64 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1998

Transactions Brief

Design of FIR Hilbert Transformers andDifferentiators in the Complex Domain

Michael Z. Komodromos, Steve F. Russell, and Ping Tak Peter Tang

Abstract—This paper presents a method for the design of FIR Hilberttransformers and differentiators in the complex domain. The methodcan be used to obtain conjugate–symmetric designs with smaller groupdelay compared to linear-phase designs. Non-conjugate symmetric Hilberttransformers are also designed. This paper is an extension of our previouswork [1], which presented the algorithm for the design of standardfrequency selective filters. The minimax criterion is used and the Cheby-chev approximation is posed as a linear optimization problem. Theprimal problem is converted to its dual and is solved using an efficientquadratically convergent algorithm developed by Tang [2]. When aconstant group delay is specified, the filter designs have almost linearphase in the passbands. When the specified group delay is half the filterlength, the algorithm results in exactly linear-phase designs.

Index Terms— Chebychev approximation, differentiators, filters,Hilbert transform, Remez algorithm.

I. INTRODUCTION

Recently, a new method has been presented for the design of FIRfilters in the complex domain [1]. The method can approximate a fre-quency response with specification of both the phase and magnitudefunctions. In [1], the algorithm was presented and its employmentfor the design of frequency selective filters was demonstrated. In thispaper, we discuss the method for the design of Hilbert transformersand differentiators.

The Parks–McClellan algorithm [3], [4] is the most commonmethod used to design FIR Hilbert transformers and differentiators.Their widely used program produces conjugate-symmetric linear-phase designs based on Chebychev approximation and the real Remezexchange algorithm. This method is popular because it is efficientand gives optimal designs in the Chebychev sense with constantgroup delay, which is one half the filter length. However, onlyconjugate–symmetric linear-phase designs can be obtained with thismethod.

Our approach to designing Hilbert transformers and differentiatorsis to use constrained weighted complex approximation. The methodresults in optimal designs in the minimax sense. It can be usedto obtain designs with smaller group delay compared to linear-phase designs. Also, by using complex approximation, nonconjugatesymmetric responses can be approximated that result in complex coef-ficients. Later we examine the case of one-sided Hilbert transformers.The design of FIR filters in the complex domain was considered in[1] for the design of frequency selective filters. We base the designof Hilbert transformers and differentiators on that work.

Recently, another approach was presented for the design of bothFIR and IIR Hilbert transformers by Kollaret al. [5]. Their approachis based on a parameter estimation method for linear systems. A

Manuscript received July 7, 1993; revised June 14, 1994 and December12, 1995. This paper was recommended by Associate Editor I. Pitas.

The authors are with the Department of Electrical and ComputerEngineering, Iowa State University, Ames, IA 50011 USA (e-mail:www.public.iastate.edu/sfr.html).

Publisher Item Identifier S 1057-7122(98)01573-6.

first approximation to a frequency response is performed in the leastsquares sense in the complex domain. An extension of this method isto weight the approximation such that a solution close to the minimaxsolution is obtained. When compared to the linear-phase minimaxdesigns, the Hilbert transformers in [5] have a smaller error in themiddle of the frequency range but a larger error at the edges of thepassband. Another approach to design FIR Hilbert transformers anddifferentiators was proposed by Pei and Shyu [6], [7] using the ideaof the eigenfilter. The method minimizes a quadratic measure of theerror in the passband and stopband.

Other authors have considered the design of FIR filters in thecomplex domain. One of these methods was developed by Chenand Parks [8]. Their approach was to convert the primal complexapproximation problem into a real approximation problem which,in itself, approximates the complex problem. A standard linearprogramming algorithm was used to solve the real approximationproblem. The major disadvantage of the method of Chen and Parks isthat the problem they solve is a discrete approximation to the originalproblem. Also, as pointed out in [8], the execution time increasessignificantly as the discretization is made finer. Our technique doesnot have this limitation.

