IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 4, APRIL 2005 289 New Tight Bounds on the Pairwise Error Probability for Unitary Space-Time Modulations Rong Li, Student Member, IEEE, and Pooi Yuen Kam, Senior Member, IEEE Abstract—We present two upper bounds and one lower bound on the pairwise error probability (PEP) of unitary space-time modulation (USTM) over the Rayleigh fading channel. The two new upper bounds are the tightest so far, and the new lower bound is the tightest at low signal-to-noise ratio. Some implications for USTM constellation design are also pointed out. Index Terms— Unitary space-time modulation, performance bound, pairwise error probability. I. I NTRODUCTION U NITARY space-time modulation (USTM) [1] achieves the full channel capacity promised by multiple antenna systems when the receiver does not know the wireless channel gains. Here we propose two upper bounds and one lower bound on the pairwise error probability (PEP) for USTM and demonstrate that they are tighter than existing bounds. II. UNITARY SPACE-TIME MODULATION Consider a system with N T transmit antennas and N R receive antennas. The fading channels are assumed to remain constant in one time-block of N C symbol intervals, and N C > N T . Thus, we have the received signals given by [2, eq. (1)] ˜ X =  γN C /N T ˜ Φ ˜ H + ˜ W. (1) Here, overhead ˜ denotes a complex quantity, and superscript † denotes its conjugate transpose. Each entry ˜ x tn of the N C × N R matrix ˜ X is the received signal at receive antenna n for symbol interval t, and γ is the expected signal-to-noise ratio (SNR) at each receive antenna. The N C × N T matrix ˜ Φ is a unitary matrix from the constellation { ˜ Φ l | ˜ Φ † l ˜ Φ l = I,l = 1,...,L}, where I is the N T × N T identity matrix. Each entry ˜ h mn of the N T × N R matrix ˜ H is the path gain from transmit antenna m to receive antenna n. Each entry ˜ w tn of the N C × N R matrix ˜ W is the additive white Gaussian noise sample at receive antenna n for symbol interval t. We assume that all the ˜ h mn ’s and ˜ w tn ’s are independent, complex, Gaussian random variables with zero mean and unit variance. If ˜ H is unknown to the receiver, and ˜ Φ l ’s are transmitted with equal probability, the maximum likelihood receiver is given in [1, eq. (15)], and the PEP of deciding in favor of ˜ Φ j given that Manuscript received August 2, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Giorgio Taricco. The authors are with the Department of Electrical and Com- puter Engineering, National University of Singapore (e-mail: {LiRong, elekampy}@nus.edu.sg). Digital Object Identifier 10.1109/LCOMM.2005.04009. ˜ Φ l is transmitted is given by [1, eq. (B.10)], i.e., P e, lj = 1 4π  ∞ −∞  NT m=1 [1 + γ (1 − d 2 m, lj )(ω 2 +1/4)] −NR ω 2 +1/4 dω (2) where γ =( γN C /N T ) 2 / (1 + γN C /N T ). Each d m, lj ,m = 1,...,N T is a singular value of the matrix ˜ Φ † j ˜ Φ l , and we have 0 ≤ d m, lj ≤ 1. The corresponding Chernoff upper bound (CUB) on the PEP is given by [1, eq. (18)] P e, lj ≤ 1 2 NT  m=1  1+ γ ( 1 − d 2 m, lj ) /4  −NR . (3) III. NEW BOUNDS ON THE PEP By letting ω = tan(θ)/2, we rewrite the PEP expression in (2) as P e, lj = 1 π  π/2 0 N d  m=1 ⎡ ⎣ 1+ γ  1 − d 2 m, lj  4 sin 2 θ ⎤ ⎦ −NR dθ (4) where N d ≤ N T is the number of non-unity singular values. This expression leads to the following new bounds on the PEP. A. Tight Upper Bound-1 Applying the inequality (1 + x) −1 ≤ x −1 ,x > 0 to the integrand in (4), we can upper bound the PEP by P e, lj ≤ 1 π  π/2 0 N d  m=1 ⎡ ⎣ γ  1 − d 2 m, lj  4 sin 2 θ ⎤ ⎦ −NR dθ. Using the integral formula in [3, eq. 2.513 1], i.e.,  sin 2n xdx = 1 2 2n  2n n  x + (−1) n 2 2n−1 n−1  k=0 (−1) k  2n k  sin 2(n − k)x 2(n − k) , this upper bound can be reduced to P e, lj ≤ (γ d gm, lj ) −N d NR  2N d N R − 1 N d N R  . (5) This is our new, tight, upper bound-1 on the PEP. In (5), d gm, lj is the geometric mean of the singular values defined as d gm, lj =  N d  m=1 ( 1 − d 2 m, lj )  1/N d . 1089-7798/05$20.00 c  2005 IEEE