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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
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.1
Introduction
5
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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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.
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPO
NSE
4.2
FrequencyResponse
Now
we
usethetim
econvolutionproperty(or
convolut
iontheorem
)tomapth
eoutputy(
n)in
frequenc
ydomain
Or
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPO
NSE
4.2
FrequencyResponse
Figure1
:Mappingtimedo
maintofrequencyd
omainusingthetim
e
con
volutionproperty
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPO
NSE
4.2
FrequencyResponse
Thef
requency
response
H()
is
usually
acomple
xquantity,sowewrite
where
and
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPO
NSE
4.2
Fre
quencyRe
sponse
,
,
responseandthe
phasere
sponse.I
fthe
impulseresponseh(n)isreal-valued
then,
8/14/2019 Chapter Four: Frequency Response
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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
,
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.2
FrequencyResponse
Solution:
Firsttheimpu
lseresponseh(n
)isjusttheoutputy(n)whentheinputis
x(n)=(n)
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.2
FrequencyResponse
hencethefrequen
cyrespon
seofthes
ystem
is
A
Thesystem
isalow-pass
filter.
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPO
NSE
4.2
FrequencyRe
sponse
So
lution
:
8/14/2019 Chapter Four: Frequency Response
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8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.2
FrequencyResponse:R
esult
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.2
FrequencyResponse
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.3
Ge
neralizedFrequency
Response
Eigen-functionand
eigen-value
in
DSP
systems
represe
ntedbyth
eimpulse
response
h(n)
is
8/14/2019 Chapter Four: Frequency Response
23/40
CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.3
Ge
neralizedFrequency
Response
Eigen-functionand
eigen-value
in
DSP
systems
Nowlet
stestwith
acomplex
exponential
8/14/2019 Chapter Four: Frequency Response
24/40
CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.3
Ge
neralizedFrequency
Response
Eigen-functionand
eigen-value
in
DSP
systems
Theout
putis
8/14/2019 Chapter Four: Frequency Response
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
8/14/2019 Chapter Four: Frequency Response
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
8/14/2019 Chapter Four: Frequency Response
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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
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.4
Fr
equency
response
ofsystem
s
in
cascade
orinpara
llel
System
sincascade
System
sinparallel
8/14/2019 Chapter Four: Frequency Response
29/40
CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.4
Fr
equency
response
ofsystem
s
in
cascade
orinpara
llel
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
Fromthedifference
equationofgenerall
inear
Forinput
theou
tputis
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
Werepla
cethisintothedifferenceequation:
Thus
b
8/14/2019 Chapter Four: Frequency Response
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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.
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
The
normalway
to
compute
the
frequ
ency
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
Themagnitudeand
phaseresponsesare
then
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
Solutio
n:
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
Because
allthecoeff
icientsarer
ealhencew
ecan
imaginar
yparts:
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
phasesp
ectra:
TheMatlabspectraareshownif
Figure9
.W
ecan
seethehighpasscharacteristic
ofthefilter
with
transitio
nfrequency
at/2
.
8/14/2019 Chapter Four: Frequency Response
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CHAPTERFOUR
:
FREQUE
NCYRESPONSE
4.5
Fr
equencyresponsein
termsoffilter
coefficients
8/14/2019 Chapter Four: Frequency Response
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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=