Integral Signal Representations (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 Electronic ISBN 0-470-84183-4Chapter 2IntegralSignal Represent at ionsThe integral transform is one of the most important tools in signal theory.The best known example is the Fourier transform,buttherearemanyother transforms of interest.

In the following, W will first discuss the basicconcepts of integral transforms. Then we will study the Fourier, Hartley, andHilbert transforms. Finally, we will focus on real bandpass processes and theirrepresentation by means of their complex envelope.2.1IntegralTransformsThe basic idea of an integral representation is to describe a signal ~ ( tvia) itsdensity $(S) with respect to an arbitrarykernel p(t, S):$(S)p(t, S) ds,t E T.(2.1)Analogous to the reciprocal basis in discrete signal representations (seeSection 3.3) a reciproalkernel O(s,t) may be found such that the densityP(s) can be calculated in the form*(S)=S,~ ( te ()s , t ) d t ,22SE S.232.1.

Integral TransformsContrary to discrete representations, we do not demandthat thekernels cp(t,S )and @(S, t ) be integrable with respect to t.From (2.2) and (2.1), we obtainInorder tostatethecondition for the validity of (2.3) in a relativelysimple form the so-called Dirac impulse d(t) is required. By this we meana generalized function with the propertyLcc~ ( t=)d(t - T ) ).(XdT,X E&(R).(2.4)The Dirac impulse can be viewed as the limit of a family of functions g a ( t )that has the following property for all signals ~ ( tcontinuous)at the origin:An example is the Gaussian functionConsidering the Fourier transform of the Gaussian function, thatisccGCY(u) = l c c g a ( t ) ,-jut-dte- _201 ,W 2we find that it approximates the constant one for a + 0, that is G,(w) M1, W E R.For the Dirac impulse the correspondence d ( t ) t)1 is introducedso that (2.4) can be expressed as X(W)= 1 X(W)in the frequency domain.Equations (2.3) and (2.4) show that the kernel and the reciprocal kernelmust satisfyS,e(s, T) p(t, S) ds = d ( t - T ) .By substituting (2.1) into (2.2) we obtain2(s)=S,L 2 ( a ) cp(t,a) d a e ( s , t ) dt(2.8)24Chapter 2.

Integral Signal Representationswhich implies thatrp(t,c) O(s, t ) d t = S(s - 0).(2.10)ITEquations (2.8) and (2.10) correspond to the relationship (cpi,8j)=the discrete case (see Chapter 3).SijforSelf-Reciprocal Kernels. A special category is that of self-reciprocalkernels. They correspond to orthonormal basesin the discrete case and satisfy= e*(s, t ) .p(t,(2.11)Transforms that contain a self-reciprocal kernel are also called unitary,because they yield 11511 = 1 1 ~ 1 1 .The Discrete Representation as a Special Case.

The discrete representation via series expansion, which is discussed in detail in the next chapter,can be regarded as a special case of the integral representation. In order toexplain this relationship, let us consider the discrete setpi(t) = p(t,si),i = 1 , 2 , 3 , ..

. .(2.12)For signals ~ ( tE )span {p(t,si); i = 1 , 2 , . . .} we may write(2.13)iiInsertion into (2.2) yields*(S)= L Z ( t ) O ( s ,t ) d t(2.14)The comparison with (2.10) shows that in the case of a discrete representationthe density ?(S) concentrates on the values si:*(S)=CQi&(S - Si).1.(2.15)252.1. Integral TransformsParseval’s Relation. Let the signals z ( t ) and y(t) besquareintegrable,z,y E L2 ( T ) .For the densities let=?(S)lz(t)O(s, t ) d t ,(2.16)where O(s, t ) is a self-reciprocal kernel satisfyingS,O(s, t ) @ * ( S ,7) d s==S,@(S,t ) ( ~ ( 7S ), d s(2.17)6 ( t - 7).Now the inner products(X7 U)=/T(2.18)z ( t ) Y * ( t ) dtare introduced.

