Design of IIR multiple notch filters based on all-pass filters

Design of IIR Multiple Notch Filters Based on All-Pass Filters

Yashwant V. Joshi and S. C. Dutta Roy, Fellow, IEEE

Abstract—A new method for the design of multiple notch filters is presented using an all-pass filter of order 2N, N being the number of notch frequencies. The all-pass filter is realized as a cascade of N second-order all-pass sections. Besides meeting the notch frequency specifications exactly, the method ensures that the realized 3-dB bandwidths are lower than those specified.

Illustrative examples are given.

Index Terms—All-pass filter-based design, digital filters, mul- tiple notch filters.

I. INTRODUCTION

N

OTCH filters have a wide variety of applications in the field of signal processing for removing a single frequency or a narrow-band sinusoidal interference. Multiple notch filters are used for the removal of multiple narrow-band or multiple frequency interference, one popular application being in harmonic cancellation. Several ways of designing finite-impulse response (FIR) and infinite-impulse response (IIR) single frequency/narrow-band digital notch filters are available in the literature [1]-[4]. A cascade of such single frequency notch filters can be used for realizing multiple notches, but this solution has the demerits of higher bandwidth and higher sensitivity to coefficient quantization.

The ideal multiple notch filter magnitude characteristics satisfy

where

H(eP") = 1, w/

0 , u> = (1)

where ui^oi (i = 1 to TV) are the N specified notch frequencies.

However, zero bandwidths cannot be realized in practice.

Hence, besides Lo^oi, it is also required to satisfy the specified 3-dB rejection bandwidths B;t (i = 1 to TV). As shown in [4], the transfer function of a single frequency notch filter having a notch at w⁰ of bandwidth B can be expressed in the form

(2)

Manuscript received April 28, 1997; revised October 21, 1998. This paper was recommended by Associate Editor P. E. Allen.

Y. V. Joshi is with the Department of Electrical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi, 110016, India, on leave from Shri Guru Gobind Singhji College of Engineering and Technology, Vishnupuri, Nanded 431602, India.

S. C. Dutta Roy is with the Department of Electrical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi, 110016, India.

Publisher Item Identifier S 1057-7130(99)01750-4.

A(z) = ^k2

- 1 _L .v-2

k2)z~l +z

represents a second-order all-pass filter, with

ki = — COS

and

k2 = [1 - tan B/2]/[l + tan B/2].

(3)

(4a) (4b) The multiple notch filter design problem has not been seriously pursued in literature. However in [4], a simple modification of the second-order notch filter of (2) with UJ⁰ = ir/2 is given, where if z~l is replaced by z~N, then one obtains multiple notches at the frequency n/(2N) and all its odd harmonics.

This allows the removal of symmetric periodic signals with period T = 47V of otherwise arbitrary waveshape. In [5], the specifications for the design of multiple notch filters are first transformed to those of an all-pass filter of order 27V. From the phase characteristics of the all-pass filter, a set of linear equations in the filter coefficients is generated and solved.

This method can satisfy the notch frequency specifications exactly. The bandwidth specifications are met approximately, but only for very small bandwidths because of the use of a first-order approximation while determining the 3-dB cutoff frequencies. For larger bandwidth specifications, the realized bandwidth is always greater than the specified one. Another problem with this design method is that, in some cases, the filter coefficients cannot be obtained because of the possibility of an ill-conditioned coefficient matrix in the linear system of equations.

In this paper, we present a new approach by which the notch frequencies are realized exactly, and the realized 3- dB rejection bandwidths are always less than the specified ones, provided the specified rejection bands do not overlap, as is usually the case. To be specific, nonoverlapping rejection bands imply that the lower 3-dB cutoff frequency of the «th notch should be greater than the upper 3-dB frequency of the (i - l)th notch. Fig. 1 shows a typical amplitude response of a multiple notch filter with N = 2 and nonoverlapping rejection bands, along with the notations for the lower and upper 3- dB cutoff frequencies [(wn, ZU²i) for the first notch and (u>i², tJ22) for the second notch]. As in [5], our method also converts the multiple notch filter design problem to that of an all-pass filter design problem, but the coefficients are determined by a different method and the filter is realized as a cascade of second-order all-pass filters.

