Probability Distributions of Products of Rayleigh and Nakagami-m Variables Using Mellin Transform Sohail Ahmed, Lie-Liang Yang and Lajos Hanzo 1
Dept. of Avionics, National University of Sciences & Technology (NUST), Pakistan. 2 School of ECS, University of Southampton, SO17 1BJ, UK. Tel: +44-23-8059 3364, Fax: +44-23-8059 4508 Email: 1
[email protected],2{lly,lh}@ecs.soton.ac.uk; http://www-mobile.ecs.soton.ac.uk
Abstract— In this contribution, we employ the Mellin transform to derive the expressions for probability density function (PDF) of the product of Rayleigh and Nakagami-m distributed random variables. We exploit the fact that the Mellin transform of a product of independent and identically distributed random variables is the product of the Mellin transforms of the individual random variables. Using this approach, the PDF of the product of random variables is expressed in the form of an easily computable infinite series.
I. INTRODUCTION Determining the probability density function (PDF) of product of independent and identically distributed random variables is a problem often encountered in fields as diverse as social statistics, biological and physical sciences, econometrics and classification. See [1], for example, and the citations therein. Products of random variables of various types of distributions have been investigated and their distributions determined in [2], [3]. This problem is of interest in the field of wireless communications as well [4]–[7]. A type of diversity combining known as product combining which is employed in Fast Frequency Hopping (FFH) [6] receivers also involves product of random variables. In [4], an approximation of the product of two independent Nakagami-m distributed random variables is developed for the sake of performance analysis of Automatic Repeat Request (ARQ) schemes with code combining. The product of random variables is an area of study that has attracted renewed attention since the advent of the idea of cooperative communication, which involve cascaded channels that can be modelled by the product of multiple fading amplitudes [8], [9]. Various techniques have been employed to investigate scenarios involving product of random variables. In [6], [10], [11], the authors have analyzed the product combining based FFH receiver, employing the characteristic function [12] and using natural logarithm to convert the multiplication operation into summation, for the sake of deriving the PDF of the product of random variables. The problem associated with The financial support of the National University of Sciences & Technology (NUST) and Higher Education Commission (HEC), Pakistan is gratefully acknowledged.
the characteristic function based analysis is that closed form expressions for PDF of the product cannot be readily obtained. In [13], the employment of Fox’s H-functions was proposed to derive the PDF of the product the same FFH receiver. In their analysis, the authors of [13] have exploited the fact that a product of H-functions is also an H-function [3]. The authors [13] have also employed another technique which consists of generalized F -variates for the sake of deriving the corresponding PDF expressions. However, this later method is more computationally demanding. In this paper, we employ the Mellin transform [3], [14], [15] to derive the expression for PDF of the product of Nakagamim distributed random variables. The Mellin transform is an integral transform, similar to Laplace and Fourier transform. The Mellin transform of a random variable is related to its PDF and the Mellin transform of a product of random variables is the product of the Mellin transforms of the individual random variables [3], [15]. This fact allows us to derive the PDF of the product of the random variables. The PDF of the product of Nakagami-m distributed random variables has also been investigated in [4] using Meijer’s G function, while we employ the Mellin transform which results in a simpler analysis of the problem. The Mellin transform has been employed previously in similar applications such as determining the PDF of product of beta, gamma and Rayleigh distributed variables [3], [7], [15]; however, the inverse Mellin Transform and the resultant PDF is expressed in these contributions in the form of Meijer’s G function. By contrast, we employ the residues method, similiar to that used in [2], [16] to derive the inverse Mellin transform, expressing the PDF in the form of an infinite series which is easier to compute. In order to demonstrate the agreement of our proposed method with previous results e.g. those given in [7], we also derive the expression for the PDF of product of Rayleigh distributed variables. It will be shown that the proposed Mellin transform based technique and the subsequent method of determining the inverse transform substantially simplifies the derivation of the PDF of the product of random variables. The remainder of this paper is structured as follows. In Section II, the proposed Mellin transform based technique is discussed and employed for the derivation of the PDF of product of Rayleigh distributed random variables. In Section III, the
same technique is employed, assuming product of Nakagamim distributed random variables. In Section IV our numerical results are discussed. Finally, in Section V, our conclusions are presented.
QL−1 Thus, if Y = l=0 Xl , represents the product of L independent and identically distributed (i.i.d.) Rayleigh distributed variables Xl , then we have M [fY (y), z] =
II. PDF OF P RODUCT
OF
R AYLEIGH VARIABLES
0
where z is a complex variable. In terms of probability theory, the f (x) in (1) denotes the PDF of the random variable X. The inverse Mellin transform is defined as [3] Z c+i∞ fX (x) = x−z M [fX (x), z]dz, (2) c−i∞
√ where i = −1 and the integration is along any path Re(z) = c, such that M [f (x), z] exists and is an analytic function of the complex variable z for c1 ≤ Re(z) ≤ c2 and c lies between two real points c1 and c2 . We begin with the PDF of a Rayleigh distributed variable which is given by [12] x −x22 e 2σ , σ2
x ≥ 0,
(3)
where σ 2 is the variance of the Rayleigh distributed variable. From [17], we know that 2
M [e−ax ] =
a−z/2 z , Γ 2 2
Hence, the Mellin transform of the PDF given by (3) may be expressed as M [fX (x), z] = a(1−z)/2 Γ(
z+1 ) 2
If we replace (z + 1)/2 by z in the above equation, this does not affect the path of integration [3]. Hence, following this replacement and after some simplification, we have Z 2yaL c+i∞ −2z −Lz L fY (y) = y a Γ (z)dz. (9) 2πi c−i∞ From the Residue Theorem [3], [17], [19], we know that the complex integral at the right-hand side of the above equation can be computed by summing the residues of the integrand associated with all its poles. Thus, we have X Res a−Lz y −2z ΓL (z) , (10) fY (y) = 2yaL
(6)
where we have defined a = 1/2σ 2 . Note that the results of (6) may also be obtained from the definition of the Mellin transform given by (1).
(z=−j)
j
where Res[.]j represents the residue at the jth pole of the integrand and the summation is carried out over all possible values of j. The PDF of Y can be determined numerically from (10) by using symbolic mathematics based softwares such as Maple or Mathematica, employing the appropriate function for finding residues of an integrable expression. However, it is insightful to derive analytical expressions for (10), which we undertake below. In (10), all the poles are contributed by the function ΓL (z). It can be shown with the aid of [18], [19] that the function Γ(z) has an infinite number of poles, i.e. it has poles at z = −j for j = 1, 2, . . .. The residue of Γ(z) at z = −j is given by [18], [19] Γ(z)(z + j)|(z=−j) =
(5)
z+1 ). (7) 2
The PDF of Y can now be obtained as the inverse Mellin transform of the expression given in (7). Thus, from (2) and (7), we have Z c+i∞ z+1 )dz. (8) fY (y) = y −z aL(1−z)/2 ΓL ( 2 c−i∞
(4)
where Γ(.) denotes the Gamma function [18]. Here, we use the property of Mellin transform whereby if M [f (t), z] = F (z), then [3], [17] M [tb f (t), z] = F (z + b).
M [fXl (x), z] = aL(1−z)/2 ΓL (
l=0
Our technique for deriving the PDF of the product of random variables is similar to that employed in [16] and is as follows. From the PDF of the random variable, we determine its Mellin transform. Next, according to the properties of the Mellin transform, the transform of the product of random variables is equal to the product of the Mellin transforms of the individual random variables [3], [15]. Therefore, the Mellin transform of the product is given by the product of the transforms of the individual variables. The inverse transform, hence, yields the PDF of the product of the variables. The Mellin transform of a function f (x) is defined as [17] Z ∞ xz−1 fX (x)dx, (1) M [fX (x), z] =
fX (x) =
L−1 Y
(−1)j . j!
(11)
Now, ΓL (z) in (10) has an Lth-order pole at each integer value of z = −j. Consequently, using the corresponding relationship characterizing the residues of multiple poles [3], [18], [19] and the Leibnitz’ rule [18] for higher order derivatives of a product of functions, the PDF of Y may be expressed as ∞ L−1 2yaL X X L − 1 fY (y) = (L − 1)! j=0 r=0 r × U (r) (z)V (L−1−r)(z) , (12) (z=−j)
where we have defined U(z) = (aL y)−z and L L (r) (r) V(z) = Γ (z)(z + j) , while U (z) and V (z) denote the
rth derivatives of U(z) and V(z), respectively. The rth derivative of U, when evaluated at z = −j, can be readily expressed as U (r) (z)|(z=−j) = [− ln(aL y)]r (aL y)j .
