Cooperative beamforming is a potentially useful technique for enhancing link reliability in modern wireless relay communications. To aid the beamforming design, the signal-To-noise ratio (SNR) of the source-To-relay (S-R) channel links must be known at the destination node. Abdallah and Papadopoulos in IEEE Trans. Signal Processing, vol. 56, no. 10, 2008, proposed a beamforming system with relay-Assisted SNR acquisition; that is, the S-R link SNR is first quantized at each relay and is then forwarded to the destination. An optimal quantizer design problem was proposed in that paper, aiming at reducing the system bit error rate; however, only the binary quantization case was considered therein, with the quantization threshold computed by a suggested rule of thumb. In this paper, we study the aforementioned quantizer design problem in the general multiple-bit setting. We show that the solutions to the first-order necessary condition for optimality can be obtained as a fixed point of a certain nonlinear map over the feasible threshold set. The existence and uniqueness of the fixed point are established by using, respectively, the Brouwer's and Schauder's fixed point theorems. Finally, we show that the fixed point thus obtained yields the globally optimal set of quantization thresholds. To the best of our knowledge, this paper is the first which leverages fixed point theories to rigorously solve SNR quantization problems in the context of wireless relay communication.
- Brouwer's fixed point theorem
- Relay communications
- Schauder's fixed point theorem
- cooperative beamforming
- cooperative communications
- fixed point theory