274 IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 2, MAY 2005

A Simple Derivation of the SpectralTransformations for IIR Filters

S. C. Dutta Roy, Fellow, IEEE

Abstract—A simple method is given for deriving the spectraltransformations for infinite-impulse response (IIR) filters, whichcan be used to transform a prototype low-pass (LP) digital filterto another LP, high-pass (HP), bandpass (BP), or band-stop(BS) digital filter with prescribed passband edge(s) and the sametolerances as those of the prototype. The method is based on acombination of bilinear transformation with the analog frequencytransformation and is simpler—conceptually, as well as from thecalculation point of view—than the conventional method based onall-pass transformation functions.

Index Terms—Digital filters, digital signal processing, infinite-impulse response (IIR) filters, spectral transformation.

I. INTRODUCTION

THE problem considered in this paper is the following:given a prototype infinite-impulse response (IIR) digital

low-pass filter (DLPF) with the transfer function , one isrequired to obtain the unique transformation so that

represents the transfer function of a low-pass(LP), high-pass (HP), bandpass (BP), or a band-stop (BS)digital filter with prescribed passband edge(s) and the same tol-erances as those of the prototype. The solution to the problem,briefly described hereafter, was given by Constantinides [1] in1970.

The requirements on are as follows:

1) the function must be real and rational, if so required;2) points inside the unit circle in the plane must be mapped

into points inside the unit circle in the plane to guaranteestability;

3) points on the unit circle in the plane must be mappedinto points on the unit circle in the plane.

Combining these requirements with the maximum modulus the-orem, the function must then be an all-pass one, of the form

(1)

where is real if 1, as required for LP-to-LP/HP trans-formation; and if 2, as required in LP-to-BP/BStransformation. For each case, one observes the correspondencebetween the passband edge(s) of the prototype DLPF and those

Manuscript received September 14, 2004.The author is with the Department of Electrical Engineering, Indian Institute

of Technology, New Delhi 110016, India (e-mail: scdroy@ee.iitd.ernet.in).Digital Object Identifier 10.1109/TE.2004.842913

of the required filter and thereby evaluates the value(s) of . Forexample, for LP-to-BP transformation, one has to use

(2)

where and are real, with the following correspondences:

(3)

In (3), is the center frequency of the digital bandpass filter(DBPF); and are its cutoff frequencies; is the cutofffrequency of the prototype DLPF; and the signs may not cor-respond to each other. The last entry in (3) dictates that the signin (2) should be negative. In the second entry, however, whichsign on the left-hand side will correspond to which one on theright-hand side is to be determined by trial and error. Even afterfinding the correct correspondences, a considerable amount oftrigonometric and algebraic manipulations are needed to arriveat the final neat result, as given in [1]. Note that the first entryin (3) is not required for finding and ; this point will be dis-cussed subsequently.

Since the publication of [1], the digital spectral transforma-tions have formed an integral part of any course on digital signalprocessing (DSP). A scan of a large number of text/referencebooks on DSP reveals that many of them only cite the Table ofTransformations without any derivation [2]–[10], while someof them derive the relatively easy LP-to-LP transformation[11]–[14] or the LP-to-HP one [15]. As exceptions, Antoniou[16] derives the LP-to-LP and BS transformations, while Bose[17] derives LP-to-LP, -HP, and -BP transformations and leavesthe LP-to-BS case to the reader to derive. Wherever derivationsare given, the method followed is that of [1] or some minormodification of the same.

Well recognized in the literature are the two routes for de-signing IIR digital filters, one using analog-to-digital, and theother using digital-to-digital frequency transformations, as illus-trated in Fig. 1, and succinctly given in Mitra [13, p. 437], whichyield the same result for the desired digital filter. Therefore,the digital-to-digital frequency transformations should clearlybe derivable by a combination of the bilinear transformation andanalog frequency transformations, instead of following the con-ventional procedure of [1]. This paper shows that this alterna-tive procedure is simpler, not only conceptually, but also fromthe calculation point of view.

0018-9359/$20.00 © 2005 IEEE

DUTTA ROY: A SIMPLE DERIVATION OF THE SPECTRAL TRANSFORMATIONS FOR IIR FILTERS 275

Fig. 1. Illustrating two routes for designing IIR digital filters.

In teaching IIR filter design, the author has experimented withboth the methods for a number of years and has found the stu-dent response to be much more favorable toward the alternativemethod presented here.

Following are the derivations of all the four types of transfor-mation by the alternative method.

