One-dimensional (1D) methods

It has been shown in [33] that CFR can be described by the low-order parametric model due to the analytical relationship between the power spectral density (PSD) and the power delay profile (PDP) of the multipath channel. Hence one does not need many CFR samples to compute the unknown underlying parameters. This allows for the development of efficient frequency-domain interpolation techniques, utilising only a few pilot symbols per OFDM block.

Various types of interpolation techniques, applied to the elementary frequency-domain least squares (LS) estimates of CFR at the equally-spaced pilot subcarriers, have been investigated [34][35]. Here these interpolation techniques are listed in ascending performance order, namely: linear (first-order), quadratic, Gaussian, ideal lowpass (DFT-based) and cubic spline. The DFT-based interpolator has been found more prone to ICI in the continuously varying fast fading channels. It has been established by the mean square error (MSE) analysis [36] that the linear (first-order) interpolator gives a better performance gain for nearly-flat fading channels, where the noise prevails over interpolation error. However, in the case of channel environments characterised by large delay spreads, it is more preferable to adopt an interpolator based on the Fourier decomposition.

Classical interpolation techniques (e.g., polynomial, spline, etc.) are not designated specifically for noise reduction. Hence, in filtering applications they cannot be regarded as optimal solutions. For receivers employing frequency-domain signal processing, transform-domain estimators have been found attractive from both the complexity and performance standpoint. The principle of transform-domain techniques is to project CFR observations at the pilot subcarrier positions by means of IDFT (or alternative fast filterbank) onto (time-domain) parameter subspace of a smaller dimension, where the filtering (denoising) is performed. The final estimates are then found by the linear combinations applied to the subspace, which are implemented through the DFT (or alternative) conversion. Implementation of the DFT filterbanks is known to be very attractive from the complexity standpoint due to the availability of the FFT hardware chips in standard MC transceiver design.

Adopting the low-complexity DFT-based transform-domain architecture, Deneire et al. [37] have developed a classical maximum likelihood (ML) estimator, which is known as optimal (attaining the Cramer-Rao lower bound [38]) in the deterministic sense when there is no way to acquire correlation and SNR statistics of the channel. The only channel parameter, utilised by the algorithm, is the maximum excess delay, which determines the model order (parametric subspace dimension). Care should be taken when selecting its magnitude, as setting the anticipated delay too high limits the filtering gain. A reasonable value is somewhat close to the CP duration. It should be noted that in

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the case of the system model affected by the additive white Gaussian noise (WGN), which is described by the spike correlation function of the same magnitude across the entire band, the ML solution coincides with the constrained LS (CLS) solution [39]. CLS, as the least complex and the most robust estimator, will be referred to in more detail in Chapter 3.

For the channels with a priori known CIR correlation and SNR, the transform-domain architecture can be enhanced by applying the linear MMSE filtering in the time domain [38][40]. The resultant MMSE estimator is known to be optimal in the Bayesian sense, i.e. when the prior statistical information about the unknown parameter set is available. It has been shown that the greatest performance benefit of the MMSE estimator in comparison with ML is present in the low SNR operational modes [38].

Minn and Bhargava [41] have introduced a solution with a lower complexity than the DFT-based MMSE filtering. It is based on identifying the set of the most significant (in the average power sense) CIR samples for estimation, and exhibits a minor performance loss with regard to the optimal MMSE estimator in the case of the sample-spaced multipath channels. However, the authors do not explain how to implement tracking of the selected samples. Based on a similar idea, in our related work [42], we present a modified order-recursive LS (MORLS) algorithm, which incorporates tracking functionality by picking up the most significant CIR samples while reducing the sum of squared estimation errors to the noise floor. The reason for not including this work as a part of this thesis is the minor complexity reduction with regard to the transform-domain MMSE estimator. In fact, the complexity decrease is true only for the channels with a few uncorrelated multipath components, whereas for the rich scattering environments and/or correlated impulse response samples no benefits are observed.

The frequency-domain linear MMSE (LMMSE) estimator is known to be optimal in the Bayesian sense for non-sample-spaced channels, for which the transform-domain estimators [38][40] exhibit performance deterioration due to time-domain interpolation errors in higher SNR operational modes. But implementation complexity of the frequency-domain LMMSE estimator is usually high. Edfors et al. [43] propose the use of a suboptimal estimator based on the singular value decomposition (SVD) of the CFR correlation matrix to save on computational effort. The works by Senol et al. [44] and Noh et al. [45] are closely related to Edfors’s [43]. Assuming that the estimation subspace dimension does not exceed the CP length, there is a proposition to diagonalise the channel correlation matrix with the Karhunen-Loeve expansion, which is also known as the eigenvalue decomposition (EVD) [44]. Noh et al. [45] suggest lowering complexity of the frequency-domain LMMSE estimator by partitioning the channel correlation matrix into small overlapped submatrices. Then CFR estimates can be obtained, taking into account only strongly correlated subcarriers, which fall into the coherence bandwidth, and ignoring the weakly correlated ones. Similarly to the low-rank approximation methods ([43] and [44]), this scheme suffers from performance degradation in comparison with the optimal MMSE. It is also questionable whether or not this technique can be applied to the pilot-assisted systems where the estimates are constructed based on the pilot subcarriers, which can be scattered in the band with a higher spacing than the coherence bandwidth.

