fir filters-window method

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Part 1: FIR Filters Window Method

Tutorial ISCAS 2007

Copyright 2007 Andreas AntoniouVictoria, BC, Canada

Email: [email protected]

July 24, 2007

Frame # 1 Slide # 1 A. Antoniou Part 1: FIR Filters Window Method
http://find/http://goback/


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Fourier Series Method

A simple method for the design of FIR filters is through the

use of the Fourier series in conjunction with the application

of a class of functions known as window functions.

Arbitrary specifications can be achieved by using a method

proposed by Kaiser.

Note: The material for this module is taken from Antoniou,Digital Signal Processing: Signals, Systems, and Filters

Chap. 9.)

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Fourier Series Method

A fundamental property of digital filters in general is that

they have a periodic frequency response with period equal

to the sampling frequency s, i.e.,

H(ej(+ks)T) =H(ejT)

Therefore, an arbitrary desired frequency response,

H(ejT), can be represented by a Fourier series as

H(ejT) =

n=h(nT)ejnT (A)

where h(nT) = 1

s

s/2

s/2

H(ejT)ejnT d

are the Fourier series coefficients.

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Fourier Series Method Contd

H(e

jT)=

n=

h(nT)ejnT

(A)

If we letejT =zin Eq. (A), we get

H(z)=

n=

h(nT)zn

This is the transfer function of an FIR filter with impulse

responseh(nT).

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Since the Fourier series coefficients are defined over the range


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Since the Fourier series coefficients are defined over the range


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A finite filter length can be achieved by truncating the

impulse response such that

h(nT) =0 for |n|>M

whereM =(N 1)/2.

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A finite filter length can be achieved by truncating the

impulse response such that

h(nT) =0 for |n|>M

whereM =(N 1)/2.

On the other hand, a causal filter can be obtained bydelaying the impulse response by a periodMTseconds or

byMsampling periods.

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Since delaying the impulse response by Msampling

periods amounts to multiplying the transfer function byzM,

the transfer function of the causal filter assumes the form

H(z) =zMM

n=M

h(nT)zn



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Since delaying the impulse response by Msampling

periods amounts to multiplying the transfer function byzM,

the transfer function of the causal filter assumes the form

H(z) =zMM

n=M

h(nT)zn

The frequency response of the causal filter is obtained by

lettingz =ejT in the transfer function, i.e.,

H(ejT)=ejMTM

n=M

h(nT)ejnT

and since |ejMT| =1, delaying the impulse response by

Msampling periods does not change the amplitude

response.

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Example

Design a lowpass filter with a desired frequency response

H(ejT)

1 for || c

0 for c


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Example Contd

0 2 4 6 8 1050

40

30

20

1.0

0

1.0

, rad/s

Gain,d

B

N=11 N=41

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The amplitude response of the filter (magnitude of the

frequency response) exhibits oscillations in the passband

as well as the stopband, which are known as Gibbs

oscillations. They are caused by the truncation of the

Fourier series.

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Fourier series.

As the filter length is increased, the frequency of the

oscillations increases but the amplitude stays constant.


F i S i M h d
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Fourier series.

As the filter length is increased, the frequency of the

oscillations increases but the amplitude stays constant.

In other words, we do not seem to be able to reduce the

passband and stopband errors below a certain limit by

increasing the filter length.

Therefore, the filters that can be designed with the Fourierseries method are of little practical usefulness.


F i S i M th d
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The standard technique for the reduction of Gibbs

oscillations is to truncate the infinite-duration impulse

response,h(nT), through the use of a discrete-time

window functionw(nT).


F i S i M th d
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The standard technique for the reduction of Gibbs

oscillations is to truncate the infinite-duration impulse

response,h(nT), through the use of a discrete-time

window functionw(nT).

If we let

hw(nT)=w(nT)h(nT)

then a modified transfer function is obtained as

Hw(z) = Z[w(nT)h(nT)]

=1

2j

H(v)Wzv v1 dv

whereH(z)is the original transfer function andW(z)is the

ztransform of the window function.


F i S i M th d
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EvaluatingHw(z)on the unit circle z =ejT gives the

frequency response of the modified filter as

Hw(ejT)=

T

2

2/T0

H(ejT)W(ej( )T) d (C)

W(ejT)is thefrequency spectrumof the window function

and the integral at the right-hand side is a convolutionintegral.


