[Paper Review] Extended Generalized-K (EGK): A New Simple and General Model for Composite Fading Channels

Ferkan Yılmaz, Mohamed‐Slim Alouini|arXiv (Cornell University)|Dec 12, 2010

Millimeter-Wave Propagation and Modeling25 references22 citations

TL;DR

This paper proposes the Extended Generalized-K (EGK) distribution, a five-parameter composite fading model for millimeter wave (60 GHz and above) and free-space optical (FSO) channels, combining multipath fading and shadowing. It derives closed-form expressions for key statistics—PDF, CDF, level crossing rate, average fade duration, fractional moments—and performance metrics like average bit error rate, outage probability, and average capacity, offering a unified, tractable framework that generalizes existing models with high accuracy as validated by simulations.

ABSTRACT

In this paper, we introduce a generalized composite fading distribution (termed extended generalized-K (EGK)) to model the envelope and the power of the received signal in millimeter wave (60 GHz or above) and free-space optical channels. We obtain the first and the second-order statistics of the received signal envelope characterized by the EGK composite fading distribution. In particular, expressions for probability density function, cumulative distribution function, level crossing rate and average fade duration, and fractional moments are derived. In addition performance measures such as amount of fading, average bit error probability, outage probability, average capacity, and outage capacity are offered in closed-form. Selected numerical and computer simulation examples validate the accuracy of the presented mathematical analysis.

Motivation & Objective

• To develop a generalized, mathematically tractable composite fading model for millimeter wave and free-space optical (FSO) channels that captures both multipath fading and shadowing effects.
• To derive closed-form expressions for first- and second-order statistics of the EGK distribution, including PDF, CDF, level crossing rate (LCR), and average fade duration (AFD).
• To evaluate key performance metrics such as amount of fading, average bit error probability (ABEP), outage probability (OP), average capacity (AC), and outage capacity (OC) under the EGK fading model.
• To demonstrate the model's generality by showing that it includes widely used distributions like Nakagami-m, Rayleigh, and generalized-K as special or limiting cases.
• To validate the analytical accuracy of the derived expressions through numerical results and Monte Carlo simulations.

Proposed method

• The EGK distribution is constructed as a product of two gamma-distributed random variables representing multipath fading and shadowing, with five parameters: fading figure (m), fading shaping factor (ξ), shadowing figure (ms), shadowing shaping factor (ξs), and average SNR (γ̄).
• The probability density function (PDF) of the EGK distribution is derived using the Fox's H-function representation, enabling the derivation of all statistical and performance metrics.
• The cumulative distribution function (CDF) is obtained via integration of the PDF, and the level crossing rate (LCR) and average fade duration (AFD) are derived using the first and second moments of the envelope envelope and its derivative.
• Performance metrics such as average bit error probability (ABEP) are derived using the moment generating function (MGF) approach and generalized H-function representations, with special cases simplified using hypergeometric functions.
• Average capacity (AC) and outage capacity (OC) are derived using the Fox's H-function representation of the SNR PDF and the Shannon capacity formula, with simplifications for special cases like Nakagami-m and generalized-K.
• The model's validity is confirmed through numerical evaluation and comparison with Monte Carlo simulations across various fading and shadowing conditions.

Experimental results

Research questions

• RQ1 Can a unified composite fading model be developed that accurately captures both multipath fading and shadowing in millimeter wave and FSO channels?
• RQ2 What are the closed-form expressions for key statistical metrics—PDF, CDF, LCR, AFD—under the proposed EGK distribution?
• RQ3 How do performance metrics like ABEP, outage probability, and average capacity behave under the EGK model, and can they be expressed in closed form?
• RQ4 To what extent does the EGK model generalize existing fading models such as Nakagami-m, Rayleigh, and generalized-K?
• RQ5 How accurate are the analytical results when compared to simulation results across diverse fading and shadowing conditions?

Key findings

• The EGK distribution provides a unified, five-parameter model that generalizes Rayleigh, Nakagami-m, Rice, and generalized-K fading distributions as special or limiting cases.
• Closed-form expressions are derived for the PDF, CDF, level crossing rate (LCR), and average fade duration (AFD), enabling precise statistical characterization of composite fading channels.
• The average bit error probability (ABEP) is expressed in closed form using the Fox's H-function and generalized hypergeometric functions, with simplifications for well-known fading models like Nakagami-m.
• Outage probability (OP) and outage capacity (OC) are derived using the CDF and SNR distribution, showing that both decrease with increasing fading figure (m) or shaping factor (ξ), indicating improved reliability.
• Average capacity (AC) is derived in closed form using the Fox's H-function, with simplifications matching known results for generalized-K and Nakagami-m channels, confirming consistency and correctness.
• Simulations confirm the analytical accuracy of the derived expressions, showing tight agreement between theoretical predictions and numerical results across varying fading and shadowing parameters.