Second-Order Statistics of $\kappa-\mu$ Shadowed Fading Channels

In this paper, novel closed-form expressions for the level crossing rate (LCR) and average fade duration (AFD) of $\kappa-\mu$ shadowed fading channels are derived. The new equations provide the capability of modeling the correlation between the time derivative of the shadowed dominant and multipath components of the $\kappa-\mu$ shadowed fading envelope. Verification of the new equations is performed by reduction to a number of known special cases. It is shown that as the shadowing of the resultant dominant component decreases, the signal crosses lower threshold levels at a reduced rate. Furthermore, the impact of increasing correlation between the slope of the shadowed dominant and multipath components similarly acts to reduce crossings at lower signal levels. The new expressions for the second-order statistics are also compared with field measurements obtained for cellular device-to-device and body centric communications channels which are known to be susceptible to shadowed fading.


I. INTRODUCTION
The κ − µ shadowed fading model first appeared in the literature in [1] and immediately after this in [2]. It has been proposed as a generalization of the popular κ−µ fading model [3]. In this model clusters of multipath waves are assumed to have scattered waves with identical powers, alongside the presence of elective dominant signal components -a scenario which is identical to that observed in κ − µ fading [3]. The key difference between the κ − µ shadowed fading model and that of classical κ−µ fading is that the dominant components of all the clusters can randomly fluctuate because of shadowing. In particular it is assumed that the shadowing fluctuation follows a Nakagami distribution [4]. Like the κ − µ distribution, the κ − µ shadowed distribution is an extremely versatile fading model which also contains as special cases other important distributions such as the One-Sided Gaussian, Rice (Nakagami-n), Nakagami-m and Rayleigh distributions. In addition to this, it also contains as a special case Abdi's signal reception model [5] which considers Ricean fading where the dominant component also undergoes shadowed fading which follows the Nakagami distribution. Due to the ability of the Nakagami probability density function (PDF) to approximate the lognormal PDF [6], the κ−µ shadowed fading model can also be used to estimate Loo's well-known model for land mobile satellite communications [7].
While the research of composite fading models such as the κ − µ / gamma model [8] and its associated second-order statistics including the level crossing rate (LCR) and average fade duration (AFD) [9] have been advanced, unfortunately, at present, similar closed-form expressions for the second-order statistics of the κ − µ shadowed model are currently unavailable in the literature. The LCR and AFD of a fading signal are of great importance in the design of mobile radio systems and in the analysis of their performance [10]. Among their many potential applications are the design of error correcting codes, optimization of interleaver size and system throughput analysis as well as channel modeling and simulation. In this paper, convenient closedform expressions for the level crossing rate and average fade duration of κ − µ shadowed fading channels are derived and subsequently verified by reduction to known special cases. An important empirical validation is also performed through comparison with field measurements from two different types of wireless channel which are known to suffer from shadowed fading, namely cellular device-to-device (D2D) communications channels [2] and body centric communications channels [11].
The remainder of this paper is organized as follows. Section II provides a brief overview of the κ − µ shadowed fading model. Important relationships between the κ − µ shadowed fading envelope and its time derivative, which underlie the second-order equations proposed here, are established in Section III. Also presented in Section III is the derivation of the LCR, while the derivation of the AFD is given in Section IV. The new expressions for the LCR and AFD of the κ − µ shadowed fading model are compared with some empirical data obtained from field measurements in Section V. Lastly, Section VI finishes the paper with some concluding remarks.

