Talk:Video/The imaginary number i and the Fourier Transform

Trigonometry

Fourier Transform and the notion of $i$ (imaginary number), cannot be separated from Trigonometry.

Fourier Series

There are functions that are hard to work with, for example, discontinuous and fractal functions. On the other hand, there are functions that have wonderful properties for example sin, cos, and the linear function. In the Fourier series, we can use functions with wonderful properties to approximate functions that are hard to work with, for example, we use sin and cos to approximate fractal functions. Moreover, we can also approximate the functions by adding up functions together and the Fourier series will tell us what coefficient to use in our combination. However, you can only approximate a function on an interval.

Fourier transform

Fourier transform is the next level of the Fourier Series. Fourier transform comes up with a way to approximate functions on the hole real line by using exponentials. $e^{ikx}$

Fourier transforms integral equation : ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx,\quad \forall \ \xi \in \mathbb {R} .$

Example for Fourier transform: We have a signal called $x(t)$ in time we will represent it in terms of the time domain. We also can represent it in another way which is called $x(f)$ we will represent it in terms of the frequency domain and This is why we called transformation. Fourier Transform is an equivalent representation of the signal.

Convolution equation : $(f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .$