E303: Communication Systems

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1 E303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London Principles of PCM Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 1 / 49

2 Table of Contents 1 Introduction 2 PCM: Bandwidth & Bandwidth Expansion Factor 3 The Quantization Process (output point-a2) Uniform Quantizers Comments on Uniform Quantiser Non-Uniform Quantizers max(snr) Non-Uniform Quantisers Companders (non-uniform Quantizers) Compression Rules (A and mu) The 6dB Law Differential Quantizers 4 Noise Effects in a Binary PCM Threshold Effects in a Binary PCM Comments on Threshold Effects 5 CCITT Standards: Differential PCM (DPCM) 6 Core and Access Networks Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 2 / 49

3 Introduction Introduction Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 3 / 49

4 Introduction PCM = sampled quantized values of an analogue signal are transmitted via a sequence of codewords. i.e. after sampling & quantization, a Source Encoder is used to map the quantized levels (i.e. o/p of quantizer) to codewords of γ bits i.e. quantized level codeword of γ bits and, then a digital modulator is used to transmit the bits, i.e. PCM system There are three popular PCM source encoders (or, in other words, Quantization-levels Encoders). Binary Coded Decimal (BCD) source encoder Folded BCD source encoder Gray Code (GC) source encoder Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 4 / 49

5 Introduction g(input) g q (output) g q : occurs at a rate F s (N.B: F s 2 F g ) Q = quantizer levels; γ = log 2 (Q) bits level samples sec Note: codeword rate (point B) γ bit codewords sec = quant. levels rate levels sec = sampling rate samples sec = F s = 2F g (1) Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 5 / 49

6 Introduction bit rate: r b = bits sec γ bits level F s levels sec e.g. for Q = 16 levels then r b = γ=4 γ=4 {}}{{}}{ (e.g. transmitted sequ. = }{{} ) γ=4 4 γ F s versions of PCM: Differential PCM (DPCM) PCM with differential Quant. Delta Modulation (DM): PCM with diff. quants having 2 levels i.e. + or are encoded using a single binary digit Note: DM DPCM Others Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 6 / 49

7 PCM: Bandwidth & Bandwidth Expansion Factor PCM: Bandwidth & Bandwidth Expansion Factor we transmit several digits for each quantizer s o/p level B PCM > F g { BPCM denotes the channel bandwidth where represents the message bandwidth F g PCM Bandwidth baseband bandwidth: bandpass bandwidth: B PCM B PCM channel symbol rate 2 Hz (2) channel symbol rate 2 2 Hz (3) Note that, by default, the Lower bound of the baseband bandwidth is assumed and used in this course Bandwidth expansion factor β : β channel bandwidth message bandwidth Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 7 / 49 (4)

8 PCM: Bandwidth & Bandwidth Expansion Factor Example - Binary PCM Bandwidth: B PCM = channel symbol rate 2 = bit rate 2 = γf s 2 = γ log 2 Q F g Hz B PCM = γf g (5) Bandwidth Expansion Factor: B PCM = γf g B PCM F g = γ β = γ (6) Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 8 / 49

9 The Quantization Process (output point-a2) at point A2 : a signal discrete in amplitude and discrete in time. The blocks up to the point A2, combined, can be considered as a discrete information source where a discrete message at its output is a level selected from the output levels of the quantizer. Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17 9 / 49

10 analogue samples finite set of levels where the symbol denotes a map In our case this mapping is called quantizing i.e. Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

11 quantizer parameters: Q : number of levels b i : input levels of the quantizer, with i = 0, 1,..., Q (b 0 = lowest level): known as quantizer s end-points m i : outputs levels of the quantizer (sampled values after quantization) with i = 1,..., Q; known as output-levels rule: connects the input of the quantizer to m i RULE: the sampled values g(kt s ) of an analogue signal g(t) are converted to one of Q allowable output-levels m 1, m 2,..., m Q according to the rule: g(kt s ) m i (or equivalently g q (kt s ) = m i ) iff b i 1 g(kt s ) b i with b 0 =, b Q = + Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

12 quantization noise at each sample instance: n q (kt s ) = g q (kt s ) g s (kt s ) (7) If the power of the quantization noise is small, i.e. P nq = E { nq(kt 2 s ) } = small, then the quantized signal (i.e. signal at the output of the quantizer) is a good approximation of the original signal. quality of approximation may be improved by the careful choice of b i s and m i s and such as a measure of performance is optimized. e.g. measure of performance: Signal to quantization Noise power Ratio (SNR q ) signal power SNR q = quant. noise power = P g P nq Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

