The Fourier Series

# Continuous Time

### Definition 19

A function
$x(t)$
is periodic if
$\exists T$
such that
$\forall t, x(t-T)=x(t)$
.

### Definition 20

The fundamental period is the smallest such
$T$
which satisfies the periodicity property in Definition 19

### Theorem 1

If
$x(t)$
and
$y(t)$
are functions with period
$T_1$
and
$T_2$
respectively, then
$x(t)+y(t)$
is periodic if
$\exists m, n \in \mathbb{Z}$
such that
$mT_1 = nT_2$
.

### Definition 21

Given a periodic function
$x(t)$
with fundamental period
$T$
and fundamental frequency
$\omega_0=\frac{2\pi}{T}$
, the Fourier Series of
$x$
is a weighted sum of the harmonic functions.
$x(t) = \sum_{k=-\infty}^{\infty}{a_ke^{jk\omega_0t}}$
To find the coefficients
$a_k$
:
Rearranging this, we can see that
$a_n = \frac{1}{T}\int_{T}{x(t)e^{-jn\omega_0t}dt}.$
For
$a_0$
, the DC offset term, this formula makes a lot of sense because it is just the average value of the function over one period.
$a_0 = \frac{1}{T}\int_{T}{x(t)dt}$
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
$\forall \tau, \ \lim_{M \rightarrow \infty}{x_M(\tau) = x(\tau)} \qquad x_M(t) = \sum_{k=-M}^{M}{a_k e^{jk\omega_0t}}$
will converge.

### Theorem 2

The Fourier Series of a continuous time periodic function
$x(t)$
will converge when
$x$
is piecewise continuous and
$\frac{d}{dt}x$
is piecewise continuous.
• If
$x$
is continuous at
$\tau$
,
$\lim_{M \rightarrow \infty}x_M(\tau) = x(\tau)$
• If
$x$
is discontinuous at
$\tau$
, then
$\lim_{M\rightarrow \infty}x_M(\tau) = \frac{1}{2}(x(\tau^-) + x(\tau^+))$
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
$N \in \mathbb{Z}$
if
$\forall n, x[n+N]=x[n]$
However, there are some differences. For example,
$x[n] = cos(\omega_0 n)$
is only periodic in discrete time if
$\exists N, M \in \mathbb{Z}, \omega_0 N = 2 \pi M$
.

### 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.
$\phi_k[n] = e^{jk\frac{2\pi}{N}n}$
• $\phi_k[n]$
is perioidic in n (i.e
$\phi_k[n+N] = \phi_k[n]$
)
• $\phi_k[n]$
is periodic in k (i.e
$\phi_{k+N}[n] = \phi_k[n]$
)
• $\phi_k[n]\cdot \phi_m[n] = \phi_{k + m}[n]$
Notice that with this basis, there are only N unique functions that we can use. An additional property of the
$\phi_k[n]$
is that
$\sum_{n=}{\phi_k[n]} = \begin{cases} N & \text{if } k = 0, \pm N, \pm 2N, \cdots\\ 0 & \text{otherwise.} \end{cases}$

### 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.
$x[n] = \sum_{k=0}^{N-1}{a_k\phi_k[n]}$
In order to find the values of
$a_k$
, we can perform a similar process as in continuous time.
\begin{aligned} x[n] &= \sum_{k=0}^{N-1}{a_k\phi_k[n]}\\ x[n]\phi_{-M}[n] &= \sum_{k=0}^{N-1}{a_k\phi_k[n]\phi_{-M}[n]}\\ \sum_{n=}{x[n]\phi_{-M}[n]} &= \sum_{n=}{\sum_{k=}{a_k\phi_{k-M}[n]}} = \sum_{k=}{a_k\sum_{n=}{\phi_{k-M}[n]}}\\ \sum_{n=}{x[n]\phi_{-M}[n]} &= a_MN\\ a_M &= \frac{1}{N}\sum_{n=}{x[n]\phi_{-M}[n]}\end{aligned}

