Modulated waveforms for harmonic minimization of far-field signals in amplitude-modulated heating of the ionosphere

This paper establishes an amplitude modulation heating model, simulating the far-field radiation of ELF/VLF signals generated by modulation heating, as well as the specific location and longitudinal extent of the radiation source. We consider various modulation waveforms and find that square-wave modulation has the highest excitation efficiency for ELF/VLF signals, and that square-wave modulation with a smaller duty cycle (<50%) exhibits higher excitation efficiency for ELF/VLF signals, while the sin²t waveform modulation yields the lowest proportion of harmonic energy in the generated signals. The amplitude of the second harmonic generated by the sin²t waveform is less than one-tenth that of the fundamental frequency, and the energy of higher-frequency harmonics can be negligibly small compared with those of the fundamental wave. It is a challenging task to achieve a balance between enhancing the excitation efficiency of ELF/VLF signals and also suppressing harmonics generated by the modulated heating process. This is because the harmonics are correspondingly enhanced as the excitation efficiency of the signals is increased. However, we find that under conditions of varying effective radiant power and modulation frequency, as long as the modulation waveform is unchanged, the energy ratio between the fundamental frequency signal generated by modulated heating and each harmonic is relatively fixed, with changes only in signal intensity and the location of the radiation source zone. This implies that one can first select modulation waveforms that make the signal less prone to distortion, then increase the effective radiated power to enhance the signal strength, without concern for harmonic interference of the fundamental signal.