UNIT IUNIT I

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SIGNALSIGNAL

Signal is a physical quantity that varies with Signal is a physical quantity that varies with

respect to time , space or any other respect to time , space or any other

independent variableindependent variable

EgEg x(tx(t)= sin t.)= sin t.

the major classifications of the signal the major classifications of the signal

are:are:

(i) Discrete time signal (i) Discrete time signal

(ii) (ii) Continuous time signal Continuous time signal

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Unit Step &Unit ImpulseUnit Step &Unit Impulse

Discrete time Unit impulse is defined asDiscrete time Unit impulse is defined as

[n]= {0, n[n]= {0, n 00

{1, n=0{1, n=0

Unit impulse is also known as unit sample.Unit impulse is also known as unit sample.

Discrete time unit step signal is defined by Discrete time unit step signal is defined by

U[nU[n]={0,n=0]={0,n=0

{1,n>= 0{1,n>= 0

Continuous time unit impulse is defined as Continuous time unit impulse is defined as

(t)={1, t=0(t)={1, t=0

{0, t {0, t 00

Continuous time Unit step signal is defined as Continuous time Unit step signal is defined as

U(t)={0, t

Periodic Signal & Periodic Signal & AperiodicAperiodic SignalSignal

A signal is said to be periodic ,if it exhibits A signal is said to be periodic ,if it exhibits periodicity.i.eperiodicity.i.e., ., X(tX(t +T)=+T)=x(tx(t), for all values of t. Periodic signal has the ), for all values of t. Periodic signal has the property that it is unchanged by a time shift of T. A signal property that it is unchanged by a time shift of T. A signal that does not satisfy the above periodicity property is that does not satisfy the above periodicity property is called an called an aperiodicaperiodic signalsignal

even and odd signal ?even and odd signal ? A discrete time signal is said to be even when, A discrete time signal is said to be even when, x[x[--nn]=]=x[nx[n]. ].

The continuous time signal is said to be even when, The continuous time signal is said to be even when, x(x(--tt)= )= x(tx(t) For ) For example,Cosexample,Cosnn is an even signal.is an even signal.

SIGNALSIGNAL

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Energy and power signalEnergy and power signal

A signal is said to be energy signal if it A signal is said to be energy signal if it

have finite energy and zero power.have finite energy and zero power.

A signal is said to be power signal if it A signal is said to be power signal if it

have infinite energy and finite power.have infinite energy and finite power.

If the above two conditions are not If the above two conditions are not

satisfied then the signal is said to be satisfied then the signal is said to be

neigtherneigther energy nor power signal energy nor power signal

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Fourier SeriesFourier SeriesThe Fourier series represents a periodic signal in terms of The Fourier series represents a periodic signal in terms of

frequency components:frequency components:

We get the Fourier series coefficients as followsWe get the Fourier series coefficients as follows::

The complex exponential Fourier coefficients are a sequence of The complex exponential Fourier coefficients are a sequence of

complex numbers representing the frequency component complex numbers representing the frequency component 00k.k.

=

p

0

tikk dte)t(xp

1X 0

=

=

1p

0n

nikk 0e)n(xp

1X

=

=

1p

0k

nikk 0eX)n(x

=

=

k

tikk 0eX)t(x

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Fourier seriesFourier series

Fourier series: a complicated waveform analyzed into a Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine functionsnumber of harmonically related sine and cosine functions

A continuous periodic signal A continuous periodic signal x(tx(t) with a period T0 may be ) with a period T0 may be represented by: represented by: X(tX(t)=)=kk=1=1 ((AAkk coscos kk t + Bt + Bkk sin sin kk t)+ t)+ AA00

DirichletDirichlet conditions conditions must be placed on must be placed on x(tx(t) ) for the series for the series to be valid: the integral of the magnitude of to be valid: the integral of the magnitude of x(tx(t) ) over a over a complete period must be finite, and the signal can only complete period must be finite, and the signal can only have a finite number of discontinuities in any finite have a finite number of discontinuities in any finite intervalinterval

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Trigonometric form for Fourier seriesTrigonometric form for Fourier series

