EC2204-NOL

UNIT IUNIT I

SIGNALSIGNAL

►Signal is a physical quantity that Signal is a physical quantity that varies with respect to time , space or varies with respect to time , space or any other independent variableany other independent variable

Eg x(t)= sin t.Eg x(t)= sin t.►the major classifications of the the major classifications of the

signal are:signal are:

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

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

Unit Step &Unit ImpulseUnit Step &Unit Impulse

Discrete time Unit impulse is defined asDiscrete time Unit impulse is defined as δ [n]= {0, n≠ 0δ [n]= {0, n≠ 0{1, n=0{1, n=0Unit 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[n]={0,n=0U[n]={0,n=0

{1,n>= 0{1,n>= 0Continuous time unit impulse is defined as Continuous time unit impulse is defined as

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

Continuous time Unit step signal is defined as Continuous time Unit step signal is defined as U(t)={0, t<0U(t)={0, t<0

{1, t≥0{1, t≥0

► Periodic Signal & Aperiodic SignalPeriodic Signal & Aperiodic Signal A signal is said to be periodic ,if it exhibits A signal is said to be periodic ,if it exhibits

periodicity.i.e., X(t +T)=x(t), for all values of t. periodicity.i.e., X(t +T)=x(t), for all values of t. Periodic signal has the property that it is unchanged Periodic signal has the property that it is unchanged by a time shift of T. A signal that does not satisfy the by a time shift of T. A signal that does not satisfy the above periodicity property is called an aperiodic above periodicity property is called an aperiodic signalsignal

► even and odd signal ?even and odd signal ? A discrete time signal is said to be even when, x[-A discrete time signal is said to be even when, x[-

n]=x[n]. The continuous time signal is said to be n]=x[n]. The continuous time signal is said to be even when, x(-t)= x(t) For example,Cosωn is an even when, x(-t)= x(t) For example,Cosωn is an even signal.even signal.

SIGNALSIGNAL

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 neigther energy nor power signal neigther energy nor power signal

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

of frequency components:of frequency components:

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

The complex exponential Fourier coefficients are a The complex exponential Fourier coefficients are a

sequence of complex numbers representing the sequence of complex numbers representing the

frequency component frequency component ωω00k.k.

p

0

tikk dte)t(x

p1

X 0

1p

0n

nikk

0e)n(xp1

X

1p

0k

nikk

0eX)n(x

k

tikk

0eX)t(x

Fourier seriesFourier series

► Fourier series: a complicated waveform analyzed into Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine a number of harmonically related sine and cosine functionsfunctions

► A continuous periodic signal x(t) with a period T0 may A continuous periodic signal x(t) with a period T0 may be represented by: be represented by: X(t)=ΣX(t)=Σ∞∞

k=1k=1 ( (AAkk cos cos kkω ω t + Bt + Bkk sin sin kkω ω t)+ t)+ AA00

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

Trigonometric form for Fourier Trigonometric form for Fourier seriesseries

► If the two fundamental components of a If the two fundamental components of a periodic signal areB1cosω0t and periodic signal areB1cosω0t and C1sinω0t, then their sum is expressed by C1sinω0t, then their sum is expressed by trigonometric identities:trigonometric identities:

►X(t)= X(t)= AA00 + + ΣΣ∞∞k=1 k=1 (( BBk k

22++ AAk k 22))1/21/2 (C (Ckk cos cos kkω ω t- t-

φφkk) or ) or

►X(t)= X(t)= AA00 + + ΣΣ∞∞k=1 k=1 (( BBk k

22++ AAk k 22))1/21/2 (C (Ckk sin k sin kω ω

t+ t+ φφkk))

UNIT IIUNIT II

Fourier TransformFourier Transform

► Viewed periodic functions in terms of frequency components Viewed periodic functions in terms of frequency components (Fourier series) as well as ordinary functions of time(Fourier series) 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 time-domain Viewed LTI systems in terms of what they do to time-domain signals (convolution with impulse response)signals (convolution with impulse response)

