ec1252 signals & systems general overview (courtesy rec)

8/14/2019 EC1252 Signals & systems General Overview (Courtesy REC)

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SIGNALS & SYSTEMSSIGNALS & SYSTEMSMrs.S.Suganthi., SeniorMrs.S.Suganthi., Senior

Lecturer/ECELecturer/ECE..

Mrs. A. Ahila Senior Lecturer/ECEMrs. A. Ahila Senior Lecturer/ECE



UNIT IUNIT I



SIGNALSIGNAL

►Signal is a physical quantity thatSignal is a physical quantity thatvaries with respect to time , space orvaries with respect to time , space orany other independent variableany other independent variable

Eg x(t)= sin t.Eg x(t)= sin t.

►the major classifications of thethe major classifications of the

signal are:signal are:

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

(ii)(ii) Continuous time signalContinuous 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=0

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

Discrete time unit step signal is defined byDiscrete time unit step signal is defined byU[n]={0,n=0U[n]={0,n=0

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

δ (t)={1, t=0δ (t)={1, t=0

{0, t ≠ 0{0, t ≠ 0Continuous time Unit step signal is defined asContinuous time Unit step signal is defined asU(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 exhibitsA signal is said to be periodic ,if it exhibitsperiodicity.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 isPeriodic signal has the property that it isunchanged by a time shift of T. A signal that doesunchanged by a time shift of T. A signal that doesnot satisfy the above periodicity property is callednot satisfy the above periodicity property is calledan aperiodic signalan aperiodic signal

► 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 ben]=x[n]. The continuous time signal is said to beeven when, x(-t)= x(t) For example,Cosωn is aneven when, x(-t)= x(t) For example,Cosωn is aneven signal.even signal.

SIGNALSIGNAL



Energy and power signalEnergy and power signal

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

it have finite energy and zero power.it have finite energy and zero power.► A signal is said to be power signal if A signal is said to be power signal if

it have infinite energy and finiteit have infinite energy and finitepower.power.

► If the above two conditions are notIf the above two conditions are notsatisfied then the signal is said to besatisfied then the signal is said to beneigther energy nor power signalneigther energy nor power signal



Fourier SeriesFourier Series The 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 thesequence of complex numbers representing the

frequency componentfrequency component ωω00k.k.

∫ ω−

=

p

0

tikk dte)t(xp

1

X 0

∑

−

=

ω−=

1p

0n

nikk 0e)n(xp

1X

∑−

=

ω=1p

0k

nikk

0eX)n(x ∑∞

−∞=

ω=k

tikk

0eX)t(x



Fourier seriesFourier series

► Fourier series: a complicated waveform analyzedFourier series: a complicated waveform analyzedinto a number of harmonically related sine andinto a number of harmonically related sine andcosine functionscosine functions

► A continuous periodic signal x(t) with a period T0A continuous periodic signal x(t) with a period T0may be represented by:may be represented by: X(t)=ΣX(t)=Σ∞∞

k=1k=1 (( A Ak k coscos k k ωω t + Bt + Bk k sinsin k k ωω t)+t)+ AA00

► Dirichlet conditionsDirichlet conditions must be placed onmust be placed on x(t) x(t) for thefor theseries to be valid: the integral of the magnitude oseries to be valid: the integral of the magnitude of

x(t) x(t) over a complete period must be finite, and theover a complete period must be finite, and thesignal can only have a finite number osignal can only have a finite number of

discontinuities in any finite intervaldiscontinuities in any finite interval



Trigonometric form for Fourier Trigonometric form for Fourierseriesseries

► If the two fundamental components of aIf the two fundamental components of aperiodic signal areB1cosω0t andperiodic signal areB1cosω0t andC1sinω0t, then their sum is expressed byC1sinω0t, then their sum is expressed bytrigonometric identities:trigonometric identities:

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

22++ A Ak k 22 ) )1/21/2 (C(Ck k coscos k k ωω t-t-

φφk k ) or ) or

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

22++ A Ak k 22 ) )1/21/2 (C(Ck k sin k sin k ωω

t+t+ φφk k ) )



UNIT IIUNIT II



Fourier TransformFourier Transform

► Viewed periodic functions in terms of frequency componentsViewed 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 frequencyViewed 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-domainViewed 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 componentsView aperiodic functions in terms of frequency components

via Fourier transformvia Fourier transform

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

► Gain insight into the meaning of Fourier transform throughGain insight into the meaning of Fourier transform throughcomparison with Fourier seriescomparison with Fourier series



