OPTIMAL POWER ALLOCATION IN A MIMO-OFDM TWISTED

PAIR TRANSMISSION SYSTEM WITH FAR-END CROSSTALK

Andreas Ahrens, Christoph Lange

Institute of Communications Engineering, University of Rostock

Richard-Wagner-Str. 31, 18119 Rostock, Germany

Keywords:

Twisted-pair Cable, OFDM, Power Allocation, Multiple-Input Multiple-Output System, Lagrange Multiplier

Method, Singular-Value Decomposition.

Abstract:

Crosstalk between neighbouring wire pairs is one of the major impairments in digital transmission via multi-

pair copper cables, which essentially limits the transmission quality and the throughput of such cables. For

high-rate transmission, often the strong near-end crosstalk (NEXT) disturbance is avoided or suppressed and

only the far-end crosstalk (FEXT) remains as crosstalk inﬂuence. If FEXT is present, signal parts are trans-

mitted via the FEXT paths from the transmitter to the receiver in addition to the direct transmission paths.

Therefore transmission schemes are of great practical interest, which take advantage of the signal parts trans-

mitted via the FEXT paths. Here a SVD (singular-value decomposition) equalized MIMO-OFDM system

is investigated, which is able to take advantage of the FEXT signal path. Based on the Lagrange multiplier

method an optimal power allocation schema is considered in order to reduce the overall bit-error rate at a ﬁxed

data rate and ﬁxed QAM constellation sizes. Thereby an interesting combination of SVD equalization and

power allocation is considered, where the transmit power is not only adapted to the subchannels but rather to

the symbol amplitudes of the SVD equalized data block. As a result it can be seen that the exploitation of

FEXT is important for wireline transmission systems in particular with high couplings between neighbouring

wire pairs and the power allocation is possible taking the different subcarriers into account.

1 INTRODUCTION

OFDM (orthogonal frequency division multiplex) is a

widely accepted transmission schema in both, wire-

line and wireless transmission. Examples include

digital subscriber line (DSL) (Bingham, 2000), Eu-

ropean digital video broadcast (DVB), digital audio

broadcast (DAB) and wireless local area networks

(WLAN) such as 802.11a and HIPERLAN/2. A lot

of publications have been published in the literature

where the resilience of multicarrier transmission sys-

tems against the delay spread was highlighted, provid-

ing a sufﬁcient guard interval length (Bingham, 2000;

van Nee and Prasad, 2000).

In a multiuser scenario considered here, a resilience

against intersymbol interference (ISI) isn’t sufﬁcient

to fulﬁll given quality criteria. In local cable net-

works, crosstalk is one of the most limiting dis-

turbances (Valenti, 2002). Since the NEXT is a

very strong disturbance several techniques have been

developed in order to avoid or suppress it (Honig

et al., 1990). In this case only the FEXT remains as

crosstalk inﬂuence. Often optical ﬁbre transmission is

used up to a building’s entrance and the last few hun-

dred metres within the building are bridged by cop-

per cables. For such short cables used in high-data

rate systems in the local cable area, the FEXT is par-

ticularly strong (Valenti, 2002) and as a result heavy

multiuser interference arise. This has lead to a great

interest in transmission systems which are capable to

take such disturbances into account.

In different publications, e.g, in (Lange and

Ahrens, 2005), it was theoretically shown, that gains

are possible by FEXT exploitation. In this contribu-

tion an interesting approach for the practical exploita-

tion of the FEXT signal parts is presented: On each

wire pair the multicarrier technique OFDM (Hanzo

et al., 2000) is used and in addition the mutual impact

of the wire pairs in a cable binder via far-end crosstalk

is taken into account. Therefore the n-pair cable

is modelled as a (n, n) MIMO transmission system

and the combination of singular-value decomposition

(SVD) and optimal power allocation using the La-

grange Multiplier method is considered with the aim

164

Ahrens A. and Lange C. (2006).

OPTIMAL POWER ALLOCATION IN A MIMO-OFDM TWISTED PAIR TRANSMISSION SYSTEM WITH FAR-END CROSSTALK.

