Performance Enhancement of Discrete Multi-Tone Systems with a Trigonometric Transform ()
the CP and 
is the 
channel impulse response. Let 
be the
Let Δ be the transmission delay of the signal between the transmitter and the receiver and let
(1)
be the convolutional matrix of the DMT symbols
, 
and
. Define the 
matrix
, where 
and 
are,
respectively, given by:
(2)
![]()
(3)
Matrices ![]()
and ![]()
of size ![]()
are, respectively, given by:
![]()
(4)
![]()
(5)
Let ![]()
be the ![]()
convolution matrix of the CIR and TEQ given by:
![]()
(6)
For the conventional DFT-DMT system, the channel input sequence ![]()
at the output of the IFFT block, as shown in Figure 1, can be presented as [17], where ![]()
is the encoded bit stream.
So, we can define the vector
![]()
(8)
such that the inner product of ![]()
with an N-points vector gives the kth FFT coefficient of that vector, where ![]()
is the Hermitian conjugate transpose operator.
Let the near end crosstalk (NEXT) or additive white
![]()
Figure 1. DFT-DMT system model. xCP means remove the CP.
Gaussian noise (AWGN) vector be:
![]()
(9)
Then the (N + M ‒ 1) × M AWGN or NEXT convolu- tion matrices with the TEQ, GAWGN or GNEXT is:
![]()
(10)
![]()
(11)
The received data contains the noise due to the ISI, ICI, AWGN, NEXT, and suffers from the effects of the channel. Now, we see the dependence of the received signal on the TEQ. The ideal received signal has no noise and is formatted to fit the demodulation scheme. In DMT modulation, this means that the received symbol has mi- nimal noise due to AWGN, NEXT, and ISI. We can de- sign the TEQ to process the received samples to achieve this target.
Next, we will express the desired signal as a function of the TEQ taps. The desired circular convolution of the ![]()
symbol and the CIR in the kth sub-channel, after the TEQ and FFT, can be written as:
![]()
(12)
and the ![]()
circulant matrix ![]()
is:
![]()
(13)
So, the received data ![]()
can be rewritten as:
![]()
(14)
We then write ![]()
for all k as
![]()
(15)
where E[・] is the statistical expectation operator and
![]()
stands for model. The proposed SNR model is
the ratio of the desired data, which excludes the effects of the noise including the ISI and ICI, to the difference be- tween the received data and the desired data.
Derive ![]()
as:
![]()
(16)
where ![]()
is an ![]()
matrix given by:
![]()
(17)
and![]()
, ![]()
and![]()
.
Similarly, derive ![]()
as,
![]()
(18)
where ![]()
is the noise variance, which is measured by the power of the noise with respect to![]()
, ![]()
is the toeplitz variance matrix of the NEXT and ![]()
is the ![]()
identity matrix. Without loss of generality, define constraint set
![]()
, so that ![]()
becomes inde-
pendent of ![]()
over this constraint set. Matrix ![]()
of size ![]()
is defined as:
![]()
(19)
where ![]()
are members of the vector ![]()
defined by Equation (8). ![]()
is a ![]()
upper diagonal matrix de- fined as:
![]()
(20)
![]()
is a lower diagonal
![]()
matrix defined as:
![]()
(21)
![]()
and ![]()
are Hermitian symmetric matrices. Now (15) becomes:
![]()
(22)
The ![]()
is a ratio of quadratic functions of![]()
. The SNR model becomes equivalent to the SNR that could be measured at the output of the FFT in an ADSL system, when the ISI and ICI have been removed from the received signal.
Using the SNR model, the number of bits per symbol that can be supported is:
![]()
(23)
where ![]()
and![]()
, k is the sub- channel index, ![]()
is the set of the indices of the used Ñ sub-channels out of N/2 + 1 sub-channels, ![]()
is the number of bits per data symbol in the sub-channel k, Г is the SNR gap, and it is a function of several factors, in- cluding modulation method, allowable probability of error, gain of any coding applied, and desired system margin.
