PCM & DPCM & DM

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.

3. 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

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 :

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

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

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)

8. Companding: Compression and Expanding Original Signal After Compressing, Before Expanding

9. 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.

10. Companding

11. Companding • The logarithmic compressor used in European telecommunications system is called A-law and is defined as

12. Companding

13. DPCM : • A Sampled sequence u(m), m=0 to m=n-1. • Letbe the value of the reproduced (decoded) sequence.

14. 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:

15. DPCM : • If is the quantized value of e(n), then the reproduced value of u(n) is: • Note:

16. Communication Channel Quantizer Σ Σ Predictor Σ Predictor Coder Decoder DPCM CODEC:

17. 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).

18. 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.

19. 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

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

21. 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

22. ADPCM with adaptation of the predictor Step-size adaptation Communication Channel Quantizer Encoder Decoder Σ Σ Σ Predictor Predictor Predictor adaptation Coder Decoder

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

24. 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

25. DM: Unit Delay Integrator Coder Unit Delay Decoder

26. DM: Step size effect : Step Size(i) slope overload (sampling frequency)(ii) granular Noise

27. Adaptive Function Unit Delay Adaptive DM: • This adaptive approach simultaneously minimizes the effects of both slope overload and granular noise

28. Vector Quantization (VQ)

29. Vector Quantization : • Quantization is the process of approximating continuous amplitude signals by discrete symbols. • Partitioning of two-dimensional Space into 16 cells.

30. 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.

31. 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.