In Proceedings of the International Conference on Signal Processing and Multimedia Applications, pages 164-169

DOI: 10.5220/0001568001640169

Copyright

c

SciTePress

of a bit-error minimization at a given data rate. Con-

trary to other publications considering a similar topic,

here the focus lies on the combination of singular-

value decomposition and power allocation. Thereby

the optimal power allocation solution is presented for

given boundary conditions (ﬁxed QAM constellation

size and limited total transmit power).

The remaining part of this contribution is organized

as follows: Section 2 introduces the cable charac-

teristics and the considered system model including

the MIMO-OFDM transmission systems with SVD-

based equalization. In section 3 possible optimiza-

tion objectives for MIMO transmission systems are

discussed and the underlying optimization criteria are

brieﬂy reviewed. In section 4 the transmit power al-

location scheme is explained and in section 5 the ob-

tained results are presented and discussed. Finally,

section 6 provides some concluding remarks.

2 SYSTEM MODEL

The distorting inﬂuence of the cable on the wanted

signal is modelled by the cable transfer function

G

k

(f) = e

−l

j

f

f

0

, (1)

where l denotes the cable length (in km) and

f

0

represents the characteristic cable frequency (in

MHz · km

2

) (Kreß and Krieghoff, 1973).

The far-end crosstalk coupling is covered by the

transfer function G

F

(f) with

|G

F

(f)|

2

= K

F

· l · f

2

, (2)

whereby K

F

is a coupling constant of the far-end

crosstalk, which depends on the cable properties such

as the type of isolation, the number of wire pairs and

the kind of combination of the wire pairs within the

binders (Valenti, 2002).

The considered cable binder consists of n wire

pairs and therefore a (n, n) MIMO transmission sys-

tem arises. The mapping of the transmit signals

u

s µ

(t) onto the received signals u

k µ

(t) (with µ =

1, . . . , n) can be described accordingly to Fig. 1.

On each wire pair of the cable binder OFDM

(orthogonal frequency division multiplexing) is used

as transmission technique to combat the effects of

the frequency-selective channel (Bahai and Saltzberg,

1999; Bingham, 2000). In such a (n, n)-MIMO-

OFDM system, an N -point IFFT (N subchannels)

has to be performed on every wire pair. By inserting

a guard interval (GI) in front of the transmit signal

and removing it at the receiver side an interchannel

interference (ICI) and intersymbol interference (ISI)

free transmission can be established, assuming that

the length of the GI is longer than the temporal ex-

tension of the channel impulse response (van Nee and

u

s 1

(t)

u

s 2

(t)

u

k 1

(t)

u

k 2

(t)

G

k

(f)

G

k

(f)

G

F

(f)

G

F

(f)

Figure 1: MIMO cable transmission model system with

FEXT (n = 2).

Prasad, 2000). After removing the GI at the receiver

an N-point FFT has to be performed. The arising sys-

tem model is depicted in Fig. 2.

The kth data block a

µ

[k] of the length n

b

= N

transmitted by the wire pair µ (with µ = 1, . . . , n) is

denoted by

a

µ

[k] = (a

1 µ

[k], a

2 µ

[k], . . . , a

N µ

[k])

T

(3)

and results in a received vector (µ = 1, . . . , n)

u

µ

[k] = (u

1 µ

[k], u

2 µ

[k], . . . , u

N µ

[k])

T

. (4)

To get an adequate system model, it is necessary to

rearrange the symbols of the data vector a

µ

[k] (with

µ = 1, . . . , n) deﬁned in (3). Combining all symbols

which are transmitted via the same subcarrier in one

vector results in

e

a

κ

[k] = (a

κ 1

[k], . . . , a

κ µ

[k], . . . , a

κ n

[k])

T

. (5)

Stacking all subcarriers in one vector leads to

e

a[k] =

e

a

T

1

[k], . . . ,

e

a

T

κ

[k], . . . ,

e

a

T

N

[k]

T

, (6)

where

e

a

κ

[k], deﬁned in (5), contains all symbols

which are transmitted via the subcarrier κ. Therefore

(6) contains all symbols which are transmitted in one

time slot simultaneously since n

b

equals N.

After removing the GI at the receiver side a N-

point FFT has to be performed and leads to the re-

ceived vector similarly to (6)

e

u[k] =

e

u

T

1

[k], . . . ,

e

u

T

κ

[k], . . . ,

e

u

T

N

[k]

T

, (7)

a

1

.

.

.

a

n

u

1

.

.

.

u

n

u

s 1

(t)

u

s n

(t)

u

k 1

(t)

u

k n

(t)

IFFT

IFFT

FFT

FFT

GI

GI

GI

−1

GI

−1

MIMO

cable

n

1

(t)

n

n

(t)

Figure 2: MIMO-OFDM cable transmission model.

OPTIMAL POWER ALLOCATION IN A MIMO-OFDM TWISTED PAIR TRANSMISSION SYSTEM WITH

FAR-END CROSSTALK

165

where

e

u

κ

[k] contains the ISI- and ICI-free receive

symbols of the subcarrier κ. However

e

u

κ

[k] still con-

tains the crosstalk between neighboring wire pairs on

each subcarrier. Due to the GI this vector has the same

length as the data vector deﬁned in (6).

Additionally a white Gaussian noise n

µ

(t) (with

µ = 1, 2, . . . , n) with power spectral density Ψ

0

is

assumed, which results after receive ﬁltering in the

vector n and can be deﬁned similar to (7) as

n[k] =

n

T

1

[k], . . . , n

T

κ

[k], . . . , n

T

N

[k]

T

. (8)

Thereby it is assumed that the noise components are

independently from each other, which can be justiﬁed

by the rectangular shape of the receive ﬁlter functions.

The block oriented transmission system description

is given by

e

u = R ·

e

a + n . (9)

The matrix R has a block diagonal structure

R =

R

1

0 ··· 0

0 R

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0 0 ··· R

N

. (10)

In equation (10) zero-matrices are denoted by 0 and

for the matrices R

κ

(with κ = 1, . . . , N) the follow-

ing syntax is used

R

κ

=

r

(κ)

1 1

··· r

(κ)

1 n

.

.

.

.

.

.

.

.

.

r

(κ)

n 1

··· r

(κ)

n n

, (11)

with the elements describing the couplings of the data

symbols on the subchannel κ. Based on the symme-

try of the considered transmission system r

(κ)

ν µ

(for

ν = µ) can be determined taking the FFT of g

k

(t) =

F

−1

{G

k

(f)} into account. The elements r

(κ)

ν µ

(for

ν 6= µ) consider the coupling between neighbouring

wire pairs and can be ascertained calculating the FFT

of g

k fn

(t) = F

−1

{G

F

(f) · G

k

(f)}. The κth value

of this vector represents r

(κ)

ν µ

. The elements r

(κ)

ν µ

(for

ν 6= µ) are assumed to be identical for each κ, al-

though in practical systems the coupling between the

wire pairs is slightly different and it depends on their

arrangement in the binder (Valenti, 2002).

The remaining interferences on each subcarrier can

now be eliminated by an efﬁcient equalization strat-

egy. A popular strategy is represented by the singular

value decomposition (SVD), which can be done on

each subcarrier separately. The SVD of the matrix

R

κ

can be written as

R

κ

=

e

U

κ

·

e

V

κ

·

f

W

H

κ

, (12)

where

e

U

κ

and

f

W

H

κ

are unitary matrices and

e

V

κ

is a

real diagonal matrix (Kovalyov, 2004).

The rearranged data vector

e

a

κ

(5) is multiplied by

the matrix

f

W

κ

and results in the transmit data vector

e

b

κ

. The received vector

e

u

κ

= R

κ

·

e

b

κ

+ n

κ

is multi-

plied by the matrix

e

U

H

κ

. Thereby neither the transmit

power nor the noise power is enhanced. The over-

all transmission relationship for the subcarrier κ (with

κ = 1, . . . , N) is deﬁned as

e

y

κ

=

e

U

H

κ

·

e

u

κ

=

e

U

H

κ

·

R

κ

·

f

W

κ

·

e

a

κ

+ n

κ

=

e

V

κ

·

e

a

κ

+

e

U

H

κ

· n

κ

. (13)

3 OPTIMIZATION OBJECTIVES

AND QUALITY CRITERIA

Current signal processing strategies for MIMO sys-

tems typically fall into two categories: data through-

put maximization at a given transmission quality or

bit-error rate minimization at a ﬁxed data rate. In this

contribution we have restricted ourselves to the BER

minimization at a ﬁxed data rate. Thereby optimal but

highly complex or suboptimal solutions with reduced

complexity can be found, e.g. (Krongold et al., 2000;

Jang and Lee, 2003; Park and Lee, 2004; Ahrens and

Lange, 2006).

