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Wireless Information Transmission System Lab.
National Sun Yat-sen University Institute of Communications Engineering
Signal Design for Band-Limited Channels
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IntroductionWe consider the problem of signal design when the channel isband-limited to some specified bandwidth of W Hz.
The channel may be modeled as a linear filter having an equivalentlow-pass frequency response C ( f ) that is zero for | f | >W .
Our purpose is to design a signal pulse g(t ) in a linearly modulatedsignal, represented as
that efficiently utilizes the total available channel bandwidth W .When the channel is ideal for | f | ≤W , a signal pulse can be designed that
allows us to transmit at symbol rates comparable to or exceeding thechannel bandwidth W .
When the channel is not ideal, signal transmission at a symbol rate equal toor exceeding W results in inter-symbol interference (ISI) among a number of adjacent symbols.
( ) ( )nn
v t I g t nT = −∑
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For our purposes, a band-limited channel such as a telephone
channel will be characterized as a linear filter having an
equivalent low-pass frequency-response characteristic C ( f ),and its equivalent low-pass impulse response c(t ).
Then, if a signal of the form
is transmitted over a band-pass telephone channel, the
equivalent low-pass received signal is
where z(t ) denotes the additive noise.
( ) ( ) ( ) ( )( ) ( ) ( )t zt ct v
t zd t cvt r l
+∗=
+−= ∫∞
∞−
τ τ τ
( ) ( )t f j c
et vt s
π 2
Re=
Characterization of Band-Limited
Channels
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Alternatively, the signal term can be represented in thefrequency domain as V ( f )C ( f ), where V ( f ) = F [v(t )].
If the channel is band-limited to W Hz, then C ( f ) = 0 for | f | >W .
As a consequence, any frequency components in V ( f ) above
| f | = W will not be passed by the channel, so we limit thebandwidth of the transmitted signal to W Hz.
Within the bandwidth of the channel, we may express C ( f ) as
where |C ( f )|: amplitude response.θ( f ): phase response.
The envelope delay characteristic:
( ) ( ) ( ) f je f C f C
θ =
( )( )
df
f d f
θ
π τ
2
1−=
Characterization of Band-Limited
Channels
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A channel is said to be nondistorting or ideal if the amplituderesponse |C ( f )| is constant for all | f | ≤ W andθ( f ) is a linear
function of frequency, i.e.,τ( f ) is a constant for all | f | ≤ W .If |C ( f )| is not constant for all | f | ≤ W , we say that the channeldistorts the transmitted signal V ( f ) in amplitude.
If τ( f ) is not constant for all | f | ≤ W , we say that the channel
distorts the signal V ( f ) in delay.As a result of the amplitude and delay distortion caused by thenonideal channel frequency-response C ( f ), a succession of pulses transmitted through the channel at rates comparable tothe bandwidth W are smeared to the point that they are nolonger distinguishable as well-defined pulses at the receivingterminal. Instead, they overlap, and thus, we have inter-symbolinterference (ISI).
Characterization of Band-Limited
Channels
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Fig. (a) is a band-limited pulse having zeros periodically spaced
in time at ±T , ±2T , etc.
If information is conveyed by the pulse amplitude, as in PAM, forexample, then one can transmit a sequence of pulses, each of
which has a peak at the periodic zeros of the other pulses.
Characterization of Band-Limited
Channels
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However, transmission of the pulse through a channel modeled as
having a linear envelope delayτ( f ) [quadratic phaseθ( f )]
results in the received pulse shown in Fig. (b), where the zero-crossings that are no longer periodically spaced.
Characterization of Band-Limited
Channels
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A sequence of successive pulses would no longer be
distinguishable. Thus, the channel delay distortion results in ISI.
It is possible to compensate for the nonideal frequency-responseof the channel by use of a filter or equalizer at the demodulator.
Fig. (c) illustrates the output of a linear equalizer that
compensates for the linear distortion in the channel.
Characterization of Band-Limited
Channels
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The equivalent low-pass transmitted signal for several different
types of digital modulation techniques has the common form
where { I n}: discrete information-bearing sequence of symbols.
g(t ): a pulse with band-limited frequency-
response G( f ), i.e., G( f ) = 0 for | f | > W .This signal is transmitted over a channel having a frequency
response C ( f ), also limited to | f | ≤ W .