Maximizing the number of bits allocated to a single channel, ![]()
involves maximizing the argument of the log function. Since the log function is a monotoni- cally increasing function of a non-negative argument, maximizing its non-negative argument will also maxim- ize the function. Mathematical notation for this statement is:
![]()
(24)
This is the well-known generalized eigenvalue prob- lem [16] and the solution is the generalized eigenvector ![]()
corresponding to the largest generalized eigenvalue ![]()
of ![]()
![]()
(25)
Hence, ![]()
where ![]()
denotes the real part. The number of bits per symbol that can be supported in the case of the conven- tional DFT-DMT system with TEQ filter bank is:
![]()
(26)
Bits/symbol.
3. The proposed DST-DMT System
For the proposed DST-DMT system with TEQ filter bank, the channel input sequence ![]()
at the output of the IDST block, as shown in Figure 2, can be presented as [18]:
![]()
(27)
where
![]()
(28)
and ![]()
is the encoded bit stream.
So, the defined vector in Equation (8) will be:
![]()
(29)
such that the inner product of ![]()
with an N-points vector gives the kth DST coefficient of that vector. Note
that in the above equation ![]()
will be ![]()
for
![]()
.
Depending on the defined vector in Equation (29) and
using the above algorithm, we can derive a new ![]()
corresponding to the largest generalized eigenvalue
![]()
of ![]()
![]()
Figure 2. The proposed DST-DMT system model. xCP means remove the CP.
![]()
(30)
Hence,
![]()
(31)
where ![]()
is the number of bits per data symbol in the sub-channel k. The number of bits per symbol that can be supported with the proposed DST-DMT systems implementing a TEQ filter bank is:
![]()
Bits/symbol (32).
4. TEQ Design Algorithm
The first derivative of (23) is:
![]()
(33)
where
![]()
(34)
Notice that![]()
, thus increa-
sing ![]()
increases![]()
. Now we can write:
![]()
. (35)
Let
![]()
(36)
where![]()
.
Set iteration counter![]()
, smoothing factor ![]()
and values ![]()
and ![]()
to zero for all![]()
. The algorithm proceeds as follows:
1) ![]()
2) ![]()
3) Compute ![]()
4) ![]()
5) If ![]()
or ![]()
then return![]()
.
6) If ![]()
set ![]()
else![]()
.
7)![]()
.
8)![]()
.
9) Go back to Step 1 and repeat.
5. Simulation Parameters
We use the eight CSA loops as our test channels [4]. All channel impulse responses consist of 512 samples sam- pled at a rate of 2.208 MHz. We add a fifth-order Che- byshevhighpass filter with cutoff frequency of 5.4 kHz and passband ripples of 0.5 dB to each CSA loop to take into account the effect of the splitter at the transmitter. The DC channel (channel 0), channels 1 - 5, and the Ny- quist channel are not used. We model the channel noise as −140 dBm AWGN distributed over the bandwidth of 1.104 MHz plus near-end-cross-talk (NEXT) noise. The NEXT noise consists of 8 ADSL disturbers as described in the ANSI T1.413 - 1995 standard [4]. The input signal power is 23 dBm distributed equally over all used sub- channels and both the FFT size, and the DST size are set to N = 512. M = 17 and v = 32. The coefficients of the FIR filters are obtained from the MATLAB Discrete Multitone Time-Domain Equalizer (DMTTEQ) Toolbox that was implemented by the Embedded Signal Proces- sing Lab at the University of Texas [19].
Bandwidth optimization is applied by shutting down (not assigning any transmit power to) sub-channels with initial SNR lower than the required SNR to transmit two bits with a given SNR gap of 9.8 + 6 − 4.2 = 11.6 dB. This corresponds to a system margin of 6 dB and a coding gain of 4.2 dB. We are not using any bit loading algorithm, so all bit rate results are calculated from the SNR distribution after the TEQ filter bank is placed into the system. We assume that the power allocation is constant over all used sub-channels and that it is not changed after the TEQ filter bank is placed in the system.
6. Simulation Results
We present simulation results to analyze and compare the performance of the proposed DST-DMT system with TEQ filter bank with the conventional DFT-DMT system with a TEQ. Figure 3 compares the SNR achieved with the proposed DST-DMT system and the conventional DFT-DMT system for CSA loop 1 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT modeled as 8 ADSL disturbers. The figure gives insight into why the performance of the proposed DST-DMT system with a TEQ filter bank out- performs the conventional DFT-DMT system with a TEQ filter bank. The proposed system has a flat magni- tude response over most of the spectrum except at the positions of the highest ISI, while for the conventional system, the SNR decreases as the frequency increases, as displayed in Figures 3-5 for CSA loops 1, 4 and 8, re- spectively.
