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E303: Communication Systems E303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London An Overview of Fundamentals: Principles of PCM Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 1 / 52 Table of Contents 1 Glossary 3 2 Introduction 4 3 PCM: Bandwidth & Bandwidth Expansion Factor 8 4 The Quantisation Process (output point-A2) 10 Uniform Quantisers 16 Comments on Uniform Quantiser 18 Non-Uniform Quantisers 21 max(SNR) Non-Uniform Quantisers 23 Companders (non-Uniform Quantisers) 26 Compression Rules (A and mu) The 6dB Law Differential Quantisers 34 Type-1 Type-2 (mse Diff Quant) Examples 5 Noise Effects in a Binary PCM 44 Threshold Effects in a Binary PCM 45 Threshold Point 46 Comments on Threshold Effects 47 6 CCITT Standards: Differential PCM (DPCM) 48 7 Problems of DPCM 49 8 Appendix-1: Alex Reeves - the "Father of PCM" 50 Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 2 / 52 Glossary Glossary CCITT = Comite Consultatif Internationale de Telegraphie et Telephonie This is an international committee based in Geneva, Switzerland, that recommends telecommunications standards, including the audio compression/decompression standards (codecs) and the famous V. standards for modem speed and compression (V.34 and so on). Although this organization changed its name to ITU-T (International Telecommunications Union-Telecommunication), the old French name lives on. Related standards: CCITT A-law, CCITT m-law, codec, ITU-T, V. standards CCITT A-law = This is a CCITT-ratified audio encoding and compression technique supported by Windows and Web phones. Among other implementations, A-law was originally intended as a phone-communications standard. Related standards: CCITT m-law, ITU-T. GSM = Groupe Speciale Mobile (Global System for Mobile Communications) This set of standards is widely used in Europe for cellular communications. The audio encoding subset of the GSM standard is best known to computer users because its data compression and decompression techniques are also being used for Web-phone communication and encoding WAV and AIFF files. Related standards: AIFF, codec, WAV Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 3 / 52 Introduction Introduction Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 4 / 52 Introduction PCM = sampled quantised values of an analogue signal are transmitted via a sequence of codewords. i.e. after sampling & quantisation, a Source Encoder is used to map the quantised levels (i.e. o/p of quantiser) to codewords of g bits i.e. quantised level codeword of g bits 7! 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, quantisation-levels Encoders). I Binary Coded Decimal (BCD) source encoder I Folded BCD source encoder I Gray Code (GC) source encoder Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 5 / 52 Introduction g(input) gq (output) 7! samples gq : occurs at a rate Fs sec (N.B: Fs 2 Fg ) ≥ · Q = quantiser’slevels; bits g = log2(Q) level N.B.: codeword rate (point B) = quant. levels rate = sampling rate g bit codewords" levels" samples" sec sec sec = Fs = 2Fg (1) Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 6 / 52 Introduction bit rate: rb = g Fs e.g. for Q = 16 levels then rb = 4 Fs " bits" levels" g level sec bits sec g=4 g=4 # # (e.g. transmitted sequ. = 101011001101 ...) z}|{ z}|{ g=" 4 |{z} versions of PCM: I Differential PCM (DPCM),PCM with differential Quant. I Delta Modulation (DM): PCM with diff. quants having 2 levels i.e. +D or D are encoded" using a single binary digit I Note: DM DPCM 2 I Others Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 7 / 52 PCM: Bandwidth & Bandwidth Expansion Factor PCM: Bandwidth & Bandwidth Expansion Factor we transmit several bits for each quantiser’so/p level BPCM > Fg ) B denotes the channel bandwidth where PCM F represents the message bandwidth g Definition (PCM Bandwidth) baseband bandwidth: channel symbol rate BPCM Hz (2) ≥ 2 bandpass bandwidth: channel symbol rate BPCM 2 Hz (3) ≥ 2 Note that, by default, the Lower bound of the ‘baseband’bandwidth is assumed and used in this course Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 8 / 52 PCM: Bandwidth & Bandwidth Expansion Factor Definition (Bandwidth Expansion Factor b) channel bandwidth b (4) , message bandwidth N.B. for Binary PCM Channel Bandwidth: channel symbol rate B = PCM 2 bit rate gFs = = = g F Hz 2 2 g " log2 Q BPCM = gFg (5) ) Bandwidth Expansion Factor: BPCM BPCM = gFg = g b = g (6) ) Fg ) Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 9 / 52 The Quantisation Process (output point-A2) The Quantisation 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 quantiser. Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 10 / 52 The Quantisation Process (output point-A2) Definition The following mapping is called quantising analogue samples finite set of levels 7! where the symbol denotes a “map” 7! N.B.: ADC Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 11 / 52 The Quantisation Process (output point-A2) quantiser parameters: Q : number of levels bi : input levels of the quantiser, with i = 0, 1, ... , Q 8 > (b0 = lowest level): known as quantiser’s end-points > > mi : outputs levels of the quantiser > <> (sampled values after quantisation) with i = 1, ... , Q; known as output-levels > rule: connects the input of the quantiser to m > i > > :RULE: the sampled values g(kTs ) of an analogue signal g(t) are converted to one of Q allowable output-levels m1, m2, ... , mQ according to the rule: g(kTs ) mi (or equivalently gq (kTs ) = mi ) 7! iff bi 1 g(kTs ) bi with b0 = ¥, bQ = +¥ ≤ ≤ Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 12 / 52 The Quantisation Process (output point-A2) quantisation noise at each sample instance: nq (kTs ) = gq (kTs ) gs (kTs ) (7) 2 If the power of the quantisation noise is small, i.e. Pnq = nq (kTs ) = small, then the quantised signal (i.e. signal at the outputE of the quantiser) is a good approximation of the original signal. quality of approximation may be improved by the careful choice of bi ’sand mi ’sand such as a measure of performance is optimised. e.g. measure of performance: Signal to quantisation Noise power Ratio (SNRq) signal power Pg SNRq = = quant. noise power Pnq Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 13 / 52 The Quantisation Process (output point-A2) N.B.: Types of Quantisation uniform non-uniform quantisers :8 uniform, or non-uniform > differential = < plus a differential circuit > :> Transfer Function: uniform quantiser non-uniform quantiser for signals with CF = small for signals with CF = large Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 14 / 52 The Quantisation Process (output point-A2) The following figure illustrates the main characteristics of different types of quantisers Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 15 / 52 The Quantisation Process (output point-A2) Uniform Quantisers Uniform Quantisers N.B.: Uniform quantisers are appropriate for uncorrelated samples let us change our notation: gq (kTs ) to gq and g(kTs ) to g the range of the continuous random variable g is divided into Q intervals of equal length D (value of g) (midpoint of the quantising interval in which the value of g falls) 7! bi 1 + bi or equivalently m = for i = 1, 2, ... , Q (8) i 2 Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 16 / 52 The Quantisation Process (output point-A2) Uniform Quantisers step size D: bQ b0 D = (9) Q rule: bi = b0 + i D rule: gq = mi iff bi 1 < g bi where bi 1 +b·i ≤ mi = (10) 2 for i = 1, 2, ... , Q Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 17 / 52 The Quantisation Process (output point-A2) Comments on Uniform Quantiser Comments on Uniform Quantiser 2 Since, in general, Q = large Pg Pg g ) q ' E Furthermore, large Q implies that Fidelity of quantiser = " gq g ' Q = 8 16 are just suffi cient for good intelligibility of speech; (but quantising noise can be easily heard at the background) voice telephony: minimum 128 levels; (i.e. SNRq 42dB) ' N.B.: 128 levels 7-bits to represent each level ) transmission bandwidth = ) " Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 18 / 52 The Quantisation Process (output point-A2) Comments on Uniform Quantiser quantiser = UNIFORM if ( pdf of the input signal = UNIFORM then 2 2g SNRq = Q = 2 (11) Quantisation Noise Power Pnq : D2 quantisation Noise Power: P = (12) nq 12 rms value of Quant. Noise: D rms value of Quant. Noise = fixed = = f g (13) p12 6 f g ) if g(t) = small for extended period of time SNRq < the design value (14) ) this phenomenon" is obvious if the signal waveform has a large CREST FACTOR Prof. A. Manikas (Imperial College) E303: Principles of PCM v.19 19 / 52 The Quantisation Process (output point-A2) Comments on Uniform Quantiser SNRq as a function of the Crest Factor Remember: peak CREST FACTOR (15) rms By using variable spacing CREST FACTOR effects = ) # small spacing" near 0 and large| spacing{z at the extremes} I = this leads to NON-UNIFORM quantisers Prof.
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