# Pulse Amplitude Modulation (PAM)

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```					     Pulse Amplitude Modulation (PAM)
In pulse-amplitude modulation (PAM) the amplitude of a
train of constant-width pulses is varied in proportion to the
sample values of the modulating (message) signal. The
pulses are usually spaced at equal time interval.

Pulse Modulation                    1
Sample-and-Hold Filter
Let s(t) be the sequence of flat-top pulses (PAM signal) generated as
shown in the previous slide.
∞
s (t ) =   ∑ m ( nT ) h ( t − nT )
n =−∞
s             s

where Ts is the sampling period and m(nTs) is the sample value of
m(t) obtained at t = nTs. The h(t) is a standard rectangular pulse of
unit amplitude and duration T, defined as the following:

⎛ T     ⎞ ⎧    1,          0<t <T
⎜t− 2   ⎟ ⎪⎪   1
h ( t ) = rect ⎜       ⎟ =⎨     ,    t = 0, and t = T
⎜ T     ⎟ ⎪    2
⎝       ⎠ ⎪⎩   0,          otherwise

Pulse Modulation                     2
Sample-and-Hold Filter
The instantaneously sampled version of m(t) is given as in Eq. (5.1) :
∞
mδ ( t ) =      ∑ m ( nT ) δ ( t − nT )
n =−∞
s                  s

where δ ( t – nTs ) is a time-shifted delta function.

The PAM signal s( t ) can be obtained by convolving mδ ( t ) with the
pulse h( t ).
∞
s ( t ) = mδ ( t ) ⊗ h ( t ) = ∫ mδ (τ ) h ( t − τ ) dτ
−∞

∞      ∞
=∫
−∞
∑ m ( nT ) δ (τ − nT ) h ( t − τ ) dτ
n =−∞
s                  s

∞                 ∞
=   ∑ m ( nT ) ∫
n =−∞
s   −∞
δ (τ − nTs ) h ( t − τ ) dτ
∞
=   ∑ m ( nT ) h ( t − nT )
n =−∞
s                 s
Pulse Modulation             3
Sample-and-Hold Filter
The convolution of two functions in time domain is equivalent to the
multiplication of those two functions in frequency domain:

s ( t ) = mδ ( t ) ⊗ h ( t ) ⇔ S ( f ) = M δ ( f ) H ( f )

where               Mδ ( f ) is the Fourier transform of mδ ( t )
S ( f ) is the Fourier transform of s ( t )
H ( f ) is the Fourier transform of h ( t )
Since
∞                                                     ∞
mδ ( t ) =   ∑ m ( nT ) δ (τ − nT )
n =−∞
s            s     ⇔ Mδ ( f ) = fs      ∑ M ( f −k f )
n =−∞
s

∞
s ( t ) = mδ ( t ) ⊗ h ( t )             ⇔ S ( f ) = fs   ∑ M ( f −k f ) H ( f )
n =−∞
s

where     fs = 1/Ts is the sampling rate.
Pulse Modulation                                 4
Pulse Modulation   5
ℑ{ h(t) }

⎛ T        ⎞ ⎧      1,            0<t <T
⎜t− 2      ⎟ ⎪⎪     1
Since   h ( t ) = rect ⎜          ⎟ =⎨       ,        t = 0, and t = T
⎜ T        ⎟ ⎪      2
⎝          ⎠ ⎪⎩     0,           otherwise

T
∞                           T                   ⎡ e − j 2π f t ⎤
Then    H ( f ) = ∫ h (t ) e − j 2π f t
dt = ∫ 1 ⋅ e − j 2π f t
dt = ⎢              ⎥
−∞                         0
⎣ − j 2π f ⎦ t =0
e − j 2π f T   1     e − j 2π f T      1        1     e − j 2π f T
=              −    =−              +        =        −
− j 2π f − j 2π f     j 2π f        j 2π f   j 2π f    j 2π f
e − jπ f T jπ f T
=           ⎡e      − e − jπ f T ⎤
j 2π f ⎣                       ⎦

