Sampling PAM- Pulse Amplitude Modulation (continued)
Transcript of Sampling PAM- Pulse Amplitude Modulation (continued)
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SamplingPAM- Pulse Amplitude Modulation
(continued)
EELE445-14Lecture 14
Sampling
Properties will be looking at for:
•Impulse Sampling•Natural Sampling•Rectangular Sampling
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Figure 2–18 Impulse sampling.
Impulse Sampling
sT2
( )
( ) [ ]L++++=
====
++⇒−= ∑∑∞
=
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−∞=
)3cos(2)2cos(2)cos(211
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)cos()(
0
10
tttT
t
TD
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T
tnDDnTtt
ssss
T
sn
sssn
nnsn
nsT
s
s
ωωωδ
πωϕ
ϕωδδ
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Impulse Sampling
( ) )(∑∞
−∞=
−=n
sT nTtts
δδ
)(tws
( ) [ ]
[ ] )(1)()(
)3cos(2)2cos(2)cos(21)()()(
∑∞
−∞=
−==
++++==
ns
sss
ssss
Ts
nffWT
twFfW
tttT
twttwtws
Lωωωδ
Impulse Sampling- text
∑
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−∞=
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−=
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Impulse SamplingThe spectrum of the impuse sampled signal is the spectrum of the unsampled signal that is repeated every fs Hz, where fs is the sampling frequency or rate (samples/sec). This is one of the basic principles of digital signal processing.
Note:This technique of impulse sampling is often used to
translate the spectrum of a signal to another frequency band that is centered on a harmonic of the sampling frequency, fs.
If fs>=2B, (see fig 2-18), the replicated spectra around each harmonic of fs do not overlap, and the original spectrum can be regenerated with an ideal LPF with a cutoff of fs/2.
Impulse SamplingUndersampling and aliasing.
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Natural SamplingGeneration of PAM with natural sampling (gating).
Natural Sampling
Duty cycle =1/3
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Natural Sampling
ss f
df 3= at Null
PAM and PCM
• PAM- Pulse Amplitude Modulation:– The pulse may take any real voltage value that is
proportional to the value of the original waveform. No information is lost, but the energy is redistributed in the frequency domain.
• PCM- Pulse Code Modulation:– The original waveform amplitude is quantized with
a resulting loss of information
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–4 Demodulation of a PAM signal (naturally sampled).
nth Nyquist region recovery
Comb Oscillatorn=1,2,3….
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–5 PAM signal with flat-top sampling.Impulse sample and hold
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–6 Spectrum of a PAM waveform with flat-top sampling.
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–7 PCM trasmission system.
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PCM Transmission
• Negative: The transmission bandwidth of the PCM signal is much larger than the bandwidth of the original signal
• Positive: The transmission range of a PCM signal may be extended with the use of a regenerative repeater.
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–8 Illustration of waveforms in a PCM system.
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–8 Illustration of waveforms in a PCM system.
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Illustration of waveforms in a PCM system.
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Illustration of waveforms in a PCM system.
•The error signal, (quantization noise), is inversely proportional to the number of quantization levels usedto approximate the waveform x(t)
•Companding is used to improve the SQNR (signal toquantization noise ratio) for small amplitude x(T) signals
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–9 Compression characteristics (first quadrant shown).
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–9 Compression characteristics (first quadrant shown).
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–10 Output SNR of 8-bit PCM systems with and without companding.
( )[ ]
( )[ ] compandinglawAAnNS
compandinglawnNS
quantionuniformx
Vn
NS
dB
dB
rms
rangepeak
dB
−+−+=⎥⎦⎤
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⎢⎣⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=⎥⎦
⎤⎢⎣⎡
ln1log2002.677.4
1lnlog2002.677.4
log2002.677.4
μμ
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Quantization Noise,Analog to Digital Converter-A/D
EELE445Lecture 16
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–7 PCM trasmission system.
q(x)
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Quantization
V
-V
Δ
x(t)
Q(x)
n
n
M
VMV
2
22
1
=
==Δ −
)(ˆ xQxn =
Quantization – Results in a Loss of Informationx(t)
Q(x)
2Δ
2Δ
−
Lost Information
After sampling, x(t)=xi
R∈ix
After Quantization:
R∈= x,x̂Q(x)
122
−==Δ nV
MV
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Quantization Noise
Quantization function:
Define the mean square distortion:
2)(
~))(()( 22
Δ≤−
=−=
xQx
andxxQxxq
Quantization
( ) noiseonquantizatithePMV
dqqq
nq==
Δ=
Δ= ∫
Δ
Δ−
2
2
2
2
2
22
3
12
1
where M=2n, V is ½ the A/D input range,and n is the number of bits
n
n
M
VMV
2
22
1
=
==Δ −
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Quantization Noise from the expectation operator:
)~(xf
2Δ
−2Δ
Δ1
x~
( )[ ] [ ]
)(~
))((ˆ,
~ˆ
2
XQXX
XQXEXXdED
XandXX
−=
−==
:ddistribute uniformly is error the of pdf The
:as n)(distortio error squared mean the define can weSo are so variable, random a is Since
⎪⎩
⎪⎨⎧ Δ
≤≤Δ
−Δ
=otherwise
xxf
02
~22)~(
SQNR – Signal to Quantization Noise Ratio
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SQNR – Signal to Quantization Noise Ratio
Px may be found using:
SQNR – Signal to Quantization Noise Ratio
The distortion, or “noise”, is therefore:
Where Px is the power of the input signal
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SQNR – Linear Quantization[ ]
12max
2max
2
<
≤
xP
xXE
x
The SQNR decreases asThe input dynamic rangeincreases
U-Law Nonuniform PCMused to increase SQNR for given Px , xmax, and n
U=255 U.S
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A-Law Nonuniform PCM
a=87.56 U.S
u-Law v.s. Linear Quantization8 bitPx is signal powerRelative to full scale
Px
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Pulse Code Modulation, PCM, Advantagecompared with analog systems
• b is the number of bits
• γ is (S/N)basebandRelative to full scale
• PPM is pulse position modulation