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A New Formula for the BER of Binary Modulations with Dual-Branch Selection over Generalized-K Composite Fading Channels
Imran Shafique Ansari, Saad Al-Ahmadi, Ferkan Yilmaz, Mohamed-Slim Alouini, Halim Yanikomeroglu
TL;DR
Deriving BER expressions for dual-branch selection combining is difficult, motivating a unified treatment across binary modulation schemes. The paper derives an exact expression using the extended generalized bivariate Meijer G-function and evaluates its computability against simulations.
Problem
BER is an important performance measure, but deriving its expression for dual-branch selection combining is difficult.
Method
The paper derives an exact unified conditional-error-probability formulation using H-functions and applies parameters covering different binary modulation schemes.
Results
The implemented extended generalized bivariate Meijer G-function evaluates fast and accurately, matching MATLAB simulations; BPSK outperforms the other modulation schemes.
Takeaways & Limitations
The formulation provides a unified BER analysis for dual-diversity independent non-identically distributed generalized-K channels.
Takeaways & Limitations
All discussed scenarios assume equal average SNR per bit.
Abstract
from arXiv · showhide
Error performance is one of the main performance measures and derivation of its closed-form expression has proved to be quite involved for certain systems. In this letter, a unified closed-form expression, applicable to different binary modulation schemes, for the bit error rate of dual-branch selection diversity based systems undergoing independent but not necessarily identically distributed generalized-K fading is derived in terms of the extended generalized bivariate Meijer G-function.
I. INTRODUCTION
The paper motivates BER analysis for dual-branch selection combining over generalized-K fading and addresses difficulties in obtaining closed-form expressions for non-identically distributed branches.
- Selection combining chooses the diversity branch with the highest signal-to-noise ratio.
- Generalized-K fading models composite multipath fading and shadowing using Gamma-based components.The model includes K-distribution as a special case and approximates Nakagami-m and Rayleigh-Lognormal fading.
- The derivation uses H-functions and represents products of Meijer G-functions through the extended generalized bivariate Meijer G-function.
- Prior BER analyses considered related fading models, i.i.d. generalized-K branches, or integral-form expressions.
- Deriving the probability density function for dual-branch selection combining was identified as difficult, motivating direct cumulative-density-based alternatives.
- This work derives an exact closed-form BER expression for binary modulations with dual-branch selection combining over independent, not necessarily identically distributed generalized-K fading.
II. THE GENERALIZED-K FADING SYSTEM AND CHANNEL MODEL
The system models a source-destination selection-combining link with independent, non-identically distributed generalized-K branches, whose instantaneous SNR depends on received amplitude.
- The considered system uses selection combining with independent, non-identically distributed fading branches.
- Generalized-K fading arises from combining Gamma-distributed Nakagami multipath fading and Gamma-distributed shadowing.
- The parameters m_m and m_s quantify multipath-fading and shadowing severity, respectively.Smaller values indicate more severe corresponding fading or shadowing conditions.
- The instantaneous SNR of branch n is γ_n = (E_b/N_0)x_n^2, where x_n is signal amplitude, E_b is average bit energy, and N_0 is AWGN power spectral density.
III. STATISTICAL CHARACTERISTICS
The statistical characterization rewrites generalized-K random-variable distributions using Meijer G-functions, enabling subsequent cumulative-distribution and BER derivations.
- The generalized-K probability density function can be expressed using a Meijer G-function.
- The PDF representation follows from the product distribution of independent Gamma random variables and its H-function form.
- Applying Meijer G-function identities yields the stated generalized-K PDF expression.
- Substituting the PDF into the relevant relation produces a generalized-K cumulative distribution function in Meijer G form.
IV. BER ANALYSIS
The BER analysis starts from a unified conditional error probability for binary modulation schemes and averages it over the selected-branch SNR distribution. Applying selection-combining distributions, EGBMGF representations, and an integral lemma yields a closed-form average BER expression.
- Selection combining chooses the branch with the highest SNR in the dual-diversity system.
- The conditional error probability is unified across coherent and non-coherent binary modulation schemes over an AWGN channel.The parameters p and q account for different modulation schemes.
- The average BER is obtained by integrating the conditional error probability over the selected-branch SNR distribution.The derivation uses integration by parts and the CDF of the selected SNR.
- The product of branch CDFs is represented using the extended generalized bivariate Meijer G-function before evaluating the remaining integral.A lemma gives the relevant integral involving the EGBMGF and an exponential term.
- Substitution of the EGBMGF representations and further manipulations produce the desired closed-form average BER expression for SC.
V. RESULTS AND DISCUSSION
The EGBMGF-based BER formulation is evaluated for multiple binary modulations and compared with Monte Carlo simulations under i.i.d. and i.n.i.d. generalized-K fading. Results show expected sensitivity to shadowing and modulation-dependent BER differences.
- Numerical evaluation: The EGBMGF implementation is evaluated against Monte Carlo simulations for dual-branch selection over i.n.i.d. generalized-K fading.The average SNR per bit is assumed equal across the discussed scenarios.
- I.I.D. channels: The implemented EGBMGF produces a perfect match to MATLAB simulated results for i.i.d. BPSK cases.
- I.I.D. channels: BER increases as the shadowing effect increases, represented by decreasing ms while keeping mm = 1 constant.
- Scope of results: Similar outcomes and exact closed-form BER results are reported for other shadowing parameters and for BFSK and DPSK.
- I.N.I.D. channels: BPSK outperforms the other modulation schemes under the stated i.n.i.d. channel parameters.The parameters are mm1 = 1, mm2 = 2, ms1 = 0.5, and ms2 = 4.
- I.N.I.D. channels: BFSK and DPSK perform similarly at lower SNR, whereas DPSK performs better than BFSK as SNR increases.
VI. CONCLUDING REMARKS
The paper derives an exact BER expression for binary modulations using dual-branch selection over independent, non-identically distributed generalized-K fading. It uses the EGBMGF and illustrates the formulation through numerical examples covering fading, shadowing, and channel unbalance.
- An exact closed-form BER expression is derived for different binary modulations using dual-branch selection over i.n.i.d. generalized-K fading.
- The analytical calculations use the extended generalized bivariate Meijer G-function.
- Numerical examples illustrate the formulation and show effects of fading severity, shadowing severity, and channel unbalance on BER.
EXTENDED GENERALIZED BIVARIATE MEIJER G-FUNCTION (EGBMGF)
This section presents implementation details for evaluating the EGBMGF, including contour limits, parameter assignments, numerical integration, and returning the computed value.
- Contour limits depend on the numerator Gamma arguments and must be half of the least-valued Gamma argument.
- The implementation assigns z_s, z_t, and a working value W before evaluating the function by numerical integration.
- After numerical integration, the implementation returns the computed value.
!End of EGBMGF"
The EGBMGF implementation is tested with specified modulation and generalized-K parameters, producing a numerical output, while the accompanying figures address i.i.d. BPSK and i.n.i.d. modulation cases.
- Testing: The implementation is tested with p = 0.5, q = 1, mm1 = 1, ms1 = 2, mm2 = 1, and ms2 = 2 over SNR values from 0 to 20 dB.
- Testing: The tested EGBMGF configuration produces the numerical result 0.00102393 − 7.09829 × 10^-16 i.
- Figures: Figure 1 concerns i.i.d. BPSK BER with mm = 1 and varying ms, while Figure 2 concerns different modulation schemes under i.n.i.d. channels.