CHAPTER 3 PCM AND TDM

Aircraft Radio and Optical Communications

___________________________________________________________________________

ET0424 Page 3 - 1

CHAPTER 3

PULSE CODE MODULATION &

MULTIPLEXING

Learning Outcomes

• Name the Sampling Theorem for Bandlimited & Bandpass Signals.

• Understand the Effects of Oversampling & Undersampling.

• Name the Two Different Types of Quantisation Techniques.

• List the Different Blocks of PCM Encoder & Decoder.

• List the Advantages of Multiplexing .

• Name the Two Different Types of Multiplexing Techniques.


___________________________________________________________________________

ET0424 Page 3 - 2

PULSE-CODE MODULATION (PCM)

3.1 INTRODUCTION

On – Board communications in an aircraft involves transmission of voice information

between the pilot & the cabin crew, between the crew & passengers and video

transmissions in the form of movies. In order to reduce the number of wires

(transmission lines) used in connecting various devices, signals are often combined as

one composite signal and transmitted on a signal transmission line. This process is

known as multiplexing. Prior to this multiplexing, signals goes through pulse code

modulation. In this chapter we go into details of both pulse code modulation &

multiplexing.

Pulse code modulation (PCM) is one of the most common techniques used today for

digitising an analog signal. PCM is used in many applications, such as telephone

system, compact disc (CD) recording, PC audio – wav format, voice mail, and many

other applications.

The essential operations in the transmitter of a PCM system are sampling, quantising

and encoding, as shown in Figure 3.1. The quantising and encoding operations are

usually performed in the same circuit, which is called an analog-to-digital converter.

The essential operations in the receiver are regeneration of corrupted signals,

decoding and demodulation of the train of quantised samples. Regeneration usually

occurs at intermediate points along the transmission path in the channel.


___________________________________________________________________________

ET0424 - 3

Fig 3.1

3.2 SAMPLING

3.2.1 SIGNALS IN TIME DOMAIN AND FREQENCY DOMAIN

The usual description of a electrical signal x(t) is in the time domain, where the

independent variable is time.

In communication applications, it is more convenient to describe signals in the

frequency domain, where the independent variable is f (frequency). That is, we think

of the signal as being composed of a number of frequency components, each with

appropriate amplitude and phase. Thus, while a signal physically exists in the time

domain, it consists of frequency components in the frequency domain, known as its

frequency spectrum.

Received

signal

PCM

waveform Analog

Signal Low Pass

Filter Quantiser Encoder

Transmitter

Repeater Repeater Distorted

PCM

waveform

Regenerated

PCM

waveform

Transmission channel

Repeater Decoder Recontruction

filter Destination

Receiver

Sampler


___________________________________________________________________________

ET0424 Page 3 - 4

3.2.2 SPECTRAL ANALYSIS OF SIGNALS IN LINEAR SYSTEMS

Any periodic signal can be fully represented in the frequency domain by a set of

components in amplitude and phase.

Time Domain Frequency Domain

spectrum Phase

spectrum Amplitude

The frequency domain representation of a periodic signal can be obtained using

Fourier Series. It defines any periodic signal as a sum of sinusoidal waveforms. The

Fourier series will consists of a summation of harmonics of the fundamental

frequency

T

f o

1= (3.2.1)

where T is the fundamental period

For example,

where T is the period

Fig 3.2.a Time domain representation of a periodic signal

Amplitude

s(t)

Signal waveform

(steady state)

0 T 2T 3T time


___________________________________________________________________________

ET0424 Page 3 - 5

φ1

φ2

φ3 φ4

φ6

φ5

frequency

0 f0 2f0 3f0 4f0 5f0 6f0 …………

A0

A1

A2

A3

A4 A6

A5

frequency

0 f0 2f0 3f0 4f0 5f0 6f0 …………

where fo =1/T and An = amplitude of signal component at nfo.

where fo =1/T and φn = phase of signal component at nfo.

Fig 3.2.b Frequency domain representation of a periodic signal

Non-periodic signals have a ‘period’ that tends to infinity. The spectrum of a non-

periodic signal is continuous. For example,

Fig 3.2 c Time domain representation of a non periodic signal

Phase

Spectrum

|φ(f)|

Amplitude

Spectrum

|S(f)|

2π

. . .

time

Noise-like

signal, n(t)

Amplitude


___________________________________________________________________________

ET0424 Page 3 - 6

Fig 3.2.d Frequency domain representation of a non periodic signal

3.2.3 SAMPLING

It is possible under certain circumstances for a continuous signal to be completely

represented by its instantaneous values (or samples) taken at equal and sufficiently

short intervals in time. The original continuous signal can then be fully recovered

from these train of samples by processing them through an appropriate filter.

Reasons for sampling an analog signal

• To convert an analog signal into a digital signal that is compatible with digital

transmission. For example, PCM transmission.

• To allow an analog signal to be digitally processed.

• To allow TDM (time division multiplexing) which is the simultaneous

transmission of several signals over the same channel.

|N(f)|

Continuous

spectrum

Frequency


___________________________________________________________________________

ET0424 Page 3 - 7

• Certain signal processing devices (for example high-power microwave tubes;

laser) can operate better on a pulse basis.

• Reduction in power needed to transmit a signal as with sampling, signal is

transmitted in burst.

IDEAL SAMPLING

Ideally, the sampling train should be one that takes instantaneously the signal value

without any modification. Such is an unit impulse train.

The unit impulse train p(t) has a period, T which is the sampling interval. The

fundamental frequency of p(t), fs = 1/T is referred to as the sampling frequency.

Consider the case of impulse-train sampling as follows :

p(t)

x(t) xp(t)

Fig 3.3 a Time domain representation of impulse sampling


___________________________________________________________________________

ET0424 Page 3 - 8

T x(0) x(t)

Fig 3.3 b Time domain representation of impulse sampling

T

0

x(t)

t

p(t)

t

Ideal

sampler

t

xp(t)


___________________________________________________________________________

ET0424 Page 3 - 9

If fs > 2 fm,

If fs < 2 fm,

Fig 3.3 c Frequency Domain representation of impulse sampling

-fm 0 fm

f

-2fs -fs 0 fs 2fs

P(f)

f

1/T

-2fs -fs -fm 0 fm fs 2fs

Xp(f)

1/T

f

(fs - fm)

Xp(f)

1/T

f

-2fs -fs -fm 0 fm fs 2fs f


___________________________________________________________________________

ET0424 Page 3 - 10

Following observations can be made from figure 3.2.2c :

1. When fs > 2fm, that is (fs - fm) > fm there is no overlapping between the

shifted replicas of X(f). X(f) is reproduced at integer multiples of the sampling

frequency. Consequently, the original signal x(t) can be recovered exactly

from the sampled signal, xp(t) by passing it through an ideal LPF with

constant gain of T and cut-off frequency fc in the range, fm < fc < (fs - fm).

The output signal will exactly be equal to x(t).

2. When fs < 2fm, that is (fs - fm) < fm there is overlap and the original signal

cannot be recovered from the sampled signal. Such overlap of replicas of X(f)

is known as aliasing.

These observations form the basis of sampling theorem.

3.2.2.1 UNIFORM SAMPLING THEOREM I

For bandlimited baseband signals

A baseband signal is a signal with frequency range from dc to a few megahertz.

If a baseband signal x(t) is bandlimited to fm then x(t) is completely characterised by

its samples taken at uniform intervals of less than 1/2fm seconds apart. The sampling

frequency fs must be greater than or equal to 2fm. That is if a signal is sampled at

fs ≥ 2fm, it can be completely recovered from its samples.

Example 1

A baseband signal has frequency components up to 3.6 kHz. What is the minimum

sampling frequency to ensure that the original signal can be recovered from its

samples?


___________________________________________________________________________

ET0424 Page 3 - 11

Using sampling theorem I,

Min. sampling frequency = 2fm = 2 x 3.6 kHz = 7.2 kHz.

NYQUIST FREQUENCY AND NYQUIST RATE

The minimum sampling frequency, fs = 2fm is referred to as Nyquist Frequency or

Nyquist Rate and minimum sampling interval, Ts = 1/2fm as Nyquist Interval.

BANDLIMITED SIGNAL

The term bandlimited means that there are absolutely no frequency components in its

spectrum above the frequency fm Hz. However, normal practical message signals do

not have a sharp frequency cut-off, and will contain frequency components at higher

frequencies. That is,

Ideal case :

Practical Case :

0 fm

f

X(f)

0 fm f

X(f)


___________________________________________________________________________

ET0424 Page 3 - 12

Normally, the frequency components in the high frequency region tends to be quite

small in amplitude and can in most cases be neglected in practice. For instance in

telephony, speech is bandlimited to 3.4kHz even though there can be frequency

components in the 7kHz range or higher.

Amplitude

Anti-aliasing filter

LPF

Typical speech spectrum

f

0 3.4kHz

Most of the useful information is captured by the 3.4kHz LPF, that is why telephony

speech is highly intelligible although it is by no means hi-fi in quality.

Hence in applying sampling theorem where fs ≥ 2fm, fm must be well defined to

avoid the occurrence of aliasing. For this reason, it is common in communication

system design, to perform an anti-aliasing filtering before any sampling operation.

THE EFFECT OF USING NON-IDEAL RECONSTRUCTION FILTER

-2fs -fs -fm 0 fm fs 2fs

Xp(f)

1/T

f

(fs - fm)

H(f), recontruction filter


___________________________________________________________________________

ET0424 Page 3 - 13

In practice, it is very difficult to build filters with a very sharp roll-off at the cut-off

frequency. As shown in figure above, some spurious frequency components are

allowed by the reconstruction filter to reach the output.

Note that these components are considerably attenuated compared to the baseband

signal. To minimise these spurious frequencies, a higher sampling frequency should

be used.

THE EFFECTS OF OVERSAMPLING AND UNDERSAMPLING

Consider x(t) as a pure sinusoidal signal of frequency fm. Assume also an ideal

reconstruction LPF with cut-off frequency, fc between fm and (fs - fm).

Oversampling

a) If fs >> 2fm

0 fm fs - fm

original signal samples

t f

0 fm

reconstructed signal samples

t f


___________________________________________________________________________

ET0424 Page 3 - 14

b) If fs > 2fm

Note : In either case the reconstructed waveform is identical to the original. It can be

seen that the original signal is fully recovered whether the sampling frequency fs >>

2fm or fs > 2fm.

It is therefore wasteful to sample a signal at too high rate, since it increases the

bandwidth requirement for the same task without any gain as most of the samples

transmitted are redundant and do not carry any extra information about the signal.

0 fm fs - fm

original signal samples

t f

0 fm

reconstructed signal samples

t f


___________________________________________________________________________

ET0424 Page 3 - 15

Undersampling

If fs < 2fm

In this case, an effect known as aliasing occurs. The reconstructed signal takes on an

identity of a lower frequency signal, which is the difference between fs and fm,

ie (fs - fm). The original signal cannot be recovered from its samples.

In the case when fs = fm, the reconstructed signal is a constant (i.e. dc). This is

consistent with the fact that when sampling once per cycle of the original signal, the

samples are all equal. The result is as though a dc signal was being sampled.

0 (fs - fm) fm

original signal sample

t f

0 (fs - fm)

reconstructed signal sample

t f


___________________________________________________________________________

ET0424 Page 3 - 16

Aliasing

Aliasing causes higher frequencies to be translated as lower frequencies in the

recovered signal. In voice transmission, aliasing will cause serious degradation in the

intelligibility.

Aliasing can occur under two conditions :

1. When the signal to be sampled is not bandlimited. For example,

To remedy this, anti-aliasing filter is used to bandlimit the signal before sampling.

0

0 f

X(f)

-fs 0 aliasing fs

f


___________________________________________________________________________

ET0424 Page 3 - 17

2. When the bandlimited signal is sampled at a frequency less than twice the highest

frequency present. (Undersampling).

aliasing

Summarising

The uniform sampling theorem allows us to completely reconstruct a bandlimited

signal from instantaneous samples taken at a rate of at least twice the highest

frequency present in the signal.

The signal can be totally recovered by processing the samples with an ideal LPF

having a bandwidth equal to the highest frequency present in the signal.

On the other hand, if the sampling theorem conditions are not satisfied, aliasing will

occur.

-fm 0 fm

f

X(f)

-fs -fm 0 fm fs

f

Xs(f)


___________________________________________________________________________

ET0424 Page 3 - 18

3.2.3.2 UNIFORM SAMPLING THEOREM II

For bandlimited bandpass signals

For a signal x(t) whose highest frequency spectral component is fm, the sampling

frequency fs must be greater or equal to 2fm, only if the lowest frequency spectral

component of x(t) is fL = 0. If the signal has a fL that is non-zero, i.e. bandpass signal,

a lower sampling frequency can often be found.

In general for a bandpass signal, the minimum sampling frequency

M

ff m

s

2= where

−=

Lm

m

ff

fM

M is the largest integer that does not exceed the value in the [ ] bracket.

Example 2

A signal x(t) has spectrum as shown :

What is the minimum sampling frequency of x(t) such that the original signal x(t) can

be recovered from the sampled version xs(t).

-5 -3 0 3 5

f(kHz)

X(f)


___________________________________________________________________________

ET0424 Page 3 - 19

fm = 5 kHz fL = 3 kHz fm - fL = 2 kHz

fs = 2fm/M M = [fm/(fm - fL)] = [5/2] = 2

Therefore, fs = 2fm/2 = fm = 5 kHz

The recovery filter is a BPF of passband 3 kHz - 5 kHz.

-10 -5 0 5 10

-5 -3 0 3 5 f(kHz)

X(f)

f(kHz)

… …

S(f)

-10 -8 -7 -5 -3 -2 0 2 3 5 7 8 10 f(kHz)

X(f)*P(f)


___________________________________________________________________________

ET0424 Page 3 - 20

3.2.3.3 PRACTICAL SAMPLING

The ideal sampling techniques discussed earlier, used impulse sampling train which

do not exist in practice. Besides, practical filters cannot be perfectly rectangular in

frequency response. Signal recovery is still possible from a sampled signal where the

sampling signal is a pulse train.

Therefore in practical sampling :

a) The sampling waveform consists of pulses of finite amplitude and duration.

b) Practical reconstruction filters do not possess ideal characteristics.

There are two forms of practical sampling,

1. Natural sampling

2. Flat-top sampling

Natural Sampling

In natural sampling the signal to be sampled is multiplied by the sampling pulse train.

p(t)

xp(t) x(t)

τ

T


___________________________________________________________________________

ET0424 Page 3 - 21

a) message signal

b) sampling train

c) Naturally sampled signal

Fig 3.4 a Waveforms of natural sampling

Observation on natural sampling

1. In time domain, the sampled signal is a pulse train whose pulse amplitude

follows the message signal over the duration of the pulse width, τ.

2. x(t) can be reconstructed from xp(t) by processing it through an ideal LPF.

3. We have assumed that the sampling frequency fs ≥ 2fm and hence no aliasing

occurs.

t

x(t)

τ T

p(t)

t

1

t

xp(t)


___________________________________________________________________________

ET0424 Page 3 - 22

Flat-top sampling

This type of sampling is also known as 'instantaneous sampling'.

The amplitude of each pulse in the sampled pulse train is constant during the complete

duration of the pulse. The amplitude of the pulse is determined by the instantaneous

sample of the analog signal x(t). It is generated by a Sample-and- Hold circuit.

Fig 3.4 b Examples of flat-top sampled signals

Fig 3.4 c Sample-and-Hold circuit

Unlike natural sampling, a low pass filter operating on the flat-top sampled signal will

not give a distortion-free output. However, this distortion may be corrected by adding

a second filter - the equalising filter, in cascade to the reconstruction filter. This

distortion is called the Aperture Distortion or Effect.

τ

T

t

flat-top samples xq(t)

t

sampling switch

discharging

switch capacitor input- analog

signal output - flat-top sampled

signal


___________________________________________________________________________

ET0424 Page 3 - 23

3.3.3.4 SUMMARY

1. Important fact - A continuous waveform representing an information source

can be completely reconstructed in a receiver from periodic samples only of

the waveform.

2. Conditions that must be satisfied - the waveform should be bandlimited and

that the instantaneous samples be taken at a high enough rate. (minimum rate

is known as Nyquist rate)

3. In practice, waveform are never bandlimited. (there is always some spectral

components outside the frequency band of interest)

4. Sampling theorem II applies to bandpass signals.

5. Practical sampling methods - Natural Sampling and Flat-top Sampling.


___________________________________________________________________________

ET0424 Page 3 - 24

3.3 QUANTISATION :

3.3.1 Quantising of Analog Signals The concept of quantisation is represented in Figure 3.5. The amplitude of the signal

m(t) is confined to the range VL to VH. We have divided this total range into M equal

intervals each of size q. Accordingly q, called the step-size, is

q = ( VH - VL ) / M (3.3.1)

In Figure 3.5 we show the specific example in which M = 8. In the centre of each of

these steps we locate quantisation levels mo, m1, ..., m7. The quantised signal mq(t) is

generated in the following way:

At any time, mq(t) has the value of quantisation level to which m(t) is closest.

Thus the signal mq(t) will at all times be found at one of the levels mo, m1, ..., m7. The

transition in mq(t) from mq(t) = mo to mq(t) = m1 is made abruptly when m(t) passes

the transition level L01 which is midway between mo and m1 and so on. Thus the

signal mq(t) does not change at all with time or it makes a "quantum" jump of step

size q.

The quantisation error, defined as mq(t)-m(t), has a magnitude which is equal or less

than q/2.

We see therefore, that the quantised signal is an approximation to the original signal.

The quality of the approximation may be improved by reducing the size of the steps,

thereby increasing the number of allowable levels. With small enough steps, the

human ear or eye will not be able to distinguish the original from the quantised signal.


___________________________________________________________________________

ET0424 Page 3 - 25

q/2

m(t)

mq(t)

VL

q

q

q

q

q

q

q

q

m(t), mq(t)

m7

m6

m5

m4 m4

m3

m2

m1

m0

time

Fig 3.5 Concept of Quantisation

VH

L67

L56

m4 L45

L34

L23

L12

L01


___________________________________________________________________________

ET0424 Page 3 - 26

3.3.2 Uniform Quantisation

The quantisation ranges and levels may be chosen in a variety of ways depending on

the intended application of the digital representation, that is whether the source signal

to be quantised is voice, video, music or any other type of analog source, each of

which possesses unique statistical characteristics. When the digital representation is

to be processed by a digital system, the quantisation levels and ranges are generally

distributed uniformly. Thus to define a uniform quantiser using the example of

Figure 3.6, we set

qxx

qxx

ii

ii

=−

=−

−

−

1

1

ˆˆ

where q is the quantisation step size. A common uniform quantiser characteristic is

shown in Figure 3.7 for the case of eight quantisation levels. Here the origin appears

in the middle of a rising part of the staircase-like function. This class of quantisers is

called the "mid-riser" class. It can be seen that the mid-riser quantiser has the same

number of positive and negative levels, and these are symmetrically positioned about

the origin.

For a uniform quantiser there is only two parameters: the number of levels and the

quantisation step size, as we discussed above. The number of levels is generally

chosen to be of the form 2B so as to make the most efficient use of B-bit binary code

words. Together, q and B must be chosen so as to cover the range of input samples.

Then we should set the peak-to-peak signal amplitude to be equal to the quantiser

input range i.e.

2 Xmax = q2B (3.3.2)

where Xmax is the peak signal amplitude.


