Walsh function

from Wikipedia, the free encyclopedia

Walsh functions, named after mathematician Joseph L. Walsh , are a group of periodic mathematical functions used in digital signal processing . Orthogonal Walsh functions are used in the context of the Walsh transformation , a variation of the discrete Fourier transformation , where they replace the trigonometric functions .

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

definition

Walsh functions in a sequential arrangement (Walsh-Kaczmarz) of order 0 to 7 in the interval [0.1] (in red), in light blue for comparison of the real part of the Fourier functions

Different systems of Walsh functions are common. The sequentially arranged Walsh functions are important ; this arrangement has an analogy to the Fourier transform, and the Walsh functions in a natural arrangement . The order , also called “generalized frequency”, expresses the number of zero crossings in the base interval [0.1]. For the definition one divides this interval [0,1] into equally long sub-intervals. The sub-interval number can be expressed as a binary number with digits. An arrangement of the Walsh functions from order 0 to order in a natural order forms a Hadamard matrix .

Walsh-Kaczmarz functions

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

With:

where represents the exclusive-or-link (XOR). forms into an orthonormal system of functions , since with the Kronecker delta the following applies:

Walsh-Paley functions

The Walsh functions in a natural arrangement, also known as Walsh-Paley functions, are easier to form, but do not have any analogy to the Fourier transform. In the -th sub-interval the function value is:

With:

properties

  • The Walsh functions are reciprocal to themselves.
  • The variables of the Walsh functions can be exchanged.
  • The product of two Walsh functions gives a new Walsh function.

application

Orthogonal functions play an important role in digital signal processing for signal approximation. The Walsh functions are non-harmonic functions (i.e. rectangular) and are therefore very well suited to describing rectangular input signals. For this purpose, a finite number of Walsh functions are placed over the signal to be approximated. The difference between the integrals of the signal and the Walsh function gives the corresponding coefficient.

literature

  • Eugen Gauß: Walsh functions for engineers and natural scientists . Teubner, 1994, ISBN 3-519-02099-8 .

Web links