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8/13/2019 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
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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.)
Frame # 2 Slide # 2 A. Antoniou Part 1: FIR Filters Window Method
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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.
Frame # 3 Slide # 3 A. Antoniou Part 1: FIR Filters Window Method
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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).
Frame # 4 Slide # 4 A. Antoniou Part 1: FIR Filters Window Method
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Fourier Series Method Contd
Since the Fourier series coefficients are defined over the range
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Fourier Series Method Contd
Since the Fourier series coefficients are defined over the range
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Fourier Series Method Contd
A finite filter length can be achieved by truncating the
impulse response such that
h(nT) =0 for |n|>M
whereM =(N 1)/2.
Frame # 6 Slide # 7 A. Antoniou Part 1: FIR Filters Window Method
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Fourier Series Method Contd
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.
Frame # 6 Slide # 8 A. Antoniou Part 1: FIR Filters Window Method
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Fourier Series Method Contd
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
Frame # 7 Slide # 9 A. Antoniou Part 1: FIR Filters Window Method
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Fourier Series Method Contd
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.
Frame # 7 Slide # 10 A. Antoniou Part 1: FIR Filters Window Method
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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
Frame # 9 Slide # 12 A. Antoniou Part 1: FIR Filters Window Method
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Fourier Series Method Contd
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.
Frame # 10 Slide # 13 A. Antoniou Part 1: FIR Filters Window Method
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Fourier Series Method Contd
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.
As the filter length is increased, the frequency of the
oscillations increases but the amplitude stays constant.
Frame # 10 Slide # 14 A. Antoniou Part 1: FIR Filters Window Method
F i S i M h d
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Fourier Series Method Contd
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.
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.
Frame # 10 Slide # 15 A. Antoniou Part 1: FIR Filters Window Method
F i S i M th d
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Fourier Series Method Contd
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).
Frame # 11 Slide # 16 A. Antoniou Part 1: FIR Filters Window Method
F i S i M th d
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Fourier Series Method Contd
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.
Frame # 11 Slide # 17 A. Antoniou Part 1: FIR Filters Window Method
F i S i M th d
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Fourier Series Method Contd
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.
Frame # 12 Slide # 18 A. Antoniou Part 1: FIR Filters Window Method
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
Frame # 13 Slide # 19 A. Antoniou Part 1: FIR Filters Window Method
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.
Frame # 14 Slide # 20 A. Antoniou Part 1: FIR Filters Window Method
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.
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.
Frame # 14 Slide # 21 A. Antoniou Part 1: FIR Filters Window Method
Window functions Contd
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Window functions Contd
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
H(ejT)W(ej( )T) d (C)
Frame # 15 Slide # 22 A. Antoniou Part 1: FIR Filters Window Method
Window functions Contd
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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
H(ejT)W(ej( )T) d (C)
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.
Frame # 15 Slide # 23 A. Antoniou Part 1: FIR Filters Window Method
Window functions Contd
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Window functions Cont d
1
c
H(ejT
)
W(ejT
)
s
s/2
c
s
Frequency response of ideal lowpass filter
Frequency spectrum of window
Frame # 16 Slide # 24 A. Antoniou Part 1: FIR Filters Window Method
Window functions Contd
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Window functions Cont d
1
c
s
s/2
s/2
s/2
s
s
Hw(ejT
)
c c
H(ejT
)
W(ej()T
)
c
Frame # 17 Slide # 25 A. Antoniou Part 1: FIR Filters Window Method
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Window functions Contd
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Window functions Cont d
1
c
s
s/2
s/2
s/2
c
s
s
Hw(ejT
)
c c
H(ejT
)
W(ej()T
)
Frame # 19 Slide # 27 A. Antoniou Part 1: FIR Filters Window Method
Window functions Contd
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Window functions Cont d
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
Frame # 20 Slide # 28 A. Antoniou Part 1: FIR Filters Window Method
Window functions Contd
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Window functions Cont d
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.
Frame # 20 Slide # 29 A. Antoniou Part 1: FIR Filters Window Method
Kaiser Window function
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One of the most important windows is the Kaiser window.
Frame # 21 Slide # 30 A. Antoniou Part 1: FIR Filters Window Method
Kaiser Window function
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One of the most important windows is the Kaiser window.
This is a parametric window which has an independentcontrol parameter .
Frame # 21 Slide # 31 A. Antoniou Part 1: FIR Filters Window Method
Kaiser Window function
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One of the most important windows is the Kaiser window.
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.
Frame # 21 Slide # 32 A. Antoniou Part 1: FIR Filters Window Method
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
Frame # 22 Slide # 33 A. Antoniou Part 1: FIR Filters Window Method
Kaiser Window function Contd
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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
Frame # 23 Slide # 34 A. Antoniou Part 1: FIR Filters Window Method
Kaiser Window function Contd
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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
Frame # 24 Slide # 35 A. Antoniou Part 1: FIR Filters Window Method
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
Frame # 25 Slide # 36 A. Antoniou Part 1: FIR Filters Window Method
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
Frame # 27 Slide # 39 A. Antoniou Part 1: FIR Filters Window Method
Design of FIR Lowpass Filters Contd
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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
Frame # 27 Slide # 40 A. Antoniou Part 1: FIR Filters Window Method
Design of FIR Lowpass Filters Contd
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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
Frame # 28 Slide # 41 A. Antoniou Part 1: FIR Filters Window Method
Design of FIR Lowpass Filters Contd
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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
6. FormwK(nT)using Eq. (D).
Frame # 28 Slide # 42 A. Antoniou Part 1: FIR Filters Window Method
Design of FIR Lowpass Filters Contd
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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
6. FormwK(nT)using Eq. (D).
7. Form
Hw(z) =zMHw(z) whereHw(z) = Z[wK(nT)h(nT)]
andM =(N 1)/2.
Frame # 28 Slide # 43 A. Antoniou Part 1: FIR Filters Window Method
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)
Frame # 29 Slide # 44 A. Antoniou Part 1: FIR Filters Window Method
Design of FIR Bandpass Filters Contd
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Gai
n
1+
1
c1 c2
p1
p2a1a2
1.0
s2
Ap
Aa
Frame # 30 Slide # 45 A. Antoniou Part 1: FIR Filters Window Method
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.
Frame # 31 Slide # 46 A. Antoniou Part 1: FIR Filters Window Method
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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
Frame # 34 Slide # 49 A. Antoniou Part 1: FIR Filters Window Method
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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).
Frame # 36 Slide # 51 A. Antoniou Part 1: FIR Filters Window Method
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.
Frame # 36 Slide # 52 A. Antoniou Part 1: FIR Filters Window Method
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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.
Frame # 38 Slide # 55 A. Antoniou Part 1: FIR Filters Window Method
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.
Frame # 38 Slide # 56 A. Antoniou Part 1: FIR Filters Window Method
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.
Frame # 39 Slide # 57 A. Antoniou Part 1: FIR Filters Window Method
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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