On Mutual Information of Stacked OSTBC

It is well known, that the Alamouti scheme is the only space-time code from orthogonal design achieving the capacity of multiple-input multiple-output (MIMO) wireless communication systems with nT = 2 transmit antennas and nR = 1 receive antenna. In this work, we propose the n-times stacked Alamouti scheme for nT = 2n transmit antennas and show that this scheme achieves the capacity in the case of nR = 1 receive antenna. For the more general case of more than one receive antenna, we show that if the number of transmit antennas is higher than the number of receive antennas we achieve a high portion of the capacity with this scheme. Further, we show that the MIMO capacity is at most twice the rate achieved with the proposed scheme for all SNR. We derive lower and upper bounds for the rate achieved with this scheme and compare it with upper and lower bounds for the capacity. Finally, we illustrate the theoretical results by numerical simulations. Recent information theoretic results have demonstrated that the ability of a system to support a high link quality and higher data rates in the presence of Rayleigh fading improves significantly with the use of multiple transmit and receive antennas [1], [2]. Since then there has been considerable work on a variety of schemes which exploit multiple antennas at both the transmitter and receiver in order to either obtain transmit and receive diversity and therefore increase the reliability of the system, e.g., orthogonal space-time block codes (OSTBC) and space-time trellis codes [3]–[5] or achieve the theoretical bounds [6] derived in [1], [2]. The performance of OSTBC with respect to mutual information has been analyzed (among others) in [7]–[9] and it was shown that the capacity is achieved only in the case of nT = 2 transmit, the well known Alamouti scheme [4], and nR = 1 receive antennas due to the rate loss inherent in OSTBC with higher number of transmit antennas. Recently, it was shown in [10] that due to this rate loss, OSTBC with odd number of antennas are always outperformed by OSTBC with even number of antennas, restricting even more the deployment of OSTBC. On the one hand, we have the OSTBC with low complexity and low rates. On the other hand, we have the space-time trellis codes, which achieve higher spectral efficiency in addition to high performance with respect to frame error rates. However, the decoding complexity of space-time trellis codes is increasing exponentially with the number of transmit antennas and the transmission rate. In order to achieve higher spectral efficiency combined with low complexity maximum likelihood detectors, [11]–[15] designed quasi-orthogonal space-time block codes (QSTBC) with transmission rate one for more than two transmit antennas. Other approaches aimed at reducing the decoding complexity of space-time trellis codes. For instance, a layered spacetime architecture was proposed in [16], where the transmit antennas were partitioned into two-antenna groups and on each group space-time trellis codes were used as component codes. In order to further decrease the complexity of this layered space-time architecture, [17]–[19] used the Alamouti scheme as component code for each group in combination with a suboptimal successive group interference suppression detection strategy. The outage probability of this scheme was analyzed in [20]. For n = 2, this transmission scheme is also referred to as double-space-time transmit diversity (DSTTD) and was proposed as one possible candidate for high speed downlink packet access (HSDPA) in 3GPP and beyond [21]. In this work, we show that the stacked Alamouti scheme is capable to achieve the capacity in combination with the optimal maximum likelihood detector for the case of nT = 2n transmit antennas and nR = 1 receive antennas. Furthermore, we show that in the case of more than one receive antenna and if nT > nR the stacked Alamouti scheme is capable to achieve a significant portion of the capacity and approaches the capacity if nT À nR. For any nT , nR, we show that the MIMO capacity is at most twice the rate achieved with the proposed scheme for all SNR. I. SYSTEM MODEL We consider a system with nT transmit and nR receive antennas. Our system model is defined by Y = GnT H T + N , (1) where GnT is the (T × nT ) transmit matrix, Y = [y1, . . . ,ynR ] is the (T × nR) receive matrix, H = [h1, . . . ,hnT ] is the (nR × nT ) channel matrix, and N = [n1, . . . ,nnR ] is the complex (T × nR) white Gaussian noise (AWGN) matrix, where an entry {nti} of N (1 ≤ i ≤ nR) denotes the complex noise at the ith receiver for a given time t (1 ≤ t ≤ T ). The real and imaginary parts of nti are independent and N (0,nT /(2SNR)) distributed. An entry of the channel matrix is denoted by {hij}. This represents the complex gain of the channel between the jth transmit (1 ≤ j ≤ nT ) and the ith receive (1 ≤ i ≤ nR) antenna, where the real and imaginary parts of the channel gains are independent and normal distributed random variables with N (0,1/2) per dimension. The channel matrix is assumed to be constant for a block of T symbols and changes independently from block to block. The average power of the symbols transmitted from