학습에앞서

§ 학습목표– FIR 필터의개념및특성을이해한다.– FIR 필터의구현방법을학습한다.– FIR 필터의주파수응답을구한다.

2

3

Discrete-time System

§ Operate on x[n] to get y[n]§ A general class of systems

– ANALYZE the systeml Tools: time-domain & frequency-domain

– SYNTHESIZE the system§ Example

– Running average l RULE: “the output at time n is the average of three consecutive

input values”

Computery[n]x[n]

General FIR Filter

§ Filter coefficients {bk}

– Filter order is M– Filter length is L = M+1– Number of filter coefficients is L

§ Example– 3-pt Average System

–

4

å=

-=M

kk knxbny

0][][

]3[]2[2]1[][3

][][3

0

-+-+--=

-=å=

nxnxnxnx

knxbnyk

k}1,2,1,3{ -=kb

])2[]1[][(][ 31 ++++= nxnxnxny

Example: filtered stock signal

§ 50-pt Averager

58/26/2019 © 2003, JH McClellan & RW Schafer

OUTPUTINPUT

Special Input Signals (1/2)

§ Unit impulse– The mathematical notation is that of the Kronecker delta function.

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ïî

ïíì

¹

==

00

01][

n

nnd

1

n

LL +-+++-+=-=å ]1[]1[][]0[]1[]1[][][][ nxnxnxknkxnxk

dddd

]4[2]3[4]2[6]1[4][2][ -+-+-+-+= nnnnnnx ddddd

Special Input Signals (2/2)

§ Unit Impulse response– When the input to the FIR filter is a unit impulse sequence, x[n]=d[n],

the output is, by definition, the unit impulse response, which is denoted by h[n].

7

å=

-=M

kk knxbny

0][][

å=

-=M

kknxkhny

0][][][

å=

-=M

kk knbnh

0][][ d

Filtery[n]x[n]

Convolution

Since h[n]=0 for n

Convolution

§ Convolution and FIR Filters– Notation:

– FIR case:

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Filtery[n]x[n]

][][][ nxnhny *=

å=

-=M

kknxkhny

0][][][

Finite limits

Finite limitsSame as bk

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Example: Convolution

n -1 0 1 2 3 4 5 6 7x[n] 0 1 1 1 1 1 1 1 ...h[n] 0 1 -1 2 -1 1 0 0 0

0 1 1 1 1 1 1 1 10 0 -1 -1 -1 -1 -1 -1 -10 0 0 2 2 2 2 2 20 0 0 0 -1 -1 -1 -1 -10 0 0 0 0 1 1 1 1

y[n] 0 1 0 2 1 2 2 2 ...

][][]4[]3[]2[2]1[][][

nunxnnnnnnh

=-+---+--= ddddd

]4[]4[]3[]3[]2[]2[

]1[]1[][]0[

----

nxhnxhnxhnxhnxh

å=

-=M

kknxkhny

0][][][

Implementation of FIR filters

§ Recall the general definition of an FIR filter

§ The basic building-block systems we need are the multiplier, the adder, and the unit-delay operator.

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Filtery[n]x[n]å

=

-=M

kk knxbny

0][][

][][ nxny b=]1[][ -= nxny

][][][ 21 nxnxny +=

FIR Structure

§ Direct Form

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Signal flow graph

å=

-=M

kk knxbny

0][][

Block-diagram structure for the Mth order FIR filter

§ Time Invariance– “Time-Shifting the input will cause the same time-shift in the output”

Linear Time-Invariant (LTI) Systems (1/2)

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§ Linearity = Two Properties– SCALING: “Doubling x[n] will double y[n]”– SUPERPOSITION: “Adding two inputs gives an output that is the

sum of the individual outputs”

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Linear Time-Invariant (LTI) Systems (2/2)

LTI systems

§ Completely characterized by– Impulse response h[n]

– Convolution y[n] = x[n]*h[n]

– FIR Example: h[n] is same as bk

§ Properties of LTI systems

– Commutative

– Associative

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][*][][*][ nxnhnhnx =

])[*][(*][][*])[*][( 321321 nxnxnxnxnxnx =

Cascaded LTI systems

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Frequency Response of an FIR System

§ Sinusoidal Response of FIR systems

§ Frequency-response

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¥

Example 1 (1/2)

§ Show the frequency response of an FIR filter with coefficients

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}1,2,1{}{ =kb

)ˆcos22()2(

21)(

ˆ

ˆˆˆ

ˆ2ˆˆ

wwwww

www

+=++=

++=

-

--

--

j

jjj

jjj

eeee

eeeH

ww

www ˆ)( is Phaseand )ˆcos22()( is Magnitude

0)ˆcos22( Sinceˆˆ -=Ð+=

³+jj eHeH

Example 1 (2/2)

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)ˆcos22()( ˆ ww +=jeH

ww ˆ)( ˆ -=Ð jeH

Example 2

§ Find y[n] for the following inputs when

1)

2)

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njj eenx )3/(4/2][ pp=

ww w ˆˆ )ˆcos22()( jj eeH -+=

( ) njjj eeeny )3/(4/3/ 23][ ppp ´= - njj ee )3/(12/6 pp-=

)4/3/()4/3/(43 )cos(2][

pppppp +-+ +=+= njnj eennx

)4/3/()3/()4/3/(3/2

)4/3/()3/()4/3/(3/1

3)(][3)(][

pppppp

pppppp

+-+--

+-+

==

==njjnjj

njjnjj

eeeeHnyeeeeHny

)cos(633][ 123)12/3/()12/3/( pppppp -=+= --- neeny njnj

Cascaded LTI systems

§ Multiply the frequency responses

– Equivalent system

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y[n]x[n])( ŵjeH

x[n])( ˆ1

wjeHy[n]

)( ˆ2wjeH

)()()( ˆ2ˆ

1ˆ www jjj eHeHeH =

L-pt Averager

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§ Dirichlet function

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11-pt Averager

NULLS or ZEROS

pw 5.0ˆ =pw 05.0ˆ =

LPF

Example: L-pt Averager

§ B&W Image

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Filtered by 11-pt averagerOriginal

Summary

§ This lecture introduced the concept of FIR filtering.– The weighted running average of a finite number of input sequence

values defines a discrete-time system.

§ The impulse response of an FIR system completely defines the system.

§ The frequency-response function is a complete characterization of the behavior of the system for any input that can be represented as a sum of sinusoids.

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å=

-=M

kk knxbny

0][][

][)(][ ˆ nxeHny jw=