Computes the real Walsh Hadamard transform of a sequence.

## x

The input sequence.

The length of the sequence has to be a power of 2, otherwise this node returns an error.

## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error

The Walsh Hadamard transform of the input sequence.

## error out

Error information. The node produces this output according to standard error behavior.

## Algorithm Definition

The Walsh Hadamard transform is based on an orthogonal system consisting of functions of only two elements -1 and 1. For the special case of n = 4, the Walsh Hadamard transform of the signal $X=\left\{{x}_{0},{x}_{1},{x}_{2},{x}_{3}\right\}$ can be noted in the following matrix form:

$WH\left\{X\right\}=\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& 1& -1\\ 1& 1& -1& -1\\ 1& -1& -1& 1\end{array}\right]\left[\begin{array}{c}{\begin{array}{c}x\end{array}}_{0}\\ {x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]$

If WHn and WHn+1 denote the Walsh Hadamard matrices of dimension 2n and 2n + 1 respectively, the rule is

${WH}_{n+1}=\left[\begin{array}{cc}{WH}_{n}& {WH}_{n}\\ {WH}_{n}& {-WH}_{n}\end{array}\right]$

where -WHn is meant in the element wise sense.

Note

The Walsh Hadamard transform fulfills the Convolution Theorem: WH{X*Y} = WH{X}WH{Y}.

## Comparing the Walsh Hadamard Transform with the Fourier transform

The Walsh Hadamard transform has similar properties to the Fourier transform, but the computational effort is considerably smaller.

The following diagram shows the Walsh Hadamard transform of a pulse pattern signal of length 256, delay 32, and width 64.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported