4. kha dc02 baseband transmission
TRANSCRIPT
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Baseband TransmissionHa Hoang Kha, Ph.D
Ho Chi Minh City University of Technology
Email: [email protected]
Chapter 2
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Baseband Transmission 2 H. H. Kha, Ph.D.
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Content
1) Discrete PAM signals
2) Power Spectra of Discrete PAM Signals
3) InterSymbol Interference
4) Nyquist’s Criterion For Distortionless Baseband Binary
Transmission
5) Correlative Coding
Baseband Transmission 3 H. H. Kha, Ph.D.
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1. Discrete PAM Signals
The use of an appropriate for baseband
representation of digital is basic to itstransmission from a source to a destination
There are some different formats for therepresentation of the binary data sequence
Unipolar format (on-off signaling) Polar format
Bipolar format (also known as pseudoternarysignaling)
Manchester format (also known as biphase baseband
signaling)
Baseband Transmission 4 H. H. Kha, Ph.D.
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Discrete PAM Signal
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2. Power Spectra of Discrete PAM Signals
Data signaling rate (or data rate) is defined as
the rate, measured in bits per second (bps), at
which data are transmitted.
It is also common practice to refer to the data
signaling rate as the bit rate, denoted by
where T b
is the bit duration
b
b
T R
1
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Power Spectra of Discrete PAM Signals
In contrast, the modulation rate is defined as the
rate at which signal level is changed, dependingon the nature of the format used to represent thedigital data
The modulation rate is measured in bauds or
symbol per secondFor an M-ary format (with M an integer power of
two) used to represent binary data, the symbolduration of the M-ary format is related to the bitduration T b by
M T T b 2log
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Power Spectra of Discrete PAM Signals
Discrete amplitude-modulated pulse train may
be described as different realizations (sample
functions) of a random process X(t)
The coefficient Ak is a discrete random variable
v(t) is basic pulse shape, centered at the origin, t = 0,
and normalized such that v(0) = 1
T is the symbol duration
k
k kT t v At X )(
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Power Spectra of Discrete PAM Signals
Baseband Transmission 9 H. H. Kha, Ph.D.
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Power Spectra of Discrete PAM Signals
To proceed with the analysis, we model the
mechanism responsible for the generation of thesequence { Ak }, defining as a discrete stationaryrandom source
The source is characterized as having
ensemble-averaged autocorrelation function
where E is the expectation operator
nk k A
A A E n R
)(
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Power Spectra of Discrete PAM Signals
The power spectral density of the discrete PAM
signal X(t) is given by
V(f) is the Fourier transform of the basic pulse v(t)
The values of the functions V(f) and R A(n) depend on
the type of discrete PAM signal being considered
)2exp()()(1
)( 2
nfT jn R f V T
f S A X
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Power Spectra of Discrete PAM Signals
NRZ Unipolar Format
Suppose that the 0s and 1s of a random binary
sequence occur with equal probability
For n = 0, we may write
21)()0( a A P A P k k
2)()()0()0(][
2222 aa A P a A P A E
k k k
Baseband Transmission 12 H. H. Kha, Ph.D.
= Sum(x^2*f(x))
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Power Spectra of Discrete PAM Signals
NRZ Unipolar Format
Consider next the product Ak Ak-n for n≠ 0
The autocorrelation function R A(n) may be expressed as
follows
44
14
1032
2 aa A A E nk k
0n
4
2)(2
2
a
a
n R A
0
0
n
n
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Power Spectra of Discrete PAM Signals
NRZ Unipolar Format
For the basic pulse v(t), we have a rectangular pulse
of unit amplitude and duration T b. The Fourier
transform of v(t) equals
The power spectral density of NRZ unipolar format
)(sin)( bb fT cT f V
n
bbb
bb
X nfT j fT cT a
fT cT a
f S )2exp()(sin4)(sin4)( 2
22
2
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Power Spectra of Discrete PAM Signals
NRZ Unipolar Format
Use Poison’s formula written in the form
We may simplify the expression for the power spectral
density S X (f) as
m bbn
bT
m f
T nfT j
1)2exp(
)(4)(sin4)(
2
2
2
f a
fT cT a
f S bb X
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Power Spectra of Discrete PAM Signals
NRZ Polar Format
Similar to that described for the unipolar format, we findthat
The basic pulse v(t) for the polar format is the same asthat for unipolar format
The power spectral density of the NRZ polar format is
0
)(2
an R
A0
0
n
n
)(sin)( 22
bb X fT cT a f S
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Power Spectra of Discrete PAM Signals
NRZ Bipolar Format
The successive 1s in the bipolar format be assigned
pulses of alternating polarity
The bipolar format has three level: a, 0, -a
Assume that the 1s and 0s in the input binary data
occur with equal probability, we find the respective
probabilities of occurrence of these level are
4
1
210
41
a A P
A P
a A P
k
k
k
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Power Spectra of Discrete PAM Signals
NRZ Bipolar Format
For n = 0, we may write
For n = 1, the dibit represented by the sequence
( Ak Ak-1) can assume only four possible forms:
(0,0), (0,1), (1,0), (1,1). Hence we may write
2
00
2
2222 aa A P a A P a A P a A E
k k k k
44
1
4
103
2
2
1
aa A A E
k k
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Power Spectra of Discrete PAM Signals
NRZ Bipolar Format
For n > 1, we find that
For the NRZ Bipolar format, we have
0nk k A A E
0
4
2
)( 2
2
a
a
n R A
otherwise
1
0
n
n
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Power Spectra of Discrete PAM Signals
NRZ Bipolar Format
The basic pulse v(t) for the NRZ bipolar format
has its Fourier transform as in previous cases
The power spectral density of the NRZ bipolar
format is given
)2exp(2exp(
42)(sin)(
222
bbbb X fT j fT jaa
fT cT f S
)(sin)(sin
)2cos(1)(sin2222
22
bbb
bb
b
fT fT cT a
fT fT c
T a
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Power Spectra of Discrete PAM Signals
Manchester Format
In Manchester format, the input binary data consists ofindependent, equally likely symbol
The autocorrelation function R A(n) for the Manchester format is
the same as for the NRZ polar format
0
)(2
an R
A
0
0
n
n
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Power Spectra of Discrete PAM Signals
Manchester Format
The basic pulse v(t) for the Manchester formatconsists of a doublet pulse of unit amplitude and total
duration T b.The Fourier transform of the pulse equals
The power spectral density of the Manchester format
is given
2sin2sin)(
bb
b
fT fT
c jT f V
2sin
2sin)( 222 bb
b X fT fT cT a f S
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Power Spectra of Discrete PAM Signals
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3. InterSymbol Interference
Consider basic elements of a baseband binary PAM
system The input signal consists of a binary data sequence {bk } with a
bit duration of T b seconds
This sequence is applied to a pulse generator, producing the
discrete PAM signal
• v(t) denotes the basic pulse, normalize such that v(0) = 1
• The coefficient ak depends on the input data and the type of
format used• The waveform x(t) represents one realization of the random
process X(t)
k
bk kT t vat x )(
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InterSymbol Interference
Baseband Transmission 25 H. H. Kha, Ph.D.
Thi t k Ht v Hr sao cho ko g y ra ch ng phP(t) là kết quả sau khi qua kênh truyền <-- đkiện
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InterSymbol Interference
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InterSymbol Interference
The receiving filter output may be written as
is scaling factor
The pulse p(t) is normalized such that
k
bk kT t pat y )(
1)0( p
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InterSymbol Interference
The output y(t) is produced in response to binary data
waveform applied to the input of the transmitting filter.Especially, the pulse is response of the cascade
connection of the transmitting filter, the channel, and the
receiving filter, which is produced by the pulse v(t) applied
to the input of this cascade connection
P(f) and V(f) are Fourier transform of p(t) and v(t)
)(t p
)()()()()( f H f H f H f V f P RC T
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InterSymbol Interference
The receiving filter output y(t) is sampled at time t i = iT b
The first term is produced by the ith transmitted bit.
The second term represents the residual effect of all
other transmitted bits on the decoding of the ith bit; thisresidual effect is called intersymbol interference (ISI)
k
bbk i kT iT pat y )(
ik
k
bbk i kT iT paa
Baseband Transmission 29 H. H. Kha, Ph.D.
t n p n giao t oa
4 Nyquist’s Criterion For Distortionless
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4. Nyquist s Criterion For Distortionless
Baseband Binary Transmission
Typically, the transfer function of the channel
and the transmitted pulse shape are specified,and the problem is to determine the transferfunctions of the transmitting and receiving filtersso as to reconstruct the transmitted datasequence {bk }
The receiver does this by extracting and thendecoding the corresponding sequence ofweights, {ak }, from the output y(t).
Except for a scaling factor, y(t) is determined by
the ak and the received pulse p(t)
Baseband Transmission 30 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For Distortionless
Baseband Binary Transmission
The extraction involves sampling the output y(t)
at some time t = iT b
The decoding requires that the weighted pulse
contribution ak p(iT b-kT b ) for k = i be free form ISI
due to the overlapping tails of all other weightedpulse contributions represented by k ≠ i
Baseband Transmission 31 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For Distortionless
Baseband Binary Transmission
This, in turn, require that we control the received
pulse p(t), as shown by
where, by normalization, p(0) = 1
0
1bb kT iT p
k i
k i
Baseband Transmission 32 H. H. Kha, Ph.D.
