ECCTD’01 - European Conference on Circuit Theory and Design, August 28-31, 2001, Espoo, Finland
Digital Multiple Notch Filters Performance
Miroslav Vlček∗
Pavel Zahradnı́k†
duced the design of almost equiripple double-notch
FIR filters. The procedure is based on the observation that the odd part of a Zolotarev polynomial
has two extra lobes for which
1
(Zp,q (w) − Zp,q (−w)) > 1 ,
(1)
2
Abstract — The performance of digital IIR and FIR
multiple stopband notch filters is presented. Provided the frequency responses of both IIR and FIR
notch filters are comparable we propose that almost
equiripple FIR notch filter exhibits constant group
delay and superior output responses over its IIR
counterpart. The group delay and the step response
of a selective filter are essential for filtering spurious
harmonics of the power supply from weak EEG and
ECG signals. The responses for a couple of input
pulse train signals together with the step responses
of both IIR and FIR notch filters are included.
Q(w) = Tr (Zp,q (w)) .
Introduction
We would like to recall the fact that in numerous
applications of digital signal processing it is of main
interest to process pulse-like signals by a selective
algorithm which does not change substantially time
sequence at the output. In apart of frequency specifications which guarantee selectivity of a filter we
have to control either group delay or step response
of the filter. The time domain properties of a filter
are often omitted emphasizing frequency responses
only.
In the paper [1] a design technique for multiple
IIR notch filter was investigated. A multiple notch
IIR filter was then used to eliminate power line
(AC) interference in the recording of electrocardiograms (ECG). However, removing of harmonics by
the multiple IIR notch filter produces a substantial
distortion of the output response due to the group
delay variation.
We confirm that due to the relation between impulse response and group delay of a filter, the selectivity of the frequency response provides distortions
of flat regions in the output signal. In order to reduce these ripples in the output signal we propose
FIR multiple notch filters which inherits a constant
group delay.
In [2] we made a study of FIR notch filters and
their abridging which led to the efficient length of
a corresponding impulse response. In [3] we intro∗ Czech Technical University in Prague, Faculty of Transportation Sciences, Konviktská 20, CZ-110 00 Praha 1,
Czech Republic. E-mail:[email protected], Tel: +420-224890720, Fax: +420-2-24890702
† Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, CZ-166 27 Praha 6, Czech Republic. E-mail:[email protected], Tel: +420-2-24352089,
Fax: +420-2-24310784
(2)
The transfer function of a double-notch FIR filter
of length 2M + 1 = r(p + q) + 1 is then
Q(w)
H(z) = z −M 1 −
.
(3)
Q(wmax )
The ripples in the passband are not exactly equal
but they fall within the limit of ripples of an
optimal single notch filter. Frequency response of
such FIR double-notch filter is shown in Fig. 1.
0
−10
−20
dB
1
and which are of the same magnitude. Substituting
the odd part of a Zolotarev polynomial in a Chebyshev polynomial we generate the transfer function
of a double-notch FIR filter using
−30
−40
−50
−60
0
0.1
0.2
0.3
0.4
0.5
ωT/π
0.6
0.7
0.8
0.9
1
Figure 1: Frequency response of the FIR doublenotch filter is generated by T4 (x) where argument
x is equal to the odd part of Zolotarev polynomial Z3,6 (u|0.682) of degree 9, with ωs T = 0.3771π,
ωm T = 0.3342π and ωp T = 0.2912π . Frequency
response 20log|H(ejω )| with notch frequencies specified by ωN 1 T = 0.3327π and ωN 2 T = 0.6673π is
plotted versus the normalised frequency. The ripples in the passband are less than 1 dB.
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1.2
or the step response
1
s(n) =
n
h(k) = 1 − (an + an+1 )
(8)
k=0
0.8
The sequence an which drives the impulse response
function appears in expression for corresponding
group delay as follows
0.6
0.4
∞
1 − a2
τ (w)
=
=
1
+
2
an Tn (w)
T
1 − 2aw + a2
n=1
0.2
0
−0.2
0
50
100
150
(9)
where Tn (w) are Chebyshev polynomials of the first
kind
1
Tn (w) = Tn ( (z + z −1 )).
2
Figure 2: Step response of the FIR double-notch
filter from Fig. 1.
For the second order allpass section the sequence
which controls both the impulse response and
group delay in a similar manner is based on radius
This class of FIR filters can play an important role
of complex conjugate poles z∞ = a ± jb. We can
in filtering of the sinusoidal interference harmonics
conclude that the impulse response (step response,
maintaining the shape of the pulse-like signals.
resp.) and group delay are related to each other
and they are both equivalently responsible for the
distortion of the output signal.
2
Group Delay and Impulse Response
The transfer function of an IIR notch filter is given
3 Examples
as
1
(4)
Hnotch (z) = [1 + A(z)] .
2
Several experiments were performed. Here we
We note that the group delay τ (ω) of an IIR notch present a couple of responses (see Fig. 4, 5, 7
filter is directly related to the group delay of a con- and 8) to the input rectangular pulse train signals
stituting allpass section A(z)
with the different duty cycle. Both FIR and IIR
double-notch filters exhibit comparable frequency
τ notch (ω) = 21 τ allpass (ω) .
(5) responses (see Fig. 1 and 3) with identical notch
frequencies ωN 1 T = 0.3327π and ωN 2 T = 0.6673π.
It simplifies the evaluation of the group delay func- The experiments confirm that FIR filter preserves
tion of a digital IIR notch filter for which zeros of the shape of a processed pulse train better then its
IIR counterpart.
the transfer function appear on the unit circle.
By simple algebra we can conclude that an allpass section which is the fundamental building
block of IIR notch filter has the impulse response 4 Conclusion
sequence related to its group delay. For the first
order allpass
The performance of digital IIR and FIR multiple
stopband notch filters was presented. Provided the
frequency responses of both IIR and FIR notch
∞
z −1 − a
1 − a2 n −n
filters are comparable, the almost equiripple FIR
=
−a
+
a
z
(6)
A1 (z) =
1 − az −1
a n=1
notch filter exhibits constant group delay and superior output responses over its IIR counterpart.
we obtain the impulse response
Two examples confirm that FIR filter preserves the
shape of a processed pulse train better then its IIR
1
h(n) = − δ(n) + (an−1 − an+1 )u(n)
(7) counterpart.
a
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1.4
0
1.2
−10
1
−20
dB
0.8
−30
0.6
−40
0.4
−50
−60
0.2
0
0.1
0.2
0.3
0.4
0.5
ωT/π
0.6
0.7
0.8
0.9
1
0
0
50
100
150
Figure 3: Frequency response 20log|H(ejω )| of a Figure 6: Step response of the IIR double-notch
competitive IIR double-notch filter designed by the filter from Fig. 3.
method in [1].
12
12
10
10
8
8
6
6
4
4
2
2
0
0
−2
−2
0
50
100
150
200
250
300
350
0
50
100
150
200
250
300
350
Figure 4: Double-notch FIR filter response to the Figure 7: Double-notch IIR filter response to the
input pulse train signal with duty cycle 12 : 108 . input pulse train signal with duty cycle 12 : 108 .
12
12
10
10
8
8
6
6
4
4
2
2
0
0
−2
−2
0
50
100
150
200
250
300
350
0
50
100
150
200
250
300
350
Figure 5: Double-notch FIR filter response to the Figure 8: Double-notch IIR filter response to the
input pulse train signal with duty cycle 18 : 102 . input pulse train signal with duty cycle 18 : 102 .
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References
[1] S. C. Pei, C. C. Tseng, IIR Multiple Notch
Filter Design Based on Allpass Filter, IEEE
Transactions on Circuits and Systems, vol. 44,
No. 2, February 1997, pp. 133-136.
[2] M. Vlček, L. Jireš, Fast Design Algorithms for
FIR Notch Filters, Proc. of IEEE International
Symposium on Circuits and Systems ISCAS’94,
London 1994, Vol.2, pp. 297 - 300.
[3] M. Vlček, R. Unbehauen, Zolotarev Polynomials and Optimal FIR Filters, IEEE Trans. on
Sig. Proc. , vol. 47, pp. 717-730, March 1999.
[4] M. Vlček, P. Zahradnı́k, R. Unbehauen, Analytical Design of FIR Filters, IEEE Trans. on
Sig. Proc. , September 2000, pp. 2705-2709.
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