1

PCM & DPCM & DM

2

Pulse-Code Modulation (PCM) :

In PCM each sample of the signal is quantized to one of the amplitude levels, where B is the number of bits used to represent each sample.

The rate from the source is bps.

The quantized waveform is modeled as :

q(n) represent the quantization error, Which we treat as an additive noise.

B2

sBF

)()()(~ nqnsns

3

Pulse-Code Modulation (PCM) :

The quantization noise is characterized as a

realization of a stationary random process q in

which each of the random variables q(n) has

uniform pdf.

Where the step size of the quantizer is22

q

B 2

2

/1

2

4

Pulse-Code Modulation (PCM) :

If :maximum amplitude of signal,

The mean square value of the quantization

error is :

Measure in dB, The mean square value of the

noise is :

B

A

2

max

maxA

122

A

12

Δ|(n)q

3Δ

1

(n)dqqΔ

1 (n)q

2B

2

max

2Δ/2

Δ/2

3

Δ/2

Δ/2

22

.dB 8.10612

2log10

12log10

2

10

2

10

BB

5

Pulse-Code Modulation (PCM) : The quantization noise decreases by 6 dB/bit.

If the headroom factor is h, then

The signal to noise (S/N) ratio is given by

(Amax=1)

In dB, this is

hh

AX

B

rms

2max

2

2

2

22

1212/

SNRh

X

N

S B

rms

hBh

B

102

2

10dB log208.106212

log10SNR

6

Pulse-Code Modulation (PCM) :

Example :

We require an S/N ratio of 60 dB and that a

headroom factor of 4 is acceptable. Then the

required word length is :

60=10.8 + 6B – 20

If we sample at 8 KHZ, then PCM require

bit 112.10 B

4log10

bit/s. 8800011 8 k

7

Pulse-Code Modulation (PCM) :

A nonuniform quantizer characteristic is

usually obtained by passing the signal

through a nonlinear device that compress

the signal amplitude, follow by a uniform

quantizer.

Compressor A/D D/A Expander

Compander

(Compressor-Expander)

Companding: Compression and Expanding

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Original Signal

After Compressing, Before Expanding

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Companding

A logarithmic compressor employed in

North American telecommunications

systems has input-output magnitude

characteristic of the form

is a parameter that is selected to give the

desired compression characteristic.

)1log(

|)|1log(||

sy

Companding

10

11

Companding

The logarithmic compressor used in

European telecommunications system is

called A-law and is defined as

A

sAy

log1

|)|1log(||

Companding

12

13

DPCM :

A Sampled sequence u(m), m=0 to m=n-1.

Let be the value of the

reproduced (decoded) sequence.

),...2(~),1(~ nunu

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DPCM:

At m=n, when u(n) arrives, a quantify ,

an estimate of u(n), is predicted from the

previously decoded samples

i.e.,

”prediction rule”

Prediction error:

)(~ nu

),...2(~),1(~ nunu

),...);2(~),1(~()(~ nununu

)(~)()( nunune

:(.)

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DPCM :

If is the quantized value of e(n), then

the reproduced value of u(n) is:

Note:

)(~ ne

)(~)(~)(~ nenunu

)(in error on Quantizati The :)(

)(~)(

))(~)(~())()(~()(~)(

)()(~)(

nenq

nene

nenunenununu

nenunu

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DPCM CODEC:

)(~ nu

)(~ nuΣ Quantizer

Σ

ΣCommunication

Channel

PredictorPredictor

)(nu )(ne )(~ ne

)(~ nu

)(~ nu

)(~ ne

Coder Decoder

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DPCM:

Remarks:

The pointwise coding error in the input

sequence is exactly equal to q(n), the

quantization error in e(n).

With a reasonable predictor the mean

sequare value of the differential signal e(n) is

much smaller than that of u(n).

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DPCM:

Conclusion:

For the same mean square quantization error,

e(n) requires fewer quantization bits than u(n).

The number of bits required for transmission

has been reduced while the quantization error

is kept the same.

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DPCM modified by the addition of

linearly filtered error sequence

)(~ nu

)(~ nuΣ Quantizer

Σ

Communication

Channel

Linear filter

)(nu )(ne )(~ ne

)(~ nu

)(~ nu

)(~ ne

Coder Decoder

(i)} a{

Linear filter

(i)} b{

Σ

Linear

filter

(i)} a{

Linear

filter(i)} b{

Σ

Σ

20

Adaptive PCM and Adaptive DPCM

Speech signals are quasi-stationary in nature

The variance and the autocorrelation function of the source output vary

slowly with time.

PCM and DPCM assume that the source output is stationary.

The efficiency and performance of these encoders can be improved

by adaptation to the slowly time-variant statistics of the speech

signal.

Adaptive quantizer

feedforward

feedbackward

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Example of quantizer with an

adaptive step size

∆ 2∆ 3∆-∆-2∆-3∆

∆/2

3∆/2

5∆/2

7∆/2

-∆/2

-3∆/2

-5∆/2

-7∆/2

M (1)

M (2)

M (3)

M (4)

M (1)

M (2)

M (3)

M (4)

000

001

010

011 0

100

101

110

111 Previous Output

Multiplier

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ADPCM with adaptation of the predictor

)(~ nu

)(~ nuΣ Quantizer

Σ

ΣCommunication

Channel

Predictor

Predictor

)(nu )(ne )(~ ne

)(~ nu

)(~ ne

Coder Decoder

DecoderEncoder

Step-size

adaptation

Predictor

adaptation

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Delta Modulation : (DM)

Predictor : one-step delay function

Quantizer : 1-bit quantizer

)1(~)()(

)1(~)(~

nunune

nunu

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Delta Modulation : (DM)

Primary Limitation of DM

Slope overload : large jump region

Max. slope = (step size)X(sampling freq.)

Granularity Noise : almost constant region

Instability to channel noise

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DM:

Unit Delay

Unit Delay

Integrator

)(nu )(ne )(~ ne

)(~ nu)(~ nu

)(~ ne )(~ nu

)(~ nu

Coder

Decoder

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DM:

Step size effect :

Step Size (i) slope overload

(sampling frequency ) (ii) granular Noise

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Adaptive DM:

1kX

1kE1ks

Adaptive

Function

Unit DelaykX 1k

Stored

k mink ,E,

11

min1min

min11

11

|| if

|| if ]2

[||

][sgn

kkk

kk

kk

kkk

kKk

XX

E

EE

XSE

This adaptive approach simultaneously minimizes the effects of both

slope overload and granular noise

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Vector Quantization

(VQ)

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Vector Quantization :

Quantization is the process of

approximating continuous amplitude

signals by discrete symbols.

Partitioning of

two-dimensional

Space into 16 cells.

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Vector Quantization :

The LBG algorithm first computes a 1-

vector codebook, then uses a splitting

algorithm on the codeword to obtain the

initial 2-vector codebook, and continue the

splitting process until the desired M-vector

codebook is obtained.

This algorithm is known as the LBG

algorithm proposed by Linde, Buzo and

Gray.

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Vector Quantization : The LBG Algorithm :

Step 1: Set M (number of partitions or cells)=1.Find the centroid of all the training data.

Step 2: Split M into 2M partitions by splitting each current codeword by finding two points that are far apart in each partition using a heuristic method, and use these two points as the new centroids for the new 2M codebook. Now set M=2M.

Step 3: Now use a iterative algorithm to reach the best set of centroids for the new codebook.

Step 4: if M equals the VQ codebook size require, STOP; otherwise go to Step 2.