<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43067_2025_256_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$N$</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation> and using <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43067_2025_256_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> nodes per interval, the approximation error is significantly reduced to a factor of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43067_2025_256_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{N^{2n}}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <msup> <mi>N</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mfrac> </math></EquationSource> </InlineEquation>. 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-<i>m</i>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43067_2025_256_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa$</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43067_2025_256_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu$</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43067_2025_256_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\eta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43067_2025_256_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu$</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> 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.</p>

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

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2025年8月12日

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Abdulrahman Faris,
Peter O. Akuon,
Vitalice Kalecha Oduol

摘要

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 $N$ $N$ and using $n$ $n$ nodes per interval, the approximation error is significantly reduced to a factor of $\frac{1}{N^{2n}}$ $\frac{1}{N^{2 n}}$ . 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, $\kappa$ $κ$ – $\mu$ $μ$ , and $\eta$ $η$ – $\mu$ $μ$ 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.