104615_Pulse Code Modulation(Sampling,Quantizing,Encoding)

7/31/2019 104615_Pulse Code Modulation(Sampling,Quantizing,Encoding)

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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)



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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=



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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



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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



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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



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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

104615_Pulse Code Modulation(Sampling,Quantizing,Encoding)

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