Skip to main content

Walsh functions

Walsh functions A complete set of functions that form an orthonormal basis for Walsh analysis: they take only the values +1 and –1, and are defined on a set of 2n points for some n. For purposes of computer representation, and also for their use in coding, it is usual to represent “+1” by “0”, and “–1” by “1”. As an example, the 8-point Walsh functions are then as follows: wal(8,0) = 00000000 wal(8,1) = 11110000 wal(8,2) = 00111100 wal(8,3) = 11001100 wal(8,4) = 10011001 wal(8,5) = 01101001 wal(8,6) = 01011010 wal(8,7) = 10101010

Note that the Walsh functions (usually denoted wal) consist alternatively of even and odd functions (usually denoted cal and sal by analogy with cos and sin). Furthermore, within the set of 2n functions there is one function of zero sequency, one of (normalized) sequency 2n–1, and one pair (odd and even) of each (normalized) sequency from 1 to 2n–1 – 1.

A set of Walsh functions corresponds, with some permutation of columns, to a Reed–Muller code and, with a column deleted, to a simplex code. See also Hadamard matrices.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"Walsh functions." A Dictionary of Computing. . 20 Mar. 2019 <>.

"Walsh functions." A Dictionary of Computing. . (March 20, 2019).

"Walsh functions." A Dictionary of Computing. . Retrieved March 20, 2019 from

Learn more about citation styles

Citation styles gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, cannot guarantee each citation it generates. Therefore, it’s best to use citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

The Chicago Manual of Style

American Psychological Association

  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.