II. AN OVERVIEW OF THE κ − µ SHADOWED FADING MODEL
The κ − µ shadowed fading model was originally proposed in [1]. A slight variant of the underlying signal model was also developed independently and appeared shortly after this in [2]. In [1] a rigorous mathematical development of the model was performed, while in [2] the model was the result of channel measurements conducted to characterize the shadowed fading observed in device-to-device communications channels. Both papers have developed important statistics related to the κ − µ shadowed fading model such as the probability density function and moment generating function. In [1], the cumulative distribution function (CDF) and the sum and maximum distributions of independent but arbitrarily distributed κ − µ shadowed variates were also derived, while, the moments of this model were presented in [2]. In the sequel, the model presented in [1] is used to develop the LCR and AFD equations proposed here. It is worth highlighting that the models proposed in [1] and [2] are related by a simple scaling factor applied to the dominant signal component and thus either could be used to arrive at the second-order equations presented here.
The received signal envelope, R, of the κ − µ shadowed fading model may be expressed in terms of the in-phase and quadrature components of the fading signal such that [1] where µ is the number of multipath clusters, which is initially assumed to be a natural number 1 , X i and Y i are mutually independent Gaussian random processes with mean E[ the power of the scattered waves in each of the clusters). Here p i and q i are the mean values of the in-phase and quadrature phase components of multipath In this model, all of the dominant components are subject to the same common shadowing fluctuation, ξ, which is a Nakagami-m random variable with the shaping parameter m used to control the amount of shadowing experienced by the dominant components and E[ξ 2 ] = 1. As with the κ − µ model [3], κ > 0 is simply the ratio of the total power of the dominant components (d 2 ) to the total power of the scattered waves (2µσ 2 ) and the mean power is given by E[R 2 ] =r 2 = d 2 + 2µσ 2 . While the PDF of R, f R (r ), could be obtained from [2, eq. (8)] by expressing the mean power of the dominant component (Ω) in terms of κ andr, that is Ω = κr 2 /(κ + 1), for the purposes of this derivation it is obtained from the PDF of the instantaneous signal-to-noise ratio (γ) given in [1, eq. (4)] via a transformation of variables where Γ(•) is the gamma function and 1 F 1 (•; •; •) is the confluent hypergeometric function [12].
In this model, m is allowed to take any value in the range m ≥ 0 where m = 0 corresponds to complete shadowing of the resultant dominant component and m → ∞ corresponds to no shadowing of the resultant dominant component. Of course when m = ∞, the PDF given in (2) becomes equivalent to the κ − µ PDF given in [3], whereas when m = 0 and hence κ = 0, the PDF given in (2) reduces to the Nakagami PDF [4]. 1 Note, this restriction is later relaxed by allowing µ to assume any positive real value.

