Complex envelope representationAll banpass waveforms can be represented by theircomplex envelope forms.Theorem:Any physical banpass waveform can berepresented by:v(t)=Re(g(t)ejoct)Ret.j:real part of .1.g(t) is called the complex envelope ofv(t),and f is the associated carrier frequency.Two otherequivalent representations are:v(t)=R(t)cos[0ct+0(t)]andv(t)=x(t)cos oct-y(t)sin Octwhere g(t)=x(t)+jy(t)=R(t) eje(t)
• All banpass waveforms can be represented by their complex envelope forms. • Theorem:Any physical banpass waveform can be represented by: v(t)=Re{g(t)ejωc t} Re{.}:real part of {.}.g(t) is called the complex envelope of v(t),and fc is the associated carrier frequency.Two other equivalent representations are: v(t)=R(t)cos[ωc t+θ(t)] and v(t)=x(t)cos ωc t-y(t)sin ωc t where g(t)=x(t)+jy(t)=R(t) ejθ(t) Complex envelope representation
Representation of modulated signals:The modulated signals→ a special type of bandpasswaveformSo we haves(t)=Re(g(t)ejoctthe complex envelope is function of the modulating signalm(t):g(t)=g[m(t)]g[.]: mapping functionAll type of modulations can be represented by a specialmapping function g[.]
• Representation of modulated signals • The modulated signals a special type of bandpass waveform • So we have s(t)=Re{g(t)ejωc t} the complex envelope is function of the modulating signal m(t): g(t)=g[m(t)] g[.]: mapping function All type of modulations can be represented by a special mapping function g[.]
. Complex envelope functions for various types ofmodulationType of modulationmapping functions g(m)AMlinear(?)A.[1+m(t)]linearDSB-SCAcm(t)SSBlinearAc[m(t)±jm'(t)]AcejDpm(t)PMnon-linearI m(t)dtjDfFMnon-linearA.e
• Complex envelope functions for various types of modulation • Type of modulation mapping functions g(m) AM Ac[1+m(t)] linear(?) DSB-SC Acm(t) linear SSB Ac[m(t)±jm’(t)] linear PM Ace jDpm(t) non-linear FM non-linear − t jDf m t dt c A e ( )
Spectrum of bandpass signals·Bandpass signal's spectrum complex envelope'sspectrumTheorem:If a bandpass waveform is represented byv(t)=Re(g(t)ejoct)then the spectrum of the bandpass waveform isV(f)=1/2[G(f-f.)+G*(-f-f.))and the PSD of the waveform isPv(f)=1/4[Pg(f-f.)+Pg(-f-f)]where G(f)=F[g(t)], Pg(f) is the PSD of g(t)Proof: v(t)=Re(g(t)ejoct)=1/2 (g(t)ejoct+g*(t)e-joct)V(f)=1/2F (g(t)ejact)+1/2 F (g*(t)e-jact)
• Bandpass signal’s spectrum complex envelope’s spectrum • Theorem:If a bandpass waveform is represented by: v(t)=Re{g(t)ejωc t} then the spectrum of the bandpass waveform is V(f)=1/2[G(f-fc)+G*(-f-fc)] and the PSD of the waveform is Pv(f)=1/4[Pg(f-fc)+Pg(-f-fc)] where G(f)=F[g(t)], Pg(f) is the PSD of g(t). Proof: v(t)=Re{g(t)ejωc t}=1/2{g(t)ejωc t+g*(t)e-jωc t} V(f)=1/2F{g(t)ejωc t}+1/2F{g*(t)e-jωc t} Spectrum of bandpass signals
We have F(g*(t))=G*(-f)Then V(f)=1/2(G(f-f.)+G*[-(f+f)])The PSD for v(t) is obtained by first evaluating theautocorrelation for v(t)R,(t)=<v(t)V(t+t)>=< Re(g(t)ejgct) Re(g(t+ t)ejoc( t)Using the identity: Re(c2)Re(ci)=1/2Re(c*2c1)+ 1/2Re(c2c*)So we have:R(t)=1/2Re<(g*(t)g(t+ t)ejoc>)+ 1/2Re<(g(t)g(t+ t) ejoct ej20ct>)negligible?But Rg(t)= <(g*(t)g(t+ t)>R;(t)=1/2Re<(g*(t)g(t+ t)ejoc>)=1/2 Re(Rg(t) ejoct)Pv(f)-F(R,(t))=1/4[Pg(f-f)+ Pg*(-f-f)]But Pg*(f)= Pg(f),so Pv(f) is real
We have F{g*(t)}=G*(-f) Then V(f)=1/2{G(f-fc)+G*[-(f+fc)]} • The PSD for v(t) is obtained by first evaluating the autocorrelation for v(t). Rv(τ)=<v(t)V(t+τ)>=< Re{g(t)ejωc t} Re{g(t+ τ)ejωc (t+ τ)} Using the identity: Re(c2)Re(c1)=1/2Re(c*2c1)+ 1/2Re(c2c*1) So we have: Rv(τ)=1/2Re<{g*(t)g(t+ τ)ejωc τ>} + 1/2Re<{g(t)g(t+ τ) ejωc t e j2ωc τ>} negligible? But Rg(τ)= <{g*(t)g(t+ τ)> Rv(τ)=1/2Re<{g*(t)g(t+ τ)ejωc τ>}=1/2 Re{Rg(τ) ejωc τ} Pv(f)=F{Rv(τ)}=1/4[Pg(f-fc)+ Pg*(-f-fc)] But Pg*(f)= Pg(f),so Pv(f) is real