f !
BASEBAND PULSETRANSMISSION
448
of the pulse-amplitude modulator and the decisionomaking device in an M-ary PAM are more complex than those in a binary PAM system. Intersymbol intert ference, noise, and imperfect synchronization cause errors to appear at the receive output. The transmit and receive filters are designed to minimize these errors. Procedures used for the design of these filters are similar to those discussed in Sections 7.4 and 7.5 for baseband binary PAM systems.
7.8
TAPPED-DELAY-LINE EQUALIZATION
.
A communication channel that readily comes to mind for the transmission of digital data (e.g., computer data) is a telephone channel, which is characterized by a high signal-to-noise ratio. However, the telephone channel is bandwidth-limited, as illustrated in Fig. 7.21 for a typical toll connection. Figure 7.21a shows the insertion loss of the channel plotted versus frequency; insertion loss (in dB) is def!ned as 10 10giO (Pol P2), where P2 is the power delivered to a load by the channel, and Po is the power delivered to the same load when it is connected directly to the source (i.e., the channel is removed). Figure 7.21 b shows the corresponding plots of the phase response and envelope (group) delay versus frequency; for the definition of envelope delay, see Section' 2.14. Figure 7.21 clearly illustrates the dispersive nature of the telephone channel. ~ efficient approach to high-speed transmission of digital data over such a channel uses a
']). . tol -\-t, 10'"
~~-:t\~'hcw combination of two basic signal-processIng 1..t ~iscrete PAM, which involves encoding
operatlOps: the amplitudes of successive pulses in a periodic pulse train with a discrete set of possible amplitude levels. /A linear modulation scheme, which offers bandwidth conservation to transmit the encoded pulse tra~er the telephone channel. At the receiving end of the system, the received signal is demodulated and synchronously sampled, and then decisions are made as to which particular symbols were transmitted. As a result of dispersion of the pulse shape by the telephone channel, we find that the number of detectable amplitude levels is often Ii . by intersymbol interference rat er t an by additive noise. In principle, if the channel is known precisely, it is virtually always possible to make the intersymbol interference at the sampling instants arbitrarily small by using a suitable pair of transmit and receive filters, so as to control the overall pulse shape in the manner
~
~
~ described previously. The transmit filter is placed directly before the modulator, .1.~X 11 ( whereas the receive filter is placed dIrectly after the demodulator. Thus, insofar ~ k "",,,, ek ".,...J as intersymbol interference is concerned, we may consider the data transmission ~\ U",k."-,,, Ch"'.'-L\ over the telephone channel as being baseband. ~Q..C-k.,.;.'-\.;l~ In practice, however, we seldom have prior knowledge of the exact channel Ov;t"ftkl>~""
characteristics. Also, there is the unavoidable problem of imprecision that arises in the physical implementation of the transmit and receive filters. The net result of all these effects is that there will be some residual distortion for ISI to be a limiting factor on the data rate of the system. To compensate for the intrinsic residual distortion, ~e may use a process known as equalization. The filter used to perform such a process is called an equalizer, A device well-suited for the design of a linear equalizer is the J:.a~E.e2-<:!~L line filter, as depicted in Fig. 7.22. For symmetry, the total nunfb'(;rv of taps IS ('./\/V"Vvv
tII
~
7.8
TAPPED-DELAY-LINE
449
EQUALIZATION
20
15
cc :3'"'" 0
c:
10
.g Q; '" E
5
2
4
3
5
Frequency (kHz) (a)
10
4
3
..-'" ..-..-
Envelope delay
,..-"-\.
Ideal
I>
linear phase
'"
Q;
"Q.
"0
I
2
T 0
0 Q; > c w
en " ""13 >
.t:
:.c:
'" "'" '" .J:: c..
-5
10 3
2
4
5
Frequency (kHz) (b)
Figure 7.21 (a) Amplitude response of typical toll connection. (b) Envelope delay and phase response of typical toll connection. (Bellamy, 1982.)
chosen to be (2N + 1), with the weights denoted by W-N>"" W_-!, WO,W!,..., WN' The impulse response of the tapped-delay-line equalizer is therefore N k(t)
=
2:
h= -N
Wh
8(t
-
kT)
(7.85)
where 8(t) is the Dirac delta function, and the delay T is chosen equal to the symbol duration. Suppose that the tapped-delay-line-equalizer is connected in cascade with a linear system whose impulse response is c(t), as depicted in Fig. 7.23. Let pet)
III
110
...: 0)
~ ;;:: 0) c >. .!!!
H
0) "0 h' .:, " 0) ';;::, 0. 0. nJ I-
N "'! r0O) ... ::I CI u::
450
...
, 7.8
TAPPED-DELAY-LiNE EQUALIZATION
451
Linear
T apped-delay-I ine equalizer, h(t)
system, c(l)
Impulse response, p(t)
Figure 7.23 Cascade connection of linear system and tapped-delay-line equalizer.
system. Then P(t) is equal to the
denote the impulse response of the equalized convolution of c(t) and h(t), as shown by
* h(t) * k=L-N
P(t) = c(t)
N
= c(t) Interchanging
the order of summation
Wk
8 (t
-
kT)
and convolution:
N
P(t)
=
L
WkC(t)
N
.
k= -N
L
* 8(t
kT) (7.86)
WkC(t -
k= -N
-
kT)
where we have made use of the sifting property of the delta function. Evaluating Eq. (7.86) at the sampling times t = nT, we get the discrete convolution sum N
p(nT)
=
L k= -N
wk c((n -
(7.87)
k)T)
Note that the sequence {p(nT)} is longer than {c(nT)}. To eliminate intersymbol interference completely, we must satisfy the Nyquist criterion for distortionless transmission described in Eq. (7.49), with T used in place of Tb' It is assumed that the P( t) is defined in such a way that the normalized condition P(O) = 1 is satisfied in accordance with Eq. (7.46). Thus, for no intersymbol interference we require that
n=°
I, p(nT)
=
{ 0,
n=/=O
But from Eq. (7.87) we note that there are only (2N + I) adjustable coefficients at our disposal. Hence, this ideal condition can only be satisfied approximately as follows:
p(nT)
=
I, {.°,
n =
°
n = ::tl, :t2,'...,
:tN
(7.88) I
I I
L
-,--
= ..,,-
..
BASEBAND
452
To simplify the notation, wri tten as
PULSE TRANSMISSION
we let the nth sample of the impulse
Cn
=
response
(7.89)
c(nT)
Then, imposing the condition of Eq. (7.88) on the discrete convolution Eq. (7.87), we obtain a set of (2N + 1) simultaneous equations:
f
k= -N
Equivalently,
I, Wk Cn-k
=
n
c(t) be
sum of
0 (7.90)
= { 0,
n = ::!:1, j:2, . . ., ::!:N
in matrix form we may write (Q
CN-I
CN CN+I
...
C-N+I
C-N
. ..
(Q
...
CI
(Q
. ..
l".z
Cl
G.2N ...
C-l
CN+l
CN
C-N-I
C-2 C-I
(Q
CN-I
. .. ...
. .. ...
...
C-2N
C-N-I
W-N
0
W-I
0
=
1
C-N
Wo
C-N+I
WI
0
WN
0
(Q
I
(7.91)
A tapped-delay-line equalizer described by Eq. (7.90) or, equivalently Eq. <7.9.1), is referred to as a zf;:j1:.!j!!}hl1r.§!lJ!::~ Such an equalizer is optimum in the sense that it minimizes the peak distortion (intersymbol interference). It also has the nice feature of being relatively simple to implement. In theory, the longer we make the equalizer (i.e., permit N to approach infinity), the more closely will the equalized system approach the ideal condition specified by the Nyquist criterion for distortlonless transmission.
7.9
ADAPTIVE EQUALIZATION The zero-forcing strategy described abrnce works well in the 1aboratory, where we have access to the system to be equalized, in which case we know the system coefficients L N' . . . , L l' (Q, Cl' . . . , CN that are needed for the solution of Eq. (7.91). In a telecommunications environment, however, the channel is usually time varying. For example, in a switched telephone network, we find that two factors contribute to the distribution of pulse distortion on different link
/.I ~
connections: ~'lDifferences in the transmission characteristics of the individual be switched together. .../ Differences in the number of links in a connection.
links that ,may
The result is that the telephone channel is random in the sense of being one of van ensemble of possible physical realizations. Consequently, the use of a fixed equalizer designed on the basis of average channel characteristics may not ade-
...
7.9
-
!IF.!:
Jm
453
ADAPTIVE EQUALIZATION
quately reduce
.
intersyrnbol
interference.
To realize the full transmission
capa-
bility of a telephone channel, there is need for ~n. 5 The process ~L of equalization is said to be adaptivp when tlie equalizer adjusts itself continu=r-
equalization
zation can be achieved,
at the receiving
prior to data transmission,
guidance of a suitable ~ce
end of the system. This~iby training
~
the filter with the
""
R 7w:.~
transmitted through the channel so as to r
adjust the filter parameters to °E!imum values. The typical telephone channel changes little during an average data call, so that precall equalization with a training sequence is sufficient in most cases encountered in practice. As mentioned previously, the equalizer is positioned after the receive filter in the receiver. In this section we study an adaptive equalizer based on the tapped-delay-line filter, which is synchronous in the sense that the tap spacing of the equalizer is the same as the symbol duration T of the transmitted signal (i.e., the reciprocal of the signaling rate). This equalizer is not only simple to implement but is also capable of realizing a satisfactory performance.
Least-Mean-Square Algorithm Consider a tapped-delay-line equalizer, whose tap-weights are adjustable as indicated in Fig. 7.24. The input sequence Ix( nT) I applied to this equalizer is produced by the transmission of a binary sequence through an unknown channel that is both dispersive and noisy. I.t is assumed that some form of pulse shaping is included in the design of the transmission system. The requirement is to co~rect for the combined effec!S- oLresidual dis.!Qrtion and noise in the system t.hrough the use of an ada2tive equalizer. To simplifY notational matters, we let
it, oft)
-
C\tYIS~''''c\;CM
Xn = x(nT}
(7.92)
Yn = y(nT}
(7.93)
Then, the output Yn of the tapped-delay-line equalizer in response to the input sequence IXnl is defined by the discrete convolution sum (see Fig. 7.24) N Yn
=
2:
k= -N
(7.94)
WkXn-k
where Wk is the weight at the kth tap, a!1d 2N + 1 is the t~t~l nu_mber of t!lps. The tap-weights constitute the adaptive filter coefficients. We assume that the input sequence IXnl pas finite ene!lO'. The adaptation may be achieved by observing the error between the desired pulse shape and the actual pulse shape at the filter output, measured at the sampling instants, and then using this error to estimate the direction in which the tap-weights of the filter should be changed so as to approach an optimum set of values.
Eor
the adaptation,
ipizes the peak distortion,
defined
'!!...e I1!..a'y~e
a pea!!:...!!isjortion criterion that
as the worst-case intersymbol
I!!ini-
interference
at
II
IIIJ!I!!I
7.9
455
ADAPTIVE EQUALIZATION
(q..",.;~
"",,~~~t
.the output of the equalize!:;. The development of an adaptive equalizer using such a cnterion builds on the zero-forcing cp(lcept described in the preceding section. However, the equalizer is optimum only when the peak distortion at its input is less than 100 percent (i.e., the intersymbol interference is not too se-
II
vere). ~~~~!)S ~e may use a mean-square error criterion, which is more generafin application; also an adaptive equalizer based on the mean-square error criterion appears to be less sensitive to timing perturbationsJhan one based on th.e peak distortion criterion. Accordingly, in what follows we will use the meansquare error criterion for the development of the adaptive equalizer. Let an denote the desired response defined as the polar representation of the nth transmitted binary symbol. Let en denote the error signal defined as the difference between the desired response an and the actual response Yn of the equalIzer, as shown by en
=
an -
(7.95)
Yn
Then, a criterion commonly used in practice (because of its mathematical ability) is the mean-square error, defined by the cost function
M~£
tract-
(7.96)
'i8 = E[e;]
(J,\\Qsv\1ffl
~
where E is the statistical expectation operator. Using Eqs. (7.94) to (7.96), the gradient of the mean-square error 'i8with respect to the kth tap-weight Wkmay be expressed as a'i8
aen
2 E [ enaw~]
aWk
aYn = -2 E [ enaWk]
(7.97)
= - 2 E[enxn-k] The expectation on the right-hand side of Eq. (7.97) is the ensemble-averaged cross-correlation between the error signal en and the input signal Xn for a lag of k samples; that is,
.
Rex(k) = E[enxn-k]
(7.98)
We may thus simplifY Eq. (7.97) to
I I
a'i8 aWk
The optimality presse.d simply as
./
condition
a'i8= 0
aWk
2 Rex(k)
for minimum
mean-square
for k = 0, ::!::1,...,
(7.99) error may now be. ex-
::!::N
(7.100)
.
~
,
.
~
.
-
;- -. -
- .. '.
0lIl' '-.'.
-
.:;;;;;;;jiI
BASEBAND PULSE TRANSMISSION
456
In light ofEq.
(7.99), this condition Rex(k)
°
=
is equivaJent
for k = 0, :t
to the requirement
1,...,
-tN
that
(7.101)
That is, for minimum
mean-square error, the cross-correlation between the output error c~mpo;'ents with integer lags corresponding -to the Tndix values of the available tap-weights of thejilter. This important result is known as the principle of orthogonality. Substituting Eqs. (7.94) and (7.95) in (7.96) and expanding terms, we find that the mean-square error <&is precisely a second-order function of the tapweights W-N' . . . , Ui-I' WO,WI'. . . , WN'The mean-sq\,lareerror performanceof the equalizer may therefore be visualized as a multidimensional bowl-shared surface that is a parabolic function of the tap-weights. The adaptive process, through successive adjustments of the tap-weights, has the task of continually seeking t.he bottom of the bowl; at this unique point, the mean-square error <&attains its mtnimum value <&min'It is therefore intuitively reasonable that successive adjustments to the tap-weights be in the direction of steepest descent of the error surface (i.e., in a direction opposite to the vector of gradients aW,/aWk' - N:so k .:5N), which should lead to the minimum mean-square error <&min' This is the basic idea of the steepestdescentalgoritlJ!!t,described by the recursive fortllilla
sequence{enI and the znput sequence{xnl must have zerosfor the (2N +1)
.
~
1
a<&
wk(n + 1) = wk(n) - '2J-LaWk' I
"
where J-Lis a small positive constant
k = 0, :t "1,. . ., :t N
(7.102)
called the step-size parameter, and the factor
1/2 has been introduced to cancel the factor 2 in the defining equation for a'f: faWk' The index n is the iteration number. Thus the use of Eq. (7.99) in (7.102) yields Wk(n + 1)
=
wk(n) + J-LRex(k),
k = O,:t
1,...,:tN
(7.103)
The use of the steepest-descent algorithm requires knowledge of the crosscorrelation function Rex (k). However, this knowledge is not available when operating in an unknown environment. We may overcome this difficulty by using an instantaneous estimate for the cross-correlation function Rex (k). Specifically, on the basis of the defining Rex
(k)
equation
=
(7.98), we may use the following estimate:
enxn-, k'
k = O,:t
1,...,:tN
(7.104)
In a corresponding fashion, we use the estimate Wk(n) in place of the tap-weight wk(n). Naturally, the use of these estimates in Eq. (7.103) results in an approximation to the steepest-descent algorithm. We may express the new recursive formula for updating the tap-weights of the equalizer as follows:
Wk(n + 1)
=
wk(n) + J-Lenxn-k'
k = 0, :t 1, , . . , :t N
(7.105)
This algorithm is known as the least-mean-square (LMS) algorithm.6 Viewing n as index for the previous IteratIon, wk(n) is the "old value" ot the kth tap-weight. and J-Lenxn-k is the "correction" applied to it to compute the "updated value" uJk(n + 1).
.1
-- -
t II
7.9
457
ADAPTIVE EQUALIZATION
The LMS algorithm is an example of a feedback system, as illustrated in the block diagram of Fig. 7.25. It is therefore possible for the algorithm to diverge (i.e., for the adaptive equalizer to become unstable). Unfortunately, the convergence behavior of the LMS algorithm is difficult to analyze. Nevertheless, proevided that the step-size parameter JLis assigned a small value, we find that after alarge number of iterations the benavlOr ot the LMS algorithm is roughly similar to that of the steepest-descent algorithm, which uses the actual gradient rather than a noisy estimate for the computation of the tap-weights. We may simplify the formulation of the LMS algorithm using matrix notation. Let the (2N + 1)-by-l vector Xn denote the tap-inputs of the equalizer: Xn =
[Xn+N"'"
xn+I'
Xn, Xn-I"'"
Xn-N]T
where the superscript T denotes matrix transposition. Correspondingly, (2N + 1)-by-l vector wn denote the tap-weights of the equalizer: Wn
= [w_~n),...,
w-I(n), wo(n), wI(n),...,
wN(n)]T
(7.106) let the
(7.107)
We may then use matrix"notation to recast the convolution sum ofEq. (7.94) in the compact form Yn -
T.A XnWn
(7.108)
where x~ wn is referred to as the inner product of the vectors xn and wn. We may now summarize the LMS algorithm as follows: the algorithm
by setting
=
1.
