This module describes the type of signals acted on by the Continuous Time Fourier Series.
The Continuous-Time Fourier Series maps finite-length (or
to infinite-length, discrete-frequency signals in
When a function repeats
itself exactly after some given period, or cycle, we say it's periodic.
A periodic function can be
mathematically defined as:
represents the fundamental period of the signal, which is the smallest positive value of T for the signal to repeat. Because of this,
you may also see a signal referred to as a T-periodic signal.
Any function that satisfies this equation is said to be periodic with period T.
We can think of periodic functions (with period
) two different ways:
as functions on all of
or, we can cut out all of the redundancy, and think of them
as functions on an interval
(or, more generally,
). If we know the signal is T-periodic then all the
information of the signal is captured by the above interval.
An aperiodic CT function
, on the other hand,
does not repeat for any
i.e. there exists no such that this equation holds.
Here's an example demonstrating a periodic sinusoidal signal with various frequencies, amplitudes and phase delays:
To learn the full concept behind periodicity, see the video below.
Khan Lecture on Periodic Signals
A periodic signal is completely defined by its values in one period, such as the interval [0,T].