Ch5.4_5

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EE3054Signals and Syst em s

Lec t u re 2L inear and T im e Invar iant

Sys temsYao Wang

Poly tec hn ic Un iversi t y

Most of the slides included are extracted from lecture presentations prepared by

McClellan and Schafer

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1/30/2008 2003, JH McClellan & RW Schafer 2

L ic ense In fo fo r SPFi rst Sl ides

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General FIR System IMPULSE RESPONSE

FIR case: h[n]=b_n

CONVOLUTION

For any FIR system: y[n] = x[n] * h[n]

Review o f Las t Lec t u re

][][][ nxnhny =

][nh

}{ kb

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GENERAL FIR FILTER

FILTER COEFFICIENTS {bk}

DEFINE THE FILTER

For example,

=

=

M

k

k knxbny0

][][

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

][][

3

0

++=

==

nxnxnxnx

knxbnyk

k

}1,2,1,3{ =kb

Feedforward Difference Equation

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GENERAL FIR FILTER

SLIDE a Length-L WINDOW over x[n]

When h[n] is not symmetric, needs to flip h(n) first!

x[n]x[n-M]

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7-pt AVG EXAMPLE

CAUSAL: Use Previous

LONGER OUTPUT

400for)4/8/2cos()02.1(][:Input ++= nnnx n

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=

= 00

01

][ n

n

n

Uni t Im pulse Signa l

x[n] has only one NON-ZERO VALUE

1

n

UNIT-IMPULSE

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},0,0,,,,,0,0,{][41

41

41

41

=nh

4-p t Avg Im pulse Response

[n] READS OUT the FILTER COEFFICIENTS

n=0h in h[n] denotes

Impulse Response

1

n

n=0

n=1

n=4

n=5

n=1

NON-ZERO

When window

overlaps [n]

])3[]2[]1[][(][41 +++= nxnxnxnxny

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What is Im pulse Response?

Impulse response is the output signal when theinput is an impulse

Finite Impulse Response (FIR) system:

Systems for which the impulse response has finiteduration

For FIR system, impulse response = Filter

coefficients h[k] = b_k

Output = h[k]* input

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FIR IMPULSE RESPONSE

Convolution = Filter Definition

Filter Coeffs = Impulse Response

=

=

M

k

knxkhny0

][][][

CONVOLUTION

=

=

M

k

k knxbny0

][][

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Convolut ion Operat ion

Flip h[n]

SLIDE a Length-L WINDOW over x[n]

x[n]x[n-M]

=

=

M

k

knxkhny0

][][][

CONVOLUTION

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More on s ignal ranges and

lengt hs a f t e r f i l t e r ing

Input signal from 0 to N-1, length=L1=N Filter from 0 to M, length = L2= M+1

Output signal?

From 0 to N+M-1, length L3=N+M=L1+L2-1

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DCONVDEMO: MATLAB GUI

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Go through the demo program for differenttypes of signals

Do an example by hand

Rectangular * rectangular

Step function * rectangular

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CONVOLUTION via Synt het ic

Po lynom ia l Mul t ip l ic a t ion

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Convolut ion v ia Synt het ic

Po lynom ia l Mul t ip l ic a t ion

More example

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MATLAB fo r FIR FILTER

yy = conv(bb,xx)

VECTORbb contains Filter Coefficients

DSP-First: yy = firfilt(bb,xx)

FILTER COEFFICIENTS {bk} conv2()for images

=

=

M

k

k knxbny0

][][

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]1[][][ = nununy

POP QUIZ

FIR Filter is FIRST DIFFERENCE

y[n] = x[n] - x[n-1]

INPUT is UNIT STEP

Find y[n]

Ch5.4_5

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