Fourier Series

Youtube Video: example 1, example 2,

We use Fourier series for any periodic signals ( it depends) and decompose the signal in to infinitive sum of sinusoidal. This is easier to present mathmatically and graphically. More importantly the use of the Fourier series is to understand how periodic signal can sum up to.

Condition for Fourier series:

Existence Condition of Fourier Series

• A periodic signal x(t) has a Fourier series if it satisfies the Dirichlet condition given by

1. x(t) is absolutely integrable over any period; that sum of integration need to be less than infinity.

2. x(t) has only a finite number of maximum and minimum over any period

3. x(t) has only a finite number of discontinuities over any period

Convergence of Fourier Series

• If x(t) satisfies the above condition, x(t) can be Fourier series representation.

• At these values, the Fourier series converges to the average of the values on either side of the discontinuity

Gibbs phenomenon.

If x(t) is discontinuous, xN(t) exhibits ripples in the vicinity of the discontinuity and the peak amplitude of these ripples does not decrease with increasing.

Before that learn  complex number

Some summary of Fourier series properties By convention, the coefficients of the cosine components are labeled "a", and the coefficients of the sine components are labeled with a "b". A few important facts can then be mentioned:
 * If the function has a DC offset, a0 will be non-zero. There is no B0 term.
 * If the signal is even, all the b terms are 0 (no sine components).
 * If the signal is odd, all the a terms are 0 (no cosine components).
 * If the function has half-wave symmetry, then all the even coefficients (of sine and cosine terms) are zero, and we only have to integrate half the signal.
 * If the function has quarter-wave symmetry, we only need to integrate a quarter of the signal.
 * The Fourier series of a sine or cosine wave contains a single harmonic because a sine or cosine wave cannot be decomposed into other sine or cosine waves.
 * We can check a series by looking for discontinuities in the signal or derivative of the signal. If there are discontinuities, the harmonics drop off as 1/n, if the derivative is discontinuous, the harmonics drop off as 1/n2.

Youtube video for complex exponential Fourier series

LTI response to exponential
Using Eulers Equation:

Convert the standard Rectangular Fourier Series into an exponential form. Even though complex numbers are a little more complicated to comprehend, we use this form for a number of reasons:

Steps to do

Construct the exponential series from the rectangular series using Euler's formulae: Example Substituting Euler's formulae:
 * 1) Only need to perform one integration
 * 2) A single exponential can be manipulated more easily than a sum of sinusoids
 * 3) It provides a logical transition into a further discussion of the Fourier Transform.
 * Substittue.png

Splitting into "positive n" and "negative n" parts gives us:

Finally collapse this into a single expression:

Side note:

Examples and Practice Problems
Practice problem Solution

MIT practice problem, Solution

Complex exponential practice problem Solution

Very useful website and a lot very good example from this website : http://tutorial.math.lamar.edu/Classes/DE/FourierSeries.aspx

Extra Resources
Continuous Fourier series : MIT course and video