Natural sampling(gating): Definition:If w(t) is an analog bandlimited waveform(BHz),the PAM signal that uses natural sampling isw,(t)=w(t)s(t)where s(t)=ZII[(t-kT,)/t] and f=1/T,≥2BTheorem:The spectrum for a natural sampling PAM signalw,(t) is:W,(f)=F[ws(t)]=dE(sinnd/πnd)W(f-nf,)where f= Ts,d is the duty cycle of s(t) (d= t/ T,),and W(f) isthe FT of w(t).Proof: Ws(f)=W(f)*S(f)and we have s(t)'s Fourier series:s(t)=cnej2元nfs and cn=d (sin元nd/元nd)
• Definition:If w(t) is an analog bandlimited waveform (BHz),the PAM signal that uses natural sampling is ws(t)=w(t)s(t) where s(t)=∑∏[(t-kTs)/τ] and fs=1/Ts≥2B • Theorem:The spectrum for a natural sampling PAM signal ws(t) is: Ws(f)=F[ws(t)]=d∑(sinπnd/πnd)W(f-nfs) where fs= Ts ,d is the duty cycle of s(t) (d= τ/ Ts),and W(f) is the FT of w(t). Proof: Ws(f)=W(f)*S(f) and we have s(t)’s Fourier series: s(t)=∑cne j2πnfs and cn=d (sinπnd/πnd) Natural sampling(gating)
So:S(f)=F[s(t)]-Zcn8(f-nf,)Ws(f)=W(f)*S(f)=d(sinnd/元nd)W(f-nf,)w(t)s(t)Baseband analog waveformSwitchingwaveform (d-1/3)Ws(t)Resulting PAM signal
So: S(f)=F[s(t)]=∑cnδ(f-nfs) Ws(f)=W(f)*S(f)=d∑(sinπnd/πnd)W(f-nfs) w(t) t Baseband analog waveform s(t) T t s τ Switching waveform (d=1/3) ws(t) t Resulting PAM signal
Generation of natural sampling PAM signal and itsspectrum:Analog bilateral switchw(t)ws(t)s(t)Generation ofPAMClockWs(f)[ W() |d|sin(元tf)/元tfd=1/3fsffB-BMagnitude spectrum of ws(t)Magnitude spectrum of w(t)
• Generation of natural sampling PAM signal and its spectrum: w(t) ws(t) s(t) Clock Analog bilateral switch Generation of PAM f │W(f)│ -B B 1 Magnitude spectrum of w(t) Ws(f) f fs d│sin(πτf)/πτf│ d=1/3 Magnitude spectrum of ws(t)
: For this example with d=1/3,the spectrum is zero forf-±3fs, ±6fs....So the choice of d will infect the resultingspectrum.The null bandwidth is 12B (3f.).So PAM signal sbandwidth is much larger than the bandwidth of theoriginal analog waveform .If f≥2B, no overlap of spectrumAccording to Ws(f),we can use an ideal low-pass filter torecover the original waveform w(t)LPFW(f)d / sin(πtf)/元tf |d=1/3fsfRecovering w(t) from ws(t)
• For this example with d=1/3,the spectrum is zero for f=±3fs ,±6fs ,.So the choice of d will infect the resulting spectrum. • The null bandwidth is 12B (3fs).So PAM signal ‘s bandwidth is much larger than the bandwidth of the original analog waveform . • If fs≥2B, no overlap of spectrum • According to Ws(f),we can use an ideal low-pass filter to recover the original waveform w(t). Ws(f) f fs d│sin(πτf)/πτf│ d=1/3 Recovering w(t) from ws(t) LPF
Demodulation of a natural sampling PAM signalAnalogmultiplierWs(t) PAMCw(t)(Gating)LPFH(f)H(f)Localoscillatorffco-fco
• Demodulation of a natural sampling PAM signal LPF H(f) Local oscillator ws(t) PAM (Gating) Analog multiplier Cw(t) f H(f) -fco fco