The signal-to-noise ratio (SNR) is a reasonable per-

formance criterion for noise-dominated scenarios. A

signal-to-noise ratio

̺ =

(Half vertical eye opening)

2

Disturbance Power

=

(U

A

)

2

(U

R

)

2

(14)

is often deﬁned as a quality parameter (Kreß et al.,

1975) with the half vertical eye opening U

A

and the

noise disturbance power U

2

R

per quadrature compo-

nent. Between the signal-to-noise ratio ̺ = U

2

A

/U

2

R

and the bit-error probability in the general case of M-

ary quadrature amplitude modulation (QAM) the in-

terrelationship

P

f

=

2

ld(M)

1 −

1

√

M

erfc

r

̺

2

(15)

holds (Kalet, 1987; Proakis, 2000). The SVD-based

equalization on each subcarrier leads to a different

half vertical eye opening

U

(ε)

A

=

p

ξ

ε

· U

s

(16)

for each data symbol. Here, U

s

denotes the half-level

transmit amplitude and

√

ξ

ε

are the positive square

roots of the eigenvalues of the matrix R

H

κ

R

κ

, de-

scribing the distortions on each subcarrier. Further-

more, each symbol of the data vector a is disturbed

by a noise with identical disturbance power in the

SIGMAP 2006 - INTERNATIONAL CONFERENCE ON SIGNAL PROCESSING AND MULTIMEDIA

APPLICATIONS

166

quadrature components, which is assumed to be un-

correlated with power U

2

R

each. The bit-error proba-

bility per symbol for QAM is deﬁned as

P

f µ

=

2

ld(M)

1 −

1

√

M

erfc

U

(µ)

A

√

2 U

R

!

.

(17)

The aggregate bit-error probability

P

f

=

1

N

b

N

b

−1

X

µ=0

P

f µ

(18)

is derived by averaging over the error probabilities of

all N

b

= n · n

b

symbols of the data block, since all

eye openings occur with the same probability.

4 POWER ALLOCATION

The half vertical eye opening of each symbol posi-

tion is weighted by the factor

√

p

µ

and therefore all

eye openings of the data block are in general different

from each other. Assuming an identical noise power

for all symbol positions, the symbol positions with the

smallest half vertical eye openings dominate the bit-

error rate. Here a transmit power partitioning scheme

would be necessary in order to minimize the overall

bit-error rate under the constraint of a limited total

transmit power.

In a power allocation scheme each symbol of the

data block is weighted by a real factor

√

p

µ

. This

setup leads to the half vertical eye opening

U

(µ)

A,PA

=

√

p

µ

·

p

ξ

µ

· U

s

(19)

per symbol. The power allocation evaluates the half-

level amplitude U

s

of the µth symbol by the factor

√

p

µ

. This causes in general a modiﬁed transmit am-

plitude U

s

√

p

µ

for each symbol of the transmit data

vector and the signal constellation changes. Together

with the noise disturbance per quadrature component

a BER per symbol and block can be calculated:

P

f µ

=

2

1 −

1

√

M

ld(M)

erfc

r

p

µ

ξ

µ

2

·

U

s

U

R

!

.

(20)

The aggregate bit-error probability per block yields

P

f

=

2

1 −

1

√

M

ld(M) N

b

N

b

−1

X

µ=0

erfc

r

p

µ

ξ

µ

2

·

U

s

U

R

!

.

(21)

In the subchannels of the multicarrier system inves-

tigated in this contribution M-ary square QAM with

transmit power (Proakis, 2000)

P

s QAM

=

2

3

U

2

s

(M − 1) (22)

is used (Proakis, 2000). Using a parallel transmission

over N subchannels the overall mean transmit power

per wire yields to

P

s

= N · P

s QAM

= N

2

3

U

2

s

(M − 1) , (23)

and results in a total transmit power of n P

s

by taking

n wire-pairs into account.

