PCM & PSTN - Imperial College London · PCM & PSTN Professor A. Manikas ... (PDH) Synchronous...

PCM & PSTN

Professor A. Manikas

Imperial College London

EE303 - Communication Systems

Prof. A. Manikas (Imperial College) EE303: PCM & PSTN 7 Dec 2011 1 / 64

Table of Contents1 Introduction2 PCM: Bandwidth & Bandwidth Expansion Factor3 The Quantization Process (output point-A2)

Uniform QuantizersComments on Uniform QuantiserNon-Uniform Quantizersmax(SNR) Non-Uniform QuantisersCompanders (non-Uniform Quantizers)Compression Rules (A and mu)The 6dB Law

Di§erential Quantizers4 Noise E§ects in a Binary PCM

Threshold E§ects in a Binary PCMComments on Threshold E§ects

5 CCITT Standards: Di§erential PCM (DPCM)6 Introduction to Telephone Network

CCITT recommendations for PCM (24-channels and 30-channels)Single-Channel Path of 2nd CCITT rec. (30-channels PCM)Implementation of 2nd PCM CCITT RecommPlesiochronous digital hierarchies (PDH)Synchronous digital hierarchies (SONET/SDH)


Introduction

Introduction


Introduction

PCM = sampled quantized values of an analogue signal aretransmitted via a sequence of codewords.

i.e. after sampling & quantization, a Source Encoder is used to mapthe quantized levels (i.e. o/p of quantizer) to codewords of γ bits

i.e. quantized level 7! codeword of γ bits

and, then a digital modulator is used to transmit the bits, i.e. PCMsystem

There are three popular PCM source encoders (or, in other words,Quantization-levels Encoders).

I Binary Coded Decimal (BCD) source encoderI Folded BCD source encoderI Gray Code (GC) source encoder


Introduction

g(input) 7! gq(output)

gq : occurs at a rate Fssamplessec

(N.B: Fs # 2 · Fg )

Q = quantizer levels;

γ = log2(Q)bitslevel

Note:

codeword rate (point B)

"γ%bit codewords

sec

= quant. levels rate

"levelssec

= sampling rate

"samplessec

= Fs = 2Fg (1)


Introduction

bit rate: rb = γ"bitslevel

Fs"

levelssec

e.g. for Q = 16 levels then rb = 4"γ

Fs

bitssec

(e.g. transmitted sequ. =

γ=4#z}|{

10101100|{z}"

γ=4

γ=4#z}|{

1101 . . .)

versions of PCM:I Di§erential PCM (DPCM),PCM with di§erential Quant.I Delta Modulation (DM): PCM with di§. quants having 2 levelsi.e. +∆ or % ∆

"are encoded usinga single binary digit

I Note: DM2DPCMI Others


PCM: Bandwidth & Bandwidth Expansion Factor

PCM: Bandwidth & Bandwidth Expansion Factorwe transmit several digits for each quantizer’s o/p level) BPCM > Fg

where%BPCM denotes the channel bandwidthFg represents the message bandwidth

PCM Bandwidthbaseband bandwidth:

BPCM # channel symbol rate2 Hz (2)

bandpass bandwidth:

BPCM # channel symbol rate2 ) 2 Hz (3)

Note that, by default, the Lower bound of the ‘baseband’ bandwidthis assumed and used in this courseBandwidth expansion factor β :

β , channel bandwidthmessage bandwidth

(4)


PCM: Bandwidth & Bandwidth Expansion Factor

Example - Binary PCMI Bandwidth:

BPCM =channel symbol rate

2=bit rate2

=γFs2= γ

"log2 Q

FgHz

) BPCM = γFg (5)

I Bandwidth Expansion Factor:

BPCM = γFg )BPCMFg

= γ

) β = γ (6)


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 adiscrete information source where a discrete message at its output is a“level” selected from the output levels of the quantizer.



analogue samples 7! finite set of levels

where the symbol 7! denotes a “map”

In our case this mapping is called quantizing

i.e.



quantizer parameters:8>>>>>>>><

>>>>>>>>:

Q : number of levelsbi : input levels of the quantizer, with i = 0, 1, . . . ,Q

(b0 = lowest level): known as quantizer’s end-pointsmi : 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 mi

RULE:

the sampled values g(kTs ) of an analogue signal g(t) areconverted to one of Q allowable output-levels m1,m2, . . . ,mQaccording to the rule:

g(kTs ) 7! mi (or equivalently gq(kTs ) = mi )i§ bi%1 * g(kTs ) * bi with b0 = %∞, bQ = +∞



quantization noise at each sample instance:

nq(kTs ) = gq(kTs )% gs (kTs ) (7)

If the power of the quantization noise is small, i.e. Pnq = E*n2q(kTs )

+= 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 ofbi ’s and mi ’s and such as a measure of performance is optimized.e.g. measure of performance: Signal to quantization Noise power Ratio (SNRq)

SNRq =signal power

quant. noise power =PgPnq



Types of quantization:

8>><

>>:

uniformnon-uniform

di§erential =%

uniform, or non-uniform

plus a di§erential circuit

Transfer Function:uniform quantizer non-uniform quantizer

for signals with CF = small for signals with CF = large



The following figure illustrates the main characteristics of di§erent types ofquantizers


The Quantization Process (output point-A2) Uniform Quantizers

Uniform Quantizers

Uniform quantizers 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 Qintervals of equal length ∆

(value of g) 7! (midpoint of the quantizing interval in which the value of g falls)

or equivalently mi =bi%1 + bi

2for i = 1, 2, . . . ,Q (8)


The Quantization Process (output point-A2) Uniform Quantizers

step size ∆:

∆ =bQ % b0Q

(9)

rule:

rule: gq = mi i§ bi%1 < g * bi where%bi = b0 + i · ∆mi =

bi%1+bi2

for i = 1, 2, . . . ,Q(10)


"Output-levels"

of the quantiser"End-points"

of the quantiser

The Quantization Process (output point-A2) Comments on Uniform Quantiser

Comments on Uniform Quantiser

Since, in general, Q = large ) Pgq ' Pg , E*g2+

Furthermore, large Q implies that Fidelity of Quantizer = "

gq ' q

Q = 8% 16 are just su¢cient for good intelligibility of speech;

(but quantizing 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 = "



if

(Quantizer = UNIFORMpdf of the input signal = UNIFORM

thenSNRq = Q2 = 22γ (11)

Quantisation Noise Power Pnq :

Quantization Noise Power: Pnq =∆2

12(12)

rms value of Quant. Noise:

rms value of Quant. Noise = fixed =∆p126= f {g} (13)

) if g(t) = small for extended period of time

) SNRq < the design value

"this phenomenon is obvious

if the signal waveform has a large CREST FACTOR

(14)



SNRq as a function of the Crest Factor

Remember:

CREST FACTOR !peakrms

(15)

By using variable spacing| {z }"

small spacing near 0 andlarge spacing at the extremes

) CREST FACTOR e§ects = #

I =)this leads to NON-UNIFORM QUANTIZERSProf. A. Manikas (Imperial College) EE303: PCM & PSTN 7 Dec 2011 19 / 64

The Quantization Process (output point-A2) Non-Uniform Quantizers

Non-Uniform QuantizersNon-Uniform quantizers are (like unif. quants) appropriate foruncorrelated samples

step size = variable = ∆i

if pdfi/p 6= uniformthen non-uniform quants yield higher SNRq than uniform quants

rms value of nq is not constant but depends on the sampled valueg(kTs ) of g(t)


The Quantization Process (output point-A2) Non-Uniform Quantizers

rule: gq = mi i§ bi%1 < g * bi

where b0 = %∞, bQ = +∞ ∆i = bi % bi%1 = variable

example:


"End-points"

of the quantiser"Output levels"

of the quantiser

The Quantization Process (output point-A2) max(SNR) Non-Uniform Quantisers

max(SNR) Non-Uniform Quantisers

bi , mi are chosen to maximize SNRq as follows:I since Q = large ) Pgq ' Pg , E

*g2+) SNRq = max if Pnq = min

where

Pnq =Q

∑i=1

Z bi

bi%1(g %mi )2 · pdfg · dg (16)

I Therefore:

minmi ,bi

Pnq (17)

(17) ()

8<

:

dPnqdbj

= 0dPnqdmj

= 0(18)

)

((bj %mj )2·pdfg (bj )% (bj %mj+1)2·pdfg (bj ) = 0 for j = 1, 2, . . . ,Q % 1%2 ·

R bjbj%1(g %mj )·pdfg (g) · dg = 0 for j = 1, 2, . . . ,Q

(19)

In the second branch of Equation-19 the parameter mj can be seen asthe statistical mean of the j th quantizer interval



Note:

the above set of equations (i.e. (19)) cannot be solved in closedform for a general pdf. Therefore for a specific pdf an appropriatemethod is given below in a step-form:

METHOD:

1. choose a m12. calculate bi ’s, mi ’s3. check if mQ is the mean of the interval [bQ%1, bQ = ∞]

if yes ! STOPelse ! choose a new m1 and then goto step-2



A SPECIAL CASEmax(SNR) Non-Uniform Quantizer of a Gaussian Input Signal

if the input signal has a Gaussian amplitude pdf, that is pdfq= N(0, σg ) then it can be proved that:

Pnq = 2.2σ2gQ%1.96

"not easy to derive

(12)

In this case the Signal-to-quantization Noise Ratio becomes:

SNRq =PgqPnq

=σ2g

2.2σ2g Q%1.96= 0.45Q1.96 (13)


The Quantization Process (output point-A2) Companders (non-Uniform Quantizers)

Companders (non-Uniform Quantizers)Their performance independent of CF

Non-unif. Quant =SAMPLE

COMPRESSION+

UNIFORM

QUANTIZER+

SAMPLE

EXPANDER

Compressor + Expander , Compander

gf7! gc i .e. gc =f{g} :

"means

”such that"

pdfgc = uniformf%17! gc



Popular companders: use log compression



Two compression rules (A-law and µ-law) which are used in PSTNand provide a SNRq independent of signal statistics are givenbelow:

µ-law (USA) A-law (Europe)

In practice%A ' 87.6µ ' 100



Compression-Rules (PCM systems)The µ and A laws

µ-law A-law

gc =ln(1+µ·| g

gmax |)ln(1+µ)

gmax gc =

8><

>:

A·| ggmax |

1+ln(A) · gmax 0 */// ggmax

/// < 1A

1+ln(A·| ggmax |)

1+ln(A) gmax 1A *

/// ggmax

/// < 1

where

gc = compressor’s output signal

(i.e. input to uniform quantiser)

g = compressor’s input signal

gmax = maximum value of the signal g



The 6dB LAW

uniform quantizer:

SNRq = 4.77+ 6γ% 20 log(CF) dB (20)

remember CF = peakrms

µ-law:SNRq = 4.77+ 6γ% 20 log(ln(1+ µ)) dB (21)

A-law:SNRq = 4.77+ 6γ% 20 log(1+ lnA) dB (22)



REMEMBER the following figure (illustrates the main characteristics ofdi§erent types of quantizers)



COMMENTS

uniform & non-uniform quantizers:

use them when samples are uncorrelated with each other (i.e. thesequence is quantized independently of the values of the precedingsamples)

practical situation:

the sequence {g(kTs )} consists of samples which are correlated witheach other. In such a case use di§erential quantizer.



Examples

PSTN

Fs = 8kHz, Q = 28 (A = 87.6 or µ = 100), γ = 8 bits/level

i.e. bit rate: rb = Fs ) γ = 8k ) 8 = 64 kbits/sec

Mobile-GSM

Fs = 8kHz, Q = 213 uniform ) γ = 13 bits/level,

i.e. bit rate: rb = Fs ) γ = 8k ) 13 = 104 kbits/sec

which, with a di§erential circuit, is reduced to rb = 13 kbits/sec


The Quantization Process (output point-A2) Di§erential Quantizers

Di§erential Quantizers

Di§erential quantizers are appropriate for correlated samplesnamely they take into account the sample to sample correlation in thequantizing process;

e.g.Transmitter (Tx) Receiver (Rx)

input0

@currentmessagesymbol

1

A

The weights w are estimated based on autocorr. function of the input

The Tx & Rx predictors should be identical.I Therefore, the Tx transmits also its weights to the Rx (i.e. weights ware transmitted together with the data)



In practice, the variable being quantized is not g(kTs ) but thevariable d(kTs )

where d(kTs ) = g(kTs )% g(kTs ) (14)

i.e.

Because d(kTs ) has small variations, to achieve a certain level ofperformance, fewer bits are required. This implies that DPCM canachieve PCM performance levels with lower bit rates.6dB law:

SNRq = 4.77+ 6γ% a in dBwhere %10dB < a < 7.77dB

(15)



A Better Di§erential Quantiser: mse Di§. Quant.the largest error reduction occurs when the di§erential quantizer operates on the di§erences

between g(kTs ) and the minimum mean square error (min-mse) estimator g(kTs ) ofg(kTs ) - (N.B.: but more hardware)



g(kTs ) = wT g

where(g = [g((k % 1)Ts ), g((k % 2)Ts ), . . . , g((k % L)Ts )]T

w = [w1,w2, . . . ,wL]T

rule:

%choose w to minimize E

*(g(kTs )% g(kTs ))2

+. . . for the Transmitter

choose w to minimize E*(dq(kTs ) + g(kTs ))2

+. . . for the Receiver



Di§erential Quantisers: Examples

The power of d(kTs ) can be found as follows:

σ2d = E*d2+

= E*g2(kTs )

+| {z }

=σ2g

+ E*g2((k % 1)Ts )

+| {z }

=σ2g

% 2E {g(kTs )g((k % 1)Ts )}| {z }2·Rgg (Ts )

+

σ2d = 2 · σ2g % 2 · Rgg (Ts ) = 2 · σ2g · (1%Rgg (Ts )

σ2g) (23)



e.g.

disadvantages : unrecoverable degradation is introduced by thequantization process.

I (Designer’s task is to keep this to a subjective acceptable level)



Remember

1 σ2g = Rgg (0)

2Rgg (τ)

σ2g= is known as the normalized autocorrelation function

3 DPCM with the same No of bits/sample ! generally gives betterresults than PCMwith the samenumber of bits.



Example of mse DPCM

assume a 4-level quantizer:I/P O/P

+5 * input * +255 +70 * input * +4 +1%4 * input * %1 %1%255 * input * %5 %7



From the last 2 figures we can see thatsmall variation to the i/p signal (25V () 26V)

+large variations to o/p waveforms


Noise E§ects in a Binary PCM

Noise E§ects in a Binary PCMIt can be proved that the Signal-to-Noise Ratio at the output of abinary Pulse Code Modulation (PCM) system, which employs a BCDencoder/decoder and operates in the presence of noise, is given by thefollowing expression

SNRout =E*g0(t)2

+

E {n0(t)2}+ E {nq0(t)2}=

22γ

1+ 4 · pe · 22γ(24)

where

pe = f(type of digital modulator)

pe = T%q

(1% ρ) · EUE5

e.g. if the digital modulator is a PSK-mod. then

pe = Tnp

2 · EUEo


Desired signal

Output noise

(due to channel noise)