Figure 6 shows the bit allocation to each sub-chan- nelfor the conventional DFT-DMT system with a TEQ filter bank as compared to that of the proposed DST- DMT system with a TEQ filter bank for CSA loop 1 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT modeled as 8 ADSL disturbers. The proposed system achieves a higher bit allocation for each sub-channel over most of the spec- trum except at the positions of the highest ISI, because each sub-channel carries different numbers of bits de- pending on its SNR. The number of bits assigned to each sub-channel in the case of the conventional system de-
![]()
(a)![]()
(b)
Figure 3. SNR achieved using the proposed system and the conventional system for CSA loop 1 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT is modeled as 8 ADSL disturbers. (a) The con- ventional DFT-DMT system with TEQ filter bank; (b) The proposed DST-DMT system with TEQ filter bank.
![]()
![]()
(a) (b)
Figure 4. SNR achieved using the proposed system and the conventional system for CSA loop 4 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT modeled as 8 ADSL disturbers. (a) Conventional DFT- DMT system with TEQ filter bank; (b) The proposed DST-DMT system with a TEQ filter bank.
![]()
![]()
(a) (b)
Figure 5. SNR achieved using the proposed system and the conventional system for CSA loop 8 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT modeled as 8 ADSL disturbers. (a) Conventional DFT- DMT system with a TEQ filter bank; (b) the proposed DST-DMT with a TEQ filter bank.
creases as the frequency increases, as displayed in Fig- ures 6-8 for CSA loops 1, 4 and 8, respectively.
Table 1 lists the data rates achieved with the proposed DST-DMT system with TEQ filter bank for the CIR in- cluding CSA loops 1-8, as well as that of the convention- al DFT-DMT system. The proposed DST-DMT system achieves higher data rates for each CIR than the conven- tional DFT-DMT system in the range of (2.899 ? 5.369) Mbps.
7. Conclusions
In this paper, we proposed a new TEQ filter bank along with the use of the DST in order to achieve higher bit rates in DMT systems. Simulations experiments have
shown that the proposed DST-DMT system with TEQ filter bank provides a better performance than the con- ventional DFT-DMT system. The results indicate that the proposed DST-DMT system achieves a higher SNR in each sub-channel (over most of the spectrum except at the positions of the highest ISI) than the conventional DFT-DMT system over the eight CSA loops. It was also concluded that the proposed DST-DMT system with TEQ filter bank achieves higher bit rates than the con- ventional DFT-DMT system. We can say that the major advantage of the DST-DMT system with TEQ filter bank is in the energy compaction property of the DST. This property leaves most of the samples at the end of each symbol close to zero, which reduces the ISI, dramatical-
![]()
![]()
(a) (b)
Figure 6. Bit allocation to each sub-channel using the proposed system and the conventional system for CSA loop 1 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT modeled as 8 ADSL disturbers. (a) Conventional DFT-DMT system with a TEQ filter bank; (b) The proposed DST-DMT system with a TEQ filter bank.
![]()
![]()
(a) (b)
Figure 7. Bit allocation to each sub-channel achieved using the proposed system and the conventional system for CSA loop 4 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT modeled as 8 ADSL distur- bers. (a) Conventional DFT-DMT system with a TEQ filter bank. (b) The proposed DST-DMT system with a TEQ filter bank.
![]()
![]()
(a) (b)
Figure 8. Bit allocation to each sub-channel achieved using the proposed system and the conventional system for CSA loop 4 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT modeled as 8 ADSL distur- bers. (a) Conventional DFT-DMT system with a TEQ filter bank; (b) The proposed DST-DMT system with a TEQ filter bank.
![]()
Table 1. The achievable bit rates in Mbps for the proposed DST-DMT system with TEQ filter bank and the conven- tional DFT-DMT system for the CIR involving standard CSA loops 1-8 with M = 17, N = 512, v = 32, input power = 23 dBm, AWGN power = −140 dBm/Hz, and NEXT mod- eled as 8 ADSL disturbers.
ly, leading to a great performance enhancement.
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