Two Interval Upper-Bound Q -Function Approximations with Applications

Zoran Perić, Aleksandar Marković, Nataša Kontrec (), Jelena Nikolić, Marko D. Petković and Aleksandra Jovanović
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Zoran Perić: Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18115 Niš, Serbia
Aleksandar Marković: Faculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, Lole Ribara 29, 38220 Kosovska Mitrovica, Serbia
Nataša Kontrec: Faculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, Lole Ribara 29, 38220 Kosovska Mitrovica, Serbia
Jelena Nikolić: Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18115 Niš, Serbia
Marko D. Petković: Faculty of Science and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia
Aleksandra Jovanović: Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18115 Niš, Serbia

Mathematics, 2022, vol. 10, issue 19, 1-15

Abstract: The Gaussian Q -function has considerable applications in numerous areas of science and engineering. However, the fact that a closed-form expression for this function does not exist encourages finding approximations or bounds of the Q -function. In this paper, we determine analytically two novel interval upper bound Q -function approximations and show that they could be used efficiently not only for the symbol error probability (SEP) estimation of transmission over Nakagami- m fading channels, but also for the average symbol error probability (ASEP) evaluation for two modulation formats. Specifically, we determine analytically the composition of the upper bound Q -function approximations specified at disjoint intervals of the input argument values so as to provide the highest accuracy within the intervals, by utilizing the selected one of two upper bound Q -function approximations. We show that a further increase of the accuracy, achieved in the case with two upper-bound approximations composing the interval approximation, can be obtained by forming a composite interval approximation of the Q -function that assumes another extra interval and by specifying the third form for the upper-bound Q -function approximation. The proposed analytical approach can be considered universal and widely applicable. The results presented in the paper indicate that the proposed Q -function approximations outperform in terms of accuracy other well-known approximations carefully chosen for comparison purposes. This approximation can be used in numerous theoretical communication problems based on the Q -function calculation. In this paper, we apply it to estimate the average bit error rate (ABER), when the transmission in a Nakagami- m fading channel is observed for the assumed binary phase-shift keying (BPSK) and differentially encoded quadrature phase-shift keying (DE-QPSK) modulation formats, as well as to design scalar quantization with equiprobable cells for variables from a Gaussian source.

Keywords: Q -function; approximation; Nakagami- m fading; modulation formats (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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