baseband transmission.pdf
TRANSCRIPT
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BASEBAND TRANSMISSION
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Discrete PAM Signals
The use of an appropriate for basebandrepresentation of digital is basic to itstransmission from a source to a destination
There are some different formats for the
representation of the binary data sequence Unipolar format (on-off signaling) Polar format Bipolar format (also known as pseudoternary
signaling) Manchester format (also known as biphase
baseband signaling)
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Discrete PAM Signal
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Power Spectra of DiscretePAM Signals
Data signaling rate (or data rate) is definedas the rate, measured in bits per second(bps), at which data are transmitted.
It is also common practice to refer to thedata signaling rate as the bit rate, denoted by
where Tb is the bit duration
bb T
R1
=
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Power Spectra of DiscretePAM Signals
In contrast, the modulation rate is defined asthe rate at which signal level is changed,depending on the nature of the format usedto represent the digital data
The modulation rate is measured in bauds orsymbol per second
For an M-ary format (withMan integerpower of two) used to represent binary data,the symbol duration of the M-ary format isrelated to the bit duration Tb by
MTT b 2log=
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Power Spectra of DiscretePAM Signals
Discrete amplitude-modulated pulse train maybe described as different realizations (samplefunctions) of a random process X(t)
The coefficientAk is a discrete random variable
v(t) is basic pulse shape, centered at the origin,t = 0, and normalized such that v(0) = 1
Tis the symbol duration
( )
=
=k
k kTtvAtX )(
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Power Spectra of DiscretePAM Signals
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Power Spectra of DiscretePAM Signals
To proceed with the analysis, we model themechanism responsible for the generation ofthe sequence {Ak}, defining as a discretestationary random source
The source is characterized as havingensemble-averaged autocorrelation function
whereEis the expectation operator[ ]nkkA AAEnR =)(
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Power Spectra of DiscretePAM Signals
The power spectral density of the discretePAM signalX(t) is given by
V(f) is the Fourier transform of the basic pulse v(t)
The values of the functions V(f) andRA(n) depend
on the type of discrete PAM signal beingconsidered
= )2exp()()(1
)(2
nfTjnRfVT
fS AX
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Power Spectra of DiscretePAM Signals
NRZ Unipolar Format Suppose that the 0s and 1s of a random
binary sequence occur with equal
probability
For n = 0, we may write2
1)()0( ==== aAPAP kk
2)()()0()0(][
2222 aaAPaAPAE kkk ==+==
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Power Spectra of DiscretePAM Signals
NRZ Unipolar Format Consider next the productAkAk-n for n 0
The autocorrelation functionRA(n) may beexpressed as follows
[ ] ( )( ) ( )( ) 44141032
2 aaAAE nkk =+= 0n
=
4
2)(2
2
a
a
nRA
0
0
=
n
n
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Power Spectra of DiscretePAM Signals
NRZ Unipolar Format For the basic pulse v(t), we have a rectangular
pulse of unit amplitude and duration Tb. TheFourier transform ofv(t) equals
The power spectral density of NRZ unipolar format
)(sin)( bb fTcTfV =
=
+=n
bbb
bb
X nfTjfTcTa
fTcTa
fS )2exp()(sin4
)(sin4
)( 22
22
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Power Spectra of DiscretePAM Signals
NRZ Unipolar Format Use Poisons formula written in the form
We may simplify the expression for thepower spectral density S
X(f) as
=
=
= m bbnb
T
mfTnfTj
1)2exp(
)(4
)(sin4
)(2
22
fa
fTcTa
fS bb
X +=
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Power Spectra of DiscretePAM Signals
NRZ Polar Format Similar to that described for the unipolar
format, we find that
The basic pulse v(t) for the polar format is
the same as that for unipolar format The power spectral density of the NRZ
polar format is
=0
)(2
anRA 00
=
nn
)(sin)(22
bbX fTcTafS =
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Power Spectra of DiscretePAM 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 therespective probabilities of occurrence of these
level are( )
( )
( ) 41
210
41
==
==
==
aAP
AP
aAP
k
k
k
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Power Spectra of DiscretePAM Signals
NRZ Bipolar Format For n = 0, we may write
For n = 1, the dibit represented by thesequence (AkAk-1) can assume only four
possible forms: (0,0), (0,1), (1,0), (1,1). Hencewe may write