13 Types of quantization: Transfer Function: uniform quantizer uniform non-uniform { differential = uniform, or non-uniform plus a differential circuit non-uniform quantizer for signals with CF = small for signals with CF = large Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

14 The following figure illustrates the main characteristics of different types of quantizers Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

15 Uniform Quantizers Uniform Quantizers Uniform quantizers are appropriate for uncorrelated samples let us change our notation: g q (kt s ) to g q and g(kt s ) to g the range of the continuous random variable g is divided into Q intervals of equal length (value of g) (midpoint of the quantizing interval in which the value of g falls) or equivalently m i = b i 1 + b i 2 for i = 1, 2,..., Q (8) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

16 Uniform Quantizers step size : rule: = b Q b 0 Q { bi = b 0 + i rule: g q = m i iff b i 1 < g b i where m i = b i 1+b i 2 for i = 1, 2,..., Q (9) (10) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

17 Comments on Uniform Quantiser Comments on Uniform Quantiser Since, in general, Q = large P gq P g E { g 2} Furthermore, large Q implies that Fidelity of Quantizer = g q q Q = 8 16 are just suffi cient for good intelligibility of speech; (but quantizing noise can be easily heard at the background) voice telephony: minimum 128 levels; (i.e. SNR q 42dB) N.B.: 128 levels 7-bits to represent each level transmission bandwidth = Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

18 Comments on Uniform Quantiser { Quantizer = UNIFORM if pdf of the input signal = UNIFORM then SNR q = Q 2 = 2 2γ (11) Quantisation Noise Power P nq : rms value of Quant. Noise: Quantization Noise Power: P nq = 2 12 rms value of Quant. Noise = fixed = (12) 12 = f {g} (13) if g(t) = small for extended period of time SNR q < the design value this phenomenon is obvious if the signal waveform has a large CREST FACTOR (14) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

19 Comments on Uniform Quantiser SNRq as a function of the Crest Factor Remember: CREST FACTOR peak rms (15) By using variable spacing }{{} small spacing near 0 and large spacing at the extremes CREST FACTOR effects = = this leads to NON-UNIFORM QUANTIZERS Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

20 Non-Uniform Quantizers Non-Uniform Quantizers Non-Uniform quantizers are (like unif. quants) appropriate for uncorrelated samples step size = variable = i if pdf i/p = uniform then non-uniform quants yield higher SNR q than uniform quants rms value of n q is not constant but depends on the sampled value g(kt s ) of g(t) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

21 Non-Uniform Quantizers rule: g q = m i iff b i 1 < g b i where b 0 =, b Q = + i = b i b i 1 = variable example: Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

22 max(snr) Non-Uniform Quantisers max(snr) Non-Uniform Quantisers b i, m i are chosen to maximize SNR q as follows: since Q = large P gq P g E { g 2} SNR q = max if P nq = min where Q bi P nq = (g m i ) 2 pdf g dg (16) i =1 b i 1 Therefore: min P nq m i,b i (17) dp nq db (17) j = 0 dp nq dm j = 0 (18) { (bj m j ) 2 pdf g (b j ) (b j m j+1 ) 2 pdf g (b j ) = 0 for j = 1, 2,..., Q 2 b j b j 1 (g m j ) pdf g (g) dg = 0 for j = 1, 2,..., Q (19) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

23 max(snr) Non-Uniform Quantisers Note: the above set of equations (i.e. (19)) cannot be solved in closed form for a general pdf. Therefore for a specific pdf an appropriate method is given below in a step-form: METHOD: 1. choose a m 1 2. calculate b i s, m i s 3. check if m Q is the mean of the interval [b Q 1, b Q = ] if yes STOP else choose a new m 1 and then goto step-2 Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

24 max(snr) Non-Uniform Quantisers A SPECIAL CASE max(snr) Non-Uniform Quantizer of a Gaussian Input Signal if the input signal has a Gaussian amplitude pdf, that is pdf q = N(0, σ g ) then it can be proved that: P nq = 2.2σ 2 g Q 1.96 not easy to derive (12) In this case the Signal-to-quantization Noise Ratio becomes: SNR q = P gq P nq = σ 2 g 2.2σ 2 g Q 1.96 = 0.45Q 1.96 (13) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

