Instantaneous sampling (Flat-top PAM): Using flat-top signaling to convert an analog waveform to→impulse sampling's generalizationpulse signalingDefinition:if w(t) is an analog bandlimited waveform(BHz),the instantaneous sampled PAM signal is given byWs(t)=Zw(kT,)h(t-kT,)where h(t) denotes the sampling-pulse shape,and for flat-topsampling the pulse shape ish(t)= II(t/t)t≤T,and f,≥2BTheorem:the spectrum for a flat-top PAM signal isWs(f)=(1/T,)H(f) ZW(f-kf)where H(f)= t sin(元 t f)/(元 t f)
• Using flat-top signaling to convert an analog waveform to pulse signaling impulse sampling’s generalization • Definition:if w(t) is an analog bandlimited waveform (BHz),the instantaneous sampled PAM signal is given by: ws(t)=∑w(kTs)h(t-kTs) where h(t) denotes the sampling-pulse shape,and for flat-top sampling the pulse shape is h(t)= Π(t/τ) τ≤Ts and f s≥2B • Theorem:the spectrum for a flat-top PAM signal is Ws(f)=(1/Ts)H(f) ∑W(f-kfs) where H(f)= τ sin(π τ f)/(π τ f) Instantaneous sampling (Flat-top PAM)
ProofiwehaveWs(t)=Zw(kT,)h(t-kT,) =h(t)*Zw(kT,)(t-kT,)=h(t)*[w(t)E8(t-kT,)]then the spectrum is:W,(f)=H(f)[W(f)* e-j2rfkTs]Here we have(1/T,)Z8(f-kf,)= (1/T,)Zcne-j2元fnTsand cn is the Fourier series' coefficient of the impulse train:Cn=1/fsandWs(f)-H(f)[W(f)* (1/T)Zs(f-kf)] =H(f)[(1/T)ZW(f)*$(f-kf))=(1/T,)H(f)ZW(f-kf.)
• Proof:we have ws(t)=∑w(kTs)h(t-kTs) =h(t)*∑w(kTs)δ(t-kTs) =h(t)*[w(t)∑δ(t-kTs)] then the spectrum is: Ws(f)=H(f)[W(f)* ∑e-j2πfkTs] Here we have (1/Ts)∑δ(f-kfs)= (1/Ts)∑cne -j2πfnTs and cn is the Fourier series’ coefficient of the impulse train: cn=1/fs and Ws(f)=H(f)[W(f)* (1/Ts)∑δ(f-kfs)] =H(f)[(1/Ts)∑W(f)*δ(f-kfs)] =(1/Ts)H(f)∑W(f-kfs)
Spectrum diagram:W(f)W,(f)t /T, / sin(元tf)/元tffB-BfsMagnitudespectrumofw(t)Magnitude spectrum of ws(t)f. Demodulation of flat-top PAM waveform?AnalogmultiplierWs(t) PAMCw(t)(flat-top)LPFH(f)H(f)Localoscillatorffco-fco
• Spectrum diagram: • Demodulation of flat-top PAM waveform: f │W(f)│ -B B 1 Magnitude spectrum of w(t) Ws(f) f fs τ /Ts│sin(πτf)/πτf│ Magnitude spectrum of ws(t) LPF H(f) Local oscillator ws(t) PAM (flat-top) Analog multiplier Cw(t) f H(f) -fco fco
Pulse Code Modulation--PCM: Definition:PCM is essentially analog-to-digital conversionof a special type where the information contained in theinstantaneous samples of an analog bandlimited signal isrepresented (encoded) by digital words in a serial bitstream.: If a digital words has n binary digits,we can have M=2npossible unique code words,each code word correspondingto a certain amplitude level. Question: the sample from analog signal can be any one ofan infinite number of levels,how can we represent thepossible infinite number of levels by only M levels?Answer:approximation--quantification
• Definition:PCM is essentially analog-to-digital conversion of a special type where the information contained in the instantaneous samples of an analog bandlimited signal is represented (encoded) by digital words in a serial bit stream. • If a digital words has n binary digits,we can have M=2n possible unique code words,each code word corresponding to a certain amplitude level. • Question: the sample from analog signal can be any one of an infinite number of levels,how can we represent the possible infinite number of levels by only M levels? • Answer:approximation-quantification Pulse Code Modulation-PCM
The analog sample value will be replaced by one of the closestallowed M values.And each allowed value corresponds one ofM allowed n bits digital wordPCM's advantages:digital circuit IC cheep PCM can accommodateall type of analog sourceseasytorealizeTDMlong-distance communicationnoise performance of a digital system betterthan the analog main disadvantage:amuchmorewiderbandwidth EX.:for voice telephone:analog signal 300-3400HzPCM system:minimum bandwidth 34kHz
• The analog sample value will be replaced by one of the closest allowed M values.And each allowed value corresponds one of M allowed n bits digital word. • PCM’s advantages: • digital circuit IC cheep • PCM can accommodate all type of analog sources • easy to realize TDM • long-distance communication • noise performance of a digital system better than the analog • main disadvantage: • a much more wider bandwidth • EX.:for voice telephone:analog signal 300-3400Hz • PCM system:minimum bandwidth 34kHz