Discrete Signal Representations (779441)
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Signal Analysis: Wavelets, Filter Banks, Time-Frequency Transfomzs andApplications. Alfred MertinsCopyright 0 1999 John Wiley & Sons LtdPrint ISBN 0-471-98626-7 ElectronicISBN 0-470-84183-4Chapter 3DiscreteSignal RepresentationsIn this chapter we discuss the fundamental concepts of discrete signal representations. Such representations are also known as discrete transforms, seriesexpansions, or block transforms.
Examplesof widely used discrete transformsare given in the next chapter. Moreover, optimal discrete representations willbe discussed in Chapter 5. The term “discrete” refers to the fact that thesignals are representedby discrete values,whereas the signals themselvesmay be continuous-time. If the signals that are to be transformed consistof a finite set of values, one also speaks of block transforms. Discrete signalrepresentations are of crucial importance in signal processing. They give acertain insight into the properties of signals, and they allow easy handling ofcontinuous and discrete-time signals on digital signal processors.3.1IntroductionWe consider a real or complex-valued, continuous or discrete-time signalassuming that z can be represented in the formni=l47X,48RepresentationsChapterSignal3.
DiscreteThe signal X is an element of the signal space X spanned by {pl,. . . ,p,}.The signal space itself is the set of all vectors which can be represented bylinear combination of { p l , . . . ,p,}. For this, the notationwill be used henceforth. The vectors pi, i = 1,.. . ,n may be linearly dependent or linearly independent of each other. If they are linearly independent,we call them a basis for X .The coefficients ai, i = 1,.
. . ,n can be arranged as a vectorwhichis{cpl,.Xreferred to as therepresentation ofXwith respect to the basis. . >P,>.One often is interested in finding the best approximation of a given signalby a signal 2 which has the series expansioni=lThis problem will be discussed in Sections 3.2 and 3.3 in greater detail. Forthe present we will confine ourselves to discussing some general concepts ofdecomposing signal spaces.
We start by assuming a decomposition of X intowhereSignal x1 is an element of the linear subspace' X1 = span { p l , . . . ,p,}andis an element of the linear subspace X2 = span {pm+l,. . . , p,}. The spaceX is called the sum of the subspacesX1 and X2. If the decomposition of X E X2 2lDefinition of a linear subspace: let M be a non-empty set of elements of the vectorspace X .
Then M is a linear subspace of X , if M itself is a linear space. This means thatall linear combinations of the elements of M must be elements of M . Hence, X itself is alinear subspace.49Series3.2. Orthogonalinto X I E X1 and x z E X2 is unique,2 we speak of a direct decomposition ofX into the subspaces X1 and X Z ,and X is called the direct sum of X1 andX,.
The notation for the direct sum isX = X1 ex2.(3.8)A direct sum is obtained if the vectors that span X1 are linearly independentof the vectors that span X,.If a space X is the direct sum of two subspaces X1 and X2 and x1 E X1and xz E X2 are orthogonal to one another for all signals X E X , that is if(x1,xz)= 0 V X E X , then X is the orthogonal sum of the subspaces X1 andX,. For this we writeIe x2.X =X1(3.9)Orthogonal Series Expansions3.23.2.1Calculation of CoefficientsWe consider a signalXthat can be represented in the formnX =C"i ui,(3.10)i= 1where the vectors ui satisfy the orthonormality conditionsij.(3.11)for i = j ,otherwise.(3.12)( U i ,U j )Here,6ij=is the Kronecker symbol'' - { 01For all signals X in (3.10) we have X E X with X = span ( u 1 , u2,.
. . , un}.Because of (3.11), u1,u2,. . . , un form an orthonormal basis for X . Each vectorui, i = 1,. . . ,n spans a one-dimensional subspace, and X is the orthogonalsum of these subspaces.The question ofhow the coefficients ai can be calculated if X and theorthonormal basis ( u 1 , . . . , U,} are given is easily answered. By taking theinner product of (3.10) with u j , j = 1,.. . ,n and using (3.11) we obtain"j=(X&),j = 1,.. . ,n.(3.13)50Chapter 3. Discrete Signal RepresentationsFigure 3.1. Orthogonal projection.3.2.2Orthogonal ProjectionIn (3.10) we assumed that X can be represented by means of n coefficientsa1 ,a l , .
. . ,a,. Possibly, n is infinitely large, so that for practical applicationswe are interested in finding the best approximationm(3.14)i= 1in the sense ofd ( z , f ) = 11%1-1211 = (z- f,z- f)z = min.The solution to this problem is3(3.15)= (z,u i ) , which means thatm(3.16)i dThis result has a simple geometrical interpretation in terms of an orthogonalprojection. Each basis vector ui spans a subspace that is orthogonal to thesubspaces spanned by u j , j # i, which means that the signal space X isdecomposed as follows:IX=Mm$Mk(3.17)Xx E+XME, mM, : .(3.18)withz=f++,The subspace M A is orthogonal to M m , and + = z - f is orthogonal to f(notation: +l&).Because of +l63 we call D the orthogonal projection of zonto M,.
Figure 3.1 gives an illustration.As can easily be verified, we have the following relationship between thenorms of X ,X and +(3.19)1 1 4 1 2 = llf112 + ll+112.3The proof is given in Section 3.3.2 for general, non-orthogonal bases.513.2. OrthogonalSeriesExpansions3.2.3The Gram-Schmidt OrthonormalizationProcedureGiven a basis {vi;i = 1,.. . , n } , we can construct an orthonormalbasis{ui;i = 1,. . . ,n } for the space span {vi;i = 1,. . .
, n } by using the followingscheme:(3.20)This method is known as the Gram-Schmidt procedure. It is easily seen thatthe result is not unique. A re-ordering of the vectors pi before the applicationof the Gram-Schmidt procedure results in a different basis.3.2.4 Parseval’s RelationParseval’s relation states that the inner product of two vectors equals theinner product of their representations with respect to an orthonormal basis.GivennX =x ai= 1i ui(3.21)and(3.22)we have(3.23)52Chapter 3.
Discrete Signal Representationswith(3.24)This is verified by substituting (3.21) into (3.23) and by making use of thefact that the basis is orthogonal:(3.25)For z = y we get from (3.23)llzll = l l a l l .(3.26)It is important to notice that the inner product of the representations isdefined as ( a ,P ) = p H a ,whereas the inner product of the signals may havea different definition.
The inner product of the signals may even involve aweighting matrix or weighting function.3.2.5Complete Orthonormal SetsIt canbe shown thatthe space Lz(a, b ) is complete. Thus, any signalz(t) E Lz(a,b) can be approximated with arbitraryaccuracy by means ofan orthogonal projectioncn$(t>=(2,V i )cpi(t),(3.27)i= 1where n is chosen sufficiently large and the basis vectors cpi(t)are taken froma complete orthonormal set.According to (3.19) and (3.23) we have for the approximation error:533.2.
OrthogonalSeriesExpansionsFrom (3.28) we concluden(3.29)i= 1(3.29) is called the Bessel inequality. It ensures that the squared sum of thecoefficients ( X ,vi)exists.An orthonormalset is said tobe complete if noadditional non-zeroorthogonal vector exists which can be added to the set.When an orthonormalset is complete, the approximationerrortendstowards zero with n + CO.The Bessel inequality (3.29) then becomes thecompleteness relationcIcc(X,(Pi)l2 = 1lXIl2 vE Lz(a,b).(3.30),=lHere, Parseval's relation states(3.31)(3.32)3.2.6Examples of Complete Orthonormal SetsFourier Series. One of the best-known discrete transforms is the Fourierseries expansion.
The basis functions are the complex exponentials(3.33)which form a complete orthonormalset. Theinterval considered is T = [-l, l].The weighting function is g ( t ) = 1. Note that any finite interval can be mappedonto the interval T = [-l, +l].Legendre Polynomials. The Legendre polynomials Pn(t),n = 0 , 1 , .. . aredefined asPn(t)= -l P ( t 2 - 1)n(3.34)2"n! dtnand can alternatively be computed accordingto the recursion formula1Pn(t)= -[(2n - 1)t Pn-l(t) - (n - 1) Pn-z(t)].n(3.35)54Chapter 3. Discrete Signal RepresentationsThe first four functions areA set pn(t), n = 0 , 1 , 2 , .