___________________________________________________________________________

ET0424 Page 3 - 27

That is,

q = 2 Xmax / 2B mid riser (3.3.3)

Fig 3.6 Input-output characteristics of a 3-bit quantiser

4ˆ

−− x

3ˆ

−− x

2ˆ

−− x

3x

4x

2x

1x

x-3 x-2 x-1 x0 x1 x2 x3

- 1ˆ

−x

input

output

q

q


___________________________________________________________________________

ET0424 Page 3 - 28

Fig .3.7 Uniform mid-riser 3-bit quantiser

Quantising Distortion

Performance of a quantiser can be described by a signal-to-noise ratio that takes both

quantising and overload distortion into account. The error introduced by the

quantising process is categorised into two types: quantising distortion and

overloading (or clipping. As long as the input signal lies within the quantiser

permitted range of -V to +V, the only form of error introduced is quantising

distortion, limited to a maximum value equal to q/2 for the linear quantiser. If the

input signal exceeds the allowed range, the quantiser output will remain in the

2

q

-2

7q

- 2

5q

-2

3q

2

3q

2

5q

2

7q

-4q -3q -2q -q q 2q 3q 4q input

output

-2

q

Peak-to-peak range


___________________________________________________________________________

ET0424 Page 3 - 29

maximum allowed level, resulting in the input signal being clipped. Careful choice

of overload thesholds xo and xN controls a tradeoff between the relative amounts of

quantising and overload distortion.

[Note that the terms distortion and noise are used interchangeably here when

describing quantisation performance. Note also that three notations are usually used

to denote the quantising step size, namely, "S", "q" and "∆". They are commonly

used in the literature].

Quantisation Noise Power

In Figure 3.8(a) linearly increasing signal is fed into the uniform quantiser and the

result is a staircase-like waveform (the quantised signal) while the difference between

the output and input waveforms is a sawtooth waveform with a peak-to-peak

magnitude equal to q. The error waveform can be described mathematically as

e = -x + q/2 (3.3.4)

It can be shown that the quantisation noise power is given by

12

2q

N q = (3.3.5)

It should be pointed out that this result (quantisation noise power) is not limited to the

case of a ramp input to an uniform quantiser. The same result can be obtained if we

make the assumption that the quantisation noise is a random signal.

Signal-to-Quantisation noise ratio

To compute the ratio of signal to quantising noise S/Nq, the input signal

characteristics must also be specified. Quite often performance for a quantiser is

based on sinusoidal inputs, because S/Nq for speech and sinusoidal inputs compare


___________________________________________________________________________

ET0424 Page 3 - 30

favourably and use of sinusoids facilitates measurements and calculations of S/Nq.

For the case of a full range (peak voltage = V) sinusoidal input that has zero overload

error, the average signal power is

21

2

2

2

2

VVandRwhere

V

VR

VS

rms

rms

rms

===

==

Ω

Equation (3.3.6) thus gives the normalised average signal power where R is

normalised to 1 Ω to simplify the mathematics.

Now, the peak value of the sinusoid can be expressed in terms of step size (q) and

number of levels in the quantiser (M), as follows

V = (q x M)/2 (3.3.7)

The rms value is therefore,

22

x MqVrms = (3.3.8)

Hence, the average signal power is

8

22Mq

S = (3.3.9)

(3.3.6)


___________________________________________________________________________

ET0424 Page 3 - 31

(a) Linear quantiser characteristic

e = y –x =error

(b) Error characteristic

Fig 3.8 Linear Quantiser

Combining eqns. (3.3.5) and (3.3.9), the signal-to-quantisation noise ratio is

25.1 MN

S

q

= (3.3.10)

x0

xN xi-1

q

xi

x = input

y = output

xN

x0

Quantising

error Overloading

error Overloading

error


___________________________________________________________________________

ET0424 Page 3 - 32

Expressing the ratio in dB's, we have

( )2

10 5.1log10 MN

S

q

=

(3.3.11)

= 1.76 + 20 log10M

Since M = 2B,

BN

S

q

676.1 +=

(3.3.12)

Note that eqn.(3.3.12) is the signal-to-noise ratio for a sinusoid whose amplitude range

coincides with the range of the quantiser. For sinusoidal inputs whose amplitude, Vx, is less

than the full range of the quantiser, V, then

V

VB

N

S x

q

10log20676.1 ++=

(3.3.13)

3.3.3 NON-UNIFORM QUANTISATION (LOGARITHMIC QUANTISATION)

One of the problems with the uniform quantiser is that the speech signal changes with time

and the variance can be quite different from one speech segment to another. If the quantiser

is designed to accommodate strong signals with large variance (like vowels), the quantisation

step size will be large and weaker signals (like consonants) will be subjected to a larger

quantisation error which in turn will have an effect on the speech quality. To solve this

problem, we abandon the uniform quantiser and design a quantiser whose step size increases

with the signal amplitude. This technique is called non-uniform quantising, since a variable


___________________________________________________________________________

ET0424 Page 3 - 33

step size is used. The effect of non-uniform quantising can be obtained by first passing the

analog signal through a compression (non-linear) amplifier and then into a uniform quantiser.

For speech signal, the uniform quantiser will usually operate on the logarithm of the speech

signal, which is a compressed version of the original signal.

