Chapter Four: Frequency Response

8/14/2019 Chapter Four: Frequency Response

1/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.1

Introduction

This

chapter

addresses

ones

uch

alterna

tivedesc

ription,b

asedon

the

transfo

rmations

of

periodicin

put

signals

effectedbyLTI(LSI)systems

.

1


2/40

CHAPTERFOUR:

FREQUE

NCYRESPONSE

4.1

Introduction

Frequen

cy

response

is

the

measure

of

any

system's

spectrum

response

at

the

output

to

a

amplitude)atitsinput.

(From

Wikipe

dia

,the

freeencyc

lope

dia)

Frequen

cyrespons

eis

a

specification

used

in

amplifiers,pre-amplifiers,

CD

pla

yers,

tape

decks

and

other

audio

co

mponents

to

measure

how

uniformly

itreproduces

sounds

from

thelowest

tonesto

thehighest.

(B

yGary

Altun

ian,

Abou

t.com

) 2


3/40

CHAPTERFOUR

:

FREQUE

NCYRESPO

NSE

4.1

Introduction

Example

of

Biomedical

engineering

perspectiveofthet

ransformationpropertiesof

TheAud

itoryEvoke

dPotential

(AEP)isawave

thatisrecordedin

an

Electroencephalogram

(EEG)le

adinresponsetoanau

ditoryclick

.

3


4/40

CHA

PTERFOU

R:

FREQUENCYRESP

ONSE

4.1

Introduction

AuditoryE

vokedRespon

seisarespon

seto

anacousticstimulus.

Theresponse,se

enas

waveforms,

isoftenreferredtointhree

sections:

Middlelatency(earlyco

rtical)response

Latecorticalresponse

TheBrainstem

responsewav

esoccurwithinth

efirst10msafte

rtheclick

stimulus.

T

hese

response

s

are

relative

insensitive

to

general

anaesthetics.

TheMiddlelate

ncywavesoccur

10to80msafte

rtheclick

stimulus.

Th

eyshow

graded

changeswithgeneralanaestheticsoverthe

clinicalconc

entrationrange.

Thelatecorticalchangesoccur80msafter

theclickstim

ulusandlater.

4


5/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.1

Introduction

5


6/40

CHAPTERFOUR

:

FREQUE

NCYRESPO

NSE

4.2

Frequency

Response

of

LTI(

LSI)

Systems

U

ton

owthed

iscussion

hasbeen

on

discrete-t

imesignals

.As

amatterof

fact,m

ostthediscussion

sofaralso

applies

tosystem

s(assum

edtobe

LTI

or

LSI)

.However

there

aresome

differences,e.g.

themea

ningoft

ime

convolution.

6


7/40

CHAPTERFOUR:

FREQUE

NCYRESPONSE

4.2

Frequency

Response

of

LTI(

LSI)

Systems

Asyste

m

ischar

acterizedbyitsimp

ulse

h(n)is

H()=h(n)e

jn

n=

Andthe

inverseis


8/40

CHAPTERFOUR:

FREQUE

NCYRESPONSE

4.2

FrequencyResponse

H()is

calledthe

frequenc

yresponseor

frequen

cycharacteristicof

thesystem.

It

isthe

frequency

character

izationo

the

system

whereas

theimpul

seresponseis

thetimecharacterization.


9/40

CHAPTERFOUR

:

FREQUE

NCYRESPO

NSE

4.2

FrequencyResponse

Now

we

usethetim

econvolutionproperty(or

convolut

iontheorem

)tomapth

eoutputy(

n)in

frequenc

ydomain

Or


10/40

CHAPTERFOUR

:

FREQUE

NCYRESPO

NSE

4.2

FrequencyResponse

Figure1

:Mappingtimedo

maintofrequencyd

omainusingthetim

e

con

volutionproperty


11/40

CHAPTERFOUR

:

FREQUE

NCYRESPO

NSE

4.2

FrequencyResponse

Thef

requency

response

H()

is

usually

acomple

xquantity,sowewrite

where

and


12/40

CHAPTERFOUR

:

FREQUE

NCYRESPO

NSE

4.2

Fre

quencyRe

sponse

,

,

responseandthe

phasere

sponse.I

fthe

impulseresponseh(n)isreal-valued

then,


13/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.2

FrequencyResponse

Thefrequencyresponse

ofasystem

ex

ists

ifthesys

tem

isbo

un

de

d-inp

ut

bound

ed-ou

tpu

t(BIBO)sta

ble

,


14/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.2

FrequencyResponse

Solution:

Firsttheimpu

lseresponseh(n

)isjusttheoutputy(n)whentheinputis

x(n)=(n)


15/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.2

FrequencyResponse

hencethefrequen

cyrespon

seofthes

ystem

is

A

Thesystem

isalow-pass

filter.


