3F4 Pulse Amplitude Modulation (PAM)

1
3F4 Pulse Amplitude Modulation (PAM)

• Dr. I. J. Wassell

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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
Serial data symbols
analogue channel pulses
Recovered data symbols
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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

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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

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PAM Scheme
HC(w) hC(t)
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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
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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)

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PAM

• Filtering of impulse train in transmit filter

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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)

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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

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PAM- Data Detection

• Sampling y(t), usually at the optimum instant
  tnTtd when the pulse magnitude is the greatest
  yields

Where vnv(nTtd) 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

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PAM- Data Detection
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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.

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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.

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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.

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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

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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
0
2
4
6
0
2
4
6
Time (bit periods)
Time (bit periods)

• For this example we will assume that a binary 0
  is sent as 0V.

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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

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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
0
2
4
6
time (bit periods)
Note non-zero values at ideal sample instants
corresponding with the transmission of binary 0s

• Clearly the ease in making decisions is data
  dependant

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Intersymbol Interference

• Matlab generated plot showing pulse superposition
  (accurately)

Decision threshold
amplitude
time (bit periods)
Received signal
Individual pulses
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Intersymbol Interference

• Sending a longer data sequence yields the
  following received waveform at the slicer input

Decision threshold
(Also showing individual pulses)
Decision threshold
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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

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

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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

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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 td0, so the slicer input is

• If the received pulse is such that

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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

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Nyquist Pulse Shaping
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Why?

• Sample h(t) with a train of d pulses at times kT

• Consequently the spectrum of hs(t) is

• Remember for zero ISI

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Why?

• Consequently hs(t)d(t)
• The spectrum of d(t)1, therefore

• Substituting fw/2p gives the Nyquist pulse
  shaping criterion

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Nyquist Pulse Shaping

• No pulse bandwidth less than 1/2T can satisfy the
  criterion, eg,

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

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Nyquist Pulse Shaping

• Hard to design practical brick-wall filters,
  consequently filters with smooth spectral
  roll-off are preferred
• Pulses may take values for tlt0 (ie non-causal).
  No problem in a practical system because delays
  can be introduced to enable approximate
  realisation.

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Causal Response
Non-causal response T 1 s
Causal response T 1s Delay, td 10s
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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, 0ltblt1/2T
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RC Pulse Shaping

• The general RC function is as follows,

H(f)
T
0
f (Hz)
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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,

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RC Pulse Shaping

• With b0 (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).

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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.

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RC Pulse Shaping
Normalised Spectrum H(f)/T
Pulse Shape h(t)
x 0
x 0.5
x 1
f T
t/T
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RC Pulse Shaping- Example 1

• Pulse shape and received signal, x 0 (b 0)

• Eye diagram

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RC Pulse Shaping- Example 2

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

• Eye diagram

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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

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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,
• PePrA received symbol is incorrectly detected
• For a binary system, Pe is known as the bit error
  probability, or the bit error rate (BER)

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BER

• The received signal at the slicer is

Where Vi is the received signal voltage and ViVo
for a transmitted 0 or ViV1 for a transmitted
1

• With zero ISI and an overall unity gain, Vian,
  the current transmitted binary symbol

• Suppose the noise is Gaussian, with zero mean and
  variance

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

• The slicer detects the received signal using a
  threshold voltage VT
• For a binary system the decision is

For equiprobable symbols, the optimum threshold
is in the centre of V0 and V1, ie VT(V0V1)/2
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BER
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BER

• The probability of error for a binary system can
  be written as
• PePr(0sent and error occurs)Pr(1sent and
  error occurs)

• For 0 sent an error occurs when yn VT
• let vnyn-Vo, so when ynVo, vn0 and when ynVT,
  vnVT-Vo.
• So equivalently, we get an error when vn VT-V0

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BER
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.

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BER

• Similarly for 1 sent an error occurs when
  ynltVT
• let vnyn-V1, so when ynV1, vn0 and when ynVT,
  vnVT-V1.
• So equivalently, we get an error when vn lt VT-V1

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BER
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BER

• Hence the total error probability is
• PePr(0sent and error occurs)Pr(1sent and
  error occurs)

Where Po is the probability that a 0 was sent
and P1 is the probability that a 1 was sent

• For PoP10.5, the min error rate is obtained
  when,

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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, VoAo and V1A1. Assuming unity
  overall gain.
• More complex with ISI. Worst case performance if
  h is taken to be the eye opening

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BER Example

• The received pulse h(t) in response to a single
  transmitted binary 1 is as shown,

Bit period T
Where,
h(0) 0, h(T) 0.3, h(2T) 1, h(3T) 0,
h(4T) -0.2, h(5T) 0
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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).

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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.

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BER Example

• The indicated data gives rise to the worst case
  1 eye opening. Dont care about data marked X
  as their pulses are zero at the indicated sample
  instant

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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,

Where sv is the rms noise at the slicer input
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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