction type Sum Difference Product Quotient

Both even Even Even Even Even

Both odd Odd Odd Even Even

Even and odd Neither Neither Odd Odd


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Product of Two Even Functions

Product of Even and Odd Functions


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Product of Even and Odd Functions Contd.

Product of an Even Function and an Odd Function


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Product of an Even Function and an Odd Function



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Product of Two Odd Functions



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Derivatives and Integrals of Functions

Function type Derivative Integral

Even Odd Odd + constant

Odd Even Even


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Discrete Time Even and Odd Signals

g g

g2

e

n nn

g gg

2o

n nn

g gn n g gn n


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Combination of even and odd

function for DT SignalsFunction type Sum Difference Product Quotient

Both even Even Even Even Even

Both odd Odd Odd Even Even

Even and odd Even or Odd Even or odd Odd Odd


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Products of DT Even and Odd

FunctionsTwo Even Functions

P d t f DT E d Odd


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Functions Contd.An Even Function and an Odd Function


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Proof Examples

Prove that product of two evensignals is even.

Prove that product of two oddsignals is odd.

What is the product of an evensignal and an odd signal? Prove it!

)()()(

)()()(

)()()(

21

21

21

txtxtx

txtxtx

txtxtx

Eventx

txtxtx

txtxtx

txtxtx

)(

)()()(

)()()(

)()()(

21

21

21

Change t-t


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Functions Contd.Two Odd Functions


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Energy and Power Signals

Energy Signal A signal with finite energy and zero power is called

Energy Signal i.e.for energy signal

0


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Energy and Power Signals Contd.

Power Signal Some signals have infinite signal energy. In that

caseit is more convenient to deal with average

signal power.

For power signals

0


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Energy and Power Signals Contd.

For a periodic signal x(t) the average signal

power is

Tis any period of the signal.

Periodic signals are generally power signals.

2

x

1

xTP t dtT


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Signal Energy and Power for DT

Signal

The signal energyof a for a discrete time signal x[n] is

2

x xn

E n

A discrtet time signal with finite energy and zero

power is called Energy Signal i.e.for energy signal

0


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Signal Energy and Power for DT

Signal Contd.

The average signal power of a discrete time power signal

x[n] is

1

2

x

1lim x

2

N

Nn N

P nN

2

x

1x

n N

P nN

For a periodic signal x[n] the average signal power is

The notation means the sum over any set of

consecutive 's exactly in length.

n N

n N


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What is System?

Systems process input signals to produce output

signals

A system is combination of elements that

manipulates one or more signals to accomplish a

function and produces some output.

system outputsignal

input

signal


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System - Example

Consider an RL series circuit

Using a first order equation:

dt

tdiLRtitVVtV

dt

tdiLtV

LR

L

)()()()(

)()(

LV(t)

R


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Mathematical Modeling of Continuous

Systems

Most continuous time systems represent how continuous

signals are transformed via differential equations.

E.g. RC circuit

System indicating car velocity

)(1

)(1)(

tvRC

tvRCdt

tdvsc

c

)()()( tftvdt

tdvm


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Mathematical Modeling of Discrete

Time Systems

Most discrete time systems represent how discrete signals are

transformed via difference equations

e.g. bank account, discrete car velocity system

][]1[01.1][ nxnyny

][]1[][ nfm

nvm

mnv


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Order of System

Order of the Continuous System is the highest

power of the derivative associated with the output

in the differential equation

For example the order of the system shown is 1.

)()()(

tftvdt

tdvm


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Order of System Contd.

Order of the Discrete Time system is the

highest number in the difference equation

by which the output is delayed

For example the order of the system shown

is 1.