If the two fundamental components of a If the two fundamental components of a

periodic signal areB1cosperiodic signal areB1cos0t and C1sin0t and C1sin0t, 0t,

then their sum is expressed by trigonometric then their sum is expressed by trigonometric

identities:identities:

X(tX(t)= )= AA00 + +

kk=1 =1 (( BBk k 22++ AAk k

22))1/21/2 (C(Ckk coscos kk tt--kk) or ) or

X(tX(t)= )= AA00 + +

kk=1 =1 (( BBk k 22++ AAk k

22))1/21/2 (C(Ckk sin sin kk t+ t+ kk))

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UNIT IIUNIT II

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Fourier TransformFourier Transform

Viewed periodic functions in terms of frequency components (FourViewed periodic functions in terms of frequency components (Fourier ier

series) as well as ordinary functions of timeseries) as well as ordinary functions of time

Viewed LTI systems in terms of what they do to frequency Viewed LTI systems in terms of what they do to frequency

components (frequency response)components (frequency response)

Viewed LTI systems in terms of what they do to timeViewed LTI systems in terms of what they do to time--domain signals domain signals

(convolution with impulse response)(convolution with impulse response)

View View aperiodicaperiodic functions in terms of frequency components via functions in terms of frequency components via

Fourier transformFourier transform

Define (continuousDefine (continuous--time) Fourier transform and DTFT time) Fourier transform and DTFT

Gain insight into the meaning of Fourier transform through Gain insight into the meaning of Fourier transform through

comparison with Fourier seriescomparison with Fourier series

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The Fourier TransformThe Fourier Transform

A transform takes one function (or signal) A transform takes one function (or signal)

and turns it into another function (or signal)and turns it into another function (or signal)

Continuous Fourier Transform:Continuous Fourier Transform:

( ) ( )( ) ( )

=

=

dfefHth

dtethfHift

ift

pi

pi

2

2

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Continuous Time Fourier TransformContinuous Time Fourier Transform

We can extend the formula for continuousWe can extend the formula for continuous--time Fourier series time Fourier series

coefficients for a periodic signalcoefficients for a periodic signal

to to aperiodicaperiodic signals as well. The continuoussignals as well. The continuous--time Fourier time Fourier

series is not defined for series is not defined for aperiodicaperiodic signals, but we call the signals, but we call the

formulaformula

the (continuous time)the (continuous time)

Fourier transformFourier transform..

==

2/p

2/p

tikp

0

tikk dte)t(xp

1dte)t(xp1X 00

= dte)t(x)(X ti

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Inverse TransformsInverse TransformsIf we have the full sequence of Fourier coefficients for a perioIf we have the full sequence of Fourier coefficients for a periodic dic

signal, we can reconstruct it by multiplying the complex signal, we can reconstruct it by multiplying the complex

sinusoids of frequency sinusoids of frequency 00k by the weights k by the weights XXkk and summing:and summing:

We can perform a similar reconstruction for We can perform a similar reconstruction for aperiodicaperiodic signalssignals

These are called the These are called the inverse transformsinverse transforms..

=

=

1p

0k

nikk 0eX)n(x

=

=

k

tikk 0eX)t(x

pi

= de)(X21)t(x ti

pi

pi

pi

= de)(X21)n(x ni

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Fourier Transform of Impulse FunctionsFourier Transform of Impulse FunctionsFind the Fourier transform of the Find the Fourier transform of the DiracDirac delta function:delta function:

Find the DTFT of the Find the DTFT of the KroneckerKronecker delta function:delta function:

The delta functions contain all frequencies at equal amplitudes.The delta functions contain all frequencies at equal amplitudes.

Roughly speaking, thatRoughly speaking, thats why the system response to an impulse s why the system response to an impulse

input is important: it tests the system at all frequencies.input is important: it tests the system at all frequencies.

