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744 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 6, JUNE 1999

Almost Equiripple FIR Half-Band Filters

Pavel Zahradnk, Miroslav Vlcek, and Rolf Unbehauen

AbstractBased on Chebyshev polynomials a novel analytical designprocedure of almost equiripple FIR half-band filters is developed. Theclosed form solution provides a direct computation of the frequencyresponse. A formula for the impulse response coefficients is also derived.Several examples are included.

Index TermsEquiripple, filter design, FIR half-band.

I. INTRODUCTION

Half-band filters play an important role in the design of filter banks.

A number of procedures for the design of linear phase FIR half-

band filters are recently available [6], [8]. The methods which lead

to feasible filters are generally derived by iterative approximation

techniques or by noniterative but still numerical procedures. The well-

known McClellanParks program [5] which is used in the design

of FIR filters for a wide application belongs to the former case,

while the windowing technique and frequency sampling method cover

standard noniterative procedures. Analytic methods are available for

the maximally flat FIR filters [1], [3], [7].In our paper we are primarily concerned with a completely ana-

lytical approach to the almost equiripple FIR half-band filter design.

The analytic technique is based on polynomials that are generated by

a function closely related to the frequency responses of Chebyshev

window functions, as emphasized in [11]. The term almost equiripple

expresses the fact that the frequency response of the filter exhibits

equiripple behavior with the exception of some ripples near the

transition band. An interesting feature of the ripples near the transition

band is the fact that they always have smaller amplitude compared

to the remaining ripples (see examples). Thus, the resulting filters

are slightly suboptimal. However, the advantage of the new method

over the numerical ParksMcClellan design procedure consists in the

fact that the coefficients of the impulse response are given by simple

explicit formulas. Note that the design process is an analytical oneand it does not require any DFT algorithm, nor do we need any

iterative technique, resulting in exceptionally fast design method.

Moreover, the design time is predictable and, for a given filter order,

the number of arithmetic operations is constant, making the design

method suitable in the adaptive filtering, as well.

II. FREQUENCY RESPONSE AND PSEUDOAMPLITUDE

Here and in the following we use the transformed variablew =

( 1 = 2 ) ( z + z

0 1

)

[10] which transforms thez

-plane onto a two-leaved

w

-plane so that the unit circle itselfj z j = 1

is mapped into the

Manuscript received July 2, 1997; revised August 21, 1998. This workwas supported in part by the A. von Humboldt Foundation. This paper was

recommended by Associate Editor V. Tavsanoglu.P. Zahradnk is with the Czech Technical University, Faculty of Elec-trical Engineering, Prague, Czech Republic. He is now on sabbatical atthe Lehrstuhl fur Allgemeine und Theoretische Elektrotechnik, UniversityErlangen-Nurnberg, D-91058 Erlangen, Germany.

M. Vlcek is with the Czech Technical University, Faculty of TransportationSciences, Prague, Czech Republic. He is now on sabbatical at the Lehrstuhl furAllgemeine und Theoretische Elektrotechnik, University Erlangen-Nurnberg,D-91058 Erlangen, Germany.

R. Unbehauen is with the Lehrstuhl fur Allgemeine und TheoretischeElektrotechnik, University Erlangen-Nurnberg, D-91058 Erlangen, Germany.

Publisher Item Identifier S 1057-7122(99)04743-1.

real interval0 1 w = c o s ! T 1

along which both leaves are

interconnected. LetH ( z )

denote the transfer function of an FIR filter

of the orderN 0 1

H ( z ) =

N 0 1

= 0

h ( ) z

0

:

(1)

Assuming an odd length of the impulse responseN = 2 ( 2 n + 1 ) + 1

and even symmetry of the impulse response coefficientsh ( ) =

h ( N 0 1 0 )

then by imposing the symmetry constraint

h ( 2 n + 1 ) = a ( 0 ) = 0 : 5

2 h ( 2 n + 1 0 2 m ) = a ( 2 m ) = 0

2 h ( 2 n + 1 0 2 m 0 1 ) = a ( 2 m + 1 )

(2)

we can write the half-band transfer function as

H ( z ) = z

0 ( 2 n + 1 )

1

2

+

n

m = 0

a ( 2 m + 1 ) T

2 m + 1

( w )

(3)

whereT

2 m + 1

( w )

are Chebyshev polynomials of the first kind. The

frequency responseH ( e

j ! T

)

is then of the form

H ( e

j ! T

) = e

0 j ( 2 n + 1 ) ! T

Q ( c o s ! T )

