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Dirichlet Conditions

Module by: Ricardo Radaelli-Sanchez. E-mail the author

Summary: The Dirichlet conditions are the sufficient conditions to guarantee existence and convergence of the Fourier series or the Fourier transform.

Named after the German mathematician, Peter Dirichlet, the Dirichlet conditions are the sufficient conditions to guarantee existence and convergence of the Fourier series or the Fourier transform.

The Weak Dirichlet Condition for the Fourier Series

Condition 1: The Weak Dirichlet Condition

For the Fourier Series to exist, the Fourier coefficients must be finite. The Weak Dirichlet Condition guarantees this existence. It essentially says that the integral of the absolute value of the signal must be finite. The limits of integration are different for the Fourier Series case than for the Fourier Transform case. This is a direct result of the differing definitions of the two.

Proof

The Fourier Series exists (the coefficients are finite) if

Weak Dirichlet Condition for the Fourier Series
∫0T|f⁢t|d t <∞ t 0 T f t
(1)
This can be shown from the initial condition that the Fourier Series coefficients be finite.
| c n |=|1T⁢∫0Tf⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t |≤1T⁢∫0T|f⁢t|⁢|e−(i⁢ ω 0 ⁢n⁢t)|d t c n 1 T t 0 T f t ω 0 n t 1 T t 0 T f t ω 0 n t
(2)
Remembering our complex exponentials, we know that in the above equation |e−(i⁢ ω 0 ⁢n⁢t)|=1 ω 0 n t 1 , which gives us
1T⁢∫0T|f⁢t|d t =1T⁢∫0T|f⁢t|d t 1 T t 0 T f t
(3)
∞
(4)

Note:

If we have the function: ∀ t ,0<t≤T:f⁢t=1t t 0 t T f t 1 t then you should note that this functions fails the above condition.

The Weak Dirichlet Condition for the Fourier Transform

Condition 2

The Fourier Transform exists if

Weak Dirichlet Condition for the Fourier Transform
∫−∞∞|f⁢t|d t <∞ t f t
(5)
This can be derived the same way the weak Dirichlet for the Fourier Series was derived, by beginning with the definition and showing that the Fourier Transform must be less than infinity everywhere.

The Strong Dirichlet Conditions

The Fourier Transform exists if the signal has a finite number of discontinuities and a finite number of maxima and minima. For the Fourier Series to exist, the following two conditions must be satisfied (along with the Weak Dirichlet Condition):

  1. In one period, f⁢t f t has only a finite number of minima and maxima.
  2. In one period, f⁢t f t has only a finite number of discontinuities and each one is finite.
These are what we refer to as the Strong Dirichlet Conditions. In theory we can think of signals that violate these conditions, sinlogt t for instance. However, it is not possible to create a signal that violates these conditions in a lab. Therefore, any real-world signal will have a Fourier representation.

Example

Let us assume we have the following function and equality:

f′t=limit   N → ∞ d f N ⁢td f t N f N t
(6)
If f⁢t f t meets all three conditions of the Strong Dirichlet Conditions, then f⁢τ=f′τ f τ f τ at every ττ at which f⁢t f t is continuous. And where f⁢t f t is discontinuous, f′t f t is the average of the values on the right and left. See Figure 1 as an example:

Figure 1: Discontinuous functions, f⁢t f t .
(a) (b)
Figure 1(a) (dircond1.png)Figure 1(b) (dircond2.png)

Note:

The functions that fail the Dirchlet conditions are pretty pathological - as engineers, we are not too interested in them.

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