25 Companders (non-uniform Quantizers) Companders (non-uniform Quantizers) Their performance independent of CF Non-unif. Quant = SAMPLE COMPRESSION Compressor + Expander Compander g f g c i.e. g c =f{g} + : means such that" UNIFORM QUANTIZER + SAMPLE EXPANDER pdf gc = uniform f 1 g c Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

26 Companders (non-uniform Quantizers) Popular companders: use log compression Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

27 Companders (non-uniform Quantizers) Two compression rules (A-law and µ-law) which are used in PSTN and provide a SNR q independent of signal statistics are given below: µ-law (USA) A-law (Europe) In practice { A 87.6 µ 100 Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

28 Companders (non-uniform Quantizers) Compression-Rules (PCM systems) The µ and A laws µ-law g c = ln(1+µ g gmax ) g ln(1+µ) max g c = A-law g A gmax 1+ln(A) g max 0 g g max < 1 A g 1+ln(A gmax ) 1 g 1+ln(A) max A g g max < 1 where g c = compressor s output signal (i.e. input to uniform quantiser) g = compressor s input signal g max = maximum value of the signal g Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

29 Companders (non-uniform Quantizers) The 6dB LAW uniform quantizer: µ-law: A-law: SNR q = γ 20 log(cf) db (20) remember CF = peak rms SNR q = γ 20 log(ln(1 + µ)) db (21) SNR q = γ 20 log(1 + ln A) db (22) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

30 Companders (non-uniform Quantizers) REMEMBER the following figure (illustrates the main characteristics of different types of quantizers) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

31 Companders (non-uniform Quantizers) COMMENTS uniform & non-uniform quantizers: use them when samples are uncorrelated with each other (i.e. the sequence is quantized independently of the values of the preceding samples) practical situation: the sequence {g(kt s )} consists of samples which are correlated with each other. In such a case use differential quantizer. Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

32 Companders (non-uniform Quantizers) Examples PSTN F s = 8kHz, Q = 2 8 (A = 87.6 or µ = 100), γ = 8 bits/level i.e. bit rate: r b = F s γ = 8k 8 = 64 kbits/sec Mobile-GSM F s = 8kHz, Q = 2 13 uniform γ = 13 bits/level, i.e. bit rate: r b = F s γ = 8k 13 = 104 kbits/sec which, with a differential circuit, is reduced to r b = 13 kbits/sec Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

Differential Quantizers Differential Quantizers Differential quantizers are appropriate for correlated samples namely they take into account

process; e.g. Transmitter (Tx) Receiver (Rx) input current message symbol The weights w are estimated based on autocorr.

33 Differential Quantizers Differential Quantizers Differential quantizers are appropriate for correlated samples namely they take into account the sample to sample correlation in the quantizing process; e.g. Transmitter (Tx) Receiver (Rx) input current message symbol The weights w are estimated based on autocorr. function of the input The Tx & Rx predictors should be identical. Therefore, the Tx transmits also its weights to the Rx (i.e. weights w are transmitted together with the data) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

34 Differential Quantizers In practice, the variable being quantized is not g(kt s ) but the variable d(kt s ) i.e. where d(kt s ) = g(kt s ) ĝ(kt s ) (14) Because d(kt s ) has small variations, to achieve a certain level of performance, fewer bits are required. This implies that DPCM can achieve PCM performance levels with lower bit rates. 6dB law: SNR q = γ a in db (15) where 10dB < a < 7.77dB Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

35 Differential Quantizers A Better Differential Quantiser: mse Diff. Quant. the largest error reduction occurs when the differential quantizer operates on the differences between g(kt s ) and the minimum mean square error (min-mse) estimator ĝ(kt s ) of g(kt s ) - (N.B.: but more hardware) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

36 Differential Quantizers ĝ(kt s ) = w T g where { g = [ g((k 1)T s ), g((k 2)T s ),..., g((k L)T s )] T w = [w 1, w 2,..., w L ] T rule: { choose w to minimize E { (g(kts ) ĝ(kt s )) 2}... for the Transmitter choose w to minimize E { (d q (kt s ) + ĝ(kt s )) 2}... for the Receiver Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

