4.1 Complex Envelope Representation of Bandpass Waveforms (1)(2)3)are low-pass-to-band-pass transformation, the factor eJoct in Eq (1)-3 )translate the spectrum of the baseband signal g(t) up to the carrier frequency fc. in communication terminology, the frequencies in the baseband signal g(t) are said to be heterodyned up to f In Cartesian coordinates, the complex envelope can be represented as: g(t)=x(t+iy(t), where x(t) is the in-phase modulation associated with v(t), and y(t is the quadrature modulation associated with v(t). Alternatively, the polar form of g(t), represented by r(t and A(t,r(t) is said to be the amplitude modulation (AM) on v(t, e(t is said to be the phase modulation(Pm) on v(t) g(t) is the baseband equivalent of the bandpass signal. 16
16 4.1 Complex Envelope Representation of Bandpass Waveforms • (1)(2)(3) are low-pass-to-band-pass transformation, the factor ejωct in Eq.(1)-(3) translate the spectrum of the baseband signal g(t) up to the carrier frequency ƒc. in communication terminology, the frequencies in the baseband signal g(t) are said to be heterodyned up to ƒc . • In Cartesian coordinates, the complex envelope can be represented as: g(t)=x(t)+jy(t), where x(t) is the in-phase modulation associated with v(t), and y(t) is the quadrature modulation associated with v(t). • Alternatively, the polar form of g(t), represented by R(t) and θ(t) , R(t) is said to be the amplitude modulation (AM) on v(t), θ(t) is said to be the phase modulation (PM) on v(t). • g(t) is the baseband equivalent of the bandpass signal
4.2 Representation of Modulated Signals Modulation is the process of encoding the source information m(t)(modulating signal) into a band-pass signal s(t)(modulated signal). Consequently, the modulated signal is just a special application of the bandpass representation. The modulated signalis given by s(t)=Reg(t)e o. e Where o=2nf, fc is carrier frequency. The complex envelope g(t)is a function of the modulating signal m(t) That is g(t)=g[m()] gl. performs a mapping operation on m(t) 17
17 4.2 Representation of Modulated Signals • Modulation is the process of encoding the source information m(t) (modulating signal) into a band-pass signal s(t) (modulated signal). Consequently, the modulated signal is just a special application of the bandpass representation. The modulated signalis given by: j t c s t g t e ( ) =Re ( ) • Where ωc=2πƒc , ƒc is carrier frequency. The complex envelope g(t) is a function of the modulating signal m(t). That is g(t) = g[m(t)] • g[.] performs a mapping operation on m(t)
4.3 Spectrum of Bandpass signals The spectrum of a bandpass signal is directly related to the spectrum of its complex envelope. Theorem: If a bandpass waveform is represented by v(t)=Reg(telo l hen the spectrum of the bandpass waveform is V(O)= 2k(f-f)+G(-f-)( (4-12) and the psd of the waveform is (f-f+p,(f-f (4-13) Where g(=Flg(t] and is the Psd of g(t) 18
18 4.3 Spectrum of Bandpass Signals • The spectrum of a bandpass signal is directly related to the spectrum of its complex envelope. Theorem: If a bandpass waveform is represented by ( ) Re ( ) (4 -11) j t c v t g t e = then the spectrum of the bandpass waveform is ( ) ( ) (4 -12) 2 1 ( ) c c V f = G f − f + G − f − f and the PSD of the waveform is ( ) ( ) (4 -13) 4 1 ( ) v g c g c P f = P f − f +P − f − f Where G(ƒ)=F[g(t)] and Pg (ƒ) is the PSD of g(t)
4.3 Spectrum of Bandpass Signals Proof: because v(t)=Reig(t)e ot g(teJo +=8(te o Then: V(O=v(l)=- 218(0+=F18(Deja. lif we use Flg(tIG(-f, and the frequency translation property of Fourier transforms, we get ()=(-f)+G[(f+fc e Which is Equation(4-12); The PSD for v(t)is obtained by first evaluating the autocorrelation for v(t: R()=(0)c8ekr) 19
19 4.3 Spectrum of Bandpass Signals • Proof: because j t j t j t c c c v t g t e g t e g t e − = = + ( ) 2 1 ( ) 2 1 ( ) Re ( ) • Then: [ ( ) ] 2 1 [ ( ) ] 2 1 ( ) [ ( )] j t j t c c V f v t g t e g t e − =F = F + F If we use F[g* (t)]=G* (- ƒ), and the frequency translation property of Fourier transforms , we get: ( ) [ ( )] 2 1 ( ) c c V f = G f − f + G − f + f Which is Equation(4-12); The PSD for v(t) is obtained by first evaluating the autocorrelation for v(t): ( ) ( ) ( ) ( ) Re ( ) Re ( ) + = + = + j t j t v c c R v t v t g t e g t e
4.3 Spectrum of Bandpass signals Using the identity: Re(c2)rel ci=re(c2 ci+Re(c2 ci and t 2=g(te C1=g(+r)e/(+) ° we can get: 28(8(t+)e-i!t(t+r) R, ([=Re Re g(g(t+r)ejoejo.(+r) Realizing that both < and ref are linear operators, we may exchange the order of the operators without offecting the result, and the autocorrelation becomes 20
20 4.3 Spectrum of Bandpass Signals • Using the identity: Re( ) 2 1 Re( ) 2 1 Re( )Re( ) C2 C1 = C2 C1 + C2 C1 • and j t c C g t e ( ) 2 = ( ) 1 ( ) + = + j t c C g t e • we can get: ( ) ( ) Re ( ) ( ) 2 1 Re ( ) ( ) 2 1 ( ) + − + + + = + j t j t j t j t v c c c c g t g t e e R g t g t e e • Realizing that both < > and Re{ } are linear operators, we may exchange the order of the operators without offecting the result , and the autocorrelation becomes :