► View aperiodic functions in terms of frequency components View aperiodic functions in terms of frequency components via Fourier transformvia Fourier transform

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

The Fourier TransformThe Fourier Transform

►A transform takes one function (or A transform takes one function (or signal) and turns it into another signal) and turns it into another function (or signal)function (or signal)

►Continuous Fourier Transform:Continuous Fourier Transform:

dfefHth

dtethfH

ift

ift

2

2

Continuous Time Fourier TransformContinuous Time Fourier TransformWe can extend the formula for continuous-time Fourier We can extend the formula for continuous-time Fourier series coefficients for a periodic signalseries coefficients for a periodic signal

to aperiodic signals as well. The continuous-time to aperiodic signals as well. The continuous-time Fourier series is not defined for aperiodic signals, but Fourier series is not defined for aperiodic signals, but we call the formulawe call the formula

the (continuous time)the (continuous time)

Fourier transformFourier transform..

2/p

2/p

tikp

0

tikk dte)t(x

p1

dte)t(xp1

X 00

dte)t(x)(X ti

Inverse TransformsInverse TransformsIf we have the full sequence of Fourier coefficients for a If we have the full sequence of Fourier coefficients for a

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

complex sinusoids of frequency complex sinusoids of frequency ωω00k by the weights Xk by the weights Xkk and and

summing:summing:

We can perform a similar reconstruction for aperiodic We can perform a similar reconstruction for aperiodic

signalssignals

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

1p

0k

nikk

0eX)n(x

k

tikk

0eX)t(x

de)(X21

)t(x ti

de)(X21

)n(x ni

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

Find the DTFT of the Kronecker delta function:Find the DTFT of the Kronecker delta function:

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

amplitudes.amplitudes.

Roughly speaking, that’s why the system response to an Roughly speaking, that’s why the system response to an

impulse input is important: it tests the system at all impulse input is important: it tests the system at all

frequencies.frequencies.

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

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

n

ni

n

ni

Laplace TransformLaplace Transform► Lapalce transform is a generalization of the Fourier transform in Lapalce transform is a generalization of the Fourier transform in

the sense that it allows “complex frequency” whereas Fourier the sense that it allows “complex frequency” whereas Fourier analysis can only handle “real frequency”. Like Fourier transform, analysis can only handle “real frequency”. Like Fourier transform, Lapalce transform allows us to analyze a “linear circuit” problem, Lapalce transform allows us to analyze a “linear circuit” problem, no matter how complicated the circuit is, in the frequency domain no matter how complicated the circuit is, in the frequency domain in stead of in he time domain.in stead of in he time domain.

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

► Most importantly, Laplace transform lifts the limit of Fourier Most importantly, Laplace transform lifts the limit of Fourier analysis to allow us to find both the steady-state and “transient” analysis to allow us to find both the steady-state and “transient” responses of a linear circuit. Using Fourier transform, one can responses of a linear circuit. Using Fourier transform, one can only deal with he steady state behavior (i.e. circuit response only deal with he steady state behavior (i.e. circuit response under indefinite sinusoidal excitation). under indefinite sinusoidal excitation).

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

Laplace TransformLaplace Transform

Application of Laplace Application of Laplace Transform to Circuit AnalysisTransform to Circuit Analysis

system

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

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

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”.

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]

UNIT IIIUNIT III

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

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.

