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
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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
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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