The Fourier Transform The Fourier Transform

►A transform takes one function (orA transform takes one function (orsignal) and turns it into anothersignal) and turns it into anotherfunction (or signal)function (or signal)

►Continuous Fourier Transform:Continuous Fourier Transform:

( ) ( )

( ) ( )∫

∫ ∞

∞−

−

∞

∞−

=

=

df e f H t h

dt et h f H

ift

ift

π

π

2

2



Continuous Time Fourier TransformContinuous Time Fourier TransformWe can extend the formula for continuous-time FourierWe 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-timeto aperiodic signals as well. The continuous-time

Fourier series is not defined for aperiodic signals, butFourier series is not defined for aperiodic signals, but

we call the formulawe call the formula

the (continuous time)the (continuous time)

Fourier transformFourier transform..

∫ ∫ −

ω−ω− ==2/p

2/p

tikp

0

tikk dte)t(x

p

1dte)t(x

p

1X 00

∫ ∞∞−

ω−=ω dte)t(x)(Xti



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

periodic signal, we can reconstruct it by multiplying theperiodic signal, we can reconstruct it by multiplying thecomplex sinusoids of frequencycomplex sinusoids of frequency ωω00k by the weights Xk by the weights Xkk

and summing:and summing:

We can perform a similar reconstruction for aperiodicWe can perform a similar reconstruction for aperiodicsignalssignals

∑

−

=

ω=1p

0k

nik

k

0eX)n(x

∑

∞

−∞=

ω=k

tikk

0eX)t(x

∫ ∞∞−

ω ωωπ

= de)(X21)t(x ti∫ π

π−ω ωω

π= 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 anRoughly speaking, that’s why the system response to an

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

1edte)t(dte)t(x)(X 0ititi ==δ==ω ω−∞

∞−

ω−∞

∞−

ω− ∫ ∫

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

n

ni

n

ni ==δ==ω ω−∞

−∞=

ω−∞

−∞=

ω− ∑∑



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

the sense that it allows “complex frequency” whereas Fourierthe 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 domainno matter how complicated the circuit is, in the frequency domainin 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 algebraicdifferential equations into a corresponding set of algebraicequations, which are much easier to solve. Physically, it producesequations, 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 analysisphase, and transfer characteristics important for circuit analysisand design.and design.

► Most importantly, Laplace transform lifts the limit of FourierMost importantly, Laplace transform lifts the limit of Fourieranalysis 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 canresponses of a linear circuit. Using Fourier transform, one canonly deal with he steady state behavior (i.e. circuit responseonly 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 anyUsing Laplace transform, one can find the response under anytypes 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 LaplaceApplication of Laplace Transform to Circuit Analysis Transform to Circuit Analysis



system

►• A system is an operation thattransforms input signal x into outputsignal y .



LTI Digital Systems

►Linear Time Invariant

• Linearity/Superposition:

►

If a system has an input that can beexpressed as a sum of signals, thenthe response of the system can beexpressed as a sum of the individualresponses to the respective systems.

►LTI



Time-Invariance&Causality

► If you delay the input, response is just adelayed 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 isno input signal” or “output at current timedoes not depend on future values of theinput”.



Convolution

► The input and output signals for LTIsystems have special relationship interms 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 continuousdomains (incident radiance defined over the film

plane) and then using those samples to reconstructnew 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 justincrease the sampling rate

►• However, few of interesting functionsin computer graphics are band limited,in particular, functions withdiscontinuities.

►• It is because the discontinuity alwaysfalls between two samples and thesamples provides no information of the

discontinuity.



Sampling theory



Aliasing



Region of ConvergenceRegion of Convergence

► Region of theRegion of thecomplexcomplex z z -plane for-plane forwhich forwardwhich forward z z --

transformtransformconvergesconverges

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 possibilitiesFour possibilities(( z z =0 is a special=0 is a specialcase and may orcase and may or

may not bemay not beincluded)included)



Z Z -transform Pairs-transform Pairs

►hh[[nn] =] = δ δ [[nn]]

Region of Region of convergence: entireconvergence: entire z z -plane-plane

►

hh

[[nn

] =] =δ δ

[[n-1n-1

]]

Region of Region of

convergence: entireconvergence: entire--

[ ] [ ] 1][0

0

=== ∑∑=

−∞

−∞=

−

n

n

n

n z n z n z H δ δ

[ ] [ ] 11

1

11][ −

=

−∞

−∞=

− =−=−= ∑∑ z z n z n z H n

n

n

nδ δ

[ ]

