In this letter, we consider the product of two independent and not identically distributed $\alpha $ - $\cal {F}$ variates. Specifically, novel closed-form expressions for the probability density function, the cumulative distribution function, and the $n$ -th moments for the envelope as well as the instantaneous signal-to-noise ratio (SNR) are derived. Subsequently, closed-form expressions for the outage probability, the average bit error rate, and the ergodic channel capacity are derived alongside their asymptotic ones. The analytical expressions are validated through numerical and Monte-Carlo simulations, where excellent agreements are observed over a wide range of SNR values. Furthermore, the asymptotic curves follow the slope of the exact curves at high SNR values.

中文翻译：

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两个α-ℱ的乘积的统计随应用而变化

在这封信中，我们考虑两个独立且非同分布的乘积 $\alpha $ —— $\cal {F}$ 变量。具体来说，概率密度函数、累积分布函数和 $n$ 导出包络的第 -th 个矩以及瞬时信噪比 (SNR)。随后，与它们的渐近表达式一起导出了中断概率、平均误码率和遍历信道容量的封闭式表达式。解析表达式通过数值和蒙特卡洛模拟得到验证，在广泛的 SNR 值范围内观察到了极好的一致性。此外，渐近曲线在高 SNR 值时遵循精确曲线的斜率。