which satisfies the periodicity property in Definition 19
are functions with period
is periodic if
Given a periodic function
with fundamental period
and fundamental frequency
, the Fourier Series of
is a weighted sum of the harmonic functions.
To find the coefficients
Rearranging this, we can see that
, the DC offset term, this formula makes a lot of sense because it is just the average value of the function over one period.
Because the Fourier Series is an infinite sum, there is a worry that for some functions
, it will not converge. The Dirichlet Convergent Requirements tell us when the Fourier Series converges. More specificially, they tell us when
The Fourier Series of a continuous time periodic function
will converge when
is piecewise continuous and
is piecewise continuous.
is continuous at
is discontinuous at
These convergence requirements are for pointwise convergence only. They do not necessarily imply that the graphs of the Fourier Series and the original function will look the same.
The definition for periodicity in discrete time is the exact same as the definition in continuous time.
is periodic with period
However, there are some differences. For example,
is only periodic in discrete time if
The sum of two discrete periodic signals is periodic
Theorem 3 is not always true in continuous time but it is in discrete time.
The Fourier Series in discrete time is the same idea as the Fourier series in continuous time: to express every signal as a linear combination of complex exponentials. The discrete time basis that we use are the Nth roots of unity.
is perioidic in n (i.e
is periodic in k (i.e
Notice that with this basis, there are only N unique functions that we can use. An additional property of the
Given a periodic discrete-time function
, the Fourier series of the function is a weighted sum of the roots of unity basis functions.
In order to find the values of
, we can perform a similar process as in continuous time.