sCorrespondence to: Prof. B. Kumar.

CCC 0098—9886/97/020135—05$17.50 Received 4 April 1995( 1997 by John Wiley & Sons, Ltd. Revised 8 February 1996

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 25, 135—139 (1997)

LETTERS TO THE EDITOR

DESIGN OF FIR NOTCH FILTERS BY USING BERNSTEINPOLYNOMIALS

SHAIL B. JAIN,1 BALBIR KUMAR2s AND S. C. DUTTA ROY3

1Department of Electronics and Communication Engineering, Delhi College of Engineering, Kashmere Gate, Delhi 110 006,India

2Department of Electronics and Communication Engineering, Delhi Institute of Technology, Kashmere Gate, Delhi 110 006,India

3Department of Electrical Engineering, Indian Institute of Technology, Hauzkhas, New Delhi 110 016, India

SUMMARY

In this paper, Bernstein polynomials have been used to derive an explicit formula for the coefficients of linear phase FIRnotch filters which are maximally flat at u"0 and n. The approach is relatively simple and enables us to design the filterfor a specific notch frequency and bandwidth. ( 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., vol. 25,135—139 (1997)

(No. of Figures: 2; No. of Tables: 1; No. of Refs: 5)

1. INTRODUCTION

Several design approaches are available for IIR as well as FIR notch filters. We consider here a new methodfor FIR notch filters which have exact linear phase and are of course unconditionally stable. In an earlierwork1 we proposed a design technique for such a filter with a specified notch frequency u

$and a specified

(maximum) 3 dB rejection bandwidth B¼ by constraining the response to be maximally flat at u"0 and n.A recursive and somewhat involved formula for computing the necessary weights was obtained through theuse of Crout’s method. In this paper we show that the design can be simplified by using Bernsteinpolynomials to derive an explicit formula for the weights. This has been possible by expressing the transferfunction as a polynomial in cosu. Transfer functions expressed in this form are particularly convenient forimplementing variable cut-off filters.2

2. DESIGN

Let the frequency response of a notch filter be given by

H(u)"n+i/0

ai(cosu)i (1)

Our aim is to approximate an ideal notch filter H$(u) with 180° phase shift at the notch frequency

u$, i.e.

H$(u)¢G

#1, DuD(u$

!1, u$(DuD)n

(2)

sThe choice of Bernstein polynomials in preference to other competitors has been dictated by their attractive properties of (i) zero errorsat extrema, (ii) smooth approximation and (iii) analytical elegance.

Figure 1. Functions f (x) and f (k/n) used to approximate notch filters

We use Bernstein polynomialss to approximate the response given by (2). Consider a function f (x) defined inthe interval [0, 1], as shown in Figure 1, with functional values given by

f (k/n)¢G#1, 0)k)¸

!1, ¸#1)k)n(3)

where ¸#1 and n!¸ give the numbers of successive discrete points at which the function f (k/n) is #1 and!1 respectively. The nth-order (n*1) Bernstein polynomial for the function f (x) is given by3

Bn(x)"

n+k/0

f (k/n) An

kB xk(1!x)n~k (4)

An alternative expression to (4) is3

Bn(x)"

n+k/0

*kf (0) An

kB xk (5)

where *k f (0) is the kth forward difference of f (k/n) at k"0 and is determined from its functional values atk"0, 1, 2,2, n.

By using f (k/n) as defined in (3) and carrying out some algebraic manipulations, we have arrived at thefollowing values of *kf (0):

*kf (0)"

1, k"0

0, 1)k)¸

2(!1)k~L Ak!1

k!¸!1B, ¸#1)k)n

(6)

It is seen from (6) that ¸ forward differences of f (x) Dx/k@n

are zero at x"0. Therefore ¸ signifies the order offlatness of f (x) at x"0 in the Butterworth sense. Using the transformation

x"(1!cosu)/2 (7)

136 LETTERS TO THE EDITOR

( 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 135—139 (1997)

Table I. Weights ai, i"0, 1, 2,2 , 9, for n"9 and ¸"1 to 9 computed by using (10b). The normalizing factor is

2n ("29 here)

¸ a0

a1

a2

a3

a4

a5

a6

a7

a8

a9

1 !510 18 72 168 252 252 168 72 18 22 !492 144 432 672 504 0 !336 !288 !108 !163 !420 504 1008 672 !504 !1008 !336 288 252 564 !252 1008 1008 !672 !1512 0 1008 288 !252 !1125 0 1260 0 !1680 0 1512 0 !720 0 1406 252 1008 !1008 !672 1512 0 !1008 288 252 !1127 420 504 !1008 672 504 !1008 336 288 !252 568 492 144 !432 672 !504 0 336 !288 108 !169 510 18 !72 168 !252 252 !168 72 !18 2

in (5), we obtain

H (u)¢Bn(x) D

x/(1~#04u)@2, 0)u)n

"

n+k/0

*k f (0) An

kB A1!cosu

2 Bk

(8)