16/40

CHAPTERFOUR

:

FREQUE

NCYRESPO

NSE

4.2

FrequencyRe

sponse

So

lution

:


17/40


18/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.2

FrequencyResponse:R

esult


19/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.2

FrequencyResponse


20/40

CHAPTERFOUR:

FREQUE

NCYRESPONSE

4.2

Fre

quencyRe

sponse

For

themagn

itude

response

H()we

d

be

ttern

otgo

from

these

two

express

iono

fH(

):

Then

Thepha

serespon

se

is


21/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.3

Ge

neralizedFrequency

Response

Eigen-functionand

eigen-value

in

DSP

systems

,

preserves

its

tim

e

identity

wheng

oing

through

asystem.

Letsstart

withadisc

rete

cosine


22/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.3

Ge

neralizedFrequency

Response

Eigen-functionand

eigen-value

in

DSP

systems

represe

ntedbyth

eimpulse

response

h(n)

is


23/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.3

Ge

neralizedFrequency

Response

Eigen-functionand

eigen-value

in

DSP

systems

Nowlet

stestwith

acomplex

exponential


24/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.3

Ge

neralizedFrequency

Response

Eigen-functionand

eigen-value

in

DSP

systems

Theout

putis


25/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.3

Ge

neralizedFrequency

Response

Eigen-functionand

eigen-value

in

DSP

systems

theoutput,itst

imevariation

does

not

change.

Thefacto

rinbrack

etsisjustthe

frequen

cyresponseH(),so


26/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.3

Ge

neralizedFrequency

Response

Eigen-functionand

eigen-value

in

DSP

systems

eigen-va

lueand

theeige

n-function

.

Actually,thephaseoftheinp

utsignal

hasbee

nchange.Forthiswe

write


27/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.4

Fr

equency

response

ofsystem

s

in

cascade

orinpara

llel

In

vario

ussituationsfiltersareconnected

in

cascaeornparae

.econ

aspresene

s

matterw

ithrespecttosystemim

pulserespo

nses.

Now

we

treatthe

problem

with

respectto

frequenc

yresponses.Byusing

theassocia

tivity

andthe

distributivit

yofimpuls

eresponses

,and

theconvolutiontheo

remofDTFT

wecanobtain


28/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.4

Fr

equency

response

ofsystem

s

in

cascade

orinpara

llel

System

sincascade

System

sinparallel


29/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.4

Fr

equency

response

ofsystem

s

in

cascade

orinpara

llel


30/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

Fromthedifference

equationofgenerall

inear

Forinput

theou

tputis


31/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

Werepla

cethisintothedifferenceequation:

Thus

b


32/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

Noticethatifthesignaldifferenceequationisw

ritten

differently

(assomeaut

horsdo)thea

boveexpressionfor

H()does

notapply

.

Whenweknowthecoefficientsofafilterwecanwritethe

expression

of

its

fre

quency

resp

onse

immed

iately

.

Conversely,ifweknow

theexpressionofthefreq

uency

responseofasystemwe

canwriteits

differenceequ

ation.

Alsonotic

ethatfornon

recursivefilter,thedenom

inator

isjust1.


33/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

The

normalway

to

compute

the

frequ

ency


34/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

Themagnitudeand

phaseresponsesare

then


35/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients


36/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

Solutio

n:


37/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

Because

allthecoeff

icientsarer

ealhencew

ecan

imaginar

yparts:


38/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients

phasesp

ectra:

TheMatlabspectraareshownif

Figure9

.W

ecan

seethehighpasscharacteristic

ofthefilter

with

transitio

nfrequency

at/2

.


39/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

4.5

Fr

equencyresponsein

termsoffilter

coefficients


40/40

CHAPTERFOUR

:

FREQUE

NCYRESPONSE

umma

ry

hischapteraddresses

onesuchalte

rnativedescription,

asedon

thetransform

ationsofperiodicinputs

ignals

ffectedbyLTI(LSI)systems.H()isc

alledthefreq

uency

.

hefreque

ncycharacter

izationofthesystemwhere

asthe

mpulseresponseisthe

timecharacte

rization.A

sys

temis

haracterizedbyitsimp

ulseh(n)is

()=h(n)e

jn

n=

hase=

Magnitude=

Documents

Chapter Four: Frequency Response