1 if

1

1

][

00

<−

=

==

=

∑∑

∑∞

=

∞

=

−

∞

−∞=

−

z

a

z

a

z a z a

z nua z H

n

n

n

nn

n

nn

►hh[[nn] =] = aann uu[[nn]]

Region of Region of convergence: |convergence: |

z z | > || > |aa| which| whichis theis the



[ ] a z z a

nua Z

n >−↔ − for 1

1 1

StabilityStability

►Rule #1: For a causal sequence, polesRule #1: For a causal sequence, polesare inside the unit circle (applies to z-are inside the unit circle (applies to z-transform functions that are ratios of transform functions that are ratios of

two polynomials)two polynomials)►Rule #2: More generally, unit circle isRule #2: More generally, unit circle is

included in region of convergence. (Inincluded in region of convergence. (Incontinuous-time, the imaginary axiscontinuous-time, the imaginary axiswould be in the region of convergencewould be in the region of convergenceof the Laplace transform.)of the Laplace transform.)

This is stable if a < 1 b rule #1.



InverseInverse z z -transform-transform

► Yuk! Using the definition requires a Yuk! Using the definition requires a

contour integration in the complexcontour integration in the complex z z --plane.plane.

►Fortunately, we tend to be interestedFortunately, we tend to be interested

in only a few basic signals (pulse, step,in only a few basic signals (pulse, step,etc.)etc.) Virtually all of the signals we’ll see can beVirtually all of the signals we’ll see can be

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

--

[ ] [ ] dz z z F j

n f n

jc

jc

1

2

1 −∞+

∞−∫ =

π



ExampleExample

► Ratio of polynomialRatio of polynomialz-domain functionsz-domain functions

► Divide through byDivide through by

the highest power of the highest power of zz

► Factor denominatorFactor denominatorinto first-orderinto first-orderfactorsfactors

► Use partial fractionUse partial fractiondecomposition to getdecomposition to get

first-order termsfirst-order terms

2

1

2

3

12][

2

2

+−

++=

z z

z z z X

21

21

21

231

21][

−−

−−

+−

++=

z z

z z z X

( )11

21

12

11

21][

−−

−−

−

−

++=

z z

z z z X

1

2

1

10

1

2

11

][ −− −

+−

+= z

A

z

A B z X



Example (con’t)Example (con’t)

►FindFind BB00 byby

polynomialpolynomialdivisiondivision

►Express in termsExpress in termsof of BB

00

►Solve forSolve for A A11 andand A A22

15

23

2

1212

3

2

1

1

12

1212

−

+−

+++−

−

−−

−−−−

z

z z

z z z z

( )11

1

12

11

512][

−−

−

−

−

+−+= z z

z z X

8

2

1

121

2

11

21

921

441

1

21

1

1

21

2

21

21

1

1

1

=++

=−

++=

−=−

++=

−

++=

=

−

−−

=−

−−

−

−

z

z

z

z z A

z

z z A



Example (con’t)Example (con’t)

►ExpressExpress X X [[ z z ]] in terms of in terms of BB00,, A A11, and, and A A22

►Use table to obtain inverseUse table to obtain inverse z z -transform-transform

►With the unilateralWith the unilateral z z -transform, or the-transform, or thebilateralbilateral z z -transform with region of -transform with region of convergence, the inverseconvergence, the inverse z z -transform-transform

is uniqueis unique

11 1

8

2

11

92][ −

− −+

−−=

z z

z X

[ ] [ ] [ ] [ ]nununn x

n

82

1 92 +

−= δ



Z Z -transform Properties-transform Properties

►LinearityLinearity

►Right shift (delay)Right shift (delay)

[ ] [ ] [ ] [ ] z F a z F an f an f a 22112211 +⇔+

[ ] [ ] [ ] z F z mnumn f m−⇔−−

[ ] [ ] [ ] [ ]

−+⇔− ∑=

−−−

m

n

nmm z n f z z F z numn f 1



Z Z -transform Properties-transform Properties

[ ] [ ] [ ] [ ]

[ ] [ ]{ } [ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]

[ ] [ ] z F z F

z r f z m f

z r f m f

z mn f m f

z mn f m f

mn f m f Z n f n f Z

mn f m f n f n f

r

r m

m

m r

mr

m n

n

n

n

m

m

m

21

21

21

21

21

2121

2121

=

=

=

−=

−=

−=∗

−=∗

∑∑∑ ∑

∑ ∑

∑ ∑

∑

∑

∞

−∞=

−−∞

−∞=

∞

−∞=

∞

−∞=+−

∞

−∞=

∞

−∞=

−

∞

−∞=

−∞

−∞=

∞

−∞=

∞

−∞=

► Convolutiondefinition

► Take z -transform

► Z -transformdefinition

► Interchangesummation

► Substitute r = n - m

► Z -transformdefinition



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 LTIsystem, 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(( z z ).).