Sampling theory

Aliasing

ZZ-transforms-transforms

►For discrete-time systems, For discrete-time systems, zz-transforms -transforms play the same role of Laplace transforms play the same role of Laplace transforms do in continuous-time systemsdo in continuous-time systems

►As with the Laplace transform, we As with the Laplace transform, we compute forward and inverse compute forward and inverse zz-transforms -transforms by use of transforms pairs and propertiesby use of transforms pairs and properties

n

nznhzH ][

Bilateral Forward z-transform

R

n dzzzHj

nh 1 ][ 2

1][

Bilateral Inverse z-transform

Region of ConvergenceRegion of Convergence

► Region of the Region of the complex complex zz-plane for -plane for which forward which 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 is a special =0 is a special case and may or case and may or may not be may not be included)included)

ZZ-transform Pairs-transform Pairs

►hh[[nn] = ] = [[nn]]

Region of convergence: Region of convergence: entire entire zz-plane-plane

►hh[[nn] = ] = [[n-1n-1]]

Region of convergence: Region of convergence: entire entire zz-plane-plane

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

1 ][0

0

n

n

n

n znznzH

11

1

1 1][

zznznzHn

n

n

n

1 if 1

1

][

00

z

a

za

z

aza

znuazH

n

n

n

nn

n

nn

►hh[[nn] = ] = aan n uu[[nn]]

Region of Region of convergence: |convergence: |zz| > || > |aa| which is the | which is the complement of a complement of a diskdisk

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 z-transform inside the unit circle (applies to z-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 continuous-time, the imaginary axis would be continuous-time, the imaginary axis would be in the region of convergence of the Laplace in the region of convergence of the Laplace 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.

Inverse Inverse zz-transform-transform

►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 Fortunately, we tend to be interested in only a few basic signals (pulse, step, etc.)only a few basic signals (pulse, step, etc.) Virtually all of the signals we’ll see can be built Virtually all of the signals we’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 Lathi, Table 5.1)have been tabulated (see Lathi, Table 5.1)

dzzzFj

nf njc

jc

1

2

1

ExampleExample

► Ratio of polynomial z-Ratio of polynomial z-domain functionsdomain functions

►Divide through by the Divide through by the highest power of zhighest power of z

► Factor denominator Factor denominator into first-order factorsinto first-order factors

► Use partial fraction Use partial fraction decomposition to get decomposition to get first-order termsfirst-order terms

21

23

12][

2

2

zz

zzzX

21

21

21

23

1

21][

zz

zzzX

11

21

121

1

21][

zz

zzzX

12

1

10 1

21

1][

z

A

z

ABzX

Example (con’t)Example (con’t)

►Find Find BB00 by by polynomial polynomial divisiondivision

►Express in terms Express in terms of of BB00

►Solve for Solve for AA11 and and AA22

15

23

2121

2

3

2

1

1

12

1212

z

zz

zzzz

11

1

121

1

512][

zz

zzX

8

21

121

21

1

21

921

441

1

21

1

1

21

2

2

1

21

1

1

1

z

z

z

zzA

z

zzA

Example (con’t)Example (con’t)

►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-transform-transform

►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

21

1

92][

zzzX

nununnxn

82

1 9 2

ZZ-transform Properties-transform Properties

►LinearityLinearity

►Right shift (delay)Right shift (delay)

zFazFanfanfa 22112211

zFzmnumnf m

m

n

nmm znfzzFznumnf1

ZZ-transform Properties-transform 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-transform-transform

► ZZ-transform definition-transform definition

► Interchange Interchange summationsummation

► SubstituteSubstitute r r = = nn - - mm

► ZZ-transform definition-transform definition

UNIT IVUNIT IV

IntroductionIntroduction

► Impulse responseImpulse response hh[n] can fully characterize a LTI [n] can fully characterize a LTI system, and we can have the output of LTI system, and we can have the output of LTI system assystem as

► The z-transform of impulse response is called The z-transform of impulse response is called transfer or system functiontransfer or system function HH((zz).).

► Frequency responseFrequency response at is valid at is valid if ROC includes and if ROC includes and

nhnxny

.zHzXzY

1

z

j zHeH

,1z

jjj eHeXeY

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 We will model and analyze LTI systems based on the magnitude and phase responses. the magnitude and phase responses.

)()()( jeXjjj eeXeX )()()(

jeHjjj eeHeH

)()()( jjj eHeXeY

)()()( jjj eHeXeY

System FunctionSystem Function

►General form of LCCDEGeneral form of LCCDE

►Compute the z-transformCompute the z-transform

knxbknyaM

kk

N

kk

00

zXzbzYza kM

kk

N

k

kk

00

)(

N

k

kk

kM

kk

za

zb

zX

zYzH

0

0

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 Location of zeros and poles

determines the frequency response determines the frequency response

and phaseand phase

N

kk

M

kk

zd

zc

a

bzH

1

1

1

1

0

0

1

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

.,...,,:poles 21 Nddd

Second-order SystemSecond-order 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 To find the difference equation that is satisfied by the input and out of this systemby the input and out of this system

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

.)

43

1)(21

1(

)1()(

11

21

zz

zzH

)(

)(

83

41

1

21

)43

1)(21

1(

)1()(

21

21

11

21

zX

zY

zz

zz

zz

zzH

]2[2]1[2][]2[8

3]1[

4

1][ nxnxnxnynyny

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.

n

nh ][

.1 when][

zznhn

n

System Function: CausalitySystem Function: Causality

► If the system is causal, it follows that If the system is causal, it follows that hh[[nn] must be ] must be a right-sided sequence. The ROC of a right-sided sequence. The ROC of HH((zz) must be ) must be outside the outside the outermostoutermost pole. pole.

► If the system is anti-causal, it follows that If the system is anti-causal, it follows that hh[[nn] must ] must be a left-sided sequence. The ROC of be a left-sided sequence. The ROC of HH((zz) must be ) must be inside the inside the innermostinnermost pole. pole.

1a

Im

Re 1a

Im

Re ba

Im

Re

Right-sided(causal)

Left-sided(anti-causal)

Two-sided(non-causal)

Determining the ROCDetermining the ROC

►Consider the LTI systemConsider the LTI system

►The system function is obtained asThe system function is obtained as

][]2[]1[2

5][ nxnynyny

)21)(21

1(

1

25

1

1)(

11

21

zz

zzzH

System Function: Inverse System Function: Inverse SystemsSystems

► is an inverse system for , ifis an inverse system for , if

► The ROCs of must overlap.The ROCs of must overlap.

► Useful for canceling the effects of another systemUseful for canceling the effects of another system

► See the discussion in Sec.5.2.2 regarding ROCSee the discussion in Sec.5.2.2 regarding ROC

zH i zH

1)()()( zHzHzG i

)(

1)(

zHzH i

)(

1)(

j

ji eHeH

nnhnhng i

)( and )( zHzH i

All-pass SystemAll-pass System

►A system of the form (or cascade of A system of the form (or cascade of these)these)

1

1

1

az

azZH Ap

1jAp eH

j

jj

j

jj

Ap ae

eae

ae

aeeH

1

*1

1

j

j

era

rea1*/1 :zero

:pole

All-pass System: General FormAll-pass System: General Form

► In general, all pass systems have formIn general, all pass systems have form

cr M

k kk

kkM

k k

kAp zeze

ezez

zd

dzzH

11*1

1*1

11

1

)1)(1(

))((

1

Causal/stable: 1, kk de

real poles complex poles

All-Pass System ExampleAll-Pass System Example

0.8

0.5

z-planeUnit circle

4

3

3

4 2

Re

Im

1 and 2 cr MM

zeros. and poles 42 has system pass-all This rc MMNM

jj erre 1conjugate & reciprocal :zero:pole

Minimum-Phase SystemMinimum-Phase System

► Minimum-phase system:Minimum-phase system: all zeros and all poles are all zeros and all poles are inside the unit circle.inside the unit circle.

► The name The name minimum-phaseminimum-phase comes from a property of comes from a property of the phase response (minimum phase-lag/group-delay).the phase response (minimum phase-lag/group-delay).

