The Fisher-Snedecor F distribution: A Simple and Accurate Composite Fading Model

—We consider the use of the Fisher–Snedecor F distribution, which is deﬁned as the ratio of two chi-squared variates, to model composite fading channels. In this context, the root-mean-square power of a Nakagami-m signal is assumed to be subject to variations induced by an inverse Nakagami-m random variable. Comparisons with physical channel data demonstrate that the proposed composite fading model provides as good, and in most cases better, ﬁt to the data compared to the generalized-K composite fading model. Motivated by this result, simple novel expressions are derived for the key statistical metrics and performance measures of interest.

The Fisher-Snedecor F Distribution: A Simple and Accurate Composite Fading Model More recently, the inverse gamma distribution was introduced to model the random variation of the mean signal power in composite fading channels [1]. Based on this, we now recall that the square root of a gamma random variable (RV) follows a Nakagami-m distribution. Exploiting this property, we consider a composite fading channel, in which the scattered multipath contribution follows a Nakagami-m distribution, while the root-mean-square (rms) signal is shaped by an inverse Nakagami-m distribution. It is demonstrated that this composite probability density function (PDF) is functionally equivalent to the Fisher-Snedecor F distribution [3]. We also derive simple expressions for a range of channel related statistics, such as the cumulative distribution function (CDF), moments, moment generating function (MGF) and amount of fading (AF). Novel analytical expressions for the corresponding outage probability (OP) and average bit error probability (ABEP) for the case of binary phase-shift keying (BPSK) and differential phase-shift keying (DPSK) are also obtained.

A. Signal Model
In an F composite fading channel, multipath fading is manifested by the same propagation mechanisms responsible for Nakagami-m fading whereas, the rms power of the received signal is subject to variation induced by shadowing. Based on this, the composite signal envelope, R, can be defined as where m represents the number of multipath clusters, X n and Y n are independent Gaussian RVs, which denote the in-phase and quadrature components of the cluster n, and E [X n ] = E [Y n ] = 0 and E X 2 n = E Y 2 n = σ 2 , with E[·] denoting statistical expectation. Moreover, A is an inverse Nakagami-m RV with the shape parameter m s and scale parameter, 1/ s , set equal to unity (i.e. s = 1) whose PDF is given by where (·) denotes the gamma function [4, eq. (8.310.1)].

B. Envelope PDF and CDF
The PDF of the composite signal envelope can be obtained by averaging the infinite integral of the conditional PDF of the Nakagami-m process with respect to the random variation of the rms signal power, Using the aforementioned signal model, this insinuates that where m is the fading severity parameter and = E r 2 is the mean power. By performing a simple transformation of variables and using [4, eq. (3.326.2)] along with some algebraic manipulations, it follows that where B(·, ·) denotes the beta function [4]. The form of the PDF in (3) is functionally equivalent to the F distribution, 1 where m s controls the corresponding amount of shadowing of the rms signal power. Thus, as m s → 0, the rms signal power in F fading channels undergoes heavy shadowing. Conversely, as m s → ∞, there is no shadowing of the rms signal power and (3) reduces to the Nakagami-m PDF. Based on the envelope PDF in (3), the corresponding CDF of the composite signal envelope, F R (r ) where 2 F 1 (·, ·; ·; ·) is the Gauss hypergeometric function [4, eq. (9.111)]. Additionally, the corresponding PDF of the instantaneous SNR, γ , of the F fading model is also readily obtained by As an intermediate step, we find the PDF, f W (w), of W = Z 2 which follows a chi-square distribution. The variate Z can be written as Z = sgn(Z ) × |Z | where sgn(·) denotes the sign function and |Z | = √ W . Consequently, the PDF f Z (z) of Z can be obtained as follows Using the standard statistical procedure of transformation of variables, we find that f R, (r, θ) = |J | f X,Y (x, y) where |J | is the Jacobian of the transformation. By carrying out the required mathematical procedure with |J | = r and we can obtain the conditional PDF of an F envelope-phase process as follows To this effect, the joint envelope-phase PDF of an F fading process can be obtained using f R, (r, θ) = ∞ 0 f R|A, (r |α, θ ) f A (α)dα, which after some algebraic manipulations can be expressed as Evidently, integrating (8) with respect to r yields the Nakagami-m phase PDF given in [5], as expected.

