A Spectrum-Amplitude Approximation for Slowly Varying Nonlinear Frequency-Modulated Pulses of Long Duration

A spectrum-amplitude approximation is derived for slowly varying amplitude-modulated and nonlinearly frequencymodulated pulses of long duration (by "long duration" we mean, essentially, "containing many oscillations"-perhaps thousandsirrespective of the absolute length of the pulse in seconds). The derivation consists of the application of Parseval's theorem to the waveform of interest, approximated by a sequence of linearly frequencymodulated pulses of constant amplitude. It is found that successful application of the approximation depends on the individual pulses being long enough for the changing frequency so that there is little overlap between the spectrum of one segment and that of another. Actual application of the technique can be made with a nominal average frequency from each segment. The technique is demonstrated on an experimental pulse of approximately 20 000 oscillations varying from 500 MHz down to 13 MHz in 960, μs.