In this paper, we consider filter-and-forward beamforming (FF-BF) for two-way relay networks employing single-carrier transmission over frequency-selective channels. In FF-BF, the relay nodes filter the received signal using finite impulse response (FIR) or infinite impulse response (IIR) filters. For the processing at the transceivers, we investigate two different cases: (1) simple slicing without equalization and (2) linear equalization. For both cases, we optimize FF-BF filters for maximization of the minimum transceiver signal-to-interference-plus-noise ratio (SINR) subject to a relay transmit power constraint. For case (1), we show that the optimization problem for FIR filters can be transformed into a convex second-order cone programming problem, which can be efficiently solved using standard tools. For case (2), leveraging results from IIR FF-BF for one-way relaying, we establish an upper and an achievable lower bound for the max-min problem. Since the gap between the upper and the lower bound is small, a close-to-optimal solution is obtained. Our simulation results reveal that the performance of FF-BF without equalization at the transceivers crucially depends on the slicer decision delay and transceivers with slicers can closely approach the performance of transceivers with equalizers provided that the FIR FF-BF filters are sufficiently long.

In this paper, we consider beamforming (BF) for cooperative networks with one multi-antenna source, multiple multi-antenna amplify-and-forward (AF) relays, and one single-antenna destination. The source BF vector and the AF-BF matrices at the relays are optimized for maximization of the signal-to-noise ratio (SNR) at the destination under three different power constraints. In particular, we consider individual relay power constraints, a joint relay power constraint, and a joint power constraint for the source and the relays. We solve the associated optimization problems in two stages. In the first stage, we find the optimal AF-BF matrices for a given BF vector at the source. For the cases of individual and joint relay power constraints, closed-form solutions for the AF-BF matrices are provided, respectively. Furthermore, for the case of a joint source-relay power constraint, the AF-BF matrices are obtained in closed form up to a scalar power allocation factor and an efficient numerical algorithm for the power allocation between the source and the relays is given. In the second stage, the optimal source BF vectors are computed. Thereby, we show that for the joint relay and the joint source-relay power constraints, the resulting problem can be transformed into a non-convex polynomial programming problem which allows for an exact solution for small-scale networks. For large-scale networks and networks with individual relay power constraints, we propose efficient suboptimal optimization methods for the source BF vector. Simulation results show the benefits of having multiple antennas at the source and/or multiple multi-antenna relays and illustrate the performance differences introduced by the three different power constraints.

Most of the existing literature on cooperative relay networks has focused on frequency-nonselective channels or frequency-selective channels with multi-carrier transmission. However, several practical systems employ single-carrier transmission over frequency-selective channels and the design of corresponding relaying schemes is a largely under-explored topic. 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 or decision feedback 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 unified expression for the filter frequency response valid for linear equalization, decision feedback equalization, and an idealized matched filter receiver. A numerical algorithm with guaranteed convergence is developed for optimization of the power allocation factor included in the expression for the IIR FF-BF filter frequency response. 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.

In this paper, we consider filter-and-forward beamforming (FF-BF) for two-way relay networks employing single-carrier transmission over frequency-selective channels. In FF-BF, the relay nodes filter the received signal using finite impulse response (FIR) or infinite impulse response (IIR) filters. For the processing at the transceivers, we investigate two different cases: (1) simple slicing without equalization and (2) linear equalization (LE) or decision-feedback equalization (DFE). For the first case, we optimize FIR FF-BF filters, respectively, for maximization of the minimum transceiver signal-to-interference-plus-noise ratio (SINR) subject to a relay transmit power constraint and for minimization of the total relay transmit power subject to two quality of service (QoS) constraints. We show that both problems can be transformed into a convex second-order cone programming (SOCP) problem, which can be efficiently solved using standard tools. For the second case, we optimize IIR and FIR FF-BF filters for max-min optimization of the SINR, and for transceivers with zero-forcing LE, also for minimization of the sum mean-squared error (MSE) at the equalizer outputs of both transceivers. Leveraging results from FF-BF for one-way relaying, we establish an upper and an achievable lower bound for the max-min problem and an exact solution for the sum MSE problem. Since the gap between the upper and the lower bound for the max-min problem is small, a close-to-optimal solution is obtained. Our simulation results reveal that the performance of FF-BF without equalization at the transceivers crucially depends on the slicer decision delay and transceivers with slicers can closely approach the performance of transceivers with equalizers provided that the FF-BF filters are sufficiently long and a sufficient number of relays is deployed.

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 closed-form 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.

In this paper, we consider beamforming (BF) for cooperative networks with one multi-antenna source, multiple multi-antenna amplify-and-forward (AF) relays, and one single-antenna destination. The source BF vector and the AF-BF matrices at the relays are optimized for maximization of the signal-to-noise ratio at the destination under a joint power constraint for the source and the relays. We solve the associated optimization problem in two stages. In the first stage, we find the optimal AF-BF matrices for a given BF vector at the source. In particular, the direction of the AF-BF matrices is derived in closed form and an efficient numerical algorithm for the convex power allocation problem between the source and the relays is provided. In the second stage, the optimal source BF vectors are computed. Thereby, we show that the resulting problem can be transformed into a non-convex polynomial programming problem which allows for an exact solution for small-scale networks. For large-scale networks, we propose efficient suboptimal optimization methods for the source BF vector.