► Frequency responseFrequency response at is validat is valid

if ROC includes andif ROC includes and

[ ] [ ] [ ]nhn xn y ∗=

( ) ( ) ( ). z H z X z Y =

( ) ( ) 1== z j z H e H ω

,1= z

( ) ( ) ( )ω ω ω j j j e H e X eY =

f



5.1 Frequency Response of LIT5.1 Frequency Response of LITSystemSystem

► Consider andConsider and, then, then

magnitudemagnitude

phasephase

► We will model and analyze LTI systems based onWe will model and analyze LTI systems based onthe magnitude and phase responses.the magnitude and phase responses.

)()()(ω

ω ω je X j j j ee X e X ∠= )()()(ω

ω ω je H j j j ee H e H ∠=

)()()(ω ω ω j j j e H e X eY

=

)()()( ω ω ω j j j e H e X eY ∠+∠=∠



System FunctionSystem Function

►General form of LCCDEGeneral form of LCCDE

►Compute the z-transformCompute the z-transform

[ ] [ ]k n xbk n yaM

k

k

N

k

k −=− ∑∑== 00

( ) z X z b z Y z a k M

k

k

N

k

k

k

−

==

− ∑∑ =00

)(

( )( )

( ) ∑

∑

=

−

−

===N

k

k

k

k M

k

k

z a

z b

z X

z Y z H

0

0

i lS F i P l /



System Function: Pole/zeroSystem Function: Pole/zeroFactorizationFactorization

►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 determinesLocation of zeros and poles determines

the frequency response and phasethe frequency response and phase

( )

( )

( )∏

∏

=

−=

−

−

−

= N

k

k

M

k

k

z d

z c

a

b z H

1

1

1

1

0

0

1

1 .,...,,:zeros 21 M ccc

.,...,,: poles 21 N d d d



Second-order SystemSecond-order System

► Suppose the system function of a LTI systemSuppose the system function of a LTI systemisis

► To find the difference equation that is To find the difference equation that issatisfied by the input and out of this systemsatisfied by the input and out of this system

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

.

)4

31)(

2

11(

)1()(

11

21

−−

−

+−

+=

z z

z z H

)(

)(

8

3

4

11

21

)

4

31)(

2

11(

)1()(

21

21

11

21

z X

z Y

z z

z z

z z

z z H =

−+

++=

+−

+=

−−

−−

−−

−

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

3]1[

4

1][ −+−+=−−−+ n xn xn xn yn yn y



System Function: StabilitySystem Function: Stability

►Stability of LTI system:Stability of LTI system:

► This condition is identical to the This condition is identical to thecondition thatcondition that

The stability condition is equivalent to the The stability condition is equivalent to thecondition that the ROC of condition that the ROC of HH(( z z ) includes) includes

the unit circle.the unit circle.

∑∞

−∞=

∞<n

nh ][

.1 when][ =∞<∑∞

−∞=

− z z nhn

n



System Function: CausalitySystem Function: Causality

► If the system is causal, it follows thatIf 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(( z z ) must be) must be

outside theoutside the outermost outermost pole.pole.

►

If the system is anti-causal, it follows thatIf the system is anti-causal, it follows that hh[[nn] must] mustbe a left-sided sequence. The ROC of be a left-sided sequence. The ROC of HH(( z z ) must be) must be

inside theinside the innermost innermost pole.pole.

1a

Im

Re 1a

Im

Re ba

Im

Re

Right-sidedcausal

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 as The system function is obtained as

][]2[]1[2

5][ n xn yn yn y =−+−−

)21)(2

11(

1

2

51

1)(

11

21

−−

−−

−−=

+−=

z z

z z

z H

S t F ti IS t F ti I



System Function: InverseSystem Function: InverseSystemsSystems

► is an inverse system for , if is 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

