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Communication system
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Block diagram of an Binary/M-ary signaling
scheme
Channel noise
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+
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Block diagram Description
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
For
Tb
For
Tb
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Block diagram Description (Continue - 1)
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
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Block diagram Description (Continue - 2)
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
100110
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Block diagram Description (Continue - 3)
{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}
Tb
Timing
100110
t
t
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Block diagram Description (Continue - 4)
Timing
HT(f)
(X(t
Information
source
Trans
filter
Pulse
generator
HR(f)
+
Receiver
filter
Channel noise n(t)
Tb
2Tb
5Tb
3Tb
4Tb
6Tb
t
t
t
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Block diagram Description (Continue - 5)
Tb
2Tb
5Tb
3Tb
4Tb
6Tb
t
t
t
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Block diagram of an Binary/M-ary signaling
scheme
Timing
+
+
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Block diagram Description
Tb
2Tb
5Tb
3Tb
6Tb
4Tb
t
t
t
t
1
0
0
0
1
0
1
0
0
1
1
0
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Typical waveforms in a binary PAM system
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Block diagram of an Binary/M-ary signaling
scheme
Timing
+
+
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Explanation of Pr(t)
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The element of a baseband binary PAM system
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The input to the A/D converter is
For
and
is the total time delay in the
system, we get.
t
t
Yt m 
t2
t1
t3
tm
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t
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The output of the A/D converter at the sampling time
tm =mTb+td
Tb
Yt m 
2Tb
5Tb
3Tb
t2
t1
4Tb
t3
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6Tb
t
t
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ISI - Inter Symbol
Interference
Yt m 
t2
t1
t3
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tm
t
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Explanation of ISI
t
t
f
f
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Explanation of ISI - Continue
t
t
f
f
Tb
2Tb
5Tb
3Tb
4Tb
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6Tb
t
21
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-The pulse generator output is a pulse waveform
If kth input bit is 1
if kth input bit is 0
-The A/D converter input Y(t)
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5.2 BASEBAND BINARY PAM SYSTEMS
- minimize the combined effects of inter symbol
interference and noise in order to achieve minimum
probability of error for given data rate.
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5.2.1
Baseband pulse shaping
The ISI can be eliminated by proper choice
of received pulse shape pr (t).
Doe’s not Uniquely Specify Pr(t) for all
values of t.
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To meet the constraint, Fourier Transform Pr(f) of Pr(t), should
satisfy a simple condition given by the following theorem
Theorem
Proof
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p r (nTb ) 
1 / 2 Tb
T
b
exp( j2fnT b )df 
1 / 2 Tb
Which verify that the Pr(t) with a transform Pr(f)
Satisfy ______________
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The condition for removal of ISI given in the theorem is called
Nyquist (Pulse Shaping) Criterion
1
-2Tb -Tb
Tb
2Tb
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The Theorem gives a condition for the removal of ISI using a Pr(f) with
a bandwidth larger then rb/2/.
ISI can’t be removed if the bandwidth of Pr(f) is less then rb/2.
Tb
2Tb
5Tb
3Tb
4Tb
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6Tb
t
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Particular choice of Pr(t) for a
given application
pr (t)
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A Pr(f) with a smooth roll - off characteristics is preferable
over one with arbitrarily sharp cut off characteristics.
Pr(f)
Pr(f)
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In practical systems where the bandwidth available for
transmitting data at a rate of rb bits\sec is between rb\2 to rb
Hz, a class of pr(t) with a raised cosine frequency
characteristic is most commonly used.
A raise Cosine Frequency spectrum consist of a flat amplitude portion and a roll off
portion that has a sinusoidal form.
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raised cosine frequency characteristic
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Summary
The BW occupied by the pulse spectrum is B=rb/2+b.
The minimum value of B is rb/2 and the maximum value is rb.
Larger values of b imply that more bandwidth is required for a
given bit rate, however it lead for faster decaying pulses, which
means that synchronization will be less critical and will not
cause
large ISI.
b =rb/2 leads to a pulse shape with two convenient properties.
The half amplitude pulse width is equal to Tb, and there are zero
crossings at t=3/2Tb, 5/2Tb…. In addition to the zero crossing
at Tb, 2Tb, 3Tb,…...
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5.2.2
Optimum transmitting and receiving
filters
The transmitting and receiving filters are chosen to provide
a proper
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-One of design constraints that we have for selecting the filters
is the relationship between the Fourier transform of pr(t) and
pg(t).
Where td, is the time delay Kc normalizing constant.
In order to design optimum filter Ht(f) & Hr(f), we will assume that Pr(f),
Hc(f) and Pg(f) are known.
Portion of a baseband PAM system
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If we choose Pr(t) {Pr(f)} to produce Zero ISI we are left
only to be concerned with noise immunity, that is will choose
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Noise Immunity
Problem definition:
For a given :
•Data Rate •Transmission power •Noise power Spectral Density •Channel transfer function •Raised cosine pulse Choose
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Error probability Calculations
At the m-th sampling time the input to the A/D is:
We decide:
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A=aKc
The noise is assumed to be zero mean Gaussian at the receiver input
then the output should also be Zero mean Gaussian with variance No
given by:
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b
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0
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-A
A
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Q(u)
dz=
U
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A

