Cooperative Filter-and-Forward Beamforming with Linear Equalization

Abstract

Abstract-In this paper, we investigate filter-and-forward beamforming (FF-BF) for relay networks employing single-carrier transmission over frequency-selective channels. In contrast to prior work, we assume that the destination node is equipped with a simple linear equalizer. The FF-BF filters at the relays are optimized for maximization of the signal-to-noise ratio at the equalizer output under a joint relay power constraint. For infinite impulse response (IIR) FF-BF filters, we derive a closedform expression for the filter frequency response, and a numerical algorithm with guaranteed convergence is developed for optimization of the power allocation factor included in the expression. We also provide an efficient gradient algorithm for recursive calculation of near-optimal finite impulse response (FIR) FF-BF filters. Simulation results show that, in general, short FIR FF-BF filters are sufficient to closely approach the performance of IIR FF-BF filters even in severely frequency-selective channels and that the proposed FF-BF scheme with equalization at the destination achieves substantial performance gains compared to a previously proposed FF-BF scheme without equalization.

Cooperative Two-Way Filter-and-Forward Beamforming for Frequency-Selective Channels

“…Let a opt j (f ) denote the optimum solution for problem (29). Since (29) is equivalent to the optimization of the FF-BF filters for two one-way relaying systems, we can draw from the results in [3], [4]. In particular, a opt j (f ), j ∈ {1, 2, }, can be efficiently obtained with the algorithms summarized in [3, Algorithm I] (for LE) and [4, Table 1] (for LE and MF).…”

“…In [2], only a simple slicer was employed at the destination requiring the FF-BF filters at the relays to equalize both the source-relay and the relay-destination channels. In [3], [4], it was shown that substantial performance gains are possible if simple linear equalization (LE) or decision-feedback equalization (DFE) is performed at the destination at the expense of an increase in complexity. For two-way relaying, the current literature has concentrated on the frequency-nonselective case.…”

“…Thereby, we consider two cases for the receive processing at the tranceivers: (1) a simple slicer is used without equalization and (2) LE is performed. The resulting FF-BF filter design problems are substantially more challenging than those for one-way relaying in [2], [3], [4], since one filter at the relay has to be optimized to achieve a certain level of performance at two receivers. In particular, we consider the optimization of the FF-BF filters at the relays for a signal-to-interference-plus-noise ratio (SINR) balancing objective under a relay transmit power constraint, i.e., maximization of the worst transceiver SINR.…”

“…However, wireless channels are typically frequency selective and multi-carrier modulation is not applicable in still evolving legacy systems such as GSM and EDGE and in simple sensor networks which cannot cope with the high peak-to-average power ratios of multi-carrier modulation. For one-way relaying systems, FF beamforming (FF-BF) for frequency-selective channels was considered in [2], [3], [4]. In [2], only a simple slicer was employed at the destination requiring the FF-BF filters at the relays to equalize both the source-relay and the relay-destination channels.…”

“…Amplify-and-forward (AF) and decode-and-forward (DF) relaying are the two most common forms of relaying [1]. More recently, filter-andforward (FF) relaying has been introduced for frequencyselective channel [2], [3], [4]. Similar to traditional AF relaying [1], no decoding/re-encoding is needed at the relays and the relays merely perform linear processing in form of infinite impulse response (IIR) or finite impulse response (FIR) filtering of the received signals.…”