In this paper, a Non Orthogonal Multiple Access (NOMA) downlink multiuser communication system is, composed of a single base station (BS) over a wireless channel to M users equipped with single antenna is considered. The BS is located at the center of a disc denoted by D with radius R. A Gaussian-Chebyshev quadrature approximation of the cumulative density function, of unordered channel gain is applied in the case of two users independent and identically distributed, located in a disk of radius R, under two techniques OMA and NOMA, in a downlink network, to evaluate ergodic capacities of NOMA and OMA users at the bounds of the fairness region of NOMA are calculated, and from a comparison between NOMA and OMA ergodic capacities as functions of SNR ratio, one can observe that NOMA always outperforms OMA technique. In this system model, NOMA users are sorted according to their locations from the BS, so that an mth user is able to detect its own signal, after performing Successive Interference Cancellation (SIC) algorithm based on an iterative cancellation of the ith user information. In the case of the failure of mth user to detect the ith user signal, mth user meets outage probability. This outage probability under an increasing SNR ratio of two NOMA users is evaluated for different values of N parameter, disk radius R, path loss exponent and power allocation coefficients. One can note that, NOMA users achieve more outage probability with increasing values of disk radius R, and the path loss exponent tends to increase, as there are more obstructions, and NOMA users tend to achieve more outage probability.
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