Pulse Modulation                           6
The Fourier transform of { h(t) }
e jπ f t − e − jπ f t
Since   sin (π f T ) =
2j

e − jπ f T jπ f T                  ⎡e jπ f T − e − jπ f T ⎤ e − jπ f T
then    H(f )=           ⎡e      − e − jπ f T ⎤ = ⎣                      ⎦×
j 2π f ⎣                       ⎦             2j              πf
sin (π f T ) − jπ f T      sin (π f T ) − jπ f T
=               e         =T ⋅             ⋅e        = T ⋅sinc ( f T ) ⋅ e − jπ f T
πf                        π fT
H(f)                                             ⎧ sin (π f T ) − jπ f T ⎫
= Lim {sinc ( f T ) ⋅ e − jπ f T } = Lim ⎨              ⋅e        ⎬
T T →0 T →0                                T →0
⎩    π fT               ⎭
d
sin ( 0 ) 0             dT
{sin (π f T )} π f cos (π f T )
Since               =   ⇒                            =
d                   πf
0       0
{π f T }
dT
⎧ sin (π f T ) − jπ f T ⎫ π f cos (π f T ) − jπ f T
Lim ⎨              ⋅e        ⎬=                ⋅e                        =1
T →0
⎩    π fT               ⎭      πf                            T →0

Pulse Modulation                                7
Quantization Process

Pulse Modulation   8
Quantization Process
• Two type of Quantizers : uniform or non-uniform
• Two type of uniform quantizers : mid-tread or mid-rise
• For mid-tread type, the origin lies in the middle of a tread
of the staircase-like graph, as shown in Fig. 5.10a.
• For mid-rise type, the origin lies in the middle of a rising
part of the staircase-like graph, as shown in Fig. 5/10b.
• Both the mid-tread and mid-rise types of uniform
quantizers are symmetric about the origin.

Pulse Modulation                       9
Pulse-Code Modulation (PCM)
• In PCM, a message signal is represented by a sequence of
coded pulses, which is accomplished by representing the
signal in discrete form in both time and amplitude.
• The basic operations performed in the transmitter of a
PCM system are sampling, quantization, and encoding, as
shown in Fig. 5.11a; the low-pass filter prior to sampling
is included only to prevent aliasing of the message signal.
• The quantizing and encoding operations are performed in
an analog-to-digital converter.
• The basic operations in the receiver are regeneration of
impaired signals, decoding and reconstruction of the train
of quantized samples, as shown in Fig. 5.11c.

Pulse Modulation                   10
The basic element of a PCM System

Pulse Modulation        11
Operations in the Transmitter
i. Sampling:
The incoming message signal is sampled with a train of
rectangular pulses, narrow enough to closely approximate
the instantaneous sampling process. The sampling rate
must be greater than twice the highest frequency
component W of the message signal. An anti-alias (low-
pass) filter is used at the front end of the sampler in order
to exclude frequencies higher than W prior to sampling.

Pulse Modulation                     12
Operations in the Transmitter
ii. Non-uniform Quantization:
The use of a non-uniform quantizer is equivalent to
passing the message signal through a compressor and then
applying the compressed signal to a uniform quantizer. A
particular form of compression law is called μ-law defined
by
log (1 + μ m )
v =
log (1 + μ )

where the logarithm is the natural logarithm, m and v are
respectively the normalized input and output voltages, and
μ is a positive constant.

Pulse Modulation             13
Compression Laws: μ-law & A-law

Pulse Modulation       14
Compression law: μ-law

•   In Fig. 5.12a, the input to the quantizer and its output are
both normalized to occupy a range of values from zero to
one.
•   Practical values of μ tend to be in the vicinity of 255.
•   The uniform quantization corresponds to μ = 0.

•   Since         log (1 + μ m )
v =
log (1 + μ )
d ⎡log (1 + μ m ) ⎤     d ⎡log (1 + μ m ) ⎤        μ
log (1 + μ ) ⎣                ⎦       ⎣               ⎦
dv                          dm                      dm              1+ μ m
=                                       =                     =
dm               ⎡log (1 + μ ) ⎤
2
log (1 + μ )      log (1 + μ )
⎣             ⎦

Pulse Modulation                                  15
Compression law: μ-law
μ
dm   1+ μ m        log (1 + μ ) log (1 + μ )
•   Then       =
d v log (1 + μ )
=
μ
=
μ
(1 + μ m )
1+ μ m

•   It is approximately linear at low input levels if μ m << 1.

•   It is also approximately logarithmic at high input levels if
μ m >> 1.