The received signal can be represented as
where and z(t ) is the AWGN.
( ) ( ) ( )t znT t h I t r n
nl +−= ∑∞
=0
( ) ( )∑
∞
= −= 0n
n nT t g I t v
( ) ( ) ( )
∫
∞
∞−
−= τ τ τ d t cgt h
Signal Design for Band-Limited Channels
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Suppose that the received signal is passed first through a filter and
then sampled at a rate 1/ T samples/s, the optimum filter from the
point of view of signal detection is one matched to the receivedpulse. That is, the frequency response of the receiving filter is
H*( f ).
We denote the output of the receiving filter as
where
x(t ): the pulse representing the response of the receivingfilter to the input pulse h(t ).
v(t ): response of the receiving filter to the noise z(t ).
( ) ( ) ( )∑∞
=
+−=0n
n t vnT t x I t y
Signal Design for Band-Limited Channels
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If y(t ) is sampled at times t = kT +τ0 , k = 0, 1,…, we have
or, equivalently,
whereτ0: transmission delay through the channel.
The sample values can be expressed as
( ) ( ) ( )00
00 τ τ τ +++−=≡+
∑
∞
= kT vnT kT x I ykT y nnk
0
, 0,1,...k n k n k
n
y I x v k ∞
−
=
= + =∑
0
00
1, 0,1,...
k k n k n k
nn k
y x I I x v k x
∞
−=≠
⎛ ⎞⎜ ⎟= + + =⎜ ⎟⎜ ⎟⎝ ⎠
∑
Signal Design for Band-Limited Channels
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We regard x0 as an arbitrary scale factor, which we arbitrarily set
equal to unity for convenience, then
where
I k : the desired information symbol at the k-thsampling instant.
: ISI
vk : additive Gaussian noise variable at the k-th
sampling instant.
k
k nn
nk nk k v x I I y ++= ∑∞
≠=
−0
∑∞
≠
=−
k n
n
nk n x I 0
Signal Design for Band-Limited Channels
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The amount of ISI and noise in a digital communication system
can be viewed on an oscilloscope.
For PAM signals, we can display the received signal y(t ) on thevertical input with the horizontal sweep rate set at 1/ T .
The resulting oscilloscope display is called an eye pattern.
Eye patterns for binary and quaternary amplitude-shift keying:
The effect of ISI is to cause the eye to close.
Thereby, reducing the margin for additive noise to cause errors.
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Effect of ISI on eye opening:
ISI distorts the position of the zero-crossings and causes areduction in the eye opening.
Thus, it causes the system to be more sensitive to a
synchronization error.
Signal Design for Band-Limited Channels
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For PSK and QAM, it is customary to display the “eye pattern” as
a two-dimensional scatter diagram illustrating the sampled values
{ yk } that represent the decision variables at the sampling instants.
Two-dimensional digital “eye patterns.”
Signal Design for Band-Limited Channels
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Assuming that the band-limited channel has ideal frequency-
response, i.e., C ( f ) = 1 for | f | ≤ W , then the pulse x(t ) has a
spectral characteristic X ( f ) = |G( f )|2
, where
We are interested in determining the spectral properties of the
pulse x(t ), that results in no inter-symbol interference.
