Pulse Modulation
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Introduction
• In Continuous Modulation C.M. a parameter in
the sinusoidal signal is proportional to m(t)
• In Pulse Modulation P.M. a parameter in the
pulse train is proportional to m(t)
• In analog P.M. the parameter (amplitude,
position, duration) is varied in a continuous
manner
• In digital P.M. the values are discrete values
• P.M. is a transition between analog modulation
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and
digital modulation
Kinds of Pulse Modulation
• PAM Pulse Amplitude Modulation
• PDM Pulse Duration Modulation
(or) PWM
Pulse Width Modulation
• PPM Pulse Position Modulation
• PCM Pulse Code Modulation
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Natural Sampling
• The sampled signal consists of a
sequence of pulses of varying amplitude
whose tops are not flat but follow the
shape of the waveform of the signal m(t).
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Flat-Top Sampling
• The sampled signal consists of a
sequence of pulses of flat tops amplitude.
• It will make distortion for recovered signal,
but the distortion will not be noticeable
when the number of samples are large,
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Pulse Amplitude Modulation
(PAM)
• The amplitudes of regularly spaced pulses
are varied in proportion to the
corresponding sampling values of a
continuous message
• Similar to natural sampling: the message
signal is multiplied by a train of rectangular
pulses. The top of each modulated
rectangle is maintained not flat
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• Two operations evaluated in the
generation of PAM:
- Instantaneous sampling every Ts
- Lengthening the duration of the
sample to some constant value T.
• To reconstruct the signal, we need an
equalizer (since the sample and hold filter
used at the transmitter alter the shape of
the signal) plus the reconstruction filter
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PAM
signal
Reconstruction
filter
Equalizer
• The reconstruction filter is an ideal L.P.F
having a cut off frequency equals to the
signal bandwidth.
• The equalizer frequency response:
Heq(f) = 1/ | H0(f)| = 1 / [ T sinc (fT )
= π f / [ sin (π f T) ]
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Quantization Process
• Quantization is to approximate each sample
to the nearest level.
• Amplitude quantization: The process of
transforming the sample amplitude m(nTs) of
a message signal m(t) at time t = nTs into a
discrete amplitude v(nTs) taken from a
finite set of possible amplitudes
Continuous
sample m
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Quantizer
g(.)
Discrete
sample v
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Define:
• L Total number of amplitude levels used in
the quantizer
• mk discrete amplitudes k = 1,2 …L
(decision levels, thresholds )
• vk representation (or reconstruction) levels
• Ik quantizer interval
vk-2
vk-1
vk
vk+1
mk-1
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mk
Ik
mk+1
mk+2
• Step size (quantum): spacing between two
representation levels
• The quantizer O/P v = vk if the input signal
m belongs to the interval Ik
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Uniform (linear) Quantization
The representation levels are uniformly spaced
 Mid-tread:
Origin in the middle of staircase
 Mid-rise:
Origin in the middle of rising part of staircase
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Mid-tread
Mid-rise
Quantization Noise for Uniform
Quantization
• The use of quantization introduces an error
between the I/P signal m(t) and the signal at
quantizer output mq(t).
• This error is known as:
quantization error or (e), where:
e = m(t) – mq(t)
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Quantization Noise for Uniform
Quantization
• If m(t) is in the range ( - mmax, mmax )
Then the Step size of the quantizer :
S = 2 mmax / L
• Assume that the quantization error is
uniformly distributed within each
quantization range, then its p.d.f f(m) will
be:f1(m)+f2(m)+ f3(m)+….+ fL(m) = 1 / S
f(m)
f1(m)
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f1(m)
f2(m)
f3(m)
s
f4m)
fL(m)
m
• σQ2 =Quantization noise
= mean square value of the error

v1  s / 2
v2  s / 2
v1  s / 2
v2  s / 2
2
2
f
(
m
)(
m

v
)
dm

f
(
m
)(
m

v
)
dm ......
 1
 2
1
2
1 s/2 2
1 s3 s2

x dx 


s s / 2
s 12 12
= mmax2 / [ 3 L2 ]
[S=2mmax/L]
• If ‘n’ is the number of bits per sample
L = 2n
S = 2mmax / 2n
2 = m
2 2-2n / 3
σ
Q
max
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• Let P = average power of the message
signal m(t) P  mmax m 2 (t ) 1 dm  m 2 max

 mmax
2m max
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• Assuming m(t) is uniformly distributed from
–mmax to mmax
• Output signal to noise ratio =
SNRo/p = P / σQ2 = 3 P 22n / mmax2
= 3 K (2)2n
• Where:
•
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K = P / mmax2 = 1/3
[SNRo/p]= 10 log [(2)2n] = 6n
(dB)
Non-Uniform quantization
(Companding)
• Why?
• Low amplitudes happens more frequently
than larger ones
• Since the mean square error is proportional
to the step size, we need to decrease the
step size for lower values than larger values.
• This is accomplished by using a compressor
at the transmitter and an expander at the
receiver
• The combination of a compressor and an
expander is called compander
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Non-Uniform quantization
(Companding)
• Two types of compressors may be used:
• A law and μ law refer to the parameter which
appear in the equation of compression and
expansion.
y
y
•
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Ax
1  log A
1  log A x
1  log A
1
x
A
for
for
A-Law
1
 x 1
A
y
log(1   x )
log(1   )
-Law
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Non-Uniform quantization
(Companding)
• Dynamic Range: is the difference in dB
between max. signal level, and min. signal
level having accepted output SNR .
• To improve dynamic range Companding is
used to keep SNR high for low signal level
as well as for high signal level
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Non-Uniform quantization
(Companding)
Output SNR
(dB)
Companded
48
10 dB
30
-48
Un-Companded
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-18
0
Si (dB)
normalized input signal
power in dB
Non-Uniform quantization
(Companding)
• Example:
• For acceptable voice transmission the received
signal have a ratio SNR > 30 dB.
• The companded system has dynamic range of
input signal=48 dB.
• The un-companded system has dynamic range
of input signal=18 dB for the same condition of
SNR>30 dB
• The penalty paid is at max. amplitude of input
signal, SNR is less 10 dB after companding.
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