Pulse Modulation                          16
Compression law: A-law
(European Standard)
•   The A-law is as shown in 5.11b and is defined as:
⎧      Am                       1
⎪             ,       0≤ m ≤
⎪ 1 + log A                     A
v =⎨
⎪1 + log ( A m )      1
⎪ 1 + log A ,           ≤ m ≤1
⎩                     A

⎧        A                           1
⎪ 1 + log A ,            0≤ m ≤
dv ⎪                                    A
=⎨
dm ⎪        1                1
,          ≤ m ≤1
⎪ (1 + log A ) m
⎩                         A

Pulse Modulation         17
Compression law: A-law
•   The derivative of m with respect to v :
⎧ 1 + log A                           1
,         0≤ m ≤
dm ⎪
⎪       A                             A
=⎨
dv ⎪                       1
⎪
(1 + log A) m ,          ≤ m ≤1
⎩                       A

•   Practical values of A tend to be in the vicinity of 100.
•   The uniform quantization corresponds to A = 1.

Pulse Modulation                     18
Operations in the Transmitter
iii. Encoding:
An encoding process is to translate the discrete set of
sample values to a more appropriate form of signal. If R
denotes the number of bits per sample, we can represent
a total of 2R distinct numbers. For example, a sample
quantized into one of 256 levels may be represented by
an 8-bit code word.

Pulse Modulation                  19
Regeneration Along the Transmission Path
The most important feature of a PCM system lies in the
ability to control the effects of distortion and noise
produced by transmitting a PCM signal over a channel.
This capability is accomplished by reconstructing the PCM
signal by means of a chain of regenerative repeaters located
sufficiently close spacing along the transmission route.

Pulse Modulation                   20
Regeneration Along the Transmission Path
Three basic functions are performed by a regenerative
repeater: equalization, timing, and decision making.
The equalizer shapes the received pulses so as to
compensate for the effects of amplitude and phase
distortion produced by the transmission characteristic of
the channel.
The timing circuitry provides a periodic pulse train, derived
from the received pulses; this is done for renewed sampling
of the equalized pulses at the instants of time when the
signal-to-noise ratio (SNR) is a maximum.

Pulse Modulation                    21
Regeneration Along the Transmission Path
The extracted sample is compared to a predetermined
threshold in the decision-making device. In each bit
symbol is a 1 or 0 on the basis of whether the threshold is
exceeded or not. If the received signal is above the
threshold, a clean new pulse representing symbol 1 is
transmitted to the next repeater, otherwise, another clean
new pulse representing symbol 0 is transmitted. Thus, the
accumulation of distortion and noise in a repeater span is
removed, provided the disturbance is not too large to cause
an error in the decision-making process.

Pulse Modulation                  22
i. Decoding and Expanding
The first operation in the receiver is to regenerate the
received pulses. These clean pulses are then regrouped
into code words and decoded into a quantized PAM
signal. The sequence of decoded samples represent an
estimate of the sequence of compressed samples
produced by the quantizer in the transmitter. In order to
restore the sequence of decoded samples to their correct
relative level, we use an expander with a characteristic
complementary to the compressor, used in the
transmitter. Ideally, the compression and expansion are
exactly inverse, except for the effect of quantization.
The combination of a compressor and an expander is
referred as a compander.
Pulse Modulation                   23

ii.   Reconstruction
This operation is achieved by passing the expander
output through a low-pass reconstruction filter whose
cutoff frequency is equal to the bandwidth of the
message signal. The recovery of the message signal is
only the estimation not exact reconstruction.

Note: The term “modulation” in pulse-code modulation is a misleading
terminology. Actually PCM is a source-coding strategy which
an analog signal emitted by a source is convicted into digital
form.

Pulse Modulation                        24
Quantization Noise

In PCM, the transmitted information is contained in the code. The
transmission noise may cause errors in recognition of the code symbols
sent. But if the signal pulses are received above a certain signal-to-noise
threshold, the average error rate can be kept very low.

To encode a continuous signal, it must be quantized into a finite number
of discrete amplitude levels. Once quantized, the instantaneous values
of the continuous signal can never be reconstructed exactly. The
introduced errors are called quantization noise. Quantization noise is
generated in the quantization process prior to being transmitted. It can
be reduced to any desired degree by choosing the number and
distribution of the quantizing levels.

Pulse Modulation                            25
Quantization Noise

To calculate the mean-square quantization noise, we assume equal
amplitude increments between levels. Specifically, let the input signal
be quantized into n levels, each spaced by an amplitude increment, a.
For binary PCM, n is chosen to be a power of 2. If we assume that the
signal amplitude is bipolar and there is no dc level to offset the
amplitude, a reasonable distribution of the quantizing levels is:

a   3a  5a
± , ± , ± , … ,±
( n − 1) a
2    2   2          2

Pulse Modulation                          26
Quantization Noise

7a
2
5a
2      a
Aj +
3a      2
Aj
2      a
Aj −
ε                           a       2
2
a
−
2
3a
−
2
5a
−
2
7a
−
2

Pulse Modulation               27
Quantization Noise
Assuming that the number of signal levels is even and that all signal
levels are equiprobable, the probability density function of the levels
is:
n
⎛1⎞ ⎛   ia ⎞
p ( x) = ∑ ⎜ ⎟ δ ⎜ x − ⎟                    where n=2k
i =− n ⎝ n ⎠ ⎝  2⎠
odd

The mean-square signal after quantization is
∞                    ∞
S = ∫ x 2 p ( x ) dx = 2 ∫ x 2 p ( x ) dx
−∞                   0

2⎪⎧⎛ a ⎞ 2 ⎛ 3a ⎞ 2 ⎡ ( n − 1) a ⎤ ⎫
2
⎪
= ⎨⎜ ⎟ + ⎜ ⎟ + … + ⎢              ⎥ ⎬
n ⎪⎝ 2 ⎠ ⎝ 2 ⎠      ⎣     2 ⎦ ⎪
⎩                                 ⎭
2 n ( n − 1)( n + 1) ⎛ a ⎞ ( n − 1) a
2   2       2

= ×                  ×⎜ ⎟ =
n          6         ⎝2⎠        12

Pulse Modulation                           28
Quantization Noise
The same method can be applied if not all levels are
equiprobable, but the results cannot be written in
compact form.
In the quantization process, each sample of the
continuous input signal is approximated to the nearest
allowed level. This is shown in Fig. 1, where the input
signal level is ε units from the nearest quantizing level
Aj. At the eventual receiver output, the quantized level Aj
could have been due to any signal amplitude in the range
Aj− (a/2) to Aj+(a/2) . This uncertainty could just as well
have been due to additive noise and therefore we treat it

Pulse Modulation                       29
Quantization Noise
The quantizing error ε is the difference between the
signal level and the nearest allowed quantizing level as
shown in Fig. 1. Lacking any information to the
contrary, we assume that all values of are equally likely
anywhere in the range −a/2 ≤ ε ≤ a/2, so that it can be
described by the uniform probability density function,

⎧1 a      −a 2≤ε<a 2
p (ε ) = ⎨
⎩ 0        elsewhere

Actually, the result is not highly dependent on this
assumption as long as the probability density function is
not strongly peaked somewhere in the interval.

Pulse Modulation                    30
Quantization Noise
The mean-square quantization noise now follows
a 2
∞                   1 2
a 2     1 ε3                  a2
ε = ∫ ε p (ε ) d ε = ∫
2      2
ε dε = ×                  =
−∞               −a 2 a       a 3                   12
−a 2

Because the peak signal is (na/2) , the peak signal-to-
noise (power) ratio is
( na 2 ) = 3 n 2
2
⎛S⎞
⎜ ⎟          = 2
⎝ N ⎠ pk qnt    a 12
Expressed in decibels, this becomes
⎡⎛ S ⎞        ⎤
⎢⎜ ⎟          ⎥ = 10log10 3 + 20log10 n = 4.8 + 20 log10 n
⎣⎝ N ⎠ pk qnt ⎦ dB

Pulse Modulation                      31
Quantization Noise
For a binary code, we have n=2k so that, for the binary
case
⎡⎛ S ⎞         ⎤
⎥ = 4.8 + 20log10 2 = 4.8 + 6 k
k
⎢⎜ ⎟
⎣ ⎝ N ⎠ pk qnt ⎦ dB

Thus the peak-signal-to-quantization noise ratio
increases by 6 dB for every additional bit used in a
binary system.

