- Email: [email protected]
Remarks on and correction to the impulse invariant method for the design of IIR digital filters
Signal Processing 80 (2000) 1687}1690
Fast communication
Remarks on and correction to the impulse invariant method for the design of IIR digital "lters Wolfgang F.G. MecklenbraK uker* Institut fu( r Nachrichtentechnik und Hochfrequenztechnik, Technische Universita( t, Wien, Gusshausstrasse 25/389, A-1040 Wien, Austria Received 21 February 2000
Abstract It is shown that the correct application of the impulse invariant method (i.i.m.) for the design of IIR digital "lters leads to a better approximation for "rst-order "lters than the conventionally used procedure. If the impulse response of the analog prototype "lter is also continuous at t"0 then the conventional approach and the correct i.i.m. give identical results. 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Design of IIR digital "lters; Impulse invariant method
1. Introduction Already in the classical contribution on digital "lters by Kaiser [2] the impulse invariant method (i.i.m.) has been proposed for the design of IIR digital "lters. Since then this technique is extensively discussed in nearly all current textbooks on DSP [1,3}5]. For further information on the i.i.m. these references should be consulted. The purpose of this contribution is to call the attention to the fact that sampling the impulse response h (t) of a property analog "lter with frequency response H ( ju) leads only then to uniquely determined sample values h (n¹) if h (t) is continuous at the sampling instances t"n¹. If however h (t) is discontinuous at t"n¹ the sample value
* Tel.: #43-1-58801-38929; fax: #43-1-58801-38999. E-mail address: [email protected] (W.F.G. MecklenbraK uker).
can be determined by several di!erent choices. Two of these choices will be discussed next.
2. Conventional choice for 5rst-order 5lters The basic building block for application of the i.i.m. is the simple "rst-order analog "lter with frequency response 1 H ( ju)" a#ju
(a'0).
(1)
The corresponding causal and stable impulse response is h (t)"e\?Rp(t) (2) where p(t) is the continuous time unit step function. Usually [1}5], sampling ¹h (t) with sampling period ¹ leads to the impulse response f [n] of a "rst-order IIR digital "lter with f [n]"¹e\?2Lp[n],
0165-1684/00/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 0 0 ) 0 0 1 1 3 - 4
(3)
W.F.G. Mecklenbra( uker / Signal Processing 80 (2000) 1687}1690
1688
where p[n] is the discrete time unit step function. The corresponding IIR digital "lter has the frequency response
(4)
Observe that h (t) in Eq. (2) is discontinuous at t"0 with a jump discontinuity of height equal to one. As sample value at the jump the conventional choice takes the right-hand limit, i.e. in this case the maximum of the jump. Since sampling in time of a non-band-limited function leads to aliasing contributions in frequency it is usually concluded that the deviations between H ( ju) and F (e S2) have to be inter preted as aliasing errors. It will be shown next that this conclusion is only partially true.
3. Fourier's choice for 5rst-order 5lters The relation between the samples of h (t) and its Fourier transform H ( ju) is given by Poisson's sum formula [6]
. (5)
Assuming convergence of both sums in Eq. (5) it is important to observe that now the sample values of h (t) at jump discontinuities have to be taken as the arithmetic means of the corresponding limiting values of the left-hand and right-hand side of the discontinuities, respectively. De"ning the samples of ¹h (t) therefore by 2 for n"0, g [n]" ¹e\?2Lp[n] for nO0,
2p G (e S2)"H ( ju)# H j u!k ¹ I$
¹ F (e S2)" ¹e\?2L e\ LS2" . 1!e\?2 e\ S2 L
2p ¹h (n¹)e\ SL2" H j u!k ¹ L\ I\
With this de"nition, Eq. (5) can be rewritten in the form
(6)
,
(8)
which clearly shows that the deviations between G (e S2) and H ( ju) for 0)"u¹")p are purely caused by aliasing in contrast to the behavior of F . 4. Comparison of the two choices Since the impulse responses f [n] and g [n] are related by ¹ f [n]"g [n]# d[n], 2
(9)
the corresponding frequency responses are connected by ¹ F (e S2)"G (e S2)# . 2
(10)
The di!erence between F and G is therefore a constant o!set for all frequencies. For easy comparison, gains and phases of the three frequency responses H , F and G are shown in Figs. 1}3 for di!erent values of the parameter a¹ (3 dB cuto! frequency). All gain functions are normalized to the gain of unity at u¹"0, so that the gain di!erences are most pronounced at u¹"p, i.e., in the stopband. It is clearly seen that G is a far better approximation to H than F which always has a stopband behavior which is 4 dB worse than H at u¹"p. Since the i.i.m.'s `bad aliasing error behaviora is usually demonstrated by this "rst-order example F [3] it should be clari"ed now that the correct i.i.m. leads in fact to G which has a better behavior than F .
