104615_pulse code modulation(sampling,quantizing,encoding)
TRANSCRIPT
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3.3 Pulse code modulation
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Coding & Streaming1000 0101=-5
0001 0101=21
Pulse Code Modulation
Samplingt1: -4.88 v
t2: +21.43 v
Analog waveform
Quantizing-5 v
21 v
, 0001 0101, 1000 0101,
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sampler quantizer encoder channel
Low-pass
filterdecoder
x(t)
xn
x(t)
PCM transmitter (A/D conversion)
~
Three basic operations
Sampling
Quantizing
Encoding
xn~ s(t)
Pulse Code Modulation
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Quantizing
Approximating the analogsample values by using finite
number of levels
Uniform quantizing
Quantizing error
quantizing noise
PCM
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Encoding
The quantized analog sample values are
replaced by n-bit binary code
Quantized Sample Voltage Gray Code Word (PCM output)
+7 110
+5 111
+3 101
+1 100
-1 000
-3 001
-5 011
-7 010
E.g. three-bit Gray code for M=8 levelsn2=
Pulse Code Modulation
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Bandwidth of PCM signals
The bit rate of PCM signal is
snfR=
Bit rate: R= fs (samples/s) n(bits/sample)
= 8 k sample/s 8 bits/sample = 64 kbps
Example. Design of a PCM signal for telephone system
Assume:
An analog audio voice-frequency (VF) telephonesignal band: 300Hz ~ 3400 Hz
The minimum sampling frequency is 2 3.4 = 6.8
ksample/sec. actually, using sampling frequency of 8ksamole/sec.
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For rectangular pulse, the first null bandwidth is
Example. the result for the case of the minimum sampling:
The bandwidth of (serial) binary PCM waveformsdepends on:
The bit rate The waveform pulse shape used to
represent the data
sPCM nfRB ==
16B8256
6B38
4B24
2B12
Bandwidth of PCM signal
(the first null bandwidth)
Length of the PCM,n(bit)
Number of quantizer levels,M
Bandwidth of PCM signals
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Effects of noise
Two main effects produce noise or
distortion: Bit errors in the recovered PCM signal .
(channel noise, improper channel
filtering, ISI etc. )
Quantizing noise that is caused by the
M-step quantized at PCM transmitter
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sampler quantizer encoder channel
Low-passfilter
decoder
x(t)xn
x(t)
PCM transmitter (A/D conversion)
~
0101110
0101010
xn~
Effects of noise
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Under certain assumptions, the ratioofthe recover analog peak signal
power to the total average noisepower is:
The ratio of the average signal powerto the average noise power is
eoutpk PM
M
N
S
)1(41
32
2
+=
eout PM
M
N
S
)1(41 2
2
+=
Effects of noise
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If Pe=0 (no ISI), the peak SNRresulting form only quantizing errors
is
The average SNRdue only toquantizing error is
23MN
S
outpk
=
2MN
S
out
=
+=
n
N
S
dB
02.6
6-dB rule
=4.77 for the peak SNR,
=0 for the average SNR.
Effects of noise
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This equation points out the significantperformance characteristic for PCM:
An additional 6-dB improvement in SNR is
obtained for each bit added to the PCM word.
Assumptions:
No bit errors
the input signal level is large enough to range
over a significant number of quantizing levels
Effects of noise
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Performance
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Example SA3-2 (page 219)
In a communications-quality audio system,
an analog voice-frequency (VF) signal witha bandwidth of 3200 Hz is converted into aPCM signal by sampling at 7000 samples/sand by using a uniform quantizer with 64steps. The PCM binary data are transmittedover a noisy channel to a receiver that hasa bit error rate (BER) of 10-4.
What is the null bandwidth of the PCMsignal if a polar line code is used?
What is the average SNRof the recovered
analog signal at the receiving end?
PCM signal bandwidth and SNR
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Nonuniform Quantizing
Characteristic voice analog signal
Nonuniform amplitude distribution
The granular quantizing noise will be aserious problem if uniform quantizing isused.
Solution: nonuiform quantizing
Nouniform Quantizing: a variable stepsize is used
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Method:
passing the analog signal through a
compression (nonlinear) amplifier and theninto a PCM circuit that uses uniform
quantizer.
-Law and A-Low
Analog
Signal
A Compression
(nonlinear)
Amplifier
PCM
(uniform quantized)
Nonuniform
Quantizing signal
Nonuniform Quantizing
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-Law
( )
+
+=
1ln
)(1ln)(
1
2
twtw
Compression
quantizercharacteristic
Uniform quantizer
characteristic
(a) M=8 Quantizer Characteristic
0
0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
=0=1
=5
=100
=225
1)(0 1 tw
Nonuniform Quantizing
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A-Law( )
( )
+
+
+
=
1)(1,ln1
)(ln1
1)(0,
1ln
)(
)(
1
1
1
1
2
twAA
twA
Atw
twA
tw
0 0.2 0.4 0.6 0.8 1.0
0 0.2 0.4 0.6 0.8 1.0
0
0.2
0.4
0.6
0.8
1.0
A=1A=2
A=5
A=87.6
A=100
Nonuniform Quantizing
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In practice, the
smooth
nonlinear
characteristicsof -Law and
A-Low
are
approximated
by piecewise
linear chords
Compressioncharacteristics
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Output SNR
Following 6-dB law:
+=
n
N
S
dB
02.6
( )rmsxV /log2077.4 =
( )[ ]+=
1lnlog2077.4
[ ]Aln1log2077.4 +=
whereUniform quantizing
-Law companding
A-Law companding
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Comparison of output SNR