SigRep&Ana DigComn

8/12/2019 SigRep&Ana DigComn

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SIGNAL REPRESENTATION &ANALYSIS AND

DIGITAL COMMUNICATION


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CREDITS


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REFERENCES

Signals & Systems (2ndEdition) Alan V Oppenheim, Allan S Willsky with S Hamid Nawab.

Communication Systems (4thedition) Simon Haykin.

Electronic Communication Systems (3rdEdition) George Kennedy

Modern Digital & Analog Comn Systems BP Lathi

Principals of Digital & Analog Communications Jerry D Gibson


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SIGNAL REPRESENTATION &

ANALYSIS

PHASEI:


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INTRODUCTION

Used in wide variety of fields: Comn.

Acoustics.

Speech & video processing.

Two important components: Signal.

System.

Application: Characterization of a given cct.

Design of cct for a particular application.


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ANALYTICAL FRAMEWORK

Language to describe signals & systems.

Set of tools for analyzing signals & systems


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SIGNALS DESCRIPTION


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TYPES OF SIGNAL

Continuous Time Signal. Independent

variable is continuous.

Discrete Time Signals. Defined at Discrete

times only.


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10

x(t)

x[n]

0 t

0 12 3 4 5 6

7

x[0]

x[1]

CONTINUOUS TIME SIGNAL

n

DISCRETE TIME SIGNAL


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SIGNAL ENERGY & POWER

Signal energy & power represents size of the

signal

Used to compare different signals.


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SIGNAL ENERGY & POWER

Continuous signal

Energy measured over [tt t]

Avg Power

Discrete signal

Energy measured over [nn n]

Avg Power

2

1

2|)(|t

tx dttxE

2

1

2

12

|)(|1 t

tx dttxttP

2

1

2|][|

n

nn

x nxE

2

1

2|][|1 n

nn

x nxN

P

)t

t(

21 ntonervalin

betweenspoinofNoN


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SIG ENERGY & POWER (infinite duration)

Continuous signal

Energy [-t

]

Avg Power

Discrete signal Energy [-n ]

Avg Power

dttxdttxE

T

TTx

22|)(||)(|lim

T

TTx dttx

TP 2|)(|

21lim

n

N

NnN

x nxnxE 22 |][||][|lim

betweenptsofNoN

nxN

PN

NnN

x

(

|][|12

1lim 2

)interval 21 nton


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MODIFICATION OF INDEPENDENT VARIABLE

Signals related by a modification of the

independent variable.

Reversal of signal (audio tape reversal).

Linear scale change of independent variable(Variable speed audio tape).

Displaced or shifted signal (differnce in

propogation time).

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15

x(t)

0 t

x(t-to)

0 t

Time shifted continuous time signal

continuous time signal


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16

x(t)

0 t

x(-t)

0 t

Reversed Continuous time signal

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Continuous time signal


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17

x[n]

0 12 3 4 5 6

7

x[0]

x[1]

Reversed discrete time signal

x[-n]

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Discrete time signal


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ODD & EVEN SIGNAL

Even sig: Identical with its reflection about

the origin :

x(-t) = x(t) or x[-n] = x[n]

Odd sig:

Identical with its reflection about the origin with

a sign change. Odd sig must necessarily be zero at t =0 or n =

0

x(-t) = -x(t) or x[-n] = -x[n]3/17/2014 19


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x(t)

0 t

Even continuous time signal

x(t)

0 t

Odd continuous time signal 3/17/2014 20


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Any sig can be broken down into a sum of

two sig one of which is odd & one of which

is even:

Even part of x(t): Ev{x(t)} = [x(t) + x(-t)] Odd part of x(t) : Od{x(t)} = [x(t) - x(-t)]

Even part is in fact even & odd part is odd.

x(t) is the sum of the two.Analogous definitions hold for discrete

case.

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PERIODIC SIGNALS

There is a positive value of To for which:x(t) = x(t + mTo)

x(t) is periodic with period To.

For x(t) = const; Tois not const as x(t) periodic

for any value of To. There is no smallest value of To as we have for

other periodic sig.

Signal x(t) that is not periodic is known asaperiodic signal.

Periodic sig are defined analogously in discretetime:

x[n] = x[n + mNo)

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QUIZ

A sig x(t) will be represented as _______ after

time to.

Even signal must necessarily be zero at origin.

(T/F). Odd sig is Identical with its reflection about the

origin.(T/F).

A square wave shown below will have

(a) Only even part (b) Only odd part

(c) Both even & odd parts (d) None of the

above3/17/2014 24-5/4 - -3/4 -/2 -/4 0 5/43/4/2/4

f(t)


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A discrete signal can have all possible values.(T/F).

Any signal can be broken down into an odd &

an even signal given by ________. A periodic signal has a positive value of T for

which x(t) = ________.

A sig is a ___________.

A discrete sig is characterized by_________. Draw a signal x(t) = A cos (w0t+) showing its intersection with y axis

Its nearest intersections on x axis from the origin.

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BASIC CONTINUOUS TIME SIGNALS

Occur frequently in nature.

Serve as basic building blocks from which

we can construct many other signals.

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CONTINUOUS TIME COMPLEX

EXPONENTIAL SIGNALS

Takes the form:

C & a are in general complex no,

depending upon the values of which, the

complex exp can take several differentcharacteristics.

Real Exp Fn: C & a are real.

Represents many real life phenomenons

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atCetx )(


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x(t)

0 t

Continuous Time exponential signal

x(t)

0 t

C

C

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+ve a

-ve a


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IMAGINARY EXPONENTIAL SIGNALS

Second imp case is when a is purely

imaginary

It is a periodic signal Proof: For the sig to be periodic

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tj

etx

)(

2]1)([(0 Tortx

)1()( TjtjTjtjTtjtj eifeeeee


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SINUSOIDAL SIGNAL

A sig closely related to the periodic

complex exponential

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)cos()( tAtx

fT

T

2

22

x(t) = A cos(t + )

0

A cos

A T = 2/


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SOME IMP IDENTITIES

Eulers identity:

Fundamental period of a constant sig is

undefined, whereas its freq is zero.

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tjte tj sincos

tjjtjj eeA

eeA

tA 22

)cos(

}{)cos( )( tjeAtA

HARMONICALLY RELATED COMPLEX


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HARMONICALLY RELATED COMPLEX

EXP

Set of periodic exponentials with fundamental

freq that are all multiples of a single positive

freq .

For k = 0; is a const.

For non zero k values is periodic withfundamental pd 2/|k|.

