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FourierTransform

# Basics

Significance: The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions, with different frequencies.

## Derivation

### Fourier Series

Periodic Function
$$f(t+T) = f(t)$$ for the fundamental period $T$. Fundamental period $T$ means that it is the smallest possible $T\gt 0$, since $2T$, $3T$, $\ldots$ are all periods.

#### Sum of periodic functions are not always periodic

Suppose we have $f(t+T)=f(t)$ and $g(t+S)=g(t)$. If we want $f(t)+g(t)$

$$g(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos \left( \frac{2\pi mt}{T} \right) + \sum_{n=1}^{\infty} b_n \sin \left( \frac{2\pi nt}{T} \right)$$ $$g(t) = \sum_{n=0}^{\infty} a_n \cos \left( \frac{2\pi mt}{T} \right) + \sum_{n=1}^{\infty} b_n \sin \left( \frac{2\pi nt}{T} \right)$$

### Approximation

To approximate $f(t)$ with fourier series $g(t)$: $$a_0 = \frac{1}{T} \int_0^\pi f(t)dt$$ Correlate $f(t)$ with basis sine function: $$b_1 = \int_0^\pi f(t) \sin \left( \frac{2\pi t}{T} \right)$$

https://www.quora.com/Can-the-integral-displaystyle-int-x-x-mathrm-d-x-be-evaluated

Question: Is the sum of two periodic functions periodic?

No if there is irrational period. For example, $\cos(t)$ and $\cos(\sqrt{2}t)$ are each periodic, with periods $2\pi$ and $2\pi/\sqrt{2}$ respectively, but the sum $\cos(t) + \cos(\sqrt{2}t)$ is not periodic.