For the case in which all signal levels are equiprobable,
the mean-square signal is given by Eq. (1); dividing this
by Eq. (3)
⎛S⎞      ( n2 − 1) a 2 × 12 = n2 − 1 ≈ n2
⎜ ⎟ =
⎝ N ⎠ pk      12         a2
Pulse Modulation            32
Quantization Noise
For the case in which all signal levels are equiprobable,
the mean-square signal is given by Eq. (1); dividing this
by Eq. (3) The performance quality, on a mean-square
basis, increases as the square of the number of levels
used and differs only by a constant from the peak S/N
relation given in Eq. (4). In both cases, the signal-to-
quantization-noise ratio increases about 6 dB for each
•   For speech, n=8 to 16 levels are adequate for intelligibility (3~4
bits).
•   For telephony, n=128 to 256 levels is the standard (7 ~8 bits).
•   For music-quality audio, n=4,096 to 65,536 levels are used (12~16
bits).

Pulse Modulation                           33
Differential PCM (DPCM)
When a voice or video signal is sampled at a rate slightly
higher than the Nyquist rate, the resulting sampled signal is
found to exhibit a high degree of correlation between
adjacent samples. That means, the signal does not change
rapidly from one sample to the next. When these highly
correlated samples are encoded as in a standard PCM
system, the resulting encoded signal contains redundant
information. Redundancy means that symbols that are not
absolutely essential to the transmission of information are
generated as a result of the encoding process. By removing
this redundancy before encoding, we obtain a more
efficient encoded signal, compared to PCM.

Pulse Modulation                    34
Differential PCM (DPCM)
Suppose a message signal m(t) is sampled at the
rate fs=1/Ts to produce a sequence of correlated
samples Ts seconds apart; this sequence is denoted
by { m( nTs ) }.
m ( nTs ) is the output of a prediction filter whose
input is the quantized version of { m( nTs ) }.
e ( nTs ) = m ( nTs ) − m ( nTs )
where
e ( nTs ) is called the prediction error.

Pulse Modulation         35
Differential PCM (DPCM)
A simple approach to implement the prediction filter is to
use a tapped-delay-line filter or digital filter, with the basic
delay set equal to the sampling period.

Pulse Modulation                      36
Differential PCM (DPCM)
The quantizer output may be expressed as:
eq ( nTs ) = e ( nTs ) + q ( nTs )                        (5.35)
where q ( nTs ) is the quantization error.
The prediction filter input is expressed as:
mq ( nTs ) = m ( nTs ) + eq ( nTs )                       (5.36)

Substituting Eq. (5.35) into (5.36)
mq ( nTs ) = m ( nTs ) + e ( nTs ) + q ( nTs )             (5.37)
= m ( nTs ) + q ( nTs )                       (5.38)

The quantized signal mq(nTs ) at the prediction filter input differs
from the sampled message signal m ( nTs ) by the quantization error
q ( nTs ).

Pulse Modulation                   37
Prediction Filter
The prediction m ( nTs ) is modeled as a linear combination
of p past sample values of the discrete input m(nTs ), where
p is the prediction order (filter order = p−1) .

Pulse Modulation                   38
Delta Modulation (DM)
• Unlike PCM, the difference between the input signal and
its approximation is quantized into ±Δ, corresponding to
positive and negative differences.
• Thus, if the approximation falls below the input signal at
any sampling epoch, it is increased by Δ.
• If the approximation lies above the signal, it is diminished
by Δ.
• Provided the input signal does not change too rapidly from
sample to sample, so that the staircase approximation
remains within ±Δ of the input signal.

Pulse Modulation                   39
Delta Modulation (DM)
We denoted the input signal by m(t) and its staircase
approximation by mq(t). The principle of Delta Modulation
can be represented in the following three discrete-time
relations:
e ( nTs ) = m ( nTs ) − mq ( nTs − Ts )                 (5.27)
eq ( nTs ) = Δ ⋅ sgn ⎡e ( nTs ) ⎤
⎣          ⎦                         (5.28)
mq ( nTs ) = mq ( nTs − Ts ) + eq ( nTs )                 (5.29)

where Ts is the sampling period
e(nTs) is an error signal between m(nTs) & mq(nTs – Ts)
eq(nTs) is the quantized version of e(nTs)
sgn[.] is the signum function, assuming the value of +1 or -1.