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,.........2,1,0,)( ket tkjk

)(tk

)(tk


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COMPLEX EXPONENTIAL

jraeCC J ;||

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ttjrttjrjat eeCeeCCe )()(

)sin()cos( teCjteC trtr

)2cos()cos(

teCjteC trtr

)()( |||| tjtrtjrj eeCeeC


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Case I: r = 0

Case I: r > 0

Case I: r < 0

treC||

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Unit Step Function

u(t) =

Function is discontinuous at t = 0

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0, t0

u(t)

1

0 t


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UNIT IMPULSE FUNCTION

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t

dtu )()(dt

tdut

)()(

)(tu1

1

)(t

1k

Approx unit step Derivative of)(tu )(tu Unit Impulse Scaled Impulse


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QUIZ

For an exponential sig of the form ,

where C & a are both positive and real, the

signal will never intersect with the time

axis.(T/F) Set of periodic exponentials with fundamental

freq that are all multiples of a single positive

freq is called______. For a signal where a = r + j w draw

the waveforms for the case when r = 0, r < 0,

r > 03/17/2014

37

atCetx )(

atCetx )(


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Draw the signal when C is complex withboth real and imaginary part whereas a is

purely imaginary indicate the amplitude &

phase of this sig.

Derive

from Euler's identity.

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tjjtjj eeA

eeA

tA 22

)cos(


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STEMS


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DEFINITION OF SYSTEM

Any process that results in the

transformation of the signal.

A system has an I/P & an O/P signal.

O/P sig is related to the I/P through sys

transformation.

Continuoustime& Discretetimesystem.

Cascade, parallel, complex, & F/Bconnection of sys to create new sys out of

existing ones.3/17/2014

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PROPERTIES OF SYSTEMS

These are basic properties of continuous &

discrete time systems.

These properties have both physical &

mathematical interpretation.

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SYSTEMS WITH & WITHOUT MEMORY

Memoryless sys: o/p for each value ofindependent variable is dependent only on thei/p at that time. (e.g. A pure resistance wherethe instantaneous o/p voltage is a fn of

instantaneous i/p current y(t) = R x(t).

Systems with memory: o/p dependent on pastvalues of i/p(s) may be along with the presenti/p.(e.g. Capacitance where the voltagedeveloped across it is a fn of running integral ofcurrent )

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t

dxC

ty )(1

)(


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INVERTIBILITY & INVERSE SYSTEMS

Distinct i/p leads to distinct o/p.

By observing sys o/p we can determine its i/p.

For every invertible system there is an inversesystem, which when cascaded to its invertiblesystem, gives an identity system. (e.g. invertible

system y[n] = x(k) has an inverse systemz [n] = y[n]y[n-1]

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CAUSALITY

o/p at any time depends only on values of i/p at

present time & in the past.

These are non anticipative systems.

Sys o/p does not anticipate future values of i/p. If two o/p to sys are identical upto time to, their

o/ps must also be equal upto this time.

All memoryless systems are causal.

e.g. y[n]=x[n]-x[n+1] ; y(t) = x(t+1)

Non causal (averaging) system

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M

Mk

knxM

ny ][12

1][


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STABILITY

For a stable system, if the i/p is bounded, its

o/p will also be bounded.

The o/p cantdiverge for a bounded i/p.

e.g. For the system if x[n]

is bounded to a value B then the o/p is also

bounded to a value B & the system is stablewhereas for a system y[n] = x(k) the o/p

becomes unbounded.

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M

Mk

knxM

ny ][12

1][


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TIME INVARIANCE

For a time-invariant system, a time shift in

the i/p signal causes a time shift in the o/p

signal.

If y[n] is the o/p of a sys for i/p x[n] then

y[nno] is the o/p for i/p x[n-no].

Eg y(t) = sin[x(t)] is time invariant sys

whereas y[n] = nx[n] is time variant sys.3/17/2014

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If y(t) is the resp to x(t) & y(t) is the response to

x(t) then

Response to x(t) + x(t) is y(t) + y(t) (additive property).

Response to ax(t) is ay(t) (scaling property). Extending further if x(t), o=1,2,3.. are a set of i/p with

corresponding o/p y(t), o=1,2,3.. Then response to

x(t)=ax(t)= ax(t) + ax(t) + ax (t) + is

y(t)=ay(t)= ay(t) + ay(t) + ay(t) +....(Superpositionproperty).

Zero i/p yields zero o/p.


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INCREMENTALLY LINEAR SYSTEMS

Responds linearly to the changes in i/p.

Difference in the responses to any two i/ps

to an incrementally linear sys is linear.

Y[n] = 2x[n] + 3 is an ex of incrementallylinear sys such that y[n] - y[n] = 2{x[n] -

x[n]}

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Linear

System+

x(t)

y(t)

y(t)


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QUIZ - 3 A Unit step function is obtained by differentiating a unit

impulse function.(T/F)

Amplitude of a unit impulse function is unity.(T/F)

Sys is a process that transforms the input sig.(T/F)

A sys with memory must have some means of storing the

previous values of output.(T/F)

An inverse system transforms the output of its invertible

system to the original input. (T/F)3/17/2014

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All memoryless systems are non causal (T/F)


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All memoryless systems are non causal. (T/F)

Averaging systems are causal. (T/F)

A system is stable even if the i/p isunbounded. (T/F)

For time-invariant system, time shift in the i/p sig causesa time shift in the o/p signal.(T/F).

A sys can be linear w/o being time invariant & it can be

time invariant w/o being linear.(T/F)

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M

Mk

knxM

ny ][12

1][


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SIGNALS & VECTORS


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SIGNALS & VECTORS

Signals are exactly like vectors.

A vector can be represented as a sum of its

components in a variety of ways, dependingupon the choice of coordinate system.

A signal can also be represented as a sumof its components in a variety of ways.

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COMPONENT OF A VECTOR

Vector has magnitude & direction.

Component of vector galong another vector xiscx = |g|cos, where c is chosen to minimize thelength of error vector e= g- cx.

When g &x are perpendicular then g has a zerocomponent along x (c = 0);

Dot product: g.x = |g||x|cos , is the anglebetween g & x. g.x = 0 implies g & x are

orthogonal. Mag/length can be obtained from the relation

|x| = x.x

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COMPONENT OF A SIGNAL

Concept of vector component &orthgonality can be extended to signals.

Approximating a real sig g(t) in terms of

another sig x(t) over an interval tto tg(t) cx(t); t t t

Error in approximation e(t) = g(t)cx(t).

For best approx, energy in error sigmust be minimized.

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2

2

)(2t

te dtteE


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For the error to be min its differentiation wrt

c must be min i.e.

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0dc

dEe

0)(2 dttedc

d

2

1

2)()(

t

tdttcxtg

dc

d= 0

0)()()(2)( 2

1

2

1

2

1

222

t

t

t

t

t

tdttxc

dc

ddttxtgc

dc

ddttg

dc

d

2

1

2

1

0)(2)()(2 2t

t

t

tdttxcdttxtg


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This gives

Area under the product of the two signals

corresponds to the inner (scalar/dotproduct).

Energy of a sig is the inner product of the

sig itself & corresponds to the vector lengthsquared.

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2

12

1

2

1 )()(1

)(

)()(2

t

tx

t

t

t

tdttxtg

Edttx

dttxtgc


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EXAMPLE

For a square sig g(t) shown in fig find thecomponent in g(t) of the form sint. In other

words approx g(t) in terms of sint:

g(t) c sint; 0 t 2; so that the energyof the error signal is min.

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0

-1

1 g(t) t

2


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LTI SYSTEMS

Representation of Discrete time sig interms of Impuses

Discrete time unit impulse response & the

convolution sum representation of LTIsystem

Continuous time


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REPRESENTATION OF COMPLEX SIG

Aim is to represent non std sigs in terms of std sigs. Define a set of basic fns which can be combined in

some way to produce expression for less familiar w/f.

Properties of these basic set of fns are inferred fromanalogy with 2/3 dimension vector spaces.

One imp property is that the fundamental buildingblocks of these geometrical spaces are unit vectors in x,y, z dir.

These unit vectors are orthogonal & indicate that ourbasic func must also be orthogonal to for them to be

able to be combined to represent a complex sig.

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EXAMPLES

With definition of orthogonality in place we

need to find some familiar func that posses

this property.

Ex.1: Set of func {1, t, t, t,.} in theinterval (t, t).

Ex.2: {cosnt, sinmt) over the interval

(t t 2/). Ex. 3: { } over the interval

(t t 2/).

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tjmtjnee 00 &


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RECAP

Continuous & Discrete Signals.

Systems.

Signals & Vectors.

Orthogonal Signal.

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TODAY

Combining the functions in each orthogonalset to represent sigs that occur in comn

sys.

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TRIGONOMETRIC FOURIER SERIES

Representation of a signal in terms of sineand cosine functions.

We will limit the discussion to periodic

functions.Any periodic function f(t) with period 2/

can be represented by infinite trigonometric

series given by

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}sincos{)( 001

0 tnbtnaatf nn

n


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EXAMPLE

Obtain Trigonometric Fourier seriesrepresentation of the signal given below.

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-5/4 - -3/4 -/2 -/4 0 5/43/4/2/4

f(t)


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IMPORTANT OBSERVATIONS

Fourier Series expansion of a periodicfunction is valid for all time, - < t < ,

even though the integration, when

computing the coefficients, is carried overonly one pd.

To obtain a Fourier series representation of

a non periodic function over a given finite

interval, the appch is to let the time interval

of interest be the period T & proceed as in

case of a periodic signal of period T.3/17/2014

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EXPONENTIAL (COMPLEX) FOURIER


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SERIES

Complex Fourier Series can be obtained fromthe trigonometric series using Eulers identity.

The exponential or complex form of a Fourierseries is given as:

Where

n = ,-2, -1, 0, 1, 2,

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tjnnectf 0)(

Tt

t

tjn

n dtetfTc

0

0

0

)(

1


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EXAMPLE

Obtain Exponential Fourier seriesrepresentation of the signal given below

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f(t)


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IMPORTANT OBSERVATIONS

Trigonometric & Exponential Fourier seriesare simply two different form of the same

series.

Two forms provide flexibility in signalanalysis, since there are situation where one

form may be preferred over the other.

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SIMPLIFICATION OF FOURIERCOEFFICIENT EVALUATION

USING SIGNAL PROPERTIES


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ODD & EVEN SIGNALS

Trigonometric Fourier series representationof an even signal contains no sine term.

Trigonometric Fourier series representation

of an odd signal contains no const & cosineterm.

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LEAST SQUARE APPROXIMATION &


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LEAST SQUARE APPROXIMATION &

GENERALIZED FOURIER SERIES

For some appln an appx to a w/f may proveadequate.

In such cases we use t runcated tr igonometr ic

Four ier Series

This truncated Fourier series expansion of fN(t) is the

only one out of all possible trigonometric sums hN(t),

of order N or less, that minimizes the integral

squared error (ISE) given by:

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}sincos{)( 01

00 tnbtnaatf n

N

n

nN

Tt

t N thtfISE

0

0

)]()([

APPROXIMATION ACCURACY


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APPROXIMATION ACCURACY

The error in the approximation is given bythe ISE by replacing hN(t) by fN(t)

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Tt

t N dttftfISE

0

0

2)]()([

]}[2

{)( 2

1

22

0

20

0n

Tt

t

N

n

n baTTadttf

EXAMPLE


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EXAMPLE

Obtain a truncated trigonometric seriesapproximation to f(t) such that the ISE in

the approx is less than 2% of the integral

squared value of f(t)

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1

0-

f(t) = |sin t|

-2-3-4 32 t



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N

nn ttf )()(

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nmforK

nmfordtttwhere

n

n

Tt

t n

0)(*)(

0

0

Tttotfromtbysidesbothgmultiplyin m 00* int&)(

Tt

t n

n

n dtttfK

0

0

)()(1 *

TRUNCATED GENERALIZED


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FOURIER SERIES

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N

Nn

nnN ttg )()(

Minimizes the integral squared error ISE given by

Tt

t N dtthtfISE

0

0

2

|)()(|

dtthtfthtf NTt

t N

*)]()([)]()([0

0

formtheofhallfor N

N

Nn

nnN tpth )()(

Tt

t

N

Nn

nnKdttfISE 0

0

22

min |||)(|

PARSEVALS THEOREM


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PARSEVALS THEOREM

N

nn ttf )()(

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83

dtttfdttfTt

t

Tt

tn

nn

*

20

0

0

0

)()(|)(|

n

Tt

t nn dtttf

0

0 )()(

**

Tt

t n

n

n dtttf0

0)()(

**

=

n

nnnK *

nnnK

2|| =

0lim

ISE

N

thatshowsThis

COMPLETENESS & UNIQUENESS


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COMPLETENESS & UNIQUENESS

Completeness: Completeness isconcerned with the fact that ISE0as

N.

Uniqueness: Related to thereqmt of having enough functions to

represent a given waveform.

If a set of orthogonal functions is complete

it is necessarily unique but not vice versa.

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SPECTRAL CONTENT OF A PERIODIC

SIGNAL


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SIGNAL

Fourier Series separates a periodic timefunction f(t) into its various components.

The set of complex Fourier coefficients cn

constitute the line spectrum of f(t). Coeff Cn specifies the complex amp of the

freq component.

Coeff c0 is the amp of DC comp, c1is thefundamental freq comp, &cn is the amp of

nthharmonic.

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SPECTRUM


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SPECTRUM

Amplitude Spectrum

Phase Spectrum

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21

22)()(|| nnn ccc

)(

)(tan 1

n

nn

c

cc

nnnn cccc &||||

EXAMPLE


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EXAMPLE

Sketch the amplitude & phase spectra of thefollowing sq wave

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0 T/2-T/2

A

EXAMPLE


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EXAMPLE

Sketch the amplitude & phase spectra of thefollowing sq wave

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-5/4 - -3/4 -/2 -/4 0 5/43/4/2/4

f(t)

QUIZ 4


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QUIZ-4

Complex functions f(t) & g(t) are orthonormalif____.