= e

0 j ( 2 n + 1 ) ! T

Q ( w )

(4)

whereQ ( w )

represents the real valued pseudoamplitude of the zero-

phase FIR filter of the real variable

w =

1

2

( z + z

0 1

)

z = e

= c o s ! T :

(5)

For the equiripple approximation [Fig. 1(a)] the pseudoamplitude

Q ( w )

[Fig. 1(b)] of the half-band filter has alternating local min-

ima and maximaw

0 m

symmetrically distributed over the stopband

( 0 1 ; 0 w

p

)

and passband( w

p

; 1 )

, respectively. It means that the

first derivative of the quantityQ ( w )

, as shown in Fig. 2, has real

zerosw

0 m

within these two disjoint intervals only. The half-band

symmetry imposed on the impulse response coefficients (2) implies

that the zeros of ( d = d w ) Q ( w ) are symmetrically distributed aboutthe origin

w = 0

. The first derivative of the pseudoamplitude can be

always factorized as

d

d w

Q ( w ) = ( 2 n + 1 ) a ( 2 n + 1 )

n

m = 1

[ U

2

( w ) 0 U

2

( w

0 m

) ] :

(6)

The distribution of alternating minima and maximaw

0 m

is not known

for the equiripple case. The equiripple approximation has a natural

limit when the distribution of the zerosw

0 m

is simplified to the set

of zeros which are confluent at6 1

, i.e.,w

2

0 m

= 1

, leading to the

maximally flat half-band filters [11].

III. DIFFERENTIAL EQUATION AND GENERATING FUNCTION

We assume the functionU

n +

( x )

defined as

U

n +

( x ) = U

n

( x ) + U

n 0 1

( x )

(7)

whereU

n

( x )

andU

n 0 1

( x )

are Chebyshev polynomials of the second

kind. The functionU

n +

( x )

fulfills the differential equation

( 1 0 x

2

)

d

2

d x

2

U

n +

( x ) 0 ( 2 x + 1 )

d

d x

U

n +

( x )

+ n ( n + 1 ) U

n +

( x ) = 0 :

(8)

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(a)

(b)

Fig. 1. (a) Amplitude frequency response j H ( e j ! T ) j . (b) PseudoamplitudeQ ( w ) .

The differential equation (8) is in fact a differential equation defining

Jacobis polynomials [2]. Considering relations

x =

2 w

2

01

0k

0 2

1

0k

0 2

; w =

1 + k

0 2

+ ( 1

0k

0 2

) x

2

(9)

the differential equation (8) can be converted to the form

( 1

0w

2

) ( w

2

0k

0 2

)

d

2

d w

2

U

n +

( w )

+

k

0 2

( 1

0w

2

)

03 w

2

( w

2

0k

0 2

)

w

d

d w

U

n +

( w )

+ 4 w

2

n ( n + 1 ) U

n +

( w ) = :

(10)

Further, we assign the derivative( d = d w ) Q ( w )

toU

n +

( w )

. Consid-

ering the same mapping of the variablew

as in the IIR equiripple

filter design [10], using Jacobis elliptic functions

w = d n ( u

jk )

(11)

we can write

2 w

2

01

0k

0 2

1

0k

0 2

= 2 c n

2

( u

jk )

01

(12)

Fig. 2. Derivative of the pseudoamplitude Q 0 ( w ) .

wherec n ( u

jk )

is a Jacobian elliptic function. Because of the multiple-

angle formula for Chebyshev polynomials [11] we obtain

U

2 n

( c n ( u

jk ) )

= U

n

( 2 c n

2

( u

jk )

01 ) + U

n 0 1

( 2 c n

2

( u

jk )

01 )

= U

n

2 w

2

01

0k

2

1

0k

2

+ U

n 0 1

2 w

2

01

0k

2

1

0k

2

:

(13)

We call the functionU

2 n

( c n ( u

jk ) )

the generating function and it

is of fundamental importance in our design procedure. Due to the

differential equations (8) and (9), the functionU

n +

( x )

is from the

family of Jacobi polynomials, namely

U

2 n

( c n ( u

jk ) ) =

n !

p

0 n +

1

2

P

( 1 = 2 ; 0 ( 1 = 2 ) )

n

( 2 c n

2

( u

jk )

01 )

(14)

whereP

( ; )

n

( x )

are Jacobis polynomials.