The signal is reconstructed at the receiver by expanding it. This process of compression-

expansion is called COMPANDING (COMpressing-exPANDING).

An example of a compression characteristic is shown in Figure 3.9. Note that at low

amplitudes the slope is larger than at large amplitudes. A signal transmitted through such a

network will have the extremities of its waveform compressed. The peak signal which the

system is intended to accommodate will, as before, range through all available quantisation

regions. But now, a small amplitude signal will range through more quantisation regions than

would be the case in the absence of compression.

Figure 3.10 gives a comparison of the variation of output signal-to-noise ratio as a function

of the input signal power when companding is used, to the case of an uncompanded system.

Note that the companded system has a far greater dynamic range than the uncompanded

system and that theoretically the companded system has an output signal-to-noise ratio which

exceeds 30 dB over a dynamic range of input signal power of 48 dB, while the uncompanded

system has a dynamic range of 18 dB for the same conditions.


___________________________________________________________________________

ET0424 Page 3 - 34

Fig 3.9 Input-output characteristic for

compression

compression

Output voltage

Input voltage


___________________________________________________________________________

ET0424 Page 3 - 35

The ITU-T standard for digital coding of voice signals with companding is 8000

samples/s, 8-bit/sample giving a PCM bit rate of 64 kbits/s. The companding

characteristic is either A-law (used in Europe) or µ-law (used in America).

3.4 PCM ENCODER

Figure 3.11 shows a block diagram of a typical PCM encoder which essentially

comprises the following,

a) Anti-aliasing filter

b) Sample and Hold

c) Analogue to Digital

Converter

d) Control Unit

e) Clock

f) Parallel to Serial Converter

S/N(dB) at quantiser output

0 -44 -40 -36 -32 -28 -24 -20 -16 -12 -8 -4

36

40

44

48

20

24

28

32

4

8

12

16

Si (dB) signal power

at quantiser

input

non-companding

companding

Fig 3.10 Comparison of companded and uncompanded systems


___________________________________________________________________________

ET0424 Page 3 - 36

Fig. 3.11 Typical PCM encoder

v

sample

& hold

ADC P/S

control unit clock

i iii

ii

iv


___________________________________________________________________________

ET0424 Page 3 - 37

Figure. 3.12. PCM Transmitter waveforms

0 1 0 1 0 0 0 0 0 1 1 0 1 0 1

b0 start stop

… …

-15.5

-21.0

-25.1

-10.3

1

0

4

6

8

2

0

levels

(i)

analog input signal volts

16

24

32

8

0 t

t

(ii) sampling clock

16

24

32

8

0 t

levels

analog samples

Sample-and-hold output

(iii)

-16

-21

-25

-10

16

24

32

8

0 t

levels

quantised samples

1 1 … 0 1

0 t

0 0 … 1 1

0 t

0 1 … 0 1

0 t

0 1 … 0 1

0 t

0 0 … 1 1

0 t

1

0 t

b4

PCM

output

5-bit

PCM

encoder

output (iv)


___________________________________________________________________________

ET0424 Page 3 - 38

The operation of the PCM decoder is essentially the reverse of the encoder, as shown in Figure 3.13 & Figure 3.14 shows the

waveforms at the output of various blocks.

(iii) (ii) PCM in

(i)

Clock

Recovery

Control

Unit

Serial-

to-

parallel DAC

(iv)

Fig 3.13 Typical PCM Decoder


___________________________________________________________________________

ET0424 Page 3 - 39

4

6

8

2

0

volts

4

6

8

2

0

volts

1 1 … 0 1

0 t

0 0 … 1 1

0 t

0 1 … 0 1

0 t

0 1 … 0 1

0 t

0 0 … 1 1

0 t

0 1 0 1 0 0 0 0 0 1 1 0 1 0 1

b0 start stop

… … 1

0 t

b4

PCM

input

16

24

32

8

0 t

levels

PCM decoder output (iii)

16

24

32

8

0 t

levels

Analog output signal

(iv)

(ii)

(i)

…

Fig 3.14 PCM Receiver Waveforms


___________________________________________________________________________

ET0424 Page 3 - 40

MULTIPLEXING TECHNIQUES

3.5 Introduction

Multiplexing is a process of simultaneously transmitting two or more individual signals over a

single communications channels. Multiplexing has the effect of increasing the number of

communications channels so that more information can be transmitted.

There are many instances in communications where it is necessary or desirable to transmit more

than one voice or data signal.Telemetry and telephone are good examples.

The concept of a simple multiplexer is illustrated in Figure 3.15. Multiple input signals are

combined by a multiplexer into a single composite signal that is transmitted over the

communications medium. At the other end of the communication link, a demulitplexer is used to

sort out the signals in their original form.

MUX DEMUX

Single Communications channel (wire or radio)

Multiple input

signals

Original input

signals

Multiplexer combines all input into a single signal.

Demultiplexer processors input signal by sorting it out into the original individual signals.

Figure 3.15 Concept of Multiplexing


___________________________________________________________________________

ET0424 Page 3 - 41

There are two basic types of multiplexing:

• Frequency Division Multiplexing (FDM)

• Time Division Multiplexing (TDM).

