Fourier Series - The Fourier Coefficients

What are the optimal Fourier coefficients (a_m, b_n) of equation [1]? Let's start by looking at a one term approximation of the above:
Here we have a one-term approximation of the function. What is the best value for a0 that we can choose in this
case? Without proof (which will come later), the optimal value is:
How do we get this? Well, to answer intuitively, the integral of the function over the period is a formal mathematical
way of writing "the average value". So the first term in the Fourier series is a constant, and it is the average value
of the function. For the square wave of Figure 1 on the previous page, the average value is
0.5, and the one term expansion along with the function is shown in Figure 2:
What is the optimal value of b1? Intuitively, we want to know how correlated the function f(t) is with
sin(2*pi*t/T). In a sense, we want to correlate the function f(t) and this basis sine function. It turns out
the optimal value for b1 is given by:
This is the mathematical way of writing the correlation between two functions over the interval [0, T] (the derivation for the factor of 2 will
be shown later). Working through
the math of the above equation, the optimal value
for b1 is:
The first two terms from equation [4], evaluated with the optimal values for a0 and b1 now add together to produce:
Instead of going term by term, let's give the optimal coefficients for all am and bn:
All of these represent the correlation of the function f(t) with the basis sine and cosine functions.
For the specific square-wave f(t) of Figure 1, the optimal values come out to be:
The first three non-zero term expansion (a0, b1, b3) is given in Figure 4:
The main takeaway point from this page is to understand that any periodic function (it should be somewhat continuous) can be represented
by the sum of sinusoidal functions, each with a frequency some integer multiple of the fundamental period (T).
In the next section, we'll look at the complex form of the Fourier Series.

We want to approximate a periodic function f(t), with fundamental period *T*, with the Fourier Series:
[Equation 1]
[Equation 2]
[Equation 3]
**Figure 2. The square waveform and the one term (constant) expansion.**
[Equation 4]
[Equation 5]
[Equation 6]
**Figure 3. The square waveform and the two term expansion.**
*[Equation 7]*
**Figure 4. The square waveform and the three term expansion.****Figure 5. The square waveform and the seven term expansion.**

Next: The Complex Fourier Series Coefficients

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