► Minimum-phase systems have some special properties.Minimum-phase systems have some special properties.

► When we design a filter, we may have multiple choices When we design a filter, we may have multiple choices to satisfy the certain requirements. Usually, we prefer to satisfy the certain requirements. Usually, we prefer the minimum phase which is unique.the minimum phase which is unique.

► All systems can be represented as a minimum-phase All systems can be represented as a minimum-phase system and an all-pass system.system and an all-pass system.

UNIT VUNIT V

ExampleExample

►Block diagram representation ofBlock diagram representation of nxb2nya1nyany 021

Block Diagram Block Diagram RepresentationRepresentation

►LTI systems with rational system LTI systems with rational system function can be represented as function can be represented as constant-coefficient difference constant-coefficient difference equationequation

►The implementation of difference The implementation of difference equations requires delayed values equations requires delayed values of theof the inputinput outputoutput intermediate results intermediate results

►The requirement of delayed The requirement of delayed elements implies need for storageelements implies need for storage

►We also need means of We also need means of additionaddition multiplicationmultiplication

Direct Form IDirect Form I

►General form of difference equationGeneral form of difference equation

►Alternative equivalent formAlternative equivalent form

M

0kk

N

0kk knxbknya

M

0kk

N

1kk knxbknyany

Direct Form IDirect Form I

►Transfer function can be written asTransfer function can be written as

►Direct Form I RepresentsDirect Form I Represents

N

1k

kk

M

0k

kk

za1

zbzH

zVza1

1zVzHzY

zXzbzXzHzV

zbza1

1zHzHzH

N

1k

kk

2

M

0k

kk1

M

0k

kkN

1k

kk

12

nvknyany

knxbnv

N

1kk

M

0kk

Alternative RepresentationAlternative Representation

►Replace order of cascade LTI systemsReplace order of cascade LTI systems

zWzbzWzHzY

zXza1

1zXzHzW

za1

1zbzHzHzH

M

0k

kk1

N

1k

kk

2

N

1k

kk

M

0k

kk21

M

0kk

N

1kk

knwbny

nxknwanw

Alternative Block DiagramAlternative Block Diagram

►We can change the order of the We can change the order of the cascade systemscascade systems

M

0kk

N

1kk

knwbny

nxknwanw

Direct Form IIDirect Form II

► No need to store the same No need to store the same data twice in previous systemdata twice in previous system

► So we can collapse the delay So we can collapse the delay elements into one chainelements into one chain

► This is called Direct Form II or This is called Direct Form II or the Canonical Formthe Canonical Form

► Theoretically no difference Theoretically no difference between Direct Form I and IIbetween Direct Form I and II

► Implementation wise Implementation wise Less memory in Direct IILess memory in Direct II Difference when using Difference when using

finite-precision arithmeticfinite-precision arithmetic

Signal Flow Graph Signal Flow Graph RepresentationRepresentation

►Similar to block diagram representationSimilar to block diagram representation Notational differencesNotational differences

►A network of directed branches connected at nodesA network of directed branches connected at nodes

►Example representation of a difference equationExample representation of a difference equation

ExampleExample

►Representation of Direct Form II with Representation of Direct Form II with signal flow graphssignal flow graphs

nwny

1nwnw

nwbnwbnw

nwnw

nxnawnw

3

24

41203

12

41

1nwbnwbny

nx1nawnw

1110

11

Determination of System Determination of System Function from Flow GraphFunction from Flow Graph

nwnwny

1nwnw

nxnwnw

nwnw

nxnwnw

42

34

23

12

41

zWzWzY

zzWzW

zXzWzW

zWzW

zXzWzW

42

134

23

12

41

zWzWzY z11zzX

zW

z11zzX

zW

42

1

1

4

1

1

2

nu1nunh

z1z

zXzY

zH

1n1n

1

1

Basic Structures for IIR Basic Structures for IIR Systems: Direct Form ISystems: Direct Form I

