2N according to (1) before IFFT. Filtering of IFFT output by the filter bank is the final step on
the transmitter side. The filter bank was built-up from polyphase components of prototype
filter mentioned above.
378
Discrete Time Systems
The procedure in the receiver is inverted. Firstly the received signal is filtered by receiver
filter bank and after that, FFT is performed. In the next step the added symbols are removed
i.e. the symbols 0 and N...2N. Through the characteristics of FMT modulation mentioned in
chapter 5 is necessary to use equalization. DFE equalization with RLS adaptive algorithm is
used in our model. Equalized symbols are then demodulated by a bank of QAM
demodulators.
Transmitter - Channel
Bank of
Channel
Bernoulli
QAM
Transmitter
Receiver
FFT
30
REC_OUT
selection
Binary
modulators
FilterBank
FilterBank
Goto2
Bernoul i Random
Subsystem7
Binary Generator
To
BIT_SEQ
Frame
Goto4
Original
TRANS_INP
Goto3
Receiver
Equalized - DFE
Bank of
REC_OUT
IFFT
To
Input
RLS DFE
Equalized
To
QAM
BIT_SEQ_OUT
Frame
Terminator
Frame
From2
Desired
per-channel
demodulators
Goto1
equalization
Err
Mode
TRANS_INP
Before_DFE
Decision Directed
0
1
Training Mode
Fig. 15. FMT system in Matlab-Simulink
The resistance of narrowband noise on chosen carrier was tested on this model. This type of
interference is very common in real conditions. Narrowband noise on 10th carrier was
applied in our case.
The evaluation was done by measuring the signal to noise ratio, SNR. For the half-overlap
FMT modulation the measurements were performed only for variants with subchannel
crossing at level -3dB. The results of the measurements are presented in Fig.17. It is obvious
that the DMT modulation has the worst properties, where the narrowband interference on
10th carrier degrades SNR on a large number of surrounding carriers. The opposite case is
FMT modulation, in both variants the SNR is degraded only on the carrier with narrowband
interference. For the half-overlap FMT modulation degradation on two nearby carriers was
expected, but the measurement shows degradation only on 10th carrier.
The model described above can be adjusted and implemented on DSP. The chosen
development kit uses the TI C6713 floating-point digital signal processor. The model is
divided into part of transmitter and part of receiver. The signal processing procedure is
identical to the model. After generation of pseudorandom binary sequence, QAM
modulation is performed. The number of bits transmitted on the sub-carrier is chosen before
the actual implementation. After it the IFFT modulation is performed and each output is
filtered by the transmitter filter bank. The last step is to adjust the amplitude of the
transmitted signal to the range of DAC converter. In this way modified model was then
Half-overlap Subchannel Filtered MultiTone Modulation and Its Implementation
379
compiled and implemented on digital signal processor with the help of the Link for CCS
toolbox.
This way of generating code is fully functional and they allow measuring the proposed
algorithm directly in the digital signal processor but they definitely cannot be considered
optimized. It is convenient to use libraries that are optimized for a given processor and
replace the standard Simulink blocks by optimized ones. It is also possible to replace the
original number formats by formats corresponding to the processor. Also the filterbank can
be designed in two ways. The first way is independent filtering in each branch of filterbank
(Sysel, Krajsa 2010).
a)
h1 h2 h3
h2N h1 h2 h3
h2N h1 h2 h3
h2N
h1 h2 h3
h2N
1
1
1
1
2
2
2
2
3
3
3
3
γ
γ
γ
γ
b)
Xi+1 Xi+1 Xi+1
X i+1 Xi+2 Xi+2 Xi+2
X i+2 Xi+3 Xi+3 Xi+3
X i+3
Xi+2N Xi+2N Xi+2N
Xi+2N
1
2
3
2N
1
2
3
2N
1
2
3
2N
1
2
3
2N
i
c)
o i
o i
2
3
o i
o1
2N
Fig. 16. Efficient filterbank implementation
The second one is described on Fig. 16, where mn
h is n-th coefficient of m-th filter, i
Xm is i-th
IFFT symbol in m-th branch and im
o is i-th output sample in m-th branch. We have three
buffers, one (a) for prototype coefficients, one (b) for input symbols from IFFT, and the last
one (c) for output frame. Buffer b is FIFO buffer, samples are written in frames of 2N
samples. This way of filtering is more effective, because we need only one for cycle for
computing one output frame.