For a complex function f(t)=_________.

To obtain a Fourier series representation of a nonperiodic function over a given finite interval, theappch is _____________.

Trigonometric Fourier series representation of aneven signal contains no sine term.(T/F)

A truncated generalized Fourier series with N+1component for closest approx is givenas_________.

Parsevals theorem shows that_____________.

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2

1

)()( *

t

t

dttftf


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If a set of orthogonal functions is unique it is

necessarily complete but not vice versa.(T/F) The amplitude and phase spectra of a

periodic signal are given by _______ &

______ resp.

The Fourier transform of func below will

have only sine components

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90

0 T/2-T/2

A

FOURIER TRANSFORM


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FOURIER TRANSFORM

Fourier Series gives spectrum of periodicsignal & approx spectrum of non periodic

sig.

Uniqueness is lost in the process. Need to develop a unique representation

for non periodic sig that is valid for all time

& therefore has a unique spectrum.

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EQUIVALENT FOURIER SERIES

EXPRESSIONS


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EXPRESSIONS

T

nnLet n 20

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92

0000 1)( nnnn(which is an incremental change in freq variable n)

nn Tcc /&

tjnn

ectf 0)(

Tt

t

tjn

n dtetf

Tc

0

0

0)(1

& becomes

n

n

tj

nTnectf

/

2

1)( dtetfc

tjT

T Tn

n)(2/

2/

/

& Where t = -T/2

FOURIER TRANSFORM PAIR


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FOURIER TRANSFORM PAIR

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93

Holding the shape of fT(t )in the intervalT/2 to T/2 fixed,if we take the limit of fT(t)as T, we obtain

dtectftf tjnTT

/

2

1)(lim)(

dtetfcwhere tjn

)(/

dlethavewewhere nn &

As complex Fourier series coefficients defined the spectrum of the signal we define

deFtf tj)(

21)(

dtetfF

tj )()(&

Correction dt = dw ve sign in exp of 2ndeqn


93/221

DIRICHLET CONDITIONS


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DIRICHLET CONDITIONS

These are the suf f ic ientbut not necessaryconditions for a Fourier Transform to exist.

These conditions are as under:

Ist Condition: Fourier Transform must exist orconverge i.e. Which

can be further reduced to the condition

2ndCondition: f(t) must have finite number of

maxima & minima in any finite interval.

3rdCondition: f(t) must have finite number of

discontinuities in any finite interval.

3/17/2014

95

dtetfF tj )()(

dttf |)(|

EXAMPLE


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EXAMPLE

Check convergence of the Fourier integralfor a signal f(t)specified by

3/17/2014

96

0,00,)(

/

tfortforVetf

t

EXAMPLE


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EXAMPLE

Check convergence of the Fourier integralfor a signal f(t)specified by

3/17/2014

97

otherwise

tfortu

,0

0,1)(

GENERALIZED FUNCTIONS


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GENERALIZED FUNCTIONS

Fourier transform for functions that are notabsolutely integrable by allowing the transform tocontain impulses or Delta functions

Delta Function: Defined as

(t) = 0 for t 0(t) = for t = 0

& the function is infinite at the origin in a very spl way that

Delta function is usually considered to be thederivative of unit step fn, obtained by first taking aramp fn & then letting its slope go to infinity.

3/17/2014

98

( ) 1t dt

PROPERTIES OF DELTA FUNCTION


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Sifting property: for(t) continuous at t = t0

Lim may not be necessarily from - to ,

however tmust lie betn the lim. In case tis o/s the lim the integration becomes zero.

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99

0 0( ) ( ) ( )t t t dt t



99/221


Derivative property:for (t) continuous at t = 0 & notation

superscript (n) denotes the nth derivative

of the fn. The first derivative of the delta fn

(n=1) occurs somewhat frequently & &

hence is given the spl name doublet.

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100

0

( ) ( )

0( ) ( ) ( 1) ( ) |n n n

t tt t t dt t



100/221


Scaling property: for(t) continuous at t = 0 & notation

Observe that

3/17/2014

101

1

( ) ( ) (0)at t dt a

1( ) ( ) ( ) ( )at t dt t t dt a

EXAMPLE


101/221

EXAMPLE

Evaluate(a)

(b)

(c)

3/17/2014

102

( )[ 1]t t dt

3

( )

t

t e dt

2( 2)[ 3 2]t t t dt

SINGULARITY FUNCTION


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SINGULARITY FUNCTION

A singularity function is defined as functionthat does not posses ordinary derivative of

all orders.

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103

FOURIER TRANSFORM & IMPULSEFUNCTIONS


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FUNCTIONS

To find Fourier transform of functions thatare not absolutely integrable.

Singularity functions are considered limits

of ordinary functions. Non rigorous but gives correct results more

easily & transparently

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104

DELTA FUNCTION


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DELTA FUNCTION

1|)()( 0

ttjtj edtettF

3/17/2014

105

Since this equation holds for all values of , the delta functionthus contain the same amt of all freq.

Using the fact that the Fourier transform is unique it is evidentthat the inverse Fourier transform is (t).

Thus the Fourier Transform pair of Delta function is given as

(t)1


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CONSTANT FUNCTION

)sin(2|

Aee

jAe

jAdtAeAF jjtjtj

1|)()( 0

t

tjtj

edtettF

)()1(2

111 tdeF tj

)(2 tde tj

3/17/2014

106

The fn is an oscillating fn & does not converge.Employing the unit impulse, we know that

As the Fourier Transform of a unit impulse is unique hence

dtAeAF tj

)(2 AAF

Now once again coming back to the Fourier Transform of a fn A


106/221


107/221

UNIT STEP FUNCTION

A Unit Step Function may be written as

u(t) = + sgn(t)

Thus

j

tFFtuF 1

)()sgn(2

1

2

1)(

3/17/2014

108

EXPONENTIAL FUNCTION FOR (-


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< t < )

Making change of variable y = -t & using the

result obtained for Constant Function

)(2 00

tjeF

dtedteeeF tjtjtjtj )( 000

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109

SINE & COSINE FUNTIONS


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SINE & COSINE FUNTIONS

tjtj eet 00

21cos 0

)()(cos 000 tF

)()(sin 000 jtFSimilarily

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110

GENERAL PERIODIC TIME

FUNCTION


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FUNCTION

n

n

n

tjn

n

n

tjn

n nceFcecFtfF )(2)( 000

3/17/2014

111

EXAMPLE


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EXAMPLE

Find the Fourier transform of the periiodicfunction shown below

3/17/2014

112

-5/4 - -3/4 -/2 -/4 0 5/43/4/2/4

f(t)

FOURIER TRANSFORM PROPERTIES


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Symmetry Property:

If g(t)G() Then G(t)2g(- )

Proof: Use Fourier Transform equation &

change of variables

3/17/2014113

1

)()( Gtg


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deGtg tj)(

2

1)(

3/17/2014114

dxexGtg tjx)()(2

xletting

tletting

dxexGg xj )()(2

txletting

dtetGg tj )()(2

)()(2)}({)(2 tGgORtGFg

EXAMPLE


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EXAMPLE

V, for -/2 t /2

g(t)= 0, otherwise

Has the Fourier Transform

Find the Fourier Transform of

3/17/2014115

)2/(

)2/sin()}({)(

VtgFG

)2/(

)2/sin()(

vt

vtvVtG



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Linearity Property:

If g1(t)G1() & g2(t)G2()

Then for arbitrary constants a & b

ag1(t) + bg2(t) aG1() + bG2()

3/17/2014116



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Scaling Property:

If g(t)G() Then for a real const b

g(bt) (1/|b|) G(/b)

Proof: Use change of variables

3/17/2014117

dtetgtgF tj)()}({


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3/17/2014118

xatletting

dteatgatgF tj

)()}({

dx

a

dt

a

xt

1&

dxexga

atgF xaj )()(1

)}({

)(1)()(1)}({ aGa

atgORaGa

atgF

EXAMPLE


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V, for -/2 t /2g(t)= 0, otherwise

Has the Fourier Transform

Find the Fourier Transform of

(i) g(2t)=

(ii) g(t/2) = -do- 3/17/2014119

)2/(

)2/sin()}({)(

VtgFG

V, for -/2 t /20, otherwise

FOURIER TRANSFORM

PROPERTIES


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PROPERTIES

Time Shifting Property:

If g(t)G()

Then g(t - to) G()

Proof: Use change of variables

0tj

e

3/17/2014

120

FOURIER TRANSFORM

PROPERTIES


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PROPERTIES

Frequency Shifting Property:

If g(t)G()

Then g(t) G(- o)

Proof: Direct substitution into F() equation

0tj

e

3/17/2014

121

EXAMPLE


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Find the Fourier Transform of the resultantsignal when a signal So + mt is multiplied by

cos t

3/17/2014122


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FOURIER TRANSFORM

PROPERTIES


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PROPERTIES

Time Integration Property:

If g(t)G()

Then

Proof: Requires Time Convolution Theorem

3/17/2014

124

)()0()(1

)(

FFj

dft

GRAPHICAL PRESENTATION OFFOURIER TRANSFORM


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Fourier Transform of a time signal specifies thespectral contents of the signal.

Fourier transform, in general , is a complex

function & hence two different graphs arenecessary to present all of the info completely.

The amp & spectrum are defined by the relations

3/17/2014125

2122

)()(|| nnn ccc

)(

)(tan 1

n

nn

c

cc

EXAMPLE


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Find the Fourier Transform of the signal f(t)specified by

Plot the amplitude & phase spectra of f(t)

3/17/2014126

0,0

0,)(

/

tfor

tforVetf

t

TRANSFORM FOR VARIOUS FORMS OFf(t)


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( )

If f(t) is a: Then F() is a:

Real & even fn of

t.Real & odd.

Imaginary & even.

Complex & even.Complex & odd.

Real & even fn of

.Imaginary & odd.

Imaginary & even.

Complex & even.Complex & odd.

3/17/2014127

EXAMPLE


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Find the Fourier transform of the periodicfunction shown below.

Sketch the real & imaginary part of the

Fourier transform. Sketch the amplitude & phase spectra of f(t).

3/17/2014128

-/2 0 t/2

f(t)

V

EXAMPLE


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Find the Fourier transform of f(t) =sint

f(t) =cost.

Sketch the real & imaginary part of the Fourier

transform. Sketch the amplitude & phase spectra of f(t).

3/17/2014129

f(t)


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LINEAR SYSTEMS, CONVOLUTION,

AND FILTERING

3/17/2014130

INTRODUCTION


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It is important to be able to analyze &specify cct & systems which op on, or

process, signals to achieve a desired

result.

Study of techniques for system

representation & analysis which are

necessary for the study of comn system.

3/17/2014131

LINEAR SYSTEM


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Given that y(t) & y(t) are the system o/presponses to i/ps r(t) & r(t), respectively, a

system is said to be linear if the i/p signal

ar(t) + brt) produces the system o/p

response ay(t) + byt) , where a & b are

any arbitrary constants.

Also the differential eqn representing such

system will have all derivatives of the i/p &o/p raised only to the first power & there are

no products of i/p & o/p or their derivatives3/17/2014

132

TIME INVARIANT SYSTEM


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Given that the o/p response of the system isy(t) for an i/p r(t) applied at t =t, a system issaid to be time invariant if the o/p is y(t) = y(t-t ) for an i/p signal r(t-t ) applied at time (t+t).

Also none of the coefficients in the differentialeq governing such system are a fn of time i.e.the coefficients are all const wrt time.

Shape of o/p response is same regardless ofwhen in time the the i/p is applied.

3/17/2014133

MATHEMATICAL MODEL OF THESYSTEM


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A mathematical model of a system predicts its o/p forany given i/p.

There are four mathematical operations which are the

basic building block of mathematical models. These

models are:

Scalar Multiplication:

Differentiation:

Integration

Time Delay Each of these operations describe a linear, Time

invariant system.

3/17/2014134

)()( trdt

dty

)()( trty

t

tdrty

0

)()(

)()( 1ttrty

IMPLICATION OF LINEARITY


134/221

For a Linear system, the o/p response dueto a sum of several i/ps is the sum of

responses due to each indl i/p.

For a Linear time invariant system the o/pcan contains only those freq which are

present at the i/p i.e. no new freq are

generated

3/17/2014135

LINEAR SYSTEM RESPONSE:CONVOLUTION


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Consider a general w/f defined for - < t < .

3/17/2014136

0 t

General Excitation signal

r(t)

r(kt)

-t

kt


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Sig r(t) may be written as an infinite sum of

pulses of widtht

3/17/2014137

0 t

Approximation of r(t) by a sum of pulses of width t

r(t)

r(kt)

- t

kt

Assuming the response of the system any

pulse has a shape denoted by h(.)


137/221

p p y ( )

The o/p response to one pulse at time t =

kt will be given as

Linearity results in complete response as

For the sequence of pulses of widtht to

exactly represent r(t) we must havet 0.

3/17/2014138

y(t) =r(kt) h(tkt)t

k ttkthtkrty )()()(

k

tttkthtkrty )()(lim)(

0

dthrty )()()(

INFERENCES


138/221

This is the Convolution Integral.

Derived by lettingt0 hence the system

response h(.) is the response to an impulsefn.

Hence if the i/p & impulse response of the

system is known the o/p of the sys can be

calculated using convolution operation

above.

Convolution op is denoted by y(t) = r(t) *3/17/2014139

dthrty )()()(

EXAMPLE


139/221

Impulse response of an RC cct is given by

Find the o/p response of this cct to the i/ppulse

3/17/2014140

0,0

0,)/1()(

/

tfor

tforeRCth

RCt

otherwise

tfortr

,0

20,1)(

SOLUTION


140/221

)(

1

0,0

0,1)(

tueRCtfor

tforeRCth

RCt

RCt

3/17/2014141

)2()(,0

20,1)(

tutuotherwise

tfortr

)(1

)( )( tueRC

th RCt

)2()()( uur


141/221

3/17/2014142

0

U(t-)

PROPERTIES OF CONVOLUTIONOPERATION


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Commutative Law: r(t)*h(t) = h(t)*r(t)Proof: Use change of variable

Distributive Law:

r(t)*[h1(t)+h2(t)]= r(t)*h1(t) + r(t)*h2(t)Proof: Straightforward

Associative Law

r(t)*[h1(t)*h2(t)] = [r(t)*h1(t) ]*h2(t)Proof: Will be considered later

3/17/2014143

TIME CONVOLUTION THEOREM


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Used to find sys o/p when convolutionintegral is not very straightforward.

If r(t) R() & h(t) H()

then F{r(t) * h(t)} = R()H()Proof:

3/17/2014144

dtedthrthtrF tj

)()()()(

ddehr j

)()()(

Interchange the order of integration & make the change of variable = t -

dedehr jj

)()(

)()( HR

INFERENCES


144/221

Time convolution theorem states that Fourier Transform ofconvolution of two time fns is product of their Fourier

Transform.

If we know the impulse response of a sys h(t) & we want to

calculate the o/p response to a given i/p, we can computethe Fourier transform of the i/p & f(t), form the prod of

these & then find the inverse transform. Often the last step

is not necessary since all we need to know is the freq

contents of the o/p.

For a cascade connection of two systems with impulse

responses h1(t) & h2(t) with i/p r(t), the Fourier Transform

of the o/p will be Y() = R()H1()H2() & y(t) =

F{Y()} 3/17/2014145

FREQ CONVOLUTION THEOREM


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If f(t) F() & g(t) G()then F{f(t) g(t)} = (1/2)[F()*G()]

Proof: Assignment

3/17/2014146

INFERENCES


146/221

Freq convolution theorem states that FourierTransform of prod of two time fns is fn of

convolution of their Fourier Transform.

Most of the comn sys req multiplication of twotime sigs. Freq convolution theorem is

extremely useful for analysis of such sys.

Associative Law of convolution can be proved

with the help of Freq convolution theorem.

3/17/2014147

GRAPHICAL CONVOLUTION


147/221

Graphical convolution is performed by carryingout following op:

Consider the convolution of the two time fns

Replace t in h(t) by which is its reflection about = 0axis. Result is then shifted by t secs.

Let t = in r(t).

Finally integrate this prod over all values of , which is

equivalent to calculating the area under the prod.

These op are repeated for all possible values of t to

produce the total convolution waveform

3/17/2014148

dthrty )()()(


148/221

SOLUTION


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Step I

3/17/2014150

3

0 t1

f1()

f2(-)

-0.5-2

SOLUTION


150/221

Step II

3/17/2014151

3

0 t1

f1()

f2(t-)

SOLUTION


151/221

Step III

3/17/2014152

3

0 t1

f1()

f2(t-)f2(t-)

SOLUTION


152/221

Step III

3/17/2014153

3

0 t1

f1()

f2(t-)f2(t-)

SOLUTION


153/221

Step IV

3/17/2014154

3

0 t1

f1()

f2(t-)f2(t-)

INFERENCES


154/221

There are sit when graphical appch is easier thanan analytical appch.

Generally a combination of both is used.

Graphical appch is first used to find the rg of t

over which convolution gives non zero values & to

get an idea about the shape of the resulting w/f.

Analyltical convolution is then performed precisely

using the limits deduced using graphicalconvolution.

3/17/2014155


155/221

DISTORTIONLESS TXN


156/221

)t-Ar(ty(t) d

3/17/2014157

)()()( propertyshiftingtimeeARY dtj

)()()()( theoremnconvolutioTimeRHY

'' FunctionTransferSystemtheeqnstwotheComparing

ionconsideratunderrangefreqtheoverAeH dtj

)(

dh tAHwhere )(&)(

INTUITIVE EXPLANATION OFDISTORTIONLESS TXN


157/221

For a distortionless system o/p sig is i/p sigmultiplied by A & delayed by time td.

From the sys tfr fn for this sys it is seen that

which indicates that higher freq comp have greaterphase lag.

This is because The o/p sig will have same compas i/p, with each comp mult by A & delayed bytd. i.e. if the i/p has comp cost o/p will havecos(t-td)= cos(t-td) implying addl ph shiftprop to freq

3/17/2014158

dh tAHwhere )(&)(

This can be explaned graphically


158/221

This varifies the fact that to achieve the sametime delay, higher freq sinusoids must undergoproportionately higher phase shifts.

3/17/2014159

/2

td

The time delay resulting from sig txn through

a sys is theve of slope of sys ph resp i.e.d


159/221

If the slope h is const is const all comp aredelayed by same time int. but if this slope is

not const td varies with freq & o/p will be

distorted. Thus only having a flat amp spectrum

doesnt guarantee a distortionless txn. The

phase response has to be linear (const td)

for a sys to be distortionless.

3/17/2014160

dh t )(

d

dt hd )(

FILTERS


160/221

Filters are usually characterized as Low pass,HighPass,BandPass or Bandstop.

These terms refer to the shape of Amplitude

Spectrum of filters Impulse Response or

Transfer Function.

Using the result obtained for Distortionless txn, a

filter is defined by System Transfer Function

3/17/2014161

elseeverywhere

forAeH

dtj

filter

0

,)( 21


161/221

3/17/2014 162

LOW PASS FILTER

LOW PASS FILTER


162/221

Amplitude Spectrum:

Phase Spectrum:

3/17/2014163

otherwise

forAeH S

tj

LPF

d

,0

||,)(

|H()|A

s-s

Slope = -td

0

0

/_H()

IMPULSE RESPONSE OF ANIDEAL LPF


163/221

Taking inverse Fourier Transform we get

3/17/2014164

)(

)(sin)(

d

dSLPF

tt

ttAth

td

As/

td-1/2fs td-1/2fs

EXAMPLE


164/221

f(t)=3 + sin3t + sin12tcos30t + 5cos47t +sin85t + 2sin102t + cos220t + sin377t

Ideal LPF with A = 2, td= 0, s = 40

rad/sec.