Basic Structures for IIR Basic Structures for IIR Systems: Direct Form IISystems: Direct Form II

Basic Structures for IIR Basic Structures for IIR Systems: Cascade FormSystems: Cascade Form

►General form for cascade implementationGeneral form for cascade implementation

► More practical form in 2More practical form in 2ndnd order systems order systems

21

21

N

1k

1k

1k

N

1k

1k

M

1k

1k

1k

M

1k

1k

zd1zd1zc1

zg1zg1zf1AzH

1M

1k2

k21

k1

2k2

1k1k0

zaza1zbzbb

zH

ExampleExample

► Cascade of Direct Form I subsectionsCascade of Direct Form I subsections

► Cascade of Direct Form II subsectionsCascade of Direct Form II subsections

1

1

1

1

11

11

21

21

z25.01z1

z5.01z1

z25.01z5.01z1z1

z125.0z75.01zz21

zH

Basic Structures for IIR Basic Structures for IIR Systems: Parallel FormSystems: Parallel Form

► Represent system function using partial fraction expansionRepresent system function using partial fraction expansion

► Or by pairingthe real polesOr by pairingthe real poles

P PP N

1k

N

1k1

k1

k

1kk

1k

kN

0k

kk zd1zd1

ze1Bzc1

AzCzH

SP N

1k2

k21

k1

1k1k0

N

0k

kk zaza1

zeezCzH

ExampleExample►Partial Fraction ExpansionPartial Fraction Expansion

►Combine poles to getCombine poles to get

1121

21

z25.0125

z5.0118

8z125.0z75.01

zz21zH

21

1

z125.0z75.01z87

8zH

Transposed FormsTransposed Forms

►Linear signal flow graph property:Linear signal flow graph property: Transposing doesn’t change the input-output relationTransposing doesn’t change the input-output relation

►Transposing:Transposing: Reverse directions of all branchesReverse directions of all branches Interchange input and output nodesInterchange input and output nodes

►Example:Example:

Reverse directions of branches and interchange input Reverse directions of branches and interchange input and outputand output

1az1

1zH

ExampleExample

Transpose

►Both have the same system function Both have the same system function or difference equationor difference equation

2nxb1nxbnxb2nya1nyany 21021

Basic Structures for FIR Systems: Direct Basic Structures for FIR Systems: Direct FormForm

►Special cases of IIR direct form structuresSpecial cases of IIR direct form structures

► Transpose of direct form I gives direct form II Transpose of direct form I gives direct form II ► Both forms are equal for FIR systemsBoth forms are equal for FIR systems

►Tapped delay lineTapped delay line

Basic Structures for FIR Basic Structures for FIR Systems: Cascade FormSystems: Cascade Form

►Obtained by factoring the polynomial Obtained by factoring the polynomial system functionsystem function

M

0n

M

1k

2k2

1k1k0

nS

zbzbbznhzH

Structures for Linear-Phase Structures for Linear-Phase FIR SystemsFIR Systems

► Causal FIR system with generalized linear phase are Causal FIR system with generalized linear phase are

symmetricsymmetric::

► Symmetry means we can half the number of Symmetry means we can half the number of multiplicationsmultiplications

► Example: For even M and type I or type III systemsExample: For even M and type I or type III systems::

IV)or II (type M0,1,...,n nhnMh

III)or I (type M0,1,...,n nhnMh

2/Mnx2/MhkMnxknxkh

kMnxkMh2/Mnx2/Mhknxkh

knxkh2/Mnx2/Mhknxkhknxkhny

12/M

0k

12/M

0k

12/M

0k

M

12/Mk

12/M

0k

M

0k

Structures for Linear-Phase Structures for Linear-Phase FIR SystemsFIR Systems

►Structure for even MStructure for even M

►Structure for odd MStructure for odd M