1.0

Fig. 1. Typical amplitude response of the multiple notch filter with N = 2,

{u>oi, U J0 2} = {O.2TT, 0.75TT}, and {Bx, B2] = {O.25TT, O.2TT}.

II. THE NEW METHOD

We use the transformation of (2), with A(z) replaced by an all-pass filter of order 27V, having the transfer function

k²ⁱ + k^u(l + k²ⁱ) ^-1

k2i)z 1 r - 2 (5)

where the coefficients ku and k²ⁱ are real and are to be determined. Note that we have used a cascade of second-order all-pass filters, instead of a composite one as in [5]. We let

- tan(_Bi/2)

k2i = (6)

1 + tan(Bi/2)

as in (4b) where Bi is the specified 3-dB bandwidth of the ith notch.

For determining the coefficients ku, we use the elementary fact that H{e^j<JJai) = 0, i = 1 to N. Substituting (5) in (2) and letting H{z) = N{z)/D{z), we get

+ ku{l + k2i)z-1 + k2i

z~2]

Clearly, N(z) will be a mirror image polynomial of the form N(z) = «0( l + fiiz'1 + ••• + (3N-iz-N+1 + (3Nz-N

+ fiN-iz-N~l + •••+ fhz~2N+l + z~2N) (8) where a⁰ = | ( 1 + llf=i k²i) and /Vs can be expressed in terms of ku's and k2i's. Since H(ejoJoi) = 0 implies N(e^jUl'^Oi) = 0, we obtain the following set of N linear equations in fa's (which are nonlinear in fci/s):

/iAr + 2 / iAr _ i COS UlOi + 2/iAr_2 COS 2uioi

-\ h 2 cos Nuj^oi = 0.(9) These can now be solved for the required coefficients ku, by using a standard software package like Mathematica (of

Wolfram Research). For N = 2, however, it is possible to obtain an explicit analytic solution, as shown in Appendix A.

III. BANDWIDTH CONSIDERATIONS

It will now be shown that for all i, the realized 3-dB rejection bandwidth B^ir by our method is less than the specified value Bi7 as well as the value BiP realized by employing the method of [5]. We start by reviewing the properties of the phase function 6A{W) of an all-pass filter of order 27V in relation to the parameters of the multiple notch:

1) 6A{W) is a monotonically decreasing function of u, from 0 at u> = 0 to -2N-K at w = TT;

2) 0A(uoi) = ~(2i - 1)TT, for i = 1 to N;

3) 0^A(Uii, U²ⁱ) = -(2i - l)?r ± TT/2 for i = 1 to N with wii > W2,i_i for i = 2 to N to ensure nonoverlapping rejection bands.

For the «th second-order all-pass filter, let 9i(ui) denote its phase and let 6i(u)u) = -vr/2 and 8i(uj2i) = -3vr/2, where u>u and uj²i are obtained from the specified 3-dB rejection bandwidth B;t and the notch frequency wOi as (see Appendix B)

Li, <^2i)

= COS Woi COS =F Sill A / 1 — COS2 Woi COS2 . (10)

Clearly, u>u and w²i would be the cutoff frequencies if the

«th all-pass filter was used to derive a second-order notch filter. It is obvious that for our design to succeed, wu's and w²iS should also satisfy the nonoverlapping constraint, viz.

uu > w2,i_i f o r « = 2 to N.

At this point, it is instructive to consider the specific case of N = 2 and examine the relevant phase excursions. Let Ai (z) and A2(z) denote the transfer functions of the component second-order all-pass filters. Also, let A^t(z) = Ai(z)A²(z) and Ap{z) be the transfer function of the fourth-order all- pass filter designed by using [5] for the same multiple notch specifications. Fig. 2 shows the plots of the corresponding (7) phase functions 0i, 02, 0t, and 0P of these all-pass filters for typical specifications, and Table I quantifies the data at four critical frequencies, where all Si's are positive.

