Spectral and Energy Efficiency of Optimal Signals with Increased Duration, Providing Overcoming “Nyquist Barrier”

Abstract

Optimal signals with increased duration, providing overcoming “Nyquist Barrier”, are considered. Optimization task of searching envelopes of signals is presented and numerically solved. Solutions of optimization task for different signal durations and correlation coefficients are presented. Corresponding energy spectra are also checked. The dependence of occupied frequency bandwidth for different level of energy spectra is obtained. BER performance of proposed optimal signals was checked by simulation. It is shown, that energy losses can be reduced by reducing correlation coefficient in optimization task and in resulting optimal signals. Application of optimal signals with increased duration allows to be closer to Shannon boundary and provides realization of data transfer with overcoming “Nyquist Barrier”. It was done by increasing spectral efficiency for different level of energy spectra.

Appendix A

The expansion coefficients of the envelope a ( t )

Form of envelope a(t) depends on number of expansion coefficients m of the finite Fourier series. We choose m by next expression:

$$ \varepsilon (m) = \sqrt {\int\limits_{ - T/2}^{T/2} {\left( {a_{m} \left( t \right) - a_{m - 1} \left( t \right)} \right)^{2} dt} } \le 0.01, $$

where a _m(t) and a _m−1(t) – functions, which have m and (m − 1) expansion coefficients of the finite Fourier series.

Results of solutions of optimization task are presented in form of expansion coefficients of the finite Fourier series in Table A.1.

Table A.1. The expansion coefficients of the envelope a(t) for n = 2.