Điều chế xung biên độ PAM

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

Introduction
•
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
Serial data
symbols
a
k
‘analogue’
channel pulses
Recovered
data symbols

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

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

PAM Scheme
H
C
(
ω
)
h
C
(t)

Symbol
clock
H
T
(
ω
) h
T
(t)
Noise N(
ω
)
Channel
+
Pulse
generator
a
k Transmit
filter
∑
∞
−∞=
−=
k
ks
kTtatx )()(
δ
∑
∞
−∞=

−=
k
Tk
kTthatx )()(
Receive
filter
H
R
(
ω
), h
R
(t)
Data
slicer
Recovered
symbols
Recovered
clock
)()()( tvkTthaty
k
k
+−=
∑
∞
−∞=
Modulator
Demodulator

PAM

•
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
}
•
Signalling period, T
•
Each transmitted pulse is given by
)( kTtha
Tk
−
Where h
T
(t) is the time domain pulse shape

PAM

•
To generate the PAM output signal, we may
choose to represent the input to the transmit
filter h
T
(t) as a train of weighted impulse
functions
∑
∞
−∞=
−=
k
ks
kTtatx )()(
δ
•
Consequently, the filter output x(t) is a train of
pulses, each with the required shape h
T
(t)
∑
∞
−∞=
−=
k
Tk
kTthatx )()(

PAM
•

Filtering of impulse train in transmit filter
Transmit
Filter
∑
∞
−∞=
−=
k
Tk
kTthatx )()(
∑
∞
−∞=
−=
k
ks
kTtatx )()(
δ
)(th
T
)(tx
s
)(tx

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 H
C
(
ω
) and the receive filter H
R
(
ω
).
•
The overall frequency response is
H(
ω
) = H
T
(
ω
) H
C
(
ω
) H
R
(
ω

)

PAM
•
Hence the signal at the receiver filter output is
)()()( tvkTthaty
k
k
+−=
∑
∞
−∞=
Where h(t) is the inverse Fourier transform of H(
ω
)
and v(t) is the noise signal at the receive filter
output
•
Data detection is now performed by the Data
Slicer

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

dkdn
vtTknhatnTyy
++−=+=
∑
∞
−∞=
))(()(
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
0
Signal at data
slicer input, y(t)
Sample clock
Sampled signal,
y

n
= y(nT+t
d
)
Ideal sample instants
at t = nT+t
d
0
TX data
TX symbol, a
k
‘1’ ‘0’ ‘0’ ‘1’ ‘0’
+A -A -A +A -A
Detected data ‘1’ ‘0’ ‘0’ ‘1’ ‘0’
Τ
t
d

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.

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.

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.

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
nk
dkdnn
vtTknhathay
++−+=
∑
≠
))(()(
–
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
•
Example
–
Response 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
amplitude
0.5
1.0
Time (bit periods)
0 2 4 6
amplitude
0.5
1.0
Modulator input Slicer input
Binary ‘1’ Binary ‘1’

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
amplitude
0.5
1.0
‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘0’ ‘0’ ‘1’

Intersymbol Interference
•
For the assumed data the signal at the slicer
input is,

•
Clearly the ease in making decisions is data
dependant
time (bit periods)
0 2 4 6
amplitude
0.5
1.0
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
amplitude

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 70
0
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 70
0
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 response
Eye 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

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