**1. Introduction**

An all-digital implementation of multicarrier modulation called *discrete multitone (DMT) modulation* has been standardised for asymmetric digital subscriber line (ADSL), ADSL2, ADSL2+ and VDSL [1]- [3]. ADSL modems rely on DMT modulation, which divides a broadband channel into many narrowband subchannels and modulated encoded signals onto the narrowband subchannels [4], [5]. With advanced digital signal processing algorithms, DMT system is to fight the impairments for wired communications such DSL-based technology. 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.

ISI and ICI caused by the length of channel impulse response can be eliminated by the use of cyclic prefix (CP) adding a copy of the last *ν* time-domain samples between DMT-symbols at the part of transmitter. The conventional equalisation in DMT-based systems consists of a (real) time-domain equaliser (TEQ) and the (complex) one-tap frequency-domain equalisers [6]. For a more sophisticated equalisation technique, a frequency-domain equaliser (FEQ) for each tone, called *per-tone equaliser (PTEQ)* has been introduced in order to give the bit rate maximising compared with existing equalisation schemes [7], [8].

The basic structure of the DMT transceiver is shown 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 between symbols and then fed to the channel. The transmitted signal sent over

©2012 Sitjongsataporn, licensee InTech. This is an open access chapter distributed under the terms of the

the channel with impulse response is generally corrupted by the additive white Gaussian noise and near-end crosstalk. The received signal is also equalised by PTEQs without TEQ concerned. The per-tone equalisation structure is based on transferring the TEQ-operations into the frequency-domain after FFT demodulation, which results in a multitap PTEQ for each tone separately. Then, the parallel of received symbols are eventually converted into serial bits in the frequency-domain.

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*<sup>T</sup>*. A

*<sup>T</sup>*. The

**2. System model and notation**

**y** = **H** · **X** + **n** ,

� � � � � � � � [ **h**¯ *<sup>T</sup>* ] 0 ··· ... ... ··· 0 [ **h**¯ *<sup>T</sup>* ]

 **0**(1)

 

**x***k*−1,*<sup>N</sup>* **x***k*,*<sup>N</sup>* **x***k*+1,*<sup>N</sup>*

� �� � **X***k*−1:*k*+1,*<sup>N</sup>*

 

P*<sup>ν</sup>* = �

+ 

is presented as [7]

cross-talk (NEXT).

**3. Per-tone equalisation**

 

*yk*,*l*+<sup>∆</sup> . . . *yk*,*N*−*l*+<sup>∆</sup>

� �� � **<sup>y</sup>***k*,*l*+∆:*N*−1+<sup>∆</sup>

 

=

In this section, the basic structure of the DMT transceiver is illustrated in Fig. 1. We describe that the data model and notation based on an FIR model of the DMT transmission channel

> � � � � � � � �

*ηk*,*l*+<sup>∆</sup> . . . *ηk*,*N*−*l*+<sup>∆</sup>

� �� � <sup>η</sup>*k*,*l*+∆:*N*−1+<sup>∆</sup>

where *l* denotes as the first considered sample of the *k*-th received DMT-symbol. This depends on the number of tap of equaliser (T) and the synchronisation delay (∆). The vector **y***k*,*i*:*<sup>j</sup>* of received samples *i* to *j* of *k*-th DMT-symbol is as **y***k*,*i*:*<sup>j</sup>* = [*yk*,*<sup>i</sup>* ··· *yk*,*j*]

size *N* is of inverse discrete Fourier transform (IDFT) and DFT. The parameter *ν* denotes as the length of cyclic prefix. The matrices **0**(1) and **0**(2) are also the zero matrices of size (*N* − *l*) × (*N* − *L* + 2*ν* + ∆ + *l*) and (*N* − *l*) × (*N* + *ν* − ∆). The vector **h**¯ is the **h** channel impulse responce (CIR) vector in reverse order. The (*N* + *ν*) × *N* matrix P*<sup>ν</sup>* is denoted by

> **0***ν*×(*N*−*ν*) **I***<sup>ν</sup>* **I***N*

which adds the cyclic prefix. The I*<sup>N</sup>* is *N* × *N* IDFT matrix and modulates the input symbols. The η*k*,*l*+∆:*N*−1+<sup>∆</sup> is a vector with additive white Gaussian noise (AWGN) and near-end

Some notation will be used throughout this chapter as follows: *E*{·} is the expectation operator and diag(·) is a diagonal matrix operator. The operators (·)*T*, (·)*H*, (·)<sup>∗</sup> denote as the transpose, Hermitian and complex conjugate operators, respectively. The parameter *k* is the DMT symbol index and **I***<sup>a</sup>* is an *a* × *a* identity matrix. A tilde over the variable indicates the frequency-domain. The vectors are in bold lowercase and matrices are in bold uppercase.

In this section, we show the concept of per-tone equaliser (PTEQ). We refer the readers to [7] for more details. The per-tone equalisation structure is based on transferring the

� ,

sequence of the *N* × 1 **x***k*,*<sup>N</sup>* transmitted symbol vector is as **x***k*,*<sup>N</sup>* = [*xk*,0 ··· *xk*,*N*−1]

**0**(2)

 ·

> 

 

Adaptive Step-Size Orthogonal Gradient-Based Per-Tone Equalisation in Discrete Multitone Systems

� �� � **H**

P*<sup>ν</sup>* **0 0 0** P*<sup>ν</sup>* **0 0 0** P*<sup>ν</sup>*

 · 

I*<sup>N</sup>* **0 0 0** I*<sup>N</sup>* **0 0 0** I*<sup>N</sup>*

. (1)

 ·

**Figure 1.** Block Diagram of a Discrete Multitone System.

It is a well known issue that in DMT theory, there is no overlapping between tones due to orthogonality derived from the discrete Fourier transformation among them. In practice, frequency-selective fading channel generally destroys such orthogonal structure leading to information interfering from adjacent tones as commonly known as ICI. In such case, information supposedly belonging to a particular tone generally smear into adjacent tones and leave some residual energy in them. The idea of a mixed-tone PTEQ for DMT-based system has been proposed in [9]. By recovering adaptively the knowledge of residual interfering signal energy from adjacent tones, the mixed-tone exponentially weighted least squares criterion can be shown to offer an improved signal to noise ratio (SNR) of the tone of interest.

In order to improve the convergence properties, the orthogonal gradient adaptive (OGA) has been presented by introducing orthogonal projection to the filtered gradient adaptive (FGA) algorithm. When the forgetting-factor is optimised sample by sample whereas a fixed forgetting-factor is used for FGA algorithm [10]. A normalised version of the OGA (NOGA) algorithm that has been introduced with the mixed-tone cost function and fixed step-size presented in [11]. With the purpose of the good tracking behaviour and recovering to a steady-state, it is necessary to let the step-size automatically track the change of system. Consequently, the concept of low complexity adaptive step-size approach based on the FGA algorithm is introduced for the per-tone equalisation in DMT-based systems in [12].

In this chapter, the focus is therefore to present low complexity orthogonal gradient-based algorithms for PTEQ based on the adaptive step-size approaches related to the mixed-tone criterion. The convergence behaviour and stability analysis of proposed algorithms will be investigated based on the mixed-tone weight-estimated errors. The convergence analysis of mechanisms will be carried out the steady-state and mean-square expressions of adaptive step-size parameter relating to the mean convergence factor.
