# Lecture 4: Periodic Signals and Representations

From the last lecture we learned how functions can be represented as a series
of other functions:

A signal
is said to be periodic with period
if

**periodic extension**. The integral over a single period of the function is denoted by

*does not matter*when we start. Usually it is convenient to start at the beginning of a period.

The building block functions that can be used to build up periodic functions
are themselves periodic: we will use the set of sinusoids. If the period of
is
,
let
.
The frequency
is said to be the
*
fundamental frequency
*
; the fundamental frequency
is related to the period of the function. Furthermore, let
.
We will represent the function
using the set of sinusoids

Note that for each basis function associated with
there are actually two parameters: the amplitude
and the phase
.
It will often turn out to be more useful to represent the function using both
sines and cosines. Note that we can write

**compact trigonometric Fourier series**. The second is the

**trigonometric Fourier series.**. To go from one to the other use

To complete the representation we must be able to compute the coefficients.
But this is the same sort of thing we did before. If we can show that the set
of functions
is
in fact an orthogonal set, then we can use the same formulas we did before.
Check:

It is interesting to see how the function gets built up at the pieces are
added together: