A tight approximate analytical framework for performance analysis of equal gain combining receiver over independent Weibull fading channels
- Abdelmajid Bessate^{1} and
- Faissal El Bouanani^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13638-016-0790-2
© The Author(s) 2017
Received: 24 August 2016
Accepted: 8 December 2016
Published: 3 January 2017
Abstract
In this paper, a method for approximating the probability distribution of sum of independent and identical Weibull random variables is adopted to analyze the performance of equal gain combiner (EGC) receiver over non-identical Weibull fading channel (WFC). Our main result is to derive a generalized expression of the probability density function (PDF) of the signal-to-noise ratio (SNR) at the EGC output in the case of non-identical WFC. Based on this PDF, accurate approximation of significant performance criteria, such as outage probability (OP), the amount of fading (AoF), and average symbol/bit error probability (ASEP/ABEP), are derived. In addition, we derived the analytical expressions for channel capacities under various adaptation policies such as optimal rate adaptation (ORA), optimal simultaneous power and rate adaptation (OPRA), channel inversion with fixed rate (CIFR), and truncated channel inversion with fixed rate (TCIFR). The proposed mathematical analysis is complemented by several numerical results and validated using Monte Carlo simulation method.
Keywords
1 Introduction
Antenna diversity is one of the most practical, effective, and widely employed technique in wireless communication receivers to reduce the effects of fading and to provide increased signal strength at the receiver. Different techniques are known to combine the signals received from multiple diversity branches. The most popular diversity techniques are equal-gain combining (EGC), maximal-ratio combining (MRC), selection combining (SC), and a combination of MRC and SC, called generalized-selection combining (GSC). The SC receiver chooses the branch with the strongest instantaneous signal-to-noise ratio (SNR), while MRC provides optimal performance, at the expense of implementation complexity, since it requires knowledge of all channel parameters. In EGC receiver, the signals in all branches are weighted with the same factor, irrespective of the signal amplitude. Moreover, co-phasing of all input signals is needed to avoid output signal cancellation. The performance of EGC and MRC diversity receivers has been extensively conducted in many previous works for several well-known fading statistical models, such as Rayleigh, Rice, and Nakagami—assuming independent or correlative fading [1–6]. The Weibull distribution is a well-known model for describing multipath fading channels in both indoor and outdoor radio propagation environments. In [7], novel analytical expressions for the joint probability density function (PDF), moment generating function (MGF), and cumulative distribution function (CDF) are derived for the multivariate Weibull distribution. The presented theoretical results are applied to analyze the performance of several diversity receivers such as SC, EGC, and MRC techniques operating under correlated Weibull fading channels (WFC). For these diversity receivers, several useful performance criteria such as moments of output SNR, including the amount of fading (AoF), and outage probability (OP) are analytically derived. Moreover, the average symbol error probability (ASEP) for several coherent and noncoherent modulation schemes is studied using moment generating function (MGF) approach. In [8], capitalizing on the general α- μ fading model simple and precise closed-form approximations to the PDF and CDF of the sum of independent and identically distributed (i.i.d) Weibull variates are derived. These approximations find applicability in several wireless communications issues such as signal detection and combining, linear equalizers, intersymbol interference, and phase jitter [8]. Considering related works, C. Sagias et al. [9, 10], have presented a moments-based approach to analyzing the performance of dual-branch EGC and MRC receivers, operating under either independent or correlated, but not necessary identically distributed WFC. In this respect, significant performance criteria, such as average output SNR, AoF, and spectral efficiency at low power regime, are extracted in closed-forms, using the moments of the output SNR for both independent and correlative fading cases. Using the same approach of the moment, El Bouanani [11–14] has generalized this idea to L-branches over independent and not necessary identically distributed (i.n.i.d) Weibull fading channels for both MRC and EGC receivers. Consequently, he has derived some performance criteria such as AoF, MGF, average capacity (AC), and ASEP in closed forms. In [15], we have generalized the same idea to derive the approximate expressions of MRC performance criteria over correlated WFC.
