Integral Signal Representations (779445), страница 3
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Representation of Bandpass SignalsLowpassFigure 2.5. Generating the complex envelope of a real bandpass signal.real-valued. The signal z ( t )is now described by means of its complex envelopewith respect to an arbitrarypositive center frequency W O :z ( t )= ?J3{ZLP(t) e j w o t } .(2.95)For the spectrum we haveX(W)12= - X,,(W+ -21X;,- WO)(2.96)- WO).(-WCorrespondingly, the system function of the filter can be written as11G(w)= - G,, (W - W O ) - G:,22For the spectrum of the output signal we have+Y(w) ==(2.97)- WO).(-WX ( W )G ( w ):XL, (W- WO)+$ X;,(-WG,,- WO)(WG:,(-w+:XL, - G:,+: X;,(-W-WO)(W- WO)WO)(2.98)-WO)- WO)(-WGLP(w-wo).The last two terms vanish since G,,W< -WO:Y(w) =(W)a xw(~+a X;,W- WO)(-W== 0 for;Y,,(W- W O ) G:,-WO)< -WOand X,,(W)= 0 for- WO)( --W - W O )+ ;Y,*,(-W-WO).(2.99)42Chapter 2.
Integral Signal RepresentationsAltogether this yieldsThis means that a real convolution in the bandpass domain can be replacedby a complex convolution in the lowpass domain:Y(t) = z(t)* g ( t )+1YLP( t )= 5 Z L P ( t )* QLP (t).(2.101)Note that theprefactor 1 / 2 must be takeninto account.
Thisprefactor did notappear in the combination of bandpass filtering and generating the complexenvelope discussed above. As before, a real filter gLP(t)is obtained if G(w) issymmetric with respect to WO.Inner Products. We consider the inner product of two analytic signalsz+(t)= ~ ( t+)j 2 ( t )andy+(t)= y(t)+j c ( t ) ,where z(t) and y(t) are real-valued. We have(X+,Y+> = ( X ,Y)+ (%C) + j( 2 ,Y) + j(X,C).(2.102)Observing (2.73), we get for the real part%{(X+,Y + > l = 2 (2,Y)*(2.103)If we describe ~ ( tand) y(t) by means of their complex envelope with respectto the same center frequency, we get(2.104)For the implementationof correlation operations this means thatcorrelationsof deterministic bandpass signals can be computed in the bandpass domainas well as in the equivalent lowpass domain.Group and Phase Delay.
The group and phase delay of a system C(w)are defined as(2.105)and(2.106)whereC ( W ) = IC(W)I &+).(2.107)432.5. Representation of Bandpass SignalsIn order to explain this, let us assume that C ( W is) a narrowband bandpasswith B << W O . The system function of the associated analytic bandpass maybe written asBecause of B<< W O , CLp( W )may be approximated asFor the complex envelope CLP(w) = C,,(WCLP(w) M IC(w0)l e - j W O T p ( W o )+ W O ) it follows thate-jwTg(wO),W5 B/2,(2.109)(2.110)with T~ and T~ according to (2.105) and (2.106).
If we now look at the inputoutput relation (2.100) we getyLP(W)M1~ ( wI e-jwoTp(wo)o )2-~XLP@).e--jwTg(wo)(2.111)Hence, in the time domainwhich means that the narrowband system C(W)provides a phase shift byT ~ ( W O ) and a time delay by T~( W O ) .2.5.2StationaryBandpass ProcessesIn communicationswe must assumethat noise interferes with bandpasssignalsthat are to be transmitted. Therefore the question arises of which statisticalproperties the complex envelope of a stationary bandpass process has. Weassume a real-valued, zero mean, wide-sense stationary bandpassprocess z ( t ) .The autocorrelation function of the process is given byT,, (T)= T,,(-T)+=E { ~ ( t )~ (. tT)}(2.113)Nowwe consider the transformed process 2 ( t ) .
For the power spectraldensity of the transformed process, ,522 ( W ) , we conclude from (1.105):1 forW#Oo forW=O44RepresentationsChapterSignal2. Integralwhere i ( t ) t)I?(w). Thus, the process 2 ( t ) has the same power spectraldensity, and consequently the same autocorrelation function, as the processz(t):(2.115)= Tzz (7).Tjrjr (T)For the cross power spectral densities Szj:( W ) andto (1.102):SZ&)fi(4=272% ( W )we get according&%(W),(2.116)SjrZ(W)=fi*(W& )% ( W ) .Hence, for the cross correlation functions:TZ&(T)=+ZZ(T),&(T)=Tz?(-T)(2.117)= t z z ( - T ) = -tzz(.).Now we form the analytic process z+(t):+j q t ) .X+@) = z ( t )(2.118)For the autocorrelation function we haveT,+,+(T)=E { [ z ( t+) j 2 ( t ) ] * [z(t+ T) + j=Tzz (T)+j= 2 Tzz(.)Tzjr(.l- j TB,qt+T)]}+ rjrjr (.l(2.119)+ 2 j P,,(.).This means that the autocorrelation function of the analytic process is ananalytic signal itself.
The power spectral density isS,+,+(W)={;> 0,for W < 0.SZZ((w) for W(2.120)Finally, we consider the complex process zLP(t)derived from the analyticprocesszLp( t ) = z+(t)e-jwot(2.121)= u ( t )+ j w(t).For the real part u(t)we haveu(t) = %{[z(t)+ j $ ( t ) ]e-jwot}= z ( t ) coswot+P(t)sin&=+ [z+(t)e-jwot+ [X+ ( t ) ] *ejwot],(2.122)452.5.
Representation of Bandpass Signals(2.123)In (2.123) two complex exponential functions dependent onwhose prefactors reduce to zero:t are includedE{[Z+(t)]*[z+(t+T)]*}*= E { z + ( t )z+(t+.r))=++ T) + j 2(t + T))}E { ( z ( t ) j q t ) ) (z(t(2.125)T,, (T) cos WOT=+ F,,(T) sin WOTIn a similar way we obtainfor the autocorrelation functionof the imaginary part of the complex envelope.The cross correlation functionbetween the real and theimaginary part is givenbyTuv(.l=--Tvu(.)(2.127)From (2.125) - (2.127) we conclude that the autocorrelation function ofthe complex envelope equals the modulated autocorrelation function of the46Chapter 2. Integral Signal Representationsanalytic signal:Correspondingly, we get for the power spectral density:4+&,(WWO)for W+WOfor+WOW> 0,< 0.(2.129)We notice that the complex envelope is a wide-sense stationary processwith specific properties:0eThe autocorrelation function of the real part equals that of the imaginary part.The cross correlation function between the real and imaginary part isantisymmetric with respect to r.
In particular, we haveTuv(O)= rvu(0)= 0.In the special case of a symmetric bandpass process, we have(W>= SzLPzLP ( - W ) *(2.130)Hence, we see that the autocorrelation function of xLP(t)is real-valued. Italso means that the cross correlation between the real and imaginary partvanishes:TU,(T) = 0, v r.(2.131).
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