It can be easily seen from Fig. 2 that S² > Si and that

<53 > <54. Hence the phase 0t reaches —vr/2 earlier than wu, say at wⁿ = w^u - e\. Similarly, 0^t reaches - 3 T T / 2 earlier than u>²i, say at TJ²i = LO²I — e². Now S² > Si implies that e2 > ei, and hence the realized bandwidth Bir =UJ2I-UJH =

(to²i — con) — (e² — ei) = Bi — (a + ve quantity) < B\.

In the case of the second notch, the phase 0^t reaches -57r/2 after uii2, say at Wi2 = uii2 + e3. Similarly, 0t reaches - 7 T T / 2

after w²2, say at U²² = w²²-\-e±. Because ^³ > <5⁴ implies e³ >

e⁴, we see that the realized bandwidth B^2r = U²² - Ui² =

(UJ22 - wi2) - (e3 - e4) = B2 - (a + ve quantity) < B2. Hence the realized bandwidths of both notches are less than the specified values.

TABLE I

PHASE CHARACTERISTICS OF ALL-PASS FILTERS

^"""""•^^ Phase Frequency ^ ^ \

W21

W12

W22

e,

-3n/2

-2JI+63

-2n+64

e

²

-6,

-62 -JI/2 -3JI/2

-JI/2-6,

-3n/2-62 -5JI/2+63

-7^2+64

Fig. 2. Plots of the various phase functions for the example of Fig. 1.

This result for N = 2 can be generalized as follows.1 Let 9i(ui) be the phase response of the zth all-pass section and let

MM

dco (11)

be its group delay response. Since a stable all-pass filter has a monotonically decreasing phase response, we see that

TJ(W) > 0, for all u. (12)

Now the coefficient k²ⁱ of the zth all-pass function is chosen, by design, to give

di{oJu) - &i(co2i) = i(w) dui = IT. (13) The group delay response T(U) of the cascade of TV all-pass sections is given by

N

(14)

i—l

Thus

r(uj)duj= I Ti(ui) dui + y^ / Tk(u>)du>. (15)

'This generalization was suggested by one of the reviewers.

Combining this with (12) and (13), we see that the first term on the right is TT, while the second term is positive. Hence

T(W) > 7T. (16)

Thus, the change in phase angle of the all-pass cascade over the interval [u>u, co²i] exceeds TT radians. Since the change in phase angle over the 3-dB bandwidth is precisely TT radians, we conclude that the 3-dB bandwidth Bir realized by our method must be less than Bi = u>2i — u>u.

No such conclusion can be reached in the case of the method of [5]. The approximation used for determining the lower/upper 3-dB cutoff frequencies in this method, viz, uJu = ui^Oi — Bi/2 and W²i = uj^Oi + Bi/2 is only valid for small 3- dB rejection bandwidths. For large bandwidth specifications, analytical investigation is difficult to carry out. However, it is clear from the plot of 0^p in Fig. 2 that the frequencies at which 9P assumes the values —TT/2, —3TT/2, —5TT/2, and - 7 T T / 2 are such that B^1P > S^{l r} and B²p > B^2r- To put this on firmer grounds, we have worked out another example with (cooi, ^02)

= (0.15TT, 0.8TT) and varied B = BX=B2 from 0 to 0.3TT. A plot of the realized bandwidths B^ir and B^iP from our design method, as well as that of [5], is shown in Fig. 3. As seen from this figure, BiP is always greater than Bi while B2P < B2

for some range of values of B² and B^2P > B² outside this range; however, Bir is always less than Bi.

Finally, we summarize our design procedure as follows:

1) from the given specifications of Ui and Bi, for i = 1 to N, compute k²i's from (6);

2) form N simultaneous equations using (9) and solve for ku',

3) obtain the all-pass transfer function of order 27V from (5).

IV. EXAMPLES

We illustrate the method by two examples.

