Wave digital filters with minimum multiplier for discrete Hilbert ...

ARTICLE IN PRESSSignal Processing 86 (2006) 3761–3768www.elsevier.com/locate/sigproWave digital filters with minimum multiplier for discreteHilbert transformer realizationS.A. Samad a, , A. Hussain a , D. Isa ba Department of Electrical, Electronic and Systems Engineering, Faculty of Engineering, National University of Malaysia (UKM),43600 UKM Bangi Selangor, Malaysiab Faculty of Engineering and Computer Science, University of Nottingham, Malaysia Campus, Jalan Broga, 43500 Semenyih,Selangor Darul Ehsan, MalaysiaReceived 1 December 2005; received in revised form 9 March 2006; accepted 14 March 2006Available online 5 April 2006AbstractThis paper describes the design of wave digital filters for discrete Hilbert transformer realization. A method for designingreal and complex half-band wave digital filters (WDFs) with a minimum number of multipliers is afforded where themultiplier coefficients for the adaptors are calculated directly from the digital domain transfer functions. This techniqueuses the equations derived for a class of infinite impulse response (IIR) filter and hence avoids the conventional techniqueof synthesizing a WDF from an analogue reference filter. The selection of the adaptor configuration is determined from themultiplier coefficient values. A minimum multiplier realization of real half-band WDFs can be obtained to meet the lowpassdigital filter magnitude specifications. By applying a transformation equation, a complex half-band WDF suitable forthe realization of the discrete Hilbert transformer with a minimum number of multiplier is obtained.r 2006 Elsevier B.V. All rights reserved.Keywords: Wave digital filters; Hilbert transformer; Minimum multiplier1. IntroductionWave digital filters (WDFs) represent a class ofdigital filters that are closely related to classical filternetworks, usually lossless filters inserted betweenresistive terminations. For every WDF, therecorresponds a reference analogue filter from whichit is derived. The analogy between a WDF and itsreference network is based not on the use ofvoltages and currents as the signal variables but Corresponding author. Tel.: +603 8921 6331;fax: +603 8921 6146.E-mail address: sas@ieee.org (S.A. Samad).the traveling wave quantities of classical electriccircuits. A variety of WDFs are available and manyhave found applications in systems requiring digitalfilters. This is due to the good sensitivity propertiesof the WDFs obtained from a properly designedanalogue filter that is both passive and lossless. Thisproperty is preserved in the digital domain by thestructures of the WDFs which emulate the analoguefilters. The good sensitivity properties translate toreduced accuracy requirement for the coefficientwordlength and a good dynamic range performance[1–3].Filters are classified according to the characteristicsof their frequency response. Real half-band0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.sigpro.2006.03.005

3762ARTICLE IN PRESSS.A. Samad et al. / Signal Processing 86 (2006) 3761–3768filters are usually a pair of filters with one having alow-pass response occupying the lower half of theuseful digital frequency band, and the other with ahigh-pass response occupying the upper half of theuseful digital frequency band. These filters areusually used in pairs for applications such as insubband coding. Complex half-band filters, on theother hand, are often used to realize the discreteHilbert transformer. The discrete Hilbert transformerarises in a variety of practical applications,including inverse filtering, complex representationsfor real bandpass signals, single side-band modulationtechniques and spectral analysis. Both infiniteimpulse response (IIR) and finite impulse response(FIR) filters can be used to realize these filters [4].For applications requiring strict linear phase, FIRfilters are usually used [5]. However, when the phaserequirement is not as stringent, the IIR filter is abetter choice as it requires fewer coefficients torealize the same transfer function as an FIR filter[6]. Using IIR filters are especially advantageous inrealizing low-cost and low-power devices.Wave digital filters have been shown to beapplicable to the design of real half-band filters[7–11]. Many of the methods rely on deriving theWDF structure from reference analogue latticefilters and the realizations may not be optimum interms of the number of multipliers required given afilter order. In addition, complex