Preuss [9] derived a generalized one-point Remez exchange al-gorithm that deals with the complex error directly. It uses Newtoninterpolation for the error and replaces the old set of extremalfrequencies with a new set where the error is maximum. For eachiteration the angles of the previous interpolation are used and themagnitudes are adjusted using a control constant. As mentioned in[9], large values of this factor may cause the procedure to diverge.The algorithm terminates when the magnitude deviations of the errorat the extremal frequencies are equal within the prescribed tolerance.In Preuss’ algorithm, there appears to be no theoretical proof thatthe algorithm is convergent or, when it does, that the obtainedapproximation is optimal in some sense. Schulist [10], suggesteda modified Preuss algorithm that improved the convergence of thealgorithm.

In this paper, we present the design of Hilbert transformers anddifferentiators in the complex domain based on a complex approx-imation algorithm by Tang [2], [11] and the filter design methodpresented in [1]. The algorithm solves the dual formulation of thecomplex approximation problem without any need of discretization.Furthermore, it has been proven to quadratically converge to thebest approximation. In the following sections, we will formulate thegeneral FIR filter design problem in the complex domain and describethe algorithm. More details on the formulation and the algorithm canbe found in [1] and [2]. Next, we describe the design of Hilberttransformers and differentiators using the complex algorithm. Finally,design examples are presented.

II. PROBLEM FORMULATION

The transfer function of an FIR filter is given by

H(z) =

N�1

k=0

hk z�k (1)

where the frequency responseH(f) is obtained by evaluatingH(z)on the unit circlez = ej2�f . We treat both cases of real and

1057–7122/98$10.00 1998 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1998 65

complex impulse responses using the same form of the filter transferfunction. As discussed in [1], the general frequency response for theapproximating FIR filter of lengthn is given by

H(f) =

N�1

k=0

hk bk(f) (2)

where, for the real coefficient case,N = n and the basis functions are

bk(f) = e�j2�fk

k = 0; � � � ; n� 1 (3)

and for the complex coefficient caseN = 2n

bk(f) = e�j2�fk

bk+n(f) = je�j2�fk (4)

for k = 0; � � � ; n � 1. The complex impulse response of the filteris given by

hkc = hk + jhn+k k = 0; � � � ; n� 1: (5)

We define the complex error between the desired response and theapproximating FIR filter by

E(f) =W (f)[D(f)�H(f)]; f 2 F (6)

whereF (the frequency domain of approximation) isF � [0; 0:5]

in the real coefficient case andF � [�0:5; 0:5] in the complexcoefficient case. The positive real-valued weight functionW (f) isintroduced to control the relative magnitude of the complex error inthe different frequency bands. The complex approximation problemis the following. FindN real coefficientshhh0 = [h00; h

01; � � � ; h

0

N�1]

such that

maxf2F

W (f) D(f)�

N�1

k=0

h0

k bk(f)

� maxf2F

W (f) D(f)�

N�1

k=0

hk bk(f) (7)

for all hhh = [h0; � � � ; hN�1]T in IR

N . The absolute error is smallerfor the frequency bands with larger weighting. The weight functioncan be merged with the desired response and the approximatingfunction.

An equivalent formulation of the approximation problem is to findoptimal variables�; h1; � � � ; hN�1, to minimize the linear function� subject to the constraints

D(f)�

N�1

k=0

hk bk(f) � � (8)

for all f 2 F . The constraints are equivalent to

N�1

k=0

hkRefbk(f)e�j�

g+ � � RefD(f)e�j�

g (9)

for all f 2 F and for all� 2 [0; 2�] [1]. The primal approximationproblem is converted to its dual equivalent. The dual of the primalproblem is defined as follows [1], [11], [12]. Define the(N +1)� 1

vectorsfff 2 FN+1, � 2 [0; 2�]N+1, andrrr 2 [0; 1]N+1. Find thevectorsfff; �; rrr, to maximize the dual variabled

d = cccT� rrr (10)

subject to the constraints

AAA � rrr = uuu1 (11)

where

cccT= [RefD(f1)e

�j�g � � � RefD(fN+1)e

�j�g] (12)

and thekth column of the(N + 1)� (N + 1) matrix AAA is

[1 Refb0(fk)e�j�

g � � � RefbN�1(fk)e�j�

g]T: (13)

The vectoruuu1 is the (N + 1 � 1) coordinate column vector with afirst element of one and the rest zero.

The relation between the primal problem and the dual problemis that the optimal (minimized) error of the primal is the same asthe optimal (maximized) error of the dual. Furthermore, it can beshown [13] that the impulse response and the dual variable can becalculated by

[d; hhh]T�AAA = ccc

T: (14)

III. COMPLEX REMEZ ALGORITHM

The complex Remez algorithm solves the dual problem. SupposeH(hhh0; f), f 2 F is a best approximation to a complex-valued desiredfrequency responseD(f) and letd0 � kE(hhh0; f)k be the error ofthe best approximation. Thend0 is the optimal value of the followingconstrained maximization problem: maximized = cccT � rrr over allfff 2 FN+1, ��� 2 [0; 2�]N+1, and rrr 2 [0; 1]N+1 subject to theconstraints

AAA � rrr = uuu1: (15)

The algorithm for the determination of the best FIR filter is asfollows.

STEP 1: Pick a set of initial frequency points and angles(fff; ���),fff 2 FN+1, ��� 2 [0; 2�]N+1 such that

rrr = AAA�1

� uuu1 � 0: (16)

The initial set has to be chosen in a way that thematrix AAA is nonsingular so its inverse exists. This isusually the case when the initial set is chosen at random.From (fff; ���), calculate the dual variabled and a set ofcoefficientshhh using (14).

STEP 2: Terminate the algorithm if the value of the dual variabled = cccT � rrr is within a certain tolerance ofkE(hhh; f)k,the Chebychev norm of the error between the desiredresponse and the approximating FIR filter.

STEP 3: Update (fff; ���) and the weighting vectorrrr (withoutviolating any of the constraints) to increase the valueof d = cccT � rrr. Return toSTEP 2and repeat the processuntil the algorithm terminates. The one-point exchangeis performed on the frequency points and angles of thecurrent iteration. First, the error is calculated on thedomain of approximation and the frequencyfnew, andangle �new at which the maximum magnitude of theerror is attained are located. This pair (fnew, �new)is then used to replace an appropriate pair in the oldextremal set. The details of the algorithm are discussedin [1], [2], and [13].

The computational complexity of the algorithm isO(N3) per

sweep where a sweep is equivalent toN iterations. For the design ofHilbert transformers and differentiators, four to five sweeps suffice togive an approximation within 1% of the true best solution. The designalgorithm was implemented with programs written in FORTRAN andMATLAB. The domain of approximation is the union of disjointpassbands and stopbands while transition bands are not specified.The input to the programs include the normalized frequency bandedges, the value of the magnitude in each band, the weighting foreach band, and the desired group delay in the passband. More detailson the implementation and complexity of the algorithm are discussedin [1], [11], and [13].

66 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1998

Fig. 1. Group delay plot for Example 1.

IV. HILBERT TRANSFORMERS

The frequency response of an ideal Hilbert transformer is given by

Hideal(f) =�j; f 2 BBBp f > 0j; f 2 BBBp f < 0

(17)

wheref 2 F andBp is the passband. The ideal Hilbert transformer isperiodic inf with period 1 and it is discontinuous at 0, 0.5,�0.5. Toavoid large errors in both the magnitude and phase responses, a groupdelay must be specified. Also, since the frequency response for thetwo-sided Hilbert transformer is conjugate symmetric, its magnitudeand phase need to be specified only for positive frequencies. Thedesired frequency response of a Hilbert transformer is then given by

D(f) =e�j(2�f� +�=2); f 2 BBBp

0; f 2 BBBs

: (18)

When the specified group delay�d is an integer, the Hilbert trans-former is not continuous atf = 0, 0.5, and�0.5. This is because thephase function is discontinuous at these points. The desired frequencyresponse can be made continuous atf = �0.5 andf = 0.5 if weallow a noninteger group delay of the form�d = �c + 0:5 where�c is an integer.