# Properties of the Fourier Series

Linearity: If
$a_k$
and
$b_k$
are the coefficients of the Fourier Series of
$x(t)$
and
$y(t)$
respectively, then
$Aa_k + Bb_k$
are the coefficients of the Fourier series of
$Ax(t)+By(t)$
Time Shift: If
$a_k$
are the coefficients of the Fourier Series of
$x(t)$
, then
$b_k = e^{-jk\frac{2\pi}{T}t_0}a_k$
are the coefficients of the Fourier Series of
$\hat{x}(t)=x(t-t_0)$
Time Reversal: If
$a_k$
are the coefficients of the Fourier Series of
$x(t)$
, then
$b_k=a_{-k}$
are the coefficients of the Fourier Series of
$x(-t)$
Conjugate Symmetry: If
$a_k$
are the coefficients of the Fourier Series of
$x(t)$
, then
$a_k^*$
are the coefficients of the Fourier Series of
$x^*(t)$
. This means that
$x(t)$
is a real valued signal, then
$a_k=a_{-k}^*$

### Theorem 4 (Parseval's Theorem)

$\textbf{Continuous Time: } \frac{1}{T}\int{|x(t)|^2dt} = \sum_{k=-\infty}^{\infty}{|a_k|^2}$
$\textbf{Discrete Time: } \frac{1}{N}\sum_{n=}{|x[n]|^2} = \sum_{k=}{|a_k|^2}$

# Interpreting the Fourier Series

A good way to interpret the Fourier Series is as a change of basis. In both the continuous and discrete case, we are projecting our signal
$x$
onto a set of basis functions, and the coefficients
$a_k$
are the coordinates of our signal in the new space.

## Discrete Time

Since in discrete time, signal is peroidic in
$N$
, we can turn any it into a vector
$\vec{x}\in \mathbb{C}^N$
.
$\vec{x} = \left[ \begin{array}{c} x[0]\\ x[1]\\ \vdots\\ x[N-1] \end{array} \right] \in \mathbb{C}^N$
We can use this to show that
$\phi_k$
form an orthogonal basis. If we take two of them
$\phi_k[n]$
and
$\phi_M[n]$
(
$k\ne M$
) and compute their dot product of their vector forms, then
$\phi_k[n] \cdot \phi_M[n] = \phi_M^*\phi_k = \sum_{}{\phi_{k-M}[n]} = 0$
That means that
$\phi_k$
and
$\phi_M$
are orthogonal, and they are
$N$
of them, therefore they are a basis. If we compute their magnitudes, we see
$\phi_k \cdot \phi_k = ||\phi_k||^2 = N, \therefore ||\phi_k|| = \sqrt{N}$
Finally, if we compute
$\vec{x}\phi_M$
where
$\vec{x}$
is the vector form of an N-periodic signal,
$\vec{x}\cdot \vec{\phi_M} = \left(\sum_{i=0}^{N-1}{a_i\phi_i}\right)\cdot \phi_M = Na_m$
$a_m = \frac{1}{N}\vec{x}\cdot \phi_M$
This is exactly the equation we use for finding the Fourier Series coefficients, and notice that it is a projection since
$N = ||\phi_m||^2$
. This gives a nice geometric intution for Parseval’s theorem
$\frac{1}{N}\sum{|x[n]|^2} = \frac{1}{N}||\vec{x}||^2 = \sum{|a_k|^2}$
because we know the norms of two vectors in different bases must be equal.

## Continuous Time

In continuous time, our bases functions are
$\phi_k(t) = e^{jk\frac{2\pi}{T}t}$
for
$k \in (-\infty, \infty)$
Since we can’t convert continuous functions into vectors, these
$\phi_k$
are really a basis for the vector space of square integrable functions on the interval
$[0, T]$
. The inner product for this vector space is
$ = \int_{0}^{T}{x(t)y^*(t)}.$
We can use this inner product to conduct the same proof as we did in discrete time.