SOME RESULTS ON CHANNEL CAPACITY WHEN USING MULTIPLE ANTENNAS Quentin H. Spencer & A. Lee Swindlehurst Department of Electrical & Computer Engineering Brigham Young University Provo, Utah 84602 spencerq@ee.byu.edu ABSTRACT Recent research has demonstrated that the capacity of a multipath-rich channel (such as indoors) is dramat- ically increased when antenna arrays are used at both transmitter and receiver. This is achieved by using the multipath to create parallel sub-channels. This ability to achieve parallelism in the channel is influenced by how much information about the channel is known at the transmitter, the structure of the channel, and the structure of the arrays. This paper discusses the in- fluence of the transmitter’s knowledge of the channel on capacity, presents closed-form solutions for capac- ity when there are two paths in the channel, and then presents numerical results based on simulations. For the two-pathcase, the simulation data compare capacity based on differing assumptions about the transmitter, and compare different array structures. Simulation re- sults are also presented for random channels generated using a statistical model for indoor multipath propaga- tion. 1. INTRODUCTION Recent research [1, 2, 3] has shown that dramatic in- creases in channel capacity are possible in multipath- rich environments when multiple antennas are used at both transmitter and receiver. The additional capacity is due to the fact that spatial diversity at both ends of the channel enables parallel transmission of data, some- thing not possible when spatial diversity is present only at one end of the channel. The results for capacity reported in the literature so far have subtle differences which are due to differing assumptions about the channel. In particular, what the transmitter knows about the channel has a great bearing on the transmission scheme, and ultimately the This work was supported by the National Science Foundation under Wireless Initiative Grant CCR 99-79452. achievable capacity. The purpose of this paper is to compare, analytically and numerically, the capacity of the channel under varying assumptions about informa- tion available to the transmitter, differing array struc- tures, and different models for the propagation in the channel. The next section provides more detail on these various assumptions. Section 3 presents a derivation for the channel capacity when the channel is low-rank and the transmitter has partial information about the channel. Section 4 presents a mathematical analysis of the singular values of H for a rank-2 channel, and uses the result to derive a closed-form expression for capac- ity in terms of the gains and angles of the two paths for both the full information and partial information at the transmitter scenarios. Section 5 presents simu- lation results based on the two-path analysis, and Sec- tion 6 presents simulation results based on a realistic indoor propagation environment. 2. BACKGROUND The capacity of a Multiple Input, Multiple Output (MIMO) channel is related to the singular values of the channel matrix H. Assume there are M T transmit- ters and M R receivers, and that the channel is narrow- band, where h ij represents the complex gain from the ith transmitter to the j th receiver. If s is the M T X 1 vector of transmitted signal signals and η is addi- tive noise, the received signal at the transmitter can be represented as      x 1 x 2 . . . x M R      =      h 11 ... h 1M T h 21 ... h 2M T . . . . . . . . . h M R1 ... h M R M T           s 1 s 2 . . . s M T      +      n 1 n 2 . . . n M R      (1) x = Hs + η (2)