1364
IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 7, JULY 2013
An Accurate Approximation of the Exponential Integral Function Using a Sum of Exponentials Ala Abu Alkheir, Student Member, IEEE, and Mohamed Ibnkahla, Member, IEEE Abstract—This paper proposes a novel approximation for the exponential integral function, E1 [x], using a sum of exponential functions. This approximation facilitates studying the error probability of a number of communication techniques in the presence of Co-Channel Interference (CCI). These include Hybrid Automatic Repeat Request (HARQ) with soft combining, selection relaying, incremental relaying, and opportunistic incremental relaying, just to name a few. To illustrate the usefulness and accuracy of the proposed approximation, we study the error probability of a Chase combining HARQ system operating in the presence of an unknown source of CCI where we derive an accurate closed form expression for the Moment Generating Function (MGF) of the resultant Signal to Interference plus Noise Ratio (SINR). The accuracy of the derived result is verified using computer simulation. Index Terms—Exponential integral function, co-channel interference, incremental relaying, HARQ.
I. I NTRODUCTION
O
BTAINING closed form expressions for the error probability of various communication systems is a vital step towards quantifying the usefulness of these systems. However, as this may not be feasible in certain cases, accurate approximations were always a viable resort. For instance, obtaining closed form expressions for cooperative diversity techniques employing Amplify and Forward (AF) relaying is known to be an open problem that is best approximated by dropping the effect of the thermal noise at the destination or by using the min{·, ·} approximation [1], [2]. Approximations can also be applied to some special functions encountered throughout the analysis, like the Gaussian and the Marcum Q-functions [3]–[5]. Recently, another special function, the exponential integral function, E1 [x], started to appear while calculating the error probability of a number of communication systems in the presence of Co-Channel Interference (CCI), like dualhop relaying [2], Chase combining Hybrid Automatic Repeat Request (HARQ), selection relaying, incremental relaying, opportunistic incremental relaying [6]–[9], and many other systems. The common factor between these systems is that assistance is only asked for if the directly received signal fails to meet a certain quality measure. When assistance is needed, the Moment Generating Function (MGF) of the resulting Signal to Interference plus Noise Ratio (SINR) after using Maximal Ratio Combining (MRC) can only be calculated by solving integrals that include E1 [x]. While closed form Manuscript received February 21, 2013. The associate editor coordinating the review of this letter and approving it for publication was M. Di Renzo. The authors are with the Electrical and Computer Engineering Dept., Queen’s University, Kingston, Ontario, Canada (e-mail:
[email protected],
[email protected]). Digital Object Identifier 10.1109/LCOMM.2013.060513.130403
expressions can be found for some of these integrals [10]–[13], others cannot be evaluated without numerical integration. Being widely encountered in physical applications, like heat transfer and quantum field theory, researchers have been mainly concerned about approximating E1 [x] for the sake of numerical evaluation [14]–[16]. It is noted, however, that there has been no single approximation that is valid for all x > 0. In fact, [17] was the first to propose such an approximation. However, these approximations rendered further manipulations even more challenging than E1 [x] itself. To facilitate obtaining closed form expressions for the MGF in the aforementioned cases, this article proposes a novel approximation using a weighted sum of exponential functions with constant coefficients. This approximation greatly simplifies the mathematical manipulations. Furthermore, with a controllable number of summation terms, the proposed approximation can achieve high accuracy levels with mild level of calculations. The remainder of this article is organized as follows. Section II presents the proposed approximation, while section III studies the error probability of a Chase combining HARQ system in the presence of CCI. Finally, conclusions are drawn in section IV. II. T HE P ROPOSED A PPROXIMATION The n order exponential integral function, En [x], is defined as [11] ∞ −xt ∞ −t e e n−1 En [x] dt, x > 0, (1) dt = x n t t 1 x which can also be calculated using E1 [x], E2 [x], . . . , En−1 [x] through the recursion 1 −x e − En−1 [x] , n > 1. (2) En [x] n−1 Hence, E1 [x] is sufficient to calculate an arbitrary En [x]. This special function can be accurately evaluated for arbitrary x > 0 using a wide range of scientific software, like MATLAB, MATHEMATICA and MAPLE. In addition, a number of integral formulas involving this function can be found in [10]– [13]. However, other integral identities, where E1 [x] appears either explicitly or implicitly, like e−αx E1 [βx + ν]dx, (3a) (x + τ )m e−αx ln(βx + ν)dx, (3b) (x + τ )m th
cannot be solved in closed forms1 for arbitrary positive parameters, α, β, ν, and τ and positive integer m = 1, 2, . . . . 1 We should remark that this statement is true to the best of our knowledge. In fact, we have been motivated to do the present work after spending a good amount of time trying to solve similar integrals.
