Use of the 1-D Hadamard Transform in the Computation of a 2-D Hadamard Transform

Abstract

Let X be an N(= 2ⁿ) × M (= 2^m) dimensional data matrix and let Y be the N × M Hadamard transform matrix as

$$ \begin{gathered} X = \left( \begin{gathered} {x_{{11}}}\quad {x_{{12}}}\quad \cdots \quad {x_{{12}}} \hfill \\ {x_{{21}}}\quad {x_{{22}}}\quad \cdots \quad {x_{{12}}} \hfill \\ \;\, \vdots \quad \quad \vdots \quad \;\;\; \vdots \quad \;\; \vdots \hfill \\ {x_{{N1}}}\quad {x_{{N2}}}\;\; \cdots \quad {x_{{NM}}} \hfill \\ \end{gathered} \right) \hfill \\ \quad = ({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_1}\quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_2}\quad \cdots \quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}}}_M}) \hfill \\ \end{gathered} $$

(89)

and

$$\begin{gathered} Y = \left( \begin{gathered} {y_{{11}}}\quad {y_{{12}}}\quad \cdots \quad {y_{{12}}} \hfill \\ {y_{{21}}}\quad {y_{{22}}}\quad \cdots \quad {y_{{12}}} \hfill \\ \;\, \vdots \quad \quad \vdots \quad \;\;\; \vdots \quad \;\; \vdots \hfill \\ {y_{{N1}}}\quad {y_{{N2}}}\;\; \cdots \quad {y_{{NM}}} \hfill \\ \end{gathered} \right) \hfill \\ \quad = ({{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y}}}_1}\quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y}}}_2}\quad \cdots \quad {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y}}}_M}) \hfill \\ \end{gathered} $$

(90)

where {x _i } and {y _i } are N-dimensional column vectors. The 2-D Hadamard transform is given by

$$ Y = {H_n}X\,{H_m} $$

(91)

where H _n and H _m are 2ⁿ × 2ⁿ and 2^m × 2^m Hadamard matrices. Since n may not necessarily be equal to m, we will use superscripts (n) and (m) to denote the columns h _i ⁽ⁿ⁾ and h _i ^(m) in H _n and H _m respectively.