Example 1: For a multiple notch filter with N = 2 and woi = 0.3TT, wO2 = 0.5TT, BI = O.lvr, and B2 = 0.15TT, our design gives the following parameters for the second-order all- pass filters: k^xl = -0.5397, k²¹ = 0.7265, k¹² = -0.0705, and k22 = 0.6128. The frequency response of the filter is shown in Fig. 4. The bandwidths realized are B^ir = 0.09307T and B2r = 0.14vr. By using the method of [5], this example

0 0

0 0 0-2 0-4 0 6 0 8

Specified 3-dB rejection bandwidth —*-

Fig. 3. Plots of realized versus specified 3-dB rejection bandwidths.

1.0

t 0 . 6

0.4I—

0-2

0-OL 0-0

—

V\

t

1

-f

\

\

\

1

1 / / /

I

0-2 0.4 0 6 1.0

Fig. 5. Frequency responses of the Example 2 filters (our method: ) and (method of [5]: - - - -).

achieves 3-dB rejection bandwidths which are lower than the specified values.

APPENDIX A

ANALYTICAL TREATMENT FOR TV = 2 CASE

Let (woi, ^02) and {B\, B²) be the required notch fre- quencies and the corresponding 3-dB rejection bandwidths.

Then

r i - t a n ( f l i / 2 ) l - t a n ( S²/ 2 ) ]

{k²¹, k²²) = | , , .___,„ ,„,, , , .___,„ ,„, |- (Al) _ l + t a n ( B i / 2 ) ' l + t a n ( B²/ 2 ) J '

The equations for determining kn and kX2 are, from (9)

(52 + 2/3i COS(WQI) + 2 cos(2w0i) = 0 (A2)

and

Fig. 4. Frequency response of the Example 1 filter.

leads to ill-conditioned coefficient matrix of the linear system of equations.

Example 2: For a multiple notch filter with TV = 3, w^Oi =

O.ITT, UJ⁰² = 0.2TT, w^O3 = 0.6TT, Bi = O.ITT, B² = O.ITT,

and B3 = 0.2TT, our design gives the following parameters for the second order all-pass filters: ku = -0.9182, k¹² = -0.8629, jfci3 = 0.2301, k21 = 0.7265, A;22 = 0.7265, and k23 = 0.5095. Fig. 5 shows the frequency response of this design as well as that of the design using the method of [5].

Realized bandwidths of the latter design are 0.0726TT, 0.1614TT,

and 0.2620TT, whereas our design gives the values 0.061 ITT, 0.0898TT, and 0.1818TT for the respective notch frequencies.

V. CONCLUSION

A new approach is presented for the design of multiple notch filters which realizes the notch frequencies exactly and

(32 + 2/?i cos(co02) + 2 cos(2w02) = 0. (A3) These are linear equations in fix and /i2, and can be solved to get

(h = - 2(cos uiOi + cos ui02) and

(A4) [i² = 2(2 cos woi cos W02 + 1)- (A5)

Also from (7) and (8), we have

fix = (1 + fc22)(i + k2l){kxl + kl2)/{l + k2lk22) (A6) and

2 + k21 + kxlk12{l k²¹k²²).

k22)}/

(A7) Substituting for A;21 and A;22 from (A1), and for f3i and (i2 from (A4) and (A5), and then reorganizing, we get the following two

\H{e?»)\ = k2)(k1 +cos u)

2 cos2 w + 2ki{l + k2)2 cos w + kf(l + k2)2 + (1 - k2)2

(B1)

nonlinear equations in kn and ki²:

/ T3 R \

+ &i2 = — I 1 + tan — tan — 1 (cos wOi + cos

and

A

= Xi

f Bi B2

= I 1 + tan — tan — ) cos

B2

(A8)

B\

tan — tan

A

= X2-

After solving, we get

and

X2

kn

c o s 0J02

Xi ± Vxl ~ 4X2

(A9)

(A10) (A11)

[3] H. Yu, S. K. Mitra, and H. Babic, "Linear phase FIR notch filter design,"

Sadhana, vol. 15, pp. 133-155, pt. 3, Nov. 1990.

[4] P. A. Regalia, S. K. Mitra, and P. P. Vaidyanathan, "The digital all-pass filter: A versatile signal processing building block," Proc. IEEE, vol.