Pulse Modulation                      40
Delta Modulation (DM)

Pulse Modulation   41
Delta Modulation (DM)
Eq. (5.29) is a first order difference equation.
Assuming the accumulation process starts at t = 0, the
solution to this equation yields the approximate result:

mq ( nTs ) = mq ( n − 1) Ts + eq ( nTs )
= mq ( nTs − Ts ) + eq ( nTs )
= mq ( nTs − 2Ts ) + eq ( nTs − Ts ) + eq ( nTs )
= mq ( nTs − 3Ts ) + eq ( nTs − 2Ts ) + eq ( nTs − Ts ) + eq ( nTs )
= mq ( n − 3) Ts + eq ( n − 2 ) Ts + eq ( n − 1) Ts + eq ( nTs )
=
n
= ∑ eq ( kTs )
k =1

Pulse Modulation                                 42
Delta Modulation (DM)
• At the sampling instant nTs , the accumulator increments
the approximation by Δ in a positive or negative direction,
depending on the algebraic sign of the error signal e(nTs ).
• If the input signal m(nTs ) is greater than the most recent
approximation mq(nTs ), a positive increment +Δ is applied
to the approximation.
• If the input signal m(nTs ) is smaller, −Δ is applied to the
approximation.
• The accumulator tracks the input samples one step (+Δ or
−Δ) at a time.

Pulse Modulation                   43
Delta Modulation (DM)

Pulse Modulation   44
Delta Modulation (DM)
•   Delta Modulation is subject to two types of
quantization error:
b) Granular Noise
•   Let q ( nTs ) denote the quantization error,
mq ( nTs ) = m ( nTs ) + q ( nTs )                        (5.31)

Substitute Eq. (5.31) into (5.27)
mq ( nTs ) = m ( nTs ) − m ( nTs − Ts ) − q ( nTs − Ts )   (5.32)
= m ( nTs ) − m ( n − 1) Ts − q ( n − 1) Ts

Pulse Modulation                     45
Delta Modulation (DM)
Thus, except for the delayed quantization error q(nTs-Ts ),
the quantizer input is a first backward difference of the
input signal, which may be viewed as a digital
approximation to the derivative of the input signal or,
equivalent, as the inverse of the digital integration process.
An estimate of the rate-of-rise limit, or slope-overload
condition for DM is:
Δ                 dm ( t )
= Δ⋅ f s ≥ max                               (5.33)
Ts                 dt

Pulse Modulation                  46
Delta Modulation (DM)

Pulse Modulation   47
Delta Modulation (DM)
If Δ is too small for the staircase approximation mq(t) to
follow a steep segment of the original message signal m(t),
with the result that mq(t) falls behind m(t), as shown in Fig.
5.16. This condition is called slope overload. The resulting
quantization error is called slope-overload distortion noise.
Granular noise occurs when Δ is too large relative to the
local slope characteristic of the original message signal
m(t). This causes the staircase approximation mq(t) to hunt
around a relative flat segment of m(t).
A delta modulation has to be adaptive. The step size has to
vary with respect to the incoming message signal.

Pulse Modulation                    48
Sigma-Delta Modulation (Σ-DM)
A drawback of delta modulation is in the accumulative
noise error in the demodulated signal. This drawback can
be overcome by integrating the message signal prior to
delta modulation.
The benefits of using integration:
•The low frequency content of the input signal is pre-emphasized.
• Correlation between adjacent samples is increased, that tends to
improve overall system performance by reducing the average power
of the error signal at the quantizer input.
•Design of the receiver is simplified.

Pulse Modulation                           49
Sigma-Delta Modulation (Σ-DM)

Pulse Modulation      50
Sigma-Delta Modulation (Σ-DM)
In Fig. 5.17a, the message signal m(t) is defined in
continuous-time format, which means that the pulse
modulator now consists of a hard-limiter followed by a
multiplier; the latter component is also fed from an external
clock to produce a 1-bit encoded signal.
The use of integration at the transmitter input clearly
requires differentiation at the receiver. The differentiation
is not necessary because of Delta modulation receiver.
Thus the receiver only consists a lowpass filter, as shown
in Fig. 5.17a. Since integration is a linear operator. The two
integrators in Fig. 5.17a now combine into a single
integrator placed after the comparator, as shown in Fig.
5.17b.
Pulse Modulation                    51
Line Codes

On-off signaling

Nonreturn-to-zero (NRZ)

Return-to-zero (RZ)

Bipolar return-to-zero (BRZ)

Split-phase (Manchester Code)

Differential encoding

Pulse Modulation                                   52
Line Codes
On-off signaling: symbol 1 is represented by transmitting a pulse of
constant amplitude for the duration of the symbol, and symbol 0 is
represented by switching off the pulse
Nonreturn-to-zero (NRZ): symbols 1 and 0 are represented by pulses of
equal positive and negative amplitudes.
Return-to-zero (RZ): symbol 1 is represented by a positive rectangular
pulse of half-symbol width, and symbol 0 is represented by
transmitting no pulse.
Bipolar return-to-zero (BRZ): Positive and negative pulses of equal
amplitude are used alternatively for symbol 1, and no pulse is used for
symbol 0. A useful property of BRZ signaling is that the power
spectrum of the transmitted signal has no DC component, and relatively
insignificant low-frequency components for the case when symbols 1
and 0 occur with equal probability.

