CE 40763

Digital Signal Processing

Fall 1992

Waveform Coding

Hossein Sameti

Department of Computer Engineering

Sharif University of Technology

2

PCM & DPCM & DM

3

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

4

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 is 22

q

B 2

2

/1

2

5

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

2max

maxA

122

A

12

Δ|(n)q

3Δ

1

(n)dqqΔ

1 (n)q

2B

2max

2Δ/2

Δ/23

Δ/2

Δ/2

22

.dB 8.10612

2log10

12log10

2

10

2

10

BB

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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 Brms

hBh

B

102

2

10dB log208.106212

log10SNR

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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

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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

11

12

Companding The logarithmic compressor used in

European telecommunications system is called A-law and is defined as

A

sAy

log1

|)|1log(||

Companding

13

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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

CommunicationChannel

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

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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

ΣCommunicationChannel

PredictorPredictor

)(nu )(ne )(~ ne

)(~ nu)(~ ne

Coder Decoder

DecoderEncoder

Step-sizeadaptation

Predictoradaptation

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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

Storedk 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.