Joint Equalization and Phase Noise Tracking for Doubly Selective Channels P. Pedrosa (1,2) , R. Dinis (1,3) and F. Nunes (1,2) (1) Instituto de Telecomunicações, 1049-001, Lisboa, Portugal (2) Instituto Superior Técnico, Universidade Técnica de Lisboa, 1049-001 Lisboa, Portugal (3) Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal e-mail:{ppedrosa, rdinis, nunes}@lx.it.pt Abstract—In this paper we consider the use of single car- rier modulations combined with frequency-domain equalization schemes (SC-FDE) in doubly selective channels where the channel variations are due to both phase noise and carrier frequency offset (CFO). Since carrier phase variations within the block duration can compromise significantly the performance of block transmission techniques. We propose an iterative receiver with joint symbol detection and phase-noise decision-directed esti- mation where we approximate the a posteriori phase noise probability density function by a weighted sum of Gaussian functions. Index Terms—Stochastic recursive filtering; Gaussian sum fil- ter; phase noise; carrier frequency offset; burst communications. I. I NTRODUCTION Block transmission techniques are known to be suitable for severely time-dispersive channels. In fact, by appending a suitable cyclic prefix (CP) to each block and employing fast Fourier transform (FFT)-based and frequency domain equalization (FDE) techniques the receiver complexity can be made relatively low and almost independent of the length of the channel impulse response. The most widely employed block transmission techniques are orthogonal frequency divi- sion multiplexing (OFDM) and single-carrier combined with frequency-domain equalization (SC-FDE). OFDM is particu- larly interesting for broadcasting systems [1] and the downlink of cellular systems [2], while SC-FDE is especially interesting for the uplink transmission [3]. The performance of SC-FDE can be significantly improved if the conventional linear FDE is replaced by a nonlinear FDE such as the iterative block decision feedback equalization (IB-DFE) [4]. The CP, which is employed to make the linear convolution associated to the channel equivalent to a cyclic convolution, must be longer than the overall channel impulse response (CIR) length (which includes the channel as well as the transmit and receiver filters). Since the CP is not used for detection purposes its duration should be a small fraction of the duration of the useful part of the block. As a consequence for broadband wireless systems, where the CIR can span over This work was partially supported by the FCT–Fundação para a Ciência e Tecnologia (pluriannual funding, ADCOD project PTDC/EEA-TEL/099973/2008, PEst-OE/EEI/LA0008/2011, and PhD grant SFRH/BD/40265/2007). a large number of symbols, we need long CPs and very long blocks, typically with several hundreds or even thousands of symbols. This means that in general the channel is no longer constant during the block duration. In fact, even for static conditions (i.e., with zero Doppler frequency shift) the equivalent channel can change due to phase variations between the local oscillators at the transmitter and the receiver. These phase variations are due to poor frequency alignment between oscillators and phase noise at each oscillator. Since phase variations within the block can lead to sig- nificant performance degradation [5], we should estimate and compensate for those variations before the detection. If the phase variation is due to pure carrier frequency offset (CFO) its estimation is relatively simple for SC-FDE because it leads to a constellation rotation that increases linearly along the block [6]. This means that we can design very efficient receivers where we perform the estimation and compensation of the CFO within each iteration of the IB-DFE [7], [8]. However, when phase variations are due to phase noise (PN) or a combination of PN with CFO it is harder to track their evolution [9]. A promising Bayesian approach for estimating the phase noise was proposed in [10] where the authors resort to a weighted sum of Gaussian functions to approximate a non- Gaussian probability density function (pdf). Although a similar solution is considered throughout this work, the following enhancements are considered. Firstly, we derive the solution for the general case of M -phase shift keying (M -PSK) con- stellations. Secondly, we provide a clear description on how the PN tracking block is integrated in the IB-DFE. Thirdly, we present performance results for QPSK signaling while assuming the additive white Gaussian noise (AWGN) channel and “real” channels. Finally, we compare our solution with a previously published technique [11]. In this paper we address the use of SC-FDE schemes in doubly selective channels where the channel variations during the block are due to both phase noise and a large CFO. We consider an iterative receiver with joint detection and decision- directed estimation of the phase noise where we approximate the a posteriori phase noise pdf by a weighted sum of Gaussian functions. Throughout this work arg{·} stands for the complex ar-