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The Fourier Series

Continuous Time

Definition 19

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

Definition 20

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

Theorem 1

If
x(t)x(t)
and
y(t)y(t)
are functions with period
T1T_1
and
T2T_2
respectively, then
x(t)+y(t)x(t)+y(t)
is periodic if
m,nZ\exists m, n \in \mathbb{Z}
such that
mT1=nT2mT_1 = nT_2
.

Definition 21

Given a periodic function
x(t)x(t)
with fundamental period
TT
and fundamental frequency
ω0=2πT\omega_0=\frac{2\pi}{T}
, the Fourier Series of
xx
is a weighted sum of the harmonic functions.
x(t)=k=akejkω0tx(t) = \sum_{k=-\infty}^{\infty}{a_ke^{jk\omega_0t}}
To find the coefficients
aka_k
:
Rearranging this, we can see that
an=1TTx(t)ejnω0tdt.a_n = \frac{1}{T}\int_{T}{x(t)e^{-jn\omega_0t}dt}.
For
a0a_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.
a0=1TTx(t)dta_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)x(t)
, it will not converge. The Dirichlet Convergent Requirements tell us when the Fourier Series converges. More specificially, they tell us when
τ, limMxM(τ)=x(τ)xM(t)=k=MMakejkω0t\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)x(t)
will converge when
xx
is piecewise continuous and
ddtx\frac{d}{dt}x
is piecewise continuous.
  • If
    xx
    is continuous at
    τ\tau
    ,
    limMxM(τ)=x(τ)\lim_{M \rightarrow \infty}x_M(\tau) = x(\tau)
  • If
    xx
    is discontinuous at
    τ\tau
    , then
    limMxM(τ)=12(x(τ)+x(τ+))\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]x[n]
is periodic with period
NZN \in \mathbb{Z}
if
n,x[n+N]=x[n]\forall n, x[n+N]=x[n]
However, there are some differences. For example,
x[n]=cos(ω0n)x[n] = cos(\omega_0 n)
is only periodic in discrete time if
N,MZ,ω0N=2πM\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.
ϕk[n]=ejk2πNn\phi_k[n] = e^{jk\frac{2\pi}{N}n}
  • ϕk[n]\phi_k[n]
    is perioidic in n (i.e
    ϕk[n+N]=ϕk[n]\phi_k[n+N] = \phi_k[n]
    )
  • ϕk[n]\phi_k[n]
    is periodic in k (i.e
    ϕk+N[n]=ϕk[n]\phi_{k+N}[n] = \phi_k[n]
    )
  • ϕk[n]ϕm[n]=ϕk+m[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
ϕk[n]\phi_k[n]
is that
n=<N>ϕk[n]={Nif k=0,±N,±2N,0otherwise.\sum_{n=<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]x[n]
with period
NN
, the Fourier series of the function is a weighted sum of the roots of unity basis functions.
x[n]=k=0N1akϕk[n]x[n] = \sum_{k=0}^{N-1}{a_k\phi_k[n]}
In order to find the values of
aka_k
, we can perform a similar process as in continuous time.
x[n]=k=0N1akϕk[n]x[n]ϕM[n]=k=0N1akϕk[n]ϕM[n]n=<N>x[n]ϕM[n]=n=<N>k=<N>akϕkM[n]=k=<N>akn=<N>ϕkM[n]n=<N>x[n]ϕM[n]=aMNaM=1Nn=<N>x[n]ϕM[n]\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=<N>}{x[n]\phi_{-M}[n]} &= \sum_{n=<N>}{\sum_{k=<N>}{a_k\phi_{k-M}[n]}} = \sum_{k=<N>}{a_k\sum_{n=<N>}{\phi_{k-M}[n]}}\\ \sum_{n=<N>}{x[n]\phi_{-M}[n]} &= a_MN\\ a_M &= \frac{1}{N}\sum_{n=<N>}{x[n]\phi_{-M}[n]}\end{aligned}