II. LOW-PASS-TO-LOW-PASS TRANSFORMATION

Let a prototype DLPF of cutoff frequency be required to betransformed to another DLPF of cutoff frequency , as shownin Fig. 2(a) and (b). Under bilinear transformation, these twofrequencies correspond to analog frequencies and , re-spectively, where

(4)

Let the analog frequency variable corresponding to the proto-type and the transformed filters be and , respectively. Then,

is the analog frequency variable for the normal-ized analog low-pass filter (ALPF) whose cutoff is at 1 rad/s.To convert this filter into another ALPF with cutoff at , put

, i.e.,

(5)

Now if and are the complex frequency variables for theprototype DLPF and the transformed DLPF, then

and (6)

Substituting (4) and (6) in (5), one gets

(7)

Fig. 2. Magnitude response of digital filters: (a) Prototype LP; (b) transformedLP; (c) transformed HP; (d) transformed BP; and (e) transformed BS. � and �

denote the passband and stop-band tolerances.

where

(8)

276 IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 2, MAY 2005

Adding unity to both sides of (7) and simplifying gives

(9)

Subtracting both sides of (7) from unity and simplifying, oneobtains

(10)

Dividing (10) by (9) and simplifying gives

(11)

where

(12)

the last expression being obtained by using the formulas for.

III. LOW-PASS-TO-HIGH-PASS TRANSFORMATION

The prototype DLPF of Fig. 2(a) needs to be transformed tothe digital HP filter of Fig. 2(c). Noting that the relevant analogfrequency transformation is , i.e.,

(13)

and combining with (6), one gets

(14)

where

(15)

Solving for from (14), as in the previous case, gives

(16)

where

(17)

the last result being obtained by using the formulas for.

IV. LOW-PASS-TO-BANDPASS TRANSFORMATION

The required BP characteristic is shown in Fig. 2(d). The cor-responding analog BP characteristic has passband edges at

(18)

and the relevant analog frequency transformation is

(19)

In terms of , this equation can be rewritten as

(20)

Combining (20) with (4), (6), and (18) and simplifying, one gets

(21)

where

(22a)

and

(22b)

Solving for from (21) in the usual manner gives

(23)

A couple of algebraic steps reduces (23) to the desired form

(24)

where

(25)

and

(26)

In deriving (25) and (26), the formulas for andhave been used.

V. LOW-PASS-TO-BAND-STOP TRANSFORMATION

Referring to Fig. 2(e), and recalling that the relevant analogfrequency transformation is exactly the reciprocal of (19), onegets

(27)

Combining (27) with (4), (6), and (18) and simplifying, one gets

(28)

DUTTA ROY: A SIMPLE DERIVATION OF THE SPECTRAL TRANSFORMATIONS FOR IIR FILTERS 277

where

(29)

and

(30)

Solving for from (28), the result can be easily put in theform

(31)

where is the same as that given by (26) and

(32)

VI. REMARKS

Several remarks are in order at this stage. First, note that geo-metric symmetry of the analog frequency transformation dic-tates that in the BP and BS cases, the center frequency wouldbe given by . Correspondingly, in the digital do-main, the center frequency will be given by

(33)

From (33), one can find as

(34)

Comparing with (26), note that

(35)

Second, notice that the stop-band edges in any of the filtersin Fig. 2 do not enter into the transformations. However, in theLP-to-LP and LP-to-HP cases, the stop-band edges are restrictedby the relations (5) and (13), respectively; i.e.,in the first case and in the second case. Thecorresponding digital frequency relations can be easily found bysubstituting . In the LP-to-BP and the LP-to-BScases, the stop-band edges are constrained by geometric sym-metry; i.e., and

in the first case, and inthe second case. If the specifications do not satisfy geometricsymmetry, then one of the stop-band edges needs to be adjusted(see [13, pp. 438–441]). This adjustment results in oversatis-fying the stop-band requirement at that end. Once again, the cor-responding digital frequency relations can be easily worked out.

Finally, note that in each case of transformation, the all-passcharacter of the transformation function emerges automatically.This action is a reflection of the all-pass character of the bilineartransformation and the geometric symmetry of the second-orderanalog frequency transformations.

VII. CONCLUSION

This paper shows that the spectral transformations for IIR fil-ters can be simply derived by applying the bilinear transforma-tion to the analog frequency transformations of a normalizedALPF to that of any other denormalized analog filter.

Only single passband and/or single stop-band filters havebeen considered here. Moorer [18] has extended the techniqueof [1] to the case of filters with any number of passbands andstop bands. From the results of this paper, one naturally con-cludes that multiple passband and stop-band cases can also behandled alternatively by applying the bilinear transformation tothe general analog transformation function, which, as shown inbooks on network synthesis [19], is an inductance-capacitance(LC) driving point immittance function, popularly known as areactance function.

Finally, as a sequel to [1], a solution to the reverse problemof determining the passband and stop-band edges of the pro-totype LP transfer function so as to assign prescribed values tothe passband and to the stop-band edge(s) of the desired transferfunction has been given in [20].

ACKNOWLEDGMENT

The author would like to thank the Editor-in-Chief of theIEEE TRANSACTIONS ON EDUCATION (TE) D. A. Conner and thereviewers for their insightful queries and constructive sugges-tions, which greatly helped in revising and improving the paper.The author also would like to thank the TE Editorial Adminis-trator J. A. Conner for her meticulous editing of the manuscript.

REFERENCES

[1] A. G. Constantinides, “Spectral transformations for digital filters,” Proc.Inst. Elect. Eng., vol. 117, pp. 1585–1590, Aug. 1970.

