Walsh Function

Functions consisting of a number of fixed-amplitude square pulses interposed with zeros. Following Harmuth (1969), designate those with Even symmetry $\mathop{\rm Cal}\nolimits (k,t)$ and those with Odd symmetry $\mathop{\rm Sal}\nolimits (k,t)$. Define the Sequency $k$ as half the number of zero crossings in the time base. Walsh functions with nonidentical Sequencies are Orthogonal, as are the functions $\mathop{\rm Cal}\nolimits (k,t)$ and $\mathop{\rm Sal}\nolimits (k,t)$. The product of two Walsh functions is also a Walsh function. The Walsh functions are then given by

$$\mathop{\rm Wal}(k,t)=\cases{ \mathop{\rm Cal}\nolimits (k/... ...op{\rm Sal}\nolimits ((k+1)/2,t) & for $k=1$, 3, 5, \dots.\cr}$$

The Walsh functions Cal($k, t$) for $k=0$, 1, ..., $n/2-1$ and $\mathop{\rm Sal}\nolimits (k,t)$ for $k=1$, 2, ..., $n/2$ are given by the rows of the Hadamard Matrix ${\hbox{\sf H}}_n$.

See also Hadamard Matrix, Sequency

References

Beauchamp, K. G. Walsh Functions and Their Applications. London: Academic Press, 1975.

Harmuth, H. F. ``Applications of Walsh Functions in Communications.'' IEEE Spectrum 6, 82-91, 1969.

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, p. 204, 1986.

Tzafestas, S. G. Walsh Functions in Signal and Systems Analysis and Design. New York: Van Nostrand Reinhold, 1985.

Walsh, J. L. ``A Closed Set of Normal Orthogonal Functions.'' Amer. J. Math. 45, 5-24, 1923.