]
]
B]
B
B
[d
[d
[d
R
R
SNR
SN
SN
tone [-]
tone [-]
tone [-]
a)
b)
c)
Fig. 17. SNR for a) DMT, b) Non-overlapped FMT, c) Half-overlapped FMT with
narrowband noise
380
Discrete Time Systems
In the term of testing and comparing the implementation on DSP is interesting for the
possibility of power spectral density measurement and for its characteristics inside and
outside of the transmission band of partial subchannels on real line. In Fig. 19 is measured
PSD for the considered modulations.
C 6713 D S K
[0 0 0]
Vert C
Synchronizat ion
Transmitter
QAM
C 6713 D SK
-C -
I FFT
and
data
filter bank
D A C
modulators
upsam pling
IFFT
Signal Fro m
D A C
W o rkspace
Subs ystem
[0 0]
Fig. 18. FMT transmitter adjusted for implementation
−50
−100
B/Hz]d
SD [P −150
DMT
FMT, Blackman, γ=14
FMT, Mod.Blackman, γ=6
FMT, Nuttall, γ=8
−2000 1 2 3 4 5 6 7 8 9 10
tone [−]
Fig. 19. Measured power spectral densit
It is clear that the implementation results confirm the theoretical assumptions about the
properties of implemented modulations, mainly about their spectral properties. For the half-
overlap FMT modulation the PSD measured was flat, as well as with DMT modulation, but
the side lobes are suppressed by up to 50 dB. For the non-overlap FMT modulation perfectly
separated subchannels and strongly repressed side lobes are again evident.
In the implementation the computational complexity of individual modulation was also
compared. The most common form of DMT modulation needs to implement only the 2N-
point FFT, while with FMT each FFT output must be filtered. This represents an increase in
the required computational power and in the memory used. A comparison of DMT and
FMT for different systems is shown in the table. It compares the number of MAC
instructions needed for processing one frame of length 2N.
Half-overlap Subchannel Filtered MultiTone Modulation and Its Implementation
381
7. Conclusion
Based on a comparison of DMT and non-overlapped FMT multicarrier modulations we
introduced in this contribution the half-overlap subchannel FMT modulation. This
modulation scheme enables using optimally the available frequency band, such as DMT
modulation, because the resultant power spectral density of the signal is flat. Also, the
border frequency band is used optimally, the same as in non-overlapped FMT modulation.
Compared to non-overlapped FMT modulation the subchannel width is double and the
carriers cannot be too closely shaped. That enables using a smaller polyphase filter order
and thus obtaining a smaller delay. In section 5 we demonstrated that if the prototype filter
was designed to satisfy the orthogonal condition, even in overlapped FMT modulation the
ICI interferences do not occur. Furthermore, a method for channel equalization with the
help of DFE equalizer has been presented and the computation of individual filter
coefficients has been derived.
8. Acknowledgments
This work was prepared within the solution of the MSM 021630513 research programme
and the Grant Agency of Czech Republic project No. 102/09/1846.
9. References
Akujuobi C.M.; Shen J. (2008) Efficient Multi-User Parallel Greedy Bit-Loading Algorithm
with Fairness Control For DMT Systems,In: Greedy Algorithms, Witold Bednorz,
103-130, In-tech, ISBN:978-953-7619-27-5
Cherubini G.; Eleftheriou E.; Olcer S., Cioffi M. (2000) Filter bank modulation techniques for
VHDSL. IEEE Communication Magazine, (May 2000), pp. 98 – 104, ISSN: 0163-6804
Bingham, J, A. C.(2000) ADSL, VDSL, and multicarrier modulation, John Wiley & Sons, Inc.,
ISBN 0-471-29099-8, New York
Benvenuto N.; Tomasin S.; Tomba L.(2002) Equalization methods in DMT and FMT Systems
for Broadband Wireless Communications. In IEEE Transactions on Communications,
vol. 50, no. 9(September 2002), pp. 1413-1418, ISSN: 0090-6778
Berenguer, I.; Wassell J. I. (2002) FMT modulation: receiver filter bank definition for the
derivation of an efficient implementation, IEEE 7th International OFDM workshop,
Hamburg, (Germany, September 2002)
Sandberg S. D. & Tzannes M. A. (1995) Overlapped Discrete Multitone Modulation for High
Speed Copper Wire Communications. IEEE Journal on Selected Areas in
Communications, vol. 13, no.9, (December 1995), pp. 1571 – 1585, ISSN: 0733-8716
Sayed, A.H. (2003) Fundamentals of Adaptive Filtering, John Wiley & Sons, Inc, ISBN 0-471-
46126-1, New York
Silhavy, P. (2007) Time domain equalization in modern communication systems based on
discrete multitone modulation. Proceedings of Sixth International Conference of
Networking.pp. , ISBN: 0-7695-2805-8 , Sante-Luce, Martinique, , April 2007, IARIA
Silhavy, P.(2008) Half-overlap subchannel Filtered MultiTone Modulation with the small
delay. Proceedings of the Seventh International Conference on Networking 2008, pp. 474-
478, ISBN: 978-0-7695-3106-9, Cancun, Mexico, April 2008, IARIA
382
Discrete Time Systems
Sysel, P.; Krajsa, O.(2010) Optimization of FIR filter implementation for FMT on VLIW DSP.