1edte)t(dte)t(x)(X 0ititi ====

1ee)n(e)n(x)(X 0in

nin

ni====

=

=

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LaplaceLaplace TransformTransform LapalceLapalce transform is a generalization of the Fourier transform in the stransform is a generalization of the Fourier transform in the sense ense

that it allows that it allows complex frequencycomplex frequency whereas Fourier analysis can only whereas Fourier analysis can only handle handle real frequencyreal frequency. Like Fourier transform, . Like Fourier transform, LapalceLapalce transform allows transform allows us to analyze a us to analyze a linear circuitlinear circuit problem, no matter how complicated the problem, no matter how complicated the circuit is, in the frequency domain in stead of in he time domaicircuit is, in the frequency domain in stead of in he time domain.n.

Mathematically, it produces the benefit of converting a set of dMathematically, it produces the benefit of converting a set of differential ifferential equations into a corresponding set of algebraic equations, whichequations into a corresponding set of algebraic equations, which are much are much easier to solve. Physically, it produces more insight of the cireasier to solve. Physically, it produces more insight of the circuit and cuit and allows us to know the bandwidth, phase, and transfer characterisallows us to know the bandwidth, phase, and transfer characteristics tics important for circuit analysis and design.important for circuit analysis and design.

Most importantly, Most importantly, LaplaceLaplace transform lifts the limit of Fourier analysis to transform lifts the limit of Fourier analysis to allow us to find both the steadyallow us to find both the steady--state and state and transienttransient responses of a linear responses of a linear circuit. Using Fourier transform, one can only deal with he steacircuit. Using Fourier transform, one can only deal with he steady state dy state behavior (i.e. circuit response under indefinite sinusoidal excibehavior (i.e. circuit response under indefinite sinusoidal excitation). tation).

Using Using LaplaceLaplace transform, one can find the response under any types of transform, one can find the response under any types of excitation (e.g. switching on and off at any given excitation (e.g. switching on and off at any given time(stime(s), sinusoidal, ), sinusoidal, impulse, square wave excitations, etcimpulse, square wave excitations, etc..

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LaplaceLaplace TransformTransform

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Application of Application of LaplaceLaplace Transform to Transform to

Circuit AnalysisCircuit Analysis

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system

A system is an operation that transforms input signal x into output signal y.

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LTI Digital Systems

Linear Time Invariant

Linearity/Superposition:

If a system has an input that can be expressed as a sum of signals, then the response of the system can be expressed as a sum of the individual responses to the respective systems.

LTI

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Time-Invariance &Causality

If you delay the input, response is just a delayed version of original response.

X(n-k) y(n-k)

Causality could also be loosely defined by there is no output signal as long as there is no input signal or output at current time does not depend on future values of the input.

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Convolution

The input and output signals for LTI systems have special relationship in terms of convolution sum and integrals.

Y(t)=x(t)*h(t) Y[n]=x[n]*h[n]

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UNIT IIIUNIT III

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Sampling theory

The theory of taking discrete sample values (grid of color pixels) from functions defined over continuous domains (incident radiance defined over the film plane) and then using those samples to reconstruct new functions that are similar to the original (reconstruction).

Sampler: selects sample points on the image plane Filter: blends multiple samples together

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Sampling theory

For band limited function, we can just increase the sampling rate

However, few of interesting functions in computer graphics are band limited, in particular, functions with discontinuities.

It is because the discontinuity always falls between two samples and the samples provides no information of the discontinuity.

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Sampling theory

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Aliasing

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ZZ--transformstransforms

For discreteFor discrete--time systems, time systems, zz--transforms play transforms play the same role of the same role of LaplaceLaplace transforms do in transforms do in

continuouscontinuous--time systemstime systems

As with the As with the LaplaceLaplace transform, we compute transform, we compute

forward and inverse forward and inverse zz--transforms by use of transforms by use of transforms pairs and propertiestransforms pairs and properties

[ ]

=

=

n

nznhzH ][Bilateral Forward z-transform

+

=R

n dzzzHjnh1

][ 2

1][pi

Bilateral Inverse z-transform

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Region of ConvergenceRegion of Convergence

Region of the complex Region of the complex

zz--plane for which plane for which forward forward zz--transform transform convergesconverges

