Level Crossing Rate and Average Fade Duration in MIMO Mobile Fading Channels Ali Abdi, Chunjun Gao, Alexander M. Haimovich Dept. of Elec. and Comp. Eng., New Jersey Institute of Technology, Newark, NJ 07102, USA Emails: ali.abdi@njit.edu, cxg9074@njit.edu, haimovich@adm.njit.edu Abstract – In this paper, important dynamic characteristics of MIMO mobile fading channels such as the level crossing rate (LCR) and average fade durations (AFD) are studied. Depending on the application, either a scalar crossing approach or a vector crossing approach is taken, and appropriate definitions for LCR and AFD in MIMO channels are provided. The more general concept of average stay duration (ASD) is also defined. Closed- from solutions for scalar MIMO LCR and vector MIMO ASD are presented, and illustrated via numerical examples. Finally, the utility of the new definitions and results, when applied to adaptive modulation in MIMO channels, Markov modeling, and the block fading approximation of MIMO channels are discussed as well. I. INTRODUCTION In mobile communications with time-selective fading, the level crossing rate (LCR), how often the signal crosses a certain threshold, is an important dynamic characteristic of the channel. The average fade duration (AFD), how long the signal stays below a given threshold, can be calculated from LCR and appears in a variety of applications. Choosing the frame length of coded packetized systems [1] and also interleaver optimization, to efficiently combat the burst of errors due to long fades, need the AFD information [2] [3]. In adaptive modulation schemes, the average time that a particular constellation is continuously used is related to AFD [4]. Average outage duration in multiuser cellular systems [5] [6], where interference from other users restricts the performance, is another close relative of AFD. Throughput (efficiency) of communication protocols such as automatic repeat request (ARQ) schemes can be estimated using AFD [7]. The transition probabilities between different states of a Markov model for fading channels has been calculated based on LCR at different levels [8]. In recent years, wireless communication over multiple-input multiple-output (MIMO) fading channels has become an active research area, specially due the very high data transmission rates that can be achieved in multi-antenna systems. Clearly, efficient communication over MIMO mobile fading channels requires a basic understanding of the statistical structure of such random channels, which could be different from the traditional single-input single- output (SISO) and single-input multiple-output (SIMO) channels. Some important characteristics of MIMO channels, such as space- time correlations in macrocells [9] and space-time-frequency correlations in macrocells [10], have been explored so far. However, to the best of our knowledge, the concepts of LCR and AFD for MIMO fading channels have not yet been defined and analyzed. More specifically, most of the LCR- and AFD-related research has been carried out in the context of SISO systems. The recent works on LCR and AFD for receive diversity combiners [11] [12] [13] [14], which eventually boil down to the crossing theory of a scalar process, appear in MIMO channels, as we will see in the sequel. However, in general, for an M-transmit N-receive multiantenna system, the joint dynamic behavior of MN correlated random signals is of interest (which has not been addressed in the literature). This requires a multidimensional approach to LCR and AFD problems. To show the utility of the theoretical results derived in this paper, we briefly discuss adaptive modulation, Markov modeling, the block fading model, and the concept of vector AFD in MIMO systems. In the first two cases we have a scalar crossing problem, whereas the last two require a vector crossing approach. The rest of this paper is organized as follows. In Sections II and III, the scalar and vector crossing problems in MIMO systems are discussed, respectively. Concluding remarks are given in Section IV. II. SCALAR CROSSING IN MIMO SYSTEMS In this section, first we present the mathematical formulation of the problem and its solution, followed by a numerical example. Then we highlight the applications. A. Mathematical Formulation Consider a time-selective narrowband M N × channel, with MN complex Gaussian processes, correlated in both space and time, and corrupted by a spatio-temporal white Gaussian noise. Obviously, depending on the presence or absence of line-of-sight (LOS), the envelope of subchannels could be Rice or Rayleigh, respectively. The instantaneous received signal-to-noise ratio (SNR) per symbol over the subchannel from the p-th transmitter to the l-th receiver is proportional to 2 | ()| lp h t , assuming perfect channel estimation at the receiver. The total instantaneous received SNR per symbol, () t γ , is therefore proportional to 2 , | ()| lp lp h t  , and is a useful measure for Markov modeling of diversity systems [15], MIMO channel characterization [16] [17], and design of MIMO systems [18] [19]. Now we want to determine the average stay duration (ASD) of () t γ within the region 1 2 [ , ] γ γ , 1 2 ASD{ (t),[ , ]} γ γγ , defined as the average time over which 1 2 () t γ γ γ ≤ ≤ . The concept of ASD is applicable to both scalar processes such as () t γ , as well as vector processes, discussed later, and includes AFD as the special case where 1 γ = -∞ . Similar to AFD, the ASD of () t γ within the region 1 2 [ , ] γ γ can be calculated according to 1 2 1 2 1 2 ASD{ (t),[ , ]}= Pr[ ]/ ICR{ (t),[ , ]} γ γγ γ γ γ γ γγ ≤ ≤ , where Pr[.] is the probability and the denominator is the incrossing rate (ICR), i.e., the average number of times that () t γ crosses one of the two borders and enters the region 1 2 [ , ] γγ . The numerator can be calculated via the Euler method [20]. For the denominator we use the result of [21]. Specifically, let 2 1 () () J i i xt u t = =  , where () i ut s are correlated nonzero mean real Gaussian processes. Also let 1 2 [ ... ] T J uu u = u and 1 2 [ ... ] T J uu u ′ ′′ ′ = u , where prime denotes differentiation with respect to time t and T is the transpose operator. The mean vector and the covariance matrix of [ ] T ′ T T uu are given by [ ] T T T 0 η η η , 11 12 21 22   =     Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ , (1)