
1. Introduction The wireless radio channel can
be parametrized as a combination of paths, each characterized by a delay and a complex amplitude. The amplitudes show
fast temporal variations due to the mobility of terminals while the
delays are almost constant over a large number of OFDM symbols (see
figure). A possible approach is that of explicitly estimating and
tracking the delays [Yang '01]. To avoid the typical problems
of nonlinear parameter estimation (i.e, threshold effects and
computational complexity), we propose to perform an unstructured
tracking of the delays through a subspace tracking algorithm and the
amplitudes by Kalman filtering or suboptimal (reduced complexity)
techniques such as LMS or WLMS [Lindbom '01].

The idea has been applied first to OFDM systems over fastvarying fading channels. The channel estimation (tracking) in OFDM systems is generally based on the use of pilot subcarriers in given positions of the frequencytime grid. In particular, we consider the comb pilot pattern arrangement (see figure) which has been shown to satisfy different criteria of optimality such as mean square error on the channel estimate [Negi '98] and capacity [Adireddy '02]. In this framework, the traditional approach to channel estimation consists of two steps. Firstly, the least squares (LS) estimates of the channel gains over the pilot subcarriers are obtained by simply backrotating the received signal according to the knowledge of the pilot symbols. Then, the LS estimates are interpolated/smoothed over the entire frequencytime grid. 


2. Subspace tracking channel estimation The Subspace Tracking (ST)
algorithm (see figure(a)) and the SubspaceAmplitude
Tracking (SAT) algorithm (see figure(b)) adaptively estimate the two
timevarying terms of the multipath channel:
delaysubspace and amplitudes. The ST and SAT algorithm can be
applied to MIMOOFDM or MCCDMA systems by designing the training
sequences as in [Jones '98].

Analysis and simulation show the relevant advantage of the unstructured tracking of the multipath delays compared to existing preinterpolation techniques in terms of mean square error (MSE) on the channel estimate. 


The advantage can be quantified (for a large number of symbol n) as the ratio between the number of pilot subcarriers and the dimension of the delaysubspace. The algorithms do not assume any a priori knowledge on the multipath channel. Even the dimension of the delaysubspace (which is related to degree of temporal diversity of the channel) is adaptively estimated symbolbysymbol (see figure). However, any a propri information on the statistics of the fading variation could be exploited by appropriate design of the amplitude tracker (e.g., Kalman filtering). 
3. Future work
The delaysubspace (and amplitude) tracking algorithms can also be applied to OFDM systems over channels with larger coherence time that use a pilot arrangement such as that employed in Hiperlan/2 or IEEE802.11b. Preliminary results show that the techniques provide relevant performance improvement compared to known channel estimation algorithms also in this framework. A thourough study is under way.
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