37 Differential Quantizers Differential Quantisers: Examples The power of d(kt s ) can be found as follows: σ 2 d = E { d 2} = E { g 2 (kt s ) } + E { g 2 ((k 1)T s ) } 2E {g(kt s )g((k 1)T s )} }{{}}{{}}{{} =σ 2 g =σ 2 g 2 R gg (T s ) σ 2 d = 2 σ 2 g 2 R gg (T s ) = 2 σ 2 g (1 R gg (T s ) ) (23) σ 2 g Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

38 Differential Quantizers e.g. disadvantages : unrecoverable degradation is introduced by the quantization process. (Designer s task is to keep this to a subjective acceptable level) Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

39 Differential Quantizers Remember 1 σ 2 g = R gg (0) 2 R gg (τ) σ 2 g = is known as the normalized autocorrelation function 3 DPCM with the same No of bits/sample generally gives better results than PCM with the same number of bits. Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

40 Differential Quantizers Example of mse DPCM assume a 4-level quantizer: I/P O/P +5 input input input input 5 7 Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

41 Differential Quantizers Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

42 Differential Quantizers From the last 2 figures we can see that small variation to the i/p signal (25V 26V) large variations to o/p waveforms Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

43 Noise Effects in a Binary PCM Noise Effects in a Binary PCM It can be proved that the Signal-to-Noise Ratio at the output of a binary Pulse Code Modulation (PCM) system, which employs a BCD encoder/decoder and operates in the presence of noise, is given by the following expression SNR out = E { g 0 (t) 2} E {n 0 (t) 2 } + E {n q0 (t) 2 } = 2 2γ p e 2 2γ (24) where p e = f(type of digital modulator) { } p e = T (1 ρ) EUE e.g. if the digital modulator is a PSK-mod. then { } p e = T 2 EUE Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

44 Noise Effects in a Binary PCM Threshold Effects in a Binary PCM Threshold Effects in a Binary PCM We have seen that: SNR out = 2 2γ 1+4 p e 2 2γ Let us examine the following two cases: SNR in = high and SNR in = low i) SNR in = HIGH ii) SNR in = LOW SNR in = high p e = small SNR in = low p e = large p e 2 2γ 1 SNR out = 2 2γ p e 2 2γ 4 p e 2 2γ SNR out 6γ db SNR out 1 4 p e Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

45 Noise Effects in a Binary PCM Threshold Effects in a Binary PCM Threshold Point - Definition Threshold point is arbitrarily defined as the SNR in at which the SNR out, i.e. 2 SNR out = 2γ p e 2 2γ falls 1dB below the maximum SNR out (i.e. 1dB below the value 2 2γ ). By using the above definition it can be shown (... for you... ) that the threshold point occurs when p e = γ where γ is the number of bits per level. Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

46 Noise Effects in a Binary PCM Comments on Threshold Effects The onset of threshold in PCM will result in a sudden in the output noise power. P signal = SNR in = SNR out reaches 6γ db and becomes independent of P signal above threshold: increasing signal power no further improvement in SNR out The limiting value of SNR out depends only on the number of bits γ per quantization levels Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49 Threshold Effects Comments

47 CCITT Standards: Differential PCM (DPCM) CCITT Standards: Differential PCM (DPCM) DPCM = PCM which employs a differential quantizer i.e. DPCM reduces the correlation that often exists between successive PCM samples The CCITT standards 32 kbits DPCM The CCITT standards 64 kbits sec sec speech signal - F g = 3.2kHz audio signal - F g = 7kHz F s = 8 ksamples sec Q = 16 levels (i.e. γ = 4 bits ) level F s = 16 ksamples sec bits Q = 16 levels (i.e. γ = 4 level ) DPCM Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

48 CCITT Standards: Differential PCM (DPCM) Problems of DPCM: 1 slope overload noise: occurs when outer quantization level is too small for large input transitions and has to be used repeatedly 2 Oscillation or granular noise: occurs when the smallest Q-level is not zero. Then, for constant input, the coder output oscillates with amplitude equal to the smallest Q-level. 3 Edge Busyness noise: occurs when repetitive edge waveform is contaminated by noise which causes it to be coded by different sequences of Q-levels. Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

49 Core and Access Networks Core and Access Networks Prof. A. Manikas (Imperial College) E303: Principles of PCM v / 49

Pulse-Code Modulation (PCM) :

PCM & DPCM & DM 1 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