realizations arenot shown for these real filters.For complex half-band WDFs the design techniquerelies on deriving the WDF structure fromreference analogue filters having the complexresponse [12]. For these filter types, the correspondingWDF structures are obtained using an elementto element translation from the analog to the digitaldomain and interconnecting the digital elementsusing the appropriate adaptors. A purely digitalapproach showing the resulting effects in both themagnitude and phase responses has not been shown.In this paper, a purely digital approach isintroduced to design real and complex half-bandWDFs. The transfer functions for a class of IIRfilter are used to obtain all-pass WDF structureswhich are then used to obtain real half-band filters.Next, complex half-band filter WDFs are obtainedfrom real half-band WDFs by applying a transformationequation. The adaptor selection is determinedfrom the multiplier coefficient values whichmay differ for the real and complex WDFs.This paper is organized in the following manner:Section 2 details the design methodology of real andcomplex IIR filters, Section 3 presents the designparameters of wave digital filters, Section 4 gives anexample of the proposed technique while theconclusion is presented in Section 5.2. Designing half-band IIR filters2.1. Real half-band IIR filtersReal half-band filters can be realized with eitherFIR or IIR filters. For IIR filters, it has been shown[13] that any odd order elliptic lowpass half-bandfilter, G(z), with a frequency response given by1 d p pGðe jo Þ p1 for 0popop ,(1a)Gðe jo Þ pds ; for o s popp, (1b)and satisfying the conditionso p þ o s ¼ p,(2a)ð1 2d p Þ 2 þ d 2 s ¼ 1, (2b)can be expressed asGðzÞ ¼ 1 2 ½A 0ðz 2 Þþz 1 A 1 ðz 2 ÞŠ, (3)where A 0 [z] and A 1 [z] are stable magnitude all-passfilters. d p and d s are the passband and stopbandripple, respectively, while o p and o s are thepassband and stopband normalized edge frequency,respectively.For G(z) with real coefficients, the frequencyresponse isGðe jo ÞþGðe jðp oÞ Þ¼1. (4)The above equality implies that G(e j(p/2 y) ) andG(e j(p/2+y) ) add up to unity for all y. Hence, G(e jo )exhibits symmetry with respect to the half-bandfrequency, p/2. This filter has a doubly complementaryhigh-pass transfer function H 1 (z) given byH 1 ðzÞ ¼ 1 2 ½A 0ðz 2 Þ z 1 A 1 ðz 2 ÞŠ. (5)Fig. 1 shows the realization of the half-band IIRfilter based on Eqs. (3) and (5).1/2A 0 (z 2 )z -1 A 1 (z 2 )+-1Fig. 1. Realization of a real-half band IIR filter.+G(z)H 1 (z)

ARTICLE IN PRESSS.A. Samad et al. / Signal Processing 86 (2006) 3761–3768 37632.2. Complex half-band IIR filtersA complex half-band filter can be used to realizethe discrete Hilbert transformer for the generationof an analytic signal from a discrete-time real signal.The analytic signal is a complex time functionhaving a Fourier transform that vanishes fornegative frequencies. The basic scheme for thegeneration of an analytic signal y[n] ¼ y re [n]+y im [n]from a real signal x[n] is shown in Fig. 2.A linear discrete-time system representing anideal Hilbert transformer has a frequency responsecorresponding to(j; 0ooop;H HT ðoÞ ¼(6)j; pooo0:However, Eq. (6) represents an unrealizable idealsystem with an infinite-length two-sided impulseresponse. An approach to develop a realizablesystem is by using complex half-band filters.Consider the complex analytic signaly½nŠ ¼x½nŠþj ^x½nŠ, (7)where x[n] and ^x[n] are real. Its discrete-timeFourier transform (DTFT) Y(e jo ) is given byYðe jo Þ¼Xðe jo Þþj ^Xðe jo Þ. (8)Since x[n] and ^x[n] are real, their correspondingDTFTs are conjugate symmetric, hence the followingcan be obtained:Xðe jo Þ¼ 1 2 Yðejo ÞþY ðe jo Þ , (9)j ^Xðe jo Þ¼ 1 2 Yðejo Þ Y ðe jo Þ . (10)For an analytic signal, Y(e jo ) is zero for p p op 0, hence from Eqs. (8)–(10) the following isderived:(Yðe jo Þ¼ 2Xðejo Þ; 0poop;0; ppoo0:(11)y re [n]The analytic signal can be generated by passing x[n]through a linear discrete-time system with afrequency response H(e jo ) given by(2; 0poop;Hðe jo Þ¼ (12)0; ppoo0:2.2.1. Transformation equationsTo obtain a complex half-band filter from that ofa real half-band, transformation equations may beused. A real half-band filter with a frequencyresponse(Gðe jo Þ¼z -1A 0 (-z 2 )1; 0p j o jo p 2 ;p0;2 pjojop;A 1 (-z 2 )Fig. 3. IIR realization of a complex half-band filter.y re [z]y im [z](13)is related to H(e jo ), the complex half-band filter,according toGðe jo Þ¼ 1 2 HðejðoþpÞ=2Þ Þ. (14)In the z domain, the relation between the transferfunctions of a complex half-band filter H(z) and areal half-band low-pass filter G(z) is given by thetransformation equationHðzÞ ¼jGð jzÞ. (15)Substituting Eq. (3) in (15), the complex half-bandfilter derived from a real half-band IIR filter is thusHðzÞ ¼A 0 ð z 2 Þþjz 1 A 1 ð z 2 Þ. (16)The realization of the complex half-band filterbased on the above decomposition is shown inFig. 