Window functions
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Window functions

Frequency spectrum of a typical window:

0

, rad/s

W(ejT

)

AML

Main-lobe width

Amax

s/2s/2


Window functions C td
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Window functions Contd

Windows are characterized by their main-lobe width,BML,

which is the bandwidth between the first negative and the

first positive zero crossings, and by their ripple ratio, r,

which is defined as

r =100Amax

AML% or R =20 log

Amax

AMLdB

whereAmaxand AMLare the maximum side-lobe and

main-lobe amplitudes, respectively.


Window functions C td
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Windows are characterized by their main-lobe width,BML,

which is the bandwidth between the first negative and the

first positive zero crossings, and by their ripple ratio, r,

which is defined as

r =100Amax

AML% or R =20 log

Amax

AMLdB

whereAmaxand AMLare the maximum side-lobe and

main-lobe amplitudes, respectively.

The main-lobe width and ripple ratio should be as low as

possible, i.e., the spectral energy of the window should be

concentrated as far as possible in the main lobe and the

energy in the side lobes should be as low as possible.


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The way by which Gibbs oscillations can be controlled by

using a window is illustrated in the next few slides which

are based on Eq. (C), i.e.,

Hw(ejT)=

T

2 2/T

0





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Window functions Cont d

The way by which Gibbs oscillations can be controlled by

using a window is illustrated in the next few slides which

are based on Eq. (C), i.e.,

Hw(ejT)=

T

2 2/T

0


Let us consider the application of a generic window in the

design of a LP filter, assuming that the area under the

curve in the frequency spectrum of the window is 2/T.


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1

c

H(ejT

)

W(ejT

)

s

s/2

c

s

Frequency response of ideal lowpass filter

Frequency spectrum of window


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1

c

s

s/2

s/2

s/2

s

s

Hw(ejT

)

c c

H(ejT

)

W(ej()T

)

c

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1

c

s

s/2

s/2

s/2

c

s

s

Hw(ejT

)

c c

H(ejT

)

W(ej()T

)


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From the illustrations, we conclude that

the steepness of the transition characteristic of the filterobtained depends on the main-lobe width of the window,

and


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From the illustrations, we conclude that

the steepness of the transition characteristic of the filterobtained depends on the main-lobe width of the window,

and

the amplitudes of the passband and stopband ripples

depend on the ripple ratio of the window.


Kaiser Window function
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One of the most important windows is the Kaiser window.


http://goforward/http://find/http://goback/


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This is a parametric window which has an independentcontrol parameter .


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This is a parametric window which has an independentcontrol parameter .

By choosing the value of and the filter length,N, arbitrary

specifications can be achieved in lowpass (LP), highpass

(HP), bandpass (BP), and bandstop (BS) filters.


Kaiser Window function Contd
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Ripple ratio versus:

0 1 2 3 4 5 6 7 8 9 1080

70

60

50

40

30

20

10

RR, dB


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Main-lobe width versus :

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

N=43

63

123

MB, rad


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The Kaiser window function is given by

wK(nT)=

I

0()I0()

for|n| M

0 otherwise

(D)

where

=

1

nM

2, I0(x) =1+

k=1

1

k!

x2

k2

andM =(N 1)/2. Its frequency spectrum is given by

WK(ejT)=wK(0) + 2

Mn=1

wK(nT) cos nT


Design of FIR Lowpass Filters
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An FIR LP filter that would satisfy the specifications shown can

be readily designed by using a procedure due to Kaiser as

detailed in the next three slides.

Gain

1+

1

cp a

1.0

s/2

Ap

Aa


Design of FIR Lowpass Filters Contd


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1. Determine the impulse responseh(nT)using the Fourier

series assuming an idealized frequency response

H(ejT) =

1 for|| c

0 forc


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1. Determine the impulse responseh(nT)using the Fourier

series assuming an idealized frequency response

H(ejT) =

1 for|| c

0 forc


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3. With the requireddefined, the actual stopband

attenuationAacan be calculated as

Aa= 20log10


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3. With the requireddefined, the actual stopband

attenuationAacan be calculated as

Aa= 20log10

4. Choose parameteras

=

0 forAa210.5842(Aa 21)

0.4 + 0.07886(Aa 21) for 21 50


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5. Choose parameterDas

D=0.9222 forAa21Aa 7.95

14.36 forAa>21

Then select the lowest odd value of Nthat would satisfy

the inequality

N sD

Bt+ 1 where Bt =a p


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D=0.9222 forAa21Aa 7.95

14.36 forAa>21


the inequality

N sD

Bt+ 1 where Bt =a p

6. FormwK(nT)using Eq. (D).


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D=0.9222 forAa21Aa 7.95

14.36 forAa>21


the inequality

N sD

Bt+ 1 where Bt =a p

6. FormwK(nT)using Eq. (D).

7. Form

Hw(z) =zMHw(z) whereHw(z) = Z[wK(nT)h(nT)]

andM =(N 1)/2.


Design of FIR Bandpass Filters
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The design method presented can be easily extended to

FIR HP, BP, and BS filters.

Consider the case where a BP filter is required that wouldsatisfy the following specifications:

Passband ripple Ap

Minimum stopband attenuation Aa

Lower passband edge p1

Upper passband edge p2

Lower stopband edgea1

Upper stopband edgea2

Sampling frequencys2

(Apand Aain dB and all frequencies in rad/s)


Design of FIR Bandpass Filters Contd
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Gai

n

1+

1

c1 c2

p1

p2a1a2

1.0

s2

Ap

Aa


Design of FIR Bandpass Filters Contd
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The only differences in the design of BP filters are to

use the more critical of the two transition widths for thedesign,

use the idealized frequency of a BP filter for the

determination of the initial impulse response.

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Example


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Design an FIR BP filter satisfying the following specifications:

Minimum attenuation for 0 200: 45 dB Maximum passband ripple for 400 <


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The idealized impulse response is obtained as

h(nT) = 1s

s/2s/2

H(ejT)ejnT d

= 1

n(sin c2nT sin c1nT)

The application of the design procedure described will give

=3.9754

D =2.580

N =53

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Use of Ultraspherical Window Function


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A more recent approach for the design of FIR filters that

parallels the method described is a method based on the

ultraspherical window function proposed by Bergen andAntoniou (see References).


Use of Ultraspherical Window Function
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A more recent approach for the design of FIR filters that

parallels the method described is a method based on the

ultraspherical window function proposed by Bergen andAntoniou (see References).

For certain specifications, this new approach tends to give

more efficient designs, i.e., the minimum filter length that

will achieve the required specifications is somewhat lower.

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Summary


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A method for the design of FIR filters using a procedure

proposed by Kaiser has been described.

The method is easy to apply and requires a minimal

amount of computation.

Hence it can be used to design filters on-line in real or

quasi-real time applications.

The method is implemented in a DSP software packageknown as D-Filter.


Summary
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A method for the design of FIR filters using a procedure

proposed by Kaiser has been described.

The method is easy to apply and requires a minimal

amount of computation.

Hence it can be used to design filters on-line in real or

quasi-real time applications.

The method is implemented in a DSP software packageknown as D-Filter.

The designs obtained are suboptimal, i.e., other methods

are available that would yield a lower filter order for the

same specifications, for example, the weighted-Chebyshevmethod which will be described in Part 2 of this tutorial.


References
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A. Antoniou,Digital Signal Processing: Signals, Systems,

and Filters, Chap. 9, McGraw-Hill, 2005.

J. F. Kaiser, Nonrecursive digital filter design using the

I0-sinh window function,in Proc. IEEE Int. Symp. Circuit

Theory, 1974, pp. 2023.

S. W. A. Bergen and A. Antoniou, Design of ultraspherical

window functions with prescribed spectral characteristics,EURASIP Journal on Applied Signal Processing, vol. 13,

pp. 2053-2065, 2004.

S. W. A. Bergen and A. Antoniou, Design of nonrecursive

digital filters using the ultraspherical window function,EURASIP Journal on Applied Signal Processing, vol. 13,

pp. 1910-1922, 2005.


References Contd
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T. Saramki, Adjustable windows for the design of FIR

filters A tutorial, 6th Mediterranean Electrotechnical

Conference, vol. 1, pp. 2833, May 1991. R. L. Streit, A two-parameter family of weights for

nonrecursive digital filters and antennas,IEEE Trans.

Acoust., Speech, Signal Process., vol. 32, pp. 108118,

Feb. 1984. A. Antoniou, Design of digital differentiators satisfying

prescribed specifications,Proc. Inst. Elect. Eng., Part E,

vol. 127, pp. 2430, Jan. 1980.

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fir filters-window method

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