Improved Nyquist filter characteristics using spline interpolation

Abstract

This paper presents and investigates a novel approach for constructing a family of intersymbol interference (ISI)-free pulses that shows comparable or better ISI performance in the presence of sampling errors, compared with some recently proposed pulses. We propose and discuss a new parametric method for the design of Nyquist filter characteristics using constraints in frequency characteristics construction. The method for constructing the filter characteristics uses a piecewise polynomial approximation of an ideal optimized staircase characteristic by spline functions. The spline polynomials are used to approximate a function that must pass through specified points. The performances of new ISI-free pulses are studied with respect to the ISI error probability. This family provides flexibility in designing an appropriate pulse even after the roll-off factor has been chosen. The results for error probability outperform the fourth-degree polynomial pulse [4].

Appendices

Appendix 1

Table 5 Spline coefficients for spline 3 function $C_{ij}, i = \overline{0,n - 1}, j = \overline{1,3}, \,\,n = 8$

Appendix 2: an example of the spline 3 frequency characteristic for α = 0.5, a ₁ = 0.72, a ₂ = 0.68, and a ₃ = 0.58

$$ S_3 (f) = \left\{ {\begin{array}{*{20}c} 1 \hfill & {0 < f < 0.5} \hfill \\ {134.858\;\left( { - 0.247853 + f} \right)\left( {0.405482 - 1.25215f + f^2 } \right),} \hfill & {0.5 \le f \le 0.571492} \hfill \\ { - 91.3551\;\left( { - 0.880229 + f} \right)\;\left( {0.356353 - 1.15038f + f^2 } \right),} \hfill & {0.571492 < f \le 0.714286} \hfill \\ {31.6229\;\left( { - 0.515168 + f} \right)\;\left( {0.992041 - 1.95195f + f^2 } \right),} \hfill & {0.714286 < f \le 0.857143} \hfill \\ { - 7.69659\;\left( { - 1.35878 + f} \right)\;\left( {0.822296 - 164122f + f^2 } \right),} \hfill & {0.857143 < f \le 1} \hfill \\ { - 7.69659\;\left( { - 1.35878 + f} \right)\;\left( {0.822296 - 164122f + f^2 } \right),} \hfill & {1 < f \le 1.14286} \hfill \\ {31.6229\;\left( { - 0.90243 + f} \right)\;\left( {1.75535 - 2.63045f + f^2 } \right),} \hfill & {1.14286 < f \le 1.28571} \hfill \\ { - 91.3551\;\left( { - 1.48908 + f} \right)\;\left( {1.55312 - 2.4803f + f^2 } \right),} \hfill & {1.28571 < f \le 1.42857} \hfill \\ {134.858\;\left( { - 1.68485 + f} \right)\;\left( { - 1.5 + f} \right)\;\left( { - 1.31515 + f^2 } \right)} \hfill & {1.42857 < f \le 1.5} \hfill \\ {0,} \hfill & {f > 1.5} \hfill \ \end{array}} \right. $$

Appendix 3

Table 6 Spline coefficients for spline 4 function $C_{ij}, i = \overline{0,n - 1}, j = \overline{1,3}, \,\,n = 10$

Appendix 4: an example of the spline 4 frequency characteristic for α = 0.5, a ₁ = 0.72, a ₂ = 0.68, a ₃ = 0.58, and a ₄ = 0.56

$$ S_4 (f) = \left\{ {\begin{array}{*{20}c} 1 \hfill & {0 < f < 0.5} \hfill \\ {289.504\;\left( { - 0.304617 + f} \right)\;\left( {0.365371 - 1.19538f + f^2 } \right),} \hfill & {0.5 \le f \le 0.55556} \hfill \\ { - 199.927\;\left( { - 0.792399 + f} \right)\;\left( {0.326346 - 1.11561f + f^2 } \right),} \hfill & {0.55556 < f \le 0.666667} \hfill \\ {87.3847\;\left( { - 0.520073 + f} \right)\;\left( {0.735569 - 1.6904f + f^2 } \right),} \hfill & {0.666667 < f \le 0.777778} \hfill \\ { - 47.5518\;\left( { - 1.08142 + f} \right)\;\left( {\;0.584549 - 1.47769f + f^2 } \right),} \hfill & {0.777778 < f \le 0.888889} \hfill \\ {15.3424\;\left( { - 0.631307 + f} \right)\;\left( {1.45708 - 236869f + f^2 } \right),} \hfill & {0.888889 < f \le 1} \hfill \\ {15.3424\;\left( { - 0.631307 + f} \right)\;\left( {1.45708 - 236869f + f^2 } \right),} \hfill & {1 < f \le 1.11111} \hfill \\ { - 47.5518\;\left( { - 1.35742 + f} \right)\;\left( {1.11796 - 2.08346f + f^2 } \right),} \hfill & {1.11111 < f \le 1.22222} \hfill \\ \begin{gathered} 87.3847\;\left( { - 1.07206 + f} \right)\;\left( {1.85954 - 2.71748f + f^2 } \right), \hfill \\ - 199.927\left( { - 1.49142 + f} \right)\;\left( {1.69978 - 2.60057f + f^2 } \right), \hfill \\ 289.504\;\left( { - 1.64316 + f} \right)\;\left( { - 1.5 + f} \right)\;\left( { - 1.35684 + f} \right), \hfill \\ \end{gathered} \hfill & \begin{gathered} 1.22222 < f \le 1.33333 \hfill \\ 1.33333 < f \le 1.44444 \hfill \\ 1.44444 < f \le 1.5 \hfill \\ \end{gathered} \hfill \\ {0,} \hfill & {f > 1.5} \hfill \ \end{array}} \right. $$