B.Complex baseband representation of passband signals Suppose that x(r)is a real-valued passband signal with a spectrum centered atf=f The complex baseband equivalent signal (sometimes also called complex envelope) of x(r)can be represented as x()= ()+fe (1.1) analytic passband signal) In terms of Fourier transforms X(f)=Xf+)= (x(f+f). f+f20 10. f+f<0 The original passband signal can be recovered from()by x()=x(e 1.2) The relationship between x()x()andx()is shown in Fig.1.1 in terms of their spectrum. xU) 料W f. 0 f. X①4 0 X0 0W2 Figure 1.1 Fourier transform of a passband signal()and the transform of the corresponding complex baseband signal Baseband to passband and go back 11 11 B. Complex baseband representation of passband signals „ Suppose that x(t) is a real-valued passband signal with a spectrum centered at f = fc. The complex baseband equivalent signal (sometimes also called complex envelope) of x(t) can be represented as 2 b 1 ( ) [ ( ) ( )] ˆ 2 c j ft x t x t jx t e− π = + (1.1) analytic passband signal xA(t) In terms of Fourier transforms 2 ( ), 0 () ( ) 0, 0 c c b Ac c Xf f f f Xf Xf f f f ⎪⎧ + + ≥ = += ⎨ ⎪⎩ + < „ The original passband signal can be recovered from xb(t) by 2 () 2 () c j ft b xt x te π = ⎡ ⎤ R ⎣ ⎦ (1.2) The relationship between ( ), ( ) and ( ) A b x txt xt is shown in Fig. 1.1 in terms of their spectrum. X(f) -fc f c 0 W 0 f c XA(f) 0 W/2 Xb (f) Figure 1.1 Fourier transform of a passband signal x(t) and the transform of the corresponding complex baseband signal. „ Baseband to passband and go back