Pulse-Code Modulation (PCM) :

Transcription

1 PCM & DPCM & DM 1

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 BF s bps. The quantized waveform is modeled as : q(n) represent the quantization error, Which we treat as an additive noise. B ~ s ( n) s( n) q( n )

3 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 q B 1/ 3

4 Pulse-Code Modulation (PCM) : If A max :maximum amplitude of signal, A max B The mean square value of the quantization Δ/ 1 error is : q (n) q (n)dq 1 3Δ Δ/ q (n) Amax B 1 Measure in db, The mean square value of the noise is : B Δ 3 Δ/ Δ/ Δ 1 10log10 10log10 6B 10.8 db

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 (A max =1) In db, this is X rms SNR SNR db S N A h max 10log X 10 rms /1 B h 1 h 1 h B B 6B log 10 h 5

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= B 0 log 10 4 B bit If we sample at 8 KHZ, then PCM require 8k bit/s. 6

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) 7

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

9 Companding A logarithmic compressor employed in North American telecommunications systems has input-output magnitude characteristic of the form log(1 s ) y log(1 ) is a parameter that is selected to give the desired compression characteristic. 9

10 Companding 10

11 Companding The logarithmic compressor used in European telecommunications system is called A-law and is defined as y log(1 A s 1 log A ) 11

12 Companding 1

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

14 DPCM: At m=n, when u(n) arrives, a quantify, an estimate of u(n), is predicted from the previously decoded samples u ~ ( n 1), u~ ( n i.e., ),... u ~ ( n) ( u ~ ( n 1), u~ ( n (.) : Prediction error: prediction rule ),...); e( n) u( n) u~ ( n) 14

15 DPCM : e ~ ( n ) If is the quantized value of e(n), then the reproduced value of u(n) is: Note: u ~ ( n) u~ ( n) e~ ( n ) u( n) u~ ( n) e( n) u( n) u~ ( n) ( u ~ ( n) e( n)) ( u ~ ( n) e~ ( n)) e( n) e~ ( n) q( n) :The Quantization error in e( n) 15

16 DPCM CODEC: u (n) ) Σ e (n ) Quantizer ~ e ( n Communication Channel ~ e ( n ) Σ Predictor Σ Predictor Coder Decoder 16

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). 17

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. 18

19 DPCM modified by the addition of linearly filtered error sequence u (n) e(n) Σ Quantizer ~ e ( n ) Communication Channel ~ e ( n ) Σ Σ Linear filter {bˆ (i)} Linear filter {â (i)} Σ Linear filter {bˆ (i)} Σ Linear filter {â (i)} Coder Decoder 19

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 0

21 Example of quantizer with an adaptive step size 7 / 5 / 110 M (3) 111 Previous Output M (4) Multiplier 3 / 101 M () / 011 M (1) M (1) - / M () -3 / 001 M (3) -5 / 000 M (4) -7 / 1

22 ADPCM with adaptation of the predictor Step-size adaptation u(n) Σ e(n) Quantizer ~ e ( n ) Σ Encoder Communication Channel Decoder ~ e ( n ) Σ Predictor Predictor Predictor adaptation Coder Decoder

23 Delta Modulation : (DM) Predictor : one-step delay function Quantizer : 1-bit quantizer u~ ( n) u~ ( n 1) e( n) u( n) u~ ( n 1) 3

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 4

25 DM: u(n) e(n) ~ e ( n ) Unit Delay Integrator ~ e ( n ) Coder Unit Delay Decoder 5

26 DM: Step size effect : Step Size (sampling frequency (i) slope overload ) (ii) granular Noise 6

27 Adaptive DM: s k 1 Adaptive Function E k 1 k, E, k Stored min Unit Delay X k 1 k X 1 k E X k 1 k 1 k 1 sgn[ S X k k min K 1 [ E E k 1 k 1 k 1 X k ] E k ] if if k k This adaptive approach simultaneously minimizes the effects of both slope overload and granular noise min min 7

28 Vector Quantization (VQ) 8

29 Vector Quantization : Quantization is the process of approximating continuous amplitude signals by discrete symbols. Partitioning of two-dimensional Space into 16 cells. 9

30 Vector Quantization : The LBG algorithm first computes a 1- vector codebook, then uses a splitting algorithm on the codeword to obtain the initial -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. 30

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 : Split M into M 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 M codebook. Now set M=M. 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. 31