4.1 Complex Envelope Representation of Bandpass Waveforms (Complex Envelope Represention) All bandpass waveforms, whether they arise from a modulated signal, interfering signal, or noise, may be represented in a convenient form v(t)will be used to denote the bandpass waveform such as the signal when s(t=v(t) the noise when n(t=v(t), and the filtered signal plus noise at the channel output when r(t=v(t), or any other type of bandpass waveform
11 4.1 Complex Envelope Representation of Bandpass Waveforms (Complex Envelope Represention) • All bandpass waveforms, whether they arise from a modulated signal, interfering signal, or noise, may be represented in a convenient form. • v(t) will be used to denote the bandpass waveform such as the signal when s(t)=v(t), • the noise when n(t)=v(t), • and the filtered signal plus noise at the channel output when r(t)=v(t), or any other type of bandpass waveform
4.1 Complex Envelope Representation of Bandpass Waveforms (Complex Envelope Represention) Theorem: Any physical bandpass waveform can be represented by v(t)=Reig(t)e/ (4-1) v()=R()Cost+(t)](4-1b) v(t)=x(t)cos@t-y(tsin@t (4-Ic)
12 4.1 Complex Envelope Representation of Bandpass Waveforms (Complex Envelope Represention) • Theorem: Any physical bandpass waveform can be represented by: ( ) Re ( ) (4-1) j t c v t g t e = v(t) R(t)Cos[ t (t)] (4 -1b) = c + v(t) x(t)cos t y(t)sin t (4 -1c) = c − c
4.1 Complex Envelope Representation of Bandpass Waveforms (Complex Envelope represention)p235 Where the complex envelope of ve (t)is g(t) g()=x()+jy(1)=g(1)|e1480)=R()e10() x(t)=Reg(t)=R(t)cose(t) y(t)=Img(D=R( (t)sine(t R()当8()|=yx2()+y2() 0()=∠g()=tmyY) x(t) f is associated carrier frequency(in hertz), Oc=2r f 13
13 4.1 Complex Envelope Representation of Bandpass Waveforms (Complex Envelope Represention)p235 • Where the complex envelope of v(t) is g(t): ( ) ( ) ( ) ( ) ( ) | ( )| ( ) j g t j t g t x t j y t g t e R t e = + = x(t) = Reg(t) R(t)cos(t) ( ) | ( )| ( ) ( ) ( ) Im ( ) ( )sin ( ) 2 2 R t g t x t y t y t g t R t t + = = − ( ) ( ) ( ) ( ) tan 1 x t y t t g t • ƒ c is associated carrier frequency(in hertz), ωc=2π ƒ c
4.1 Complex Envelope Representation of Bandpass Waveforms Proof: Any physical waveform (it does not have to be periodic)may be represented over all time, To >o, by the complex Fourier series: v(t)=∑cneo,o=2/ 1=- Because the physical waveform is real, c-p=Cn, and using Re{}=(1/2){}+(1/2){-}, we get: v()=Re{co+2∑cnel Because v(t)is a bandpass waveform, the cn have negligible magnitudes for n in the vicinity of o and, in particular, Co=0. Thus, with the introduction of an arbitrary parameter fc, eqation above becomes:
14 4.1 Complex Envelope Representation of Bandpass Waveforms • Proof: Any physical waveform (it does not have to be periodic) may be represented over all time, , by the complex Fourier series: T0 → 0 0 ( ) , 2 / 0 v t c e T n n jn t n = = =+ =− Because the physical waveform is real,c-n=cn , and using Re{.}=(1/2){.}+ (1/2){.}* , we get: = + = j t n n v t c c e 0 1 ( ) Re 0 2 Because v(t) is a bandpass waveform , the cn have negligible magnitudes for n in the vicinity of 0 and , in particular, c0 =0. Thus , with the introduction of an arbitrary parameter fc , Eqation above becomes:
4.1 Complex Envelope Representation of Bandpass Waveforms v(t)=Re(2)c,eJ(no-o ). e So that(1)follows, where n=oo g()≡2 ce (nC02) Because v(t)is a bandpass waveform with nonzero spectrum concentrated near f=f, the Fourier coefficients c are nonzero only for values of n in the range ±nfo≈f. Therefore, from Eq(4-8),g(t)hasa a spectrum that is concentrated near f=0. That is, g(t) is a baseband waveform The waveform g(t), x(t),y(t),r(t), and e(t are all baseband waveforms, and except for g(t), they are all real waveform. r(t) is a nonnegative real waveform 15
15 4.1 Complex Envelope Representation of Bandpass Waveforms Because v(t) is a bandpass waveform with nonzero spectrum concentrated near ƒ = ƒ c , the Fourier coefficients cn are nonzero only for values of n in the range ±nƒ0 ≈ ƒc . Therefore , from Eq(4-8), g(t) has a spectrum that is concentrated near ƒ =0 . That is , g(t) is a baseband waveform. = − = = j n t j t n n n c c v t c e e ( ) Re (2 ) ( ) 1 0 So that (1) follows ,where j n t n n n c g t c e ( ) 1 0 ( ) 2 − = = The waveform g(t), x(t),y(t), R(t), and θ(t) are all baseband waveforms, and except for g(t), they are all real waveform. R(t) is a nonnegative real waveform