Pulse Modulation                         53
Line Codes
Split-phase (Manchester Code): symbol 1 is represented by a positive
pulse followed by a negative pulse, with both pulses being of equal
amplitude and half-symbol width. For symbol 0, the polarities of these
two pulses are reversed. The Manchester code suppresses the dc
component and has relatively insignificant low-frequency components,
regardless of the signal statistics.
Differential encoding: a transition is used to designate symbol 0,
whereas no transition is used to designate symbol 1. It is apparent that a
differentially encoded signal may be inverted symbol 1. It is apparent
that a differentially encoded signal may be inverted without affecting
its interpretation. The original binary information is recovered by
comparing the polarity of adjacent symbols to establish whether or not
a transition has occurred. The differential encoding requires the use of
a reference bit.

Pulse Modulation                           54
Line Codes
Differential encoding:
The encoded differential data are generated by    en = d n ⊕ en −1

where ⊕ is a modulo 2 adder or exclusive OR gate (XOR) operation.
The received encoded data are decided by     d n = en ⊕ en −1

where the tilde denotes receiving-end data

Pulse Modulation                           55
Example of Differential Coding

Encoding:
Input sequence       dn             1     1    0    1   0   0   1
Encoded sequence     en      1      0     1    1    0   0   0   1
Reference digit

Decoding (with correct channel polarity):
Received sequence    en      1      0     1    1    0   0   0   1
Decoded sequence     dn             1     1    0    1   0   0   1

Decoding (with inverted channel polarity):
Received sequence    en      0      1     0    0    1   1   1   0
Decoded sequence     dn             1     1    0    1   0   0   1

Pulse Modulation                   56
Differential Coding

Each digit in the encoded sequence is obtained by comparing the present
input bit with the past encoded bit. A binary 1 is encoded if the present
input bit and the past encoded bit are of opposite state, and a binary 0 is
encoded if the states are the same.
This is a great advantage when the waveform is passed through thousands
of circuits in a communication system and the positive sense of the output is
lost or changes occasionally as the network changes, such as switching
between several data paths.

Pulse Modulation                           57
Time Division Multiplexing (TDM)

Pulse Modulation       58
Time Division Multiplexing (TDM)

Example:
Twelve different message signals, each with a bandwidth
of 10 KHz, are to be multiplexed and transmitted.
Determine the minimum bandwidth required if TDM-PAM
is used

Minimum bandwidth = 10 KHz × 12 = 120 KHz
or
Nyquist Rate = 2 × 10 KHz = 20 KHz
Minimum bandwidth = (20 KHz × 12) / 2 = 120 KHz

Pulse Modulation                59
Time Division Multiplexing (TDM)
Example:
Channel 1 of a two-channel PAM system handles 0~8 KHz signals;
the second channel handles 0~10 KHz signals. The two channels are
sampled at equal intervals of time using very narrow pulses at the
lowest frequency that is theoretically adequate. The sampled signals
are time-multiplexed and passed through a low-pass filter before
transmission. At the receiver the pulses in each of the two channels are
passed through appropriate holding circuits (i.e., sample-and-hold) and
low-pass filters.

What is the minimum clock frequency of the PAM system?
What is the minimum cutoff frequency of the low-pass filter used
before transmission that will preserve the amplitude information on
the output pulses?

Pulse Modulation                          60
Time Division Multiplexing (TDM)
Solution:
In order to sample channel 2 adequately, we must take samples at a rate
of 20 KHz. Therefore the commutator clock rate is 40 KHz and the
commutator must recycle at a rate of 20 KHz.
For the composite (interlaced) signal, the minimum cutoff frequency of the LPF:

Ch. 1
Ch. 2
Ch. 2    Ch. 1
Ch. 1                                    Ch. 2
Ch. 1

1                        1
20KHz                    40KHz

1 Tx = 40 KHz ⇒ Tx = 25 μ sec

Pulse Modulation     61

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