Since
the condition for no ISI is
k
k nn
nk nk k v x I I y ++= ∑∞
≠=
−0
( )( )
( )
1 0
0 0k
k x t kT x
k
=⎧⎪= ≡ = ⎨
≠⎪⎩
( ) ( )∫−=
W
W
ft jdf e f X t x
π 2
(*)
Design of Band-Limited Signals for
No ISI – The Nyquist Criterion
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Nyquist pulse-shaping criterion (Nyquist condition for
zero ISI)
The necessary and sufficient condition for x(t ) to satisfy
is that its Fourier transform X ( f ) satisfy
( )( )
( )
1 0
0 0
n x nT
n
=⎧⎪= ⎨
≠⎪⎩
( ) T T m f X m
=+∑∞
−∞=
Design of Band-Limited Signals for
No ISI – The Nyquist Criterion
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Proof:
In general, x(t ) is the inverse Fourier transform of X ( f ). Hence,
At the sampling instant t = nT ,
( ) ( )∫∞
∞−= df e f X t x ft j π 2
( ) ( )∫∞
∞−= df e f X nT x fnT j π 2
Design of Band-Limited Signals for
No ISI – The Nyquist Criterion
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Breaking up the integral into integrals covering the finite
range of 1/ T , thus, we obtain
where we define B( f ) as
( ) ( )( )( )
( )
( )
( )
2 1 2 2
2 1 2
1 22 '
1 2
1 22
1 2
1 22
1 2
' '
m T j fnT
m T m
T j f nT
T
m
T j fnT
T m
T j fnT
T
x nT X f e df
X f m T e df
X f m T e df
B f e df
π
π
π
π
∞ +
−=−∞
∞
−
=−∞∞
−=−∞
−
=
= +
⎡ ⎤= +⎢ ⎥
⎣ ⎦
=
∑ ∫
∑ ∫
∑∫
∫( ) ( )∑
∞
−∞=
+=m
T m f X f B
'
2 1 2 1: ,
2 2
1 1
' : ,2 2
'
m f f
T m m
f T T
f T T
df df
= +
− +⎡ ⎤⎢ ⎥⎣ ⎦
−⎡ ⎤
⎢ ⎥⎣ ⎦
=
Design of Band-Limited Signals for
No ISI – The Nyquist Criterion
(1) Periodicfunction.
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Design of Band-Limited Signals for No
ISI–
The Nyquist CriterionObviously B( f ) is a periodic function with period 1/ T , and,
therefore, it can be expanded in terms of its Fourier series
coefficients {bn} as
where
Comparing (1) and (2), we obtain
( )( )
( )
Recall that the conditions for
no ISI are (from ):
1 0
0 0k
k x t kT x
k
∗
=⎧⎪= ≡ = ⎨
≠⎪⎩
( ) 2 j nfT
n
n
B f b eπ ∞
=−∞
= ∑
( )1 2
2
1 2
T nfT
n
T
b T B f e df π −
−
= ∫
( )nT Txbn −=
(2)
( ) ( )1 2
2
1 2
1T j fnT
nT
x nT B f e df bT
π −
−− = =∫
(9.2-20)
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Therefore, the necessary and sufficient condition for
to be satisfied is that
which, when substituted into , yields
or, equivalently
This concludes the proof of the theorem.
( )
( )
0
0 0n
T nb
n
=⎧⎪= ⎨
≠⎪⎩
( ) ∑∞
−∞=
=n
nfT j
neb f Bπ 2
( ) T f B =
( ) T T m f X m
=+∑∞
−∞=
k
k nn
nk nk k v x I I y ++= ∑∞
≠=
−
0
Design of Band-Limited Signals for
No ISI – The Nyquist Criterion
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Suppose that the channel has a bandwidth of W . Then
C ( f )≣ 0 for | f | > W and X ( f ) = 0 for | f | > W .
When T < 1/2W (or 1/ T > 2W )
Since consists of nonoverlapping
replicas of X ( f ), separated by 1/ T , there is no choice for
X ( f ) to ensure B( f )≣ T in this case and there is no way
that we can design a system with no ISI.
( ) ( )∑+∞
∞− += T n f X f B
Design of Band-Limited Signals for
No ISI – The Nyquist Criterion
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When T = 1/2W , or 1/ T = 2W (the Nyquist rate), the
replications of X ( f ), separated by 1/ T , are shown below:
In this case, there exists only one X ( f ) that results in B( f ) = T ,
namely,
which corresponds to the pulse
( )( )
( )
0
T f W X f
otherwise
⎧ <⎪= ⎨
⎪⎩
( )( )sin
sinct T t
x t t T T
π π
π
⎛ ⎞= ≡ ⎜ ⎟
⎝ ⎠
Design of Band-Limited Signals for
No ISI–
The Nyquist Criterion
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Design of Band-Limited Signals for NoISI – The Nyquist Criterion
The smallest value of T for which transmission with zero ISIis possible is T = 1/2W , and for this value, x(t ) has to be a sincfunction.
The difficulty with this choice of x(t ) is that it is noncausaland nonrealizable.
A second difficulty with this pulse shape is that its rate of
convergence to zero is slow.The tails of x(t ) decay as 1/ t ; consequently, a small mistimingerror in sampling the output of the matched filter at thedemodulator results in an infinite series of ISI components.
Such a series is not absolutely summable because of the 1/ t rate of decay of the pulse, and, hence, the sum of the resultingISI does not converge. 1 1 1
1 ?2 4 8
+ + + + ="1 1 1
1 ?2 3 4
+ + + + ="
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When T > 1/2W (or 1/ T <2W ), B( f ) consists of overlapping
replications of X ( f ) separated by 1/ T :
In this case, there exist numerous choices for X ( f ) such that
B( f )≣ T .
Design of Band-Limited Signals for
No ISI–
The Nyquist Criterion
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A particular pulse spectrum, for the T > 1/2W case, that has
desirable spectral properties and has been widely used in
practice is the raised cosine spectrum.
Raised cosine spectrum:
β: roll-off factor. (0 ≤β≤ 1)
( )
10
2
1 1- 11 cos
2 2 2T 2
102
rc
T f T
T T X f f f
T T
f T
β
π β β β
β
β
⎧ −⎛ ⎞≤ ≤⎜ ⎟⎪
⎝ ⎠⎪⎪ ⎧ ⎫⎡ ⎤− +⎪ ⎛ ⎞ ⎛ ⎞
= + − ≤ ≤⎨ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎩ ⎭⎪
⎪+⎛ ⎞⎪ >⎜ ⎟
⎪ ⎝ ⎠⎩
Design of Band-Limited Signals for
No ISI–
The Nyquist Criterion
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The bandwidth occupied by the signal beyond the Nyquistfrequency 1/2T is called the excess bandwidth and is usuallyexpressed as a percentage of the Nyquist frequency.
β= 1/2 => excess bandwidth = 50 %.
β= 1 => excess bandwidth = 100%.
The pulse x(t ), having the raised cosine spectrum, is
x(t ) is normalized so that x(0) = 1.
( ) ( ) ( )
( )
( )222
222
41
cos
sin
41
cossin
T t
T t
T t c
T t
T t
T t
T t t x
β
πβ
π
β
πβ
π
π
−=
−=
Design of Band-Limited Signals for
No ISI–
The Nyquist Criterion
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Pulses having a raised cosine spectrum:
Forβ= 0, the pulse reduces to x(t ) = sinc(πt/T ), and the
symbol rate 1/ T = 2W .
Whenβ= 1, the symbol rate is 1/ T = W .
Design of Band-Limited Signals for
No ISI–
The Nyquist Criterion
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If the receiver filter is matched of the transmitter filter, we have X rc( f )= GT ( f ) G R( f ) = | GT ( f )|
2. Ideally,
and G R( f ) = , where t 0 is some nominal delay that isrequired to ensure physical realizability of the filter.
Thus, the overall raised cosine spectral characteristic is splitevenly between the transmitting filter and the receiving filter.
An additional delay is necessary to ensure the physicalrealizability of the receiving filter.
( ) f GT
∗
( ) ( )02 ft j
rcT e f X f G π −=
Design of Band-Limited Signals for
No ISI–
The Nyquist Criterion
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Square Root Raised Cosine Filter
The cosine roll-off transfer function can be achieved by using
identical square root raised cosine filter at the
transmitter and receiver.The pulse SRRC (t ), having the square root raised cosine
spectrum, is
( )( ) ( )
2
sin 1 4 cos 1
1 4
where is the inverse of chip rate ( 0.2604167 s)
and = 0.22 for WCDMA.
C C C
C C
C
t t t T T T
SRRC t
t t
T T
T
π β β π β
π β
μ
β
⎛ ⎞ ⎛ ⎞− + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠=⎛ ⎞⎛ ⎞⎜ ⎟− ⎜ ⎟
⎜ ⎟⎝ ⎠⎝ ⎠≈
( )rc X f
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Design of Band-limited Signals withControlled ISI -- Partial-Response Signals
It is necessary to reduce the symbol rate 1/T below the Nyquist
rate of 2W symbols/s to realize practical transmitting and
receiving filters.Suppose we choose to relax the condition of zero ISI and, thus,
achieve a symbol transmission rate of 2W symbols/s.
By allowing for a controlled amount of ISI, we can achieve this
symbol rate.
The condition for zero ISI is x(nT )=0 for n≠0.
Suppose that we design the band-limited signal to have controlled
ISI at one time instant. This means that we allow one additionalnonzero value in the samples { x(nT )}.
f d l d l h
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Design of Band-limited Signals withControlled ISI -- Partial-Response Signals
One special case that leads to (approximately) physically
realizable transmitting and receiving filters is the duobinary signal
pulse:
Using Equation 9.2-20
When substituted into Equation 9.2-18, we obtain:
( ) ( )( )⎩
⎨⎧ ==
otherwise 0
1,0 1 nnT x
( )
( )⎩⎨⎧ =
=otherwise 0
1-,0 nT bn
( ) fT jTeT f B
π 2−+=
( )nT Txb
n−=
( ) 2 j nfT
n
n
B f b eπ
∞
=−∞
= ∑
D i f B d li i d i l i h
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Design of Band-limited Signals withControlled ISI -- Partial-Response Signals
It is impossible to satisfy the above equation for 1/ T >2W .
For T =1/2W , we obtain
Therefore, x(t ) is given by:
This pulse is called a duobinary signal pulse.
( ) ( ) ( )( )
( )( )⎪⎩
⎪⎨
⎧<=
⎪⎩
⎪⎨⎧ <+=
−
−
otherwise 0
2
cos1
otherwise 0
121
2 W f W f e
W
W f eW f X
W f j
W f j
π π
π
( ) ( )1
sinc 2 sinc 22
x t Wt Wt π π ⎡ ⎤⎛ ⎞
= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
D i f B d li it d Si l ith
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Design of Band-limited Signals withControlled ISI -- Partial-Response Signals
Time-domain and frequency-domain characteristics of a
duobinary signal.
Modified duobinary signal pulse:
( )( )( )
( )
1 -1
1 12
0 otherwise
nn
x x nT nW
=⎧⎪⎛ ⎞
= = − =⎨⎜ ⎟⎝ ⎠ ⎪
⎩
The spectrum decays
to zero smoothly.
--physically realizable
D i f B d li it d Si l ith
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Design of Band-limited Signals withControlled ISI -- Partial-Response Signals
The corresponding pulse x(t ) is given as
The spectrum is given by
( )( ) ( )
sinc sinct T t T
x t
T T
π π + −= −
( )( )
⎪⎩
⎪⎨
⎧
>
≤=−=
−
W f
W f W
f
W
jee
W f X
W f jW f j
0
sin2
1 π π π
Zero at f =0.
D si f B d li it d Si ls ith
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Design of Band-limited Signals withControlled ISI -- Partial-Response Signals
Other physically realizable filter characteristics are obtained by
selecting different values for the samples { x(n /2W )} and more
than two nonzero samples.
As we select more nonzero samples, the problem of unraveling
the controlled ISI becomes more cumbersome and impractical.
When controlled ISI is purposely introduced by selecting two or
more nonzero samples form the set { x(n /2W )}, the class of band-limited signal pulses are called partial-response signals:
( ) sinc 22 2n
n n x t x W t
W W π
∞
=−∞
⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎢ ⎥
⎝ ⎠ ⎝ ⎠⎣ ⎦∑
( )( )
( )⎪
⎩
⎪⎨
⎧
>
≤⎟ ⎠
⎞⎜⎝
⎛
=
−∞
−∞=∑
W f
W f eW
n x
W f X
W f jn
n
0
22
1 π
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Data Detection for Controlled ISI
Two methods for detecting the information symbols at the
receiver when the received signal contains controlled ISI:
Symbol-by-symbol detection method.
Relatively easy to implement.
Maximum-likelihood criterion for detecting a sequence of
symbols.
Minimizes the probability of error but is a little more complex toimplement.
The following treatment is based on PAM signals, but it is easily
generalized to QAM and PSK.
We assume that the desired spectral characteristic X ( f ) for the
partial-response signal is split evenly between the transmitting
and receiving filters, i.e., |GT ( f )|=|G R( f )|=| X ( f )|1/2.
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Data Detection for Controlled ISI
Symbol-by-symbol suboptimum detection
For duobinary signal pulse, x(nT )=1, for n=0,1, and is zero
otherwise.The samples at the output of the receiving filter (demodulator)
have the form
where { I m} is the transmitted sequence of amplitudes and {vm}
is a sequence of additive Gaussian noise samples.
