which satisfies the periodicity property in Definition 19
Theorem 1
If
x(t)
and
y(t)
are functions with period
T1
and
T2
respectively, then
x(t)+y(t)
is periodic if
∃m,n∈Z
such that
mT1=nT2
.
Definition 21
Given a periodic function
x(t)
with fundamental period
T
and fundamental frequency
ω0=T2π
, the Fourier Series of
x
is a weighted sum of the harmonic functions.
x(t)=∑k=−∞∞akejkω0t
To find the coefficients
ak
:
Rearranging this, we can see that
an=T1∫Tx(t)e−jnω0tdt.
For
a0
, the DC offset term, this formula makes a lot of sense because it is just the average value of the function over one period.
a0=T1∫Tx(t)dt
Because the Fourier Series is an infinite sum, there is a worry that for some functions
x(t)
, it will not converge. The Dirichlet Convergent Requirements tell us when the Fourier Series converges. More specificially, they tell us when
∀τ,limM→∞xM(τ)=x(τ)xM(t)=∑k=−MMakejkω0t
will converge.
Theorem 2
The Fourier Series of a continuous time periodic function
x(t)
will converge when
x
is piecewise continuous and
dtdx
is piecewise continuous.
If
x
is continuous at
τ
,
limM→∞xM(τ)=x(τ)
If
x
is discontinuous at
τ
, then
limM→∞xM(τ)=21(x(τ−)+x(τ+))
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.
Discrete Time
The definition for periodicity in discrete time is the exact same as the definition in continuous time.
Definition 22
A function
x[n]
is periodic with period
N∈Z
if
∀n,x[n+N]=x[n]
However, there are some differences. For example,
x[n]=cos(ω0n)
is only periodic in discrete time if
∃N,M∈Z,ω0N=2πM
.
Theorem 3
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.
ϕk[n]=ejkN2πn
ϕk[n]
is perioidic in n (i.e
ϕk[n+N]=ϕk[n]
)
ϕk[n]
is periodic in k (i.e
ϕk+N[n]=ϕk[n]
)
ϕk[n]⋅ϕm[n]=ϕk+m[n]
Notice that with this basis, there are only N unique functions that we can use. An additional property of the
ϕk[n]
is that
∑n=<N>ϕk[n]={N0if k=0,±N,±2N,⋯otherwise.
Definition 23
Given a periodic discrete-time function
x[n]
with period
N
, the Fourier series of the function is a weighted sum of the roots of unity basis functions.
x[n]=∑k=0N−1akϕk[n]
In order to find the values of
ak
, we can perform a similar process as in continuous time.