3.3. WDF equivalentx[n]HilbertTransformery im [n]Fig. 2. Generation of an analytic signal using a Hilberttransformer.The relationship between a WDF, operating at asampling rate F ¼ 1/T, and its reference analoguefilter is established in the frequency domain by themeans of the bilinear transformations ¼ 2 z 1, (17)T z þ 1

3764ARTICLE IN PRESSS.A. Samad et al. / Signal Processing 86 (2006) 3761–3768where s is the reference domain and z the digitaldomain. Direct realization of the digital filterobtained with Eq. (17) will produce signal flowgraphswith delay-free loops. To overcome this,WDFs use wave network characterization to obtainrealizable digital structures.The signals used in WDF representation are thewave quantities A representing the forward travellingwaves and B representing the backwardtravelling waves. For a classical port characterizedby a current I, a voltage V and a port resistance R,the equations that relate these parameters to thewave quantities areA ¼ V þ IR,(18a)B ¼ V IR. (18b)Each port is thus represented by the wave quantitiesand a port resistance value. For a two-port network,the S-matrix describing the reflection and transmittanceproperties of the network is also used todescribe WDFs withB ¼ SA, (19)where"S ¼ S #11 S 12. (20)S 21 S 22By using the wave characterization, each element inthe analogue domain is represented as a two-portnetwork. To form interconnection between elements,adaptors are used. These adaptors containthe multipliers and adders of the filter.3.1. AdaptorsFor an all-pass section, one corresponding WDFstructure consists of a cascade of two-port adaptors.This structure corresponds, in the analogue domain,to a cascade of unit elements used in the design ofmicrowave filters. The two-port adaptors used forthe digital realization of the all-pass sections canhave one or two multipliers. Realization with singlemultiplieradaptors will give a canonic implementationof the digital filter. Fig. 4 shows three differentconfigurations of a single-multiplier two-port adaptor.Among the adaptors, the structure with itsmultiplier coefficient {g, 1 g, (1+g)} that is lessthan half in absolute value should be selected due toits superior finite wordlength property [1]. Theseadaptors are equivalent since they implement thea 1-1same relation asb 1 ¼ ga 1 þð1 þ gÞa 2 ,++ +b 1b 2(a)a 1b 1(b)a 1b 1-1(c)+++(21a)b 2 ¼ð1 gÞa 1 ga 2 . (21b)The multiplier coefficient, g, is related to the portresistance R 1 and R 2 derived from the analoguereference filter according tog ¼ R 1 R 2. (22)R 1 þ R 2To obtain the coefficients of the adaptor from theall-pass digital domain transfer function, SchurγType I1-γ-1Type II-(1+γ)Type IIIFig. 4. Three equivalent two-port adaptors.+++-1a 2a 2b 2a 2b 2

ARTICLE IN PRESSS.A. Samad et al. / Signal Processing 86 (2006) 3761–3768 3765γThe transfer function for this filter can be writtenin the form of Eq. (3) as(a)γ N-1γ 1GðzÞ ¼ 1 0:23680414 þ z 22 1 þ 0:23680414z 2 þ z 1 0:71490399 þ z 2 1 þ 0:71490399z 2 .(26)(b)parameterization can be used. One of the mostimportant properties of the Schur algorithm is itsorthonormality, which has been exploited tosynthesize various types of lattice filters [14].Let A N (z 1 ) be an all-pass function of order N,real, analytic inside the unit circle and such thatA N ðz 1 Þ p1; for z1 ¼ 1. (23)By definingg N ¼ A N ð0Þ, (24)A Nγ Nz -1z -1Fig. 5. (a) Two-port adaptor symbol, (b) WDF implementationof the all-pass structure.1 (z 1 ) is generated asA N 1 ðz 1 Þ¼ 1 A N ðz 1 Þ g Nz 1 1 g N A N ðz 1 Þ . (25)By repeating the Schur parameterization proceduresin Eqs. (24) and (25), N parameters {g N, g N 1,y,g 1 } can be obtained representing the adaptorcoefficients for the WDF. The symbol for a twoportadaptor is shown in Fig. 5(a) while Fig. 5(b)shows the interconnection of the adaptors obtainedusing Schur parameterization.4. Design examplesz -1To obtain the WDF equivalent, successive Schurparameterization is applied to each of the twoallpass sections. For the allpass section A 0 , theresulting adaptor coefficients are {0, 0.23680414}and for A 1 they are {0, 0.71490399}. It can beobserved that the magnitude of the