An adaptive equalizer for data signals is allowed to start up and train in a wholly decision-directed mode but is precluded from converging to an incorrect state by inhibiting equalizer updating for equalizer output values which fall within a predetermined null zone.

BACKGROUND OF THE INVENTION 
The invention relates to adaptive equalizers such as are used in voiceband 
data sets and other data transmission applications. 
Adaptive equalizers for data transmission are typically started up in a 
so-called ideal reference mode in which a stream of predetermined "ideal 
reference" values is transmitted to the equalizer over the channel being 
equalized. The ideal reference data is known a priori at the equalizer and 
the differences between the equalizer output values, on the one hand, and 
the known transmitted values, on the other hand, are used by the equalizer 
as error signals to update its tap coefficients. The latter define the 
equalizer transfer characteristic, hereinafter also referred to as the 
equalizer "state". The updating algorithm updates the tap coefficients in 
such a way as to minimize some function of the error signal--typically its 
mean-squared value over time. When the equalizer has "converged" to a 
point at which the mean-squared error is at an absolute, or global, 
minimum, the equalizer output constellation, i.e., the ensemble of 
possible equalizer output values, will be substantially congruent with the 
transmitted constellation, i.e., the ensemble of possible transmitted data 
symbol values. The channel is then said to be "equalized." 
Thereafter, the equalizer operates in response to so-called 
decision-directed errors in which quantized versions of the equalizer 
outputs are used in the place of the ideal reference data. The tap 
coefficients, and thus the equalizer state, are thus continually adapted 
over time as equalizer operation continues. Advantageously, this allows 
the equalizer to continually fine-tune its transfer characteristic and 
thereby compensate, for example, for time-varying effects, such as changes 
in the communication channel characteristics. 
Phenomena such as phase hits and channel switching can result in a 
subsequent loss of equalization, meaning that the tap coefficients then 
stored in the equalizer no longer equalize the channel. A new set of tap 
coefficients which will equalize the channel must then somehow be arrived 
at, that process being referred to as "re-training." Depending on the 
level of distortion in the channel, it may be possible to continue to 
allow the equalizer to simply continue to adapt in a decision-directed 
mode, starting, for example, with the coefficient values then stored in 
the equalizer or with some predetermined set of initial values. 
Disadvantageously, however, it is possible with this approach for the 
equalizer to converge to an incorrect state in which the decision-directed 
error function is at a local minimum, rather than being at the 
above-mentioned absolute, or global, minimum. This results from the fact 
that, in some equalizer states, the actual and decision-directed errors 
are different for particular equalizer outputs that represent particular 
transmitted symbols. The equalizer is thus "stuck" in a stable state in 
which its output constellation is different from the transmitted 
constellation and the transmitted data is not correctly recovered. 
To avoid this problem, the conventional approach is for a data set whose 
equalizer needs to be retrained--hereinafter referred to as the 
"downstream" data set--to transmit a message to the data set at the other 
end of the channel--hereinafter referred to as the "upstream" data 
set--requesting the retrain. The ideal reference data is then 
retransmitted and the equalizer in the downstream data set reconverges to 
the correct state. 
Although generally satisfactory in many applications, this approach has 
drawbacks. For example, the fact that the downstream data set must 
communicate its need to be retrained to the upstream data set means that 
communication of user data from the downstream to the upstream data set, 
which might well otherwise be able to continue, will have to be 
interrupted. Moreover, the upstream data set will require some form of 
detection circuitry to recognize the retrain request, thereby adding to 
the cost and complexity of the data set. 
An additional disadvantage occurs in multipoint networks. In such 
applications, the need for, say, the (upstream) control data set to 
transmit ideal reference data over the network for the benefit of a 
particular (downstream) tributary data set whose equalizer needs to be 
retrained means that normal communication between the control data set and 
the other tributary data sets in the network will be interrupted. 
It is thus desirable to have a scheme which allows an adaptive equalizer to 
be trained in a decision-directed mode while eliminating the possibility 
that it will converge to an incorrect state. 