Initialize
2.
equalizer to zero at n = 1, which corresponds to time t = T). For n = 1, 2, . . . , compute
WI
0 (i.e., set all the tap-weights
T.A Yn - xnwn en
=
an - Yn
= wn + JLe~n
Wn+1
where JL is the step-size parameter.
3.
Continue
the computation
until steady-state
conditions
Correction lIen Xn - k
+ Old value (;,k(n)
+
L
Updated value (;,k(n + 1)
Unit delay T
Figure7.25 Signal-flow graph representation of the LMSalgorithm.
are reached.
of the
f
so
iiiI
458
BASEBAND
JI
PULSE TRANSMISSION
Operation of the Equalizer There are two modes of operation for an adaptive equalizer, namely, the training mode and decision-directed mode, as shown in Fig. 7.26. During the training mode, a known sequence is transmitted and a synchronized version of:this signal is generated in the receiver, where (after a time shift equal to the transmission delay) it is applied to the adaptive equalizer as the desired response; the tapweights of the equalizer are thereby adjusted in accordance with the LMS algorithm. A training sequence commonly used in practice is the so-called pS!3ldonoise (PN) sequence, which consists of a deterministic sequence with noiselike ~haractenstlcs; a tull discussion of this sequence is presented in Chapter 9. When the training process is completed, the adaptive equalizer is switched to its second mode of operation: the decision-directedmode. In this mode of operation, the error signal is defined by ~ en
(7.109)
= an - Yn
where Yn is the equalizer output at time t = nT, and an is the final (not necessarily) correct estimate of the transmitted symbol an, Now, in normal oper.ation the decisions m;tde by the receiver are correct with high probability. This means that the .error estimates are correct most of the time, thereby permitting the adaptive equalizer to operate satisfactorily. Furthermore, an adaptive equalizer operating in a decision-directed rr..ode is able to track relatively slow variations in channel characteristics.
.~
~t
It turns out that the larger the step-size parameter JL, the faster the tracking capability of the adaptive equalizer. However, a large step-size parameter f.Lmay result in an unacceptably high excessmean-square error,defined as that part ofthe mean-square value of the error signal in excess of the minimum attainable value ~min (which results when the tap-weights are at their optimum settings). We therefore find that in practice the choice of a suitable value for the step-size parameter JLinvolves making a compromise between fast tracking and reducing the excess mean-square error.
Implementation An important The methods
Approaches
advantage of the LMS algorithm is that it is simple to implement. of implementing adaptive equalizers may be divided into three
Decision device
Adaptive equalizer
A Xn
I
,
Yn {wk)
1
)1
S
an 1
an
Training seq uence generator
en
L Figure 7.26
+
Illustrating the two modes of operation of an adaptive equalizer.
r I 7.9
459
ADAPTIVE EQUALIZATION
broad categories: analog, hardwired digital, and programmabledigital, as described here: .
1.
The analog approach is primarily based on the use of charge-coupleddevice (CCD) technology. The basic circuit realization 'of the CCD is a row of fieldeffect transistors with drains and sources connected in series, and the drains capacitively coupled to g<,\tes.The set of adjustable tap-weights are stored in digital memory locations, and the multiplications of the analog sample values by the digitized tap-weights take place in analog fashion. This approach has significant potential in applications where the symbol rate is too high for digital implementation. 2. In hardwired digital implementation of an adaptive equalizer, the equalizer input is first sampled and then quantized into a form suitable for storage in shift registers. The set of adjustable tap-weights is also stored in shift registers. Logic circuits are used to perform the required digital arithmetic (e.g., multiply and accumulate). In this second approach, the circuitry is hardwired for the sole purpose of performing equalization. It is the most widely used method of building adaptive equalizers and lends itself to implementation in very-large-scale integrated (VLSI) circuit form. 3. The use of a programmable digital processor in the form of a microprocessor, for example, offers flexibility in that the adaptive equalization is performed as a series of steps or instructions in the microprocessor. An important advantage of this approach is that the same hardware may be time shared to perform a multiplicity of signal-processing functions such as filtering, modulation, and demodulation in a modem (modulator-demodulator) used to transmit digital data over a telephone channel.
