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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 the 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 the ordering are the following strings:

OrderingDescription
'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

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### Walsh-Hadamard Transform of a Signal

This example shows a simple input signal and its Walsh-Hadamard transform.

```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 locations 0, 1, 3, and 6. Form the Walsh functions with the sequency values 0, 1, 3, and 6 to recreate x.

```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 = y(0+1)*w0 + y(1+1)*w1 + y(3+1)*w3 + y(6+1)*w6
```
```w =

19    -1    11    -9    -7    13   -15     5

```

## More About

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### Algorithms

The fast Walsh-Hadamard transform 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, Kenneth G. Applications of Walsh and Related Functions: With an Introduction to Sequency Theory. London: Academic Press, 1984.

[2] Beer, Tom. "Walsh Transforms." American Journal of Physics. Vol. 49, 1981, pp. 466–472.

## See Also

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