In this paper, we propose frequency-domain (FD) and time-domain (TD) beamforming (BF) schemes for cooperative orthogonal frequency division multiplexing (OFDM) networks with multiple amplify-and-forward relays. Whereas for FD-BF the BF weights are applied in the FD, for TD-BF cyclic BF filters (C-BFFs) are used on the TD signal avoiding discrete- time Fourier transform operations at the relays and drastically reducing the required amount of feedback from the receiver to the relays. Adopting the average mutual information (AMI) per sub-carrier as optimality criterion, we show that the direction of the optimal FD-BF weights can be obtained in closed-form and that the optimal sub-carrier power allocation (PA) problem is convex. For solution of the PA problem an interior point method and a bisectional search dual method are provide. Furthermore, for solution of the C-BFF optimization problem an efficient gradient algorithm is proposed. Simulation results for IEEE 802.11n channels show that TD-BF with short C-BFFs closely approaches the performance of FD-BF and outperforms direct transmission without relaying.

Transmit beamforming (BF) and receive combining are simple and popular methods for performance enhancement in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. In this paper, we propose a novel single-data stream, time-domain BF (TD-BF) scheme for MIMO-OFDM systems which uses cyclic BF filters (C- BFFs). Assuming perfect channel state information (CSI) at the transmitter, the C-BFFs are optimized for two different criteria, namely, maximum average mutual information (AMI) per sub-carrier and minimum average uncoded bit error rate (BER). If the C-BFF lengthLgis equal to the number of sub-carriers Nc,closed-form solutions to both optimization problems exist. For the practically relevant caseLg< Ncwe present numerical methods for calculation of the optimum C-BFFs for both criteria. Using a global vector quantization (GVQ) approach the C-BFFs are quantized for practical finite-rate feedback channels. Simulation results for typical IEEE 802.11 n channels confirm the excellent performance of the proposed scheme and show that TD-BF has a more favorable performance/feedback rate trade-off than previously proposed frequency-domain BF (FD-BF) schemes.

In this paper, we consider minimum bit error rate (BER) beamforming (BF) for multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. In particular, assuming perfect channel state information (CSI) at the transmitter we optimize cyclic BF filters (C-BFFs) for minimization of the average BER and the maximum sub-carrier BER, respectively. If the C-BFF length Lgis equal to the number of sub-carriers Nc, closed-form solutions to both optimization problems exist. For the practically relevant case Lgcwe present numerical methods for calculation of the optimum C-BFFs. For finite-rate feedback channels we provide a global vector quantization (GVQ) scheme for codebook design. Simulation results for typical IEEE 802.11n channels confirm the excellent performance of the proposed minimum BER BF schemes for both uncoded and coded MIMO-OFDM systems.

Transmit beamforming (BF) and receive combining are simple and popular methods for performance enhancement in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. In this paper, we propose a novel time-domain BF (TD-BF) scheme for MIMO-OFDM systems which uses cyclic BF filters (C-BFFs). Assuming perfect channel state information (CSI) at the transmitter, the C- BFFs are optimized for maximization of the average mutual information per sub-carrier. Using a global vector quantization approach the C-BFFs are quantized for practical finite-rate feedback channels. Simulation and numerical results for typical IEEE 802.11n channels confirm the excellent performance of the proposed scheme and show that TD-BF has a more favorable performance/feedback rate trade-off than previously proposed frequency-domain BF (FD-BF) schemes.

In this paper, we propose beamforming schemes for frequency-selective channels with decision-feedback equalization (DFE) at the receiver and with, respectively, perfect and quantized channel state information (CSI) at the transmitter. For beamforming with perfect CSI and infinite impulse response (IIR) beamforming Alters (BFFs) we provide a closed-form expression for the optimum BFFs. We also devise two efficient numerical methods for recursive calculation of the optimum finite impulse response (FIR) BFFs with perfect CSI. For beamforming with quantized CSI and finite-rate feedback channel, we propose a global vector quantization (GVQ) algorithm for codebook design. This algorithm is deterministic and independent of initial conditions and does not impose any constraints on the number of transmit and receive antennas, the antenna correlation, or the fading statistics. Our simulation results for typical GSM/EDGE channels show that in general short FIR BFFs are sufficient to closely approach the performance of IIR BFFs even in severely frequency-selective channels. Furthermore, finite-rate feedback beamforming with only a few feedback bits achieves significant performance gains over single-antenna transmission, transmit antenna selection, and optimized delay diversity in frequency-selective fading.

In this paper, we consider beamforming with finite-rate feedback for frequency-selective channels with decision-feedback equalization (DFE) at the receiver. Using average bit error rate as performance measure, we propose a global vector quantization (GVQ) algorithm for codebook design. This algorithm is deterministic and independent of initial conditions. The proposed design method does not impose any constraints on the number of transmit and receive antennas, the antenna correlation, or the fading statistics. Simulation results for typical GSM/EDGE channels show that finite-rate feedback beamforming with one or two feedback bits achieves significant performance gains over single-antenna transmission, transmit antenna selection, and optimized delay diversity. While beamforming filters of length one are preferable if only few feedback bits can be afforded, longer filters can improve performance if a sufficiently large number of feedback bits is available.

In this paper, we propose beamforming schemes for frequency-selective channels with decision-feedback equalization (DFE) at the receiver. We consider both finite impulse response (FIR) and infinite impulse response (IIR) beamforming filters (BFFs). In case of IIR beamforming, we are able to derive closed-form expressions for the optimum BFFs. In addition, we provide an efficient numerical method for recursive calculation of the optimum FIR BFFs. Simulation and numerical results for typical GSM/EDGE channels confirm the significant performance gains achievable with beamforming compared to single-antenna transmission and optimized delay diversity.