FDM systems are used to deal with analog information.

TDM is used for digital information.

The primary difference between these techniques is that in FDM, individual signals to be

transmitted are assigned a different frequency within a common bandwidth. In TDM, the

multiple signals are transmitted in different time slots.

3.5.1 Frequency Division Multiplexing (FDM)

Frequency division multiplexing (FDM) is based on the idea that a number of signals can

share the bandwidth of a common communications channel. The multiple signals to be

transmitted over this channel are each used to modulate a separate carrier. Each carrier is

on a different frequency. The modulated carriers are then added together to form a single

complex signal that is transmitted over the single channel.

Figure 3.16 shows a general block diagram of an FDM system.

Each signal to be transmitted feeds a modulator circuit. The carrier for each modulation

fc, is on a different frequency. The carrier frequencies are usually equally spaced from

one another over specific frequency range. Each input signal is given a portion of the

bandwidth. The result is illustrated in Figure 3.17. As for the type of modulation, any

type of modulation, any of the standard kinds can be used including AM, SSB, FM, or

PM.


___________________________________________________________________________

ET0424 Page 3 - 42

Modulator

Modulator

Modulator

Carrier fc1

Modulator

Carrier fc2

Carrier fc3

Carrier fcn

Signal 1

Signal 2

Signal 3

Signal n

Linear mixer of summer

Transmitter

Original data modulates carriers of different frequencies

All carriers are combined into a single composite signal that modulated a transmitter.

Antenna

Single communication channel

Fig 3.16 The Transmitting end of a FDM system

fc1 fc2 fc3 fc4 Frequency

Bandwidth of overall communication channel

Bandwidth of single channel

Fig 3.17 Spectrum of an FDM signal


___________________________________________________________________________

ET0424 Page 3 - 43

The modulator outputs containing the sideband information are added together in a linear

mixer. In the linear mixer, modulation and the generation of sidebands do not take place.

Instead all the signals are simply added together algebraically. The resulting output signal

is a composite of all carriers containing their modulation. This signal is then used to

modulate a radio transmitter or the composite signal may be transmitted over the single

communication channel.

Figure 3.18 shows the block diagram of a demultiplexer at the receiver.

Receiver Demodulator

BPF fc1

BPF fc2

BPF fc3

BPF fcn

Demodulator

Demodulator

Demodulator

Demodulator

1

2

3

n

Signal received is demodulated into composite signal

Bandpass filters select out individual channels

Output Signals

Demodulators recover original signals

Fig 3.18 Receiving end of an FDM system

A receiver picks up the signal and demodulates it into the composite signal. This is sent to a

group of bandpass filters (BPF) each centered on one of the carrier frequencies. Each filter

passes only its channels and rejects all others. A channel demodulator then recovers each original

input signal.


___________________________________________________________________________

ET0424 Page 3 - 44

3.5.2 Time Division Multiplexing (TDM)

In FDM, multiple signals are transmitted over a single channel by sharing the channel

bandwidth. This is done by allocating each signal a portion of the spectrum within that

bandwidth. In TDM, each signal can occupy the entire bandwidth of the channel. However, each

signal is transmitted for only a brief period of time. In the other words, the multiple take turns

transmitting over the single channel. This concept is illustrated graphically in Figure 3.19. Here,

four signals are transmitted over a single channel. Each signal is allowed to use the channel for a

fixed period of time, one after another. Once all the signals have been transmitted, the cycle

repeats again and again.

Signal 1 Signal 2 Signal 3 Signal 4 Signal 1 Signal 2 Signal 3 Signal 4

One frame

Time

Note: Equal time slots for each signal

Fig 3.19 The basic TDM concept

Time division multiplexing may be used with both digital and analog signals. To transmit

multiple digital signals, the data to be transmitted is formatted into serial data words. For

example, in Figure 3.19, each time slot might contain one byte from each channel. One channel

transmits 8 bits and then halts while the next channel transmit 8 bits. The third channel then

transmits its data word and so on. One transmission of each channel completes one cycle of

operation called a frame. The cycle repeats itself at a high rate of speed. In this way, the data

bytes of the individual channels are simply interleaved. The resulting single-channel signal is a

digital bit stream that must somehow be deciphered and reassembled at the receiving end.


___________________________________________________________________________

ET0424 Page 3 - 45

Fig 3.20(a) Time-division multiplexing of three signals

By sampling the three signals at the same rate and interleaving the pulses into separate time slots,

the three signals share the same communication channel simultaneously. Such a system is known

as a Pulse Amplitude Modulation (PAM)-TDM system.

S1(1) S1(2) S1(3)

S3(1) S3(2) S3(3)

S2(1) S2(2) S2(3)

Figure 3.20 (b) Time interlacing of three baseband signals

t

T T

T

3

2 1

3

1

2

Filter

Communication channel

decommutator

commutator

S1(t)

S2(t)

S3(t)

S3(t)

S1(t)

S2(t)


___________________________________________________________________________

ET0424 Page 3 - 46

S1's, S2's and S3's are of equal amplitude only for the sake of illustration. Note that all samples

belonging to the same baseband signal are separated by T.

