Điều chế xung biên độ PAM

• PAM is a general signalling technique whereby pulse amplitude is used to convey the message • For example, the PAM pulses could be the sampled amplitude values of an analogue signal •

3F4 Pulse Amplitude Modulation

(PAM)

Dr I J Wassell

• The purpose of the modulator is to convert

discrete amplitude serial symbols (bits in a

binary system) a k to analogue output pulses which are sent over the channel

• The demodulator reverses this process

Modulator Channel Demodulator

• Possible approaches include

– Pulse width modulation (PWM)

– Pulse position modulation (PPM)

– Pulse amplitude modulation (PAM)

• We will only be considering PAM in these lectures

• PAM is a general signalling technique

whereby pulse amplitude is used to convey the message

• For example, the PAM pulses could be the sampled amplitude values of an analogue

signal

• We are interested in digital PAM, where the pulse amplitudes are constrained to chosen from a specific alphabet at the transmitter

x( ) ( )

Receive filter

H R (), h R (t)

Data slicer

) (t a h t kT v t

• In binary PAM, each symbol a k takes only

two values, say {A 1 and A 2}

• In a multilevel, i.e., M-ary system, symbols

may take M values {A 1 , A 2 , A M}

t

• Filtering of impulse train in transmit filter

Transmit Filter

• Clearly not a practical technique so

– Use a practical input pulse shape, then filter to

realise the desired output pulse shape

– Store a sampled pulse shape in a ROM and read out through a D/A converter

• The transmitted signal x(t) passes through the channel H C () and the receive filter H R ().

• The overall frequency response is

H() = H T () H C () H R ()

• Hence the signal at the receiver filter output is

) ( )

( )

PAM- Data Detection

• Sampling y(t), usually at the optimum instant

t=nT+t d when the pulse magnitude is the

greatest yields

n k

d k

((

) (

Where v n =v(nT+t d ) is the sampled noise and t d is the

time delay required for optimum sampling

• y n is then compared with threshold(s) to

determine the recovered data symbols

PAM- Data Detection

Data Slicer decision threshold = 0V

• We need to derive an accurate clock signal at

the receiver in order that y(t) may be sampled at

the correct instant

• Such a signal may be available directly (usually not because of the waste involved in sending a signal with no information content)

• Usually, the sample clock has to be derived

directly from the received signal

• The ability to extract a symbol timing clock

usually depends upon the presence of transitions

or zero crossings in the received signal

• Line coding aims to raise the number of such

occurrences to help the extraction process

• Unfortunately, simple line coding schemes often

do not give rise to transitions when long runs of constant symbols are received

• Some line coding schemes give rise to a

spectral component at the symbol rate

• A BPF or PLL can be used to extract this component directly

• Sometimes the received data has to be linearly processed eg, squaring, to yield a component of the correct frequency

non-Intersymbol Interference

• If the system impulse response h(t) extends over

more than 1 symbol period, symbols become

smeared into adjacent symbol periods

• Known as intersymbol interference (ISI)

• The signal at the slicer input may be rewritten as

n n

k

d k

d n

n a h t a h n k T t v



) )

((

) (

– The first term depends only on the current symbol a n

– The summation is an interference term which

depends upon the surrounding symbols

Intersymbol Interference

• The received pulse at the slicer now extends over 4 bit periods giving rise to ISI

• The actual received signal is the

superposition of the individual pulses

time (bit periods)

Note non-zero values at ideal sample instants

corresponding with the transmission of binary ‘0’s

‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘0’ ‘0’ ‘1’

Decision threshold

Intersymbol Interference

• Matlab generated plot showing pulse superposition (accurately)

0 1 2 3 4 5 6 7 8 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Decision threshold

time (bit periods) Received

signal

Individual pulses

Intersymbol Interference

• Sending a longer data sequence yields the

following received waveform at the slicer input

Decision threshold

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Decision threshold (Also showing individual pulses)