By using (6), we get from (8)

H (u)"1#n+

k/L`1

2(!1)k~L An

kB Ak!1

k!¸!1B A1!cosu

2 Bk

"1#n+

k/L`1

2(!1)k~L An

kB Ak!1

k!¸!1Bn+i/0

2~k(!1)i Ak

i B (cosu)i (9)

By simple manipulations, (9) can be reduced to the form

H (u)"n+i/0

ai(cosu)i (10a)

where

ai"2~n C2n A

0

iB#n+

k/L`1

(!1)k`i~L 2n`1~k An

kB Ak!1

¸ B Ak

iB, i"0, 1, 2,2 , n (10b)

with

A0

iB¢G1, i"00, otherwise

(10c)

Table I shows the weights computed using formula (10b) for n"9 with ¸"1 to 9. It is seen from (10) that fora given n there are only n possible different notch filters corresponding to ¸"1 to n discrete notchfrequencies, say u

1, u

2,2, u

n. The desired notch frequency u

$may not be exactly one of these n values in

general. In our previous work1 we suggested a methodology for obtaining the desired notch frequency u$by

linear combination of two (out of n) adjacent notch filters mentioned above. In the present context theprocedure gets slightly modified and is given here for completeness.

LETTERS TO THE EDITOR 137

( 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 135—139 (1997)

Figure 2. Frequency responses DH(u) D for ¸

1"10 (— · — ·) and ¸

2"11 (— — —) and final response (——) for example considered in

Section 4

3. DESIGN PROCEDURE

Problem

Given a specified notch frequency u$

and 3 dB rejection bandwidth B¼, we are required to designa maximally flat FIR notch filter.

Step 1. Obtain the required value of n by using the formula

n*integer M12

[(n/B¼)2!(n/B¼)#3]N. (11)

Step 2. Obtain ¸"¸1which results in a notch frequency u

L1closest to but less than u

$. The value of ¸

1is

found by using¸1"(n#1)!integer part of [n(0·55#0·5 cosu

$)] (12)

For ¸2¢¸

1#1 the corresponding notch frequency u

L2will obviously be closest to but greater than u

$.

Step 3. Compute the coefficients a(L1)i

and a(L2)i

by using (10b) with the value of n, ¸1

and ¸2

computed insteps 1 and 2.

Step 4. The weights of the desired notch filter are given by

ai"a a(L1)

i#(1!a) a(L2)

i(13a)

wherea"(u

L2!u

$)/(u

L2!u

L1) (13b)

The empirical formulae given in (11) and (12) were arrived at after modifying the ones proposed by Kaiserand Reed4 and Herrmann5 for maximally flat lowpass filters.

4. PERFORMANCE

The suggested formula (10b) takes much less time for computing the weights than the one proposed in (8) ofReference 1. A number of notch filters were designed by using the proposed formulae (10b), (11), (12) and (13)and these did give the desired results. Figure 2, for example, shows the frequency response of a notch filter

138 LETTERS TO THE EDITOR

( 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 135—139 (1997)

designed for the specific values u$"1·2 rad and B¼"0·38 rad. The computed values of n, ¸

1, ¸

2, u

L1,

uL2

and a are 31, 10, 11, 1·177783 rad, 1·245955 rad and 0·674 respectively. The realized notch frequency andB¼ are exactly 1·2 rad and 0·38 rad.

5. CONCLUSIONS

A simple and explicit mathematical formula for computing the weights of notch filters has been obtainedthrough the use of Bernstein polynomials. Its use in designing a notch filter with given specifications has beendemonstrated.

ACKNOWLEDGEMENTS

The authors thank the reviewers for their useful comments.

REFERENCES

1. S. C. Dutta Roy, S. B. Jain and B. Kumar, ‘Design of digital FIR notch filters’, Proc. IEE »is. Image Signal Process., 141, 334—338(1994).

2. A. V. Oppenheim, W. F. G. Mecklenbrauker and R. M. Mersereau, ‘Variable cutoff linear phase digital filters’, IEEE ¹rans. Circuitsand Systems, CAS-23, 199—203 (1976).

3. P. J. Davis, Interpolation and Approximation, Dover, New York, 1975.4. J. F. Kaiser and W. A. Reed, ‘Data smoothing using low-pass digital filters’, Rev. Sci. Instrum., 48, 1447—1457 (1977).5. O. Herrmann, ‘On the approximation problem in nonrecursive digital filter design’, IEEE ¹rans. Circuit ¹heory, CT-18, 411—413

(1971).

LETTERS TO THE EDITOR 139

( 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 135—139 (1997)