Considering now generally different half-level am-

plitudes U

s

√

p

µ

after power allocation on the symbol

layers, it follows

P

s µ

=

√

p

µ

2

·

2

3

U

2

s

(M − 1) = p

µ

2

3

U

2

s

(M − 1)

(24)

for the µth symbol position.

If now a block of N

b

data symbols, transmitted

over N parallel subchannels per wire pair, is ana-

lyzed with these generally different half-level ampli-

tudes U

s

√

p

µ

after power allocation, the mean trans-

mit power of the block becomes

P

s,PA

= n N

2

3

U

2

s

(M − 1)

1

N

b

N

b

−1

X

µ=0

√

p

µ

2

.

(25)

From the requirement

P

s,PA

− n P

s

= 0 (26)

that the overall mean transmit power for the whole

binder consisting of n wire pairs is limited to n P

s

it

follows, that the auxiliary condition

n N

N

b

P

s QAM

N

b

−1

X

µ=0

√

p

µ

2

− n N P

s QAM

= 0

N

b

−1

X

µ=0

p

µ

− N

b

= 0

(27)

has to be maintained.

In order to ﬁnd the optimal

√

p

µ

the Lagrange mul-

tiplier method is used (Park and Lee, 2004; Ahrens

and Lange, 2006). Contrary to other publications,

here the power allocation has not been carried out

on each subcarrier independently from each other,

although this might be possible. To consider the

subcarrier speciﬁc distortions (e. g. increasing cable

attenuation with increasing subcarrier indices) in a

best possible way, here all subcarrier singular val-

ues are combined in one vector, in order to smooth

out the distortions. The Lagrangian cost function

J(p

0

, ··· , p

N

b

−1

) may be expressed as

J(···) =

A

N

b

N

b

−1

X

µ=0

erfc

r

p

µ

ξ

µ

2

·

U

s

U

R

!

+λ·B

N

b

,

(28)

OPTIMAL POWER ALLOCATION IN A MIMO-OFDM TWISTED PAIR TRANSMISSION SYSTEM WITH

FAR-END CROSSTALK

167

with the Lagrange multiplier λ (Park and Lee, 2004)

and

A =

2

ld(M)

1 −

1

√

M

. (29)

The parameter B

N

b

in (28) describes the boundary

condition

B

N

b

=

N

b

−1

X

µ=0

p

µ

− N

b

= 0 (30)

following from (27). Differentiating the Lagrangian

cost function J(p

0

, ··· , p

N

b

−1

) with respect to the

p

µ

and setting it to zero, leads to the optimal set of

power allocation coefﬁcients. As solution for the p

µ

we get (a computer algebra system such as MAPLE or

MATLAB can come in handy)

p

µ

=

1

ξ

µ

U

2

R

U

2

s

W

A

2

ξ

2

µ

2 π N

2

b

λ

2

U

4

s

U

4

R

!

, (31)

where W(x) describes the Lambert W function (Cor-

less et al., 1996). The parameter λ can be calcu-

lated by insertion of (31) in (30) and numeric analysis.

With calculated λ the optimal p

µ

can be determined

using (31).

Power allocation with lower complexity can be

achieved by suboptimal methods, which can on the

one hand rely on an approximation for the erfc(x)

function or which ensure on the other hand equal

signal-to-noise ratios per symbol (Ahrens and Lange,

2006).

5 RESULTS

The FEXT impact is in particular strong for short ca-

bles (Valenti, 2002). Therefore for numerical analy-

sis an exemplary cable of length l = 0.4 km with

n = 10 wire pairs is chosen. The wire diameter is

0.6 mm and hence a characteristic cable frequency of

f

0

= 0.178 MHz · km

2

is assumed. On each of the

wire pairs a multicarrier system with N = 10 sub-

carriers was considered. The actual crosstalk circum-

stances are difﬁcult to acquire and they vary from ca-

ble to cable. Therefore a mean FEXT coupling con-

stant of K

F

= 10

−13

(Hz

2

·km)

−1

is exemplarily em-

ployed (Valenti, 2002; Aslanis and Ciofﬁ, 1992). The

average transmit power on each wire pair is supposed

to be P

s

= 1 V

2

and as an external disturbance a

white Gaussian noise with power spectral density Ψ

0

is assumed. Identical systems on all wire pairs were

presumed (multicarrier symbol duration T

s

= 2 µs,

M-ary QAM, a block length of n

b

= 10 and a guard

interval length of T

g

= T

s

/2). Furthermore, the base-

band channel of the multicarrier system is excluded

from the transmission in order to provide this fre-

quency range for analogue telephone transmission.

