Integral Signal Representations (779445), страница 2
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032RepresentationsChapterSignal2. IntegralConvolution. We consider a convolution in time of two signals z ( t ) andy(t). The Hartley transforms are XH(W)and YH(w),respectively. The corre-spondence isThe expression becomes less complex for signals with certain symmetries.For example, if z ( t ) has even symmetry, then z ( t )* y(t) t)XH(W)YH(w).If z ( t ) is odd,then z(t) * y(t)XH(W)YH(-w).Pro0f .cc[z(t)* y ( t ) ] caswt dt =cc=Iccz(r)-Lccz ( r )y(t[11-r)drcc- r ) caswtdtcaswt dtdrz(r) [ c o s w ~ Y ~ (+w sinwTYH(-w))] dr.To derive the last line, we made use of the shift theorem. Using (2.48) and(2.49) we finally get (2.59). 0Multiplication.
The correspondence for a multiplication in time isProof. In the Fourier domain, we have332.3. The Haxtley TransformFor the Hartley transform this means+X g w ) * Y i ( w ) - X & ( w ) * Y i ( w ) X;;(w) * Y i ( w )+ X g w ) * Y i ( w ) .Writing this expression in terms of X H ( W )and Y H ( wyields)(2.60). 0Parseval's Relation. For signals x ( t ) and y(t) and their Hartley transformsX H ( W )and Y H ( w )respectively,,we haveccLx ( t ) y(t) dt ='127rX H ( W )Y H ( wdw.)(2.61)-ccSimilarly, the signal energy can be calculated in the time andin the frequencydomains:ccE, = I c c x z ( t )d t(2.62)These properties are easily obtained from the results in Section 2.1 by usingthe fact that the kernel (27r-5 cas wt is self-reciprocal.Energy Density and Phase.
In practice, oneof the reasons t o compute theFourier transform of a signal x ( t ) is t o derive the energy density S,",(w) =IX(w)I2and the phase L X ( w ) . In terms of the Hartley transform the energydensity becomesS,",(4 =I W w I l Z +I~{X(w))lZ(2.63)-X$@)+ X&+)2The phase can be written as(2.64)34RepresentationsChapterSignal2. Integral2.42.4.1The Hilbert TransformDefinitionChoosing the kernelp(t - S) =-1(2.65)7r(t - S ) ’~we obtain the Hilbert transform.
For the reciprocal kernel O(s - t ) we use thenotation i ( s - t ) throughout the following discussion. It ish(s - t ) =17r(s - t )~= p(t - S ) .(2.66)With i(s) denoting the Hilbert transform of z ( t ) we obtain the followingtransform pair:-1x ( t ) = 7r -cc ?(S)- t - s ds1$(2.67)03dt.Here, the integration hasto be carried out according to the Cauchy principalvalue:ccThe Fourier transforms of p(t) and i ( t ) are:@(W)@ ( O ) = 0,= j sgn(w)withB(w) =-jsgn(w)withB(0)= 0.(2.69)(2.70)In the spectral domain we then have:X(W) =@(W)X(w) =B(w)X ( W )X(w)= j sgn(w) X ( w )= -jsgn(w) ~ ( w ) .(2.71)(2.72)We observe that the spectrum of the Hilbert transform $(S) equals thespectrum of z ( t ) ,except for the prefactor - j sgn(w). Furthermore, we seethat, because of @ ( O ) = k(0)= 0, the transform pair (2.67) isvalidonlyfor signals z ( t ) with zero mean value.
The Hilbert transform of a signal withnon-zero mean has zero mean.352.5. Representation of Bandpass Signals2.4.2Some Properties of the HilbertTransform1. Since the kernel of the Hilbert transform is self-reciprocal we have2. A real-valued signal z ( t )is orthogonal to its Hilbert transform 2 ( t ) :(2.74)(X,&)= 0.We prove this by making use of Parseval’s relation:27r(z,2) = ( X ’ X )cc-LX ( w ) [ - j sgn(w)]* X * ( w ) dw(2.75)CQI X ( W ) ~sgn(w)~dw= jJ-cc= 0.3. From (2.67) and (2.70) we conclude that applying the Hilbert transformtwice leads to a sign change of the signal, provided that the signal haszero mean value.2.5Representation of Bandpass SignalsA bandpass signal is understood as a signal whose spectrum concentrates ina region f [ w o - B , WO B ] where WO 2 B > 0.
See Figure 2.1 for an exampleof a bandpass spectrum.+’-0 0IxBP(W>l*00Figure 2.1. Example of a bandpass spectrum.036RepresentationsChapterSignal2. Integral2.5.1Analytic SignalandComplexEnvelopeThe Hilbert transform allows us to transfer a real bandpass signal xBP(t) intoa complex lowpass signal zLP(t).For that purpose, we first form the so-calledanalytic signal xkP( t ) ,first introduced in [61]:xzp(t) = XBP(t)+jZBP(t).(2.76)Here, 2BP(t)is the Hilbert transform of xBP(t).The Fourier transform of the analytic signal isx ~ ~ ( w=) xBP(w)+ j JiBP(w) =2 XBp(w) forW> 0,xBP(w)forW= 0,forW< 0.l 0This means that the analyticfrequencies only.signal hasspectralcomponents(2.77)for positiveIn a second step, the complex-valued analytic signal can be shifted intothe baseband:ZLP(t) = xc,+,(t)e-jwot.(2.78)Here, the frequency WO is assumed to be the center frequency of the bandpassspectrum,as shown in Figure 2.1.
