Signal Processing - September 2017 - 151

Nonlinear crosstalk cancelers

has attracted most of the research interest for FEXT cancelIn the multiuser case, the MAC capacity can be achieved by
ation. There are several variants of linear receivers, for
detecting a single user at a time and then subtracting the
example, based on the criterion of ZF and the MMSE.
detected signal from the received signal before continuing to
These structures are much simpler due to the absence of
detect the next user. This scheme is known
feedback operations. The ZF canceler, as
as successive interference cancelation or
implied by its name, attempts to cancel
While the downstream
generalized decision feedback equalizer
the self-crosstalk, assuming that this is
(GDFE). It should be noted that DSL systhe only disturbance present [thus ignorprocessing attempts to
tems have a 6-dB noise margin, which
ing even the additive white Gaussian
cancel the crosstalk,
means that error propagation is unlikely.
noise (AWGN) and any other kind of
it also has another
SIC's optimality requires linear optimal
interference]. It does this by the applicaconsideration to deal with: tion of the inverse operator Fzf = H -1 at
detection at each stage, using a minimumthe precoded signals at
mean-square error (MMSE) receiver and
the receiver. The application of a linear
each line should operate
subtraction of the signal only after the error
ZF canceler H - 1 on the received signal y
correction code has been successfully
within the assigned power in (2) cancels the crosstalk in the upstream
decoded and the users properly ordered.
DSL system. The ith user receives a crossspectral mask.
Both of these requirements incur significant
talk-free signal Yi = X i + h iinv w, where
complexity. In the following, we present a
is the ith row of H - 1 . It can be seen
h inv
i
simplified version that is near optimal in DSL systems [9].
that ZF processing amplifies the power of additive noise
Consider a decision feedback equalizer (DFE) receiver
and interference, such that the resultant noise power
2
based on the QR decomposition of channel H, which can result
becomes R Nj = 1 [H - 1] i, j v 2w, i . Thus, the resulting perforin rates close to the MFB. The computation of the QR decommance is strongly dependent on the condition number of the
position of the matrix yields H = QR, where Q is a unitary
channel matrix and becomes very poor if the matrix is close
matrix and R is an upper triangular matrix. First, the unitary
to singular.
nature of matrix Q is made use of with the linear operation
The linear MMSE canceler F minimizes the MSE
Q H on the received vector in (2) to get a rotated version of the
between the output of the canceler and the true value, i.e.,
received vector
arg min F E 6 x - Fy 2@ [40] is given as
H

z = Q y = Rx + Q w.

Fmmse = (H H H + v w2 /Px I) -1 H H

(5)

(8)

and results in the spectral efficiency as
As the noise is Gaussian independent and identically distributed between the lines, the multiplication with the unitary
structure of matrix Q does not change the statistics of the noise.
Now the upper triangular nature of matrix R is made use of,
and the symbols are estimated, starting with the last row. The
cancelation operation on the mth signal is given as
Xt m = Z -

i = m+ 1

[R] m, i t
X i, m = N, N - 1g1.
[R] m, m

(6)

The DFE is sensitive to error propagation: any error in the
decision, based on Xt m in (6), increases the probability of error
for subsequent user detection. However, with proper matching of
the modulation and the SNR, the error probability can be made
small enough so that the error propagation is negligible. In this
case, the spectral efficiency for the ith user can be expressed as
-1
R dfe
Px, i | R ii | 2 v -w,2i h .
i = log 2 ^ 1 + C

(7)

Obviously, user performance is significantly affected by user
ordering (i.e., which user is detected first and which later). For a
detailed discussion of various user ordering schemes, see [39].

Linear crosstalk cancelers
To reduce complexity, the straightforward approach is to
use linear crosstalk cancelers. The use of linear receivers

R mmse
= log 2 e
i

C - 1 Px /v w2
o.
6^H H + v w2 /Px Ih- 1@ii
H

(9)

The implementation complexity of MMSE is almost
identical to that of ZF and is generally better than the ZF solution. The use of MMSE requires the knowledge of the noise
covariance matrix. However, any underestimate of this matrix
will still yield better performance than ZF.
Nevertheless, the advantage of MMSE is negligible for
diagonal dominant channels as well as when the SNR is
very high. Thus, the MMSE performs significantly better
than the ZF only at high frequency tones. More information can be found in "Linear Cancelers for Upstream
DSL Systems."

Crosstalk cancelation in the downstream
In the downstream, crosstalk is precompensated for before the
transmission of signals. While the downstream processing
attempts to cancel the crosstalk, it also has another consideration to deal with: the precoded signals at each line should
operate within the assigned power spectral mask. Another
important difference from the upstream is the lack of channel
estimation at the central office. In this section, we assume a

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

151

Table of Contents for the Digital Edition of Signal Processing - September 2017