III. LEVEL CROSSING RATE
The level crossing rate of a fading signal envelope, N R (r) is defined as the expected number of times that the envelope crosses a given signal level in a positive (or negative) direction per second and is given by [13] whereṙ is the time derivative of r and f R,Ṙ (r,ṙ) is the joint probability density of R andṘ. If we initially hold the shadowing fluctuation constant, the variation of the signal envelope would follow a κ − µ distribution [3]. In this instance, from (1), it is easy to see that the κ − µ signal power can be obtained as the sum of µ squared Rice variates. This is a simple but important observation as it allows us to show that the PDF of the time derivative of R, denoted asṘ, is zero-mean Gaussian distributed. Letting Z represent a Rice distributed random variable, it follows that Differentiating both sides of (4) with respect to time we find thaṫ Knowing that for the Rice case,Ż is a zero mean Gaussian distributed random variable with Most importantly though, from (5), we can see thatṘ is obtained as a linear transformation oḟ Z and thus it can be deduced in a similar fashion to [13] that the PDF of the rate of change of the envelopeṘ is uncorrelated with R and thus f R,Ṙ (r,ṙ) = f R (r) × fṘ(ṙ).
Now considering the shadowed fluctuation of the dominant component separately, which in this model is assumed to follow a Nakagami-m distribution. Using the model given in [15] it has already been shown that the slope is zero mean Gaussian distributed and its PDF independent of the envelope and thus f R,Ṙ (r,ṙ) = f R (r) × fṘ(ṙ). Knowing that the Nakagami-m distribution appears as a special case of the κ − µ distribution, the variance of the slope can also be obtained by letting κ = 0 in (6), µ = m and interchangingr 2 with Ω such thatσ 2 R = π 2 f 2 m Ω/m. As above, to remove the dependency of the formulations on the mean power of the dominant component, we substitute Ω = κr 2 /(1 + κ) which giveṡ Having demonstrated that for both the multipath and shadowing, the variation of the fading components are independent of the PDF of the time derivative of the envelope, it now becomes possible to rewrite (3) as where f R (r) is the κ − µ shadowed PDF given in (2) and fṘ(ṙ) is the PDF of the rate of change of the envelopeṘ. Following the approach taken in [7] it seems reasonable to assume that the PDF ofṘ is the result of two correlated zero-mean Gaussian random processes. Lettinġ R =Ȧ +Ḃ whereȦ is the rate of change of the envelope due to the multipath component anḋ B is the rate of change of the envelope due to the shadowed dominant component, the joint density ofȦ andḂ is given by [16] fȦ ,Ḃ (ȧ,ḃ) = 1 2πσ whereσ A andσ B are the variances of the two random variablesȦ andḂ, and ρ is the correlation between them. SubstitutingȦ =Ṙ −Ḃ into (9), the integral fṘ(ṙ) = ∞ −∞ fȦ ,Ḃ (ṙ −ḃ,ḃ)dḃ can be evaluated as [7] and therefore Substituting (2), (6), (7) and (11) 2 into (8) and performing the necessary mathematical operations, we obtain the LCR of the κ−µ shadowed fading envelope (normalized to the maximum Doppler frequency, f m ) as For the case when the slopes of the multipath and shadowed components of the received signal are uncorrelated (i.e. ρ = 0), (12) can be further reduced to the signal crosses lower levels at lower rates. Furthermore, the impact of increasing correlation between the slope of the shadowed dominant and multipath signals also acts to cause the signal to cross lower levels at lower rates. Fig. 2 shows the normalized LCR of the κ − µ shadowed fading signal for the special cases when it coincides with the normalized LCRs of the Nakagami [15], Rice [14] and κ − µ [17] fading models i.e. κ = m = 0 for Nakagami, µ = 1 and m = ∞ for Rice and m = ∞ for κ − µ.

IV. AVERAGE FADE DURATION
The average fade duration (AFD) of a fading signal envelope, T R (r), is defined as the average length of time that the signal spends below the threshold level R and is related to the LCR through the relationship [15] T As we can see, to calculate the AFD, it is necessary to have an expression for the cumulative distribution function, F R (r), of the κ−µ shadowed fading signal. As the CDF of the instantaneous SNR in κ − µ shadowed fading channels has conveniently been derived in [1, eq. (6)], to obtain the CDF of the received signal envelope, the same quadratic transformation used to yield the PDF of the fading signal is employed which gives where Φ 2 (•, •; •; •, •) is the bivariate confluent hypergeometric function. Now, the normalized AFD of a κ − µ shadowed fading signal can be straightforwardly obtained by substituting (12) and (15) into (14) which gives Again for the case when the slopes of the multipath and shadowed components of the received signal are uncorrelated (i.e. ρ = 0), (16) can be further reduced to  given in (12). All parameter estimates for the κ − µ shadowed fading model were obtained using the lsqnonlin function available in the Optimization toolbox of MATLAB along with the PDF given in (2). It should be noted that both sets of data were normalized to their respective rms signal levels prior to parameter estimation. Using these parameter estimates, the maximum Doppler frequency and correlation were then obtained by minimizing the sum of the squared error between the empirical and theoretical LCR plots. As we can quite clearly see, the normalized LCR of the κ−µ shadowed fading model provides an excellent fit to the on-body data and a very good approximation of the device-to-device channel. To allow the reader to reproduce these plots, parameter estimates for both measurement scenarios are given in Table I Table I.   Table I.