D. Moments and MGF
With the aid of (5), the n th moment of the instantaneous SNR can be derived using [4, eq. (3.194

III. EMPIRICAL COMPARISON OF F AND K G MODELS
The usefulness of the F distribution to model composite fading channels can be verified by comparing its fit to empirical data contrasted with more established models, such as generalized-K (K G ). To compare the two models, we used field measurements obtained at 5.8 GHz for deviceto-device (D2D) communications within an indoor open office area and an outdoor environment. To emulate the user equipment (UE), the hardware setup described in [6] was used. In this study, a D2D link was formed between two persons, namely person A, an adult female of height 1.65 m and weight 51 kg, and person B, an adult male of height 1.83 m and weight 73 kg. As shown in Fig. 1, during the D2D measurements, person A held the UE at her left-ear to imitate making a voice call while person B held the UE either at his left-ear or in the right-front pocket of his clothing to mimic carrying a device under two different channel conditions, namely line-of-sight (LOS) and non-LOS (NLOS). Consequently, the head-to-head (H2H) and head-topocket (H2P) channels were considered in LOS and NLOS conditions for each environment.
The parameter estimates for the F and K G fading models were obtained using a non-linear least squares function programmed in MATLAB. To allow our results to be reproduced, all parameter estimates are given in Table I. As an example of data fitting process, Fig. 2 shows the CDFs of the F and K G fading models fitted to some empirical data. These plots were typical of the fits obtained, but more importantly illustrated that the characterizations obtained by the F model provided a better fit to the empirical data at the tail of the CDFs. This is an important attribute of the proposed model as it is the deepest fades which are one of the greatest sources of degradation in wireless system performance. In a similar manner to the analysis performed in [7], here, the Akaike information criterion (AIC) was used to select the most likely model between the F and K G distributions. As a result of the ranking performed in Table I (which is based on the computed AIC), it was found that the F model was selected as the most likely model for 62.5% of the considered cases.

IV. APPLICATIONS IN DIGITAL COMMUNICATIONS A. Amount of Fading
It is recalled here that the AF is often used as a relative measure of the severity of fading encountered in wireless transmission over fading channels [8]. With the aid of (9), for the case of an F fading channel, this can be obtained as denotes the variance operator. Fig. 3 shows the estimated AF values for different multipath (0.5 ≤ m ≤ 10) and shadowed fading (3 ≤ m s ≤ 20) conditions. It is clear that the greatest AF occurs when the channel is subject to simultaneous heavy shadowing (m s → 3) and severe multipath fading (m → 1/2). On the contrary, the value of the AF approaches zero as both the multipath and shadowed fading parameters become large (i.e. m, m s → ∞). In particular, when m s → ∞ this reduces to the AF of a Nakagami-m fading channel, i.e. AF = 1/m, and consequently asymptotically approaches the AF of the Rayleigh (AF = 1) and one-sided Gaussian (AF = 2) models when m → 1 and m → 1/2 respectively.

B. Outage Probability
The OP for the case of F fading can be derived with the aid of [4, eq.
where γ th is the pre-determined SNR threshold. For the case of asymptotically large values ofγ , the OP can be expressed  Fig. 4 illustrates the exact OP for heavy (m s = 1/2), moderate (m s = 5) and light (m s = 50) shadowed fading environments along with the respective asymptotic curve. The simulation results are also shown in Fig. 4 to verify the analytical results. In this case, the data underlying the simulated plots was obtained through the straightforward calculation of the ratio of two gamma random sequences. It is clear that the OP increases rapidly as the shadowing becomes heavier, indicating that the communication performance degrades in environments which exhibit heavy shadowed fading characteristics. Moreover, it can be seen that the asymptotic OP provides a good agreement with the exact OP at high average SNR levels. It is also evident that in a lightly shadowed fading environment, the estimated OP for F distributed channels becomes practically equivalent to the OP of the Nakagamim channels, namely P Nak out = 1 − (m, mγ th /γ )/ (m) [10].

C. Average Bit Error Probability
It is recalled that the ABEP for a DPSK scheme is given by , with c 1 = 1/2 and c 2 = 1. Thus, the ABEP in the case of F fading channels can be readily deduced using (10). Fig. 5 shows the ABEP versus γ for DPSK (c 1 = 0.5 and c 2 = 1) under different multipath and shadowed fading conditions along with the corresponding simulation results. As expected, the ABEP increases rapidly as multipath fading becomes severer and shadowing becomes heavier. Furthermore, in a lightly shadowed fading environment (m s = 50), the estimated ABEP for an F fading channel approaches the ABEP of Nakagami-m fading channels, where M γ (s) = (1 − sγ /m) −1 [8]. Similarly, by setting m = 1 and m s = 50 (light shadowing), the estimated ABEP for an F fading channel corresponds to that of Rayleigh fading channels, where M γ (s) = (1 − sγ ) −1 [8].
Recalling the Craig integral representation in [11, eq. (20)], the corresponding ABEP for the case of BPSK modulation can be obtained by substituting (10) into this expression and setting g = 1 [12]. Again, it is recalled that for the case of asymptotically large values ofγ , u denoting a positive fixed constant which differs according to the chosen modulation scheme, e.g. u = 2 for a BPSK scheme. Based on this, a simple asymptotic expression is readily deduced for the ABEP of BPSK modulated signals over F fading channels V. CONCLUSION We have introduced the F distribution as an accurate and tractable composite model to characterize the combined effects of multipath and shadowing. Using some physical channel data, obtained for D2D communications, it was shown that the F fading model often provided a better fit compared to the K G model. Most importantly though, when comparing the analytical forms of the PDFs, CDFs and ABEP of the two models, the F distribution shows less complexity compared to the K G model. In particular, the PDF and CDF of the F distribution require the computation of a smaller number of special functions than the K G model, while determining the ABEP of the K G model given in [13, eq. (8)] requires the calculation of the Meijer-G function whereas the ABEP of the F fading model does not. Based on the tractability of the F distribution, novel and simple expressions have been derived for the key statistical metrics and performance measures.