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3.2 Pulse amplitude modulation

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3.2 Pulse amplitude modulation

Basis ---- The sampling theoremPAM. The amplitude of the carrier pulse varies with the analog baseband signal.Characteristics of PAM signal

Pulse-type signal, its amplitude denotes the analog information.Satisfied with sampling theoremPAM’s bandwidth is wider than that of analog waveform

Two classes of PAM signalsNatural Sampling (Gating)

Instantaneous Sampling

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3.2.1 Natural Sampling (Gating)

clock

S(t)

Analog bilateral switch

Ws(t)=w(t)s(t)W(t)

Generation of PAM with natural sampling

3.2 Pulse amplitude modulation

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PAM signal with natural sampling

)()()( tstwtws =

If w(t) is a analog waveform bandlimitedto B Hertz, The PAM signal that uses natural sampling (gating) is

where

∑ ∏∞

−∞=

−

=k

skTttsτ

)(

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Spectrum of PAM waveform

[ ] )(sin)()( sn

ss nffWdndndtwFfW −== ∑

∞

−∞= ππ

The spectrum for a naturally sampled PAM signal is

sf/=sfπω=Where sd/τ=

And is the spectrum of the original unsampledwaveform.

[ ])()( twFfW =

the spectrum of PAM signal with natural sampling is a function of the spectrum of the analog input waveform.

(3-3)

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3/1/ == sTd τ

Example.The case of an input waveform that has a rectangular spectrum, where the duty cycle of the switching waveform is

And the sampling rate is

Bfs 4=

Spectrum of PAM waveform

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Demodulation of PAM signal

Method 1 ---- Low-pass filter

Method 2 ---- Product detection

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3.2.2 Instantaneous sampling

W(t)

-τ/2 τ/2 t

h(t)=∏(---)tτ Ws(t)

Generation by using a sample-and-hold type of electronic circuit

δ(t-KTs)

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Instantaneous sampled PAM

Characteristic

At t=kTs, the sampling values w(kTs) determine the amplitude of the flat-top rectangular pulses.

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If w(t) is a analog waveform bandlimited to B hertz, the instantaneous sampled PAM signal is given by

where h(t) denotes the sampling-pulse shape.

∑∞

−∞=

−=k

sss kTthkTwtw )()()(

>

<=

= ∏ 2/,0

2/,1)(

τ

τ

τ t

ttth

BffT sss 2/1 ≥=≤τwhere and

Instantaneous sampled PAM

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Spectrum of Flat-top PAM

The spectrum for a Flat-top PAM signal is

[ ]

==

−= ∑∞

−∞=

ffthFfH

kffWfHT

fWk

ss

s

πτπττ sin)()(

)()(1)(

where

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Spectrum of Flat-top PAM

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Ws(f) Ws(f)´1/H(f) LPFWs(f)

Note;

1: Equalization filter ------ reduce the high frequency loss2: decreasing τ, The pulse width τ is called aperture.

Demodulation of flat-top PAM

Method 1 ---- Low-pass filter

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Prefilter is needed before the multiplier to compensate for the spectral loss due to the aperture effect

Note;

Method 2 ---- Product detection

Ws(t)

PAM (flat-topsampling)

PrefilterH1(f)

Cos(nwst)

Oscillatorwo=nws

Low PassFilter ｜H(f)｜

Ws(t)^

H1(f)= Sin(πτf)πτf

fco-fco f

H(f)

Where B< fco<fs-B

Demodulation of flat-top PAM

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Comparison -- in time domain

Fig3-1 PAM Signal with natural sampling Fig 3-5 PAM signal with flat-top sampling

waveform of PAM signal

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Comparison -- in frequency domain

)(sin)( sn

s nffWdndndfW −= ∑

∞

−∞= ππ

Spectrum of PAM signal

Fig3-3 The Spectrum of natural sampling PAM Fig 3-6 The Spectrum of Flat-top PAM

∑

∑∞

−∞=

∞

−∞=

−

=

−=

ns

ns

ss

nffWffd

nffWfHT

fW

)(sin

)()(1)(

πτπτ

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The bandwith required for the transmission of PAM is much large than that of the original analog signal.

noise performance of the PAM system can never be better than that analog signal. ---- not very good for long-distance transmission.

Provide a means for converting an analog signal to a PCM signal. Time-division multiplexing etc.

Summary