[ ] ( ) ( ) ( ) ( ) ( ) ( ) 200
22222 a
aAPaAPaAPaAEkkkk
==+=+==
[ ] ( )( ) ( )( )44
1
4
1032
2
1
aaAAE kk =+=
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Power Spectra of DiscretePAM Signals
NRZ Bipolar Format For n > 1, we find that
For the NRZ Bipolar format, we have
[ ] 0=nkk
AAE
=
0
4
2
)( 2
2
a
a
nRA
otherwise
1
0
=
=
n
n
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Power Spectra of DiscretePAM 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 bipolarformat is given
( )
+= )2exp(2exp(
42
)(sin)(22
2
bbbbX fTjfTjaa
fTcTfS
[ ]
)(sin)(sin
)2cos(1)(sin2
222
22
bbb
bbb
fTfTcTa
fTfTcTa
=
=
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Power Spectra of DiscretePAM Signals
Manchester Format In Manchester format, the input binary
data consists of independent, equally likely
symbol The autocorrelation functionRA(n) for the
Manchester format is the same as for the
NRZ polar format
=0
)(2a
nRA0
0
=
n
n
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Power Spectra of DiscretePAM Signals
Manchester Format The basic pulse v(t) for the Manchester format
consists of a doublet pulse of unit amplitude and
total duration Tb.The Fourier transform of thepulse equals
The power spectral density of the Manchesterformat is given
=
2sin
2sin)( bbb
fTfTcjTfV
=
2sin2sin)(
222 bb
bX
fTfT
cTafS
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Power Spectra of DiscretePAM Signals
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InterSymbol Interference
Consider basic elements of a baseband binary PAMsystem
The input signal consists of a binary data sequence {bk} witha bit duration ofTb seconds
This sequence is applied to a pulse generator, producing thediscrete 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 waveformx(t) represents one realization of the randomprocessX(t)
( )
=
=k
bk kTtvatx )(
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InterSymbol Interference
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InterSymbol Interference
The receiving filter output may be written as
is scaling factor
The pulsep(t) is normalized such that
( )
=
=k
bk kTtpaty )(
1)0( =p
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InterSymbol Interference
The outputy(t) is produced in response to binarydata waveform applied to the input of thetransmitting filter. Especially, the pulse isresponse of the cascade connection of the
transmitting filter, the channel, and the receivingfilter, which is produced by the pulse v(t) applied tothe input of this cascade connection
P(f) and V(f) are Fourier transform ofp(t) and v(t)
)(tp
)()()()()( fHfHfHfVfP RCT=
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InterSymbol Interference
The receiving filter outputy(t) is sampled at time ti =iTb
The first term is produced by the ith transmitted bit.
The second term represents the residual effect of all othertransmitted bits on the decoding of the ith bit; this residualeffect is called intersymbol interference (ISI)
( )
=
=k
bbki kTiTpaty )(
( )
=
+=
ikk
bbki kTiTpaa
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Typically, the transfer function of the channeland the transmitted pulse shape arespecified, and the problem is to determinethe transfer functions of the transmitting and
receiving filters so as to reconstruct thetransmitted data sequence {bk} The receiver does this by extracting and then
decoding the corresponding sequence of
weights, {ak}, from the outputy(t). Except for a scaling factor,y(t) is determined
by the ak and the received pulsep(t)
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
The extraction involves sampling the outputy(t) at some time t = iTb
The decoding requires that the weighted
pulse contribution akp(iTb-kTb) for k = i be freeform ISI due to the overlapping tails of allother weighted pulse contributions
represented by k
i
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
This, in turn, require that we control thereceived pulsep(t), as shown by
where, by normalization,p(0) = 1
( )
=0
1bb kTiTp
ki
ki
=
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
The receiver output
Which implies zero intersymbol interference (ISI) This condition assures perfect reception in the
absence of noise
( ) ii aty =
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Consider the sequence of samples {p(nTb)},where n = 0, 1, 2,
Sampling in the time domain produces
periodicity in frequency domain
WhereRb = 1/Tb is the bit rate P
(f) is the Fourier transform of an infinite periodic sequence
of delta functions of period Tb, and whose strengths areweighted by the respective sample values ofp(t)
( )
=
=n
bb nRfPRfP )(
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
That is
where m = i k.
Impose the condition of zero ISI on thesample values ofp(t)
[ ] ( )dtftjmTtmTpfP bb 2exp)()()( =
( )dtftjtpfP 2exp)()0()( =
)0(p=
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Sincep(0) = 1, by normalization, the conditionfor zero ISI is sastisfied if
Nyquist criterion for distortionless
baseband transmission
( ) bn
b TnRfP =
=
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Ideal solutionA frequency functionP(f), occupying the
narrowest band, is obtained by permittingonly one nonzero component in the seriesfor eachfin the range extending fromB0toB0, whereB0 denotes half the bit rate
We specifyP(f)
20
bRB =
=
00 22
1)(
B
frect
BfP
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Ideal solution Hence, signal waveform that produces zero
ISI is defined by the sinc function
( )tB
tBtp
0
0
2
2sin)(
=
( )tBc 02sin=
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InterSymbol Interference
Ideal solution
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
There are two practical difficulties that makeit an undesirable objective for system design: It requires that the amplitude characteristic ofP(f)
be flat formB0 toB0 and zero elsewhere. This is
physically unrealizable because of the abrupttransitions at B0
The function p(t) decreases as 1/|t| for large |t|,resulting in a slow rate of decay. This is caused by
the discontinuity ofP(f) at B0. Accordingly, thereis practically no margin of error in sampling timesin the receiver
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Practical solution We may overcome the practical difficulties
posed by the ideal solution by extending
the bandwidth fromB0 = Rb/2 to anadjustable value betweenB0 and 2B0
In doing so, we permit three components
as shown by( ) ( )
0
002
122)(
BBfpBfpfP =+++
00 BfB
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Practical solution A particular form ofP(f) that embodies many desirable
features is constructed by a raised cosine spectrum
Rolloff factor
( )
+=
0
22cos1
4
1
21
)(10
1
0
0
fB
ff
B
B
fP
10
101
1
2
2
fBf
fBff
ff
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Nyquists Criterion For DistortionlessBaseband Binary Transmission
Practical solution The time responsep(t), that is, the inverse Fourier
transform ofP(f), is defined
( )220
20
01612cos)2(sin)(
tBtBtBctp
=
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InterSymbol Interference
Practical solution
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Correlative Coding
It is possible to achieve a bit rate of2B0
per secondin a channel of bandwidthB0 Hertz by addingintersymbol interference to the transmitted signal in acontrolled manner
Such schemes are called correlative coding or partial-
response signaling schemes The design of these schemes is based on the premise
that since intersymbol interference introduced intothe transmitted signal is known, its effect can be
compensated at the receiver. Correlative coding may be regarded as a practical
means of achieving the theoretical maximumsignaling rate of2Bo per second in a bandwidth ofB0hertz
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Correlative Coding
Duobinary signaling Consider a binary input sequence {bk} consisting
of uncorrelated binary digits each having durationTb seconds, with symbol 1 represented by a pulse
of amplitude +1 volt, and symbol 0 by a pulse ofamplitude -1 volt
This sequence is applied to duobinary encoder, it
is converted into a three-level output, namely -2,0, and +2 volts
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Correlative Coding
Duobinary signaling
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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 uncorrelated
binary digits into a sequence {ck} pf correlateddigits This correlation between the adjacent transmitted
levels may be viewed as introducing ISI into thetransmitted signal
1+= kkk bbc
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Correlative Coding
Duobinary signaling The overall transfer function of this filter
connected in cascade with the ideal
channelHc(f) is( )[ ]bC fTjfHfH += exp1)()(
( ) ( )[ ] ( )
( ) ( )bbC
bbbC
fTjfTfH
fTjfTjfTjfH
=
++=
expcos)(2
expexpexp1)(
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Correlative Coding
Duobinary signaling For the ideal channel of bandwidthB0 = R b/2, we
have
The overall frequency response has the form of ahalf-cycle cosine function
= 0
1
)( fHC otherwise
2bRf
( ) ( )
=0
expcos2)(
bb fTjfTfH
otherwise
2bRf
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Correlative Coding
Duobinary signaling The corresponding value of the impulse
response consists of two sinc pulse, time-
displaced by Tb seconds( ) ( )[ ]
( ) bbbb
b
b
TTt
TTt
Tt
Ttth
+=
sinsin)(
( ) [ ]( )
( )
( )tTt
TtT
TTtTt
TtTt
b
bb
bb
b
b
b
=
=
sin
sinsin
2
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Correlative Coding
Duobinary signaling
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Correlative Coding
Duobinary signaling The original data {bk} may be detected from the
duobinary-coded sequence {ck} by subtracting theprevious decoded binary digit from the currently
received digit ck
It is apparent that ifck is received without error
and if also the previous estimate at time t = (k-1)Tb corresponds to a correct decision, then thecurrent estimate will be correct too
1
= kkk bcb
kb
1
kb
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Eye Pattern
One way to study ISI in a PCM or datatransmission system experimentally is toapply the received wave to the verticaldeflection plates of an oscilloscope an toapply a sawtooth wave at the transmittedsymbol rate R = 1/T to the horizontaldeflection plates
The waveforms in successive symbol intervals
are thereby translated into one interval onthe oscilloscope display The resulting display is called an eye pattern
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Eye Pattern
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Eye Pattern
The width of the eye opening defines thetime interval over which the received wavecan be sampled without error form ISI. It isapparent that the preferred time for samplingis the instant of time at which the eye isopened widest
The sensitivity of the system to timing error isdetermined by the rate of closure of the eye
as the sampling time is varied The height of the eye opening, at a specified
sampling time, defines the margin over noise
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Eye Pattern