SUMMARY OF THE INVENTION 
The present invention is directed to such a scheme. In accordance with the 
invention, the equalizer is inhibited from updating for at least one 
equalizer output value whose actual and decision-directed errors differ 
for at least a particular transmitted symbol, that particular equalizer 
output thus being more distant in the transmitted constellation from the 
transmitted symbol than from at least one other point of that 
constellation. In practical embodiments, equalizer updating is inhibited 
for a whole range of such equalizer output values falling within a 
so-called "null zone," the latter including at least one point of the 
undesired stable constellation but no points of the transmitted 
constellation. 
The invention is illustrated herein in the context of a 16-point quadrature 
amplitude modulation (QAM) system in which the undesired stable 
constellation has the same shape and orientation as the transmitted 
constellation, but is of smaller amplitude. The null zone illustratively 
lies between the four innermost points and twelve outermost points of the 
undesired constellation. In a first embodiment, the null zone is in the 
shape of a square annulus and encompasses the twelve outermost points of 
the undesired constellation. In a second embodiment, the null zone is in 
the shape of a circular annulus and encompasses the four outermost points 
of the undesired constellation.

DETAILED DESCRIPTION 
FIG. 1 depicts a receiver 10 for data signals transmitted from a 
transmitter (not shown) over a bandlimited communication channel, e.g., 
voiceband telephone circuit. The data signals are illustratively 
quadrature amplitude modulated (QAM) data signals wherein four information 
bits, after having been scrambled and differentially encoded in 
conventional fashion, are transmitted during each of a succession of 
symbol intervals of duration T. The symbol rate is thus 1/T, yielding a 
binary transmission rate of 4T/ bits per second. During each symbol 
interval, the four bits to be transmitted are encoded into two signals, 
each of which can take on one of the four values [+1, -1, +3, -3]. The two 
signals transmitted during the m.sup.th symbol interval comprise data 
symbol, or signal point, A.sub.m --a complex quantity having real and 
imaginary components a.sub.m and a.sub.m, respectively. FIG. 2 depicts the 
so-called signal constellation of all 16 possible such symbols. 
Components a.sub.m and a.sub.m, in turn, amplitude modulate respective 
in-phase and quadrature-phase carrier waves, and the combined modulated 
signals form a QAM signal s(t) of the form 
##EQU1## 
where g(t) is a real function and .omega..sub.c is the radian carrier 
frequency. Signal s(t) is then transmitted to receiver 10. 
In receiver 10, the received QAM signal s.sub.r (t) passes through 
automatic gain control circuit 8 where it emerges as signal s.sub.r.sup.' 
(t). The latter is applied to an input circuit 11 and, more particularly, 
to analog bandpass filter 12 thereof. The function of filter 12 is to 
filter out any energy in signal s.sub.r.sup.' (t) outside of the 
transmission band of interest. 
Input circuit 11 further includes a phase splitter 14, a sampler in the 
form of an analog-to-digital (a/d) converter 17 and sample clock 19. Phase 
splitter 14 responds to the output signal q(t) of filter 12 to generate 
two versions of signal q(t). One of these is q'(t), which may be identical 
to q(t) or may be a phase-shifted version of it. The other, represented as 
q'(t), is the Hilbert transform of q'(t). Signals q'(t) and q'(t) may be 
regarded as the real and imaginary components of a complex signal Q'(t). 
Signals q'(t) and q'(t) are passed to a/d converter 17. The latter is 
operated by clock 19 p times per symbol interval to generate a sampled 
signal in the form of equalizer input samples q.sub.k and q.sub.k, k=1,2 . 
. . of signals q'(t) and q'(t). (In a typical embodiment, p may take on 
the value of 2.) Equalizer input samples q.sub.k and q.sub.k may be 
thought of as components of a complex equalizer input sample Q.sub.k. 
Equalizer input sample components q.sub.k and q.sub.k pass on to 
transversal filter equalizer 22. The latter generates an output once every 
T seconds. In particular, the output of equalizer 22 during the m.sup.th 
receiver symbol interval of duration T is complex passband equalizer 
output, or output point, U.sub.m having components u.sub.m and u.sub.m. 