( ) z H i ( ) z H

1)()()( == z H z H z G i

)(

1)(

z H z H i =

)(

1)(

ω

ω

j

j

ie H

e H =⇔

[ ] [ ] [ ] [ ]nnhnhn g i δ =∗=⇔

)( and )( z H z H 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

a z Z H Ap

( ) 1=ω j

Ap e H

( )ω

ω

ω

ω

ω

ω

j

j j

j

j j

Apae

eae

ae

aee H −

−−

∗−

−−

=−

−=

1

*1

1

θ

θ

j

j

er a

rea

1*/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 k k

k k

M

k k

k Ap

z e z e

e z e z

z d

d z z H

1

1*1

1*1

1

1

1

)1)(1(

))((

1

Causal/stable: 1, <k k d e

real poles complex poles



All-Pass System ExampleAll-Pass System Example

0.8

0.5

z -plane

Unit

circle

4

3−

3

4− 2

Re

Im

1 and 2 == cr M M

zeros.and poles 42 hassystem pass-allThis =+== r c M M N M

θ θ j j er re 1conjugate&reciprocal :zero: pole − →



Minimum-Phase SystemMinimum-Phase System

► Minimum-phase system:Minimum-phase system: all zeros and all polesall zeros and all poles

are inside the unit circle.are inside the unit circle.

► The name The name minimum-phaseminimum-phase comes from a propertycomes from a property

of the phase response (minimum phase-lag/group-of the phase response (minimum phase-lag/group-delay).delay).

► Minimum-phase systems have some specialMinimum-phase systems have some special

properties.properties.

► When we design a filter, we may have multipleWhen we design a filter, we may have multiple

choices to satisfy the certain requirements. Usually,choices to satisfy the certain requirements. Usually,

we prefer the minimum phase which is unique.we prefer the minimum phase which is unique.

►

All systems can be represented as a minimum-All systems can be represented as a minimum-



UNIT VUNIT V



ExampleExample

►Block diagram representation of Block diagram representation of

[ ] [ ] [ ] [ ]nxb2nya1nyany 021 +−+−=

Block DiagramBlock Diagram



Block DiagramBlock DiagramRepresentationRepresentation

►LTI systems withLTI systems withrational systemrational systemfunction can befunction can be

represented asrepresented asconstant-coefficientconstant-coefficientdifference equationdifference equation

► The implementation The implementationof differenceof differenceequations requiresequations requires

delayed values of delayed values of



Direct Form IDirect Form I

►General form of difference equationGeneral form of difference equation

►Alternative equivalent formAlternative equivalent form

[ ] [ ]∑∑==

−=−M

0k

k

N

0k

k knxbknya

[ ] [ ] [ ]∑∑==

−=−−M

0kk

N

1kk knxbknyany





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

►Direct Form I RepresentsDirect Form I Represents

( )

∑

∑

=

−

=

−

−

=N

1k

k

k

M

0k

kk

za1

zb

zH

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )zVza1

1zVzHzY

zXzbzXzHzV

zb

za1

1zHzHzH

N

1k

kk

2

M

0k

kk1

M

0k

kkN

1k

kk

12

−==

==

−==

∑

∑

∑∑

=

−

=

−

=

−

=

−

[ ] [ ]

[ ] [ ] [ ]nvknyany

knxbnv

N

1k

k

M

0kk

+−=

−=

∑

∑

=

=



Alternative RepresentationAlternative Representation

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

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )zWzbzWzHzY

zX

za1

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 theWe can change the order of thecascade systemscascade systems

[ ] [ ] [ ]

[ ] [ ]∑∑=

=

−=

+−=

M

0kk

N

1k k

knwbny

nxknwanw



Direct Form IIDirect Form II

► No need to store the sameNo need to store the samedata twice in previousdata twice in previoussystemsystem

► So we can collapse theSo we can collapse thedelay elements into onedelay elements into one

chainchain► This is called Direct Form II This is called Direct Form II

or the Canonical Formor the Canonical Form

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

► Implementation wiseImplementation wise

Less memory in Direct IILess memory in Direct II

Difference when usingDifference when usingfinite-precisionfinite-precision

arithmeticarithmetic

Signal Flow GraphSignal Flow Graph



Signal Flow GraphSignal Flow GraphRepresentationRepresentation

►Similar to block diagramSimilar to block diagramrepresentationrepresentation Notational differencesNotational differences

►A network of directed branchesA network of directed branchesconnected at nodesconnected at nodes



ExampleExample

►Representation of Direct Form II withRepresentation of Direct Form II withsignal flow graphssignal flow graphs [ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ]

[ ] [ ]

[ ] [ ]nwny

1nwnw

nwbnwbnw

nwnw

nxnawnw

3

24

41203

12

41

=

−=

+=

=

+=

[ ] [ ] [ ]