Eye Diagrams

• Worst case error performance in noise can be

obtained by calculating the worst case ISI over all possible combinations of input symbols

• A convenient way of measuring ISI is the eye

diagram

• Practically, this is done by displaying y(t) on a

scope, which is triggered using the symbol clock

• The overlaid pulses from all the different symbol periods will lead to a criss-crossed display, with

an eye in the middle

Example R-C responseEye Diagram

Decision threshold

Optimum sample instant

Eye Diagrams

• The size of the eye opening, h (eye height)

determines the probability of making incorrect decisions

• The instant at which the max eye opening occurs

gives the sampling time t d

• The width of the eye indicates the resilience to symbol timing errors

• For M-ary transmission, there will be M-1 eyes

Eye Diagrams

• The generation of a representative eye

assumes the use of random data symbols

• For simple channel pulse shapes with binary symbols, the eye diagram may be

constructed manually by finding the worst case ‘1’ and worst case ‘0’ and

superimposing the two

Nyquist Pulse Shaping

• It is possible to eliminate ISI at the sampling instants by ensuring that the received pulses satisfy the Nyquist pulse shaping criterion

• We will assume that t d=0, so the slicer input is

n n

k

k n

n a h a h n k T v



) ) ((

) 0 (

• If the received pulse is such that

0

0 for

1 )

(

n

n nT

h

Nyquist Pulse Shaping

• Then

n n

and so ISI is avoided

• This condition is only achieved if

T T

k f

H k

Nyquist Pulse Shaping

f

0

0 for

1 )

(

n

n nT

h

1 ) 2

(

1 )

k f

Nyquist Pulse Shaping



T

• No pulse bandwidth less than 1/2T can

satisfy the criterion, eg,

Clearly, the repeated spectra do not sum to a constant value

Nyquist Pulse Shaping

• The minimum bandwidth pulse spectrum

H(f), ie, a rectangular spectral shape, has a

sinc pulse response in the time domain,

0

2 1 2T

1 - for

)

H

• The sinc pulse shape is very sensitive to

errors in the sample timing, owing to its low rate of sidelobe decay

Nyquist Pulse Shaping

• Hard to design practical ‘brick-wall’ filters, consequently filters with smooth spectral roll-off are preferred

• Pulses may take values for t<0 (ie

non-causal) No problem in a practical system because delays can be introduced to enable approximate realisation

Raised Cosine (RC) Fall-Off

Pulse Shaping

• Practically important pulse shapes which

satisfy the criterion are those with Raised

Cosine (RC) roll-off

• The pulse spectrum is given by

2

1 2

1 2

1

0

) 2

1

( 4 cos

2

1

)

T f

T

T f

T f

H

With, 0<<1/2T

1 2

1 2

1

0

) 2

1

( 4 cos

2

1

)

T f

T

T f

T f

H

2 cos

sin )

(

t

t t

T

t T t





RC Pulse Shaping

• With =0 (i.e., x = 0) the spectrum of the filter is

rectangular and the time domain response is a sinc pulse, that is,

T f

T f

t T t

h





sin )

(

• The time domain pulse has zero crossings at

intervals of nT as desired (See plots for x = 0).

RC Pulse Shaping

• With =(1/2T), (i.e., x = 1) the spectrum of the

filter is full RC and the time domain response is a pulse with low sidelobe levels, that is,

T f

Tf T

f

2

cos )

T

t

4 1

1 )

(

2 2

• The time domain pulse has zero crossings at

intervals of nT/2, with the exception at T/2

where there is no zero crossing See plots for x

= 1.