Proceedings of the 4th International Conference on Circuits, Systems and Signals
(CSS'10). ISBN: 978-960-474-208- 0, Corfu, 2010 WSEAS Press
22
Adaptive Step-size Order Statistic
LMS-based Time-domain Equalisation
in Discrete Multitone Systems
Suchada Sitjongsataporn and Peerapol Yuvapoositanon
Centre of Electronic Systems Design and Signal Processing (CESdSP)
Mahanakorn University of Technology
Thailand
1. Introduction
Discrete multitone (DMT) is a digital implementation of the multicarrier transmission
technique for digital subscriber line (DSL) standard (Golden et al., 2006; Starr et al., 1999).
An all-digital implementation of multicarrier modulation called DMT modulation has been
standardised for asymmetric digital subscriber line (ADSL), ADSL2, ADSL2+ and very high
bit rate DSL (VDSL) (ITU, 2001; 2002; 2003). ADSL modems rely on DMT modulation,
which divides a broadband channel into many narrowband subchannels and modulated
encoded signals onto the narrowband subchannels. The major impairments such as the
intersymbol interference (ISI), the intercarrier interference (ICI), the channel distortion, echo,
radio-frequency interference (RFI) and crosstalk from DSL systems are induced as a result
of large bandwidth utilisation over the telephone line. However, the improvement can be
achieved by the equalisation concepts. A time-domain equaliser (TEQ) has been suggested
for equalisation in DMT-based systems (Bladel & Moenclaey, 1995; Baldemair & Frenger, 2001;
Wang & Adali, 2000) and multicarrier systems (Lopez-Valcarce, 2004).
The so-called shortened impulse response (SIR) which is basically the convolutional result
of TEQ and channel impulse response (CIR) is preferably shortened as most as possible. By
employing a TEQ, the performance of a DMT system is less sensitive to the choice of length
of cyclic prefix. It is inserted between DMT symbols to provide subchannel independency
to eliminate intersymbol interference (ISI) and intercarrier interference (ICI). TEQs have been
introduced in DMT systems to alleviate the effect of ISI and ICI in case that the length of SIR
or shorter than the length of cyclic prefix (F-Boroujeny & Ding, 2001). The target impulse
response (TIR) is a design parameter characterising the derivation of the TEQ. By employing
a TEQ, the performance of a DMT system is less sensitive to the choice of length of the cyclic
prefix. In addition to TEQ, a frequency-domain equaliser (FEQ) is provided for each tone
separately to compensate for the amplitude and phase of distortion. An ultimate objective of
most TEQ designs is to minimise the mean square error (MSE) between output of TEQ and
TIR which implies that TEQ and TIR are optimised in the MSE sense (F-Boroujeny & Ding,
2001).
Existing TEQ algorithms are based upon mainly in the MMSE-based approach (Al-Dhahir
& Cioffi, 1996; Lee et al., 1995; Yap & McCanny, 2002; Ysebaert et al., 2003). These include
384
Discrete Time Systems
the MMSE-TEQ design algorithm with the unit tap constraint (UTC) in (Lee et al., 1995) and
the unit energy constraint (UEC) in (Ysebaert et al., 2003). Only a few adaptive algorithms
for TEQ are proposed in the literature. In (Yap & McCanny, 2002), a combined structure
using the order statistic normalised averaged least mean fourth (OS-NALMF) algorithm for
TEQ and order statistic normalised averaged least mean square (OS-NALMS) for TIR is
presented. The advantage of a class of order statistic least mean square algorithms has been
presented in (Haweel & Clarkson, 1992) which are similar to the usual gradient-based least
mean square (LMS) algorithm with robust order statistic filtering operations applied to the
gradient estimate sequence.