(13)
and we have for the product of Nakagami-m variables ∞ L−1 2y 2m−1 amL X X L − 1 fY (y) = (L − 1)!ΓL (m) j=0 r=0 r × U (r) (z)V (L−1−r)(z) ,
(20)
(z=−j)
With the aid of [16], we arrive at where U (r) (z) and V (r) (z) are as defined in Sec II and given ( r−1 X r − 1 by (13) and (14) respectively. We note that for m = 1, (20) V (t) (z)|(z=−j) (−1)r−treduces to (12). V (r) (z)|(z=−j) = L t t=0 ∞ h i X 1 IV. N UMERICAL R ESULTS AND D ISCUSSION × (r − 1 − t)! (1 + k)r−t The PDFs given by (12) and (20) have to be evaluated nuk=0 ) merically. Note that the infinite series seen in these equations j r−1 X 1 X r−1 should be convergent in order to allow the computation of the V (t) (z)|(z=−j) + . (14) t (k)r−t PDF from a finite number of terms. It has been found that t=0 k=1 residues for j ≤ 40 are sufficient for computing the PDF and For a detailed derivation of (14), the reader may refer to [16]. the BER sufficiently accurately. In Fig. 1, we have plotted the PDF of the product of L III. PDF OF P RODUCT OF NAKAGAMI -m VARIABLES independent Rayleigh variables, assuming various values of L. In this section, we demonstrate the Mellin transform based It can be seen from Fig. 1 that the PDF curve becomes flatter approach employed in Sec.II to derive the PDF of product of and its tail gets longer upon increasing the value of L. This is Nakagami-m varaibles. The PDF of a Nakagami-m distributed expected, because owing to the multiplication operation, while the probably of low values has inceased, there is a non-zero variable is given by [12] probability of Y attaining high values as well. Thus for high 2 m m 2m−1 −mx2 values of L, the PDF converges slowly. Our results in Fig. 1 Ω , x ≥ 0, (15) fX (x) = x e Γ(m) Ω agree with those given in [7]. In Fig. 2, we have plotted the PDF of the product of where Ω = E[x2 ] and m is the Nakagami-m fading parameter. Nakagami-m variables, assuming various values of L and Hence, the Mellin transform of the PDF given by (15) may fading parameter m = 2.5. In this figure, we observe trends be expressed as similar to those in Fig. 1 with increasing diversity order. In z + 2m − 1 1 a(1−z)/2 Γ , (16) Fig. 3, we have plotted the PDF of the product of L NakagamiM [fX (x), z] = Γ(m) 2 m variables, assuming diversity order L = 3 and various m values of the fading parameter m. We observe in this figure where we have defined a = Ω and the results of (16) have that with increasing value of fading parameter m, the peaks been obtained from the definition of the Mellin transform given of the PDF curve tend to shift rightward indicating a more by (1). QL−1 Gaussian characteristics. Thus, if Y = l=0 Xl , represents the product of L i.i.d. Nakagami-m distributed variables, then we have V. C ONCLUSION M [fY (y), z] =
1
ΓL (m)
aL(1−z)/2 ΓL
z + 2m − 1 . 2
(17)
To determine the PDF of Y , we have to evaluate the inverse Transform of (17), which is given by Z c+i∞ aL/2 y −z a−Lz/2 fY (y) = ΓL (m)2πi c−i∞ z + 2m − 1 dz. (18) × ΓL 2 By replacing (z + 2m − 1)/2 by z, the last equation becomes Z 2y 2m−1 amL c+i∞ −2z −Lz L fY (y) = L y a Γ (z)dz. (19) Γ (m)2πi c−i∞ We note the the integrand in (19) is similar to that in (9). Consequently, our analysis is similar to that done in Sec II
We have used the Mellin transform to derive the PDF of the product of Rayleigh and Nakagami-m distributed variables. Employing the proposed Mellin transform based technique, the PDF of the product was determined in a closed form, as seen in (12) and (20). As is well known, Nakagami-m is a very versatile distribution, which fits a wider range of wireless channel fading conditions. The proposed method can readily be applied to studies involving product of random variables such as modeling of cascaded channels or the socalled N ∗Nakagami channels in the context of cooperative communication [5]. In our future work, we extend this technique to derivation of expressions of product of Gaussian and Rician variables, two distributions that are frequently encountered in wireless communication. Moreover, we focus on determining the cumulative distribution functions of product of random variables
Fig. 1. The PDF of the product Y of of Rayleigh variables, assuming various values of diversity order L and σ2 = 1. Fig. 3. The PDF of the product Y of of Nakagami-m variables, assuming diversity order L = 3, Ω = 1 and various values of fading parameter m.
Fig. 2. The PDF of the product Y of of Nakagami-m variables, assuming various values of diversity order L, Ω = 1 and fading parameter m = 2.5.
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