Signal Processing - September 2017 - 152

Linear Cancelers for Upstream DSL Systems
To illustrate the operation of linear crosstalk cancelers, consider the simple case of two users. The users transmit with equal
power Px over twisted pairs of equal length and experience
the equal power of additive noise Pw . The direct channel
gain for both the users is H d , and the crosstalk coupling is
aH d. Thus, in this simple case we have b = b r = b c = a,
and hence a is a diagonal-dominance parameter. The transmit and received signal vectors can be expressed as
;

Y1 = H 1 a X 1 + W 1 .
E
E
E; E ;
d;
a 1 X2
W2
Y2

SNR awgn
o,
2a 2 SNR awgn + 1

2
J
N
1
K
c 1 + a2 +
m - 4a 2 O
SNR awgn
mmse
K
O.
R
= log 2 KSNR awgn
O
1
2
K
O
1+ a +
SNR
awgn
L
P
(S6)

(S1)

Without crosstalk canceling, the crosstalk signal interferes with the reception of the desired signal, which is
now received in the presence of a noise term of power Pw ,
and an interference term of power a 2 H d Px decreases the
rate. Thus, the spectral efficiency for each is
R fext = log 2 e 1 +

In contrast to the no-cancelation case, the ZF canceler
performs very well whenever a % 1 and approaches
SWP. On the other hand, the ZF canceler performs very
poorly when the channel matrix is close to singular
^ a " 1 h (see Figure S1).
The minimum-mean-square error (MMSE) canceler for
this simple matrix results in the spectral efficiency:

(S2)

Comparing (S5) and (S6), the performances of both the
MMSE and ZF coincide at a high SNR. However, if the
signal experiences situations where the attenuation is
large or where external interference dominates the receiver noise [i.e., lower additive white Gaussian noise
(AWGN) SNR], the MMSE canceler performs better than
the ZF canceler, with a minor increase in complexity.

Fzf = H

-1

1
; 1 - a E.
=
H d (1 - a 2) - a 1

Spectral Efficiency (bits/s/Hz)

where SNR awgn = H 2d Px /Pw . For purposes of comparison, note that the single-wire performance (SWP)
i s R swp = log 2 (1 + SNR awgn), w h e r e a s t h e m a t c h e d f i l t e r b o u n d ( M F B ) i s e v e n h i g h e r : R mfb =
log 2 (1 + (1 + a 2) SNR awgn). Thus, for a high signal-to-noise
ratio (SNR), even small values of a can cause significant
performance degradation.
The zero-forcing (ZF) crosstalk canceler uses the inverse
of the channel matrix as receiver preprocessing:
(S3)

8
7
6
5

AWGN SNR = 20 dB

4
3
2
1
0

AWGN SNR = 10 dB
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Diagonal Dominance Parameter (α)

Thus, the estimate of the symbol of user 1 is
Xt 1 = X 1 +

1
a
W1 W 2.
H d ( 1 - a 2)
H d (1 - a 2)

(1 - a 2) 2
o.
1 + a2

(S5)

perfect channel for precoding and deal with the cancelation
performance of various precoders. Some notes on channel
estimation are given in the "Design Considerations and
Challenges for G.fast" section.
152

MFB
Linear MMSE Canceler

(S4)

It can be seen that ZF processing removes the effect of
crosstalk but enhances the noise by folding noise from
other users. The spectral efficiency for the ZF canceler for
each user is given as

R zf = log 2 e 1 + SNR awgn

1

SWP
Linear ZF Canceler

FIGURE S1. The performance of linear cancelers versus the diagonaldominance parameter ^ah for two users with an AWGN SNR of 10 dB
and 20 dB. For low crosstalk coupling (i.e., a % 1), the MFB is close
to the SWP, and the SWP metric is sufficient. For large a , the MFB
becomes more relevant as a metric for performance comparison.
The figure shows that for an SNR of 20 dB (or higher) the MMSE has
practically no advantage over ZF, except for the case where the channel matrix is very close to singular. For lower SNRs (e.g., 10 dB), the
advantage of the MMSE is greater, and the MMSE canceler performs
better than the ZF.

BC capacity
The capacity region of the Gaussian BC has been derived by
many authors [41]-[45]. Their work was based on the
concept of dirty paper coding [37], which makes it possible

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