Estimation and Direct Equalization of Doubly Selective Channels

INTRODUCTION

Over the last decade, the mobile wireless telecommunication industry has undergone tremendous changes and experienced rapid growth. The reason behind this growth is the increasing demand for bandwidth hungry multimedia applications. This demand for even higher data rates at the user’s terminal is expected to continue for the coming years asmore and more applications are emerging. Therefore, current cellular systems have been designed to provide date rates that range from a few megabits per second for stationary or low mobility users to a few hundred kilobits per second for high mobility users. In addition to the frequency-selectivity characteristics caused by multipath propagation, the channel often exhibits time-variant characteristics caused by the user’s mobility. 

This results in the so-called doubly selective (timeand frequency-selective) channels. In linear and decision feedback equalizers have been developed for single carrier transmission over doubly selective channels. There, the time-varying channel was approximated using the basis expansion model (BEM). The BEM coefficients are then used to design the equalizer (linear or decision feedback). So far, it was assumed that the BEM coefficients are perfectly known at the receiver, and that they were obtained by a least-squares (LS) fitting to the noiseless underlying communication channel (modeled using Jakes’ model). In other words, perfect channel state information (CSI) was assumed to be known at the receiver side. This is, however, far from being realistic, since a more realistic approach is to estimate the channel or directly obtain the equalizer coefficients. 

This can be achieved by using training symbols, or blindly or semiblindly by combining training with blind techniques. In this paper we will focus on pilot-symbol-assisted-modulation- (PSAM-) based, blind, and semiblind techniques for channel estimation and direct equalization of rapidly time-varying channels. PSAM techniques rely on time multiplexing data symbols and known pilot symbols at known positions, which the receiver utilizes to either estimate the channel or obtain the equalizer coefficients directly. In this context, we first derive the optimal minimum mean-squared error (MMSE) interpolation filter. Then we derive the conventional BEM channel

estimation technique based on LS fitting. While the MMSE interpolation filter requires the channel statistics, the latter does not require a priori knowledge of the channel statistics. It was shown that the modeling error between the true channel and the BEM channel model is quite large for the case when the BEM period equals the time window. This case corresponds to a critical sampling of the Doppler spectrum. 

Reducing this modeling error can be achieved by setting the BEM period equal to a multiple of the time window. In other words, we can reduce the modeling error by oversampling the Doppler spectrum. In the authors treated the first case ignoring the modeling error. However, when BEM oversampling is used, LS fitting of the BEM channel based on pilot symbols only is sensitive to noise. Here, we show that robust-PSAM-based channel estimation can be obtained by combining the optimal-MMSE-interpolation based channel estimation with the LS fitting of the BEM.Although this can be applied to the critically  sampled case as well as to the oversampled case with oversampling factor greater than one, little gain is obtained for the critically sampled case. In addition, we show that the channel estimation step can be skipped and obtain the equalizer coef ficients directly based on the pilot symbols. This is referred to as PSAM-based direct equalization. The training overhead imposed on the system can be completely eliminated by using blind techniques for channel estimation and direct equalization.

Due to the poor performance of blind techniques and their high implementation complexity, better performance and reduced complexity semiblind techniques can be obtained. Semiblind techniques are obtained by combining blind techniques with training.For our blind techniques we focus on deterministic approaches. For time-invariant (TI) channels, a least-squaresbased deterministic channel estimation method is discussed , and deterministic mutually referenced equalization is proposed . 

Subspace-based methods have also been proposed for channel identification/equalization for TI channels. For doubly selective channels, deterministic blind identification/equalization techniques are proposed where for a zero-forcing (ZF) FIR solution to exist, the number of subchannels (receive antennas) is required to be greater than the number of basis functions used for BEM channel modeling. In blind techniques based on linear prediction are proposed for doubly selective channels, where second-order statistics of the data are used. However, these techniques also require the number of receive antennas to be greater than the number of basis functions of the BEM channel. However, we propose an approach for which the ZF solution already exists when only two subchannels (receive antennas) are used.