the new frequency response of an i.i.m. derived IIR digital "lter is given by
5. Consequences for higher-order 5lters
¹ 1#e\?2 e\ S2 G (e S2)" . 2 1!e\?2 e\ S2
For rational analog prototype frequency responses H ( ju) with stopbands at high frequencies
(7)
W.F.G. Mecklenbra( uker / Signal Processing 80 (2000) 1687}1690
1689
Fig. 1. (a) Comparison of gains of H , F and G for a¹"0.015708. (b) Comparison of phases of H , F and G for a¹"0.015708.
Fig. 2. (a) Comparison of gains of H , F and G for a¹"0.15708. (b) Comparison of phases of H , F and G for a¹"0.15708.
a partial fraction expansion leads to , A I H ( ju)" ak#ju I
and (11)
with complex-valued A and a where Re a '0 I I I and only N simple poles are assumed. With the results of Eqs. (4) and (7) the corresponding IIR digital "lters with frequency responses G and F are given by , ¹ 1#e\?I 2 e\ S2 G(e S2)" A I 2 1!e\?I 2 e\ S2 I
(12)
, 1 F(e S2)" A ¹ . I 1!e\?I 2 e\ S2 I By Eq. (10) G and F are related by
(13)
¹ , F(e S2)"G(e S2)# A . (14) I 2 I It is easily seen that the additional constant o!set vanishes if , A "0, I I
(15)
1690
W.F.G. Mecklenbra( uker / Signal Processing 80 (2000) 1687}1690
Fig. 3. (a) Comparison of gains of H , F and G for a¹"0.62832. (b) Comparison of phases of H , F and G for a¹"0.62832.
which is equivalent to h (0)"0 in the time domain, i.e. h (t) is now continuous everywhere. Therefore, if H ( ju) vanishes for "u"PR with order higher than 1 the IIR digital "lter with frequency response F(e S2) (Eq. (13)) conventionally obtained form the i.i.m. is identical to the correct result G(e S2) (Eq. (12)).
6. Conclusion It is shown that the correct application of the impulse invariant method (i.i.m.) for the design of IIR digital "lters leads to a better approximation for "rst-order "lters than the conventionally used procedure. If the impulse response of the analog prototype "lter is also continuous at t"0 then the conventional approach and the correct i.i.m. give identical results.
Acknowledgements The author thanks Gerhard Doblinger for his support and discussions on the subject of this letter.
After completion of this manuscript the author became aware of [7]. He thanks Jim Kaiser for indication of its existence and Leland Jackson for generously providing a preprint.
References [1] B. Gold, C.M. Rader, Digital Processing of Signals, McGraw-Hill, New York, 1969, p. 51. [2] J.F. Kaiser, Digital "lters, in: F.F. Kuo, J.F. Kaiser (Eds.), System Analysis by Digital Computer, Wiley, New York, 1966, p. 245 (Chapter 7). [3] A.V. Oppenheim, R.W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cli!s, NJ, 1975, p. 198. [4] L.R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing, Prentice-Hall, Englewood Cli!s, NJ, 1975, p. 216. [5] R.A. Roberts, C.T. Mullis, Digital Signal Processing, Addison-Wesley, Reading, MA, 1987, p. 194. [6] A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill, New York, 1962, p. 47. [7] L.B. Jackson, A Correction to Impulse Invariance, IEEE SP-Lett., 2000, submitted for publication.
Fast communication
Remarks on and correction to the impulse invariant method for the design of IIR digital "lters Wolfgang F.G. MecklenbraK uker* Institut fu( r Nachrichtentechnik und Hochfrequenztechnik, Technische Universita( t, Wien, Gusshausstrasse 25/389, A-1040 Wien, Austria Received 21 February 2000
Abstract It is shown that the correct application of the impulse invariant method (i.i.m.) for the design of IIR digital "lters leads to a better approximation for "rst-order "lters than the conventionally used procedure. If the impulse response of the analog prototype "lter is also continuous at t"0 then the conventional approach and the correct i.i.m. give identical results. 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Design of IIR digital "lters; Impulse invariant method
1. Introduction Already in the classical contribution on digital "lters by Kaiser [2] the impulse invariant method (i.i.m.) has been proposed for the design of IIR digital "lters. Since then this technique is extensively discussed in nearly all current textbooks on DSP [1,3}5]. For further information on the i.i.m. these references should be consulted. The purpose of this contribution is to call the attention to the fact that sampling the impulse response h (t) of a property analog "lter with frequency response H ( ju) leads only then to uniquely determined sample values h (n¹) if h (t) is continuous at the sampling instances t"n¹. If however h (t) is discontinuous at t"n¹ the sample value
* Tel.: #43-1-58801-38929; fax: #43-1-58801-38999. E-mail address: [email protected] (W.F.G. MecklenbraK uker).
can be determined by several di!erent choices. Two of these choices will be discussed next.
2. Conventional choice for 5rst-order 5lters The basic building block for application of the i.i.m. is the simple "rst-order analog "lter with frequency response 1 H ( ju)" a#ju
(a'0).
(1)
The corresponding causal and stable impulse response is h (t)"e\?Rp(t) (2) where p(t) is the continuous time unit step function. Usually [1}5], sampling ¹h (t) with sampling period ¹ leads to the impulse response f [n] of a "rst-order IIR digital "lter with f [n]"¹e\?2Lp[n],
0165-1684/00/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 0 0 ) 0 0 1 1 3 - 4
(3)
W.F.G. Mecklenbra( uker / Signal Processing 80 (2000) 1687}1690
1688
where p[n] is the discrete time unit step function. The corresponding IIR digital "lter has the frequency response
(4)
Observe that h (t) in Eq. (2) is discontinuous at t"0 with a jump discontinuity of height equal to one. As sample value at the jump the conventional choice takes the right-hand limit, i.e. in this case the maximum of the jump. Since sampling in time of a non-band-limited function leads to aliasing contributions in frequency it is usually concluded that the deviations between H ( ju) and F (e S2) have to be inter preted as aliasing errors. It will be shown next that this conclusion is only partially true.
3. Fourier's choice for 5rst-order 5lters The relation between the samples of h (t) and its Fourier transform H ( ju) is given by Poisson's sum formula [6]
. (5)
Assuming convergence of both sums in Eq. (5) it is important to observe that now the sample values of h (t) at jump discontinuities have to be taken as the arithmetic means of the corresponding limiting values of the left-hand and right-hand side of the discontinuities, respectively. De"ning the samples of ¹h (t) therefore by 2 for n"0, g [n]" ¹e\?2Lp[n] for nO0,
2p G (e S2)"H ( ju)# H j u!k ¹ I$
¹ F (e S2)" ¹e\?2L e\ LS2" . 1!e\?2 e\ S2 L
2p ¹h (n¹)e\ SL2" H j u!k ¹ L\ I\
With this de"nition, Eq. (5) can be rewritten in the form
(6)
,
(8)
which clearly shows that the deviations between G (e S2) and H ( ju) for 0)"u¹")p are purely caused by aliasing in contrast to the behavior of F . 4. Comparison of the two choices Since the impulse responses f [n] and g [n] are related by ¹ f [n]"g [n]# d[n], 2
(9)
the corresponding frequency responses are connected by ¹ F (e S2)"G (e S2)# . 2
(10)
The di!erence between F and G is therefore a constant o!set for all frequencies. For easy comparison, gains and phases of the three frequency responses H , F and G are shown in Figs. 1}3 for di!erent values of the parameter a¹ (3 dB cuto! frequency). All gain functions are normalized to the gain of unity at u¹"0, so that the gain di!erences are most pronounced at u¹"p, i.e., in the stopband. It is clearly seen that G is a far better approximation to H than F which always has a stopband behavior which is 4 dB worse than H at u¹"p. Since the i.i.m.'s `bad aliasing error behaviora is usually demonstrated by this "rst-order example F [3] it should be clari"ed now that the correct i.i.m. leads in fact to G which has a better behavior than F .
the new frequency response of an i.i.m. derived IIR digital "lter is given by
5. Consequences for higher-order 5lters
¹ 1#e\?2 e\ S2 G (e S2)" . 2 1!e\?2 e\ S2
For rational analog prototype frequency responses H ( ju) with stopbands at high frequencies
(7)
W.F.G. Mecklenbra( uker / Signal Processing 80 (2000) 1687}1690
1689
Fig. 1. (a) Comparison of gains of H , F and G for a¹"0.015708. (b) Comparison of phases of H , F and G for a¹"0.015708.
Fig. 2. (a) Comparison of gains of H , F and G for a¹"0.15708. (b) Comparison of phases of H , F and G for a¹"0.15708.
a partial fraction expansion leads to , A I H ( ju)" ak#ju I
and (11)
with complex-valued A and a where Re a '0 I I I and only N simple poles are assumed. With the results of Eqs. (4) and (7) the corresponding IIR digital "lters with frequency responses G and F are given by , ¹ 1#e\?I 2 e\ S2 G(e S2)" A I 2 1!e\?I 2 e\ S2 I
(12)
, 1 F(e S2)" A ¹ . I 1!e\?I 2 e\ S2 I By Eq. (10) G and F are related by
(13)
¹ , F(e S2)"G(e S2)# A . (14) I 2 I It is easily seen that the additional constant o!set vanishes if , A "0, I I
(15)
1690
W.F.G. Mecklenbra( uker / Signal Processing 80 (2000) 1687}1690
Fig. 3. (a) Comparison of gains of H , F and G for a¹"0.62832. (b) Comparison of phases of H , F and G for a¹"0.62832.
which is equivalent to h (0)"0 in the time domain, i.e. h (t) is now continuous everywhere. Therefore, if H ( ju) vanishes for "u"PR with order higher than 1 the IIR digital "lter with frequency response F(e S2) (Eq. (13)) conventionally obtained form the i.i.m. is identical to the correct result G(e S2) (Eq. (12)).
6. Conclusion It is shown that the correct application of the impulse invariant method (i.i.m.) for the design of IIR digital "lters leads to a better approximation for "rst-order "lters than the conventionally used procedure. If the impulse response of the analog prototype "lter is also continuous at t"0 then the conventional approach and the correct i.i.m. give identical results.
Acknowledgements The author thanks Gerhard Doblinger for his support and discussions on the subject of this letter.
After completion of this manuscript the author became aware of [7]. He thanks Jim Kaiser for indication of its existence and Leland Jackson for generously providing a preprint.
References [1] B. Gold, C.M. Rader, Digital Processing of Signals, McGraw-Hill, New York, 1969, p. 51. [2] J.F. Kaiser, Digital "lters, in: F.F. Kuo, J.F. Kaiser (Eds.), System Analysis by Digital Computer, Wiley, New York, 1966, p. 245 (Chapter 7). [3] A.V. Oppenheim, R.W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cli!s, NJ, 1975, p. 198. [4] L.R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing, Prentice-Hall, Englewood Cli!s, NJ, 1975, p. 216. [5] R.A. Roberts, C.T. Mullis, Digital Signal Processing, Addison-Wesley, Reading, MA, 1987, p. 194. [6] A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill, New York, 1962, p. 47. [7] L.B. Jackson, A Correction to Impulse Invariance, IEEE SP-Lett., 2000, submitted for publication.