3.2 Pulse amplitude modulation
3.2 Pulse amplitude modulation
1
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3.2 Pulse amplitude modulation Basis ---- The sampling theorem PAM. 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 theorem PAM’s bandwidth is wider than that of analog waveform
Natural Sampling (Gating) Two classes of PAM signals Instantaneous Sampling 2
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3.2 Pulse amplitude modulation 3.2.1 Natural Sampling (Gating) Generation of PAM with natural sampling Analog bilateral switch W(t)
Ws(t)=w(t)s(t)
S(t)
clock
3
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PAM signal with natural sampling If w(t) is a analog waveform bandlimited to B Hertz, The PAM signal that uses natural sampling (gating) is w s (t ) = w (t ) s (t ) where s (t ) =
t − kT s ∑ ∏ τ k = −∞ ∞
4
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Spectrum of PAM waveform The spectrum for a naturally sampled PAM signal is W s ( f ) = F [w s (t ) ] = d
sin n π d W ( f − nf s ) ∑ n πd n = −∞
(3-3)
∞
f= πd = τ/s s/ω
Where And W ( f ) = F [w(t )] is the spectrum of the original unsampled waveform. the spectrum of PAM signal with natural sampling is a function of the spectrum of the analog input waveform. 5
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Spectrum of PAM waveform Example. The case of an input waveform that has a rectangular spectrum, where the duty cycle of the switching waveform is d = τ / Ts = 1 / 3
And the sampling rate is
f s = 4B
6
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Demodulation of PAM signal Method 1 ---- Low-pass filter Method 2 ---- Product detection
7
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3.2.2 Instantaneous sampling Generation by using a sample-and-hold type of electronic circuit
t h(t)=∏(---)
τ
W(t)
-τ/2
τ/2
Ws(t) t
δ(t-KTs)
8
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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.
9
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Instantaneous sampled PAM If w(t) is a analog waveform bandlimited to B hertz, the instantaneous sampled PAM signal is given by ∞ w s (t ) =
∑ w ( kT ) h (t − kT )
k = −∞
s
s
where h(t) denotes the sampling-pulse shape. t 1, t < τ / 2 h (t ) = ∏ = τ 0 , t > τ / 2 where τ ≤ Ts = 1 / f s and f s ≥ 2 B 10
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Spectrum of Flat-top PAM The spectrum for a Flat-top PAM signal is ∞ 1 Ws ( f ) = H ( f ) ∑ W ( f − kf s ) Ts k = −∞
where
sin πτ f H ( f ) = F [h (t ) ] = τ πτ f
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Spectrum of Flat-top PAM
12
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Demodulation of flat-top PAM z
Method 1 ---- Low-pass filter Ws(f)
1/H(f)
Ws´(f)
LPF
Ws(f)
Note; 1: Equalization filter ------ reduce the high frequency loss 2: decreasing τ, The pulse width τ is called aperture. 13
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Demodulation of flat-top PAM z
Method 2 ---- Product detection Ws(t)
PAM (flat-top sampling)
Low Pass Filter ｜H(f)｜
Prefilter H1(f) H1(f)=
^ Ws(t)
Sin(πτf)
Cos(nwst)
πτf
Oscillator wo=nws
H(f)
-fco
fco
f
Where B< fco
1
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3.2 Pulse amplitude modulation Basis ---- The sampling theorem PAM. 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 theorem PAM’s bandwidth is wider than that of analog waveform
Natural Sampling (Gating) Two classes of PAM signals Instantaneous Sampling 2
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3.2 Pulse amplitude modulation 3.2.1 Natural Sampling (Gating) Generation of PAM with natural sampling Analog bilateral switch W(t)
Ws(t)=w(t)s(t)
S(t)
clock
3
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PAM signal with natural sampling If w(t) is a analog waveform bandlimited to B Hertz, The PAM signal that uses natural sampling (gating) is w s (t ) = w (t ) s (t ) where s (t ) =
t − kT s ∑ ∏ τ k = −∞ ∞
4
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Spectrum of PAM waveform The spectrum for a naturally sampled PAM signal is W s ( f ) = F [w s (t ) ] = d
sin n π d W ( f − nf s ) ∑ n πd n = −∞
(3-3)
∞
f= πd = τ/s s/ω
Where And W ( f ) = F [w(t )] is the spectrum of the original unsampled waveform. the spectrum of PAM signal with natural sampling is a function of the spectrum of the analog input waveform. 5
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Spectrum of PAM waveform Example. The case of an input waveform that has a rectangular spectrum, where the duty cycle of the switching waveform is d = τ / Ts = 1 / 3
And the sampling rate is
f s = 4B
6
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Demodulation of PAM signal Method 1 ---- Low-pass filter Method 2 ---- Product detection
7
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3.2.2 Instantaneous sampling Generation by using a sample-and-hold type of electronic circuit
t h(t)=∏(---)
τ
W(t)
-τ/2
τ/2
Ws(t) t
δ(t-KTs)
8
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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.
9
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Instantaneous sampled PAM If w(t) is a analog waveform bandlimited to B hertz, the instantaneous sampled PAM signal is given by ∞ w s (t ) =
∑ w ( kT ) h (t − kT )
k = −∞
s
s
where h(t) denotes the sampling-pulse shape. t 1, t < τ / 2 h (t ) = ∏ = τ 0 , t > τ / 2 where τ ≤ Ts = 1 / f s and f s ≥ 2 B 10
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Spectrum of Flat-top PAM The spectrum for a Flat-top PAM signal is ∞ 1 Ws ( f ) = H ( f ) ∑ W ( f − kf s ) Ts k = −∞
where
sin πτ f H ( f ) = F [h (t ) ] = τ πτ f
11
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Spectrum of Flat-top PAM
12
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Demodulation of flat-top PAM z
Method 1 ---- Low-pass filter Ws(f)
1/H(f)
Ws´(f)
LPF
Ws(f)
Note; 1: Equalization filter ------ reduce the high frequency loss 2: decreasing τ, The pulse width τ is called aperture. 13
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Demodulation of flat-top PAM z
Method 2 ---- Product detection Ws(t)
PAM (flat-top sampling)
Low Pass Filter ｜H(f)｜
Prefilter H1(f) H1(f)=
^ Ws(t)
Sin(πτf)
Cos(nwst)
πτf
Oscillator wo=nws
H(f)
-fco
fco
f
Where B< fco
