IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 38, NO. 4 APRIL 1990
55 1
Multichannel Signal Processing for Data
Communications in the Presence 01
Crosstalk
MICHAEL L. HONIG, KENNETH STEIGLITZ,
Abstruct- We consider transmission of data over multiple coupled
channels, such as bundles of twisted-pair copper wires in the local subscriber loop, and between central offices in the public switched telephone network. Transceiver designs for such channels typically treat the
crosstalk between adjacent twisted pairs as random noise uncorrelated
with the transmitted signal. We propose a transmitterheceiver pair that
compensates for crosstalk by treating an entire bundle of twisted pairs as
a single multiinput~ultioutputchannel with a (slowly varying) matrix
transfer function. The proposed transceiver uses multichannel adaptive FIR filters to cancel near- and far-end crosstalk, and to pre- and
postprocess the input/output of the channel. The linear pre- and postprocessors that minimize mean squared error between the received and
transmitted signal in the presence of both near- and far-end crosstalk are
derived. The performance of an adaptive near-end crosstalk canceller
using the stochastic gradient (LMS) transversal algorithm is illustrated
via numerical simulation. Plots of mean squared error versus time and
eye diagrams are presented assuming a standard transmission line model
for the channel.
A signal design algorithm that maps a vector input bit stream to a
stream of channel symbol vectors is also presented. This algorithm is
illustrated explicitly for a simple model of two coupled channels. It is
shown that the achievable rate using the proposed signaling scheme is
very close to the rate attainable in the absence of far-end crosstalk, and
is significantly greater than the achievable rate assuming that far-end
crosstalk is treated as additive noise with unknown statistics.
I. INTRODUCTION
0
NE of the major limitations on the maximum data rate that can
be achieved over the local telephone subscriber loop is crosstalk
interference from neighboring channels [ 11. Both near- and far-end
crosstalk are illustrated in Fig. 1 . The channel is assumed to consist
of a collection of single-inputkingle-output channels terminated at
both ends by a transmitterheceiver pair. As an example, the channel
might be a bundle of twisted pair copper wires. We will refer to this
multiinput/multioutput channel as the vector channel, and the constituent single-inputkingle-output channels as scalar channels. Nearend crosstalk is the signal present at a receiver caused by a transmitter
connected to a different scalar channel, and located on the same side
of the vector channel. Far-end crosstalk results from a transmitter
located on the opposite side of the vector channel.
Thus far, transceiver designs for subscriber loop channels have
I
B. GOPINATH
I
I
1 TRANSCEIVER
I
:----.
TRANSCEIVER
-..-..._
,
I
I
:
1
.
...__.
NEAR-END
CROSSTALK
2
FAR-END
CROSSTALK
Fig. 1. Near- and far-end crosstalk.
treated crosstalk as random noise uncorrelated with the transmitted
signal. It is reasonable to assume, however, that crosstalk is a linear effect [2]-[SI, and that the vector channel can be modeled as a
(slowly varying) matrix transfer function. We hope to demonstrate
that transceiver designs which exploit these facts can significantly
increase the data rate relative to designs which treat crosstalk as
random noise.
In the next section a full-duplex transmittedreceiver pair for use
over a vector channel will be described. Simulation results which
illustrate the performance of the proposed receiver in the presence of
near-end crosstalk are included in Section 111. It is assumed throughout this paper that at least one end of the vector channel is terminated
at a single physical location. For example, this assumption applies
to a cable of twisted pairs which is terminated at a central office.
In this case, the proposed transmittedreceiver pair could be used
at the central office to increase the data rate from the central office
to customer premises. In many cases of practical interest, however,
both ends of the vector channel are terminated in (different) single
physical locations. Examples are trunks between central offices and
subscriber loops between single office buildings and central offices.
In these cases, the proposed transmitter/receiver pairs could be used
on both ends of the channel.
11. TRANSCEIVER
ATTRIBUTES
It is assumed that transmission over the N constituent scalar channels is synchronous. If the N sources associated with the scalar channels are asynchronous, then, as illustrated in Fig. 2, a multiplexer is
required to map the Nsequences of source bits to a single sequence of
N-bit vectors. This sequence of N-bit vectors is subsequently mapped
to a vector waveform u ( t ) , which is the input to the vector channel.
