Signals and Signal Spaces (Mertins - Signal Analysis (Revised Edition))

Signal Analysis: Wavelets, Filter Banks, Time-Frequency TransformsandApplications. Alfred MertinsCopyright 0 1999 John Wiley & Sons LtdPrint ISBN 0-471-98626-7 ElectronicISBN 0-470-84183-4Chapter 1Signals and Signal SpacesThe goal of this chapter is to give a brief overview of methods for characterizing signals and for describing their properties. Wewill start with adiscussion of signal spaces such as Hilbert spaces, normed and metric spaces.Then, the energy density and correlation function of deterministic signals willbe discussed.

The remainder of this chapter is dedicated to random signals,which are encountered in almost all areas of signal processing. Here, basicconcepts such as stationarity, autocorrelation, and power spectral densitywillbe discussed.1.l1.1.1Signal SpacesEnergy and Power SignalsLet us consider a deterministic continuous-time signalz(t),which may be realor complex-valued.

If the energy of the signal defined byis finite, we call it an energy signal. If the energy is infinite, but the meanpower12Chapter 1 . Signals and Signal Spacesis finite, we call z ( t ) a power signal. Most signals encountered in technicalapplications belong to these two classes.A second important classification of signals is their assignmentto thesignalspaces L,(a, b ) , where a and b are the interval limits within which the signalis considered. By L,(a, b) with 1 5 p < m we understand that class of signalsz for which the integralI”lX(t)lPdtto be evaluated in the Lebesgue sense is finite.

If the interval limits a and bare expanded to infinity, we also write L p ( m )or LP@). According to thisclassification, energy signals defined on the real axis are elements of the spaceL2 (R).1.1.2NormedSpacesWhen considering normed signal spaces,we understand signals as vectorsthatare elements of a linear vector spaceX . The norm of a vector X can somehowbe understood as the length of X.

The notation of the norm is 1 1 ~ 1 1 .Norms must satisfy the following three axioms, where a is an arbitraryreal or complex-valued scalar, and 0 is the null vector:Norms for Continuous-Time Signals. The most common norms forcontinuous-time signals are the L, norms:(1.6)For p+ m, the norm (1.6) becomesllxllL, = ess sup Iz(t)l.astsbFor p = 2 we obtain the well-known Euclidean norm:Thus, the signal energy according to (1.1) can also be expressed in the form00XEL2(IR).(1.8)31.1. Signal SpacesNorms for Discrete-Time Signals. The spaces l p ( n ln2), are the discretetime equivalent to the spaces L p ( a ,b ) .

They are normed as follows:(1.9)For p+ CO,(1.9) becomes llzlleoo = sup;LnlIx(n)I.For p = 2 we obtainThus, the energy of a discrete-time signal z ( n ) ,n E Z can be expressed as:n=-cc1.1.3MetricSpacesA function that assigns a real number to two elements X and y of a non-emptyset X is called a metric on X if it satisfies the following axioms:y) 2 0,(i)d(x,(ii)(iii)d(x, y) = 0 if and only ifX= y,d(X,Y) = d(Y,X),d(x, z ) I d(x, y) d(y, z ) .(1.13)(1.14)+The metric d(x,y) can be understood as the distance between(1.12)Xand y.A normed space is also a metric space. Here, the metric induced by thenorm is the norm of the difference vector:Proof (norm + metric).

For d ( z , g) = 112 - 2 / 1 1 the validity of (1.12) immediately follows from (1.3). With a = -1, (1.5) leads to 1 1 2 - 2 / 1 1 = 119 - zlI,and (1.13) is also satisfied. For two vectors z = a - b and y = b - c thefollowing holds according to (1.4):Thus, d(a,c ) I d(a,b)+ d(b,c ) , which means that also (1.14) is satisfied.

04Chapter 1 . Signals and Signal SpacesAn example is the Eucladean metric induced by the Euclidean norm:1/2,I 4 t ) - Y,,,l2dt]Accordingly, the following distancebetweenstated:2,Y EL z ( a ,b ) .(1.16)discrete-time signals canbeNevertheless, we also find metrics which are not associated with a norm.An example is the Hamming distancend(X,Y) = CK X k+ Y k ) mod 21,k=lwhich states the number of positions where twobinarycode words X =[Q, 2 2 , . . . ,X,] and y = [ y l ,y ~. .,.,yn] with xi,yi E (0, l} differ (the space ofthe code words is not a linear vector space).Note. The normed spaces L, and l , are so-called Banachspaces, whichmeans that they are normed linear spaces which are complete with regard totheir metric d ( z , y) = 1 1 2 - y 11.

A space is complete if any Cauchy sequenceofthe elements of the space converges within the space. That is, if 1 1 2 , - zl,+0 as n and m + m, while the limit of X, for n + 00 lies in the space.1.1.4Inner Product SpacesThe signal spaces most frequently considered are the spaces L 2 ( a , b ) and&(nl,n2); for these spaces inner products can be stated. An inner productassigns a complex number to two signals z ( t ) and y ( t ) , or z(n) and y ( n ) ,respectively.