Im{z}

Re{z}Entire plane

Im{z}

Re{z}Complement of a disk

Im{z}

Re{z}Disk

Im{z}

Re{z}

Intersection of a disk and complement of a disk

Four possibilities (Four possibilities (zz=0 =0 is a special case and is a special case and

may or may not be may or may not be

included)included)

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ZZ--transform Pairstransform Pairs

hh[[nn] = ] = [[nn]]

Region of convergence: Region of convergence:

entire entire zz--planeplane

hh[[nn] = ] = [[nn--11]]

Region of convergence: Region of convergence:

entire entire zz--planeplane

hh[[nn--1] 1] zz--1 1 HH[[zz]]

[ ] [ ] 1 ][ 00

=== =

=

n

n

n

n znznzH

[ ] [ ] 111

1 1][ =

=

=== zznznzHn

n

n

n

[ ]

1 if 1

1

][

00

|> |aa| which is | which is the complement the complement of a diskof a disk

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[ ] azza

nuaZ

n >

for 1

11

StabilityStability

Rule #1: For a causal sequence, poles are Rule #1: For a causal sequence, poles are inside the unit circle (applies to zinside the unit circle (applies to z--transform transform functions that are ratios of two polynomials)functions that are ratios of two polynomials)

Rule #2: More generally, unit circle is Rule #2: More generally, unit circle is included in region of convergence. (In included in region of convergence. (In continuouscontinuous--time, the imaginary axis would time, the imaginary axis would be in the region of convergence of the be in the region of convergence of the LaplaceLaplace transform.)transform.)

This is stable if |This is stable if |aa| < 1 by rule #1.| < 1 by rule #1.

It is stable if |It is stable if |zz| > || > |aa| and || and |aa| < 1 by rule #2.| < 1 by rule #2.

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Inverse Inverse zz--transformtransform

Yuk! Using the definition requires a contour Yuk! Using the definition requires a contour

integration in the complex integration in the complex zz--plane.plane.

Fortunately, we tend to be interested in only Fortunately, we tend to be interested in only

a few basic signals (pulse, step, etc.)a few basic signals (pulse, step, etc.)

Virtually all of the signals weVirtually all of the signals well see can be built ll see can be built

up from these basic signals. up from these basic signals.

For these common signals, the For these common signals, the zz--transform pairs transform pairs have been tabulated (see have been tabulated (see LathiLathi, Table 5.1), Table 5.1)

[ ] [ ] dzzzFjnfn

jc

jc

1

21

+

= pi

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ExampleExample

Ratio of polynomial zRatio of polynomial z--

domain functionsdomain functions

Divide through by the Divide through by the

highest power of zhighest power of z

Factor denominator into Factor denominator into

firstfirst--order factorsorder factors

Use partial fraction Use partial fraction

decomposition to get decomposition to get

firstfirst--order termsorder terms

21

23

12][2

2

+

++=

zz

zzzX

21

21

21

231

21][

+

++=

zz

zzzX

( )1121

1211

21][

++=

zz

zzzX

12

1

10 1

211

][

+

+=z

A

z

ABzX

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Example (Example (concontt))

Find Find BB00 by by polynomial divisionpolynomial division

Express in terms of Express in terms of

BB00

Solve for Solve for AA11 and and AA22

1523

2121

23

21

1

12

1212

+

+++

z

zz

zzzz

( )111

1211

512][

++=

zz

zzX

8

21

121

211

21

921

441121

1

1

21

2

21

21

1

1

1

=

++=

++=

=

++=

++=

=

=

z

z

z

zzA

z

zzA

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Example (Example (concontt))

Express Express XX[[zz]] in terms of in terms of BB00, , AA11, and , and AA22

Use table to obtain inverse Use table to obtain inverse zz--transformtransform

With the unilateral With the unilateral zz--transform, or the transform, or the bilateral bilateral zz--transform with region of transform with region of convergence, the inverse convergence, the inverse zz--transform is transform is uniqueunique

11 1

8

211

92][

+

=

zzzX

[ ] [ ] [ ] [ ]nununnxn

821

9 2 +

=

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ZZ--transform Propertiestransform Properties

