Fourier seriesWe have seen that a periodic signal x:Time → Reals with period p ∈ Time is one where for all t ∈ Time
A remarkable result, due to Joseph Fourier, 1768-1830, is that such signal can (usually) be described as a constant term plus a sum of sinusoids,
x(t) = A0 + ∑(k=1 to ∞) Ak cos (kω0t + φ k )
Each term in the summation is a cosine with amplitude Ak and phase φ k. The frequency ω 0, which has units of radians per second, is called the fundamental frequency, and is related to the period by
In other words, a sinusoid with frequency ω 0 has period p. The constant term A0 is sometimes called the DC term, where "DC" stands for "direct current," a reference back to the origins of much of this theory in circuit analysis. The terms where k ≥ 2 are called harmonics.
Using the Fourier series expansion for synthesis of signals is problematic because of the infinite summation. However, for most practical signals, the coefficients Ak become very small (or even zero) for large k, so a finite summation can be used as an approximation. Even when an infinite summation is used, the expansion of a periodic waveform is not always exact. There are some technical mathematical conditions that f must satisfy for it to be exact. These conditions are beyond the scope of this course. Fortunately, these conditions are rarely an issue with practical, real-world signals.