non-zerocoefficients can in fact be obtained directly fromthe transfer function.The WDF configuration obtained by choosingthe appropriate adaptor is as shown in Fig. 6. Theadaptor as shown in Fig. 4(a) of Type I is requiredfor the nonzero coefficient of section A 0 while thatof Type III is required for section A 1 . Furthermore,the zero coefficient adaptor can be omitted whilemaintaining the required number of delays. Aminimum multiplier design for the WDF is obtainedwhere only two single-multiplier adaptors arerequired.The lowpass and highpass frequency responsesfor the minimum multiplier half-band WDFare shown in Figs. 7 and 8, respectively. Both themagnitude and phase responses are shown. Themagnitude response meets the specifications requiredfor the low-pass filter, while high-passresponse is complementary to the low-pass response.Since the WDF is derived from an IIRz -2Adaptor Type Iγ= -0.23680414To show the design process two examples areshown here, one for a real half-band WDF and theother for the corresponding complex half-bandWDF. A fifth-order IIR digital filter satisfyingEq. (2) is selected having the following digital filterspecifications:1/2z -1 +-1+G(z)H 1 (z)Normalized passband edge frequency, o s : 0.4p.Normalized stopband edge frequency, o p : 0.6p.Stopband ripple, d s : 0.01550, (R s ¼ 20 logd s ¼ 36.193366 dB).Passband ripple, d p : 6.00661 10 5 , (R p ¼ 20log(1–2d p ) ¼ 1.043518 10 3 dB).z -2Adaptor Type IIIγ= -0.71490399Fig. 6. Real Half-Band WDF.

3766ARTICLE IN PRESSS.A. Samad et al. / Signal Processing 86 (2006) 3761–3768Fig. 7. Lowpass magnitude and phase responses for the halfbandWDF.essentially equivalent to those obtained for the realhalfband WDF except for the sign change. In theselecting the adaptors for the complex half-bandWDF, the non-zero coefficient adaptor for A 0requires a Type I as shown in Fig. 4(a), while theadaptor for A 1 requires a Type II adaptor as shownin Fig. 4(b). The resulting complex half-band WDFstructure is shown in Fig. 9.The magnitude response for the complex halfbandfilter is shown in Fig. 10. Except forfrequencies near zero and p, the magnitude is unitythroughout the passband. This is the main criterionfor a complex half-band filter.Fig. 11 shows the phase difference between thereal, y re , and imaginary, y im, outputs. A 901 phasedifference is obtained in the positive frequencyranges, 0 to p, while 2701 phase difference isobtained for p to 2p. However, at frequencies near0, p and 2p the phase response is not constant. Inapplying this minimum multiplier discrete Hilberttransformer realization, careful selection of theapplied signal frequency with respect to thesampling frequency has to be observed so that itfalls between the accepted ranges. However, it hasthe advantage of requiring only two multipliers forits implementation.Fig. 8. Highpass magnitude and phase responses for the halfbandWDF.filter transfer function, strictly linear phase is notpossible. However, the phase change is monotonicallydecreasing in the passband of both filters.The equation for the corresponding complex halfbandfilter is formed by transforming Eq. (26) to theform of Eq. (15), obtaining the following:HðzÞ ¼ 1 0:2368041466 z 221 0:7149039978 zþ jz .2 1 0:2368041466z21 0:7149039978z 2(27)By applying Schur parameterization to the newallpass section A 0, the WDF multiplier adaptorcoefficients obtained are {0, 0.23680414}, and for A 1the coefficients are {0, 0.71490399}. The values arez -1 z -2z -2Adaptor Type Iγ= 0.23680414Fig. 9. Complex half-band WDF.Adaptor Type IIγ= 0.71490399y re(z)y im(z)

ARTICLE IN PRESSS.A. Samad et al. / Signal Processing 86 (2006) 3761–3768 3767Fig. 10. Magnitude response of the complex half-band WDF.Fig. 11. Phase difference between real and imaginary components of the complex half-band WDF.Conventionally, the Hilbert transformer is designedusing FIR filters [5]. The performanceof FIR filters designed using the popular Parks–McClellan algorithm [6] having the same specificationsis shown in Fig. 12 where they are comparedwith the WDF. The plotted passband magnituderesponses clearly show the superiority of theWDF in terms of filter order and hence the numberof multiplier coefficients. The phase responsesare not compared since FIR filters have linearphase while the WDF shown here have constantphase in a wide range of frequencies. Inapplications where linear phase is not required, theWDF filter is a better choice as it requires fewercoefficients to realize the same transfer function asan FIR filter.