LinearityLinearity

Right shift (delay)Right shift (delay)

[ ] [ ] [ ] [ ]zFazFanfanfa 22112211 ++

[ ] [ ] [ ]zFzmnumnf m [ ] [ ] [ ] [ ]

+

=

m

n

nmm znfzzFznumnf1

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ZZ--transform Propertiestransform Properties

[ ] [ ] [ ] [ ][ ] [ ]{ } [ ] [ ]

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

[ ] [ ][ ] [ ]zFzF

zrfzmf

zrfmf

zmnfmf

zmnfmf

mnfmfZnfnfZ

mnfmfnfnf

r

rm

m

m r

mr

m n

n

n

n

m

m

m

21

21

21

21

21

2121

2121

=

=

=

=

=

=

=

=

=

=

=

+

=

=

=

=

=

=

Convolution definitionConvolution definition

Take Take zz--transformtransform

ZZ--transform definitiontransform definition

Interchange summationInterchange summation

SubstituteSubstitute rr = = nn -- mm

ZZ--transform definitiontransform definition

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UNIT IVUNIT IV

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IntroductionIntroduction

Impulse responseImpulse response hh[n[n] can fully characterize a LTI ] can fully characterize a LTI

system, and we can have the output of LTI system assystem, and we can have the output of LTI system as

The zThe z--transform of impulse response is called transform of impulse response is called transfer or transfer or

system functionsystem function HH((zz).).

Frequency responseFrequency response at is valid if at is valid if

ROC includes and ROC includes and

[ ] [ ] [ ]nhnxny =

( ) ( ) ( ).zHzXzY =( ) ( )

1== zj zHeH

,1=z

( ) ( ) ( ) jjj eHeXeY =

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5.1 Frequency Response of LIT 5.1 Frequency Response of LIT

SystemSystem

Consider and Consider and

, then, then

magnitudemagnitude

phasephase

We will model and analyze LTI systems based on the We will model and analyze LTI systems based on the

magnitude and phase responses. magnitude and phase responses.

)()()( jeXjjj eeXeX = )()()( jeHjjj eeHeH =

)()()( jjj eHeXeY =

)()()( jjj eHeXeY +=

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System FunctionSystem Function

General form of LCCDEGeneral form of LCCDE

Compute the zCompute the z--transformtransform

[ ] [ ]knxbknya Mk

k

N

kk =

== 00

( )zXzbzYza kMk

k

N

k

kk

==

=00

)(

( ) ( )( )

=

=== N

k

kk

kM

kk

za

zb

zXzY

zH

0

0

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System Function: Pole/zero System Function: Pole/zero

FactorizationFactorization

Stability requirement can be verified.Stability requirement can be verified.

Choice of ROC determines causality.Choice of ROC determines causality.

Location of zeros and poles determines the Location of zeros and poles determines the

frequency response and phasefrequency response and phase

( )( )( )

=

=

= N

kk

M

kk

zd

zc

a

bzH

1

1

1

1

0

0

1

1 .,...,,:zeros 21 Mccc

.,...,,:poles 21 Nddd

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SecondSecond--order Systemorder System

Suppose the system function of a LTI system isSuppose the system function of a LTI system is

To find the difference equation that is satisfied by To find the difference equation that is satisfied by

the input and out of this systemthe input and out of this system

Can we know the impulse response? Can we know the impulse response?

.

)431)(

211(

)1()(11

21

+

+=

zz

zzH

)()(

83

411

21

)431)(

211(

)1()(21

21

11

21

zXzY

zz

zz

zz

zzH =

+

++=

+

+=

]2[2]1[2][]2[83]1[

41][ ++=+ nxnxnxnynyny

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System Function: StabilitySystem Function: Stability

Stability of LTI system:Stability of LTI system:

This condition is identical to the condition This condition is identical to the condition

that that

The stability condition is equivalent to the The stability condition is equivalent to the

condition that the ROC of condition that the ROC of HH((zz) includes the unit ) includes the unit circle.circle.

=