Equalizer 22 generates its outputs by forming linear combinations of the 
equalizer input sample components in accordance with the relations 
##EQU2## 
In these expressions r.sub.m and r.sub.m are (N.times.1) matrices, or 
vectors, respectively comprised of the N most recent real and imaginary 
equalizer input sample components, N being a selected integer. That is 
##EQU3## 
In addition, c.sub.m and d.sub.m are (N.times.1) vectors, each comprised of 
an ensemble of N tap coefficients having values associated with the 
m.sup.th receiver interval. The values of these coefficients define the 
transfer characteristic, or "state," of the equalizer. (The superscript 
"T" used in the above expressions indicates the matrix transpose 
operation, wherein the (N.times.1) vectors c.sub.m and d.sub.m are 
transposed into (1.times.N) vectors for purposes of matrix multiplication. 
This superscript should not be confused with the symbol interval T.) The 
values of the coefficients in these vectors are determined in the manner 
described below. Vectors c.sub.m and d.sub.m may be thought of as the real 
and imaginary components of a complex coefficient vector C.sub.m. 
Passband equalizer output U.sub.m is demodulated by demodulator 25 to yield 
baseband output, or output point, Y.sub.m. The latter and passband 
equalizer output U.sub.m are associated with, and respectively represent 
baseband and passband versions of, transmitted symbol A.sub.m. Baseband 
output Y.sub.m has real and imaginary components y.sub.m and y.sub.m, the 
demodulation process being expressed as 
EQU y.sub.m =u.sub.m cos .theta..sub.m *+u.sub.m sin .theta..sub.m * 
EQU y.sub.m =-u.sub.m sin .theta..sub.m *+u.sub.m cos .theta..sub.m *, 
.theta..sub.m * being an estimate of the current carrier phase. For 
purposes of generating y.sub.m and y.sub.m in accordance with the above 
expressions, demodulator 25 receives representations of cos .theta..sub.m 
* and sin .theta..sub.m * from a carrier source 27. 
Baseband output Y.sub.m is quantized, or sliced, in decision circuit 31. 
The resulting output A.sub.m * is a decision as to the value of the 
transmitted symbol A.sub.m, that decision being the point of the 
transmitted constellation to which baseband output Y.sub.m is closest. In 
particular, the real and imaginary parts of A.sub.m *, a.sub.m * and 
a.sub.m *, are decisions as to the data signal values represented by the 
real and imaginary components a.sub.m and a.sub.m of transmitted symbol 
A.sub.m. Decision circuit 31, more particularly, forms decision a.sub.m * 
(a.sub.m *) by identifying the one of the four possible data signal values 
[+1, -1, +3, -3] that is closest to the value of baseband output component 
y.sub.m (y.sub.m). 
Decision A.sub.m * is also used to generate an error signal for use in 
updating coefficient vectors c.sub.m and d.sub.m. In particular, decision 
components a.sub.m * and a.sub.m * are combined in decision remodulator 35 
with sin .theta..sub.m * and cos .theta..sub.m * from carrier source 27 to 
form remodulated, or passband, decision A.sub.pm *. The real and imaginary 
components of A.sub.pm *, a.sub.pm * and a.sub.pm, * are formed in 
accordance with 
EQU a.sub.pm *=a.sub.m * cos .theta..sub.m *-a.sub.m * sin .theta..sub.m * 
EQU a.sub.pm *=a.sub.m * sin .theta..sub.m *+a.sub.m * cos .theta..sub.m *. 
Passband decision A.sub.pm * is subtracted from passband equalizer output 
U.sub.m in subtractor 38 to yield passband error E.sub.pm having 
components e.sub.pm and e.sub.pm given by 
EQU e.sub.pm =u.sub.m -a.sub.pm * 
EQU e.sub.pm =u.sub.m -a.sub.pm *. 
Passband error E.sub.pm is referred to as a "decision-directed" error 
inasmuch as it is generated using decision A.sub.m *. The 
decision-directed error is the same as the actual error as long as the 
decision A.sub.m * is correct. Otherwise, as discussed in further detail 
hereinbelow, the actual and decision-directed errors will be different. 
Error signal components e.sub.pm and e.sub.pm are extended via gate 
39--whose function is described in further detail hereinbelow--to 
coefficient store and update unit 23 within equalizer 22 for purposes of 
updating the values of the coefficients in coefficient vectors c.sub.m and 
d.sub.m, and thus the state of the equalizer, in preparation for the next, 
(m+1).sup.st, symbol interval. The so-called mean-squared error stochastic 
updating algorithm--which approximates a true mean-squared error 
minimization algorithm and which thus minimizes the mean-squared value of 
the decision-directed errors--is illustratively used, the updating rules 
being 
##EQU4## 
.alpha. being a predetermined constant. These rules can be written in 
complex notation as 
EQU C.sub.m+1 =C.sub.m -.alpha.R.sub.k E.sub.pm. 
The problem to which the invention is directed will now be explained. In 
this discussion, terms such as equalizer output point and equalizer output 
constellation are used to mean baseband equalizer output point, baseband 
equalizer output constellation, etc. 
As shown in FIG. 3, the state of an equalizer can be characterized by a 
gain G and a rotation .phi. which respectively relate the amplitude and 
orientation of the associated equalizer output constellation--i.e., the 
set of complex output values generated at the output of demodulator 25 for 
all possible points of the transmitted constellation--to the transmitted, 
or ideal, constellation. When the equalizer is in the correct converged 
state, the values of G and .phi. are 1 and 0, respectively, so that the 
equalizer output constellation is substantially congruent with the 
transmitted constellation, i.e., the constellation of FIG. 2 in this 
embodiment. 
On the other hand, when an equalizer is in other than the correct converged 
state--as would occur, for example, during equalizer start-up, after a 
phase hit or upon an abrupt significant change in channel 
characteristics--G and .phi. will both, in general, have values other than 
1 and 0 respectively. Ideally, G and .phi. should thereafter ultimately 
take on the values 1 and 0 as the coefficients continue to update. 
Unfortunately, however, it is possible for the equalizer to converge to a 
state in which G and .phi. are other than 1 and 0, respectively, and thus 
in which the output constellation is other than that which is transmitted. 
Assume, for example, that at some point in the process of starting up, the 
equalizer is in the state whose associated baseband output constellation 
is as shown in FIG. 6. (Note that the scale of FIG. 6 is expanded from 
that of FIGS. 2 and 3 for drawing clarity.) The gain G is substantially 
less than unity. The rotation .phi. is assumed to be zero, however, in 
order to simplify the discussion. Also to simplify the discussion, we will 
focus attention only on the upper-right quadrant of the constellation the 
four points of which are labelled a, b, c and d. Inasmuch as the 
constellation has four-fold symmetry, this can be done without loss of 
generality. 
In particular, note that, although point a is closest in the transmitted 
constellation to the associated transmitted point which it actually 
represents, i.e., point (1,1), points b, c and d are more distant from 
their associated transmitted points--(3,1), (1,3) and (3,3), 
respectively--than from at least one other point in the transmitted 
constellation. In particular, they, too, are all closest to the point 
(1,1). The decision at the output of decision circuit 31 in response to 
any of these four equalizer output points will thus, in fact, be the point 
(1,1), i.e., a.sub.m *=a.sub.m *=1. Thus the decision-directed error 
signal generated by subtractor 38 will be generated as a function of the 
difference between the equalizer output values and the point (1,1)--the 
error signal actually being the passband version of that difference. In 
three cases out of four, then, the decision-directed error will be 
different from the actual error. 
Let us now examine what will happen as coefficient updating continues. 
Assume that the transmitted data is random so that each of the four points 
a, b, c and d occurs with the same average frequency. The 
decision-directed mean-squared error (DMSE) can then be computed by simply 
squaring the distance from the point (1,1) to each of the points a, b, c 
and d and taking the average. (The term "decision-directed mean-squared 
error" is used here to denote the fact that the mean-squared error is 
computed using decisions. The actual mean-squared error is, of course, the 
averaged squared distance from each of the points a, b, c and d to the 
points (1,1), (3,1), (1,3) and (3,3), respectively.) 
If we take the coordinates of point a to be (A,A), then the coordinates of 
points b, c and d will necessarily be (3A,A), (A,3A) and (3A,3A), 
respectively. The DMSE is then given by 
##EQU5## 
The first derivative of the DMSE, (20A-8), has the value zero at A=0.4 and 
its second derivative, 20, is positive everywhere. The DMSE thus has a 
minimum at A=0.4. Since the coefficient updating algorithm seeks to 
establish the equalizer in a state in which the mean-squared error is 
minimized, the equalizer will converge to a state in which A=0.4, meaning 
that each equalizer output component, instead of having one of the values 
[+1, -1, +3, -3], will have one of the values [+0.4, -0.4, +1.2, -1.2], 
and the equalizer output constellation is the stable, but incorrect, 
constellation of FIG. 7. The decisions at the output of decision circuit 
31 thus continue to be erroneous indefinitely. The equalizer output 
constellation for this state is shown in FIG. 7. 
Considering the problem in a more general context, FIG. 11 is a graph, 
plotted in cylindrical coordinates, of a so-called decision-directed error 
surface for the constellation of FIG. 2. This graph plots the value of the 
decision-directed mean-squared error on the z axis as a function of the 
equalizer state as represented by the gain G and rotation .phi., which are 
plotted on the r and .theta. axes, respectively. A contour version of the 
plot of FIG. 11 is shown in FIG. 12. 
The r and .theta. coordinates of, for example, the point labelled W in 
FIGS. 11 and 12 are 1 and 0, respectively. This point thus represents the 
correct converged state of the equalizer, as discussed above, and, indeed, 
is a point at which the decision-directed mean-squared error and, indeed, 
the actual mean-squared error is at its absolute, or global, minimum. In 
actuality, since the transmitted data is differentially encoded, a 
rotation of the equalizer output constellation by any multiple of .pi./2 
will have no affect on ultimate data recovery. Thus the points labelled W, 
X, Y and Z are equivalent for purposes here. 
In general, the direction in which equalizer state changes as the 
coefficient updating algorithm continues to minimize the DMSE is the 
direction of maximum gradient of the DMSE, assuming that the transmitted 
data symbols occur randomly. This can be thought of as the direction of 
steepest descent in FIGS. 11 and 12. As long as the equalizer state is in 
relatively close proximity to a particular one of the points W, X, Y and 
Z, e.g., point L (shown only in FIG. 10), the direction of maximum 
gradient is toward that particular point, e.g., toward point W. Thus, the 
equalizer converges to the correct state. 
On the other hand, if the equalizer is sufficiently removed from any of the 
points W, X, Y and Z, the direction of maximum gradient may be not toward 
any of the absolute minima at points W, X, Y and Z, but rather toward some 
other local minimum. This is just the situation discussed above in 
connection with FIG. 6. If the equalizer is at, for example, point M, for 
which G.noteq.1 and .phi.=0, which is exactly the situation depicted in 
FIG. 6, or, for example, at point N, for which G.noteq.1 and 
.phi..noteq.0, the direction of maximum gradient is toward point S, at 
which each of the equalizer output components has one of the values [+0.4, 
-0.4, +1.2, -1.2] and it is to that point that the equalizer will 
converge. Again, by symmetry, points T, U and V are equivalent to point S. 
As seen from FIGS. 11 and 12, there are other local minima, as well. 
However, changes in equalizer state are in the direction of maximum 
gradient only to the extent that the symbols in the transmitted stream are 
randomly distributed. A sufficiently long sequence of particular 
transmitted symbols can result in a short-term average DMSE which is quite 
different from the long-term average DMSE and which moves the equalizer 
state away from the nearest minimum. Local minima such as those at points 
O, P, Q and R (and the (unlabelled) corresponding points in the other 
quadrants) are relatively "shallow" and sequences of symbols that would 
cause the equalizer to "shake loose" from such points will occur 
relatively frequently. The existence of such minima does not, therefore, 
appreciably add to the time required for the equalizer to converge and 
thus such points pose no particular problem. 
