1.Impulse response
Joseph Fourier first represented Fourier integral theorem in the following DOE:
[1]
Which is then introduced into the first delta function as following:
[1]
And the end end up with what mathematicians called Dirac delta function:
[1]
2.Fourier series
The input x(t) is a function with a fundamental period x(t)= 1 from x= 0 to 1 and f(x)= -1 to 0, with a discontinuity at x=0. The following graphs from matlab represents Gibbs phenomena, as n increases the overshot decreases.
3.Filters
The upper is the Gaussian filter, while bottom is the unsharp.