10 15 20 25 30

10

−8

10

−6

10

−4

10

−2

10

0

M = 4

M = 16

M = 64

10 · lg(E

s

/Ψ

0

) (in dB) →

bit-error rate →

SISO-OFDM without PA

MIMO-OFDM without PA

MIMO-OFDM with PA

Figure 3: BER comparison analyzing SISO-OFDM (n =

1) and MIMO-OFDM (n = 10, K

F

= 10

−13

(Hz

2

·

km)

−1

) with and without PA for different QAM constel-

lation sizes.

For a fair comparison the ratio of symbol energy

to noise power spectral density at the cable output is

deﬁned for the MIMO case (n > 1) according to

E

s

Ψ

0

= (T

s

+ T

g

)

P

k

+ (n − 1)P

k fn

Ψ

0

, (32)

with

P

k

= P

s QAM

· T

s

+∞

Z

−∞

|G

s

(f) · G

k

(f)|

2

df (33)

and

P

k fn

= P

s QAM

·T

s

+∞

Z

−∞

|G

s

(f)·G

k

(f)·G

F

(f)|

2

df .

(34)

Thereby it is assumed, that the mean transmit power

tends to P

s QAM

and G

s

(f) is the transmit ﬁlter trans-

fer function describing the OFDM pulse shaping. The

results are depicted in Fig. 3 with the QAM constel-

lation sizes M as parameter. For purposes of sim-

plicity and in order to obtain meaningful results, the

QAM constellation sizes are chosen to be equal in all

subchannels of the multicarrier systems. This seems

to be reasonable in the example considered here,

since short cables do not have very strong frequency-

selective characteristics. For general cable transmis-

sion the optimization of bit loading with low com-

plexity in the MIMO context remains open for fur-

ther investigations. Furthermore it seems to be worth

mentioning that in case of different QAM constella-

tion sizes M also different overall bit rates can be

achieved. This could be used for an adaptation of

the bit rate to the user’s needs. In case of MIMO-

OFDM the signal parts, which are transmitted via the

SIGMAP 2006 - INTERNATIONAL CONFERENCE ON SIGNAL PROCESSING AND MULTIMEDIA

APPLICATIONS

168

FEXT paths are no longer disturbance: Now they are

exploited as useful signal parts. Therefore the trans-

mission quality is improved compared to the SISO-

OFDM case (OFDM transmission over a (ﬁctive) per-

fectly shielded single wire pair). Similar results are

known from MIMO radio transmission with multi-

ple transmit and/or receive antennas, where multi-

ple transmission paths are exploited, too (Raleigh and

Ciofﬁ, 1998; Raleigh and Jones, 1999).

The results show that under severe FEXT inﬂu-

ence it is worth taking the FEXT signal paths into

account (Fig. 3). At small FEXT couplings no sig-

niﬁcant gains are possible by MIMO-OFDM without

PA compared to a perfectly shielded wire pair (SISO-

OFDM), because the FEXT coupled signal parts are

very small. The results in Fig. 3 show further the po-

tential of appropriate power allocation strategies. The

absolut achievable gains depend on the actual cable

type and on the isolation of the wire pairs.

6 CONCLUSION

In this contribution, the practical exploitation of the

FEXT paths for improving the signal transmission

quality was investigated in terms of an exemplary

multicarrier transmission system on a symmetric cop-

per cable. It was shown, that the MIMO-OFDM cable

transmission enables gains in the BER performance

especially under severe FEXT inﬂuence. Thereby it

could be shown that power allocation is necessary to

achieve a minimum bit-error rate. In the exemplary

system considered here some restrictions were made,

which directly lead to some open points for further

investigations: In order to use MIMO-OFDM for ca-

bles of any length the most important open point is

the optimization of bit loading in combination with

the power allocation in the MIMO-OFDM context.

