# Fourier series

We have seen that a**periodic**signal

*x*:

*Time → Reals*with period

*p*∈

*Time*is one where for all

*t*∈

*Time*

*x*(

*t*) =

*x*(

*t*+

*p*).

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*) = *A*_{0}
+ ∑_{(k=1
to ∞)}
*A*_{k}
cos (*k*ω_{0}*t*
+ *φ *_{k}
)

Each term in the summation is a cosine with amplitude *A _{k}*
and phase φ

*. The frequency ω*

_{k}_{0}, which has units of radians per second, is called the

**fundamental frequency**, and is related to the period by

_{0}= 2π /

*p*.

In other words, a sinusoid with frequency ω
_{0} has period *p*. The constant term *A*_{0}
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 *A _{k}* 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.