Discrete Time Systems Part 14 ppt

A time-domain equaliser TEQ has been suggestedfor equalisation in DMT-based systems Bladel & Moenclaey, 1995; Baldemair & Frenger, 2001;Wang & Adali, 2000 and multicarrier systems Lopez-

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)

h2 1

h2 2

h2 3

h2 2N

h3 1

h3 2

h3 3

h3 2N

hγ 1

hγ 2

hγ 3

hγ 2N

o2N i

o1

i

o2 i

o3 i

X1 i+2

X2 i+2

X3 i+2

X2N i+2

X1 i+3

X2 i+3

X3 i+3

X2N i+3

X1 i+2N

X2 i+2N

X3 i+2N

X2N i+2N

a)

b)

c)

Fig 16 Efficient filterbank implementation

The second one is described on Fig 16, where m

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

narrowband noise

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

Synchronizat ion and upsam pling

Transmitter filter bank

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

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

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

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 transmissiontechnique 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 highbit rate DSL (VDSL) (ITU, 2001; 2002; 2003) ADSL modems rely on DMT modulation,which divides a broadband channel into many narrowband subchannels and modulatedencoded signals onto the narrowband subchannels The major impairments such as theintersymbol 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 beachieved by the equalisation concepts A time-domain equaliser (TEQ) has been suggestedfor 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 Byemploying 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 beenintroduced 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 impulseresponse (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 cyclicprefix In addition to TEQ, a frequency-domain equaliser (FEQ) is provided for each toneseparately to compensate for the amplitude and phase of distortion An ultimate objective ofmost TEQ designs is to minimise the mean square error (MSE) between output of TEQ andTIR 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

Adaptive Step-size Order Statistic LMS-based Time-domain Equalisation

in Discrete Multitone Systems

22

the MMSE-TEQ design algorithm with the unit tap constraint (UTC) in (Lee et al., 1995) andthe unit energy constraint (UEC) in (Ysebaert et al., 2003) Only a few adaptive algorithmsfor TEQ are proposed in the literature In (Yap & McCanny, 2002), a combined structureusing the order statistic normalised averaged least mean fourth (OS-NALMF) algorithm forTEQ and order statistic normalised averaged least mean square (OS-NALMS) for TIR ispresented The advantage of a class of order statistic least mean square algorithms has beenpresented in (Haweel & Clarkson, 1992) which are similar to the usual gradient-based leastmean square (LMS) algorithm with robust order statistic filtering operations applied to thegradient estimate sequence.

The purpose of this chapter is therefore finding the adaptive low-complexity time-domainequalisation algorithm for DMT-based systems which more robust as compared to existingalgorithms The chapter is organised as follows In Section 2 , we describe the overview ofsystem 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 theconstrained optimisation for TEQ and TIR are introduced We derive firstly the stochasticgradient-based TEQ and TIR design criteria based upon the well known low-complexityNLMS algorithm with the method of Lagrange multiplier It is simple and robust for ISI andICI 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 statisticnormalised averaged least mean square (AS-OSNALMS) algorithms for TEQ and TIR can beintroduced 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 simulationresults 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 isbased on parameters in (ITU, 2001) The transmitted signal sent over the channel with impulseresponse 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 particularlyused to make the shortened-channel impulse response (SIR) length, which is the desiredlength 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 thefrequency-domain

The data model is based on a finite impulse response (FIR) model of transmission channeland will be used for equaliser in DMT-based systems The basic data model is assumed thatthe 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

bit stream

[¯hT]0· · ·

· · ·0[ ¯hT]

where l determines the first considered sample of the k-th received DMT-symbol and

• ¯h is the CIR vector h with coefficients in reverse order.

η=E{ηkηT

k}

TEQ w

AWGN

y(n)

e(n)

TIR b

6

CIR h

x(n)

delay

_

Fig 2 Block diagram of MMSE-TEQ

the expectation, transpose and Hermitian operators, respectively The vectors are in boldlowercase and matrices are in bold uppercase

3 Minimum mean square error-based time-domain equalisation

The design of minimum mean square error time-domain equalisation (MMSE-TEQ) is based

on the block diagram in Figure 2 The transmitted symbol x is sent over the channel with the

impulse response h and corrupted by AWGN η The convolution of the L-tap TEQ filter

the shorten impulse response (SIR) Then the orthogonality between the tones are restoredand ISI vanishes (Melsa et al., 1996)

The result of time-domain error e between the TEQ output and the TIR output is then

minimised in the mean-square sense as

Some constraints that are added on the TEQ and TIR (Ysebaert et al., 2003) as follows

1 The unit-norm constraint (UNC) on the TIR

By solving Eq.(6) subject to

2 The unit-tap constraint (UTC) on the TEQ

A UTC on w can be calculated with the method of the linear equation

where ej is the canonical vector with element one in the j-th position By determining

the dominant generalised eigen-vector, the vector w can be obtained as the closed-form

3 The unit-tap constraint (UTC) on the TIR

Similarly, a UTC on b can be described as

The coefficients of TEQ w can be computed by Eq.(8).

4 The unit-energy constraint (UEC) on TEQ and TIR

Three UECs can be considered as

4 The proposed normalised least mean square algorithm for TEQ and TIR

We study the use of the LMS algorithm by means of the simplicity of implementationand robust performance But the main limitation of the LMS algorithm is slow rate ofconvergence (Diniz, 2008; Haykin, 2002) Most importantly, the normalised least mean square(NLMS) algorithm exhibits a rate of convergence that is potentially faster than that of thestandard LMS algorithm Following (Haykin, 2002), we derive the normalised LMS algorithmfor TEQ and TIR as follows

δw(n+1)andδb(n+1)are defined as

where e(n)is the estimation error

To formulate the complex constraint of Eq.(17) as the pair of real constraints.

4.1 The proposed normalised least mean square time-domain equalisation (NLMS-TEQ)

the both results equal to zero Hence,

∂J1(n)

∂a k(n+1) =0 ,

1The method of Lagrange multiplier is defined as a new real-valued Lagrange function h(w)

where f(w)is the real function and C(w)is the complex constraint function The parametersλ1 andλ2

are the Lagrange multipliers, whereλ=λ +j λ

∂J1(n)

∂b k(n+1) =0 The results are given by

2[wk(n+1) −wk(n)]y∗(n−k) = −λ∗wy(n−k)y∗(n−k)2

wherey(n)2is the Euclidean norm of the tap-input vector y(n).

From the definition of the estimation error e(n)in Eq.(18), the conjugate of e(n)is written as

e (n) =wT(n+1)y∗(n) −bT(n+1)d∗(n) (41)The mean-square error|e(n)|2is minimised by the derivative of|e(n)|2with respect to w(n+

4.2 The proposed normalised least mean square-target impulse response (NLMS-TIR)

set the results equal to zero Hence,

∂J2(n)

∂u k(n+1) =0 ,and

∂J2(n)

∂v k(n+1) =0 The results are

Thus, we have

2[bk(n+1) −bk(n)] +λ∗bd(n−k) =0, f or k=0, 1, , M−1 (56)

2[bk(n+1) −bk(n)]d∗(n−k) = −λ∗bd(n−k)d∗(n−k)2

where e∗(n)is given in Eq.(41)

normalised as

b(n+1) = b(n+1)

3 Minimum mean square error-based time- domain equalisation

The design of minimum mean square error time- domain equalisation (MMSE-TEQ) is based

on... respect to w(n+