N0
Perror decreases as
increase
Hence we need to maximize the signal
to noise Ratio
Thus for maximum noise immunity the filter transfer functions HT(f)
and HR(f) must be xhosen to maximize the SNR
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Optimum filters design calculations
We will express the SNR in terms of HT(f) and HR(f)
We will start with the signal:
The PSD of the transmitted signal is given by::
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And the average transmitted power ST is
The average output noise power of n0(t) is given by:
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The SNR we need to maximize is
Or we need to minimize
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Using Schwartz’s inequality
The minimum of the left side equaity is reached when
V(f)=Const*W(f)
If we choose :
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 2 is minimized when
The filter should have a linear phase response in a total time delay of td
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Finally we obtain the maximum value of the SNR to be:
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For AWGN with
and
pg(f) is chosen such that it does not change much over the
bandwidth of interest we get.
Rectangular pulse can be used at the input of HT(f).
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5.2.3 Design procedure and Example
The steps involved in the design procedure.
Example:Design a binary baseband PAM system to
transmit data at a a bit rate of 3600 bits/sec with a bit
error probability less than 10 4.
The channel response is given by:
The noise spectral density is Gn ( f )  1014 watt / Hz
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Solution:
rb  3600bits / sec
pe  10  4
B  2400 Hz
Gn ( f )  10  4 watt / Hz
If we choose a braised cosine pulse spectrum with
b  rb / 6  600
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We choose a pg(t)
We choose K1  (3600)(103 )

Dr. Uri Mahlab
p g ( f ) H T ( f ) H c ( f ) H R ( f )  pr ( f )
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Plots of Pg(f),Hc(f),HT(f),HR(f),and Pr(f).
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4
P

