Electric Circuits and Networks

14.8 FOURIER TRANSFORMS OF SOME SEMI-INFINITE DURATION WAVEFORMS

Inputs, even if they are periodic waveforms, are switched on to the circuit at some finite time-instant. This fact is usually brought out by specifying an input function as f(t) u(t), where f(t) is a periodic or aperiodic waveform. A DC voltage of V applied to the circuit from t = 0 is specified as Vu(t) and a unit amplitude cosine wave applied to the circuit from t = 0 is specified as cos(ωt)u(t).

We developed the Fourier transforms for periodic waveforms, including the one with infinite period, i.e., a constant function, in the last section. We study the effect of switching at t = 0 on their Fourier transforms in this section.

14.8.1 Fourier Transform of e^–αt u(t)

Let v₁(t) = ...