PCM &amp; DPCM &amp; DM

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# PCM &amp; DPCM &amp; DM - PowerPoint PPT Presentation

PCM &amp; DPCM &amp; DM. 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 :

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### PCM & DPCM & DM

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.
Pulse-Code Modulation (PCM) :
• The quantization noise is characterized as a realization of a stationary random processq in which each of the random variables q(n) has uniform pdf.
• Where the step size of the quantizer is
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 :
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
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
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

Original Signal

After Compressing, Before Expanding

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.
Companding
• The logarithmic compressor used in European telecommunications system is called A-law and is defined as
DPCM :
• A Sampled sequence u(m), m=0 to m=n-1.
• Letbe the value of the reproduced (decoded) sequence.
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:
DPCM :
• If is the quantized value of e(n), then the reproduced value of u(n) is:
• Note:

Communication

Channel

Quantizer

Σ

Σ

Predictor

Σ

Predictor

Coder

Decoder

DPCM CODEC:
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).
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.

Communication

Channel

Quantizer

Σ

Σ

Linear filter

Linear

filter

Linear

filter

Σ

Linear filter

Σ

Σ

Coder

Decoder

• 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.
• feedforward
• feedbackward

Previous Output

111

7∆/2

M (4)

Multiplier

110

5∆/2

M (3)

101

3∆/2

M (2)

100

∆/2

M (1)

-3∆

-2∆

-∆

0

2∆

3∆

011

-∆/2

M (1)

010

-3∆/2

M (2)

001

-5∆/2

M (3)

000

-7∆/2

M (4)

Example of quantizer with an adaptive step size

Step-size

Communication

Channel

Quantizer

Encoder

Decoder

Σ

Σ

Σ

Predictor

Predictor

Predictor

Coder

Decoder

Delta Modulation : (DM)
• Predictor : one-step delay function
• Quantizer : 1-bit quantizer
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
DM:

Unit Delay

Integrator

Coder

Unit Delay

Decoder

DM:

Step size effect :

(sampling frequency)(ii) granular Noise

Function

Unit Delay

• This adaptive approach simultaneously minimizes the effects of both slope overload and granular noise

### Vector Quantization (VQ)

Vector Quantization :
• Quantization is the process of approximating continuous amplitude signals by discrete symbols.
• Partitioning of

two-dimensional

Space into 16 cells.

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.
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.