Substituting (2.16) into (2.18) yields(2,fj) =///STO(s,r) y * ( t ) O*(s,t ) d r d t d s .).(X(2.19)TBecause of (2.17), (2.19) becomes@,G)==l x ( r )l y * ( t ) 6 ( t - r ) d t d rl).(X(2.20)y*(r)dr.From (2.20) and (2.18) we conclude that($76) = (2,!A(2.21)*Equation (2.21) is known as Parseval’s relation. For y ( t ) = z ( t ) we obtain(&,g)= ( x 7 x )+11211 = l l x l l(2.22)26RepresentationsChapterSignal2. Integral2.2The Fourier TransformWe assume a real or complex-valued, continuous-time signal z ( t ) whichisabsolutely integrable (zE Ll(IR)).For such signals the Fourier transform00X ( w ) = L m z ( t ),-jut dt(2.23)exists.

Here, W = 2 nf , and f is the frequency in Hertz.The Fourier transform X ( w ) of a signal X E Ll(IR) has the followingproperties:1. X E ~ o o ( I R )with I I X lloo I 11~111.2. X is continuous.3. If the derivative z'(t) exists and if it is absolutely integrable, then00~ ' ( tC)j w td t4. ForW+ m and W + -m= j w X(W).(2.24)we have X ( w ) + 0.If X ( w ) is absolutely integrable, z ( t ) can be reconstructed from X ( w ) viathe inverse Fourier transform00z(t) =2nX ( w ) ejWtdw(2.25)--oc)for all t where z ( t ) is continuous.The kernel used is1 '2ncp(t,W ) = -eJWt,T = (-m, m),(2.26)and for the reciprocal kernel we have'O(W,t ) = ,-jut,S = (-m, m).(2.27)In thefollowing we will use the notationz ( t ) t)X ( w ) in order to indicatea Fourier transform pair.We will now briefly recall the most important properties of the Fouriertransform.

Most proofs are easily obtained from the definition of the Fouriertransform itself. More elaborate discussions can be found in [114, 221.l A self-reciprocal kernel is obtained either in the form cp(t,w) = exp(jwt)/&integrating over frequency f , not over W = 2xf: cp(t,f ) = exp(j2xft).or by272.2. The Fourier TransformLinearity.

It directly follows from (2.23) that+az(t) Py(t)+a X ( w ) PY(w).t)(2.28)Symmetry. Let z ( t ) t)X ( w ) be a Fourier transform pair. ThenX ( t ) t)27rz(-w).(2.29)Scaling. For any real a , we have(2.30)Shifting. For any real t o , we havez(t - t o )X(w).t)e-jwto(2.31)Accordingly,(2.32)e j w o t z ( t ) t)X ( w - WO).Modulation.

For any real2WO,we have1coswot z ( t ) t)- X ( w - WO)2+ 1X ( w-+WO).(2.33)Conjugation. The correspondence for conjugate functions isz*(t)t)X * ( - W ) .(2.34)Thus, theFourier transform of real signals z ( t )= X* ( t )is symmetric: X * ( W ) =X(-W).Derivatives. The generalization of (2.24) isd"- z ( t ) t)(jw)"dt"X(w).(2.35)Accordingly,d"dw "(-jt)" z ( t ) t)- X @ ) .(2.36)Convolution.

A convolution in the time domain results in a multiplicationin the frequency domain.28Chapter 2. Integral Signal RepresentationsAccordingly,z(t) y(t)1X(w)27rt)-*Y(w).(2.38)Moments. The nth moment of z ( t ) given bycctn ~ ( td t),n = 0,1,2.(2.39)and the nth derivative of X ( w ) at the origin are related as(2.40)Parseval’s Relation. According to Parseval’s relation, inner products oftwo signals can be calculated in the time as well as the frequency domain. Forsignals z ( t )and y ( t ) and theirFourier transforms X ( w ) and Y ( w ) ,respectively,we haveccL~ ( ty*(t)) dt =27rX Y( w* d() w ).(2.41)-WThis property is easily obtained from (2.21) by using the fact that the scaledkernel (27r)-iejwt is self-reciprocal.Using the notation of inner products, Parseval’s relation may also bewritten as1( 2 ’ 9 )= # ’ V .(2.42)From (2.41) with z ( t ) = y(t) we see thatthecalculated in the time and frequency domains:signal energycanbe(2.43)This relationship is known as Parseval’s theorem.

