840 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 4, JULY 2003 New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels Marco Chiani, Senior Member, IEEE, Davide Dardari, Member, IEEE, and Marvin K. Simon, Fellow, IEEE Abstract—We present new exponential bounds for the Gaussian function (one- and two-dimensional) and its inverse, and for -ary phase-shift-keying (MPSK), -ary differential phase-shift-keying (MDPSK) error probabilities over additive white Gaussian noise channels. More precisely, the new bounds are in the form of the sum of exponential functions that, in the limit, approach the exact value. Then, a quite accurate and simple approximate expression given by the sum of two exponential func- tions is reported. The results are applied to the general problem of evaluating the average error probability in fading channels. Some examples of applications are also presented for the computation of the pairwise error probability of space–time codes and the average error probability of MPSK and MDPSK in fading channels. Index Terms—Bounds, fading, -ary differential phase-shift keying (MDPSK), -ary phase-shift keying (MPSK), function, space–time codes (STCs). I. INTRODUCTION T HE GAUSSIAN function, or, equivalently, the error function and its complement are of great importance whenever Gaussian variables occur [1], [2]. These functions are tabulated, and often available as built-in functions in mathematical software tools. However, in many cases it is useful to have closed-form bounds or approximations instead of the exact expression, to facilitate expression manipulations [3], [4]. In fact, exponential-type bounds or approximations are par- ticularly useful in evaluating the bit-error probability in many communication theory problems, such those arising in coding, fading, and multichannel reception [5]. Here, we provide new exponential-type upper bounds on the function and its inverse. The two-dimensional (2-D) case has also been considered. Moreover, a quite accurate approximation is developed in the form of the sum of two exponentials. Gener- ally, bounds or approximations are not suitable for application to average error-probability evaluation because their accuracy is not guaranteed for a wide range of values. However, we show Manuscript received June 24, 2002; revised September 10, 2002; accepted September 16, 2002. The editor coordinating the review of this paper and ap- proving it for publication is A. Svensson. The work of M. Chiani and D. Dardari was supported in part by MIUR and CNR, Italy. This paper was presented in part at the IEEE Global Telecommunications Conference, Taipei, Taiwan, November 2002. M. Chiani and D. Dardari are with DEIS, CSITE-CNR, University of Bologna, 40136 Bologna, Italy (e-mail: mchiani@deis.unibo.it; ddardari@ deis.unibo.it). M. K. Simon is with the Jet Propulsion Laboratory, Pasadena, CA 91109-8099 USA (e-mail: marvin.k.simon@jpl.nasa.gov). Digital Object Identifier 10.1109/TWC.2003.814350 that the accuracy of our results is preserved when used to eval- uate the average error probability in fading channels. Some ex- amples of applications are reported for the computation of the pairwise error probability (PEP) for space–time codes (STCs) and the average error probability of -ary phase-shift keying (MPSK) and -ary differential phase-shift keying (MDPSK). II. IMPROVED EXPONENTIAL-TYPE BOUNDS ON THE The complementary error function is usually defined as [2] (1) The tail probability of a unit variance zero mean Gaussian random variable is the function, which is related to the by (2) In the following, we will focus our attention on the , all results also being useful for the function by the relation in (2). A few years ago, the following integral of an exponential form for the function appeared in [6] (3) Although this alternative form can be obtained by trivial manip- ulations of the results given in Weinstein [7] and Pawula et al. [8], it is not explicitly stated in either paper. In the past, some exponential-type bounds have been derived. By adopting the Chernoff–Rubin bound we have, for , the exponential-type bound [1] (4) This can be improved by a factor 1/2. In fact, it is not difficult to show that the following also holds [1]: (5) In [9], it was observed that the bound in (5) can be derived from (3) by replacing the integrand with its maximum that occurs at as follows: (6) 1536-1276/03$17.00 © 2003 IEEE