3/17/2014165


165/221

3/17/2014 166

BAND PASS FILTER

BAND PASS FILTER

tj


166/221

Amplitude Spectrum:

Phase Spectrum:

3/17/2014167

otherwise

forAeH

dtj

filter ,0

||,)( 21

|H()|

A

-2

Slope = -td

0

0

/_H()

-1 -1 -2

SYSTEM TFR FN : BPF IN TERMSOF LPF


167/221

System tfr fn of BPF can be said to be aSystem tfr fn:

Shifted by amts

Thus

3/17/2014168

otherwise

forAeH

dtj

,0

2||,

)(12

1

2

21

22

211

211 HHHBPF

IMPULSE RESPONSE OF ANIDEAL BPF


168/221

Impulse Response of a BPF can be found byobs that Sys Tfr Fn of a BPF is nothing but

the sys tfr fn of an LPF shifted by &

using freq shifting property of Fourier

Transform

This comes out to be

3/17/2014169

2

21

dd

d

BPF tttt

tt

Ah2

cos2

sin

2 12

12


169/221

3/17/2014 170

HIGH PASS FILTER

HIGH PASS FILTER

fA tj ||


170/221

Amplitude Spectrum:

Phase Spectrum:

3/17/2014171

otherwise

forAeH L

tj

filter

d

,0

||,)(

|H()|

A

Slope = -td

0

0

/_H()

-L -L

SYSTEM TFR FN : HPF IN TERMSOF LPF


171/221

System Tfr Fn of HPF can be written interms of Sys Tfr Fn of LPF as:

3/17/2014172

)()(

LPF

tj

HPF HAeH d

IMPULSE RESPONSE OF ANIDEAL HPF


172/221

Impulse Response of a HPF will be

3/17/2014173

)()()( 11 LPFtj

HPFHPF HAeFHFth d

)(11 LPFtj

HFAeF d


173/221

3/17/2014 174

BAND STOP FILTER

BAND STOP FILTER

fA tj d ||


174/221

Amplitude Spectrum:

Phase Spectrum:

3/17/2014175

otherwise

forAeH L

tj

filter

d

,0

||,)(

|H()|

A

Slope = -td

0

0

/_H()

-1-2 21

SYSTEM TFR FN : BSF IN TERMSOF BPF


175/221

System tfr fn of BSF can be said to be aSystem tfr fn:

3/17/2014176

)()( BPFtjBSF HAeH d

IMPULSE RESPONSE OF ANIDEAL BPF


176/221

Impulse Response of a BSF can be found byobs that Sys Tfr Fn of a BSF is nothing but

This comes out to be

3/17/2014177

)()()( 11 BPFtjBSFBSF HAeFHFth d

)(11 BPFtj

HFAeF d

LIMITATIONS


177/221

Ideal filters are not practically realizable.

A unit impulse applied to an ideal filter at timet=0 will give an o/p even for time t


178/221

In terms of freq domain concepts the sysmust satisfy PaleyWiener criterian to be

causal.

This criterian is given by the eqn

However before applying this criterian it is

necessory to est that

3/17/2014179

dH21

)(log

dH 2)(

REQUIREMENTS OF A FILTER


179/221

A filter must satisfy three very stringentreqmts: Const gain in passband.

Linear phase response across the passband.

Perfect attn o/s the passband

It is not possible to realize a filter that exactlyacheives all of these characteristics.

Three diff types of filters are desighned each

of which provides a good approx to one of theideal LPF properties while compromisingsomewhat on others.

3/17/2014180

BUTTERWORTH FILTER


180/221

Approx the reqmt of const gain throughout thepassband.

Amplitude char of Low pass butterworth filteris expressed as

It may be seen that butterworth filter providemaximally flat amp response in the passband

but their attn o/s passband may not besufficient for many applns.

3/17/2014181

freqcutoffdBtheiswhereH C

n

C

3)(1

1)(212

systheoforderthecalledusuallyfntfrsystheinpolesofnothedenotesn ,.......,3,2,1

BUTTERWOTH LPF AMP RESPFOR >0


181/221

3/17/2014182

Ideal (n=)

5thorder(n=5)

3rdorder (n=3)

1storder (n=1)


182/221

NOISE IN COMMUNICATION SYSTEMS

* Noise is an undesirable disturbance, which isl t d ith th d i d i l Di t ti d t


183/221

uncorrelated with the desired signal. Distortions due to

nonlinearities of the system, even though undesirable ,cannot be called noise.

* Noise may be defined as an extraneous form of energywith random frequency and amplitude which tends to

interfere with reception of a signal from a distant Tx.

* A transmitting eqpt does not produce noise in general.In fact, the signal level is raised to such a high magnitudein the Tx that any noise existing in the transmitting

system can be easily ignored in comparison to the infosignals.

EFFECTS OF NOISE


184/221

Noise modifies the desired signal in anunwanted manner resulting in:

Hiss in the loudspeaker output.

In TV Snow or confetti becomes superimposed

on the picture.

Cancellation of or production of unwanted

pulses in dig comn.

IMPACT


185/221

Affects sensitivity of the rxr.

Reduction in the Bandwidth of a system

TYPES OF NOISE

--EXTERNAL NOISE

INTERNAL NOISE


186/221

--INTERNAL NOISE

* In general noise may be picked up by a signal during its txn. from a Txto a Rx. This type of noise is commonly termed as External noise.

* Alternatively noise may be produced within a receiving eqptwhile it receiving a signal and this type of noise is termed as internal noise.

EXTERNAL NOISE ATMOSPHERIC NOISE


187/221

ATMOSPHERIC NOISE

CAUSED BY LIGHTINING AND OTHERNATURAL ELECTRIC DISTURBANCES

IN THE FORM OF RANDOM IMPULSES

HENCE SPREAD OVER THE ENTIRESPECTRUM USED FOR BROADCASTING.

FD STR INVERSELY PROP TO FREQ.

FREQUENCY RANGE : 1 MHz - 30 MHz

EXTERNAL NOISE EXTRA TERRESTRIAL NOISE


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SOLAR NOISE BEING LARGE BODY AT A VERY HIGH TEMP, SUN

RADIATES NOISE OVER A VERY BROAD FREQUENCYSPECTRUM

SOLAR CYCLES EVERY 11 YRS,PEAKS AFTER 100 YRS

FOR WHICH ELECTRICAL DISTURBANCES ERUPT, SUCHAS CORONA FLARES AND SUNSPOTS.

COSMIC NOISE

RADIATION FROM OTHER STARS AND GALAXIES IN THESAME MANNER AS SUN

FREQUENCY RANGE : 8 MHz1.4 GHz

EXTERNAL NOISE INDUSTRIAL NOISE


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SOURCES ARE AUTOMOBILE, AIRCRAFT

IGNITION, ELECTRIC MOTORS ETC.

RECEIVED NOISE INCREASES AS THE

RECEIVER BANDWIDTH IS INCREASED

FREQUENCY RANGE : 1 MHz600 MHz

Internal Noise

Internal noise is the electronic noise generated by the passive and active componentsof communications equipment.