The output of the channel i s y ( t ) = H *U(?)+ n ( t ) where H ( t )is the
matrix impulse response of the vector channel, n ( t ) is a noise term,
and
H *u(t) = L'H(t
Paper approved by the Editor for Data Communications and Modulation
of the IEEE Communications Society. Manuscript received June 13, 1988;
revised February 14, 1989. This work was supported in part by the NSF under Grant MIP-8705454 and under the U.S. Army Research Office-Durham
Contract DAAG29-85-K-0191. This paper was presented in part at the IEEE
Conference on Acoustics, Speech, and Signal Processing, New York City,
NY, April 1988.
M. L. Honig is with Bellcore, Morristown, NJ 07960.
K. Steiglitz is with the Department of Computer Science, Princeton University, Princeton, NJ 08544.
B. Gopinath is with the Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08855.
IEEE Log Number 9034844.
FELLOW, IEEE, AND
- s)u(s)ds.
A demultiplexer may be needed at the receiver to convert the decoded
vector bit stream to the N constituent scalar bit streams.
Assuming that the receiver synchronously samples the output of
the vector channel, we can write the received sample vector at time
T as
W )= H * s f ( T ) + G * s " ( T ) + n ( T )
(1)
wheresf ( T )is the vector of far-end input symbols at time T ,s"(T)is
the vector of near-end input symbols at time T (assuming full-duplex
OO90-6778/90/0400-0551$01 .OO
0 1990 IEEE
552
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 38, NO. 4 APRIL 1990
ADJACENT
TRANSMITTED
SIGNALS
CHANNEL
/"
\\
,
1.... .......4
Vector
V&
:
Bit
Streams
Fig. 2. Multichannel transmitter and receiver.
Fig. 4. Crosstalk cancellation combined with multichannel equalization.
7
I
jMux
4
,
,
,............I ...t..l.............
i m/
3
/]II
i ...I..1.A.....L....21....,...Lm
...{ .....;
............................................
I
I :
j Receiver
1
_ _ :
I
L--i/
~
............................................
.................
i-
i2(
T)
Receiver 2
'
,
k
................
I
1
I
I
~
I
I
i
i
'
i imRm1
~
2
I ' t
1
' I
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transmission), and T i s discrete valued. G ( T )is the "near-end matrix
impulse response," i.e., the ikth component of G ( T ) is the response
at time T on channel i to an impulse on channel k at time zero
where the transmitter and receiver are on the same side of the vector
channel. The ith diagonal component of G ( T ) is the echo response
which leaks through the hybrid on channel i. It is assumed that H and
G contain any linear filtering (pulse shaping) at the transmitter and
receiver. Since we can safely ignore second-order crosstalk, i.e.,
crosstalk arising from crosstalk, the only nonzero terms in H ( T )
and G ( T )will correspond to those pairs of scalar channels which are
physically close to each other.
A . Near-End Crosstalk Cancellation
Since the near-end symbol vectors"(T) is available at the receiver,
it should be possible to eliminate near-end crosstalk by simply subtracting from the sampled signal vector r the output of a filter with
impulse response G ( T )and input s"(T).Such a filter can be considered a multichannel version of an echo canceller. Since the impulse
response G is initially unknown, and may vary slowly with time,
the filter must be adaptive. This is illustrated in Fig. 3 which shows
near-end crosstalk cancellers (CTC's) combined with an echo canceller (EC) and a decision feedback equalizer (DFE) for one component of a vector channel containing three scalar channels. It is
assumed for simplicity that the intersymbol interference caused by
transmitted symbols not yet detected is small compared to the intersymbol interference caused by previously detected symbols (i.e., the
precursors of the impulse response H ( T ) are small) so that a linear
preequalizer is unnecessary. The crosstalk cancellers shown in Fig.
3 are assumed to be single-input/single-outputadaptive FIR filters.
Because the duration of the near-end crosstalk impulse response is
typically much shorter than a typical echo response, the crosstalk
cancellers require many fewer taps than the echo canceller. For the
crosstalk model used in the next section, only two or three taps per
canceller are sufficient.
An important observation is that the same error can be used to
adapt the coefficients of the equalizer, echo canceller, and crosstalk
cancellers for a given scalar channel. Specifically, referring to Fig.
3, e l ( T ) = BI(T) - x l ( T ) is used to adapt the coefficients of EC 1,
DFE 1, CTC2 and CTC3 . In the absence of noise el (T) should
therefore converge to zero, assuming each filter is at least as long
as the associated impulse response to be cancelled. The combination of EC 1, DFE 1, and C T C k l rk = 2, 3, can be viewed as a
multiinput/single-outputadaptive FIR filter.
