Dirichlet conditions

Dirichlet conditions guarantee that a periodic function $x(t)$ can be exactly represented by its Fourier transform.

Readings

• Wikipedia: Dirichlet conditions

Conditions

Condition 1

The function must be absolutely integrable over a single period $T$. This is equivalent to the statement that the area enclosed between the abcissa and the function is finite over a single period.

$\underset{T}{\int }|x(t)|<\mathrm{\infty }$

Condition 2

Given any finite period of time the number of local maxima and minima of $x(t)$ within that period is finite.

Condition 3

Given any finite period of time there is a finite number of discontinuities in the function $x(t)$