A Simple Example

Consider a signal vector of N=4 elements (samples):

$${\bf x}=[x_0, x_1, x_2, x_3]^T=[0, 1, 2, 3]^T$$

The corresponding 4 by 4 WHT matrix (Hadamard ordered) is:

$${\bf H}_h=\frac{1}{2} \left[ \begin{array}{rrrr} 1 & 1 & 1 & ... ...& -1 1 & 1 & -1 & -1 1 & -1 & -1 & 1 \end{array} \right]$$

The rows (or colums) of this matrix can be reordered according to their sequency by the following mapping:

$$\begin{tabular}{ccccc} sequency order & 0 & 1 & 2 & 3 \\ bin... ...10 & 11 & 01 \\ Hadamard order & 0 & 2 & 3 & 1 \end{tabular}$$

to get the sequency ordered (also called Walsh ordered) matrix

$${\bf H}_w=\frac{1}{2} \left[ \begin{array}{rrrr} 1 & 1 & 1 & ... ...1 & 1 & -1 \end{array} \right]=\frac{1}{2}[h_0, h_1, h_2, h_3]$$

where ${\bf h}_i$ is the ith colum (or row) vector of the symmetric matrix ${\bf H}_w^T={\bf H}_w$ representing a square wave of sequency $i$ (with $i$ zero-crossings). Now the sequency spectrum of the signal can be found as

$${\bf X}={\bf H}_w {\bf x} =\frac{1}{2} \left[ \begin{array}{r... ...\left[ \begin{array}{r} 3 -2 0 -1 \end{array} \right]$$

and the inverse transform (note $H_w^{-1}=H_w$) represents the signal vector as a linear combination of a set of square waves of different sequencies:

$${\bf x}={\bf H}_w {\bf X}=\frac{1}{2}[h_0, h_1, h_2, h_3] ... ... -2 0 -1 \end{array} \right] =\frac{3h_0-2h_1-h_3}{2}$$

We can also verify that indeed the inverse transform will produce the original signal from its spectrum:

$${\bf x}={\bf H}_w {\bf X}=\frac{1}{2}\left[ \begin{array}{rrr... ... =\left[ \begin{array}{c} 0 1 2 3 \end{array} \right]$$