On the other hand, the minima at points S, T, U and V are relatively 
"deep." Thus much longer, and thus less-frequently-occurring, sequences of 
symbols are required for the equalizer to shake loose from one of those 
points. Indeed, waiting for such a sequence to occur randomly would 
inordinately delay the convergence process. 
In accordance with the invention, this problem is overcome by inhibiting 
equalizer updating for particular equalizer output values in such a way 
that the decision-directed error surface no longer has troublesome minima 
such as those at points S, T, U and V. This is accomplished by inhibiting 
equalizer updating for at least one equalizer output value whose actual 
and decision-directed errors differ for at least a particular transmitted 
symbol, that particular equalizer output thus being more distant in the 
transmitted constellation from the transmitted symbol than from at least 
one other point of that constellation. 
In practical embodiments of the invention, equalizer updating is inhibited 
for a whole range of such equalizer output values falling within a 
so-called "null zone," the latter including at least one point of the 
undesired stable constellation but no points of the transmitted 
constellation. In particular, in a first implementation of the invention 
in the present illustrative embodiment, the null zone includes the twelve 
outer points of the undesired stable constellation of FIG. 7, i.e., the 
points (.+-.0.4, .+-.1.2), (.+-.1.2, .+-.0.4) and (.+-.1.2, .+-.1.2), and 
extends throughout the range of points one of whose coordinates has a 
magnitude which is greater than unity--and, as in this embodiment, 
preferably greater than 1.2--and less than 2, and the other of whose 
coordinates has a magnitude less than 2. Note that the magnitude of each 
point in the null zone is greater than the magnitude of each inner point 
of the transmitted constellation, i.e., the points at (.+-.1, .+-.1) but 
less than the magnitude of each outer point thereof, i.e., the points 
(.+-.1, .+-.3), (.+-. 3, .+-.1), (.+-.3, .+-.3). This null zone is the 
shaded region in each of FIGS. 6-8. 
Let us now return to FIG. 6. As long as the gain G is so small that A&lt;0.4 
and points a, b, c and d are all inside the inner boundary of the null 
zione, as is, in fact, the case in FIG. 6, the equalizer, although 
adapting toward the undesired local minimum at A=0.4, is also adapting 
toward the desired minimum at A=1. This corresponds to a movement from, 
say, point K in FIG. 12 along the .theta.=0 line toward both points S and 
W. Once the equalizer reaches the state for which A=0.4, however, points 
b, c and d fall within the null zone, as shown in FIG. 7. The 
contributions to the DMSE from those points, which would otherwise tend to 
"push" the equalizer toward the undesired minimum at point S, are now 
ignored, in accordance with the invention. The DMSE is then simply given 
by 
##EQU6## 
The first derivative of the DMSE is thus (4A-4), which has a minimum at 
A=1. Thus, even if the errors associated with points b, c and d were never 
further used for updating, the equalizer would converge, as desired, to a 
state for which A=1. 
Of course, as can be seen from FIG. 8, once G becomes so large that points 
b, c and d lie outside of the null zone, their contributions to the DMSE 
must again be taken into account in computing minima in that function. By 
now, however, points b, c and d are closer to the points (1,3), (3,1) and 
(3,3), respectively, than to the point (1,1) and the expression for the 
DMSE is computed using the distance to each of those points (1,3), (3,1) 
and (3,3) rather than to the point (1,1). That is, the decision-directed 
error for each of the points b, c and d is now the same as its actual 
error. Without showing the calculation here, it suffices to note that the 
minimum for the DMSE computed this way will still be at A=1, so that the 
equalizer output constellation becomes substantially congruent with the 
transmitted constellation. 