The zeros of the generating functionU

2 n

( c n ( u

jk ) )

and, conse-

quently, the extremes of the pseudoamplitudeQ ( w )

of the almost

equiripple half-band filter are distributed along thew

axis as follows:

w

2

0 m

= k

2

+ k

2

c o s

2

m

2 n + 1

; m = 1 ;

1 1 1; n :

(15)

The distribution of alternating minima and maximaw

0 m

is not known

for the equiripple case. However, we have found that the distribution

of local extremes (15) approximates with very good accuracy the set

of local extremes of an equiripple case (see examples) resulting inthe almost equiripple frequency response of the filter.

IV. PSEUDOAMPLITUDE AND IMPULSE RESPONSE

The pseudoamplitudeQ ( w )

of the filter is based on the polynomi-

als K2 n

( u

jk )

. We define the polynomials K2 n

( u

jk )

by straightfor-

ward integration of the generating functionU

2 n

( c n ( u

jk ) )

K

2 n

( u

jk ) = U

2 n

( c n ( u

jk ) ) d ( d n ( u

jk ) ) :

(16)

According to the results obtained in [11] for the discrete prolate

window function, we can developU

n

( ( 2 w

2

01

0k

2

) = ( 1

0k

2

) )

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(a)

(b)

Fig. 3. (a) Graph 1 = 1 ( wp

; 2 n ) . (b) Graph ( 2 n + ) k 0 =( 2 n + ) k

0

( w

p

; 2 n ) .

in form

U

n

2 w

2

0 1 0 k

0 2

1 0 k

0 2

=

n

= 0

( 0 1 )

n + 1 +

n 0

2

( 1 0 k

0 2

)

0

+

n

m = 1

( 0 1 )

m

n

= m

( 0 1 )

n + 1 +

n 0

1

2

0 m

+

2

+ m

( 1 0 k

2

)

0

T

2 m

( w ) :

(17)

Using identity2 T

2 m

( w ) = U

2 m

( w ) 0 U

2 m 0 2

( w )

we can conclude

that

U

n

2 w

2

0 1 0 k

2

1 0 k

2

+ U

n 0 1

2 w

2

0 1 0 k

2

1 0 k

2

=

n

m = 0

( 0 1 )

m

n

= m

( 0 1 )

2 n + 1

2 + 1

n +

n 0

1

2 + 1

0 m

( 1 0 k

2

)

0

U

2 m

( w )

=

n

m = 0

( 0 1 )

m

c ( 2 m + 1 ) U

2 m

( w )

(18)

TABLE IEXTREMES

TABLE IICOEFFICIENTS OF THE IMPULSE RESPONSE

where coefficients

c ( 2 m + 1 )

=

n

= m

( 0 1 )

2 n + 1

2 + 1

n +

n 0

2 + 1

0 m

( 1 0 k

2

)

0

(19)

constitute, in fact, the impulse response coefficients of almost equirip-

ple half-band filter. Introducing (18) into the integral representation

of polynomials (16) we obtain the formula

K

2 n

( u j k ) =

n

m = 0

( 0 1 )

m

c ( 2 m + 1 ) U

2 m

( w ) d w

=

n

m = 0

( 0 1 )

m

c ( 2 m + 1 )

2 m + 1

T

2 m + 1

( w ) :

(20)

The pseudoamplitudeQ ( w )

is given by the normalized polynomial

K

2 n

( u j k )

Q ( w ) =

1

2

+

K

2 n

( u j k )

n o r m

=

1

2

+

K

2 n

( u j k )

K

2 n

( 1 ) + K

2 n

( w

0 1

)

(21)

wherew

0 1

denotes the extreme of the polynomialK

2 n

( u j k )

closest

to the band edgew = 1

given by

w

0 1

= k

2

+ k

2

c o s

2

2 n + 1

:

(22)

Finally, for the impulse responseh ( )

holds

h ( 2 n + 1 ) =

1

2

h [ 2 n + 1 6 ( 2 m + 1 ) ] =

( 0 1 )

m

2 ( 2 m + 1 )

c ( 2 m + 1 )

K

2 n

( 1 ) + K

2 n

( w

0 1

)

form = 0 ; 1 1 1 ; n

h [ 2 n + 1 6 ( 2 m ) ] = 0 ;

form = 1 ; 1 1 1 ; n

(23)

where = 0 ; 1 1 1 ; N 0 1

.

V. FILTER DESIGN

As in the numerical ParksMcClellan design procedure, in our

analytical design the degree equation is not known. Despite this fact,

the secondary design parametersn

,k

can be obtained from the

filter specificationsw

p

,

, using graphs or approximating formulas

with very good accuracy. From the graph in Fig. 3(a), for given

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(a)

(b)

Fig. 4. (a) Amplitude frequency responses j H ( e j ! T ) j and j Ho p t

( e

j ! T

) j .(b) Logarithmic amplitude frequency responses 2 0 l o g j H ( e j ! T ) j and2 0 l o g j H

o p t

( e

j ! T

) j .

1 = 2 0 l o g ( )

andw

p

the appropriate2 n

can be obtained. From

the graph in Fig. 3(b) for givenw

p

and previously obtained2 n

, the

k

0 can be estimated. The approximating formulas

2 n = 0

w

p

(

1

+

2

) +

3

0 1

2

1

w

p

0

[ w

p

(

1

+

2

) +

3

0 1 ]

2

0 4

1

(

4

0 1 ) w

p

2

1

w

p

(24)

k

0

=

1

2 n +

2

2 n +

w

p

+

( 2 n + 2 ) (

3

2 n +

4

)

(25)

1 = 2 0 l o g ( ) = (

1

2 n +

2

) w

p

+

3

2 n +

4

2 n +

(26)

with the constants

1

= : 0 0 5 5

,

2

= 0 0 : 4 8 6 9

,

3

= 0 : 0 2 4

,

4

= 0 : 7 3 3 0

,

1

= 0 9 : 7 6 5

,

2

= 0 2 : 0 9 0 2

,

3

= 0 9 : 2 7 6 8

,

and

4

= 3 : 9 8 3 7

were deduced by linear approximation of the

graphs for0 : w

p

0 : 3

and4 2 n 3 4

. Outside of these

limits, the approximating formulas are less accurate, especially (26)

for low-degree filters.

TABLE IIIEXTREMES

TABLE IVCOEFFICIENTS OF THE IMPULSE RESPONSE

The filter design procedure consists of the following steps.

For the givenw

p

and1

, the number2 n

is obtained (using a

formula or graph).

Fromw

p

and2 n

the appropriatek

0 is obtained.

For the values2 n

andk

0 the impulse response of the filterh ( )

is evaluated.

VI. EXAMPLES OF THE DESIGN

Example 1: Design a half-band lowpass filter with the desired

values!

p

T = 0 : 4 5

and1 = 0 3 0

dB.

From!

p

T = 0 : 4 5

followsw

p

= 0 : 5 6 4

. From (24) results

2 n = 2 : 8 0 6 8 . From (25) for adjusted 2 n = 4 (N = 3 ) followsk

0

= 0 : 2 0 0

. The extremesw

0 m

of the pseudoamplitudeQ ( w )

of the filter are summarized in the Table I. The coefficients of the

impulse responseh ( )

of the filter are given in the Table II. The

amplitude frequency responsej H ( e

j ! T

) j

is shown in Fig. 4(a), and

the logarithmic amplitude frequency response is shown in Fig. 4(b).

The actual parameters of the proposed filter are!

p

T = 0 : 4 4 9 5

and

1 = 0 3 2 : 0 0 4

dB. The filter was compared with the truly equiripple

(optimal) half-band FIR filter of the same degree and the same ripple,

designed using the Remez exchange algorithm [5] combined with

the half-band trick [9]. The extremesw

0 m

of its pseudoamplitude

Q

o p t

( w )

are given in Table I. The coefficients of the corresponding

impulse responseh

o p t

( )

are given in Table II. The amplitude

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(a)

(b)

Fig. 5. (a) Amplitude frequency responses j H ( e j ! T ) j and j Ho p t

( e

j ! T

) j .(b) Logarithmic amplitude frequency responses 2 0 l o g j H ( e j ! T ) j and2 0 l o g j H

o p t

( e

j ! T

) j .

frequency responsej H

o p t

( e

j ! T

) j

is shown in Fig. 4(a), and the

logarithmic amplitude frequency response in Fig. 4(b) (dashed). The

comparison of the filters shows that while the passband frequency

in the proposed case amounts!

p

T = 0 : 4 4 9 5

, in the optimal case

amounts!

p

T = 0 : 4 5 2 9

, i.e., the passband frequency of the

proposed filter differs by 0.75% with respect to the optimal case.