We also point out that a special training sequence, {s"(T)},can be
used initially to obtain a least squares estimate of the filter coefficients
with very little computation. This technique has been discussed in [6]
for echo cancellation. Specifically, the least squares estimate of the
coefficients of the multiinputlsingle-output FIR filter just mentioned
has the form Q - ' ( T @ (T) where Q ( T ) is a matrix that depends only
on the input sequences"(O),s"(l),...,s"(T), andp(T) is a vector.
If the training sequence {s"(T)}is selected in advance, then the
sequence of matrices {Q -'(T)} can be prestored, thereby greatly
reducing the necessary on-line computation.
E . Multichannel Equalization
Fig. 4 shows a multichannel linear equalizer followed by a multichannel decision feedback equalizer to remove intersymbol interference and far-end crosstalk. The multichannel linear equalizer compensates for the precursors of the channel impulse response H. Assuming that second-order crosstalk is negligible, the bank of singleinputlsingle-output DFE's shown in Fig. 4 removes intersymbol interference caused by previously detected symbols. Because for a typical far-end crosstalk impulse response, the energy in the tail is much
less than that of the self-impulse response of the associated scalar
channel, relatively few additional coefficients are needed in the DFE
to remove far-end crosstalk from an adjacent channel.
The near-end crosstalk cancellers and DFE's shown in Fig. 4 can
be viewed as one multiinput/single-outputadaptive FIR filter. The
error e l (T) = SI(T) - x I (T) can be used to adapt the coefficients of
all the filters shown. We point out that multichannel equalization has
also been proposed in digital radio applications [7], [8], and that a
multichannel DFE has also been proposed for use with multidimensional codes over channels with severe intersymbol interference [ 9 ] ,
[W.
C. Pre- a n d Postprocessors
Rather than use a multichannel linear equalizer to invert the channel transfer function H , it is possible to introduce linear pre- and
postprocessors, T and R , at the transmitter and receiver, respec-
553
HONIG et al.: MULTICHANNEL SIGNAL PROCESSING
where the expectation is over the probability measures of the transmitted signals s " ( t ) and sf(?). Assuming that the near-end signals
are independent of the far-end signals, but that both signals have the
same second-order statistics, the expectation in ( 3 ) can be evaluated
as
0
0
Wires Going
0
Into Page
I
0
.
P o 3
Circularly Symmetric Channel
TRANSMITTER
OUTPUTS
Fig. 5.
RECENED
SIGNALS
where the N x N matrices & and $, are the spectra of the data a$
noise signals (N-vectors), respectively. Note that the term RdT -XT
is sim ly residual near-end crosstalk after (partial) cancellation.
If P i s fixed, then it is shown in the Appendix that the R which
minimizes the MSE is
Pre- and postprocessors for circularly symmetric channel
n"(t)
R = &(HP)* +xPs.Sd(Gf)*]
[cif&(Hf)
* + ef& (GP)* +sn] - '
Crosstalk
(5)
assuming that the inverse exists. Similarly, we can fix R , and find
the f that minimizes the MSE, given by ( 3 ) , subject to the average
transmitted power constraint
Canceller
a'(t)
r(t)
Fig. 6.
Block diagram showing pre- and postprocessors and crosstalk carceller.
tively, to diagonalize H , and thereby remove far-end crosstalk.
Single-input/single-outputequalizers :an then be used at the receiver
to invert the diagonal elements of THR in the frequency domain. In
general T and R will be frequency dependent, that is, the impulse
response matrices associated with T and R will vary with time. However, if the channel impulse response matrix is circulant, which is the
case for cables with circular geometries, then T is simply the FFT
matrix W where [W],, = e--2rJ'm/N.
Similarly, R = W* where "*"
denotes complex conjugate transpose. This is illustrated in Fig. 5. In
this case the matrices T and R depend only on the number of channel components N.Although channels encountered in practice are
not likely to be circularly symmetric, the addition of the fixed preand postprocessors W and W * may reduce the amount of crosstalk
present [1 11.
Rather than diagonalize the channel matrix A, we can instead
select analog linear pre- and postprocessors T and R to minimize
the L2 norm of the difference between the analog received signal
r ( t ) and the transmitted waveformsf(t). Fig. 6 shows a model of a
multichannel communications system with crosstalk cancellation and
pre- and postprocessors. The Fourier transform of the received signal
vector is
f ( w ) = R ( w ~ ( o ) f ( w p ( o+R(w$(w)f(w)s'"(w)
)
+R(obi(w) - X ( w ) f ( o V ( w )
(2)
whereX(w) is the transfer function of the crosstalk canceller, f ( w )
is the transfer function associated with T , and the other variables are
defined similarly. For notational convenience, we will assume that all
variables underneath a " -'' are functions of the frequency variable
w . A typical choice for T and R is to minimize the mean squared
error (MSE),
(3)
for some constant K . The resulting expression is
P
=
[(RH)*@H+ (@e
-T)*(@G -8)+ p Z - l ( @ H ) * ,
(7)
assuming that s^, is nonsingular, and that the inverse in (7) exists.
