Ergodic Mutual Information for Generalized Fadings

Novel expressions for the ergodic mutual information (EMI) under BPSK modulation of single-input single-output (SISO) systems operating in generalized fading channels, namely Rayleigh, Nakagami-$m$, Rician, $\eta$-$\mu$ and $\kappa$-$\mu$, are derived. Different from the conventional results of the EUI, our developed expressions contain only the simplest numerical calculations, but no any Meijer's G-functions, which must be implemented in the specific computing software.


I. INTRODUCTION
Ergodic mutual information (EMI) is an important performance measurement metric in wireless multi-path channels, indicating the maximal transmission rate. Many researches on the EMI in fading scenarios have been presented in the past decades, including Rayleigh [1], Nakagami-m [2], Rician [3] fading channels, and generalized fading channels, namely η-µ and κ-µ [4]. However, all of these works assumed the input signals followed Gaussian distribution. In fact, a practical implementation is the case where the channel inputs are drawn from discrete constellations, the importance of which we cannot emphasize too much. Although the works in [5] have derived the recursive expression for the EMI under BPSK in Nakagami-m fading channels, the computation complexity is prohibitively high for large m.
In this paper, we derive novel expressions for the ergodic mutual information of single-input single-output systems under BPSK operating both in specific fading channels, including Rayleigh, Nakagami-m and Rician, and in generilized fading channels, including η-µ and κ-µ. It seems difficult to derive the closed-form exact formulas of EMI, thus we develop some simple closed-form approximate formulas, with high precision, to estimate the maximal transmission rate. Notably, our proposed expressions encompass no complicated structures, such as the commonly-used Meijer's G-functions in the conventional works. To the best of our knowledge, this is the first time to propose a comprehensive analysis for the ergodic mutual information with finite-alphabet inputs under general multi-path fading types.
The remaining parts of this manuscript is structured as follows: Section II presents the derived ergodic mutual information for generalized fadings. In Section III, simulation results are provided. Finally, Section IV concludes the paper. Denote f (·) as the probability density function (PDF) of the instantaneous Signal to Noise Ratio (SNR) per symbol. Then, the ergodic mutual information is defined as where I (·) represents the instantaneous input-output mutual information as a function of the SNR γ.

A. AWGN
Assume that the transmitted data stream are independent and identically distributed (i.i.d.) zero-mean binary symbols with equal probabilities, then the input-output mutual information in terms of the SNR γ under BPSK modulation over additive white Gaussian noise (AWGN) channels can be formulated as [6] An approximate formula of I (γ), with compact form and high precision, is expressed as [7] where ϑ = 0.6507. And the precision of Equ.

B. Rayleigh Fading Scenario
The PDF of received SNR in Rayleigh fading channels can be written as whereγ denotes the average SNR.
Lemma 1. The ergodic mutual information in Rayleigh fading channels is approximated aŝ Proof. On the basis of Equ. (3) and Equ. (4), the ergodic mutual information is derived as follows: The PDF of received SNR under Nakagami-m fading is given by Lemma 2. The ergodic mutual information under Nakagamim fading is approximated aŝ Proof. On the basis of Equ. (3) and Equ. (7),

D. Rician Fading Scenario
The PDF of received SNR under Rician fading is given by where n!Γ(n+1) is the 0-th order modified Bessel function of the first kind.
Lemma 3. The ergodic mutual information under Rician fading is approximated aŝ Proof. On the basis of Equ. (3) and Equ. (11), in which the step "(a)" is based on [8,Equ. (3.326.2)] and the step "(b)" is based on the Taylor series expansion of exp(x) = +∞ n=0 1 n! x n .

E. η-µ Fading Scenario
The PDF of received SNR under η-µ fading is given by [9] f ( 1 4 z 2 ) n n!Γ(ν+n+1) is the modified Bessel function of the first kind. It is well known that the η-µ fading holds two formats depending on the correlation in the main cluster. Moreover, h and H are functions of η and varies from one format to another. More specifically, in Format Lemma 4. The ergodic mutual information under η-µ fading is approximated aŝ Proof.
where the step "(a)" is due to [8, Equ. (3.326.2)], the step "(b)" is due to the Legendre duplication formula and the step "(c)" is due to the following Taylor series [10] 1 Notably, for any format of η-µ fading, it is easy to prove that h H + ϑγ 2µH > 1 with the expressions listed on the last page, thus Equ. (18) can be directly utilized.

III. SIMULATION
In this part, numerical and simulation results are provided to demonstrate the feasibility and validity of the former derivations. Notably, Rayleigh, Rician, η-µ and κ-µ fading can all be generated from Gaussian randoms, and the Nakagamim fading can be directly obtained by the Matlab function random(). Fig. 1 compares the approximated and simulated ergodic mutual information in terms ofγ for AWGN, Rayleigh and Nakagammi-m channels. Firstly, let us focus on the scenario of AWGN, whose approximated values are calculated by Equ. (3). As it shows, the simulated results meet accurately with the derivations, which suggests that Equ. (3) possesses high approximation precision and it is accurate enough to apply Equ. (3) into the estimation of mutual information. Most importantly, it can be observed that the approximation matches the empirical distribution for both Rayleigh and Nakagamim fading. Besides, it can be seen from this figure that the Nakagami-m fading channel tends to be AWGN as m grows. Then, Fig. 2 shows the ergodic mutual information as a function ofγ under Rician fading for selected value of K. It can be seen that the derivations meet tightly the results given via numerical simulations for all K and SNR. Moreover, with the increment of K, the positive effect from multi-path fading degrades, which contributes to the convergence trend toward AWGN in the graph above. Later, Fig. 3 and Fig. 4 move on to illustrate more generalized fading types, i.e. η-µ and κ-µ fadings. In Fig. 3, both Format 1 and Format 2 are presented to validate Equ. (15). As can be seen from both Fig. 3 and Fig. 4, the analytical results match well with the simulations.