The One-Bit Null Space Learning Algorithm and Its Convergence

This paper proposes a new algorithm for MIMO cognitive radio secondary users (SU) to learn the null space of the interference channel to the primary user (PU) without burdening the PU with any knowledge or explicit cooperation with the SU. The knowledge of this null space enables the SU to transmit in the same band simultaneously with the PU by utilizing separate spatial dimensions than the PU. Specifically, the SU transmits in the null space of the interference channel to the PU. We present a new algorithm, called the one-bit null space learning algorithm (OBNSLA), in which the SU learns the PU's null space by observing a binary function that indicates whether the interference it inflicts on the PU has increased or decreased in comparison to the SU's previous transmitted signal. This function is obtained by listening to the PU transmitted signal or control channel and extracting information from it about whether the PU's signal-to-interference-plus-noise power ratio (SINR) has increased or decreased. The OBNSLA is shown to have a linear convergence rate and an asymptotic quadratic convergence rate. Finally, we derive bounds on the interference that the SU inflicts on the PU as a function of a parameter determined by the SU. This lets the SU control the maximum level of interference, which enables it to protect the PU completely blindly with minimum complexity. The asymptotic analysis and the derived bounds also apply to the recently proposed blind null space learning algorithm.

I. Introduction

Multiple input multiple output (MIMO) communication opens new directions and possibilities for Cognitive Radio (CR) networks [1]–[7]. In particular, in underlay CR networks, MIMO technology enables the SU to transmit a significant amount of power simultaneously in the same band as the Primary User (PU) without interfering with it, if the SU utilizes separate spatial dimensions than the PU. This spatial separation requires that the interference channel from the SU to the PU be known to the SU. Thus, acquiring this knowledge, or operating without it, is a major topic of active research in CR [6], [8]–[15] and in other fields [16]. We consider MIMO primary and secondary systems defined as follows: we assume a flat-fading MIMO channel with one PU and one SU, as depicted in Fig. 1. Let be the channel matrix between the SU's transmitter and the PU's receiver, hereafter referred to as the SU-Tx and PU-Rx, respectively. In the underlay CR paradigm, SUs are constrained not to inflict “harmful” interference on the PU-Rx. This can be achieved if the SU restricts its signal to lie within the null space of ; however, this is only possible if the SU knows . The optimal power allocation in the case where the SU knows the matrix in addition to its own Channel State Information (CSI) was derived by Zhang and Liang [1]. For the case of multiple SUs, Scutari et al. [3] formulated a competitive game between the secondary users. Assuming that the interference matrix to the PU is known by each SU, they derived conditions for the existence and uniqueness of a Nash Equilibrium point to the game. Zhang et al. [9] were the first to take into consideration the fact that the interference matrix may not be perfectly known (but is partially known) to the SU. They proposed robust beamforming to assure compliance with the interference constraint of the PU while maximizing the SU's throughput. Another work on the case of an unknown interference channel with known probability distribution is due to Zhang and So [11], who optimized the SU's throughput under a constraint on the maximum probability that the interference to the PU is above a threshold. Fig. 1.

Our cognitive radio scheme. is unknown to the secondary transmitter and is a stationary noise (which may include stationary interference). The interference from the SU, , is treated as noise; i.e., there is no interference cancellation.