Consider the binary case where I m=±1, Bm takes on one of
three possible values, namely, Bm =-2,0,2 with corresponding
probabilities ¼, ½, ¼.
If I m-1 is the detected symbol from the (m-1)th signaling
1m m m m m m y B v I I v−= + = + +
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Data Detection for Controlled ISI
Symbol-by-symbol suboptimum detection
interval, its effect on Bm, the received signal in the mth
signaling interval, can be eliminated by subtraction, thusallowing I m to be detected.
Major problem with this procedure is error propagation: if
I m-1 is in error, its effect on Bm is not eliminated but, in fact,
is reinforced by the incorrect subtraction.
Error propagation can be avoided by precoding the data.
The precoding is performed on the binary data sequence
prior to modulation.
From the data sequence { Dn}, the precoded sequence {Pn}
is given by: 2,...,1 , 1 == − mP DP mmm
Modulo-2 subtraction
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Data Detection for Controlled ISI
Symbol-by-symbol suboptimum detection
Set I m=-1 if Pm=0 and I m=1 if Pm=1, i.e., I m=2Pm-1.
The noise-free samples at the output of the receiving filterare given by
Since , it follows that the data sequence Dm
is obtained from Bm using the relation:
Consequently, if Bm=±2, then Dm=0, and if Bm=0, Dm=1.
( ) ( ) ( )1 1 12 1 2 1 2 1m m m m m m m B I I P P P P− − −= + = − + − = + −
121
1 +=+ − mmm BPP
1m m m D P P −= ⊕
( )2 mod 12
1+= mm B D
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Symbol-by-symbol suboptimum detection
Binary signaling with duobinary pulses
The extension from binary PAM to multilevel PAM signalingThe M -level amplitude sequence { I m} results in a noise-free
sequence
Data Detection for Controlled ISI
,...2,1 ,1 =+= − m I I B mmm
reference
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Data Detection for Controlled ISI
Symbol-by-symbol suboptimum detection
which has 2 M -1 possible equally spaced levels.
The amplitude levels are determined from the relation:
where {Pm} is the precoded sequence that is obtained from
an M -level data sequence { Dm} according to the relation
where the possible values of the sequence { Dm} are 0, 1,
2,…, M -1.In the absence of noise, the samples at the output is given
by:
( )12 −−= M P I mm
( ) M P DP mmm mod 1−=
( )[ ]12 11 −−+=+= −− M PP I I B mmmmm
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Data Detection for Controlled ISI
Symbol-by-symbol suboptimum detection
Hence
Since Dm=Pm+Pm-1 (mod M ), it follows that
Four-level signal transmission with duobinary pulses
( )12
11 −+=+ − M BPP mmm
( ) ( ) M M B D mm mod 12
1−+=
(M=4)
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Data Detection for Controlled ISI
Symbol-by-symbol suboptimum detection
In the case of the modified duobinary pulse, the controlled ISI
is specified by the values x(n /2W )=-1, for n=1, x(n /2W )=1, forn=-1, and zero otherwise.
The noise-free sampled output from the receiving filter is
given as:
Where the M -level sequence { I m} is obtained by mapping a
precoded sequence according to
and
2−−= mmm I I B
( ) M P DP mmm mod 2−⊕=
( )12 −−= M P I mm
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Data Detection for Controlled ISI
Symbol-by-symbol suboptimum detection
From these relations, it is easy to show that the detection
rule for recovering the data sequence { Dm} from { Bm} inthe absence of noise is
The precoding of the data at the transmitter makes it possibleto detect the received data on a symbol-by-symbol basis
without having to look back at previously detected symbols.
Thus, error propagation is avoided.
The symbol-by-symbol detection rule is not the optimum
detection scheme for partial-response signals. Nevertheless, it
is relatively simple to implement.
( ) M B D mm mod 2
1=
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Data Detection for Controlled ISI
Maximum-likelihood Sequence Detection
Partial-response waveforms are signal waveforms with
memory. This memory is conveniently represented by a trellis.The trellis for the duobinary partial-response signal for binary
data transmission is illustrated in the following figure.
