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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 :

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pulse code modulation pcm
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 pcm1
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 pcm2
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 pcm3
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 pcm4
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 pcm5
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
Companding: Compression and Expanding

Original Signal

After Compressing, Before Expanding

companding
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.
companding2
Companding
  • The logarithmic compressor used in European telecommunications system is called A-law and is defined as
slide13
DPCM :
  • A Sampled sequence u(m), m=0 to m=n-1.
  • Letbe the value of the reproduced (decoded) sequence.
slide14
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:
slide15
DPCM :
  • If is the quantized value of e(n), then the reproduced value of u(n) is:
  • Note:
dpcm codec

Communication

Channel

Quantizer

Σ

Σ

Predictor

Σ

Predictor

Coder

Decoder

DPCM CODEC:
slide17
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).
slide18
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.
dpcm modified by the addition of linearly filtered error sequence

Communication

Channel

Quantizer

Σ

Σ

Linear filter

Linear

filter

Linear

filter

Σ

Linear filter

Σ

Σ

Coder

Decoder

DPCM modified by the addition of linearly filtered error sequence
adaptive pcm and adaptive dpcm
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
example of quantizer with an adaptive step size

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
adpcm with adaptation of the predictor
ADPCM with adaptation of the predictor

Step-size

adaptation

Communication

Channel

Quantizer

Encoder

Decoder

Σ

Σ

Σ

Predictor

Predictor

Predictor

adaptation

Coder

Decoder

delta modulation dm
Delta Modulation : (DM)
  • Predictor : one-step delay function
  • Quantizer : 1-bit quantizer
delta modulation dm1
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
slide25
DM:

Unit Delay

Integrator

Coder

Unit Delay

Decoder

slide26
DM:

Step size effect :

Step Size(i) slope overload

(sampling frequency)(ii) granular Noise

adaptive dm

Adaptive

Function

Unit Delay

Adaptive DM:
  • This adaptive approach simultaneously minimizes the effects of both slope overload and granular noise
vector quantization
Vector Quantization :
  • Quantization is the process of approximating continuous amplitude signals by discrete symbols.
      • Partitioning of

two-dimensional

Space into 16 cells.

vector quantization1
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 quantization2
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.