Properties of Walsh Functions and the Walsh Transform

Geophysics ◽  
1990 ◽  
Vol 55 (7) ◽  
pp. 843-850 ◽  
Author(s):  
R. K. Shaw ◽  
B. N. P. Agarwal

Walsh functions are a set of complete and orthonormal functions of nonsinusoidal waveform. In contrast to sinusoidal waveforms whose amplitudes may assume any value between −1 to +1, Walsh functions assume only discrete amplitudes of ±1 which form the kernel function of the Walsh transform. Because of this special nature of the kernel, computation of the Walsh transform of a given signal is simpler and faster than that of the Fourier transform. The properties of the Fourier transform in linear time are similar to those of the Walsh transform in dyadic time. The Fourier transform has been widely used in interpretation of geophysical problems. Considering various aspects of the Walsh transform, an attempt has been made to apply it to some gravity data. A procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources, viz., a sphere, a horizontal cylinder, and a 2-D vertical prism of large depth extent. The technique has been applied to data from the published literature to evaluate its applicability, and the results are in good agreement with the more conventional ones.


2010 ◽  
Vol 42 (4) ◽  
pp. 37-65
Author(s):  
Andrey N. Tereshchenko ◽  
Svetlana S. Melnikova ◽  
Lev A. Hnativ ◽  
Valeriy K. Zadiraka ◽  
Natalya V. Koshkina
Keyword(s):  

1972 ◽  
Vol C-21 (2) ◽  
pp. 187-189 ◽  
Author(s):  
Anthony C. Davies
Keyword(s):  

1981 ◽  
Vol 71 (4) ◽  
pp. 1351-1360
Author(s):  
Tom Goforth ◽  
Eugene Herrin

abstract An automatic seismic signal detection algorithm based on the Walsh transform has been developed for short-period data sampled at 20 samples/sec. Since the amplitude of Walsh function is either +1 or −1, the Walsh transform can be accomplished in a computer with a series of shifts and fixed-point additions. The savings in computation time makes it possible to compute the Walsh transform and to perform prewhitening and band-pass filtering in the Walsh domain with a microcomputer for use in real-time signal detection. The algorithm was initially programmed in FORTRAN on a Raytheon Data Systems 500 minicomputer. Tests utilizing seismic data recorded in Dallas, Albuquerque, and Norway indicate that the algorithm has a detection capability comparable to a human analyst. Programming of the detection algorithm in machine language on a Z80 microprocessor-based computer has been accomplished; run time on the microcomputer is approximately 110 real time. The detection capability of the Z80 version of the algorithm is not degraded relative to the FORTRAN version.


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