The constant p must be selected so that (6) holds. Finding a combination of f and R that satisfy (5) an$ (7) appears to be analytically
intractable. Of course, in practice T and R would most likely be
digital FIR filters. The receiver filter R can then be adapted to minimize the MSE [12]. The transmitter filter T can also be adapted
given an estimate of the matrix channel impulse response. Observe
that f should be adapted to minimize the cost function
M
~ ( f=)MSE + p
x trace (T&TT:)
(8)
i=l
where MSE is given by (4),.!?d is the covariance matrix for the data,
f i is the ith transmitter matrix tap, M is the filter order, and p can
be adjusted so that the constraint (6) holds for appropriate K . In
practice, a feedback channe! is needed to provide an error signal for
the adaptive algorithm for T .
Modification of the preceding results to the case where f and R^
are digital FIR filters, and to the case where f andR are analog pulse
shaping filters for (vector) pulse-amplitude modulation is presented
in [13]. The optimum f and R for the latter case without nearend crosstalk are derived in [14]. Related results are presented in
[15] and [16]. In addition, a minimum MSE analysis of decision
feedback equalization combined with echo cancellation for the singleinputhingle-output channel is given in [17] and [18].
III. SIMULATION
RESULTS:
CANCELLING
NEAR-END
CROSSTALK
Since near-end crosstalk is typically much more severe than far-end
crosstalk, the following simulation results assume only the presence
of near-end crosstalk. The ith channel output is therefore the ith
component of r in (1) where w(T)]ik = 0, i # k , for all T. Binary
signaling and baud-rate sampling are assumed so that the input symbols s i ( T ) are & 1. The self-impulse response of each channel h(T)
was computed from a transmission line model of 12 kft of 24 gauge
twisted pair, and is shown in Fig. 7. The crosstalk impulse response
g i k ( T )= [G(T)likr i # k , was computed from the crosstalk transfer
554
IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. 38, NO. 4 APRIL 1990
so that E,"=,glk(T)
= 0. In this model, nearly all of the crosstalk
impulse response is concentrated at T = 0. Consequently, this type
of crosstalk interference can be cancelled using an FIR filter with
only two or three taps.
The receiver configuration shown in Fig. 3 was simulated for
a particular scalar channel with six nearest neighbors. It was assumed that the crosstalk impulse response between each of the six
neighboring channels and the channel simulated was the same, i.e.,
g;k(T) = g ; / ( T ) ,k # 1 , SO that
-
15wO
10000 -
hdt)
-
5000
0
J
- 5 m b
0
'
'
200
400
'
'
'
2
'
20000
5 m
0
1
0
100
200
300
400
time (1.25e-7 see/sarnple)
500
t,, = 0.05pF
(a)
0
-100
-200
-600
-700
U
0
100
200
300
400
r=l
k=O
IV. MULTICHANNEL
SIGNAL
DESIGN
500
time (1.25-7 sec/sample)
0))
Fig. 8.
k=O
7
where g ( T ) = gjk(T), i # k , and the noise term n ( T ) in (1) is assumed to be very small relative to the crosstalk interference. It was
also assumed that the echo response has been perfectly cancelled,
so that g j l ( T ) = 0. Fig. 9 shows eye diagrams at the receiver corresponding to the cases where there are no crosstalk cancellers, and
where there are six crosstalk cancellers with 1 tap and 5 taps each
connected to the six neighboring channels. The DFE contains 60 taps,
and the standard gradient (LMS) transversal algorithm [12] with a
step-size 0 = 0.003 was used to adapt both the DFE and crosstalk
cancellers. The eye diagrams were plotted after a delay of 1500 samples to ensure convergence. The optimal sampling phase [a function
of h(T)] was determined prior to simulation so that only three samples per segment are shown in Fig. 9. The data rate for this example
is 800 kbps in each direction. The coupling capacitance P j k was selected large enough so that the eye without crosstalk cancellation is
closed. Fig. 10 shows plots of averaged squared error versus time,
that is, e:(T) averaged over 200 separate simulation runs. Because
of the large coupling capacitance, the decrease in asymptotic MSE
due to crosstalk cancellation is dramatic (approximately 8 dB).
Although transmission and reception over all scalar channels is
assumed to be synchronous, there is an arbitrary fixed phase offset between a given transmitted near-end symbol and the subsequent
receiver sampling time. This phase offset is determined by the relative phase at which the near-end and far-end symbols are transmitted. In principle this phase offset can be adjusted to minimize the
amount of near-end crosstalk present. However, since only a few
taps are needed to eliminate the near-end crosstalk caused by the
model previously described, and since adjusting the relative phase
of the near-end and far-end transmitters is difficult in practice, this
type of optimization was not attempted.
Self-impulse response,
-5000
6
I
600 800 1000 1200 1400 1600
time (1.25e-7 sec/sample)
R= 440 obms/mile, I= I d , C= 0.083pF
Fig. 7.
7
(a) Crosstalk impulse response. (b) Crosstalk impulse response for
t > 0. The impulse at t = 0 is not shown.
If pulse-amplitude modulation is used to transmit data over an NinputlN-output channel, the transmitted signal corresponding to the
sequence of N-vector symbols { s k } is
function (see [ 3 ] , [5])
P jk ( x ) dx
(9)
where L is the length of the wire, and r ( w )
=
( R + j d ) ( G + j w C ) where R , I , G and C are, respectively,
l e wire resistance, inductance, conductance, and capacitance per
unit length, and i ' ; k ( X ) is the capacitive coupling between channels i and k as a function of length. As a first-order model, we
assume that t ; k ( x ) = t ; k is a constant. For large L (9) becomes
g i k ( w ) = ( j c d t ; k ) / [ 2 r ( u ) ] and
,
the corresponding impulse response
is
where Z o ( t ) is the modified Bessel function of order zero. A plot of
the sampled response gjk ( T ) used to generate the simulation results
is shown in Fig. 8. The magnitude of the impulse at T = 0 is selected
where p ( t ) is the (scalar) pulse shape, T is the symbol interval,
and each symbol vector s k can assume one of M values. For what
follows we choose to work with a more general mapping of symbols
to transmitted signal. Specifically, we assume that the transmitted
signal vector can be written as
wherep(sk; t ) is the vector pulse corresponding to the kth symbol
vector s k , and has finite support on the time interval [0, T I . There
are M different pulse shapes corresponding to the M possible symbol vectors. The form ( 1 3 ) assumes that the baud rate on each scalar
channel is the same, which could be a disadvantage if, for example,
one scalar channel causes a large amount of attenuation relative to
the other channels. However, the number of bits per baud, or information rate, transmitted on each scalar channel need not be the
555
HONK e l al.: MULTICHANNEL SIGNAL PROCESSING
EYE DIAGRAM WITHOUT CROSSTALK CANCELLATION
I
AVERAGED SQUARED ERROR VS TIME
I
I
m7
7
I10
I5
20
25
30
n
200
IOD
600
800
tow
1200
id00
1600
time (1 2 % 6 secisample)
6 interferers 60 DFE taps. 1 Sampielbaud.bela- 0 003
Sample
60 DFE taps 1 sample baud LMS algorilhm beta- 0 003
Fig. 10. Average squared error versus time.
(a)
EYE DIAGRAM WITH CROSSTALK CANCELLATION
I
I
--
___
10
15
v
25
20
30
Sample
1 tap cancellei 1 sample haud LMS algorilhm
hem 0 0 0 3
(b)
EYE DIAGRAM WITH CROSSTALK CANCELLATION
same. Although we assume that the receiver samples the output of
the vector channel synchronously, there can be a constant phase shift
between the samples of different scalar channels.
Our problem is to map the set of symbol vectors to a set of pulse
shapes. Assume for simplicity that the components of sk are binaryvalued, so that M = 2 N . As in [19], we assume that the transmitter
output is constrained in amplitude, and that the receiver can distinguish between two (vector) outputs only if they are separated in
amplitude in at least one component by at least d, the receiver discrimination, at some time t o . This leads to the following signal design
problem (see [19]).
Problem P ) Given the N x N matrix impulse response H ( t ) , find
waveformsp(s;; t ) , i = 1,...,2N,where Ipi(t)l 5 1 for all t and
1 5 i 5 N , to maximize the minimum L , distance between pairs
of distinct outputs over the fixed time interval [0, TI.
The L , norm of a continuous vector functionf(t) over the interval
[0, TI is maxi supo<i<r [f(t)li. As an example, if N = 1, then
there are two pulse s&%pes corresponding to the input bits 1 and 0.
The solution to ( P ) in this case is simply p ( 0 ; t ) = - p ( l ; t ) =
signh(T - t).
Assume now that the vector channel consists of N identical scalar
channels (e.g., twisted pairs) randomly situated inside a single
binder group. The channel impulse response matrix is therefore H ( t )
where @(t)]ii = h ( t ) , i = 1,. . . ,N,and the channel output is
y e ; ; t ) = H y ( s i ; t ) . Furthermore, assume that t is discrete, i.e.,
t = 7 , 2 r , . . . ,K r = T. A mapping of input symbols to pulse shapes
p(si; t ) , which approximately solves Problem P ) , can be obtained by
solving the following linear program for each i = 1 , . . . ,2N :
(Pk(Sj;t)l 51, f = 7 , 2 7 , ’ ” , K T .
------I
1+
7
IO
1s
20
25
30
Sample
5 laps,canceller, 1 sample,baud. LfAS alg0r8Ihm. beta- 0 003
(C)
Fig. 9. (a) Eye diagram without crosstalk cancellation. (b) Eye diagram with
one tap per crosstalk canceller. (c) Eye diagram with 5 taps per crosstalk
canceller.
The cost function Z is the minimum difference in amplitude between
two distinct output components at time T. The linear program therefore maximizes this minimum “eye opening” over all constituent
channels. This heuristic technique for signal design seems good intuitively as long as the impulse response of each channel is roughly
the same, and the far-end crosstalk impulse responses between different pairs of scalar channels are also similar. If this is not the case,
then other signal design techniques, in which the baud rate and/or
number of bits per symbol on each scalar channel is different, must
be considered.
556
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 38, NO. 4 APRIL 1990
maximize Jy ( T )1 for each of these cases are:
PI(^) = ~ 2 ( t )= s i g n [ h I 1 ( T - t ) + h z l ( T - t ) l
(184
p l ( t ) = -p2(t) = s i g n [ h I 1 ( T - t ) - h z l ( T - t ) ]
(18b)
p l ( t ) = -p2(t) = -sign[hll(T - t ) -h21(T - t ) ]
(18c)
p l ( t ) = p 2 ( t ) = -sign[hll(T - t ) + h ~ ( T -t)]
(18d)
<
I
where 0 t 5 T. Given a sequence of binary-valued vectors {sk },
each containing two components, the corresponding transmitted signal has the form ( 13) where each sk selects one of the four preceding
cases. s k therefore determines U ( t ) for kT 5 t 5 (k 1)T.
Consider the input ( 12) to a single-inputkingle-outputchannel with
impulse response h ( t ) where s k E { f 1) (binary signaling). The
corresponding data rate is
\
+
Fig. 11. Two coupled channels.
A . Example: Two Coupled Channels
The previous signal design is now illustrated for a simple model
of two coupled channels. In what follows we ignore the presence of
near-end crosstalk. In particular, it is assumed that either near-end
crosstalk has been sufficiently attenuated via crosstalk cancellation,
or that the channel is being used in half-duplex mode. The output of
a particular channel k within a bundle can then be written as
Yk(t) =
I'
w h ( t -S)ds
+nX(t) +
1
R = -.
It is shown in [20] that if T satisfies
sup [C(T
2
then any two distinct channel outputs corresponding to source sewhere s k # s[ for at least one k , are sepaquences { s k } and
rated in amplitude by at least d at some time t o .
Assume now that the input to the two-inputltwo-output channel
previously described is given by ( 13) where the pulse vectorsp (sk ; t )
are selected from (18). If one of the four sets of inputs in (18) is
applied to the channel for 0 5 t 5 T , then the possible output values,
y~(T) and y2(T), are
C1(T) and f Cz(T) where
{SA}
+
Ci(t) =
h l l ( t-s)uI(s)ds
y2(Z) =
I'
+
h22(t - s ) u ~ ( s ) ~ s
JG'
hzl(t - s ) u ~ ( s ) ~ s( 1 6 4
I'
(16b)
+
-
(14)
where hik(t) is the impulse response on channel k due to an impulse
on channel i, and u i ( t ) is the input to channel i.
Consider the simple model shown in Fig. 11 consisting of two
coupled channels. The channel outputs are
i'
+ 6) - 2C(6)1 = d
6
where
nR(f)
where u ( t ) is the input to the channel, n x ( t ) is crosstalk noise from
physically adjacent channels, and n R ( t ) is an independent noise process assumed to represent the inherent noise of the channel and receiver. Crosstalk noise can be written as
y l ( t )=
(19)
T
h12(t -S)UI(S)ds.
.I'
Ihii(s) +hzi(S)IdS
(224
' I ~ I -h21(s)Ids.
I W
(22b)
and
C2(t) = ~
In analogy with (20), T I , T2, 61, and S, can be chosen so that
In addition, we assume that h l l ( . ) = h22(.) and h12(.) = t 1 2 ~ ( . ) ,
although the following discussion can be extended to more general
d
(234
situations.
SUP[CI(GI+ T I ) -2C1(61)1 = y
61
Assume that the input symbol s k contains two binary-valued components, then four different pulse vectorsp(si, t ) = p i ,1 i 4 , and
must be chosen. In each case, the signal design algorithm (LP) says
d
that the components of the pulse vector, p;l and pi2, are selected to
(23b)
sup [C2(62 + T2) - 2C2(62)1 = 2.
simultaneously maximize the output components lyl (T) I and ly2(T)I.
62
In particular, (LP) can be rewritten as
It is easily shown that the L , distance between any pair of (vector)
channel outputs corresponding to a pair of distinct in uts ui and uk
max Ivl(T)l subject to lul(T)I= /y2(T)I
PIrP2
given by (13) with associated symbol sequences {s, } and {sj"}
where sji' #sfk' for at least one I , is at least d. Consequently, the
where the signs of y l and y2 are determined by sk , and time is now
receiver can distinguish between channel outputs if the transmitted
assumed to be continuous. This implies that either p l ( t ) = p 2 ( 4 or
signal is given by (13) and (18) where T = m a x ( T I , T2). The rate
p l ( t ) = -p2(t), 0 t 5 T, so that (16) implies
in this case is
< <
1)
<
Y I ( T ) = L f [ h ~ ~ -( T
4 h h 2 1 ( T -s)Ip2(s)ds
(17)
+
where or - applies depending on whether p 2 = P I or p2 = - P I .
There are four possible choices for the signs of y l ( t ) and y 2 ( t )
corresponding to each of the four possible messages. The inputs that
on each of the two scalar channels.
As a specific example, consider the linear system described by the
I
557
HONIG et al.: MULTICHANNEL SIGNAL PROCESSING
state equation
RATE USING PROWSED SIGN4LS
406
d.O.04
a/A*40. A=5430
11
where A and CY are positive constants and 11 U I oi, = I(u2 ( 1 00 = 1.
In this case the output of the system (channel) is equal to the state,
i.e., y l = X I and y2 = x2. Solving this equation for x l ( t ) and x2(t)
gives (16) where
ROSS TALK TREATED AS NOSE
W
and
Fig. 12. Achievable rates versus coupling parameter E.
1
hI2(t) = h21(t) = -[e-a(l-f)'
2
-e-a(l+e)r
I.
(26b)
From (22)
cl(t>
-111
1 --E
-e-a(l-e)'
1
(27a)
and
chosen so that the achievable rate on each channel with no crosstalk
( E = 0) is 1 Mb/s. Fig. 12 indicates that the aggregate rate for the two
channels with the simple signalling scheme proposed is insensitive to
the amount of crosstalk present. If, however, the crosstalk is treated
as noise, then the throughput drops quite fast as E increases since
the minimum distance between signals must increase in almost direct
proportion to the amount of crosstalk.
V . CONCLUSIONS
For the channel models considered, the results in Sections I11 and
In this case the cS1 and
are zero, so that
82
which maximize the left-hand sides of (23)
and the rate is given by (24).
Suppose now that crosstalk is treated as random additive noise.
The output of each channel can be written as (14), for k = 1, 2
where the crosstalk noise n x ( t ) satisfies
IV suggest that the deleterious effects of crosstalk can be significantly reduced by using crosstalk cancellers and by proper design
of channel inputs. For the crosstalk impulse response shown in Fig.
8, simulation results show that only 1 tap per crosstalk canceller
is sufficient to "open the eye," that is, ensure that the worst case
interference does not cause an error in the absence of noise. Of
course, more accurate models for crosstalk, such as those proposed
in [3] and [4], could be used to generate crosstalk impulse response.
Assuming the crosstalk cancellers are adaptive, however, the exact
nature of the coupling should not greatly affect performance as long
as the cancellers contain a sufficient number of taps.