. . which is orthonormal on the interval [-l, l]with weighting function g ( t ) = 1 is obtained by(3.36)Chebyshev Polynomials. The Chebyshev polynomials are defined asn 2 0,Tn(t)= cos(n arccos t ),-15 t 5 1,(3.37)and can be computed according to the recursionTn(t)= 2t Tn-l(t) - Tn-2(t).(3.38)The first four polynomials areTo(t) = 1 7Tl(t) = t ,T2(t) = 2t2 - 1,T3(t) = 4t3 - 3t.Using the normalization~ o ( t ) for n = oV n ( t )=m ~ , ( t ) for n > owe get a set whichis orthonormal on thefunction g ( t ) = (1 -(3.39)interval [-l7+l]with weightingLaguerre Polynomials.
The Laguerre polynomialsdn~,(t=) et-(tne-t),dtn = 0,1,2,. ..(3.40)553.2. OrthogonalSeriesExpansionscan be calculated by means of the recursionL,(t) = (2n - 1 - t ) L,_l(t)-( n - 1)2 Ln-2(t).(3.41)The normalization1n!(Pn(t)= -L,(t)n = 0 , 1 , 2 , .. .(3.42)yields a setwhich is orthonormal on theinterval [0, m] with weightingfunctiong ( t ) = e c t . The first four basis vectors areAn alternative is to generate the set,-tP$n(t) = T L " ( t ) ,71.
=0 , 1 , 2 , .. . ,(3.43)which is orthonormal with weight one on the interval [0, m]. As will be shownbelow, the polynomials $"(t), n = 0 , 1 , 2 , . . . can be generated by a network.For this, lete-Ptfn(t) = $"(2Pt) = - + W ) .(3.44)The Laplace transform is given by(3.45)Thus, a function fn(t) is obtainedfromanetworkwiththetransferfunction F,(s), which is excited by an impulse. The network can be realizedas a cascade of a first-order lowpass and n first-order allpass filters.Hermite Polynomials. The Hermite polynomials are defined as(3.46)A recursive computation is possible in the formHk(t) = 2t Hk-l(t) - 2(k - 1) Hk-z(t).(3.47)56RepresentationsChapterSignal3. DiscreteWith the weighting function g ( t ) = e P t 2 the polynomialsq5k(t)= (2k Ic!&)p &(t),k = 0 , 1 , 2 , ..
.(3.48)form an orthonormal basis for L2 (R).Correspondingly, the Hermite functionscpk(t)= (2' /c!Hb(t),k = 0 , 1 , 2 , .. . ,(3.49)form an orthonormal basis with weight one. The functions (Pk(t) are also obtained by applying the Gram-Schmidt procedure to the basis {tb eCt2I2; k =0,1,. . .} [57].Walsh Functions.
Walsh functions take on the values 1 and -1. Orthogonality is achieved by appropriate zero crossings. The first two functions aregiven bycpo(t) = ( 1for O 5 t 5 I ,(3.50)Further functions can be computed by means of the recursionforO<t<;- 1) for<t 5 1i(2b)cpm+l(t)=( 2 t - 1)m= 1,2, ...,Ic= 1 , . . . ,2-1for05t<ifor < t 5 1i(3.51)Figure 3.2 shows the first six Walsh functions; they are named according totheir number of zero crossings.3.3GeneralSeries ExpansionsIf possible, one wouldchoose anorthonormal basis for signal representation. However, in practice,a given basis is often notorthonormal.
Forexample, in data transmissiona transmitted signal may have the formz(t) = Cmd(m) s ( t - m T ), where d ( m ) is the data and s(t) is an impulseresponse that satisfies s(t)s(t - mT)dt = S,O. If we now assume that z(t) istransmitted through a non-ideal channel with impulse response h(t), then we573.3. General Series Expansionshave a signal r ( t ) = C , d ( m ) g ( t- mT) with g ( t ) = s ( t ) * h @ )on the receiverside.
This new basis { g ( t - m T ) ;m E Z} is no longer orthogonal, so that thequestion arises of how to recover the data if r ( t ) and g ( t ) are given.3.3.1Calculating the RepresentationIn the following, signals 2 E X with X = span { p l , .. . ,p,} will be considered. We assume that the n vectors { p l , ..
. ,p,} are linearly independent sothat all X E X can be represented uniquely as(3.52)i= 1As will be shown, the representationa = [ a l , .. . ,a,] T(3.53)with respect to a given basis {pl,. . . , p,} can be computed by solving alinear set of equations and also via the so-called reciprocal basis. The set ofequations is obtained by multiplying (inner product) both sides of (3.52) withpj, j = 1,..
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