1. 3F4 Pulse Amplitude Modulation (PAM)
Dr. I. J. Wassell

2. Introduction
The purpose of the modulator is to convert discrete amplitude serial symbols (bits in a binary system) ak to analogue output pulses which are sent over the channel.
The demodulator reverses this process ak
Modulator
Channel
Demodulator
Serial data symbols
‘analogue’ channel pulses
Recovered data symbols

3. Introduction Possible approaches include
Pulse width modulation (PWM)
Pulse position modulation (PPM)
Pulse amplitude modulation (PAM)
We will only be considering PAM in these lectures

4. 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
We are interested in digital PAM, where the pulse amplitudes are constrained to chosen from a specific alphabet at the transmitter

5. PAM Scheme Symbol clock HT(w) hT(t) Noise N(w) Channel +
Pulse generator ak
Transmit filter
Receive filter
HR(w), hR(t)
Data slicer
Recovered symbols
Recovered clock
Modulator
Demodulator
HC(w)
hC(t)

6. PAM
In binary PAM, each symbol ak takes only two values, say {A1 and A2}
In a multilevel, i.e., M-ary system, symbols may take M values {A1, A2 ,... AM}
Signalling period, T
Each transmitted pulse is given by
Where hT(t) is the time domain pulse shape

7. PAM
To generate the PAM output signal, we may choose to represent the input to the transmit filter hT(t) as a train of weighted impulse functions
Consequently, the filter output x(t) is a train of pulses, each with the required shape hT(t)

8. PAM
Transmit
Filter
Filtering of impulse train in transmit filter

9. PAM 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 HC(w) and the receive filter HR(w).
The overall frequency response is
H(w) = HT(w) HC(w) HR(w)

10. PAM Hence the signal at the receiver filter output is
Where h(t) is the inverse Fourier transform of H(w) and v(t) is the noise signal at the receive filter output
Data detection is now performed by the Data Slicer

11. PAM- Data Detection
Sampling y(t), usually at the optimum instant t=nT+td when the pulse magnitude is the greatest yields
Where vn=v(nT+td) is the sampled noise and td is the time delay required for optimum sampling
yn is then compared with threshold(s) to determine the recovered data symbols

12. PAM- Data Detection Data Slicer decision threshold = 0V
Signal at data slicer input, y(t)
Sample clock
Sampled signal, yn= y(nT+td)
Ideal sample instants at t = nT+td
TX data
TX symbol, ak
‘1’
‘0’ +A -A
Detected data T td

13. Synchronisation
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.

14. Synchronisation
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.

15. Synchronisation
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 non-linearly processed eg, squaring, to yield a component of the correct frequency.

16. 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
The first term depends only on the current symbol an
The summation is an interference term which depends upon the surrounding symbols

17. Intersymbol Interference
Example
Response h(t) is Resistor-Capacitor (R-C) first order arrangement- Bit duration is T
Modulator input
Slicer input
Binary ‘1’
Binary ‘1’
1.0
1.0
amplitude
amplitude
0.5
0.5 2 4 6 2 4 6
Time (bit periods)
Time (bit periods)
For this example we will assume that a binary ‘0’ is sent as 0V.

18. Intersymbol Interference
The received pulse at the slicer now extends over 4 bit periods giving rise to ISI.
time (bit periods) 2 4 6
amplitude
0.5
1.0
‘1’
‘0’
The actual received signal is the superposition of the individual pulses

19. Intersymbol Interference
For the assumed data the signal at the slicer input is,
‘1’
‘1’
‘0’
‘0’
‘1’
‘0’
‘0’
‘1’
1.0
amplitude
0.5
Decision threshold 2 4 6
time (bit periods)
Note non-zero values at ideal sample instants corresponding with the transmission of binary ‘0’s
Clearly the ease in making decisions is data dependant

20. Intersymbol Interference
Matlab generated plot showing pulse superposition (accurately)
Decision threshold
amplitude
time (bit periods)
Received signal
Individual pulses

21. Intersymbol Interference
Sending a longer data sequence yields the following received waveform at the slicer input
Decision threshold
(Also showing individual pulses)
Decision threshold

22. 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

23. Example R-C response Eye Diagram h = eye height h Decision threshold
Optimum sample instant

24. 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 td
The width of the eye indicates the resilience to symbol timing errors
For M-ary transmission, there will be M-1 eyes

25. 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

26. 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 td=0, so the slicer input is
If the received pulse is such that

27. Nyquist Pulse Shaping Then and so ISI is avoided
This condition is only achieved if
That is the pulse spectrum, repeated at intervals of the symbol rate sums to a constant value T for all frequencies

28. Nyquist Pulse Shaping
H(f) T f
1/T
-1/T T
-2/T
2/T f

29. Why? Sample h(t) with a train of d pulses at times kT
Consequently the spectrum of hs(t) is
Remember for zero ISI

30. Why? Consequently hs(t)=d(t) The spectrum of d(t)=1, therefore
Substituting f=w/2p gives the Nyquist pulse shaping criterion

31. Nyquist Pulse Shaping
No pulse bandwidth less than 1/2T can satisfy the criterion, eg,
1/T
-1/T T
-2/T
2/T f
Clearly, the repeated spectra do not sum to a constant value

32. Nyquist Pulse Shaping
The minimum bandwidth pulse spectrum H(f), ie, a rectangular spectral shape, has a sinc pulse response in the time domain,
The sinc pulse shape is very sensitive to errors in the sample timing, owing to its low rate of sidelobe decay

33. 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.