Output quantisation noise

Noise E§ects in a Binary PCM Threshold E§ects in a Binary PCM

Threshold E§ects in a Binary PCM

We have seen that: SNRout = 22γ

1+4·pe ·22γ

Let us examine the following two cases: SNRin = high andSNRin = low

i) SNR in = HIGH ii) SNR in = LOW

SNRin = high ) pe = small SNRin = low ) pe = large

) 1+ 4 · pe · 22γ ' 1

) SNRout = 22γ ) 1+ 4 · pe · 22γ ' 4 · pe · 22γ

) SNRout ' 6γ dB ) SNRout ' 14·pe


Noise E§ects in a Binary PCM Threshold E§ects in a Binary PCM

Threshold Point - Definition

Threshold point is arbitrarily defined as the SNRin at which theSNRout, i.e.

SNRout =22γ

1+ 4 · pe · 22γ

falls 1dB below the maximum SNRout(i.e. 1dB below the value 22γ).

By using the above definition it can be shown (. . . for you. . . ) thatthe threshold point occurs when

pe = 116·22γ

where γ is the number of bits per level.


Noise E§ects in a Binary PCM Comments on Threshold E§ects

Threshold E§ects

Comments

The onset of threshold in PCM will result in a sudden " in the output noise power.Psignal = " ) SNRin = " ) SNRout reaches 6γ dB and becomes independent of Psignal) above threshold: increasing signal power) no further improvement in SNRoutThe limiting value of SNRout depends only on the number of bits γ perquantization levels


CCITT Standards: Di§erential PCM (DPCM)


DPCM = PCM which employs a di§erential quantizer

i.e. DPCM reduces the correlation that often exists betweensuccessive PCM samples

The CCITT standards 32 kbitssec

DPCM The CCITT standards 64 kbitssec

DPCMspeech signal - Fg = 3.2kHz audio signal - Fg = 7kHzFs = 8ksamplessec

Fs = 16ksamplessec

Q = 16 levels (i.e. γ = 4 bitslevel) Q = 16 levels (i.e. γ = 4 bits

level)


CCITT = Consulting Committee for International Telegraphy & Telephony

This is now known as ITU-T (Telecom. Standardization Sector of the International Telecommunications)


Problems of DPCM:

1 slope overload noise:occurs when outer quantization level is too small for large inputtransitions and has to be used repeatedly

2 “Oscillation” or granular noise:occurs when the smallest Q-level is not zero. Then, for constantinput, the coder output oscillates with amplitude equal to thesmallest Q-level.

3 “Edge Busyness” noise:occurs when repetitive edge waveform is contaminated by noise whichcauses it to be coded by di§erent sequences of Q-levels.


Introduction to Telephone Network


Note that, as calls are routed through the PSTN, they will be routed(multiplexed ) through a hierarchy of switching centers


Introduction to Telephone NetworkCCITT recommendations for PCM (24-channels and

30-channels)

CCITT recommendations for PCM(24-channels and 30-channels)

1960 British Post O¢ce (BPO) (currently BT) had established a24-ch PCM system with objective the system to be available in1968. Some of this work become the basis to the formation of anumber of CCITT recommendations.

In Europe, the original 24-ch PCM systems , which were designedmainly for up to 32Km transmission routes, have been replaced by30-ch PCM systems .



30-channels)

There are two di§erent CCITT recommendations for PCM. Themain di§erences between these two recommendations are shown inthe following table:



30-channels)

That is,1st CCITT rec. (24-channels PCM)

2nd CCITT rec. (30-channels PCM)



30-channels)

Note

I A-law = better than µ-law (cheaper to produce and easy equipmentmaintenance, smaller quantization error in particular within the mostsignificant part of the dynamic range).

I in 24-ch PCM the signalling information is conveyed within eachspeech time-slot (technique known as bit stealing). Result: a slightreduction in speech-coding performance.