Properties of the Fourier Series

Linearity: If
aka_k
and
bkb_k
are the coefficients of the Fourier Series of
x(t)x(t)
and
y(t)y(t)
respectively, then
Aak+BbkAa_k + Bb_k
are the coefficients of the Fourier series of
Ax(t)+By(t)Ax(t)+By(t)
Time Shift: If
aka_k
are the coefficients of the Fourier Series of
x(t)x(t)
, then
bk=ejk2πTt0akb_k = e^{-jk\frac{2\pi}{T}t_0}a_k
are the coefficients of the Fourier Series of
x^(t)=x(tt0)\hat{x}(t)=x(t-t_0)
Time Reversal: If
aka_k
are the coefficients of the Fourier Series of
x(t)x(t)
, then
bk=akb_k=a_{-k}
are the coefficients of the Fourier Series of
x(t)x(-t)
Conjugate Symmetry: If
aka_k
are the coefficients of the Fourier Series of
x(t)x(t)
, then
aka_k^*
are the coefficients of the Fourier Series of
x(t)x^*(t)
. This means that
x(t)x(t)
is a real valued signal, then
ak=aka_k=a_{-k}^*

Theorem 4 (Parseval's Theorem)

Continuous Time: 1Tx(t)2dt=k=ak2\textbf{Continuous Time: } \frac{1}{T}\int{|x(t)|^2dt} = \sum_{k=-\infty}^{\infty}{|a_k|^2}
Discrete Time: 1Nn=<N>x[n]2=k=<N>ak2\textbf{Discrete Time: } \frac{1}{N}\sum_{n=<N>}{|x[n]|^2} = \sum_{k=<N>}{|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
xx
onto a set of basis functions, and the coefficients
aka_k
are the coordinates of our signal in the new space.

Discrete Time

Since in discrete time, signal is peroidic in
NN
, we can turn any it into a vector
xCN\vec{x}\in \mathbb{C}^N
.
x=[x[0]x[1]x[N1]]CN\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
ϕk\phi_k
form an orthogonal basis. If we take two of them
ϕk[n]\phi_k[n]
and
ϕM[n]\phi_M[n]
(
kMk\ne M
) and compute their dot product of their vector forms, then
ϕk[n]ϕM[n]=ϕMϕk=<n>ϕkM[n]=0\phi_k[n] \cdot \phi_M[n] = \phi_M^*\phi_k = \sum_{<n>}{\phi_{k-M}[n]} = 0
That means that
ϕk\phi_k
and
ϕM\phi_M
are orthogonal, and they are
NN
of them, therefore they are a basis. If we compute their magnitudes, we see
ϕkϕk=ϕk2=N,ϕk=N\phi_k \cdot \phi_k = ||\phi_k||^2 = N, \therefore ||\phi_k|| = \sqrt{N}
Finally, if we compute
xϕM\vec{x}\phi_M
where
x\vec{x}
is the vector form of an N-periodic signal,
xϕM=(i=0N1aiϕi)ϕM=Nam\vec{x}\cdot \vec{\phi_M} = \left(\sum_{i=0}^{N-1}{a_i\phi_i}\right)\cdot \phi_M = Na_m
am=1NxϕMa_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=ϕm2N = ||\phi_m||^2
. This gives a nice geometric intution for Parseval’s theorem
1Nx[n]2=1Nx2=ak2\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
ϕk(t)=ejk2πTt\phi_k(t) = e^{jk\frac{2\pi}{T}t}
for
k(,)k \in (-\infty, \infty)
Since we can’t convert continuous functions into vectors, these
ϕk\phi_k
are really a basis for the vector space of square integrable functions on the interval
[0,T][0, T]
. The inner product for this vector space is
<x,y>=0Tx(t)y(t).<x, y> = \int_{0}^{T}{x(t)y^*(t)}.
We can use this inner product to conduct the same proof as we did in discrete time.