Two-dimensional (2D) methods

The optimal 2D MMSE estimator makes full use of the frequency (intrablock) and time-domain (interblock) correlations of CFR of the time-varying multipath fading channel. Li et al. [56] have derived the optimal 2D MMSE estimator for the OFDM system, where all subcarriers are used as a training reference. Similarly to [43], the estimator is designed through EVD and low-rank approximation of the frequency-domain channel correlation matrix, but the parameters comprising the subspace are subject to additional tracking in the time domain (i.e. between successive blocks). The main benefit of such an estimator architecture is the independence of the tracking filterbank’s complexity on the number of subcarriers being present in the system. The authors also introduce a robust estimator design, which is independent of the actual channel correlation function, for the case of the sample-spaced multipath channels, and estimator implementation based on the infinite-length time-domain FIR filters. It should however be noted that the robust filters are suboptimal by their nature as they cannot achieve the true MMSE in the majority of cases.

Another related work by the same author [57] considers a 2D block-type pilot-assisted CFR interpolator, which is optimal in the MMSE sense. The author deduces the robust interpolator design, which is independent of the channel statistics. Interestingly, the robust design criteria for the interpolator are the same as for the filter considered in the previous work [56]. To reduce computational complexity of the interpolator implementation, it is proposed to use a transform-domain architecture based on 2D FFT/IFFT.

Sandell and Edfors [58] investigate application of the optimal 2D Wiener filter and interpolator to the pilot- assisted OFDM channel estimation. The optimal scheme is compared to the suboptimal counterparts, characterised by much smaller computational complexity that is achieved by the low-rank approximation and the use of two separable 1D filters instead of the optimal 2D architecture. The separable filtering approach is stipulated by the

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separation property of the correlation function in the time and frequency domain [56]. Interestingly, the authors find that the estimators based on the separable filters perform even better than the optimal 2D estimator, as it appears that the latter is more susceptible to interpolation errors.

Classical interpolators have also been applied to 2D channel identification. Wang and Liu [59] propose the approximation of the doubly selective channel by means of polynomial basis functions, whereas Chang and Su [60] use polynomial regression models to express the channel response jointly in the time and frequency domain. These estimation methods do not require knowledge of the channel statistics. However, construction of the optimal polynomial estimator (with minimum model error) requires adjusting dimensions of the CFR observation window, as well as selection of the proper model order. In the work by Chang and Su [60], it is reported that performance deteriorates as the maximum Doppler frequency increases. It should also be pointed out that complexity can be higher than that of the DFT-based methods (e.g., [57]) for a large number of pilot subcarriers.

Dong et al. [61] investigate application of several kinds of 2D separable interpolators to OFDM channel estimation: lowpass-windowed sinc approach and Deslauriers-Dubuc method. Their performance is of the same order as that of the frequency-domain 2D MMSE interpolator [57], with remarkable loss in the upper SNR range due to the higher interpolation error floor, but the complexity is several orders smaller. In particular, the authors point out the drawbacks of the real-time implementation of the block-type interpolators, like [57] and [59]: large latency and memory requirements.

It should be noted that interpolators developed in the works [59], [60] and [61] are model-independent, whereas the optimal MMSE interpolator [57] relies on the underlying doubly selective channel model with a priori known time-frequency correlation function and SNR. Hence the principal advantage of the model-based MMSE interpolator (or Wiener filter) is the ability to restore 2D CFR without error in the absence of noise. This can be achieved if the number of processed samples is larger than the model order, and provided that pilot symbols are selected with the appropriate density in the time-frequency grid. On the other hand, the model-independent algorithms always produce interpolation error, but their advantage is that no prior information on channel statistics is required.

