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DESIGN OF FIR DIGITAL FILTERS

1. Lesson 17 -39 minutes

DESIGN OF FIR DIGITAL FILTERSN-iH z) = Z h(n)z-nn=o

FOR LINEAR PHASE

Basic design methodsfor FIR digitalfilters

h(n) = h(N-1-n)BASIC DESIGN METHODS:(D WINDOWS(2) FREQUENCY SAMPLING(3 EQUIRIPPLE DESIGN

DESIGN OF FIR FILTERS .USING WINDOWSDESIRED UNIT SAMPLE RESPONSE: hd(n)

h(n)= w(n) hd(n)w(n) =0

Design of FIR filtersusing the windowmethod.

n(N-1)H(e")= EfHd(ei") W[ej(w 8)ld6r7

sin(wN/2)8 sin(w/2)

N=8-2r 2 r 2vN N

Magnitude of theFourier transform foran eight pointrectangular window.

17.1

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Effect of convolvingthe Fourier trans-form of a rectangularwindow with an ideallow pass filtercharacteristic.

.6

hd(n)

o 0 -20 30 40 50.2

RECTANGULAR WINDOW

3

Unit-sample responseof an ideal low-passfilter truncated bya 51-point rectangularwindow.

Frequency responsecorresponding to theunit-sample responsein viewgraph e.

27 r )

WOOw)

7r I Al27

17.2

/W(ej(w-8))

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The Hamming andHAMMING-, Bartlett windows.

BARTLETT WINDOW

-100

HAMMING WINDOW

a)--e1

7Tr- W

Frequency response ofan FIR lowpass filterobtained bymultiplying the unit-sample response ofan ideal low passfilter by a Bartlettwindow.

Frequency response ofan FIR lowpass filterfilter obtained bymultiplying the unitsample response ofan ideal lowpassfilter by a Hammingwindow. (Note that thestopband attenuationis approximately 65 dbnot 30 db as stated inthe lecture.)

17.3

hd(n)

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Equally spacedfrequency samples ofan ideal low pass

T -****-*-* -- filter.

V1 27rH ejwk) =HD(ei"k) Wk= k k=20,rI(N-1)

h(n)

190 il. ? 9 9

A unit-sample responsethe magnitude ofwhose DFT is equal tothe frequency samplesin viewgraph j..The bottom trace isthe magnitude of theFourier transformof this unit-sampleresponse.

Another unit-sampleresponse themagnitude of whoseDFT is equal to thefrequency samplesin viewgraph j..The bottom trace isthe magnitude of theFourier transform ofthis unit-sampleresponse.

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h(n)

H(ei')

1+81

(PwS, (82/8) FIXED81 (or 82) MINIMIZED

h(n)

Equiripple approxi-mation of a lowpassfilter.

PWS

17.5

Unit sample responseand frequency responsewhen one frequencysample is moved fromthe stopband intothe transition band.

Similar to viewgraphm. with a differentvalue of the frequencysample in thetransition band.

pp 59 5? .~ IL. ?. .. p.-@5 "4 4* 59 -u

- . .. .~?..

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S I i I | 1 | 1 i I '

0.110 Example of a high. 0065 1order equiripple0.020 lowpass filter.- 00250070

0 005 0.10

0-20

-40-60

-F -80O0 -100

0.050 0.100 0.150 0.200 0.250 0.300Normalized frequency w

~IIIii~I111IIIll0.350 0.400 0.450 0.500

2. Comments

FIR filters are an important class of digital filters, and in con-trast with continuous-time FIR filters, the implementation of digitalfilters of this type is relatively straightforward.Design techniques for FIR digital filters are generally carried outdirectly in the discrete-time domain. This lecture introduces thethree primary design techniques, specifically the window method, thefrequency sampling method, and the algorithmic design of optimumfilters. The window method basically begins with a desired unit-sample response which is then truncated by means of a finite durationwindow. In the frequency sampling method, the frequency responseof the FIR filter is specified in terms of samples of the desiredfrequency response. The first two design procedures generally do notresult in optimum filter designs. There is available an algorithmicdesign procedure which generates optimum equiripple FIR filterdesigns. While I mention this technique only briefly in the lecture,it is developed in considerable detail in the text in sections 7.6 and 7.7

3. Reading

Text: Section 7.4 (page 444) and section 7.8.

17.6

'111111I IDDIIII lpi-120 1--10 0

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4. Problems

Problem 17.1

In the window method of design for lowpass FIR filters we indicatedthat the transition width of the resulting filter is dependentprimarily on the width of the main lobe of the Fourier transform ofthe window. For the purposes of this problem, we define the mainlobe as the symmetric interval between the first negative andpositive frequencies at which W(ejo) = 0. Consider the followingthree windows:

Rectangular: wR (n) = 1 In| < N - 1= 0 otherwise

Bartlett: w (n) = 1 - n In j < N - 1B N= 0 otherwise

Raised cosine:w H(n) = a+0cos(wn/(N-l)) |n l < N - 1= 0 otherwise

(If a=a=0.5 this is the Hanning window and if a=0.54 and 0=0.46 thisis the Hamming window.)Determine the Fourier transforms of each of these windows. Also,determine the width of the main lobe for each window, assumingthat N >> 1.

Problem 17.2

Let h1 (n) and h2 (n) denote the unit sample responses of two FIRfilters of length 8. The two are related by a four-point circularshift, i.e.,

h2 (n) = h1 ((n-4)) 8 R8 (n)

The Fourier transform of h1 (n) is sketched in Figure P17.2-j, andcorresponds to a low pass filter with cut-off frequency .

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H, (e j)

Figure P17.2-1

(a) Determine the relationship between the DFT of h (n) and h2 (n)and show in particular that the magnitude of the DFT of h (n) andh2 (n) are equal.

(b) Would it also be reasonable to consider h2 (n) to correspond toa lowpass filter with cut-off frequency of ?

Problem 17.3Let hd (n) denote the unit-sample response of an ideal desired systemwith frequency response Hd (e ) and let h(n) denote the unit-sampleresponse, of the length N sampled, for an FIR system with frequencyresponse H(ej). In Sec. 5.6 it was asserted that a rectangularwindow of length N samples applied to hd(n) will produce a unit-sample response H(n) such that the mean-square error

7TE= f Hd(e$) - H(e] )12 do

-7Tis minimized.(a) The error function E(el ) = Hd (el") - H(ej) can be expressed asthe power seriesE(e ) = e(n)e-jon4..dn=-oFind the coefficients e(n) in terms of hd(n) and h(n).(b) Express the mean-square error e in terms of the coefficients e(n).(c) Show that for a unit-sample response h(n) of length N samples, 2is minimized when

17.8

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h h(n)h (n) = 10A

0 < n < N-otherwise

That is, the rectangular window gives the best mean-squareapproximation to a desired frequency response for a fixed value of N.

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