It should be obvious by now that we need a way to move freely between the two domains. Any well-behaved periodic waveform can be represented as a series of sine and/or cosine waves at multiples of its fundamental frequency plus (sometimes) a dc offset.

This is known as a Fourier series. This very useful (and perhaps rather surprising) fact was discovered in 1822 by Joseph Fourier, a French mathematician, in the course of research on heat conduction.

Not all signals used in communication are strictly periodic, but they are often close enough for practical purposes. Fourier’s discovery, applied to a time-varying signal, can be expressed mathematically as follows:

ƒ(t) = A0 + A1 ωt + B1 ωt + A2 ωt + B2 ωt + A3 ωt + B3 ωt + ⋅ ⋅ ⋅

ƒ(t) = any well-behaved function of time. For our purposes, ƒ(t) will generally be either a voltage v(t) or a current i(t).

An and Bn = real-number coefficients; that is, they can be positive, negative, or zero.

ω = radian frequency of the fundamental.

The radian frequency can be found from the time-domain representation of the signal by finding the period (that is, the time T after which the whole signal repeats exactly) and using the equations:

ƒ = 1/Τ and ω = 2πƒ

The simplest ac signal is a sinusoid. The frequency-domain representation of a sine wave has already been described for a voltage sine wave with a period of 1 ms and a peak amplitude of 1 V.

For this signal, all the Fourier coefficients are zero except for B1, which has a value of 1 V. The equation becomes: v(t) = sin (2000πt) V which is certainly no surprise.

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