34. Causal Response Non-causal response T = 1 s Causal response T = 1s
Delay, td = 10s

35. 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
With, 0<b<1/2T

36. RC Pulse Shaping
The general RC function is as follows,
H(f) T
f (Hz)

37. RC Pulse Shaping
The corresponding time domain pulse shape is given by,
Now b allows a trade-off between bandwidth and the pulse decay rate
Sometimes b is normalised as follows,

38. RC Pulse Shaping
With b=0 (i.e., x = 0) the spectrum of the filter is rectangular and the time domain response is a sinc pulse, that is,
The time domain pulse has zero crossings at intervals of nT as desired (See plots for x = 0).

39. RC Pulse Shaping
With b=(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,
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.

40. RC Pulse Shaping Normalised Spectrum H(f)/T Pulse Shape h(t) x = 0
t/T

41. RC Pulse Shaping- Example 1
Pulse shape and received signal, x = 0 (b = 0)
Eye diagram

42. RC Pulse Shaping- Example 2
Pulse shape and received signal, x = 1 (b = 1/2T)
Eye diagram

43. 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

44. 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 vice-versa
In a PAM system, the probability of error is,
Pe=Pr{A received symbol is incorrectly detected}
For a binary system, Pe is known as the bit error probability, or the bit error rate (BER)

45. BER The received signal at the slicer is
Where Vi is the received signal voltage and
Vi=Vo for a transmitted ‘0’ or
Vi=V1 for a transmitted ‘1’
With zero ISI and an overall unity gain, Vi=an, the current transmitted binary symbol
Suppose the noise is Gaussian, with zero mean and variance

46. BER
Where f(vn) denotes the probability density function (pdf), that is,
and

47. BER dx vn
f(vn) b a

48. BER
The slicer detects the received signal using a threshold voltage VT
For a binary system the decision is
Decide ‘1’ if yn VT
Decide ‘0’ if yn<VT
For equiprobable symbols, the optimum threshold is in the centre of V0 and V1, ie VT=(V0+V1)/2

49. BER yn f(yn|‘0’ sent) VT f(yn|‘1’ sent) P(error|‘0’) P(error|‘1’) V1
f(yn|‘1’ sent)
P(error|‘0’)
P(error|‘1’) V1 V0

50. BER The probability of error for a binary system can be written as:
Pe=Pr(‘0’sent and error occurs)+Pr(‘1’sent and error occurs)
For ‘0’ sent: an error occurs when yn VT
let vn=yn-Vo, so when yn=Vo, vn=0 and when yn=VT, vn=VT-Vo.
So equivalently, we get an error when vn VT-V0

51. BER yn
f(yn|‘0’ sent) VT
P(error|‘0’) V0 vn
f(vn)
VT - V0
P(error|‘0’)

52. BER
Where,
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.

53. BER Similarly for ‘1’ sent: an error occurs when yn<VT
let vn=yn-V1, so when yn=V1, vn=0 and when yn=VT, vn=VT-V1.
So equivalently, we get an error when vn < VT-V1

54. BER yn VT f(yn|‘1’ sent) P(error|‘1’) V1 VT -V1 vn f(vn) P(error|‘1’)
f(yn|‘1’ sent)
P(error|‘1’) V1
VT -V1 vn
f(vn)
P(error|‘1’)
-(VT -V1)
V1-VT

55. BER Hence the total error probability is
Pe=Pr(‘0’sent and error occurs)+Pr(‘1’sent and error occurs)
Where Po is the probability that a ‘0’ was sent and P1 is the probability that a ‘1’ was sent
For Po=P1=0.5, the min error rate is obtained when,

56. BER Consequently, Notes:
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

57. BER Example
The received pulse h(t) in response to a single transmitted binary ‘1’ is as shown, t T 2T 3T 4T 5T
-0.2
0.2
0.4
0.6
0.8
1.0
h(t)
(V)
Bit period = T
Where,
h(0) = 0, h(T) = 0.3, h(2T) = 1, h(3T) = 0, h(4T) = -0.2, h(5T) =0

58. 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).

59. 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,
= 0.5
as indicated in the following diagram.

60. BER Example t T 2T 3T 4T 5T
-0.2
0.2
0.4
0.6
0.8
1.0
‘1’
‘0’
‘X’ 6T 7T 8T
-0.4
-0.6
-0.8
-1.0
Optimum sample point for circled bit, amplitude = = 0.5
-0.3
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

61. BER Example Similarly the worst case ‘0’ eye opening is
= -0.5
So, worst case eye opening h = 0.5-(-0.5) = 1V
Giving the BER as,
Where sv is the rms noise at the slicer input

62. Summary 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