Example 1—Wide-Band Hilbert Transformer:This example is awide-band Hilbert transformer with length 46 and passband [0.04,0.5]. An exhaustive search using different specifications of groupdelay has shown that the best magnitude error is achieved whenthe group delay is specified to be one-half the filter length. TheChebychev error for the linear-phase design is 0.001 08. When thegroup delay is specified to be 15.5 samples the error is 0.00140. Thegroup delay for this case is shown in Fig. 1 with maximum deviationof 0.05 samples in the passband. The Chebychev error increasesalmost linearly with decreasing specified delay. For a specified delayof 4.5 samples the error is 0.0105. The algorithm for this exampleconverged within 0.1% of the best solution in 226 iterations (aboutfive sweeps) and took 2 min and 10 s on a 66-MHz IBM-compatiblepersonal computer.

Example 2—One-Sided Hilbert Transformer:When the Hilberttransformer needs to be one-sided the desired response must bespecified in the interval [�0.5, 0.5] since it is not conjugatesymmetric. The impulse response coefficients are complex numbers.The desired response of the one-sided Hilbert transformer withnonzero response only for positive frequencies is defined as

D(f) =e�j(2�f� +�=2); f 2 BBBp

0; f 2 BBBs

0; f < 0

: (19)

Since the Hilbert transformer is periodic and a stopband is definedfor negative frequencies, a small stopband should be defined in the

Fig. 2. Magnitude in decibels of the one-sided Hilbert transformer of Ex-ample 2.

Fig. 3. Zeros of the one-sided Hilbert transformer of Example 2.

vicinity of f = 0.5 to allow discontinuity there. The example wepresent here has stopbands in the intervals [�0.5, 0.002] and [0.498,0.5] and a passband in the interval [0.04, 0.46]. The length of thecomplex impulse response is 22 and the weighting is1 : 1 : 1. Agroup delay of ten samples is specified. The magnitude in decibels isshown in Fig. 2. The optimal error is 0.0891 resulting in a passbandripple of 0.741 dB and stopband attenuation of 21 dB. Fig. 3 showsthe zeros of the design. Note that the zeros do not possess anysymmetries since the coefficients are complex and the phase is notlinear. The algorithm converged within 0.1% of the best solution in243 iterations.

V. DIFFERENTIATORS

Differentiators are used to obtain samples of the derivative of abandlimited signal from the samples of that signal. For a signals(t)and its Fourier transformS(f) the following relation exists:

ds(t)

dt$ (j2�f)S(f): (20)

Thus, the Fourier transform of the derivative of a signal is (j2�f)times the Fourier transform of the signal. Therefore, the desiredfrequency response of the differentiator is given by

D(f) =2�fe�j(2�f� ��=2); f 2 BBBp

0; f 2 BBBs:(21)

For a full-band differentiator, the desired response is not continuousat f = �0:5 and f = 0:5 when the group delay is an integer. Asbefore, it can be made continuous by specifying a group delay of theform �d = �c + 0:5 where�c is an integer. To avoid a large groupdelay error in the vicinity off = 0, an inverse weighting function

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 1, JANUARY 1998 67

Fig. 4. Magnitude response of the full-band differentiator of Example 3.

Fig. 5. Group delay of the full-band differentiator of Example 3.

of the form

W (f) =1

f +�f; f 2 F (22)

is used to force the error to be zero atf = 0 and make the filterresponse zero there. The value of�f is small and is used to avoiddividing by zero atf = 0. The use of inverse weighting has theeffect of causing the Chebychev error to increase starting atf = 0

and having the largest value atf = 0:5.Example 3—Full-Band Differentiator:This example is a full-band

differentiator of order 31 using the design specifications in [7]. In[7], a quadratic measure of the error is minimized. The passbandis defined on [0, 0.5] and the specified group delay is chosen tobe 11.5 samples. The algorithm converged within 0.1% of the bestapproximation in 92 iterations (three sweeps) and the Chebychev errorof the approximation is 0.0213. The magnitude response is shownin Fig. 4 and the group delay is shown in Fig. 5. The group delayvaries between 11.4 and 11.6 samples. In [7], the Chebychev erroris reported to be 0.0507 and the group delay varies between 11.46and 11.54. Our design gives better magnitude response and slightlyworse group-delay response.

VI. CONCLUSIONS

This paper presented a new design method for FIR Hilbert trans-formers and differentiators. The linear-phase restriction has beenremoved using complex approximation. More importantly, the con-jugate symmetry restriction is also removed. This gives enormousfreedom in specifying arbitrary magnitude and phase responses. Thecomplex approximation problem is formulated as a linear optimiza-tion problem and the dual is solved using Tang’s algorithm. Thedesign algorithm results in optimal designs in the Chebychev sense

and guarantees convergence. In most cases, the algorithm convergesin less than five sweeps for approximations within 0.1% of the bestapproximation. The method does not require any discretization and,therefore, the problem to be solved does not become excessivelylarge for long filters.

We discussed the design of conjugate symmetric Hilbert trans-formers with lower group delay than one half the filter length.The deviation of the group delay from a constant is less thatone sample in most cases, but the magnitude error is slightlylarger compared to the linear-phase design of the same length.An example of a nonconjugate symmetric Hilbert transformer withnonzero response only for positive frequencies was also discussed.The transformer necessarily has complex coefficients. Finally, a wide-band differentiator was discussed. It is important to note that nostopband needs to be specified at the frequency-domain endpointssince our method allows specification of noninteger group delays.

REFERENCES

[1] M. Z. Komodromos, S. F. Russell, and P. T. P. Tang, “Design of FIRfilters with complex desired frequency response using a generalizedRemez algorithm,”IEEE Trans. Circuits Syst.,to be published.

[2] P. T. P. Tang, “A fast algorithm for linear complex Chebychev approx-imations,” Math. Computat.,vol. 51, pp. 721–739, Oct. 1988.

[3] J. H. McClellan, T. W. Parks, and L. R. Rabiner, “A computer programfor designing optimum fir linear phase digital filters,”IEEE Trans. AudioElectroacoust.,vol. AU-21, pp. 506–526, Dec. 1973.

[4] J. H. McClellan and T. W. Parks, “A unified approach to the design ofoptimum FIR linear-phase digital filters,”IEEE Trans. Circuit Theory,vol. CT-20, pp. 697–701, May 1973.

[5] I. Kollar, R. Pintelon, and J. Schoukens, “Optimal FIR and IIR Hilberttransformer design via LS and minimax fitting,”IEEE Trans. Instrum.Meas,vol. 39, pp. 847–852, Dec. 1990.

[6] S. Pei and J. Shyu, “Design of FIR Hilbert transformers and dif-ferentiators by eigenfilter,”IEEE Trans. Circuits Syst.,vol. 35, Nov.1988.

[7] , “Eigen-approach for designing FIR filters and all-pass phaseequalizers with prescribed magnitude and phase response,”IEEE Trans.Circuits Syst. II,vol. 39, Mar. 1992.

[8] X. Chen and T. W. Parks, “Design of FIR filters in the complex domain,”IEEE Trans. Acoust., Speech, Signal Processing,vol. ASSP-35, pp.144–153, Feb. 1987.

[9] K. Preuss, “On the design of FIR filters by complex Chebychevapproximation,”IEEE Trans. Acoust., Speech, Signal Processing,vol.37, pp. 702–712, May 1989.

[10] M. Schulist, “Improvements of a complex FIR filter design algorithm,”Signal Process.,vol. 20, pp. 81–90, May 1990.

[11] P. T. P. Tang, “Chebychev approximation on the complex plane,” Ph.D.dissertation, Dept. Math., Univ. California at Berkeley, May 1987.

[12] D. G. Luenberger,Linear and Nonlinear Programming,2nd ed. Read-ing, MA: Addison-Wesley, 1984.

[13] M. Z. Komodromos, “Optimal FIR filter design,” Ph.D. dissertation,Dept. Elect. Eng. Comput. Eng., Iowa State Univ., Ames, Dec. 1993.