c 2013 IEEE 1089-7798/13$31.00
ALKHEIR and IBNKAHLA: AN ACCURATE APPROXIMATION OF THE EXPONENTIAL INTEGRAL FUNCTION USING A SUM OF EXPONENTIALS
√ √ d √ e−t √ = −2 π Q 2t ≈ 4 π dt t
N +1
an bn e−2bn t
5 Chebyshev Swamee and Ohija Allen and Hastings Barrya et. al Continued Fraction Expansion, K = 20
4 3 2
−2
Eqn.(7), N=I=2,α = 10
1
ε(x)
Nonetheless, these integrals and the similar can be accurately solved if E1 [x] is properly represented using some approximation. To obtain such an approximation, let us start by rewriting the definition of E1 [x] as ∞ −t ∞ −t 1 e e √ √ dt, dt = E1 [x] = (4) t t t x x using√this expression, it can be observed that the first term, e−t / t, is√related to the derivative of the Gaussian Q-function, 2 ∞ Q(x) ( 2π)−1 x e−t /2 dt, as
1365
0 −1 −2 −3
(5)
−4
n=1
√ 2t) where we have invoked the general approximation of Q( √ N +1 given by Q 2t ≈ n=1 an e−2bn t where an and bn are constant coefficients and N is a positive integer that controls +1 the accuracy of the approximation [3], [4]. While {an , bn }N n=1 can be calculated in a number of ways, this article adopts the results in [4] wherein Jensen’s Inequality has been used to obtain a lower bound of Craig’s representation of the Q-function. Consequently, an and bn are calculated as an = (θn −θn−1 )/π and bn = 0.5{cot(θn−1 ) − cot(θn )}/(θn − θn−1 ) while the +1 angles {θn }N n=0 can be chosen in a number of different ways as long as θ0 = 0, θ0 < θ1 < · · · < θN +1 , and θN +1 = π/2. However, by recalling that these angles capture the behavior of the monotonically increasing function exp(−t/ sin2 θ) over the range θ ∈ [0, π/2]; it can be deduced +1 that a uniformly spaced selection where {θn = nΔ}N n=0 and Δ = (θN +1 −θ0 )/(N +1) is a suitable selection. Accordingly, an = aN = Δ/π, ∀n and bn = (cot(θn−1 ) − cot(θn ))/(2Δ). Furthermore, to reduce the computational complexity while maintaining accuracy, θ0 can be shifted to start at θα > 0. This value is chosen such that the function exp(−t/ sin2 θ) reaches α, 0 < α < 1, of the maximum value exp(−t) achieved at θ = π/2, i.e., exp(−t/ sin2 θα ) = α exp(−t). After √ some ma −t−1 ln α . nipulations, it can be shown that θα = cot−1 Unfortunately, this value depends on t, which means that Δ, +1 and hence, aN and {bn }N for every n=1 , have to be recalculated −1 −1 value of t. To avoid this, we use θα = cot −t0 ln α where t0 = 10−2 when t ∈ (0, 10] and t0 = 102 when t ∈ (10, ∞), respectively. By substituting (5) in (4) and solving the integral, E1 [x] can be approximated by
−5 −40
Fig. 1.
−30
−20
−10 x (dB)
0
10
20
Relative estimation error, (x), as a function of x.
{bi }I+1 i=1 . By comparing it to previous approximations, like the Chebyshev approximation [18], the Swamee and Ohija approximation [19], the Allen and Hastings approximation [15] [19], the truncated continued fraction approximation [15], and the approximation of Barry et al. [17], it can be observed that i) this approximation uses a single expression for all x > 0 while most other approximations use different expressions for different ranges, ii) it is simpler to calculate and to manipulate, and iii) it offers a flexible complexity/accuracy tradeoff. Furthermore, by looking at Figure 1where the relative estimation ˆ 1 [x]|/E1 [x] [15] error, defined as (x) log10 |E1 [x] − E is shown, it can be seen that, unlike the aforementioned approximations, the proposed approximation gives an almost steady level of accuracy for a wide range of x > 0. III. A PPLICATION A. System Model
(7)
To illustrate the usefulness of the proposed approximation, let us consider a pair of single-antenna terminals, a source S and a destination D, communicating in the presence of an unknown source of CCI, denoted I. To mitigate the various channel impairments, this pair uses Chase combining HARQ with one possible retransmission and a decision threshold of λd [6]. The S-D and I-D channels are assumed to undergo flat Rayleigh fading with coefficients denoted by h and g, respectively, while x and s denote the unit-energy transmitted signals of S and I, respectively, and w is the zero-mean Additive White Gaussian Noise (AWGN) at D with variance 2 . Accordingly, the received signal at D is written as y = σ√ √ h Ex x + g Es s + w, where Ex and Es are the transmission energies of S and I, respectively. Assuming perfect knowledge of h and no knowledge of g, it can be shown that D perceives an SINR of [2] γ0 , (8) ψ0 = ζ0 + 1
Using this approximation, the integrals in (3) can be readily solved. Computational-wise, this approximation only requires +1 calculating the constant coefficients aN , aI , {bn }N n=1 and
where the Signal to Noise Ratio (SNR), γ0 , and the Interference to Noise Ratio (INR), ζ0 , are defined as γ0 Ex |h|2 /σ 2 and ζ0 Es |g|2 /σ 2 , respectively. Following the Rayleigh fading assumption, these two ratios are exponentially distributed
N +1
√ ˆ 1 [x] = 4 2πaN bn Q 2 bn x . E1 [x] ≈ E
(6)
n=1
This approximation is more tractable than E1 [x] itself. However, the presence of Q(x) may render evaluating certain integrals using this approximation a challenging task. Hence, we can replace this function by its exponential approximation which yields the desired approximation of E1 [x] as N +1
I+1
√ ˆ 1 [x] = 4 2πaN aI bn e−4bn bi x . E1 [x] ≈ E n=1 i=1
1366
IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 7, JULY 2013
0
10
−1
10
(9)
where Fψ0 (ψ) is the Cumulative Distribution Function (CDF) of ψ0 , while Pe,unassisted and Pe,assisted are the error probabilities of the unassisted and assisted modes, respectively. Following the definition of ψ0 given in (8), it can be shown that Fψ0 (ψ) is given by μ e−ψ/¯γ , Fψ0 (ψ) = 1 − (10) μ+ψ ¯ On the other hand, assuming BPSK where μ γ¯/ζ. modulation and using the exponential approximation of √ Q( 2ψ) given above, Pe,unassisted and Pe,assisted can be ap N +1 proximated as Pe,unassisted ≈ aN n=1 Mψ0 |ψ0 ≥λd (−2bn ) N +1 and Pe,assisted ≈ aN n=1 MΨ|ψ0