The first number on the left is the new data bit and the number on the
right is the received signal level.
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Data Detection for Controlled ISI
Maximum-likelihood Sequence Detection
The duobinary signal has a memory of length L=1. In general,
for M -ary modulation, the number of trellis states is M L
.The optimum maximum-likelihood sequence detector selects
the most probable path through the trellis upon observing the
received data sequence { ym} at the sampling instants t =mT ,
m=1,2,….
The trellis search is performed by the Viterbi algorithm.
For the class of partial-response signals, the received sequence
{ ym,1≤m≤ N } is generally described statistically by the jointPDF p(y N |I N ), where y N =[ y1 y2 … y N ]’ and I N =[ I 1 I 2 … I N ]’
and N > L.
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Data Detection for Controlled ISI
Maximum-likelihood Sequence Detection
When the additive noise is zero-mean Gaussian, p(y N |I N ) is a
multivariate Gaussian PDF, i.e.,
where B N =[ B1 B2 … B N ]' is the mean of the vector y N and Cis the N X N covariance matrix of y N .
The ML sequence detector selects the sequence through the
trellis that maximizes the PDF p(y N |I N ).
Taking the natural logarithms of p(y N |I N ):
( )( )
( ) ( )1
2
1 1| exp
22 det N N N N N N N
pπ
−′⎡ ⎤= − − −⎢ ⎥⎣ ⎦
y I y B C y BC
( ) ( ) ( ) ( ) N N N N N N N p ByCByCIy −′
−−−= −1
2
1det2ln
2
1|ln π
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Data Detection for Controlled ISI
Maximum-likelihood Sequence Detection
Given the received sequence { ym}, the data sequence { I m} that
maximizes ln p(y N |I N ) is identical to the sequence { I N } thatminimizes (y N -B N )'C-1(y N -B N ), i.e.,
The metric computations in the trellis search are complicated
by the correlation of the noise samples at the output of the
matched filter for the partial-response signal.
In the case of the duobinary signal waveform, the correlation
of the noise sequence {vm} is over two successive signal
samples.
( ) ( )⎥⎦⎤
⎢⎣⎡ −′−= −
N N N N N
N
ByCByII
1minargˆ
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Data Detection for Controlled ISI
Maximum-likelihood Sequence Detection
Hence, vm and vm+k are correlated for k =1 and uncorrelated for
k >1.If we isgnore the noise correlation by assuming that
E (vmvm+k )=0 for k >0, the computation can be simplified to
where
and xk = x(kT ) are the sampled values of the partial-response
signal waveform.
( ) ( )
2
1 0
ˆ arg min arg min N N
N L
N N N N N m k m k
m k
y x I −= =
⎡ ⎤⎛ ⎞′⎡ ⎤= − − = −⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎢ ⎥⎣ ⎦∑ ∑I I
I y B y B
∑=
−= L
k
k mk m I x B0
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52
Signal Design for Channels with Distortion
In this section, we perform the signal design under the condition
that the channel distorts the transmitted signal.
We assume that the channel frequency-response C ( f ) is know for
| f |≤W and that C ( f )=0 for | f |>W .
The filter responses GT ( f ) and G R( f ) may be selected to
minimize the error probability at the detector.
The additive channel noise is assumed to be Gaussian with powerspectral densityΦnn( f ).
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Signal Design for Channels with Distortion
For the signal component at the output of the demodulator, we
must satisfy the condition
where X d ( f ) is the desired frequency response of the casecade of
the modulator, channel, and demodulator, and t 0 is a time delay
that is necessary to ensure the physical realizability of themodulation and demodulation filter.
The desired frequency response X d ( f ) may be selected to yield
either zero ISI or controlled ISI at the sampling instants.
We shall consider the case of zero ISI by selecting X d ( f ) =
X rc( f ), where X rc( f ) is the raised cosine spectrum with an
arbitrary roll-off factor.
( ) ( ) ( ) ( )02
,
j ft
T R d G f C f G f X f e f W
π −= ≤
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Signal Design for Channels with Distortion
The noise at the output of the demodulation filter may be
expressed as
where n(t ) is the input to the filter.
Since n(t ) is zero-mean Gaussian, v(t ) is zero-mean Gaussian,
with a power spectral density
For simplicity, we consider binary PAM transmission. Then, the
sampled output of the matched filter is
where x0 is normalized to unity, I m=±d , and vm represents the
noise term.