In this paper, we propose a tight approximate expression of the output SNR PDF by generalizing the approximate CDF convolution derived in [16] to non-identical Weibull random variates (RVs). The tightness of this approximate CDF is proved and validated by using the Kolmogorov–Smirnov statistical method. Based on this approximation, closed expressions of statistical characteristic, of an EGC receiver, given in terms of well-known hypergeometric function, related to PDF, OP, and MGF are derived. Other performance criteria such as AoF, ASEP, average bit error probability (ABEP), average capacity (AC) under optimal rate adaptation with constant transmit power (ORA), capacity under optimal simultaneous power and rate adaptation (OPRA), channel inversion with fixed rate (CIFR), and truncated channel inversion with fixed rate (TCIFR) schemes are presented in tight closed-form approximation. After this introduction, Section 2 deals with the communication system model with EGC combiner over Weibull fading channels. Section 3 shows the statistical characteristics of EGC over WFC such as PDF and MGF. Section 4 presents several novel closed-expressions in terms of well-known hypergeometric functions such as that AoF, OP, AC, OPRA capacity, CIFR capacity, TCIFR capacity, and ABEP and ASEP for M-PSK and M-FSK modulation schemes. In Section 5, all results are illustrated and verified by computer simulations using Mathematica software. Finally, Section 6 contains a brief conclusion.
2 Channel model
3 Statistical characteristics
In this section, we begin by recalling the main result of Johnson [16] regarding the derivation of a tight approximate CDF of the sum of i.i.d Weibull RVs. Furthermore, we derive the approximate PDF of output SNR at EGC receiver operating under i.i.d WFC. Our main result is to generalize this approximation to be also valid in the case of i.n.i.d WFC.
3.1 PDF of the sum of i.i.d Weibull RVs
with \(T=\sum _{i=1}^{L}T_{i}, T_{i}\) is a Weibull (ω,β) RV, and \(\sigma =\frac {\Gamma (L+1/\beta)}{L!\Gamma (1+1/\beta)}\).
which represents the PDF of generalized Gamma (GG) RV [17] with parameters \(\frac {\omega }{\sigma },\beta L,\) and β.
3.2 PDF of output SNR at EGC receiver
3.2.1 Case of i.i.d WFC
Lemma 1
Let us assume that the scale parameter of γ _{ i } is constant \((\alpha _{i}=\alpha =\omega \sqrt {E_{s}/N_{0}}).\) The square root of instantaneous SNR γ _{ i } is a Weibull(α,β).
Proof
□
which concludes the Lemma proof.
Proposition 2
Proof
□
Using the expressions (4) and (11), and proceeding by Jacobian transform, we get (10) which concludes the proof of the proposition.
3.2.2 Case of i.n.i.d Weibull RV
It is known that two RVs are equal or very close, respectively if and only if their moments are equal or very close, respectively. Let us begin by computing both exact and approximate nth moment of output SNR. a. Exact n th moment of γ
b. Approximate n th moment of GG RV
Proposition 3
Proof
To compare the exact and approximate nth moment, we will proceed by two approaches. When β is sufficiently large, an analytical approach, based on approximation at infinity of both exact and approximate nth moments, is used to prove equality between them. Otherwise, plotting the two moment curves, for several values of φ and β, will be useful to show the closeness between the two expressions. □
where Ψ(.) is the digamma function that represents the first derivation of Gamma function.
Consequently, for each natural integer n, the exact and approximate nth moments can be approximated by the same expression as given by (25) and (29). Thereby, the two RV γ and γ _{ A } are very close. b. Curve-based approach
Taking into account that it is very difficult to upper bound the error between exact and approximate moment, we will opt for a graphical approach by plotting the curves of the moments versus the power delay profile φ and the shape parameter β. See the evaluation section.
3.3 MGF of the output SNR γ
Proposition 4
Proof
Using the equality ([21], /07.34.21.0012.01) the MGF can be written as mentioned in (30).
