Maximum Likelihood Frequency Detection for Discrete Hadamard Precoded OFDM

In recent years, orthogonal frequency division multiplexing (OFDM) has become very popular, particularly in the field of wireless communications. Precoded OFDM (POFDM) provides additional advantages, including greater immunity to the effects of multipath channels. However, OFDM and POFDM are more sensitive to frequency errors at the receiver than single-carrier transmission. Following earlier work by Gardner for the case of single-carrier transmission, and Daffara and Chouly for the case of conventional OFDM, we introduce a maximum likelihood frequency detector (MLFD) for POFDM. The precoder used is the discrete Hadamard transform (DHT). Existing MLFD schemes for OFDM assume QAM input symbols. However, the input symbols to the OFDM system after precoding are approximately Gaussian distributed, due to the central limit theorem. Therefore, it is not clear that the MLFD will operate correctly for POFDM. An expression for the s-curve of the DHT-POFDM MLFD is developed for the frequency offset range -0.8 < deltaT < 0.8 in an AWGN channel, and this s-curve is verified by simulation. It is shown both theoretically and by simulation that the s-curve for the DHT-POFDM MLFD is similar to that of Daffara et al.´s OFDM MLFD in AWGN channels. The tracking performance of a feedback frequency recovery loop, consisting of the MLFD and a digital phase-locked loop (DPLL), is compared for both conventional OFDM and DHT-POFDM. It is shown that the DHT-POFDM MLFD achieves lower MSE than the OFDM MLFD in the presence of multipath fading. We conclude that frequency tracking in multipath channels is a more robust process for DHT-POFDM than conventional OFDM.