10
To maintain a e

For Pr(f) with raised cosine shape
 P ( f ) df
r
1

And hence ST  (14.06)(3600)(1010 )  23dBm
Dr. Uri Mahlab
Which completes the design.
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5.3
In order to transmit data at a rate of rb bits/sec with zero ISI PAM
data transmission system requires a bandwidth of at least rb /2
HZ.
+
Binary PAM data
transmission at a
rate of rb bits/sec
with zero ISI
1.such filters are physically unrealizable
2. Any system with this filters will be extremely
sensitive to perturbations.Dr. Uri Mahlab
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204
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The duobinary scheme utilizes controlled amounts of ISI
for transmitting data at a rate of rb /2 HZ.
The shaping filters for duobinary are easier to realize
than the ideal rectangular filters.
The duobinary signaling schemes use pulse spectra
Pr(f):
The pulse response Pr(t):
Dr. Uri Mahlab
Pr t  
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5.3.1
The output Y(t) of the receive filter can be written as:
If the output is sampled at tm= mTb/2+ td than it is obvious
that in the absence of noise:
The Am' s can assume one of two values +/-A depending on
whether the m th input bit is 1 or 0. Since Y(tm) depends on
Am & Am-1 assuming no noise :
+2A if the m th and (m-1st)bits are both 1's
Y(tm) =
0 if the m th and (m-1st)bits are different
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-2A if the m th and (m-1st)bits are both zero
5.3.2
Transmitting and receiving filters for optimum performance
The receiving levels at the input to the A/D converter are 2A,
0, and -2A with probabilities 1/2, 1/4.
The probability of bit error pe is given by:
Since no is a zero mean Gaussian random variable with a
variance No we can write pe as:
For the direct binary PAM case :
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Where /2=Gn(f) is the noise power spectral density the
probability of error is:
The integral can be evaluated as :
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5.4
M-ARY SIGNALING SCHEMES
In baseband binary PAM we use pulses with one of 2 possible
amplitude,
In M-ary baseband PAM system we allowed M possible levels
(M>2) and there M distinct input symbols.
During each signaling interval of duration Ts  1 rs
the source is converted to a four-level PAM pulse train by the
the pulse generator.
The signal pulse noise passes through the receiving filter and
is sampled by the A/D converter at an appropriate rate and
phase.
the M-ary PAM scheme operating with the preceding
constraints can transmit data at a bit rate of rs log2M bit/sec
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and require a minimum bandwidth
of rs/2 HZ.
5.7
MISCELLANEOUS TOPICS
5.7.1 Eye Diagram
The performance of baseband PAM systems depends on
the amount of ISI and channel noise.
The received waveform with no noise and no distortion is
shown in Figure 5.20a the “open” eye pattern results
Figure 5.20b shows a distorted version of the waveform the
corresponding eye pattern.
Figure 5.20c shows a noise distorted version of the
received waveform and the corresponding eye
pattern.
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In the figure 5.21 we see typical eye patterns of a
duobinary signal
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If the signal-to-noise ratio at the receiver is high
then the following observations can be made from
the eye pattern shown simplified in Figure 5.22:
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1. The best time to sample the received waveform is
when the eye is opening is largest.
2. The maximum distortion is indicated by the vertical
width of the two branches at sampling time.
3. The noise margin or immunity to noise is proportional
to the width of the eye opening.
4. The sensitivity of the system to timing errors is
revealed by the rate of closing of the eye as sampling
time is varied.
5. The sampling time is midway between zero crossing.
6. Asymmetries in the eye pattern indicate nonlinearities
in the channel.
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5.7.2
synchronization
Three methods in which this synchronization can be obtained
are:
1. Derivation of clock information from a primary or secondary
standard.
2. Transmitting a synchronizing clock signal.
3. Derivation of the clock signal from the received waveform
itself.
An example of a system used to derive a clock signal from
the received waveform is shown in figure 5.23.
To illustrate the operating of the phase comparator network let
us look at the timing diagram shown in figure 5.23b
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5.7.3
Scrambler and unscrambler
Scrambler: The scrambler shown in figure 5.24a
consists a “feedback” shift register.
Unscrambler:The matching unscrambler have a ”feed
forward” shift register structure.
In both the scrambler and unscrambler the outputs of
several stages of of shift register are added together
modulo-2 and the added to the data stream again in
modulo-2 arithmetic
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Scrambler affects the error performance of the
communication system in that a signal channel error
may cause multiple error at the output of the
unscrambler.
The error propagation effect lasts over only a finite
and small number of bits.
In each isolated error bit causes three errors in the
final output it must also be pointed out that some
random bit patterns might be scrambled to the
errors or all ones.
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