Using this and the fact that
is linear, calculating
for combinations of complex exponentials is also
The action of on an input such
as those in the two equations above is easy to explain.
scales each exponential component
by a different complex number
. As such, if we can write a function
as a combination of complex exponentials it allows us to easily calculate the output of a system.
Fourier Series Synthesis
Fourier demonstrated that an arbitrary
can be written as a linear combination of harmonic
is the fundamental frequency. For almost all
of practical interest, there exists
to make Equation 2 true. If
is finite energy (
), then the equality in Equation 2
holds in the sense of energy convergence; if
is continuous, then Equation 2 holds
pointwise. Also, if
meets some mild conditions (the Dirichlet
conditions), then Equation 2 holds
pointwise everywhere except at points of discontinuity.
- called the Fourier coefficients -
tell us "how much" of the sinusoid
The formula shows
as a sum of complex exponentials, each of which is easily processed by an
LTI system (since it is an eigenfunction of
every LTI system). Mathematically,
it tells us that the set of
form a basis for the space of T-periodic continuous
We know from Euler's formula that
Synthesis with Sinusoids Demonstration
Guitar Oscillations on an iPhone
Fourier Series Analysis
Finding the coefficients of the Fourier series expansion involves some algebraic manipulation of the synthesis formula.
First of all we will multiply both sides of the equation by
Now integrate both sides over a given period,
On the right-hand side we can switch the summation and
integral and factor the constant out of the
Now that we have made this seemingly more complicated, let us
focus on just the integral,
, on the right-hand side of the above equation.
For this integral we will need to consider two cases:
we will have:
, we will have:
has an integer number of periods,
, between and
. 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
as well. What this means is
which also holds for the integral involving the sine function.
Therefore, we conclude the following about our integral of
Now let us return our attention to our complicated equation,
Equation 5, 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 Equation 5 will have a nonzero
result is when and
Finally, we have our general equation for the Fourier
Consider the square wave function given by
on the unit interval .
Thus, the Fourier coefficients of this function found using the Fourier series analysis formula are
Fourier Series Summary
Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The continuous time Fourier series synthesis formula expresses a continuous time, periodic function as the sum of continuous time, discrete frequency complex exponentials.
The continuous time Fourier series analysis formula gives the coefficients of the Fourier series expansion.
In both of these equations
is the fundamental frequency.