LOW SNR ANALYSIS OF THE NON-COHERENT MIMO CHANNEL UNDER ARBITRARY CHANNEL AND NOISE CORRELATION STRUCTURES Marko Beko * , Jo˜ ao Xavier and Victor Barroso Instituto Superior T´ ecnico – Instituto de Sistemas e Rob´ otica Av. Rovisco Pais, 1049-001 Lisboa, Portugal {marko,jxavier,vab}@isr.ist.utl.pt ABSTRACT The non-coherent single-user multiple-input multiple-output (MIMO) channel in the low signal-to-noise ratio (SNR) regime is analyzed from the capacity point of view. We investigate the impact of channel and noise correlation on the mutual information for the on-off and Gaussian signaling schemes. The novelty is that we allow an arbitrary structure for both the channel and noise correlation matrices. Our results estab- lish that, in the low SNR regime, mutual information is max- imized when the channel correlation matrix is of rank one. 1. INTRODUCTION In slowly fading scenarios, channel stability allows the re- ceiver to be trained in order to obtain the channel state infor- mation (CSI) necessary for coherent detection of the trans- mitted codeword. The scope of this paper will be fast fading scenarios, where the channel coefficients change too quickly to allow reliable channel estimation. Hence, CSI is no more available, and the receiver must operate in a non-coherent mode. Furthermore, we focus on the low signal-to-noise ratio (SNR) regime. This is due to the fact that a variety of digital communication systems (more specifically in wireless, sensor and satellite networks) operate in the power-limited region. See [2, 3] for a more thorough discussion of this topic. Previous work. In [3], low SNR MIMO systems when CSI is available at the receiver have been considered. The interplay of rate, bandwidth, and power is analyzed in the region of en- ergy per bit close to its minimum value. In [4], the scenario where no CSI is available at the receiver has been treated. It has been demonstrated that the optimal signaling at low SNR achieves the same capacity as the known channel case for sin- gle transmit antenna systems. In [5], it has been shown that knowledge of the first and second derivatives of capacity at low SNR gives us insight on bandwidth and energy efficiency for signal transmission. More precisely, these quantities tell * This work was partially supported by Fundac ¸˜ ao para a Ciˆ encia e a Tecnologia (ISR/IST plurianual funding) through the POS Conheci- mento Program that includes FEDER funds, and by FCT PhD Grant SFRH/BD/12809/2003, and by IST/ISR Plurianual Unidade 101. us how spectral efficiency grows with energy-per-bit. In [6], a formula for the second-order expansion of the input-output mutual information at low SNR is obtained, whereas in [7] the capacity and the reliability function as the peak constraint tends to zero are analyzed for a discrete-time memoryless channel with peak constrained inputs. In [8], Rao and Hassibi have shown that the on-off signaling presented in [4] general- izes to the multi-antenna setting and attains the known chan- nel capacity. The tradeoff between communication rate and average probability of decoding error using a framework of error-exponent theory has been considered in [9]. It is argued that the advantage of having multiple antennas is best realized when the fading is fully correlated, i.e., a performance gain of MN and a peakiness gain of M 2 NT can be achieved where M , N and T represent the number of transmit, receive an- tennas, and the length of the coherence interval, respectively. In [2], the approach in [8] has been extended as both noise and channel correlation have been taken in account. Contribution and paper organization. We study the non- coherent MIMO channel in the low SNR regime from the ca- pacity viewpoint. The novel aspect is that we allow the chan- nel covariance matrix to have an arbitrary correlation struc- ture. This, together with the fact that the noise covariance matrix is also allowed to have an arbitrary structure, makes this scenario the most comprehensive and challenging one. The data model is introduced in section 2. In section 3, the spatially correlated non-coherent MIMO block Rayleigh fad- ing channel is analyzed and the impact of channel and noise correlation on the mutual information is obtained for the on- off signaling. In section 4, we extend the analysis presented in section 3 to the case of Gaussian signaling. The main conclu- sion is that for both cases mutual information is maximized when the channel correlation matrix is fully correlated. We also argue that the on-off signaling is optimal for this multi- antenna setting. Some mathematical details are left to an ap- pendix.