The purpose of this chapter is therefore finding the adaptive low-complexity time-domain
equalisation algorithm for DMT-based systems which more robust as compared to existing
algorithms. The chapter is organised as follows. In Section 2 , we describe the overview of
system and data model. In Section 3 , the MMSE-based time-domain equalisation is reviewed.
In Section 4 , the derivation of normalised least mean square (NLMS) algorithm with the
constrained optimisation for TEQ and TIR are introduced. We derive firstly the stochastic
gradient-based TEQ and TIR design criteria based upon the well known low-complexity
NLMS algorithm with the method of Lagrange multiplier. It is simple and robust for ISI and
ICI. This leads into Section 5 , where the order statistic normalised averaged least mean square
(OS-NALMS) TEQ and TIR are presented. Consequently, the adaptive step-size order statistic
normalised averaged least mean square (AS-OSNALMS) algorithms for TEQ and TIR can be
introduced as the solution of MSE sense. This allows to track changing channel conditions and
be quite suitable and flexible for DMT-based systems. In Section 6 , the analysis of stability
of proposed algorithm for TEQ and TIR is shown. In Section 7 and Section 8 , the simulation
results and conclusion are presented.
2. System and data model
The basic structure of the DMT transceiver is illustrated in Fig. 1. The incoming bit stream
is likewise reshaped to a complex-valued transmitted symbol for mapping in quadrature
amplitude modulation (QAM). Then, the output of QAM bit stream is split into N parallel bit
streams that are instantaneously fed to the modulating inverse fast Fourier transform (IFFT).
After that, IFFT outputs are transformed into the serial symbols including the cyclic prefix
(CP) between symbols in order to prevent intersymbol interference (ISI) (Henkel et al., 2002)
and then fed to the channel. The transmission channel will be used throughout the chapter is
based on parameters in (ITU, 2001). The transmitted signal sent over the channel with impulse
response is generally corrupted by the additive white Gaussian noise (AWGN).
The received signal is also equalised by TEQ. The number of coefficients of TEQ is particularly
used to make the shortened-channel impulse response (SIR) length, which is the desired
length of the channel after equalisation. The frequency-domain equaliser (FEQ) is essentially
a one-tap equaliser that is the fast Fourier transform (FFT) of the composite channel of
the convolution between the coefficients of the channel (h) and the tap-weight vector (w)
of TEQ. The parallel of received symbols are eventually converted into serial bits in the
frequency-domain.
The data model is based on a finite impulse response (FIR) model of transmission channel
and will be used for equaliser in DMT-based systems. The basic data model is assumed that
the transmission channel, including the transmitter and receiver filter front end. This can
be represented with an FIR model h. The k-th received sample vector which is used for the
detection of the k-th transmitted symbol vector x k, N, is given by
Adaptive Step-size Order Statistic LMS-based
Time-domain Equalisation in Discrete Multitone Systems
385
AWGN + NEXT
bit
P / S
S / P
bit
x (n)
y (n)
stream
QAM
CIR
TEQ
stream
S / P
IFFT
+
input
+
FFT
FEQ
P / S
QAM
output
h
w
CP
CP
Fig. 1. Block diagram for time-domain equalisation.
⎡
⎤
⎡
⎤
H T
⎡
⎤
y
⎢
⎥
⎡
⎤
k, l+Δ
ηk, l+Δ
⎢
⎢
[ ¯h T ] 0 · · ·
⎥
x k−1, N
⎣
.
⎥
⎢
⎥
⎢
⎥
.
⎣
⎦
.
.
⎦ = ⎢
· (
)·
+
⎢
.
⎥ I ⊗ P ν F H
x k, N
⎣
.
⎦,
(1)
. .
⎥
N
.
. . .
0(
y
⎣0(1)
2)⎦
x k+1, N
k, N− l+Δ
ηk, N− l+Δ
· · · 0 [ ¯ T
h ]