76, pp. 19-37, Jan. 1988.

[5] S. C. Pei and C. C. Tseng, "IIR multiple notch filter design based on all-pass filter," IEEE Trans. Circuits Syst. II, vol. 44, pp. 133-136, Feb.

1997.

Yashwant V. Joshi, was born in Maharashtra, India, on September 22, 1964.

He received the B.E. degree and M.E. degrees in electronics from Marathwada University, Aurangabad, India, in 1986 and 1991 respectively. In 1994, he was appointed under the Quality Improvement Programme of the Government of India to the Indian Institute of Technology, Delhi, where he worked under the supervision of Prof. S. C. Dutta Roy and obtained the Ph.D. degree in 1998.

He was a Lecturer from 1986 to 1993 in the Electronics and Computer Science Department of SGGS College of Engineering and Technology, Nanded, India, where he has been an Assistant Professor since 1993. His research interests include digital filters, signal and image processing, and VLSI.

APPENDIX B DERIVATION OF (10)

From (2) and (3), we get (B1), shown at the bottom of the previous page. Substituting for ki and k2 from (4a) and (4b), respectively, equating \H(e>")\ to l / \ / 2 and simplifying, we get the following quadratic equation in cos u>:

2 1 B

cos to — 2 cos too cos — cos to

B B

+1 cosz cu0 cosz sin — 1 = 0 . (B2) After solving (B2) and replacing UJ⁰ and B by w^Oi and Bi, respectively, we get (10).

ACKNOWLEDGMENT

The authors thank the reviewers for their helpful suggestions and constructive criticisms.

REFERENCES

[1] R. Carney, "Design of digital notch filter with tracking requirements,"

IEEE Trans. Space, Electron., Telem., vol. SET-9, pp. 109-114, Dec.

1963.

[2] K. Hirano, S. Nishimura, and S. K. Mitra, "Design of digital notch filters," IEEE Trans. Circuits Syst., vol. CAS-21, pp. 540-546, July 1974.

S. C. Dutta Roy (SM'66-F'95) was born in Mymensingh, Bangladesh, on November 1, 1937. He received the B.Sc.(Hons.) degree in physics, the M.Sc.(Tech.) degree in radio physics and electronics, and the D.Phil, degree for research on network theory and solid state circuits, all from the University of Calcutta, Calcutta, India, in 1956, 1959, and 1965, respectively.

He has been a Professor of Electrical Engineering at the Indian Institute of Technology (IIT), New Delhi, since January 1970, and is currently an Emeritus Fellow at the same Institute. He was Head of the Department during 1970-73 and Dean of Undergraduate Studies during 1983-86. Previously, he served at IIT as Associate Professor during 1968-69, the University of Minnesota as an Assistant Professor during 1965-68, the University of Kalyani, West Bengal, India, as a Lecturer during 1961-65, and the River Research Institute, West Bengal, India, as a Research Officer (1960-61). He was a Visiting Professor at the University of Leeds, England during 1973-74, and a Visiting Fellow at the Iowa State University during 1978-79, on leave from IIT Delhi. He teaches circuits, systems, electronics and signal processing courses, and conducts and supervises research in the same areas.

Professor Dutta Roy is a Fellow of the Indian National Science Academy, the Indian National Academy of Engineering (INAE), the Indian Academy of Sciences, the National Academy of Sciences, India, and the Acoustical Society of India, and is a Distinguished Fellow of the Institution of Electronics and Telecommunication Engineers (IETE). He has served on the Editorial Boards of the International Journal of Circuit Theory and Applications, the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, Multidimensional Systems and Signal Processing, Neural Network World, and all the Indian Journals in his field. He was an Honorary Editor for the IETE Journal of Research during 1981-83, and is currently Chairman of the Editorial Board, IETE Journal of Education. He was a member of the Indian National Committee of the URSI, in charge of Commission C : Signals and Systems, during 1981-88. He has served as Editor of Publications of the INAE and as a Vice President of the IETE. He was the Technical Chair for IEEE TENCON'98 held at New Delhi.