][]1[01.1][ nxnyny


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Interconnected Systems

notes

Parallel

Serial (cascaded)

Feedback

LV(t)

R

L

C


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Interconnected System Example

Consider the following systems with 4 subsystem Each subsystem transforms it input signal

The result will be:

y3(t)=y1(t)+y2(t)=T1[x(t)]+T2[x(t)]

y4(t)=T3[y3(t)]= T3(T1[x(t)]+T2[x(t)]) y(t)= y4(t)* y5(t)= T3(T1[x(t)]+T2[x(t)])* T4[x(t)]


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Feedback System Used in automatic control

e(t)=x(t)-y3(t)= x(t)-T3[y(t)]=

y(t)= T2[m(t)]=T2(T1[e(t)])

y(t)=T2(T1[x(t)-y3(t)])= T2(T1([x(t)] - T3[y(t)] ) )=

=T2(T1([x(t)]T3[y(t)]))


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Types of Systems

Causal & Anticausal Linear & Non Linear

Time Variant &Time-invariant

Stable & Unstable Static & Dynamic

Invertible & Inverse Systems


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Causal & Anticausal Systems

Causal system : A system is said to be causal if

the present value of the output signal depends only

on the present and/or past values of the input

signal. Example: y[n]=x[n]+1/2x[n-1]


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Causal & Anticausal Systems Contd.

Anticausal system : A system is said to be

anticausalif the present value of the output

signal depends only on the future values of

the input signal.

Example:y[n]=x[n+1]+1/2x[n-1]


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Linear & Non Linear Systems

A system is said to be linear if it satisfies the

principle of superposition

For checking the linearity of the given system,

firstly we check the response due to linearcombination of inputs

Then we combine the two outputs linearly in the

same manner as the inputs are combined and again

total response is checked

If response in step 2 and 3 are the same,the system

is linear othewise it is non linear.


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Time Invariant and Time Variant

Systems

A system is said to be time invariant if a time

delay or time advance of the input signal leads to a

identical time shift in the output signal.

0

0 0

( ) { ( )}

{ { ( )}} { ( )}

i

t t

y t H x t t

H S x t HS x t

00

0 0

( ) { ( )}

{ { ( )}} { ( )}

t

t t

y t S y t

S H x t S H x t


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Stable & Unstable Systems

A system is said to be bounded-input bounded-

output stable (BIBO stable) iff every bounded

input results in a bounded output.

i.e.| ( ) | | ( ) |x yt x t M t y t M


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Stable & Unstable Systems Contd.

Example

-y[n]=1/3(x[n]+x[n-1]+x[n-2])

1[ ] [ ] [ 1] [ 2]

3

1(| [ ] | | [ 1] | | [ 2] |)

31

( )3

x x x x

y n x n x n x n

x n x n x n

M M M M


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Stable & Unstable Systems Contd.

Example:The system represented by

y(t) = A x(t) is unstable ; A1

Reason:let us assume x(t) = u(t), then atevery instant u(t) will keep on multiplying

with A and hence it will not be bonded.


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Static & Dynamic Systems

A static system is memoryless system

It has no storage devices

its output signal depends on present values of the

input signal

For example


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Example: Static or Dynamic?


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Example: Static or Dynamic?

Answer:

The system shown above is RC circuit

R is memoryless C is memory device as it stores charge

because of which voltage across it cant

change immediately Hence given system is dynamic or memory

system


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LTI Systems

LTI Systems are completely characterized by its

unit sample response

The output of any LTI System is a convolution of

the input signal with the unit-impulse response, i.e.

Properties of Convolution


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Commutative Property

][*][][*][ nxnhnhnx

Distributive Property

])[*][(])[*][(

])[][(*][

21

21

nhnxnhnx

nhnhnx

Associative Property

][*])[*][(][*])[*][(

][*][*][

12

21

21

nhnhnxnhnhnx

nhnhnx

Useful Properties of (DT) LTI Systems
http://stellar.mit.edu/S/course/6/sp08/6.003/courseMaterial/topics/topic1/lectureNotes/Lecture__3/Lecture__3.pdfhttp://stellar.mit.edu/S/course/6/sp08/6.003/courseMaterial/topics/topic1/lectureNotes/Lecture__3/Lecture__3.pdfhttp://stellar.mit.edu/S/course/6/sp08/6.003/courseMaterial/topics/topic1/lectureNotes/Lecture__3/Lecture__3.pdf


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Useful Properties of (DT) LTI Systems

Causality:

Stability:

Bounded Input Bounded Output

00][ nnh

k

kh ][

kk

knhxknhkxny

xnx

][][][][

][for

max

max


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THANKS

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Signal & Systems _EDUSAT