The commutator is performing both the sampling and multiplexing. The commutator must

operate at rate that satisfies the sampling theorem for each signal. Consequently, the signal with

highest frequency determines the commutator speed.

For example, suppose the maximum frequencies for the three input signals are f1 = 4 kHz, f2 = 12

kHz, and f3 = 4 kHz. Then the commutator speed must be 24 kHz to satisfy the worst-case

condition. We can calculate the communication channel pulse rate as follows: The commutator

completes one cycle, called a frame, every 1/24 kHz = 41.67 µs. Each time around, the

commutator picks up a pulse from each of the three signals. Hence, there are 3 pulses/cycle x 24

k cycles/s = 72 k pulses/s. In general, the gross channel output pulse rate is given by:

Gross channel output pulse rate = number of signal inputs x commutator speed.

This example illustrates that multiplexing of many signals will require relatively high-frequency

transmission systems. A little creativity, however, can help minimise the transmission

bandwidth.

A better scheme is shown in Figure 3.21 with insertion of signals S1(t), S3(t), and a dummy

input(unconnected input) in between 3 inputs of S2(t). Now, with uniform sampling the

commutator speed can be 8 kHz (i.e. (2 pulses/41.67 µs)/(5 + 1 pulses/cycle) = 8 kHz), and the

channel pulse rate will be 8 kHz x (5+1) pulses/cycles = 48 k pulses/s. Note that the sampling

frequency for S2(t) is given by:

fs2 = commutator speed x no.of inputs for S2(t)

= 8 kHz x 3 = 24 kHz

So, in general the sampling frequency for any signal in a commutator structure is given by,

fs = commutator speed x number of inputs for that signal


___________________________________________________________________________

ET0424 Page 3 - 47

Dummy

S1(t)

S2(t)

S3(t)

41.67 µs

Output pulse train

Pulse from S3(t)

Pulse from S1(t) Pulse from S2(t)

Commutator

Dummy

It should be realised that the commutator speed can be reduced further by inserting more inputs

for each of the three signals. However this will result in a more complex commutator structure.

So a trade-off between the commutator speed and the complexity of its structure is necessary.

8 kHz

Fig 3.21 TDM for minimum channel output pulse rate

Example 3

A PAM-TDM system is used to multplex four signals m1(t) = cos ω

0t,

m2(t) = 0.5cos ω

0t, m

3(t) = 2cos 2ω

0t, and m

4(t) = cos 4ω

0t where

ω0 = 2000π radians/s.

a) If each signal is sampled at Nyquist rate, sketch the commutator structure

assuming uniform sampling.

b) What is the commutator speed?

Solution

a)

Signal fm

(kHz) fs(kHz) No. of inputs

m1 1 2 1

m2 1 2 1

m3 2 4 2

m4 4 8 4


___________________________________________________________________________

ET0424 Page 3 - 48

Fig 3.22 Commutator Structure

Note that to get uniform sampling:

• all the signal inputs should be evenly spaced

• for the same signal all inputs must be evenly e.g. all 4 m4's are evenly spaced.

b) The rule for determining the commutator speed is as follows: the lowest value of fs in

column 3 is chosen to be the commutator speed i.e. commutator speed = 2 k cycles/s

The same general concepts discussed above on PAM-TDM can apply as well to the TDM

of PCM signals as just an additional encoder is needed as shown in Figure 3.23.

Figure 3.23 Commutator structure for PCM-TDM signal

The gross channel output bit rate is:

Gross channel output bit rate, R =

cmmutator speed x no. of inputs x no. of bits per sample

PCM-TDM

signal Encoder

PAM-TDM

signal

m2

m1

m4

m3

m4

m3

m4

m2

m1

m4

m3

m4

m3

m4

m4

PAM-TDM signal

m4


___________________________________________________________________________

ET0424 Page 3 - 49

Example 4

A 4 kHz signal is transmitted through a PCM system. The system employs a 8-bit quantiser

which has a step size of 5 mV.

a) Calculate the quantisation noise power and the maximum output signal-to-

quantisation noise ratio (in dBs) of the system.

b) If five similar systems are time-multiplexed, what is the minimum transmission

bandwidth required by the multiplexed system?

Solution

a) Quantisation noise power, 12

2qN q = =

12

)105(23−

× = 2.08 x 10-6 W

Signal-to-quantisation noise = 1.76 + 6B = 1.76 + 6 x 8 dB

= 49.76 dB

b) fm = 4 kHz, fs = 2fm = 8 kHz

Gross output bit rate, R = commutator speed x no. of inputs x no. of bits per

sample

= 8000 x 5 x 8 = 320 kbps

Note: Here commutator speed equals fs as the five signals are similar.

Hence minimum transmission bandwidth = R/2 = 320/2 = 160 kHz.

CHAPTER 3 PCM AND TDM

Documents

Transcript of CHAPTER 3 PCM AND TDM