Walsh function

Walsh functions, named after the mathematician Joseph L. Walsh, are a group of periodic mathematical functions that are used in digital signal processing. Orthogonal Walsh functions can be found in the Walsh transform, a variation of the Discrete Fourier Transform, Application, where they replace the trigonometric functions.

In the abstract framework of harmonic analysis, the Walsh functions are considered as characters of the Cantor group.

Definition

There are various functional systems of Walsh functions common. Most important are sequentially arranged Walsh functions, this arrangement has an analogy to the Fourier transform and the Walsh functions in a natural arrangement. Order, also referred to as " generalized frequency " expresses the number of zero crossings in the base interval [ 0,1]. To define one divides the interval [0,1] into equal subintervals. The sub-interval number can be expressed as a binary number with digits. An arrangement of the Walsh functions of order 0 to order in a natural arrangement forms a Hadamard matrix.

Walsh - Kaczmarz functions

The Walsh functions in sequential arrangement, also known as Walsh - Kaczmarz functions and as shown in the adjacent figure for 0-7 are [0,1] defined in the interval and continued periodically outside. In th sub- interval is the function value:

With:

The exclusive- OR operation is ( XOR). forms an orthonormal function system as the Kronecker delta, then:

Walsh - Paley functions

Walsh functions in the natural configuration, also known as Walsh functions Paley are easier to form, but have no analogy to the Fourier transform of. In th sub- interval is the function value:

With:

Properties

• The Walsh functions are reciprocal to itself
• The variables of the Walsh functions may be interchanged.
• The product of two Walsh functions yields a new Walsh function.

Application

Orthogonal functions play an important role in digital signal processing for the Signalapproximation. The Walsh functions are not harmonic functions (ie, rectangular) and thus very suitable to describe rectangular input signals. For this purpose, a finite number of Walsh functions are overlaid on the signal to be approximated. The difference of the integrals of signal and Walsh function returns the corresponding coefficients.