Chapter 19 2012 Troncoso Romero and Jovanovic Dolecek, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Digital FIR Hilbert Transformers:Fundamentals and Efficient Design Methods David Ernesto Troncoso Romero and Gordana Jovanovic Dolecek Additional information is available at the end of the chapter http://dx.doi.org/10.5772/46451 1. Introduction DigitalHilberttransformersareaspecialclassofdigitalfilterwhosecharacteristicisto introducea/2radiansphaseshiftoftheinputsignal.IntheidealHilberttransformerall thepositivefrequencycomponentsareshiftedby/2radiansandallthenegative frequencycomponentsareshiftedby/2radians.However,theseidealsystemscannotbe realized since the impulse response is non-causal. Nevertheless, Hilbert transformers can be designedeitherasFiniteImpulseResponse(FIR)orasInfiniteImpulseResponse(IIR) digital filters [1], [2], and they are used in a wide number of Digital Signal Processing (DSP) applications,suchasdigitalcommunicationsystems,radarsystems,medicalimagingand mechanical vibration analysis, among others [3]-[5]. IIRHilberttransformersperformaphaseapproximation.Thismeansthatthephase responseofthesystemisapproximatedtothedesiredvaluesinagivenrangeof frequencies. The magnitude response allows passing all the frequencies, with the magnitude obtained around the desired value within a given tolerance [6], [7]. On the other hand, FIR Hilberttransformersperformamagnitudeapproximation.Inthiscasethesystem magnituderesponseisapproximatedtothedesired valuesinagivenrangeoffrequencies. Theadvantageisthattheirphaseresponseisalwaysmaintainedinthedesiredvalueover the complete range of frequencies [8].WhereasIIRHilberttransformerscanpresentinstabilityandtheyaresensitivetothe roundingintheircoefficients,FIRfilterscanhaveexactlinearphaseandtheirstabilityis guaranteed.Moreover,FIRfiltersarelesssensitivetothecoefficientsroundingandtheir phase response is not affected by this rounding. Because of this, FIR Hilbert transformers are oftenpreferred[8]-[15].Nevertheless,themaindrawbackofFIRfiltersisahigher complexitycomparedwiththecorrespondingIIRfilters.Multipliers,themostcostly MATLAB A Fundamental Tool for Scientific Computing and Engineering Applications Volume 1446 elementsinDSPimplementations,arerequiredinanamountlinearlyrelatedwiththe lengthofthefilter.AlinearphaseFIRHilberttransformer,whichhasananti-symmetrical impulse response, can be designed with either an odd length (Type III symmetry) or an even length(TypeIVsymmetry).Thenumberofmultipliersmisgivenintermsofthefilter length L as~ m CL , where C = 0.25 for a filter with Type III symmetry or C = 0.5 for a filter with Type IV symmetry.The design of optimum equiripple FIR Hilbert transformers is usually performed by Parks-McClellanalgorithm.UsingtheMATLABSignalProcessingToolbox,thisbecomesa straightforwardprocedurethroughthefunctionfirpm.However,forsmalltransition bandwidth and small ripples the resulting filter requires a very high length. This complexity increaseswithmorestringentspecifications,i.e.,narrowertransitionbandwidthsandalso smaller pass-band ripples. Therefore, different techniques have been developed in the last 2 decadesforefficientdesignofHilberttransformers,wherethehighlystringent specificationsaremetwithanaslowaspossiblerequiredcomplexity.Themost representativemethodsare[9]-[15],whicharebasedinveryefficientschemestoreduce complexity in FIR filters. Methods[9]and[10]arebasedontheFrequencyResponseMasking(FRM)technique proposed in [16]. In [9], the design is based on reducing the complexity of a half-band filter. Then,theHilberttransformerisderivedfromthishalf-bandfilter.In[10],afrequency responsecorrectorsubfilterisintroduced,andallsubfiltersaredesignedsimultaneously under the same framework. The method [11] is based on wide bandwidth and linear phase FIRfilterswithPiecewisePolynomial-Sinusoidal(PPS)impulseresponse.Thesemethods offeraveryhighreductionintherequirednumberofmultipliercoefficientscomparedto thedirectdesignbasedonParks-McClellanalgorithm.Animportantcharacteristicisthat theyarefullyparallelapproaches,whichhavethedisadvantageofbeingareaconsuming since they do not directly take advantage of hardware multiplexing.The Frequency Transformation (FT) method, proposed first in [17] and extended in [18], was modifiedtodesignFIRHilberttransformersin[12]basedonatappedcascaded interconnectionofrepeatedsimplebasicbuildingblocksconstitutedbytwoidentical subfilters. Taking advantage of the repetitive use of identical subfilters, the recent proposal [13] gives a simple and efficient method to design multiplierless Hilbert transformers, where acombinationoftheFTmethodwiththePipelining-Interleaving(PI)techniqueof[19] allowsgettingatime-multiplexedarchitecturewhichonlyrequiresthreesubfilters.In[14], an