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Shot Noise

Shot noise is the result of electron-hole recombination and minority carrier randomdiffusion in semiconductor devices. The power spectral density of shot noise isproportional to the current passing through the device and is given by

Pn= 2. Is. qwhere Is= saturation current (A)

q = electron charge (1.59 X 10-19C)

Thermal Noise

Thermal noise is the result of random motion of thermally agitated free-electronswithin a resistive component. Thermally agitated electrons within a resistor collidewith the molecules of that conductor, thus setting in motion a chain reaction with all


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the other free electrons.

Maximum noise power output of a resistor is

Pn= KT BW

where T = absolute noise temperatureK = Boltzmann's constant (1.38 X 10-23 J-K)

BW= operating bandwidth (Hz)


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SIGNAL-TO-NOISE RATIO (SNR)

The signal-to-noise ratio (SNR) expresses in decibels the difference betweenbaseband signal power and noise power at the input or output of a communicationsreceiver. This ratio is perhaps the most important criteria of establishing


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performance for electronic equipment including communications receivers. The

SNR is expressed in dB by

Where SNR = signal-to-noise ratio (dB)

Ps= signal power (W)Pn= noise power (W)

NOISE FIGURE (NF)

Thermal noise has been considered to be an inherent part of all electronic devices andcircuits and is largely responsible for the degradation of the overall systemperformance.


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Noise figure is defined as the signal-to-noise ratio at the input of a network divided bythe signal-to-noise ratio at the output of that network

p

All manufacturers of communications equipment express internal noise in terms ofnoise figure (NF).


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DIGITAL COMMUNICATION

SAMPLING PROCESS


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An op that is basic to DSP & Dig Comn.

An analog sig is converted to

corresponding seq of pulses usually

spaced uniformly in time.

Sampling rate must be chosen properly to

ensure unique rep of orig analog sig.

SAMPLING RATE

Consider a band limited sig g(t) hose spectr m is


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Consider a band limited sig g(t) whose spectrum isband limited to B Hz.

Sampling g(t) at rate of Hz can be accomplished bymultiplying g(t) by an impulse train , consisting of

unit impulses repeating periodically every secs.where .

Sampled sig consists of impulses spaced at secs

The nthimpulse loc at t= , has str .

Thus

Sf

)(tST

ST SS fT /1

ST

SnT )( SnTg

)()()()( Sn

ST nTtnTgtgtg S

)(G)(tg A


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)(tST

)(tg)(G

B2B2

B

f0

t

ST

t B2B2

B

f0Sf

S

QUANTIZATION


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mp

-mp

Allo

wedquantizatio

nlevelsL

2mp/L

m(t)

Quantized samples of m(t)

QUANTIZATION PROCESS

Lim the amp of msg sig m(t)to the rg )(


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Lim the amp of msg sig m(t)to the rg .

is not necessarily the peak amp of m(t). The amp of m(t) beyond are chopped off.

Thus is not a parameter of sig m(t) but aconstant of the quantizer.

The amp rg is divided into Luniformly spaced interval each of width.

A sample value is appx by mid pt of the

interval in which it lies. The quantized samples are coded &

transmitted as bin pulses.

),(pp

mm

pm

pm

pm

),(pp

mmLmv p2


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DIGITAL CARRIER SYSTEM

DEFINITION


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Baseband Digital System: Sig are tx directly w/o change in freq.

Low freq.

Tx over pair of wire, coax cable, FOC.

Digital Carrier System:

Sig spectrum shifted to high freq rg.

Achieved by modulating a high freq sinusoidby the baseband sig.

REQMT OF DIG CARRIER SYS


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Ant Size. FDM

TECHNIQUES


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Shifting of sig spectrum to a higher freq isachieved through modulation.

Two types:

Amplitude Modulation (ASK)

Angle Modulation

Freq Modulation (FSK)

Phase Modulation (PSK)

AMPLITUDE SHIFT KEYING (ASK)


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Carrier Amp is varied in proportion to msgsig.

1 0 110 001

tCcos

)(tm

ttm Ccos)(

PHASE SHIFT KEYING (PSK)

The info resides in the phase of the pulse


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The info resides in the phase of the pulse.

1 0 110 001

tCcos

)(tm

ttm Ccos)(

FREQ SHIFT KEYING (FSK)

Data transmitted by varying carrier freq


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Data transmitted by varying carrier freq.

1 0 110 001

tCcos

)(tm

1C

0C

Q. 1.

Fill in the blanks:


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Fill in the blanks: If energy signal g(t) represents variation of voltage

with time, the unit of its energy is volt sec.

If f(t) is multiplied by an impulse function shiftedtowards positive t by an amt T the resultant g(t) isgiven as f(T)(t T).

The area under an Impulse function shifted by timeT is equal to unity

Fourier series representation of even periodicfunction contains only constant & cosineterms.

If two signals have bandwidth of B1 and B2 hz resp.The bandwidth of the product of two signals isB1+B2 Hz.

Q. 2.


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State True or False: For a finite energy signal, the signal amplitude 0;

as time |t|. True

The terms analog and Digital time qualify the nature

of signal along the time axis. False

Impulse function is has zero value at all time instants

except at t = 0 where it is undefined. True

Time compression of a signal results in its spectral

compression. False

An Ideal filter represents a causal system.False


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Q 4 (a)

In the ideal case the margin provided by the


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In the ideal case the margin provided by the

correlation coefficient Cn for distinguishing

the two pulses in antipodal scheme is 2 (from -

1 to 1) & in orthogonal it is 1 (0 to 1).

The noise & channel distortion reduce thismargin & hence it is important to start with as

large margin as possible.

Antipodal scheme offers double the margin as

compared to orthogonal scheme & hence it is

preferred over orthogonal scheme.

Q 4 (b)

Ideal filters have time domain response


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Ideal filters have time domain response

that starts even before an input is applied.

Thus it represents a non causal system.

Non causal system cannot exist in reality.

Hence it is impossible to realize an ideal

filter.


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Q 4 (d)

Reqmt of impracticably large Ant size for


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Reqmt of impracticably large Ant size for

efficient txn at low freq forbids use of low

freq.

FDM possible after modulation giving more

efficient use of channel BW.


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Q 5. (a)


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t-1 0

2

g(t)=2exp(-t/2)

844

42

)(

0

0

1

2

2

dtedt

dttgE

t

g

Since g(t)0 as tThe signal g(t) is an energy signal

Q 5. (b)

2t


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710

5)7( edtet

t

As per sifting property of impulse function

provided impulse function lies between the limits t1 to t2

)()()(1 TfdttfTtt

When limits are changed from 5 - 10 to -10 to 3 impulsefunction lies outside the limit of integration and the integralresults in 0


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2

2

limdt

xddxd

t

SigRep&Ana DigComn

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