Example 2: Design a half-band lowpass filter with desired values

!

p

T = 0 : 4 2 7

and1 = 0 8 0

dB. From!

p

T = 0 : 4 2 7

we getw

p

= 0 : 2 2 7 3

. From (24)2 n = 3 : 7 9 7 6

results. From (25) for an

adjusted2 n = 3 2

(N = 6 7

)k

0

= 0 : 2 3 7 9

follows. The extremes

w

0 m

of the pseudoamplitudeQ ( w )

are summarized in Table III.

Coefficients of the impulse responseh ( )

are given in Table IV.

The amplitude frequency responsej H ( e

j ! T

) j

is shown in Fig. 5(a)

and the logarithmic amplitude frequency response in Fig. 5(b). The

actual parameters of the proposed filter are!

p

T = 0 : 4 2 7

and

1 = 0 8 0 : 2 3 6

dB. Along with the proposed filter, an optimal filter of

the same degree and of the same ripple was designed. Its parameters

are presented in the same manner as in the previous example. The

passband frequency of the optimal filter amounts!

p

T = 0 : 4 3 0 6

,

meaning that the passband frequency of the proposed filter differs by

0.82% with respect to the optimal case.

REFERENCES

[1] T. Cooklev, S. Samadi, A. Nishihara, and N. Fujii, Efficient implemen-tation of all maximally flat FIR filters of a given order, Electron. Lett.,

vol. 29, no. 7, pp. 598599, Apr. 1, 1993.[2] G. Szego, Higher Orthogonal Polynomials. Providence, RI: American

Mathematical Society, 1939, ch. IV.[3] D. Herrmann, On the approximation problem in nonrecursive digital

filter design, IEEE Trans. Circuit Theory, vol. CT-18, pp. 411413,May 1971.

[4] B. L. Jackson, On the relationship between digital Hilbert transform-ers and certain low-pass filters, IEEE Trans. Acoust., Speech, SignalProcessing, vol. ASSP-23, pp. 381383, Aug. 1975.

[5] J. H. McClellan, T. W. Parks, and L. R. Rabiner, A computer programfor designing optimum FIR linear phase digital filters, IEEE Trans.

Audio Electroacoust., vol. AU-21, pp. 506526, Dec. 1973.[6] F. Mintzer, On half-band, third-band and n th-band filters and their

design, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30,pp. 734738, Dec. 1982.

[7] R. J. Rajagopal and S. C. Dutta Roy, Design of maximally-flat FIRfilters using the Bernstein polynomials, IEEE Trans. Circuits Syst..,

vol. CAS-34, pp. 15871590, Dec. 1987.[8] P. P. Vaidyanathan, Efficient and multiplierless design of FIR filter

with very sharp cutoff via maximally flat building blocks, IEEE Trans.Circuits Syst., vol. CAS-32, pp. 236244, Mar. 1985.

[9] P. P. Vaidyanathan and T. Q. Nguyen, A TRICK for the designof FIR half-band filters, IEEE Trans. Circuits Syst., vol. CAS-34, pp.297300, Mar. 1987.

[10] M. Vlcek and R. Unbehauen, Analytical solution for design of IIRequiripple filters, IEEE Trans. Acoust., Speech, Signal Processing, vol.37, pp. 15181531, Oct. 1989.

[11] , Note to the window function with nearly minimum sidelobeenergy, IEEE Trans. Circuits Syst., vol. 37, pp. 13231324, Oct. 1990.

Global Exponential Stability of a Class of Neural Circuits

Xue-Bin Liang and Li-De Wu

Abstract This paper obtains the global exponential stability (GES)of the class of HopfieldTank neural circuits, which can represent ageneralization of the existing stability results in the sense that only theglobal asymptotic stability (GAS) of the neural circuits was obtainedunder existing sufficient conditions in the literature. An example of aneural circuit which is globally asymptotically stable (GAS) rather thanglobally exponentially stable (GES) is also given.

Index TermsGlobal exponential stability, HopfieldTank, neural cir-

cuits.

I. INTRODUCTIONRecently, there has been considerable attention in the literature

(see, e.g., [1][13]) in the qualitative analysis of the class of Hop-

Manuscript received August 24, 1995; revised June 15, 1998. This workwas supported in part by the National Natural Science Foundation of Chinaunder Grant 69702001. This paper was recommended by Associate Editor A.Kuh.

The authors are with the Department of Computer Science, Fudan Univer-sity, Shanghai 200433, China.

Publisher Item Identifier S 1057-7122(99)04749-2.

10577122/99$10.00 1999 IEEE