A common problem with [43], [44], [45] and other similar algorithms, relying on the non-Fourier decomposition, is the dependence of the expansion basis, filtering matrices and coefficients on the channel correlation function. Their low-complexity implementation is possible only when the channel correlation function is

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known a priori so that it can be used in the fixed estimator design. Another problem is that the effective approximated model order should generally be selected in accordance with SNR at the receiver input so as to maximise transmission capacity. However, the study of the latter problem has not been reported so far.

In practical OFDM systems, part of the subcarriers, typically at the band edges and at DC, are left unmodulated in order to provide spectral shaping with out-of-band interference reduction, and to avoid DC offset, respectively. These unused subcarriers are termed as virtual subcarriers (VCs). CFR at VCs cannot be estimated directly, limiting applicability of the conventional estimators. Huang et al. [46] propose a number of modified estimators (LS and LMMSE criteria based) to address this problem.

It has already been mentioned that propagation delays of the multipath channel are generally non-sample-

spaced, i.e. not multiple integers of a system sampling period. This results in spectral leakage (aliasing effect) after DFT processing at the receiver that lengthens the effective CIR observed in a single block and hence introduces interpolation errors if the transform-domain channel estimation is used. To resolve this problem in low-complexity DFT-based solutions, Yang et al. [47] propose a suboptimal windowed MMSE estimator design, which has only a minor performance loss in comparison with the optimal frequency-domain MMSE estimator. There is, however, a difficulty with the window shape selection as the shaping parameter is a complicated function of the noise power, and can be determined only by numerical search.

An alternative proposal is to use the discrete cosine transform (DCT) instead of DFT in the transform-domain estimator architectures [48]. CFR interpolation based on DCT allows reduction of the aliasing effect, though does not eliminate it completely. Implementation of the DCT-based estimator can be realised by fast DCT algorithms and architectures, which are competitive with FFT. Taking into account that both schemes [47] and [48] are positioned as low-complexity solutions, it could be of interest to compare their performances in the worst-case aliasing conditions. In the work by Simeone et al. [49], the authors refer to a detailed model of the wireless channel that is parameterised as a linear combination of paths, each of which is characterised by an excess delay and complex amplitude. As direct estimation of the excess delays represents a nonlinear problem of high computational complexity, they propose tracking the delay-subspace, which can be regarded as the orthonormal basis obtained through EVD of the correlation matrix, by the recursive algorithm. The time-varying amplitudes of the multipath components are estimated by the least mean square (LMS) filter. One should point out that the computational load increases by an order of magnitude, stipulated by the QR factorisation at each recursion step of the subspace tracker, in comparison with the low-complexity systems without delay-subspace identification.

All the previously mentioned frequency and transform-domain estimation methods are directly applicable for CFR acquisition and subsequent frequency-domain equalisation in the case when the maximum excess delay of the channel does not exceed the guard interval (cyclic prefix) of the transmitted blocks. With regard to the wireless channel, this condition is typically satisfied due to the short propagation time and rapid echo attenuation of the radio signal. In the DMT and OFDM transmissions over the wireline channels, CIR length can be relatively large with respect to the MC block size that leads to a severe limitation of the bandwidth efficiency. It is known that if the prefix length is insufficient, IBI emerges between two successive blocks, as well as ICI between different subcarriers [50]. These interference effects can be eliminated by time-domain equalisation (TEQ), based on FIR filtering. TEQ

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performs effective compression of the impulse response energy spread, based on different criteria: minimisation of the impulse response energy outside of the prefix interval [51][52], minimisation of the mean square error between the actual and desired impulse response [53], maximisation of the signal-to-interference-plus-noise ratio (SINR) at the output of equaliser [50]. Acker et al. [54] propose a per-tone equaliser (PTEQ), which is designed by transferring TEQ to the frequency domain. The main disadvantage of all TEQ approaches is significant computational complexity, especially at the equaliser’s initialisation stage. A computationally efficient alternative is developed by Marelli and Fu [55]. It is based on the frequency-domain processing of the received MC signal using the fast analysis-synthesis filterbanks and subband equalisation.

A common problem of all TEQ-related methods is that they require transmission of the training sequence simultaneously on all the active subcarriers in the band. In the wireline DMT channels (e.g., xDSL trunks) characterised by the invariant state of the response, training needs to be performed only in the beginning of the communication session. In contrast to that, the necessity of the response variation tracking in the wireless channel forces frequent re-training that can be achieved by inserting pilot symbols into the transmit sequence at the selected frequencies. The inability to utilise the frequency-domain differentiation of training and data limits TEQ application in the pilot-assisted MC systems (e.g., wireless OFDM).