Figure 2.2 illustratestheprocedure ofobtaining the complex envelope. We observe that it is not necessary to realizean ideal Hilbert transform with system function B ( w ) = - j sgn(w) in orderto carry out this transform.The signal xLP(t)is called the complex envelope of the bandpass signalxBp(t). The reason for this naming convention is outlined below.In orderto recover a real bandpass signal zBP(t)from its complex envelopexLp( t ) ,we make use of the fact thatfor(2.80)372.5.
Representation of Bandpass Signals\ ' /IWOWI00WFigure 2.2. Producing the complex envelope of a real bandpass signal.Another form of representing zBP(t)is obtained by describing the complexenvelope with polar coordinates:(2.81)wit hIZLP( t )I = . \ / 2 1 2 ( t )+ 212 ( t ) ,v(t)tane(t) = -.u(t)(2.82)From (2.79) we then conclude for the bandpass signal:ZBP(t)=IZLP(t)lcos(uot+ e(t)).(2.83)We see that IxLP(t)l can be interpreted asthe envelope of the bandpass signal(see Figure 2.3). Accordingly, zLP(t)is called the complex envelope, and the38Chapter 2. Integral Signal RepresentationsFigure 2.3. Bandpass signal and envelope.analytic signal is called the pre-envelope.
The real part u ( t ) is referred to asthe in-phase component, and the imaginary part w ( t ) is called the quadraturecomponent.Equation (2.83) shows that bandpass signals can in general be regardedas amplitude and phase modulatedsignals. For O ( t ) = 8 0 we have a pureamplitude modulation.It should be mentionedthat the spectrumof a complex envelopeis alwayslimited to -WO at the lower bound:XLP(w)0 forW< -WO.Thispropertyimmediatelyresultsfromthefactcontains only positive frequencies.(2.84)thatananalyticsignalApplication in Communications. In communications we often start witha lowpass complex envelopezLP(t)and wish to transmit it asa real bandpasssignal zBP(t).Here, the real bandpass signal zBP(t)is produced from zLp( t )according to (2.79). In thereceiver, zLp( t )is finally reconstructed as describedabove.
However, one important requirementmustbemet, whichwill bediscussed below.The real bandpass signalzBp(t) = u ( t ) coswot(2.85)is considered. Here, u(t)is a given real lowpass signal. In order to reconstructu(t) from zBP(t),we have to add the imaginary signal ju(t)sinwot to thebandpass signal:z(p)(t):= u(t) [coswot+ j sin wot] = u ( t ) ejwot.(2.86)Through subsequent modulation we recover the original lowpass signal:u ( t )= .(P) ( t ) e-jwot.(2.87)392.5. Representation of Bandpass Signals-W0IWOWFigure 2.4. Complex envelope for the case that condition (2.88) is violated.The problem, however, is to generate u(t)sinwot from u(t)coswot in thereceiver.
We now assume that u ( t ) ejwOt is analytic, which means thatU(w)0 for w< -WO.(2.88)As can easily be verified, under condition (2.88) the Hilbert transform of thebandpass signal is given by2 ( t ) = u(t) sinwot.(2.89)Thus, under condition (2.88) the required signal z(p)(t)equals the analyticsignal ziP(t),and the complex envelope zLp( t )is identical to the given u(t).The complex envelope describes the bandpass signal unambiguously, that is,zBP(t)can always be reconstructed from zLP(t);the reverse, however, is onlypossible if condition (2.88) is met. This is illustrated in Figure 2.4.Bandpass Filtering and Generating the Complex Envelope. In practice, generating a complex envelope usually involves the task of filtering thereal bandpass signal zBP(t)out of a more broadband signal z ( t ) .This means40RepresentationsChapterSignal2. Integralthat zBP(t)= z ( t ) * g ( t ) has to be computed,where g ( t ) is the impulse responseof a real bandpass.The analytic bandpassg+@) associated with g ( t ) has the system function+G+(w)= G ( w ) [l jB(w)].(2.90)Using the analytic bandpass, the analyticsignal can be calculated as(2.91)For the complex envelope, we haveIf we finally describe the analytic bandpassenvelope of the real bandpassby means of the complex(2.93)this leads toXL,(W)=X(W+WO)GLP ( W ) .(2.94)We find that XLP(w) is also obtained by modulating the real bandpass signalwith e-jwot and by lowpass filtering the resulting signal.
See Figure 2.5 foran illustration.The equivalent lowpass G L P ( w ) usually has a complex impulse response.Only if the symmetry condition GLP(u)= GE,(-w) is satisfied, the result isa real lowpass, and the realization effort is reduced. This requirement meansthat IG(w)I musthave even symmetryaround W O and the phaseresponseof G ( w ) must be anti-symmetric. In this case we also speak of a symmetricbandpass.Realization of Bandpass Filters by Means of EquivalentLowpassFilters. We consider a signal y(t) = z ( t )* g @ ) ,where z ( t ) ,y(t), and g ( t ) are412.5.
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