Đi u ki n mongmuốn
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For Distortionless
Baseband Binary Transmission
The receiver output
Which implies zero intersymbol interference (ISI)
This condition assures perfect reception in theabsence of noise
ii
at y
Baseband Transmission 33 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For Distortionless
Baseband Binary Transmission
Consider the sequence of samples { p(nT b )},
where n = 0, ±1, ±2, …
Sampling in the time domain produces
periodicity in frequency domain
Where Rb = 1/T b is the bit rate
P δ(f) is the Fourier transform of an infinite periodic sequence of
delta functions of period T b, and whose strengths are weighted
by the respective sample values of p(t)
n
bb nR f P R f P )(
Baseband Transmission 34 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For Distortionless
Baseband Binary Transmission
That is
where m = i – k.
Impose the condition of zero ISI on the samplevalues of p(t)
dt ft jmT t mT p f P bb
2exp)()()(
dt ft jt p f P 2exp)()0()(
)0( p
Baseband Transmission 35 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For DistortionlessBaseband Binary Transmission
Since p(0) = 1, by normalization, the condition for
zero ISI is sastisfied if
Nyquist criterion for distortionless baseband
transmission
b
n
b T nR f P
Baseband Transmission 36 H. H. Kha, Ph.D.
Ph t 1 bit --> 1 xung
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For DistortionlessBaseband Binary Transmission
Ideal solution
A frequency function P(f), occupying the narrowest band, isobtained by permitting only one nonzero component in the seriesfor each f in the range extending from – B0 to B0, where B0
denotes half the bit rate
We specify P(f)
Hence, signal waveform that produces zero ISI is defined by thesinc function
2
0
b R
B
00 22
1)(
B
f rect
B f P
Baseband Transmission 37 H. H. Kha, Ph.D.
t B
t Bt p
0
0
2
2sin)(
t Bc 02sin
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InterSymbol Interference
Ideal solution
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InterSymbol Interference
Ideal solution
Baseband Transmission 39 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For DistortionlessBaseband Binary Transmission
There are two practical difficulties that make it
an undesirable objective for system design: It requires that the amplitude characteristic of P(f) be
flat form – B0 to B0 and zero elsewhere. This isphysically unrealizable because of the abrupttransitions at ± B
0 The function p(t) decreases as 1/|t| for large |t|,
resulting in a slow rate of decay. This is caused bythe discontinuity of P(f) at ± B0. Accordingly, there ispractically no margin of error in sampling times in the
receiver
Baseband Transmission 40 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For DistortionlessBaseband Binary Transmission
Practical solution
We may overcome the practical difficulties posed by the idealsolution by extending the bandwidth from B0 = Rb /2 to an
adjustable value between B0 and 2B0
In doing so, we permit three components as shown by
0
002
122)(
B B f p B f p f P
00 B f B
Baseband Transmission 41 H. H. Kha, Ph.D.
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For DistortionlessBaseband Binary Transmission
Practical solution
A particular form of P(f) that embodies many desirablefeatures is constructed by a raised cosine spectrum
• Rolloff factor
0
22cos1
4
1
2
1
)(10
1
0
0
f B
f f
B
B
f P
10
101
1
2
2
f B f
f B f f
f f
0
11 B
f
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InterSymbol Interference
Practical solutionĐ nh đ i giữa t c đ suyhao và độ rộng băng thông
Nyquist’s Criterion For Distortionless
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Nyquist s Criterion For DistortionlessBaseband Binary Transmission
Practical solution
The time response p(t), that is, the inverse Fourier
transform of P(f), is defined
A more general relationship between required
bandwidth and symbol transmission rate involves the
roll-off factor
22
0
2
00
161
2cos)2(sin)(
t B
t Bt Bct p
Baseband Transmission 44 H. H. Kha, Ph.D.
)1(2 010 B f B B
5 C l ti C di
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5. Correlative Coding
It is possible to achieve a bit rate of 2B0 per second in a
channel of bandwidth B0 Hertz by adding intersymbolinterference to the transmitted signal in a controlled manner
Such schemes are called correlative coding or partial-
response signaling schemes
The design of these schemes is based on the premise thatsince intersymbol interference introduced into the
transmitted signal is known, its effect can be compensated
at the receiver.
Correlative coding may be regarded as a practical means ofachieving the theoretical maximum signaling rate of 2Bo per
second in a bandwidth of B0 hertz
Baseband Transmission 45 H. H. Kha, Ph.D.
