Signals and Systems 06

7/29/2019 Signals and Systems 06

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2. Filtering

Idealfrequencyselectivefiltersarefiltersthatletfrequencycomponentsoveragivenfrequencybandpassthrough

undistorted,whilecomponentsatotherfrequenciesarecompletelycutoff.

Filteringisneededwhennoiseisaddedtoasignalbutthenoisehasmostofitsenergyatfrequenciesoutsideofthe

bandwidthofthesignal.Wewanttorecoverthesignalfromitsnoisymeasurement.

Ideallowpassfilter

Theoutput ( )y t ofthesystemforaninput ( )x t is

givenby,. .

( ) ( ) ( ) ( ) ( )F T

y t x t h t H j X j = .

The step responseof the ideal lowpass filter is the

running

integral

of

the

sinc

function.

Its

plot

indicates

that there are potentially undesirable oscillations

beforeandafterthediscontinuity.

Theimpulseresponseofanideallowpassfilterisnot

realizable as it is noncausal. Approximation of the

ideal lowpass filtercanberealizedbythecausalLTI

filters. Butterworth filter is one of them. The

magnituderesponseofanNthorderB>W>filterwith

cutofffrequency c isgivenby,

TheplotofthemagnituderesponseofBWfilterforN=2isshownbelow.


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3. Sampling

Samplingistheacquisitionofacontinuoussignalatdiscretetimeintervals.TheprocessisshowninFigurebelow.

Inthefrequencydomain,

Samplingtheorem:Abandlimitedsignal ( )x t with ( ) 0X = for M > isuniquelydeterminedfrom itssamples

( )x nT if 2s M > where2

sT

= isthesamplingfrequency.

Giventhesignalsamples,wecanrecover ( )x t byfiltering ( )px t usinganideallowpassfilterwithdcgainTandcut

offfrequencybetween s and s M .

Theoriginalspectrumcenteredat 0 = canbe recoveredundistorted if itdoesnotoverlapwith itsneighboringreplicas.

SamplingusingZOH:Thezeroorderhold(ZOH)retainsthevalueofthesignalsampleupuntilthefollowingsample

instant.Itbasicallyproducesstaircasesignalsfromthesamples.WecanviewZOHasafilterwithimpulseresponse

0 ( )h t asshowninFigurebelow.

ThefrequencyresponseofaZOHwithimpulseresponse 0 ( )h t isgivenby,

20( ) sinc

2

Tj T

H Te

=

2

2sin

2

Tj T

e

=

.

Theinverseof 0 ( )H is,2

1

1( )

2 sin( / 2)

Tj

H eT

= .

Thereconstructionfilteristhecascadeoftheinversefilterandthelowpassfilter.

1( ) ( ) ( )r lpH TH H =

ThemagnitudeandthephaseplotofthereconstructionfilterisshowninFigurebelow.


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This

frequency

response

can

not

be

realized

exactly

in

practice.

But

it

can

be

approximated

with

a

causal

filter.

ThewholeprocessisshowninFigurebelow.

Example

1. Aperiodicsignal ( )x t isgivenasan inputofa2ndorderButterworthfilterasshownbelow.Whatshouldbethe

output

of

the

filter?

Given,

1 / 4T T=

,

0.001 sT=

,

500

2R =

,

2

1

H4L =

and

6

10 FC

=.

Thedifferentialequationrelatingtheoutputandinputofthefilteris,2

2( ) ( )

d y L dyLC y t x t

dt R dt + + = .

Thefrequencyresponseofthefilterisgivenby,

2

( ) 1( )( )

( ) 1

YHLX

LC j jR

= =+ +

1sinc

2 2k

ka

=

2. FindtheinverseF.T. ( )x t of ( )X whosemagnitudeandphaseareshownbelow.

Ifthephasefunctionwere0,then ( )x t wouldbeastandardsincfunction 0 ( )x t .

0( ) 5 sin

W Wtx t c

=

.

Here,theslopeofthephaseplotisW

.Hence, 0( ) ( )x t x tW

= = 5 sinW W

c tW

=

.


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3. ConsiderthecausalLTIsystemshowninFigurebelow.Findthefrequencyresponseoftheoverallsystem.

1 2 2

1( ) ( ) ( ) ( 1)

( ) 2 1H H H j

j j

= = +

+ +

2

4

1( )

1H

+=+

and,1 1

2

2( ) tan tan1

H

= + + .

4. PlottheFouriertransformofthesignal ( ) sin1000 cos1200x t t t = + .Whatshouldbethesamplingperiodthat

wouldensurenoaliasing?

Noaliasingwilloccurifthesamplingfrequencyis, 2400s > i.e.,2

2400T

< 1

1200= s [Ans]

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Signals and Systems 06