4 Performance criteria
4.1 Outage probability of the output SNR γ
with θ is expressed in (17).
4.2 Amount of fading (AoF)
with μ _{2} and μ _{1} are respectively the first and second moment of the output SNR γ.
4.3 Capacity under ORA policy
4.4 Capacity under OPRA policy
Proposition 5
where \(x_{0}=\left (\frac {\gamma _{0}}{\theta ^{2}}\right)^{\frac {\beta }{2}}\) and Γ(.,.) denote the upper incomplete gamma function.
Proof
□
Now, substituting (39), (40), and (41) into (38), we obtain the expression (37) which concludes the proposition’s proof.
4.4.1 Optimal cutoff SNR γ _{0}
Using the derivative of the upper incomplete gamma function, we obtain \( \frac {\partial f(y)}{\partial y}=-\frac {2}{\beta \theta ^{2}}\Gamma \left (L,y\right) y^{\frac {-2}{\beta }-1}<0\) for all y≥0. Moreover, \(\underset {y\rightarrow 0^{+}}{\lim }f(y)=+\infty \) and \(\underset {y\rightarrow +\infty }{\lim }f(y)=-\Gamma \left (L\right) <0.\) Thus, we conclude that there is a unique x _{0}, consequently a unique γ _{0}, for which f(x _{0})=0. Besides, the value of γ _{0} can be calculated, using any calculation software, by solving (44).
4.5 Capacity under CIFR policy
Proposition 6
Proof
Thus, by substituting (48) into (47), the expression in (46) can be easily derived. □
4.6 Capacity under TCIFR policy
Proposition 7
where γ ^{∗} is a fixed cutoff fade depth.
Proof
From (50), it is obvious that 〈C〉_{TCIFR}=〈C〉_{CIFR}(1−P _{out}(γ ^{∗})). Then by substituting (31) and (46) into (50), we easily obtain (49). □
4.7 Average symbol error probability (ASEP)
Proposition 8
Values of ϱ and δ for some signaling constellations
Modulation | M | ϱ | δ |
---|---|---|---|
BPSK | 2 | 1/2 | 1 |
BFSK | 2 | 1/2 | 1/2 |
M-PSK | ≥4 | 1 | sin2(π/M) |
M-FSK | ≥4 | (M−1)/2 | 1/2 |
Proof
□
Now, using the identity ([21], /07.34.21.0012.01) we obtain (51) which concludes the proof of the proposition.
4.8 Average bit error probability (ABEP)
Proposition 9
Proof
□
Now, by substituting (58) into (57), we obtain the expression (55)
5 Evaluation
In this section, all the analytical expressions are evaluated using Mathematica software. In addition, our approximate PDF and most of other results are validated using numerical Monte Carlo simulations from (4) by generating L×10^{7} Weibull-distributed random values. Without loss of generality, we have supposed an exponentially decaying power delay profile (PDP) \(\overline {\gamma }_{i}/\overline {\gamma }_{1}=\exp \left [-\varphi \left (i-1\right) \right ],\) where φ is the average fading power decay factor [22], and \(\overline {\gamma }_{1}=1\) for all figures except Figs. 10 and 15. The value of β is assumed to be the same for each receiver branch.
In Fig. 10, the OPRA capacity, given in (37), versus optimal threshold SNR γ _{0}, computed numerically by solving (44), is traced for several values of β, and φ=0.227. From these curves, we can obviously observe that below γ _{0}=0.2, no data can be transmitted. Moreover, the greater is β, the higher is optimal simultaneous power and rate adaptation capacity.
6 Conclusions
In this paper, we have derived a new tight approximation for the PDF of output SNR at EGC receiver operating under uncorrelated but not identical Weibull fading channels. Based on this approximate PDF, the analytical expressions of many performance criteria have been derived. All results are illustrated using Mathematica software and validated by Monte Carlo simulation. The results that we have achieved are far better than all previous works.
Declarations
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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