Adaptive Modulation for Multiple Antenna Channels June Chul Roh and Bhaskar D. Rao Department of Electrical and Computer Engineering University of California, San Diego La Jolla, CA 92093-0407 E-mail: jroh@ece.ucsd.edu, brao@ece.ucsd.edu Abstract—We consider the use of adaptive modulation scheme for multiple transmit and multiple receive antenna system with in- stantaneous channel information known to both the receiver and the transmitter. We derive an efficient bit allocation algorithm which maximizes the transmission rate for a given transmit power. The algorithm is generally a greedy algorithm; however, we derive a sufficient condition for the bit allocation algorithm to be glob- ally optimal, which is found to be satisfied in all digital modula- tion schemes. It is found that when uncoded M-ary QAM is used with a target symbol error probability of 10 -5 there is about 9 dB gap between the channel capacity and the throughput of adaptive modulation. I. I NTRODUCTION The adaptive modulation for scalar channels was studied in [1]. The fundamental concept of adaptive modulation is that the system parameters in the physical layer are adaptively changed based on the channel status to increase the communication link quality, mostly, transmission rate. This paper considers using the adaptive modulation method in multiple antenna channels. In particular, we focus on the power allocation problem over multiple spatial channels. In recent years, multiple antenna communication systems have gathered much attention for high-rate transmission over wireless channels. Telatar [2] showed the information-theoretic capacity of multiple-input multiple-output (MIMO) channels with flat fading. If the channel state information is known to both the trans- mitter and the receiver, an MIMO channel can be decomposed into parallel independent single-input single-output (SISO) channels by employing appropriate operations at the transmit- ter and the receiver. The resulting decomposed channels are characterized by the channel gain matrix, i.e., the gains of the decomposed channels are determined to be the singular values of the channel gain matrix. After decomposing the channel into parallel channels, the remaining problem is how to allocate the transmit power over the decomposed channels to maximize the total transmission rate. The adaptive modulation for multiple antenna channels is concerned with adaptation of modulation parameters in spatial as well as temporal domain. This research was supported by CoRe research grant No. Cor00-10074 and by a research grant from Ericsson. The power allocation problem can be equivalently consid- ered as so-called bit allocation problem 1 over multiple spatial channels if the target link quality is fixed. The bit allocation problem involves solving an optimization problem with integer variables, in which the optimum bit allocation over the multiple channels is determined to minimize the total transmit power for a given number of transmission bits. That is, the cost function is the total required transmit power for transmitting the given number of bits with a target link quality satisfied. The channel gain matrix characterizes the cost function. One contribution of this paper is that the bit allocation prob- lem is formulated as an optimization problem, and then an effi- cient bit allocation algorithm is derived. The derived algorithm is a greedy algorithm, which generally may not be the global optimum. However, we derive a sufficient condition for the bit allocation algorithm to be globally optimal, which is found to be satisfied in all M -ary digital modulation schemes. We are also interested in the average transmission rate that can be affordable with adaptive modulation in MIMO systems and how far the average rate is away from the channel capacity. We consider an adaptive modulation scheme that changes the modulation order of M -ary QAM, M ∈{0, 2, 2 2 ,..., 2 8 }, de- pending on the channel state. The simulation results show that the average transmission rate of the adaptive uncoded M -ary QAM is about 9 dB away from the channel capacity when the target symbol error probability is set to 10 −5 . II. SYSTEM MODEL We consider a point-to-point flat fading channel with mul- tiple antennas at both the transmitter and the receiver. The number of transmit antennas is denoted by t and the number of receive antennas by r. We consider a linear discrete channel model y = Hx + w (1) where x ∈ C t×1 is the transmitted signal, y ∈ C r×1 is the received signal, H ∈ C r×t is the channel gain matrix, and w ∈ C r×1 is the zero-mean complex Gaussian noise with co- variance E{ww † } = I r . As in [2], we assume that H is a ran- dom matrix independent of x and w. H is a complex Gaussian 1 The bit allocation problem has been also studied for multi-carrier communi- cation applications; there, computationally efficient suboptimum schemes have been focused because the number of parallel channels is usually large.