PCM & DPCM & DM 1 Pulse-Code Modulation (PCM) : In PCM - - PowerPoint PPT Presentation

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PCM & DPCM & DM

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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.

B

2

s

BF ) ( ) ( ) ( ~ n q n s n s  

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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

2 2      q

B 

  2

2 

 / 1

2  

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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

 

max

A

12 2 A 12 Δ | (n) q 3Δ 1 (n)dq q Δ 1 (n) q

2B 2 max 2 Δ/2 Δ/2 3 Δ/2 Δ/2 2 2

      

 



. dB 8 . 10 6 12 2 log 10 12 log 10

2 10 2 10

    



B

B

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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

 In dB, this is

h h A X

B rms

   2

max

2 2 2 2

2 12 12 / SNR h X N S

B

rms

   

h B h

B 10 2 2 10 dB

log 20 8 . 10 6 2 12 log 10 SNR     

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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 11 2 . 10   B

4 log 10

bit/s. 88000 11 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.

) 1 log( |) | 1 log( | |      s y



Companding

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Companding

 The logarithmic compressor used in

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

A s A y log 1 |) | 1 log( | |   

Companding

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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 ( ~   n u n u

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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:

) ( ~ n u

),... 2 ( ~ ), 1 ( ~   n u n u

),...); 2 ( ~ ), 1 ( ~ ( ) ( ~    n u n u n u 

) ( ~ ) ( ) ( n u n u n e  

: (.) 

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DPCM :

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

the reproduced value of u(n) is:

 Note:

) ( ~ n e

) ( ~ ) ( ~ ) ( ~ n e n u n u  

) ( in error

• n

Quantizati The : ) ( ) ( ~ ) ( )) ( ~ ) ( ~ ( )) ( ) ( ~ ( ) ( ~ ) ( ) ( ) ( ~ ) ( n e n q n e n e n e n u n e n u n u n u n e n u n u          

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DPCM CODEC:

) ( ~ n u

) ( ~ n u

Σ Quantizer Σ Σ

Communication Channel

Predictor Predictor

) (n u

) (n e

) ( ~ n e

) ( ~ n u

) ( ~ n u

) ( ~ n e

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

) ( ~ n u

) ( ~ n u

Σ Quantizer Σ

Communication

Channel

Linear filter

) (n u

) (n e

) ( ~ n e

) ( ~ n u

) ( ~ n u

) ( ~ n e

Coder Decoder

(i)} a ˆ {

Linear filter (i)} b ˆ {

Σ

Linear filter (i)} a ˆ { Linear filter

(i)} b ˆ {

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) 000 001 010 011 100 101 110 111 Previous Output Multiplier

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ADPCM with adaptation of the predictor

) ( ~ n u

) ( ~ n u

Σ

Quantizer

Σ Σ

Communication Channel Predictor Predictor

) (n u ) (n e ) ( ~ n e ) ( ~ n u ) ( ~ n e

Coder Decoder

Decoder Encoder Step-size adaptation Predictor adaptation

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Delta Modulation : (DM)

 Predictor : one-step delay function  Quantizer : 1-bit quantizer

) 1 ( ~ ) ( ) ( ) 1 ( ~ ) ( ~      n u n u n e n u n u

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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

) (n u

) (n e

) ( ~ n e

) ( ~ n u

) ( ~ n u

) ( ~ n e

) ( ~ n u

) ( ~ n u

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:

1  k

X

1  k

E

1  k

s

Adaptive Function Unit Delay

k

X

1 

k

Stored

k min k,

E ,  

1 1 min 1 min min 1 1 1 1

| | if | | if ] 2 [ | | ] [ sgn

      

                         

k k k k k k k k k k k K k

X X E E E X S E

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

• f 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;

• therwise go to Step 2.