M a tương quan
C l ti C di
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Correlative Coding
Duobinary signaling
Consider a binary input sequence {bk } consisting of
uncorrelated binary digits each having duration Tb
seconds, with symbol 1 represented by a pulse of
amplitude +1 volt, and symbol 0 by a pulse of
amplitude -1 volt This sequence is applied to duobinary encoder , it is
converted into a three-level output, namely -2, 0, and
+2 volts
Baseband Transmission 46 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Duobinary signaling
Baseband Transmission 47 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Duobinary signaling The digit ck at the duobinary coder output is the
sum of the resent binary digit bk and its previousvalue bk-1
One of the effects of the transformation is tochange the input sequence {bk } of uncorrelatedbinary digits into a sequence {ck } of correlateddigits
This correlation between the adjacent transmittedlevels may be viewed as introducing ISI into thetransmitted signal
1
k k k bbc
Baseband Transmission 48 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Duobinary signaling
The overall transfer function of this filter connected in cascadewith the ideal channel H c(f) is
bC fT j f H f H exp1)()(
bbC
bbbC
fT j fT f H
fT j fT j fT j f H
expcos)(2
expexpexp1)(
Baseband Transmission 49 H. H. Kha, Ph.D.
2
Correlative Coding
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Correlative Coding
Duobinary signaling
For the ideal channel of bandwidth B0 = R b /2, we have
The overall frequency response has the form of a
half-cycle cosine function
0
1)( f H C
otherwise
2b R f
0
expcos2
)( bb fT j fT
f H
otherwise
2b R f
Baseband Transmission 50 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Duobinary signaling
The corresponding value of the impulse response consists of twosinc pulse, time-displaced by T b seconds
bb
bb
b
b
T T t
T T t
T t
T t t h
sinsin)(
t T t
T t T
T T t
T t
T t
T t
b
bb
bb
b
b
b
sin
sinsin
2
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Duobinary signaling
C l ti C di
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Correlative Coding
Duobinary signaling
Baseband Transmission 53 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Duobinary signaling
The original data {bk } may be detected from theduobinary-coded sequence {ck } by subtracting theprevious decoded binary digit from the currentlyreceived digit ck
It is apparent that if ck is received without error and ifalso the previous estimate at time t = (k-1)T bcorresponds to a correct decision, then the current
estimate will be correct too
1
ˆˆ
k k k bcb
k bˆ
1
ˆk
b
Baseband Transmission 54 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Duobinary signaling – Practical solution
Use precoder before the duobinary coding to avoid errorpropagation
The precoder operation performed on the input binary sequence
{bk } converts it into another sequence {ak } defined by
1
k k k aba
Baseband Transmission 55 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Duobinary signaling – Practical solution
The resulting precoder output {ak } is applied to the duobinarycoder
The sequence {c k } is related to {ak } as follows
1 k k k aac
Baseband Transmission 56 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Baseband Transmission 57 H. H. Kha, Ph.D.
Correlative Coding
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Correlative Coding
Decision rule
volt1if 1
volt1if 0
k
k
k c symbol
c symbol b
Baseband Transmission 58 H. H. Kha, Ph.D.
6 Eye Pattern
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6. Eye Pattern
One way to study ISI in a PCM or data
transmission system experimentally is to applythe received wave to the vertical deflectionplates of an oscilloscope an to apply a sawtoothwave at the transmitted symbol rate R = 1/T tothe horizontal deflection plates
The waveforms in successive symbol intervalsare thereby translated into one interval on theoscilloscope display
The resulting display is called an eye pattern
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Eye Pattern
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Eye Pattern
Baseband Transmission 60 H. H. Kha, Ph.D.
Eye Pattern
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Eye Pattern
The width of the eye opening defines the time
interval over which the received wave can besampled without error form ISI. It is apparentthat the preferred time for sampling is the instantof time at which the eye is opened widest
The sensitivity of the system to timing error isdetermined by the rate of closure of the eye asthe sampling time is varied
The height of the eye opening, at a specifiedsampling time, defines the margin over noise
Baseband Transmission 61 H. H. Kha, Ph.D.
Eye Pattern
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Eye Pattern
Baseband Transmission 62 H. H. Kha, Ph.D.
Homework
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Homework
Problems: 4.1, 4.2, 4.3
Problems: 4.7, 4.8, 4.9 Problems: 4.16, 4.18, 4.19
Problems: 4.21, 4.25, 4.26
Textbook:
Simon Haykin, Communication System , 4th Edition,
John Wiley & Son, Inc. , 2001.