104612_pulse amplitude modulation (synchronisation_intersymbol interference_eye diagrams)
TRANSCRIPT
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Introduction
The purpose of the modulator is to convertdiscrete amplitude serial symbols (bits in a
binary system) akto analogue output pulses
which are sent over the channel. The demodulator reverses this process
Modulator Channel Demodulator
Serial data
symbols
ak
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 techniquewhereby pulse amplitude is used to convey
the message
For example, the PAM pulses could be thesampled amplitude values of an analogue
signal
We are interested in digital PAM, where thepulse amplitudes are constrained to chosen
from a specific alphabet at the transmitter
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PAM Scheme
HC()
hC(t)
Symbol
clock
HT() hT(t)
Noise N()
Channel
+
Pulse
generator
ak Transmit
filter
=
=k
ks kTtatx )()(
=
=k
Tk kTthatx )()(
Receive
filter
HR(), hR(t)
Data
slicer
Recoveredsymbols
Recovered
clock
)()()( tvkTthatyk
k +=
=
Modulator
Demodulator
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PAM
In binary PAM, each symbol aktakes only
two values, say {A1andA
2}
In a multilevel, i.e., M-ary system, symbols
may take Mvalues {A1,A
2,... A
M}
Signalling period, T
Each transmitted pulse is given by)( kTtha Tk
Where hT(t) is the time domain pulse shape
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PAM
Filtering of impulse train in transmit filter
Transmit
Filter
==
k
Tk kTthatx )()(
==
k
ks kTtatx )()(
)(thT
)(txs )(tx
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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 outthrough a D/A converter
The transmitted signalx(t) passes through the
channelHC() and the receive filterHR(). The overall frequency response is
H() = HT() H
C() H
R()
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PAM Hence the signal at the receiver filter output is
)()()( tvkTthaty
k
k +=
=Where h(t) is the inverse Fourier transform ofH()
and v(t) is the noise signal at the receive filter
output
Data detection is now performed by the DataSlicer
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PAM- Data Detection Samplingy(t), usually at the optimum instant
t=nT+tdwhen the pulse magnitude is the greatest
yields
n
k
dkdn vtTknhatnTyy ++=+=
=
))(()(
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
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PAM- Data Detection
Data Slicer decision
threshold = 0V
0
Signal at data
slicer input,y(t)
Sample clock
Sampled signal,
yn=y(nT+td)
Ideal sample instants
at t= nT+td
0
TX data
TX symbol, ak
1 0 0 1 0
+A -A -A +A -A
Detected data 1 0 0 1 0
td
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Synchronisation We need to derive an accurate clock signal at
the receiver in order thaty(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 acomponent of the correct frequency.
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Intersymbol Interference
If the system impulse response h(t) extends overmore 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
nk
dkdnn vtTknhathay +++=
))(()(
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
ExampleResponse h(t) is Resistor-Capacitor (R-C) first
order arrangement- Bit duration is T
For this example we will assume that a
binary 0 is sent as 0V.
Time (bit periods)0 2 4 6
amp
litud
e
0.5
1.0
Time (bit periods)0 2 4 6
amp
litu
de
0.5
1.0
Modulator input Slicer input
Binary 1 Binary 1
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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
time (bit periods)
0 2 4 6
amp
litu
de
0.5
1.0
1 1 0 0 1 0 0 1
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Intersymbol Interference
For the assumed data the signal at the slicerinput is,
Clearly the ease in making decisions is data
dependant
time (bit periods)0 2 4 6
amp
litu
de
0.5
1.0
Note non-zero values at ideal sample instants
corresponding with the transmission of binary 0s
1 1 0 0 1 0 0 1
Decision threshold
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Intersymbol Interference
Matlab generated plot showing pulse superposition(accurately)
0 1 2 3 4 5 6 7 80
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
amp
litu
de
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Intersymbol Interference Sending a longer data sequence yields the
following received waveform at the slicer input
Decision
threshold
0 10 20 30 40 50 60 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 700
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)
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Eye Diagrams Worst case error performance in noise can be
obtained by calculating the worst case ISI over allpossible combinations of input symbols.
A convenient way of measuring ISI is the eyediagram
Practically, this is done by displayingy(t) on ascope, which is triggered using the symbol clock
The overlaid pulses from all the different symbolperiods will lead to a criss-crossed display, withan eye in the middle
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Example R-C responseEye Diagram
Decision
threshold
Optimum sample instant
h = eye height
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
h
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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 withbinary symbols, the eye diagram may be
constructed manually by finding the worst
case 1 and worst case 0 andsuperimposing 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 td=0, so the slicer input is
n
nk
knn vTknhahay ++=
))(()0(
If the received pulse is such that
=
=0for0
0for1)(
n
nnTh
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Nyquist Pulse Shaping Then
nnn vay +=
and so ISI is avoided
This condition is only achieved if
TT
kfH
k
=
+
=
That is the pulse spectrum, repeated at
intervals of the symbol rate sums to a
constant value Tfor all frequencies
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Why? Sample h(t) with a train of pulses at times kT
=
=k
s kTtthth )()()(
Consequently the spectrum ofhs(t) is
=k
s TkHT
H )2(1
)(
Remember for zero ISI
=
=0for0
0for1)(
n
nnTh
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Why?
Consequently hs(t)= (t)
The spectrum of (t)=1, therefore
1)2(1
)( == k
s TkHT
H
Substitutingf=/2 gives the Nyquist
pulse shaping criterion =k
TTkfH )(
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Nyquist Pulse Shaping
1/
1/ 0
T
2/ 2/ f
No pulse bandwidth less than 1/2Tcan
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,
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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 fort
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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
2121
210
)21(4
cos
21
)( 2
+
+
= TfT
Tf
TfT
TfT
fH
With, 0
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RC Pulse Shaping
The general RC function is as follows,H(f)
f(Hz)
T
0
T2
1+
T2
1
T2
1
T
1
2121
210
)21(4
cos
21
)(2
+
+
= TfT
Tf
TfT
TfT
fH
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RC Pulse Shaping The corresponding time domain pulse shape
is given by,
( )
=2
41
2cossin
)(t
t
tT
tT
th
Now allows a trade-off between bandwidth and the pulse decayrate
Sometimes is normalised as follows,
( )T
x
21
=
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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,
TfTfH 21)( =
=
tT
tT
th
sin
)(
The time domain pulse has zero crossings at
intervals ofnTas desired (See plots forx = 0).
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RC P l Sh i
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RC Pulse ShapingNormalised Spectrum H(f)/T Pulse Shape h(t)
x =
0
x = 0.
5
x = 1
f*T t/T
l h i l
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RC Pulse Shaping- Example 1
Eye diagram
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-1
-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)
l h i l
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RC Pulse Shaping- Example 2
Eye diagram
Pulse shape and received signal,x = 1 ( = 1/2T)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.2
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
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RC Pulse Shaping- Example
The much wider eye opening forx = 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,
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)
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BER The received signal at the slicer is
nin vVy +=
Where Viis 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=a
n,
the current transmitted binary symbol
Suppose the noise is Gaussian, with zero mean
and variance 2v
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BER
2
2
2
22
1)( v
nv
v
n evf
=
Wheref(vn) denotes the probability densityfunction (pdf), that is,
dxxfdxxvx n )(}Pr{ =+