Decision-Feedback Equalization To develop further insight into adaptive equalization, consider a baseband channel with impulse response denoted in its sampled form by the sequence {hnl where hn = h(nT). The response of this channel to an input sequence {xnl, in the absence of noise, is given by the discrete convolution sum Yn
= 2:k -
hk Xn-k
!toxn
(7.11 0)
+ k
hkxn-
k
+ k>O 2:
hkxn-
k
The first term of Eq. (7.110) represents the desired data symbol. The second term is due to the precursors of the channel impulse response that occur before the main sample !to associated with the desired data symbol. The third term is due to the postcursors of the channel impulse response that occur after the main sample !to. The 'precursors and postcursors of a channel impulse response are illustrated in Fig. 7.27. The idea of decisionjeedback equalization7 is to use data decisions made on the basis of precursors of the channel impulse response to take care of the postcursors; for the idea to work, however, the decisi9ns would obviously have to be correct. Provided that this condition is satisfier., a decision-
, I
f i
1!1,
~11
~
-
-"
.;
BASEBAND PULSETRANSMISSION
460 hO
0
Postcursors
Precursors
Impulse response of a discrete channel.
Figure 7.27
feedback equalizer is able to provide an improvement over the performance of the tapped-delay-line equalizer. A decisionjeedback equalizer consists of a feedforward section, a feedback section, and a decision device connected together as shown in Fig. 7.28. The feedforward section consists of a tapped-delay-line filter whose taps are spaced at the reciprocal of the signaling rate. The data sequence to be equalized is applied to this section. The feedback section consists of another tapped-delay-line filter whose taps are also spaced at the reciprocal of the signaling rate. The input applied to the feedback section consists of the decisions made on previously detected symbols of the input sequence. The function of the feedback section is to subtract out that portion of the intersymbol interference produced by previously detected symbols from the estimates of future samples. Note that the inclusion of the decision device in the feedback loop makes
I
the equalizer intrinsically nonlinear and therefore more difficult to analyze than an ordinary tapped-delay-line equalizer. Nevertheless, the mean-square error criterion can be used to obtain a mathematically tractable optimization of a decision-feedback equalizer. Indeed, the LMS algorithm can be used to jointly adapt both the feedforward tap-weights and the feedback tap-weights based on a common error signal. To be specific, let the augmented vector cn denote the combination of the feedforward and feedback tap-weights, as shown by w(l) Cn
--
(7.111)
n
[ w(2)n ]
A I
,
Xn
Feedforward section, A (1) Wn
Decision
an
device
Feedback
I
section, A (2)
W"
I I: I I
, I
I
L
Figure 7.28
Block diagram of decision-feedback equalizer.
. 7.10
"'"
-)
461
EYE PATTERN
where the vector w~l) denotes the tap-weights of the feedforward section, and w~2)denotes the tap-weights of the feedback section. Let the augmented vector
vn denote the combination of input samples for both sections:
Vn
=
1
I
I (7.112)
[::]
1
I
where Xn is the vector of tap-inputs in the feedforward section, and an is the vector of tap-inputs (i.e., present and past decisions) in the feedback section. The common error signal is defined by en
=
an -
I I
(7.113)
c~v n
where the superscript T denotes matrix transposition, and an is the polar representation of the nth transmitted binary symbol. The LMS algorithm for the decision-feedback equalizer is described by the update equations:
w~lll w~2]. 1
= w~l) = w~2)
+ J.Llenxn
(7.114)
+ J.L2eiln
(7.115)
where J.Lland J.L2are the step-size parameters for the feedforward and feedback sections, respectively. A decision-feedback equalizer yields good performance in the presence of moderate to severe intersymbol interference as experienced in a fading radio
~
for example.
"
"
~
.
7.10 EYE PATTERN In previous sections of this chapter we have discussed various techniques for dealing with the effects of receiver noise and intersymbol interference on the performance of a baseband pulse-transmission system. In the final analysis, what really matters is how to evaluate the combined effect of these impairments on overall system performance in an operational environment.~n experim~tal tool for such an evaluation in an insightful manner is the so':called eye pattern, which is defined as the synchronized superposition of all ossible realizations of t e signa of interest (e.g., received signal, receiver output) viewed within a partKular signaling interval. The eye pattern derives its name from the fact that it resemoles the human eye fur binary waves. The interior region of the eye pattern is called the eye opening. An eye pattern provides a great deal of useful information about the performance of a data transmission system, as described in Fig. 7.29. Specifically, we may make the following statements:
.
II
r.
,J
II, I
The width of the eye opening defines the time interval over which the received signal can be sampled without errorfrom intersymbol interference; it is apparent that the preferred time for sampling is the instant of time at which the eye is open
the widest.
'
-
--
-------
"---I
