Derivation of Fourier Coefficients Equation
2.5
2002/07/11
2003/11/04 11:52:03 US/Central
Michael
Haag
mjhaag@rice.edu
Prashant
Singh
prash@ece.rice.edu
Mariyah
Poonawala
mariyah@rice.edu
exponential
fourier
fourier coefficients
fourier series
signals
sinusoid
The module goes through the steps of deriving the Fourier coefficient equation from the general Fourier series equation.
Introduction
You should already be familiar with the existence of the
general Fourier
Series equation, which is written as:
f
t
n
c
n
ω
0
n
t
What we are interested in here is how to determine the Fourier
coefficients,
c
n
, given a function
f
t
. Below we will walk through the steps of deriving
the general equation for the Fourier coefficients of a given
function.
Derivation
To solve for
c
n
, we have to do a little algebraic manipulation.
First of all we will multiply both sides of by
ω
0
k
t
, where
k
.
f
t
ω
0
k
t
n
c
n
ω
0
n
t
ω
0
k
t
Now integrate both sides over a given period,
T:
t
T
0
f
t
ω
0
k
t
t
T
0
n
c
n
ω
0
n
t
ω
0
k
t
On the right-hand side we can switch the summation and
integral along with pulling out the constant out of the
integral.
t
T
0
f
t
ω
0
k
t
n
c
n
t
T
0
ω
0
n
k
t
Now that we have made this seemingly more complicated, let us
focus on just the integral,
t
T
0
ω
0
n
k
t
, on the right-hand side of the above equation.
For this integral we will need to consider two cases:
n
k
and
n
k
. For
n
k
we will have:
n
n
k
t
T
0
ω
0
n
k
t
T
For
n
k
, we will have:
n
n
k
t
T
0
ω
0
n
k
t
t
T
0
ω
0
n
k
t
t
T
0
ω
0
n
k
t
But
ω
0
n
k
t
has an integer number of periods,
n
k
, between 0 and
T. Imagine a graph of the
cosine; because it has an integer number of periods, there are
equal areas above and below the x-axis of the graph. This
statement holds true for
ω
0
n
k
t
as well. What this means is
t
T
0
ω
0
n
k
t
0
as well as the integral involving the sine function.
Therefore, we conclude the following about our integral of
interest:
t
T
0
ω
0
n
k
t
T
n
k
0
Now let us return our attention to our complicated equation,
, to see if we can finish
finding an equation for our Fourier coefficients. Using the
facts that we have just proven above, we can see that the only
time will have a nonzero
result is when k and
n are equal:
n
n
k
t
T
0
f
t
ω
0
n
t
T
c
n
Finally, we have our general equation for the Fourier
coefficients:
c
n
1
T
t
T
0
f
t
ω
0
n
t
Finding Fourier Coefficients Steps
To find the Fourier coefficients of periodic
f
t
:
-
For a given k, multiply
f
t
by
ω
0
k
t
, and take the area under the curve (dividing
by T).
-
Repeat step (1) for all
k
.