INTRODUCTION
One of the most talkedabout areas in the telecommunications industry today
is the Digital Subscriber Line (DSL) technology. The primary applications of
asymmetric DSL (ADSL) are the delivery of digitally encoded video and access
to digital services, particularly Internet (Bahai and Saltzberg,
2002). ADSL meets these needs over a single wirepair. Because virtually
all customers have a wirepair channel providing voice service, no additional
channel need be installed to provide this new service. DMT modulation and Orthogonal
Frequency Division Multiplexing (OFDM) are alldigital multicarrier modulation
schemes. DMT modulation is adopted as the transmission format for asymmetric
format for ADSL (Vanbleu et al., 2006).
There exist different QAM forms, but in the conventional DMT, the rectangular
QAM is used. The rectangular QAM constellation is, in general, suboptimal
in the sense that it does not maximally space the constellation points
for a given energy. In this study a new constellation is introduced that
differs from rectangular QAM and has a lower BER in DMT systems. It is
also must be noted that its implementation complexity does not increase
at all.
Due to an imperfect balance between the twisted pair equipment at the customers’
premises and at the central offices, different noises occur in the telephone
lines (Panigrahi et al., 2006). Digital noise
floor, crosstalk from other lines, impulse noise, InterSymbol Interference
(ISI) and InterChannelInterference (ICI) are the most prominent impairments
limiting throughput (Twardowski, 2006). Impulse noise
is the most corrupting of these noises in terms of errors because of its highly
random characteristics that render its prediction difficult and its high amplitudes
compared to the transmitted DSL signal. The impact of impulse noise on practical
systems depends on the impulse power, duration, interarrival and spectral characteristics
(Ghazi Maghrebi et al., 2007). The frequency response
of the channel is modeled as a lowpass filter and adds two common forms of
interference: ISI and ICI. Also six standard channel models (CSA No. 1 through
CSA No. 6) are used with AWGN and burst noise separately.
The primary advantage of multicarrier modulation is its ability to transmit
information over frequencyselective fading channels using a divideandconquer
approach. Rather than transmitting data on a single carrier at a high data rate,
information can be redistributed into several slower data streams, modulated
on several different carriers and simultaneously transmitted (Wyglinski
et al., 2008). DMT is a multicarrier modulation technique that divides
the transmission bandwidth into a large number of narrow subchannels or tones,
permitting reliable and high data rate transmission over channels where severe
ISI can occur (Zhu et al., 2007). In practice,
DMT modulation is the preferred methodology because of its advantage in computational
complexity (Starr et al., 2003).
ISI induced by the channel often significantly impairs simple receiver performance.
To alleviate this effect, block transmissions are widely adopted. In such a
transmission scheme, the transmitted data stream is divided into consecutive
equal size blocks and redundancy is added between blocks (Wu
and Chern, 2007). Unfortunately, the cost effective handling of ISI comes
with the expense of bandwidth efficiency reduction caused by CP at the transmitter
(Kim and Lee, 2007).
MATERIALS AND METHODS
This research was started in Islamic Azad University about two years
ago. Block diagram of a DMT transceiver is shown in Fig.
1. Each tone is loaded with a certain QAM constellation. The frequency
response of the channel is modeled as a lowpass filter because in wire
mediums, high frequency electromagnetic waves are quickly attenuated
while low frequency waves retain much of their power, even over long distances.
In this study, the channel impulse response h = [h_{0}…h_{L}]
is longer than the CP length (v), i.e., L>v. The received symbol can
be written as the convolution of the transmitted symbol and the channel
impulse response, so the transmitted symbol at time k1 will contribute
to the received symbol at time k. Then the matrix equation will be
and
where,
are the Nth transmitted and received sample at time k, respectively. Also
I and 0 are the identity and zero matrices, respectively. The relation
of DMT symbols vectors in frequency and time domain are:

Fig. 1: 
DMT transceiver block diagram 
where, I_{N} is NxN IDFT and F_{N} is NxN DFT. By taking
FFT of Eq. 1.
where, T^{(k1)} and T^{(k)} are both Toeplitz matrices.
Substituting Eq. 2 in Eq. 3, the demodulated
received symbol becomes:
The terms T^{(k)} P and T^{(k1)} P are not circulant, so pre
and post multiplying with F_{N} and I_{N} does not produce any
diagonal matrices. Therefore, we can not apply eigenvalue decomposition of a
circulant matrix. The eigenvectors of a circulant matrix form a DFT matrix and
its eigenvalues are equal to the FFT of the [h_{0} h_{1} …
h_{v} 0]^{T} vector. The unwanted contributions in
from subsymbols different from
are ISI and ICI interferences (Acker et al., 2001).
Different systems use different constellations, compromising between the complexity
of the implementation and efficiency of the constellation for the given transmit
power. QAM which is used in conventional DMT is a class of nonconstant envelope
schemes that can achieve higher bandwidth efficiency than MPSK with the same
average signal power (Xiong, 2000). In this study, a novel
constellation is introduced and compared with the conventional QAM constellation
in DMT system. The new one is a kind of nonrectangular QAM which is named NQAM.
These constellations are shown in Fig. 2.

Fig. 2: 
Different used constellations 
Some of the main parameters of modulated signals, for analyzing the signal
constellation diagram, are the average power per symbol, peaktoaverage
ratio (PAR), minimum Euclidean distance and the noise immunity. A 2^{N}QAM
signal is initially considered (M = 2^{N}). The average
power per symbol, P and the peaktoaverage power ratio (PAR) are:
where, I_{k} and Q_{k} are the inphase and quadrature
kth components, respectively. P_{k} = I_{k}^{2}+Q_{k}^{2}
is the power in the kth symbol and 3dB is due to the PAR of a sine wave.
Also k is number of the given constellation point, k = 1, 2,… , M.
The third parameter is the Euclidian distance d(i, j) that is the geometric
distance between points i and j of the constellation diagram. The minimum Euclidian
distance, d_{N}, normalized to the average symbol power P indicates
the relative noise immunity of the particular constellation (Golden
et al., 2006):
Taking minimum Euclidian distance of a 2QAM, d_{2}, as a reference,
the noise immunity of M points constellations, with the minimum Euclidian distance
of d_{M}, can be estimated as (Golden et al.,
2006):
This value expresses the difference in SNR necessary to ensure the same
average probability of symbol error when different constellations are
employed.
The probability of the correct detection of a rectangular QAM symbol
is
where,
is the symbol error probability of ary
amplitude modulation with onehalf the average power of the QAM signal.
Then we have:
where, E_{avg}/N_{o} is the average SNR per symbol. The
symbol error probability, P_{S}, of the rectangular QAM is:
At high SNRs,
To obtain the bit error probability from the symbol error probability,
for rectangular QAM with Gray coding, each symbol error most likely causes
one bit error at high SNRs. Thus the bit error probability will be:
RESULTS AND DISCUSSION
In this study, a new constellation introduced instead of QAM modulation
in conventional DMT. These constellations are compared based on their
specifications in Table 1. As shown their average powers
are the same and in this situation, we compared their performances as
BER.
For the first experiment, six standard channels, CSA No.1 through CSA No. 6,
are applied to the DMT system separately. The cyclic prefix length is 16 bits
and channel impulse response length is 512 taps.
Table 1: 
Main parameters of different constellations 

Also the channels have the
AWGN noise and the results are shown in Fig. 3af. It is clear
that NQAM has lower BER with respect to conventional QAM especially in high
SNRs. For the second experiment the channels have burst noise; it means that
the length, power and the position of occurrence in the channel are variables.
In this case, as shown in Fig. 4af, the new constellation
also has lower BER. The program is been run 10000 times for each channel and Fig. 3 and 4 show the average of the results.
It is clear, based on the results of simulation in Fig.
3 and 4, that the new constellation (NQAM) for all
standard channels has a better performance, in terms of BER, with respect
to the conventional rectangular QAM for different SNRs with AWGN and burst
noises. As it is clear, NQAM has slightly better performance with respect
to the rectangular QAM for high SNRs.

Fig. 3: 
The BER of QAM and NQAM on CSA No. 1 through CSA No.
6 channels with AWGN noise (a) CSA No. 1, (b) CSA No. 2, (c) CSA No.
3, (d) CSA No. 4, (e) CSA No. 5 and (f) CSA No. 6 

Fig. 4: 
The BER of QAM and NQAM on CSA No. 1 through CSA No.
6 channels with burst noise (a) CSA No. 1, (b) CSA No. 2, (c) CSA
No. 3, (d) CSA No. 4, (e) CSA No. 5 and (f) CSA No. 6 
Finally, based on these results, one can conclude that the new constellation,
NQAM, has a better performance and can replace the conventional rectangular
QAM in DMT system.