Communications and information theory use the Gaussian $Q$ -function, a positive and decreasing function, across the literature. Its approximations were created to simplify mathematical study of the Gaussian $Q$ -function expressions. This is important since the $Q$ -function cannot be represented in closed-form terms of elementary functions. In a noise model with the Gaussian distribution function and various digital modulation schemes, closed-form approximations of the Gaussian $Q$ -function are used to predict a digital communications system's symbol error probability (SEP) or bit error probability (BEP). Another significant scenario pertains to fading channels, whereby it is important to accurately determine, through a closed-form expression, the precise evaluations of complex integrals involved in the computations of SEP or BEP. In addition to the aforementioned scenarios, it is imperative for a communications system designer to ascertain the requisite operational signal-to-noise ratio for the specific application, based on the target SEP (or BEP). In this scenario, the crucial role of the explicit invertibility of the Gaussian $Q$ -function approximation is of significant importance in achieving this objective. In this paper we propose a survey of the approximations of the Gaussian $Q$ -function found in the literature, reviewing also the approximations originally given for the 4 classical special functions related to it, restricting the analysis to the explicitly invertible ones, and classifying them on the basis of their accuracy (on the significant range), simplicity, and easiness of inversion, also distinguishing the bounds from approximations. We also list the inverses of some of them, already published or newly found in this research.

Explicitly Invertible Approximations of the Gaussian Q-Function: A Survey

Soranzo A.;Vatta F.
;
Comisso M.;Buttazzoni G.;Babich F.
2023-01-01

Abstract

Communications and information theory use the Gaussian $Q$ -function, a positive and decreasing function, across the literature. Its approximations were created to simplify mathematical study of the Gaussian $Q$ -function expressions. This is important since the $Q$ -function cannot be represented in closed-form terms of elementary functions. In a noise model with the Gaussian distribution function and various digital modulation schemes, closed-form approximations of the Gaussian $Q$ -function are used to predict a digital communications system's symbol error probability (SEP) or bit error probability (BEP). Another significant scenario pertains to fading channels, whereby it is important to accurately determine, through a closed-form expression, the precise evaluations of complex integrals involved in the computations of SEP or BEP. In addition to the aforementioned scenarios, it is imperative for a communications system designer to ascertain the requisite operational signal-to-noise ratio for the specific application, based on the target SEP (or BEP). In this scenario, the crucial role of the explicit invertibility of the Gaussian $Q$ -function approximation is of significant importance in achieving this objective. In this paper we propose a survey of the approximations of the Gaussian $Q$ -function found in the literature, reviewing also the approximations originally given for the 4 classical special functions related to it, restricting the analysis to the explicitly invertible ones, and classifying them on the basis of their accuracy (on the significant range), simplicity, and easiness of inversion, also distinguishing the bounds from approximations. We also list the inverses of some of them, already published or newly found in this research.
2023
Pubblicato
https://ieeexplore.ieee.org/document/10318209
File in questo prodotto:
File Dimensione Formato  
SVCBB2023_Explicitly_Invertible_Approximations_of_the_Gaussian_Q-Function_A_Survey.pdf

accesso aperto

Tipologia: Documento in Versione Editoriale
Licenza: Creative commons
Dimensione 5.59 MB
Formato Adobe PDF
5.59 MB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/3067239
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? ND
social impact