A composite Gauss–Legendre quadrature method for the Q-function approximation and its application in 6-bit symbol sequence modulation over generalised fading distributions

Abstract This article introduces a 6-bit Symbol Sequence Modulation (6-SSM) scheme developed to enhance communication security and efficiency through sequence-based modulation. The 6-SSM is designed to exploit symbol sequences, enabling improved secrecy performance through time-slot diversity. To analytically evaluate the performance of the proposed 6-SSM system, an exponent-based Q-function approximation (QFA) is also developed. This QFA employs a composite Gauss–Legendre quadrature applied to the polar form of the Q function. By dividing the integration range into intervals $$N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> and using $$n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> nodes per interval, the approximation error is significantly reduced to a factor of $$\frac{1}{N^{2n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mfrac> </mml:math> . A practical configuration with two intervals and two nodes yields a four-exponent QFA that offers high accuracy, low relative error, and reduced complexity compared to existing approximations. The QFA is used to derive closed-form expressions for the bit error rate (BER) of the 6-SSM scheme under generalised fading channels, including Nakagami- m , $$\kappa$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> – $$\mu$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> , and $$\eta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> – $$\mu$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> distributions. The analytical results match closely Monte Carlo simulations, confirming the precision of the proposed QFA. Additionally, BER expressions are employed to analyse the secrecy rate, revealing that higher-order SSM offers improved physical layer security.