REFERENCES

Ahrens, A. and Lange, C. (2006). Transmit Power Alloca-

tion in SVD Equalized Multicarrier Systems. Inter-

national Journal of Electronics and Communications

(AE

¨

U), 60. accepted for publication.

Aslanis, J. T. and Ciofﬁ, J. M. (1992). Achievable Infor-

mation Rates on Digital Subscriber Loops: Limiting

Information Rates with Crosstalk Noise. IEEE Trans-

actions on Communications, 40(2):361–372.

Bahai, A. R. S. and Saltzberg, B. R. (1999). Multi-Carrier

Digital Communications – Theory and Applications of

OFDM. Kluwer Academic/Plenum Publishers, New

York, Boston, Dordrecht, London, Moskau.

Bingham, J. A. C. (2000). ADSL, VDSL, and Multicarrier

Modulation. Wiley, New York.

Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J.,

and Knuth, D. E. (1996). On the Lambert W Function.

Advances in Computational Mathematics, 5:329–359.

Hanzo, L., Webb, W. T., and Keller, T. (2000). Single- and

Multi-carrier Quadrature Amplitude Modulation. Wi-

ley, Chichester, New York, 2 edition.

Honig, M. L., Steiglitz, K., and Gopinath, B. (1990). Multi-

channel Signal Processing for Data Communications

in the Presence of Crosstalk. IEEE Transactions on

Communications, 38(4):551–558.

Jang, J. and Lee, K. B. (2003). Transmit Power Adapta-

tion for Multiuser OFDM Systems. IEEE Journal on

Selected Areas in Communications, 21(2):171–178.

Kalet, I. (1987). Optimization of Linearly Equalized

QAM. IEEE Transactions on Communications,

35(11):1234–1236.

Kovalyov, I. P. (2004). SDMA for Multipath Wireless Chan-

nels. Springer, New York.

Kreß, D. and Krieghoff, M. (1973). Elementare Ap-

proximation und Entzerrung bei der

¨

Ubertragung von

PCM-Signalen

¨

uber Koaxialkabel. Nachrichtentech-

nik Elektronik, 23(6):225–227.

Kreß, D., Krieghoff, M., and Gr

¨

afe, W.-R. (1975).

G

¨

utekriterien bei der

¨

Ubertragung digitaler Signale. In

XX. Internationales Wissenschaftliches Kolloquium,

Nachrichtentechnik, pages 159–162, Ilmenau. Tech-

nische Hochschule.

Krongold, B. S., Ramchandran, K., and Jones, D. L.

(2000). Computationally Efﬁcient Optimal Power Al-

location Algorithms for Multicarrier Communications

Systems. IEEE Transactions on Communications,

48(1):23–27.

Lange, C. and Ahrens, A. (2005). Channel Capacity of

Twisted Wire Pairs in Multi-Pair Symmetric Copper

Cables. In Fifth International Conference on Informa-

tion, Communications and Signal Processing (ICICS),

pages 1062–1066, Bangkok (Thailand).

Park, C. S. and Lee, K. B. (2004). Transmit Power Alloca-

tion for BER Performance Improvement in Multicar-

rier Systems. IEEE Transactions on Communications,

52(10):1658–1663.

Proakis, J. G. (2000). Digital Communications. McGraw-

Hill, New York, 4 edition.

Raleigh, G. G. and Ciofﬁ, J. M. (1998). Spatio-Temporal

Coding for Wireless Communication. IEEE Transac-

tions on Communications, 46(3):357–366.

Raleigh, G. G. and Jones, V. K. (1999). Multivariate

Modulation and Coding for Wireless Communication.

IEEE Journal on Selected Areas in Communications,

17(5):851–866.

Valenti, C. (2002). NEXT and FEXT Models for Twisted-

Pair North American Loop Plant. IEEE Journal on

Selected Areas in Communications, 20(5):893–900.

van Nee, R. and Prasad, R. (2000). OFDM for wireless

Multimedia Communications. Artech House, Boston

and London.

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FAR-END CROSSTALK

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