In vector notation it can bewritten as1(2’2)= -( X ’ X ) .(2.44)27r292.3. The Haxtley Transform2.3The HartleyTransformIn 1942 Hartley proposed a real-valued transform closely related to theFouriertransform [67]. It maps a real-valued signal into a real-valued frequencyfunction using only real arithmetic. The kernel of the Hartley transform isthe so-called cosine-and-sine (cas) function, given by+cas w t = cos w t wt.sin(2.45)+This kernel can be seen as a real-valued version of d w t = cos w t j sin wt, thekernel of the Fourier transform. The forward and inverse Hartley transformsare given bymXH(W)= l m x ( t )dtcaswt(2.46)andx(t) = I]XH(W)caswt dw,2lr -m(2.47)where both the signal x(t) and the transform XH(W)are real-valued.In the literature,one also finds a more symmetric version based on the selfreciprocal kernel (27r-+ cas wt.

However, we use the non-symmetric form inorder t o simplify the relationship between the Hartley and Fourier transforms.The Relationship between the Hartley and Fourier Transforms. Letus consider the even and odd parts of the Hartley transform, given byThe Fourier transform may be written asccX(w) = l c c x ( t ) e-jwt dtccx(t) coswt dt - j=X & ( W ) - jX&(W)-XH(W)+Xff(-W)2- j(2.50)XH(W) - X f f ( - W )230Chapter 2. Integral Signal RepresentationsThus,%{X(W)} = X % w ) ,(2.51)S { X ( W ) } = -X&(w).The Hartley transform can be written in terms of the Fourier transformasX&)= % { X ( w ) }- S { X ( w ) } .(2.52)Due to their close relationship the Hartley and Fourier transforms sharemanyproperties. However,some propertiesare entirely different.

Inthefollowing we summarize the most important ones.Linearity. It directly follows from the definition of the Hartley transformthata z ( t )+PY(t)a X H ( w )+ P Y H ( W ) .(2.53)*Scaling. For any real a, we have(2.54)Proof.Time Inversion. From (2.54) with a = -1 we getz(-t)Xff(-w).(2.55)t)Shifting. For any real t o , we havez(t - t o ) t)coswto X H ( W )+ sinwto X H ( - W ) .Proof. We may writeccLz(t - t o ) caswt dt =Lz(J) cas ( W [ [+ t o ] )dJ.Expanding the integralon the right-hand side using the propertycas ( a+ p) = [cosa + sinal cosp + COS^yields (2.56).

0- sinalsinp(2.56)312.3. The Haxtley TransformModulation. For any realWO,we have1coswot z ( t ) t)- X H ( W - WO)2+ -21+WO).(2.57)Proof. Using the property1cosa casP = - cas ( a - P)2+ 51cas ( a + P),we get001,x(t) coswot caswt dtx(t) cas ( [ W12= -X&- WO)-welt) dt ++ -21X &z(t) cas ( [ W +wo]t)dt+WO).Derivatives. For the nth derivative of a signal x(t) the correspondence isd"~dtnz ( t ) t)W" [cos(y)XH(W) sin(y)XH(-W)].(2.58)gProof. Let y ( t ) =x ( t ) .The Fourier transform is Y ( w ) = (jw)" x ( w ) .By writing jn as jn = cos(+ j sin(?), we getY(w) =W"y)[cos (y)+j=W"[cos+ j wnsin( y ) ~] (w )(y)% { X ( w ) } - sin (y)S{X(W)}][cos (y)S { X ( W )+} sin (y)%{x(w)}]For the Hartley transform, this meansyH(w)= w n [cos(?)x&((w)-sin(?)x;(w)+ cos (y)x ; ( w ) + sin (y)x&(w,].Rearranging this expression, based on (2.48) and (2.49), yields (2.58).