[ ] [ ] [ ]1nwbnwbny

nx1nawnw

1110

11

−+=

+−=

Determination of SystemDetermination of System



Determination of SystemDetermination of SystemFunction from Flow GraphFunction from Flow Graph

[ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ]nwnwny

1nwnw

nxnwnw

nwnw

nxnwnw

42

34

23

12

41

+=

−=

+=

α=−=

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )( ) ( ) ( )zWzWzY

zzWzW

zXzWzW

zWzW

zXzWzW

42

134

23

12

41

+==

+=

α=−=

−

( )( )( )

( )

( ) ( )

( ) ( ) ( )zWzWzY z1

1zzX

zW

z1

1zzXzW

42

1

1

4

1

1

2

+=α−

α−

=

α−−α

=

−

−

−

−

( )( )( )

[ ] [ ] [ ]nu1nunh

z1

z

zX

zYzH

1n1n

1

1

+−

−

−

α−−α=

α−α−

==

Basic Structures for IIRBasic Structures for IIR



Basic Structures for IIRBasic Structures for IIRSystems: Direct Form ISystems: Direct Form I



Basic Structures for IIRBasic Structures for IIRSystems: Cascade FormSystems: Cascade Form

► General form for cascade implementationGeneral form for cascade implementation

►

More practical form in 2More practical form in 2ndnd

order systemsorder systems

( )( ) ( )( )

( ) ( )( )∏∏

∏∏

=

−∗−

=

−

=

−∗−

=

−

−−−

−−−=

21

21

N

1k

1

k

1

k

N

1k

1

k

M

1k

1k

1k

M

1k

1k

zd1zd1zc1

zg1zg1zf 1

AzH

( ) ∏=

−−

−−

−−−+

=1M

1k2

k21

k1

2k2

1k1k0

zaza1

zbzbbzH



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

z25.01

z1

z5.01

z1

z25.01z5.01

z1z1

z125.0z75.01

zz21zH

−

−

−

−

−−

−−

−−

−−

−+

−+

=−−

++=

+−

++=




Basic Structures for IIRBasic Structures for IIRSystems: 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

ze1B

zc1

AzCzH

( ) ∑∑ =−−

−

=

−

−−+

+=

SP N

1k2

k21

k1

1k1k0

N

0k

k

k zaza1

zee

zCzH



ExampleExample►Partial Fraction ExpansionPartial Fraction Expansion

►

Combine poles to getCombine poles to get

( )( ) ( )1121

21

z25.0125

z5.01188

z125.0z75.01zz21zH −−−−

−−

−−

−+=

+−++=

( )21

1

z125.0z75.01

z878zH −−

−

+−+−

+=



Transposed Forms Transposed Forms

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

output relationoutput relation

► Transposing: Transposing: Reverse directions of all branchesReverse directions of all branches

Interchange input and output nodesInterchange input and output nodes

►Example:Example:

( )1az1

1zH

−−=



ExampleExample

Transpose

►Both have the same system functionBoth have the same system functionor difference equationor difference equation

[ ] [ ] [ ] [ ] [ ] [ ]2nxb1nxbnxb2nya1nyany 21021 −+−++−+−=

Basic Structures for FIR Systems: DirectBasic Structures for FIR Systems: Direct



as c S uc u es o Sys e s ecyFormForm

►Special cases of IIR direct formSpecial cases of IIR direct form

structuresstructures

► 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

Basic Structures for FIRBasic Structures for FIR



Basic Structures for FIRBasic Structures for FIRSystems: Cascade FormSystems: Cascade Form

►Obtained by factoring the polynomialObtained by factoring the polynomialsystem functionsystem function

( ) [ ] ( )∑ ∏= =

−−−

++==

M

0n

M

1k

2

k2

1

k1k0

nS

zbzbbznhzH

Structures for Linear PhaseStructures for Linear-Phase



Structures for Linear-PhaseStructures for Linear-PhaseFIR SystemsFIR Systems

► Causal FIR system with generalized linear phase areCausal 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)orII(type M0,1,...,n nhnMh

III)orI(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

−++−+−=

+−−+−+−=

−+−+−=−=

∑

∑∑∑∑∑

−

=

−

=

−

=

+=

−

==

tructures or L near-P aseruc ures or near- ase



FIR SystemsFIR Systems

►Structure for even MStructure for even M

►Structure for odd MStructure for odd M

ec1252 signals & systems general overview (courtesy rec)

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