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fwht - Fast Walsh–Hadamard transform

Syntax

y = fwht(x) y = fwht(x,n) y = fwht(x,n,ordering)

Description

y = fwht(x) returns the coefficients of the discrete Walsh–Hadamard transform of the input x. If x is a matrix, the FWHT is calculated on each column of x. The FWHT operates only on signals with length equal to a power of 2. If the length of x is less than a power of 2, its length is padded with zeros to the next greater power of two before processing.

y = fwht(x,n) returns the n-point discrete Walsh–Hadamard transform, where n must be a power of 2. x and n must be the same length. If x is longer than n, x is truncated; if x is shorter than n, x is padded with zeros.

y = fwht(x,n,ordering) specifies the ordering to use for the returned Walsh–Hadamard transform coefficients. To specify ordering, you must enter a value for the length n or, to use the default behavior, specify an empty vector [] for n. Valid values for ordering are the following strings:

Ordering	Description
`'sequency'`	Coefficients in order of increasing sequency value, where each row has an additional zero crossing. This is the default ordering.
`'hadamard'`	Coefficients in normal Hadamard order.
`'dyadic'`	Coefficients in Gray code order, where a single bit change occurs from one coefficient to the next.

For more information on the Walsh functions and ordering, see Walsh–Hadamard Transform.

Examples

This example shows a simple input signal and the resulting transformed signal.

x = [19 -1 11 -9 -7 13 -15 5];
y = fwht(x)
y =
     2     3     0     4     0     0    10     0

y contains nonzero values at these locations: 0, 1, 3, and 6. By forming the Walsh functions with the sequency values of 0, 1, 3, and 6, we can recreate x, as follows.

w0 = [1 1 1 1 1 1 1 1];
w1 = [1 1 1 1 -1 -1 -1 -1];
w3 = [1 1 -1 -1 1 1 -1 -1];
w6 = [1 -1 1 -1 -1 1 -1 1];

w = 2*w0 + 3*w1 + 4*w3 + 10*w6;
y1=fwht(w)
y1 =
     2     3     0     4     0     0    10     0

x1 = ifwht(y)
x1 =
    19    -1    11    -9    -7    13   -15     5

Algorithm

The fast Walsh-Hadamard tranform algorithm is similar to the Cooley-Tukey algorithm used for the FFT. Both use a butterfly structure to determine the transform coefficients. See the references for details.

References

[1] Beauchamp, K.G., Applications of Walsh and Related Functions, Academic Press, 1984.

[2] Beer, T., Walsh Transforms, American Journal of Physics, Volume 49, Issue 5, May 1981.