Although the square null zone--which is shown superimposed on the 
transmitted constellation in FIG. 4--is generally satisfactory, it does 
allow certain relatively minor local minima to exist in the 
decision-directed error surface. The equalizer will shake loose from such 
local minima in short order, as described above. It may be desired, 
however, to eliminate even such minor local minima. This may be 
accomplished by using a circular, rather than square, null zone, as shown 
in FIG. 5. The inner radius of the null zone is at least .sqroot.2, since 
that is the magnitude of point (1,1). Preferably, as shown in expanded 
view in FIG. 9, the inner limit of the null zone is the circle passing 
through the point (1.2, 1.2), i.e., point d of the undesired stable 
constellation, that point having a radius of about 1.70. The outer limit 
of the null zone is preferably the circle passing through the point (2,2), 
which has a radius of 2 .sqroot.2.congruent.2.83. 
Computing the DMSE without regard to point d (i.e., eliminating the 
(1-3A).sup.2 terms of the first DMSE equation set forth above and dividing 
by 3 instead of by 4) yields 
EQU DMSE=[22A.sup.2 -20A.sup.2 +6]/3, 
whose first derivative has a minimum at A=5/11.congruent.0.46. Thus even 
when the equalizer has reached the point at which poind d is in the null 
zone, it begins to adapt not toward the state for which A=1, which is the 
state ultimately desired, but rather to a state having A.congruent.0.46. 
The latter is not a stable state, however, because as the equalizer adapts 
theretoward, its output constellation expands and, as seen in FIG. 10, 
points b and c enter the null zone at the point where A.congruent.0.38. 
Thus, as in the case of the square null zone, the equalizer output 
constellation continues to expand until it reaches the state for which 
A=1. 
FIGS. 13 and 14 show the decision-directed error surface that results from 
the use of the null zone of FIG. 5. Note the absence of any significant 
local minima. 
The null zone of FIG. 5 is illustratively the one used in receiver 10 of 
FIG. 1. In particular, the receiver includes a squared sum generator 41 
which responds to equalizer baseband output components y.sub.m and y.sub.m 
to generate the squared magnitude of the current baseband equalizer 
output, i.e., .vertline.Y.sub.m .vertline..sup.2. This magnitude is then 
compared by a comparator 41 to the values [(1.2).sup.2 +(1.2).sup.2 ]=2.88 
and [(2.0).sup.2 +(2.0).sup.2 ]=8.0 to see whether .vertline.Y.sub.m 
.vertline..sup.2 is within the null zone. (Working with the squared 
magnitude of Y.sub.m rather than its magnitude per se eliminates the need 
to compute a square root but is, of course, otherwise equivalent.) 
An output bit from comparator 44 on lead 45 indicates whether or not the 
current equalizer baseband output is within the null zone. This bit 
controls gate 39. In particular, if the current equalizer baseband output 
is not within the null zone, the error components from subtractor 38 are 
simply passed on to coefficient store and update unit 23, as previously 
described. If, on the other hand, the current equalizer baseband output is 
within the null zone, gate 39 does not pass the error components through 
but rather provides error components of zero value. Inasmuch as each 
equalizer updating term is a multiplicative function of the error 
component values, this causes the updating term to be zero, thereby 
effectively inhibiting equalizer updating for the equalizer output in 
question. 
The foregoing merely illustrates the principles of the invention. For 
example, particular processing steps shown herein as being performed in 
the analog (digital) domain could be performed in the digital (analog) 
domain if desired. Moreover, although the invention is illustrated herein 
in the context of a receiver comprised of a number of discrete functional 
blocks, a receiver in which the invention is implemented may be comprised, 
for example, of one or more programmed processors which carry out the 
functions of those blocks. In addition, the invention is applicable to 
signal constellations other than QAM or other rectilinear constellations 
and, indeed, is potentially applicable to virtually any constellation in 
which the use of a null zone as described herein may be found to be 
advantageous. 
It should also be emphasized that, even in the context of a QAM system such 
as that described herein, the limits and shape of the null zone may be 
different from the two null zones shown and described herein. 
It will thus be appreciated that, although a specific implementation of the 
invention is shown and described herein, those skilled in the art will be 
able to devise numerous alternative embodiments which, although not 
explicitly shown or described herein, embody the principles of the 
invention and are thus within its spirit and scope.