APPENDIX
DERIVATION
OF THE MINIMUM
MSE EQUALIZER
00
1-2'
(29)
For reliable transmission the outputs of each channel should therefore
be separated by d+E/( 1 - e 2 ) where d is the required discrimination
which accounts for the noise inherent in the channel and receiver,
and does not depend on the amount of crosstalk interference. Let
If R^ is the minimum MSE filter, then
MSE(R + &)
-
MSE
= trace
{&
03
S _ _ [ 2 Re (*RHf&(RHf)*
-
SliCifgd
I
(30)
for small enough
is 1/T3 where
E.
In this case the achievable rate for each channel
C3(T3)=
5
( A)
d+
(31)
~
or
This is true if
trace
{
-11
lI&(Hf$d[(RHf)*
+&f$d[(R&f)*
-
( X f ) * ] +$,,R * ) dw
1
2 0 (A.2)
for all SR. Consequently, (A.I) is true if
T3
-
A 1n (1
-
5 (.+ &))
.
(32)
Fig. 12 shows a plot of achievable rates, given by (24), (28), and
(32), versus E for d = 0.01. The parameters A and CY in (25) were
fif$d[(RHf)*-13 + G f $ d [ ( k & f ) *
-
(xf)*]+Sn& * = 0,
(A.3)
and solving for R^ gives (5). A similar type of calculation gives (7).
558
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 38, NO. 4 APRIL 1990
ACKNOWLEDGMENT
The authors thank S. Boyd and P. Crespo for several helpful
discussions about this work, and J. Cioffi for his review of the
manuscript.
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Michael L. Honig was born in Phoenix, AZ, in
1955. He received the B.S. degree in electrical engineering from Stanford University, Stanford, CA,
in 1977, and the M.S. and Ph.D. degrees in electrical engineering from the University of California,
Berkeley, in 1978 and 1981, respectively.
From July 1981 to October 1983 he was a member of the Technical Staff at AT&T Information Systems, formerly part of Bell Laboratories, Holmdel,
NJ, where he worked on the design and performance analysis of local area networks, and on
voiceband data transmission. He subsequently transferred to the Systems
Principles Research Division at Bell Communications Research, where he
is currently investigating problems in data communications and signal processing.
Dr. Honig is a member of Tau Beta Pi and Phi Beta Kappa.
*
Kenneth Steiglitz (S’57-M’64-SM’79-F’81) was
born in Weehawken, NJ, on January 30, 1939. He
received the B.E.E. (magna cum laude), M.E.E.,
and Eng.Sc.D. degrees from New York University,
New York, NY, in 1959, 1960, and 1963, respectively.
Since September 1963 he has been at Princeton
University, Princeton, NJ, where he is now Professor of Computer Science, teaching and conducting research on VLSI design and implementation of
signal processing, optimization algorithms, and the
foundations of computing. He is the author of Introduction to Discrete Systems (New York: Wiley, 1974), and coauthor, with C. H. Papadimitriou,
of Combinatorial Optimization: Algorithms and Complexity (Englewood
Cliffs, NJ: Prentice-Hall, 1982).
Dr. Steiglitz is a member of the VLSI Committee of the IEEE ASSP Society, is serving his second term as member of the Administrative Committee,
and has also served on the Digital Signal Processing Committee, as Awards
Chairman of that Society, and as chairman of their Technical Direction Committee. He is an Associate Editor of the journal Networks, and is a former
Associate Editor of the Journal of the Association for Computing Machinery. A member of Eta Kappa Nu, Tau Beta Pi, and Sigma Xi, he received
the Technical Achievement Award of the ASSP Society in 1981, their Society
Award in 1986, and the IEEE Centennial Medal in 1984.
*
B. Gopinath was born in Chengannoor, India, in
1944. In 1964, he received the Master of Science
degree in mathematics and physics from the University of Bombay. In electrical engineering, he received Master’s and Ph.D. degrees in 1965 and
1968, respectively, from Stanford University, Stanford, CA.
In 1968, he joined Bell Telephone Laboratories
at Murray Hill, NJ, as a Member of Technical Staff
in the Mathematics Research Center, and continued
there until the breakup of AT&T in 1983. He then
became the manager of Communications Principles Research Group at Bellcore and later, in 1985, became the Division Manager of Systems Principles
Research Division. He has taught at the University of California, Berkeley
(as a Gordon McKay Professor in 1980), Columbia University, New York
(in 1987), and the University of Goettingen, Germany (as an Alexander von
Humbolt Fellow in 1972). He is presently the State of New Jersey Professor of
Electrical and Computer Engineering at Rutgers University, New Brunswick.
His current research interests are panoramic audio-visual environments, parallel object-oriented languages and databases, supercomputing, and signal
processing.