[2] L. R. Rabiner and B. Gold, Theory and Applications of Digital SignalProcessing. New Delhi, India: Prentice-Hall, 1975, pp. 260–263.

[3] M. Bellanger, Digital Processing of Signals: Theory and Prac-tice. Chicester, U.K.: Wiley, 1985, pp. 185–186.

[4] L. C. Ludeman, Fundamentals of Digital Signal Processing. NewYork: Harper & Row, 1986, pp. 181–183.

[5] T. W. Parks and C. S. Burrus, Digital Filter Design. New York: Wiley,1987, pp. 214–215.

[6] J. G. Proakis and D. G. Manolakis, Introduction to Digital Signal Pro-cessing. New York: Macmillan, 1989, pp. 633–635.

[7] , Digital Signal Processing: Principles, Algorithms, and Applica-tions. New York: Macmillan, 1992, pp. 648–650.

[8] W. Higgins and D. C. Munson, Jr., “Infinite impulse response digitalfilter design,” in Handbook of Digital Signal Processing, S. K. Mitraand J. F. Kaiser, Eds. New York: Wiley, 1993, pp. 321–322.

[9] A. Ambardkar, Analog and Digital Signal Processing. Pacific Grove,CA: Brooks/Cole, 1999, pp. 694–696.

[10] B. A. Shenoi, Magnitude and Delay Approximation of 1-D and 2-D Dig-ital Filters. Berlin, Germany: Springer-Verlag, 1999, pp. 60–64.

[11] A. V. Oppenheim and R. W. Schafer, Digital Signal Pro-cessing. Englewood Cliffs, NJ: Prentice-Hall, 1975, pp. 226–230.

[12] , Discrete Time Signal Processing. Englewood Cliffs, NJ: Pren-tice-Hall, 1989, pp. 430–438.

[13] S. K. Mitra, Digital Signal Processing: A Computer-Based Ap-proach. New York: McGraw-Hill, 2001, pp. 441–446.

[14] T. J. Cavicchi, Digital Signal Processing. New York: Wiley, 2002, pp.598–603.

[15] J. R. Johnson, Introduction to Digital Signal Processing. EnglewoodCliffs, NJ: Prentice-Hall, 1989, pp. 239–242.

[16] A. Antoniou, Digital Filters: Analysis and Design. New Delhi, India:Tata McGraw-Hill, 1979, pp. 186–190.

[17] N. K. Bose, Digital Filters: Theory and Applications. New York: El-sevier, 1985, pp. 194–203.

[18] J. A. Moorer, “General spectral transformations for digital filters,” IEEETrans. Acoust., Speech, Signal Processing, vol. ASSP-29, no. 5, pp.1092–1094, Oct. 1981.

278 IEEE TRANSACTIONS ON EDUCATION, VOL. 48, NO. 2, MAY 2005

[19] L. Wienberg, Network Analysis and Synthesis. New York: McGraw-Hill, 1962, pp. 534–538.

[20] B. Nowrouzian and A. G. Constantinides, “Prototype reference transferfunction parameters in the discrete time frequency transformations,” inProc 33rd Midwest Symp. Circuits Syst., Calgary, AB, Canada, Aug.1990, pp. 1078–1082.

S. C. Dutta Roy (SM’66–F’95) received the Ph.D. degree in radio physics andelectronics from Calcutta University, Calcutta, India, in 1965.

He worked previously at the River Research Institute and the University ofKalyani, both in West Bengal, India, and the University of Minnesota, Min-neapolis, before joining the Indian Institute of Technology (IIT), Delhi, wherehe was a Professor of Electrical Engineering until 1998. He was also the Headof the Department for three years and the Dean of Undergraduate Studies for an-other three years. From 1998 to 2004, he was an Emeritus Fellow, and currently,he is an Indian National Science Academy Senior Scientist at the same institute.His visiting positions include one year at the University of Leeds, Leeds, U.K.,and another year at the Iowa State University, Ames. His research interests in-clude circuits—active, passive, and distributed, systems—linear as well as non-linear, and signal processing—analog as well as digital.

Prof. Dutta Roy is a Fellow of the Acoustical Society of India and ofall the science and engineering academies of India and is a distinguishedFellow of the Institution of Electronics and Telecommunication Engineers(IETE). He received several prestigious national awards, including the ShantiSwarup Bhatnagar Prize, the Vikram Sarabhai Award, Pandit JawaharlalNehru Award, the O. P. Bhasin Award, and the University Grants CommissionLectureship Award. He is the author/coauthor of a large number of researchpapers in IEEE and Institution for Electrical Engineers (IEE) publicationsas well as in other professional journals, two book chapters, one U.S.and three Indian patents, and five full-semester-long video courses. Hehas served as an Associate Editor of IEEE TRANSACTIONS ON CIRCUITS

AND SYSTEMS—I; an Honorary Editor of the Journal of the Institution ofElectronics and Telecommunication Engineers (IETE); the Chairman of theEditorial Board of the IETE Journal of Education; and as a Member of theEditorial Board of International Journal of Circuit Theory and Applications,Multidimensional Systems and Signal Processing, Neural Network World,IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, and severalother journals.