Thenotation is ( X ,y). An inner productmust satisfy thefollowing axioms:(i)(4(iii)k,Y>=( Y A *(1.18)(aa:+Py,z) = Q ( X , . Z ) + P ( Y , 4(2,~2 )0, ( 2 , ~= )0 if and only if(1.19)(1.20)Here, a and ,B are scalars with a,@EExamples of inner products areX = 0.(E,and 0 is the null vector.(1.21)51.1. Signal SpacesandThe inner product (1.22) may also be written aswhere the vectors are understood as column vectors:'More general definitions of inner products include weighting functions orweighting matrices. An inner product of two continuous-time signals z ( t ) andy ( t ) including weighting can be defined aswhere g ( t ) is a real weighting function with g ( t ) > 0, a 5 t5 b.The general definition of inner products of discrete-time signals iswhere G is a real-valued, Hermitian, positive definite weighting matrix.

Thismeans that GH = GT = G, and all eigenvalues Xi of G must be larger thanzero. As can easily be verified, the inner products (1.25) and (1.26) meetconditions (1.18) - (1.20).The mathematical rules for inner products basically correspond to thosefor ordinary productsof scalars. However, the order in which the vectors occurmust be observed: (1.18) shows that changing the order leads to a conjugationof the result.As equation (1.19) indicates, a scalar prefactor of the left argument maydirectly precede the inner product: (az,y) = a (2,y). If we want a prefactorlThe superscript T denotestransposition.Theelementsof a and g mayberealorcomplex-valued.

A superscript H , as in (1.23), means transposition and complex conjug&tion. A vector a H is also referred to as the Herrnitian of a.If a vector is to be conjugatedbut not to be transposed, we write a * such that a H = [=*lT.6Chapter 1 . Signals and Signal Spacesof the right argument to precede the inner product, it must be conjugated,since (1.18) and (1.19) lead toDue to (1.18), an inner product(2,~is )always real: ( 2 , ~= )!I&{(%,z)}.By defining an inner product we obtain a norm and also a metric.

Thenorm induced by the inner product isWe will prove this in the following along with the Schwarz inequality, whichstatesIb , Y >I I l 1 4(1.29)IlYll.Equality in (1.29) is given only if X and y are linearly dependent, that is, ifone vector is a multiple of the other.Proof (inner product + n o m ) . From (1.20) it follows immediately that(1.3) is satisfied. For the norm of a z , we conclude from (1.18) and (1.19)llazll = ( a z , a z y= [la12(2,z)]1/2 = la1( 2 , 2 ) 1 /=2la1 l l z l l .Thus, (1.5) is also proved.Now the expression112+will be considered. We haveAssuming the Schwarz inequality is correct, we conclude112+ Y1I2 I 1 1 4 1 2 + 2 l l 4 lThis shows that also (1.4) holds.IlYll + 11YIl2 = (1 1 4 + llYll)2*0Proof of the Schwarz inequality.

The validity of the equality sign in theSchwarz inequality (1.29) for linearly dependent vectors can easily be proved71.1. Signal Spacesby substituting z = a y or y = a z , a E C,into (1.29) and rearranging theexpression obtained, observing (1.28). For example, for X = a y we haveIn order to prove the Schwarz inequality for linearly independent vectors,some vector z = z + a y will be considered. On the basis of (1.18) - (1.20) wehave0I(G.4=(z a y , X=(z,z+ay)+(ay,z+ay)=(~,~)+a*(~,Y)+a(Y,~)+aa*(Y,Y).++ay)This also holds for the special a (assumption: y(1.30)# 0)and we getThe second and the fourth termcancel,(1.32)Comparing (1.32) with (1.28) and (1.29) confirms the Schwarz inequality.0Equation (1.28) shows that the inner products given in (1.21) and (1.22)lead to the norms (1.7) and (1.10).Finally, let us remark that a linear space with an inner product which iscomplete with respect to the induced metric is called a Hilbert space.81.2Chapter 1 .

Signals and Signal SpacesEnergyDensityandCorrelation1.2.1Continuous-Time SignalsLet us reconsider (1.1):00(1.33)E, = S__lz(t)l2 dt.According to Parseval’s theorem, we may also writeE, = -(1.34)where X(W)is the Fourier transform of ~ ( t )The. ~quantity Iz(t)I2in (1.33)represents the distribution of signal energy withrespect to time t ; accordingly,IX(w)I2 in (1.34) can be viewed as the distribution of energy with respect tofrequency W. Therefore IX(w)I2 is called the energy density spectrum of z ( t ) .We use the following notation= IX(w)I2.(1.35)The energy density spectrum S,“,(w) can also be regarded as the Fouriertransform of the so-called autocorrelation functionccr,”,(r) =Jz * ( t )z(t + r ) dt = X * ( - r )* X(.).(1.36)-ccWe haveccS,”,(W)= l c c r f z ( ~e-jwT)dr.(1.37)The correspondence is denoted as S,”,(w) t)r,”,(r).The autocorrelationfunction is a measure indicating thesimilarity betweenan energy signal z(t) and its time-shifted variant z r ( t )= z ( t r ) .