RC Pulse Shaping- Example 1

• Eye diagram

-0.5 0 0.5 1 1.5 2

0 1 2 3 4 5 6 7 8 -0.6

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

• Pulse shape and received signal, x = 0 ( = 0)

RC Pulse Shaping- Example 2

• Eye diagram

• Pulse shape and received signal, x = 1 ( = 1/2T)

0 0.2 0.4 0.6 0.8 1 1.2

0 1 2 3 4 5 6 7 8 -0.2

0 0.2 0.4 0.6 0.8 1 1.2

RC Pulse Shaping- Example

• The much wider eye opening for x = 1 gives

a much greater tolerance to inaccurate

sample clock timing

• The penalty is the much wider transmitted bandwidth

Probability of Error

• In the presence of noise, there will be a finite

chance of decision errors at the slicer output

• The smaller the eye, the higher the chance that the noise will cause an error For a binary system a

transmitted ‘1’ could be detected as a ‘0’ and versa

vice-• In a PAM system, the probability of error is,

P e=Pr{A received symbol is incorrectly detected}

• For a binary system, P e is known as the bit error probability, or the bit error rate (BER)

• The received signal at the slicer is

n i

2

2

2 2

2

1 )

Where f(v n ) denotes the probability density

function (pdf), that is,

dx x

f dx

x v

Pr{

For equiprobable symbols, the optimum threshold

is in the centre of V 0 and V 1 , ie V T =(V 0 +V 1 )/2

V 1

V 0

) 0

| (error P v n V T V o

1

) 1

| (

) 0

| (error P P error P P

• For ‘0’ sent: an error occurs when yn V T

– let v n =y n -V o , so when y n =V o , v n =0 and when y n =V T , v n =V T

– So equivalently, we get an error when v n V T -V 0

• The Q function is one of a number of

tabulated functions for the Gaussian

cumulative distribution function (cdf) ie the integral of the Gaussian pdf

n

V

V Q dv

v f error

P



) ( )

0

| (

(



• Similarly for ‘1’ sent: an error occurs when yn <V T

– let v n =y n -V 1 , so when y n =V 1 , v n =0 and when y n =V T,

v n =V T -V 1.

– So equivalently, we get an error when v n < V T -V 1

) (

) (

) 1

| (error P v n V T V1 P v n V1 V T

n

V

V Q dv

v f error

P

1

1

) ( )

1

| (



• Hence the total error probability is

P e=Pr(‘0’sent and error occurs)+Pr(‘1’sent and error occurs)

1

) 1

| (

) 0

| (error P P error P P

1

1 V P

V Q P

V

V Q

P

v

T o

T





• Consequently,

o v

– Q(.) is a monotonically decreasing function of its

argument, hence the BER falls as h increases

– For received pulses satisfying Nyquist criterion, ie

zero ISI, Vo=Ao and V1=A1 Assuming unity overall gain.

– More complex with ISI Worst case performance if h

is taken to be the eye opening

BER Example

• The received pulse h(t) in response to a

single transmitted binary ‘1’ is as shown,

BER Example

• What is the worst case BER if a ‘1’ is received as

h(t) and a ‘0’ as -h(t) (this is known as a polar

binary scheme)? Assume the data are equally likely

to be ‘0’ and ‘1’ and that the optimum threshold

(OV) is used at the slicer

• By inspection, the pulse has only 2 non-zero

amplitude values (at T and 4T) away from the ideal sample point (at 2T).

BER Example

• Consequently the worst case ‘1’ occurs

when the data bits conspire to give negative non-zero pulse amplitudes at the sampling instant

• The worst case ‘1’ eye opening is thus,

1 - 0.3 - 0.2 = 0.5

as indicated in the following diagram.

BER Example

• The indicated data gives rise to the worst case ‘1’ eye

opening Don’t care about data marked ‘X’ as their pulses are zero at the indicated sample instant

BER Example

• Similarly the worst case ‘0’ eye opening is

-1 + 0.3 + 0.2 = -0.5

• So, worst case eye opening h = 0.5-(-0.5) = 1V

• Giving the BER as,

at the slicer input

• For PAM systems we have

– Looked at ISI and its assessment using eye diagrams – Nyquist pulse shaping to eliminate ISI at the

optimum sampling instants

– Seen how to calculate the worst case BER in the

presence of Gaussian noise and ISI