( ) ( ) ( ) Rv t n t g d τ τ τ ∞
−∞= −∫
( ) ( ) ( )2
vv nn R f f G f Φ = Φ
0m m m m m y x I v I v= + = +
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Signal Design for Channels with Distortion
vm is zero-mean Gaussian with variance
The error probability is given by
The probability of error is minimized by maximizing the ratiod 2 / σ2
v .
There are two possible solutions for the case in which the
additive Gaussian noise is white so thatΦnn( f )= N 0 /2.1st solution: pre-compensate for the total channel distortion at the
transmitter, so that the filter at the receiver is matched to the
received signal.
( ) ( )22
v nn R f G f df σ ∞
−∞= Φ∫
22
/ 2
2 2 /
1
2 v
y
d v
d P e dy Q
σ σ π
∞−
⎛ ⎞= = ⎜ ⎟
⎜ ⎟⎝ ⎠
∫
SNR
l D f Ch l h D
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Signal Design for Channels with Distortion
The transmitter and receiver filters have the magnitude
characteristics
The phase characteristic of the channel frequency response
C ( f ) may also be compensated at the transmitter filter.
For these filter characteristics, the average transmitted power
is
( ) ( )( )
( ) ( )
,
,
rcT
R rc
X f G f f W
C f
G f X f f W
= ≤
= ≤
( )( ) ( )
( )
( )
2 2 222
2
W W W m rc
av T T W W W
E I X f d d P g t dt G f df df
T T T C f − − −
= = =∫ ∫ ∫
(9.2-71)
Si l D i f Ch l ith Di t ti
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Signal Design for Channels with Distortion
Hence,
The noise at the output of the receiver filter isσ2v= N 0 /2 and,
hence, the SNR at the detector is
( )
( )
1
2
2
W rc
avW
X f d P T df
C f
−
−
⎡ ⎤⎢ ⎥=⎢ ⎥
⎣ ⎦
∫
( )( )
1
2
22
0
2 W rcav
W v
X f P T d df N C f σ
−
−⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦∫
(9.2-73)
(9.2-74)
Si l D i f Ch l ith Di t ti
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Signal Design for Channels with Distortion
2nd solution: As an alternative, suppose we split the channel
compensation equally between the transmitter and receiver filters,
i.e.,
The phase characteristic of C ( f ) may also be split equally
between the transmitter and receiver filter.
The average transmitter power is
( ) ( )( )
( )
( )
( )
1/ 2
1/ 2
,
,
rcT
rc
R
X f G f f W
C f
X f
G f f W C f
= ≤
= ≤
( )
( )
2W
rc
avW
X f d P df
T C f −= ∫
(9.2-75)
Si l D i f Ch l ith Di t ti
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Signal Design for Channels with Distortion
The noise variance at the output of the receiver filter is
The SNR at the detector is
From Equations 9.2-73 and 9.2-78, we observe that when we
express the SNR d 2 / σ2
v
in terms of the average transmitter power
Pav, there is a loss incurred due to channel distortion.
( )
( )
2 0
2
W rc
vW
X f N df
C f
σ −
= ∫
( )
( )
22
2
0
2 W rcav
W v
X f P T d df
N C f σ
−
−
⎡ ⎤= ⎢ ⎥
⎢ ⎥⎣ ⎦∫ (9.2-78)
Si n l D si n f Ch nn ls ith Dist ti n
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Signal Design for Channels with Distortion
In the case of the filters given by Equation 9.2-71, the loss is
In the case of the filters given by Equation 9.2-75, the loss is
When C ( f )=1 for | f |≤W , the channel is ideal and
so that no loss is incurred.
When there is amplitude distortion, |C ( f )|<1 for some range of frequencies in the band | f |≤W and there is a loss in SNR.
It can be shown that the filters given by 9.2-75 result in thesmaller SNR loss.
( )
( )
210log
W rc
W
X f df
C f −∫
( )
( )
2
210log
W rc
W
X f df
C f −
⎡ ⎤⎢ ⎥
⎢ ⎥⎣ ⎦
∫
( ) 1W
rcW
X f df −
=∫
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