The Kalman filter is known as a generalised recursive implementation of the optimal linear MMSE estimator (Wiener filter) of infinite length, which considers time-variation of the unknown parameter(s) according to the selected dynamic model. In contrast to the Wiener filter, which is restricted to stationary signals and noise, the Kalman filter has the ability to accommodate non-stationary processes described statistically in the form of state sequences. Due to its relation to Bayesian theory, the Kalman filter has the same problems as its Wiener counterpart, i.e. it relies on the second-order statistics of signal and noise. The classical vector Kalman filter [39] can be applied to track the doubly selective channel, but it suffers from high computational complexity. Lower-complexity architecture consists of a bank of scalar Kalman filters [62]. Each of them tracks variation on a separate subcarrier, and the MMSE combiner smoothes filter outputs by taking the frequency-domain correlation into account. He and Lee [63] propose Kalman filterbank architecture, which is similar to [62]. The main difference is that the frequency- domain smoothing precedes the time variation tracking stage.

The major problems with the algorithms [62] and [63] are increased complexity in the case of a large number of subcarriers (as each of them is allocated an individual Kalman filter) and a lack of scalability when pilot-assisted

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transmissions are considered. Unlike the Wiener filters, which can perform both filtering and interpolation over a fixed-length sequence of OFDM blocks, recursive implementation, inherent to the Kalman filter, processes only the current and the preceding block and therefore does not allow for interpolation of the channel response across a sequence of pilot block observations. This necessitates a separate implementation of the time-domain interpolator, bringing to naught the memory-saving benefits of the Kalman filters. It should also be noted that in contrast to the pilot interpolation, the decision-directed operation of the estimator, suggested in the works [62] and [63], is much less reliable as it is prone to uncontrolled error propagation.

Cai et al. [64] consider the same estimator structure as [56] and propose application of recursive algorithms instead of the Wiener FIR filters to track the channel response parameters in the EVD subspace. Such a tracking estimator implementation is more efficient than the approaches [62] and [63] as the complexity order is not affected by the number of subcarriers used for the reference signals. The authors point to the problem of the classical Kalman filter being sensitive to accurate information about noise variance. To overcome this challenge, they propose the use of the H∞ filtering algorithm, which is optimal in the sense of minimising the maximum disturbance effect

(instantaneous energy), instead of the Kalman estimator, which is optimal in the MMSE sense. The H∞ and Kalman

estimators have a similar structure; hence they share the same advantages and drawbacks. The main problem is still the inability of the recursive filtering structure to incorporate pilot-assisted interpolation across OFDM block sequence. The robustness extent of the H∞ algorithm is also questionable as it is not free from the uncertainty of the

design parameter selection. Furthermore, noise variance measurement is not a very big challenge, as it can be done on the virtual subcarriers [46].

All the previously mentioned channel estimation methods consider non-parameterised (or partially parameterised [49]) channel models, which are characterised by only a few deterministic and statistical properties (effective model order, correlation function and noise variance). In reality, a great variety of multipath fading channels, particular to the large cell scenarios, consist of only a few dominant propagation paths (typically two to six according to the existing ETSI and PCS channel reference models), each of which is characterised by an individual excess delay, Doppler frequency shift and complex amplitude. As the number of the underlying parameters is fixed, the channel response is uniquely identifiable through them along the observed training sequence of a certain length. This forms a basic idea behind the wireless channel sounding theory and practice [65]. Relying on such a parametric channel model, Yang et al. [66] use the minimum description length (MDL) criterion [67] to detect the number of multipath components, and employ estimation of the signal parameters by rotational invariance techniques (ESPRIT) [68] for multipath excess delay acquisition. Based on the identified set of parameters, it is further proposed to track the slow variation of the delays and the fast variation of the path gains by means of the interpath interference cancellation delay locked loops (IPIC DLLs) and the 2D Wiener filterbank (similar to [56] and [58]), respectively. Liu [69] shows how to extend ESPRIT to the 2D operation, allowing for additional identification of the Doppler frequency shift corresponding to each multipath component. The complexity of the parametric channel estimator is considerable due to the DLL bank and the EVD operation required by the ESPRIT. The results show that the performance gain, in comparison with the lower-complexity non-parametric algorithms, is achieved only when the

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number of multipath components is small, otherwise the difference is minor. The latter circumstance makes this type of estimators unsuitable from the commercial standpoint for OFDM transceivers designed to operate in indoor environments with rich scattering and a multitude of propagation paths.