Introduction to Telephone Network Single-Channel Path of 2nd CCITT rec. (30-channels PCM)

Single-Channel Path of 2nd CCITT rec.(30-channels PCM)


Introduction to Telephone Network Implementation of 2nd PCM CCITT Recomm

Implementation of 2nd PCM CCITT Recomm.1st Level Multiplexing


Introduction to Telephone Network Implementation of 2nd PCM CCITT Recomm

Based on the 24-channels and 30-channels PCM CCITTrecommendations (primary multiplex groups) the core telephonenetwork evolved from using Frequency Division Multiplex (FDM)technology to digital transmission and switching

I These two PCM CCITT recommendations have led to two PDH(Plesiochronous digital hierarchies) CCITT recommendations forassembling the TDM telephony data streams from di§erent calls.

I Plesiochronous means: “almost synchronous ”because bits are stu§ed into the frames as padding and the callslocation varies slightly - jitters - from frame to frame


Introduction to Telephone Network Plesiochronous digital hierarchies (PDH)

PDH Hierarchy

Hierarchical Level American DS-x European CEPT-x0 DS-0 64 kbits/s CEPT-0 64 kbits/s1 DS-1 1, 544 kbits/s CEPT-1 2, 048 kbits/s2 DS-2 6, 312 kbits/s CEPT-2 8, 448 kbits/s3 DS-3 44, 736 kbits/s CEPT-3 34, 368 kbits/s4 DS-4 274, 176 kbits/s CEPT-4 139, 264 kbits/s5 CEPT-5 565, 148 kbits/s



The 24-channel PDH TDM CCITT recommendation (DS-x)

The 30-channel PDH TDM CCITT recommendations (CEPT-x)



Main disadvantage of PDH Networks

PDH multiplexing was designed for point-to-point communicationsand channels cannot be added to, or extracted from, a highermultiplexing level demultiplexing down and then multiplexing upagain, through the entire PDH

For instance, to isolate a particular call from DS4, say, it must bedemultiplexed to DS1.

i.e. this is a very complex procedure and needs very expensiveequipment at every exchange to demultiplex and multiplex high speedlines

American & European Telephone Systems are incompatible(therefore very expensive equipment required to translate one formatto the other for transatlantic tra¢c)

Solution: SONET/SDH Signal HierarchyProf. A. Manikas (Imperial College) EE303: PCM & PSTN 7 Dec 2011 60 / 64

Introduction to Telephone Network Synchronous digital hierarchies (SONET/SDH)

Synchronous digital hierarchies (SONET/SDH)

The traditional PDH standards are based on the DS (USA) andCEPT (Europe) PCM systems (24-channels and 30-channels PCMCCITT recommendation)

PDH hierarchy is almost synchronous (extra bits are inserted intothe digital stream to bring them to a common rate).

In 1988 SDH (Synchronous Digital Hierarchy) was adopted byITU and ETSI (European Telecommunications Standards Institute)based on SONET (synchronous optical Networks)

SDH signals have a common external timing i.e.SDH is synchronous



The SDH standards used in Europe are

STM-1 which provides 155 Mbits/sec




etc. (increments of 155 Mbits/sec)

The most important main standards are STM-1, STM-4 andSTM-16. These are commercially available.



SONET/SDH Hierarchy

Hierarchical Level American SONET STS -x European SDH STM -x0 STS -3 = 3) DS-3 STM -1 = 1) CEPT-41 STS -12 = 12) DS-3 STM -4 = 4) CEPT-42 STS -48 = 48) DS-3 STM -16=16) CEPT-4

Key Advantages

it is simple to add and drop channels to meet customer requirements

more bandwidth is available for network management

equipment is smaller and cheaper

network flexibility

integrate and manage various types of tra¢c on a single fiber.


PCM & PSTN - Imperial College London · PCM & PSTN Professor A. Manikas ... (PDH) Synchronous...

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