optimized design was developed to minimize the overall number of filter coefficients in a modifiedFT-PI-basedstructurederivedfromtheoneof[13],whereonlytwosubfiltersare needed.Basedonmethods[13]and[14],adifferentarchitecturewhichjustrequiresone subfilter was developed in [15].Inthischapter,fundamentalsondigitalFIRHilberttransformerswillbecoveredby reviewingthecharacteristicsofanalyticsignals.Themainconnectionexistingbetween Digital FIR Hilbert Transformers: Fundamentals and Efficient Design Methods447 Hilberttransformersandhalf-bandfilterswillbehighlightedbut,atthesametime,the complete introductory explanation will be kept as simple as possible. The methods to design low-complexityFIRfilters,namelyFRM[16],FT[17]andPPS[11],aswellasthePI architecture[19],whicharethecornerstoneoftheefficienttechniquestodesignHilbert transformers presented in [9]-[15], will be introduced in a simplified and concise way. With suchbackgroundwewillprovideanextensiverevisionofthemethods[9]-[15]todesign low-complexityefficientFIRHilberttransformers,includingMATLABroutinesforthese methods.2. Complex signals, analytic signals and Hilbert transformers Arealsignalisaone-dimensionalvariationofrealvaluesovertime.Acomplexsignalisa two-dimensionalsignalwhosevalueatsomeinstantintimecanbespecifiedbyasingle complexnumber.Thevariationofthetwopartsofthecomplexnumbers,namelythereal part and the imaginary part, is the reason for referring to it as two-dimensional signal [20]. Arealsignalcanberepresentedinatwo-dimensionalplotbypresentingitsvariations againsttime.Similarly,acomplexsignalcanberepresentedinathree-dimensionalplotby considering time as a third dimension.Real signals always have positive and negative frequency spectral components, and these componentsaregenerallyrealandimaginary.Foranyrealsignal,thepositiveand negativepartsofitsrealspectralcomponentalwayshaveevensymmetryaroundthe zero-frequencypoint,i.e.,theyaremirrorimagesofeachother.Conversely,thepositive andnegativepartsofitsimaginaryspectralcomponentarealwaysanti-symmetric,i.e., theyarealwaysnegativesofeachother[1].Thisconjugatesymmetryistheinvariant nature of real signals.Complex signals, on the other hand, are not restricted to these spectral conjugate symmetry conditions. The special case of complex signals which do not have a negative part neither in their real nor in their imaginary spectral components are known as analytic signalsor also as quadrature signals [2]. An example of analytic signal is the complex exponential signalxc(t), presented in Figure 1, and described by = = + = +ee e00 0( ) ( ) ( ) cos( ) sin( ).j tc r ix t e x t jx t t j t (1)Thereal part andtheimaginarypart of theanalytic signalarerelatedtroughthe Hilbert transform.Insimplewords,givenananalyticsignal,itsimaginarypartistheHilbert transform of its real part. Figure 1 shows the complex signal xc(t), its real part xr(t) and its imaginarypart,xi(t).Figure2presentsthefrequencyspectralcomponentsofthese signals.Itcanbeseenthattherealpartxr(t)andtheimaginarypartxi(t),bothreal signals,preservethespectralconjugatesymmetry.Thecomplexsignalxc(t)doesnot havenegativepartsneitherinitsrealspectralcomponentnorinitsimaginaryspectral component.Forthisreason,analyticsignalsarealsoreferredasone-sidespectrum MATLAB A Fundamental Tool for Scientific Computing and Engineering Applications Volume 1448 signals.Finally,Figure3showstheHilberttransformrelationbetweentherealand imaginary parts of xc(t). Figure 1. The Hilbert transform and the analytic signal of xr(t) = cos(0t), 0= 2. Figure 2. From left to right, frequency spectrum of xr(t), xi(t) and xc(t). Figure 3. Hilbert transform relations between xr(t) and xi(t) to generate xc(t). -2-10120246-2-1012real timeimaginaryxr(t)xc(t)xi(t)( Hilbert transform of xr(t) ) ( )ix t( )cx t( )rx t( )rx tAnalytic signal Hilbert 0Freq0e 0e0.5 0.5( )iX eRealImaginary0Freq0e1( )cX eRealImaginary0Freq0e 0e0.5( )rX eRealImaginary Digital FIR Hilbert Transformers: Fundamentals and Efficient Design Methods449 The motivation for creating analytic signals, or in other words, for eliminating the negative partsoftherealandimaginaryspectralcomponentsofrealsignals,isthatthesenegative partshaveinessencethesameinformationthanthepositivepartsduetotheconjugate symmetrypreviouslymentioned.Theeliminationofthesenegativepartsreducesthe requiredbandwidthfortheprocessing.ForthecaseofDSPapplications,itispossibleto form a complex sequence xc(n) given as follows, = + ( ) ( ) ( ),c r ix n x n jx n(2)withthespecialpropertythatitsfrequencyspectrumXc(ej)isequaltothatofagivenreal sequencex(n)forthepositiveNyquistintervalandzeroforthenegativeNyquistinterval, i.e., s < = s