3768ARTICLE IN PRESSS.A. Samad et al. / Signal Processing 86 (2006) 3761–3768Fig. 12. Passband magnitude responses of WDF and conventional Hilbert transformer FIR filters.5. ConclusionThis paper has shown the design of WDFsfor discrete Hilbert transformer realization. TheIIR filter transfer function is used directly in theextraction of the multiplier coefficients for the twoportadaptors. Examples are shown where theconfiguration of an adaptor is selected based onthe value of the obtained multiplier coefficient. Aminimum multiplier realization of a real half-bandWDF is obtained to meet the low-pass digital filtermagnitude specifications. The doubly complementaryhigh-pass realization is simultaneously obtainedfrom the same structure giving an efficientdesign. By applying a transformation equation, acomplex half-band WDF suitable for the realizationof the discrete Hilbert transformer with a minimumnumber of multiplier is obtained.References[1] A. Fettweis, Wave digital filters: theory and practice, Proc.IEEE 74 (2) (February 1986) 270–327.[2] J.G. Chung, K.K. Parthi, Pipelined Lattice and WaveDigital Recursive Filters, Kluwer Academic Publishers,Norwell, MA, 1996.[3] S.A. Samad, Digital domain design of cascaded wave digitalfilters with tunable parameters, Iran. J. Elec. Comput. Eng. 3(2) (2004) 1–5.[4] I. Kollar, R. Pintelon, J. Schoukens, Optimal FIR and IIRHilbert tranformer design via LS and minimax fitting, IEEETans. Instrum. Meas. 39 (6) (December 1990) 847–852.[5] Y.C. Lim , Y.J. Yu, T. Saramaki, Optimum masking levelsand coefficient sparseness for Hilbert transformers and halfbandfilters designed using the frequency-response maskingtechnique, IEEE Trans. Circ. Syst. I: Regular Papers 12 (6)(November 2005) 2444–2453.[6] M.D. Lutovac, D.V. Tosic, B.L. Evans, Filter Design forSignal Processing Using MATLAB and Mathematica,Prentice-Hall, Upper Saddle River, NJ, 2000.[7] A. Antoniou, Digital Filters: Analysis Design and Applications,McGraw-Hill, New York, 1993.[8] A. Fettweis, H. Levin, A. Sedlemeyer, Wave digital latticefilters, Int. J. Circ. Theory. Appl. (June 1974) 203–211.[9] W. Wegener, Wave digital directional filters with reducednumber of multipliers and adders, Arch. Fur Elek.ubertragung33 (6) (1979) 239–243.[10] J. Yli-Kaakinen, T. Saramäki, Design of very low-sensitivityand low-noise recursive filters using a cascade of low-orderlattice wave digital filters, IEEE Trans. CAS-II 46 (7) (July1999) 906–914.[11] H. Johansson, L. Wanhammer, Wave digital filters structuresfor high speed narrow-band and wide-band filtering,IEEE Trans. CAS-II 46 (6) (1999) 726–741.[12] K. Meetkotter, Antimetric wave digital filters derived fromcomplex reference circuits, Proceedings of European Conferenceon Circuit Theory and Design, Stuttgart, W.Germany, vol. 2, September 1983, 24–25.[13] P.P. Vaidyanathan, P.A. Regalia, S.K. Mitra, Design ofdoubly complementary IIR digital filters using a singlecomplex allpass filter with multirate applications, IEEETrans. Circ. and Systems, CAS-34 (April 1987) 378–379.[14] L. Gaszi, Explicit formulas for lattice wave digital filters,IEEE Trans. Circ. Systems, CAS-32, (January 1985) 68–88.