The Fast Fourier Transform and the Hadamard Transform

Abstract

The discrete Fourier transform of a set of data, say x ₀, x _1,…, x _{N -1} is given by the transform coefficients X ₀, X ₁,…, X _{N -1} by the relation

$$ \left( \begin{gathered} \;\;{X_0} \hfill \\ \;\;{X_1} \hfill \\ \;\;{X_2} \hfill \\ \;\;\;\; \vdots \hfill \\ {X_{{N - 1}}} \hfill \\ \end{gathered} \right) = \left( \begin{gathered} 1\quad \quad 1\quad \quad \quad \quad 1\quad \quad \ldots \quad \quad \quad 1 \hfill \\ 1\quad \quad W\quad \quad \quad {W^2}\quad \;\, \ldots \quad \quad \;{W^{{N - 1}}}\quad \hfill \\ 1\quad \quad {W^2}\quad \quad {W^4}\quad \quad \ldots \quad \quad {W^{{2(N - 1)}}} \hfill \\ \vdots \quad \quad \quad \quad \quad \quad \quad \quad \quad \vdots \hfill \\ 1\quad {W^{{N - 1}}}\quad {W^{{2(N - 1)}}}\quad \;\; \ldots \quad {W^{{(N - 1)(N - 1)}}}\quad \hfill \\ \end{gathered} \right)\left( \begin{gathered} \;\;{x_0} \hfill \\ \;\;{x_1} \hfill \\ \;\;{x_2} \hfill \\ \;\;\; \vdots \hfill \\ \;{x_{{N - 1}}} \hfill \\ \end{gathered} \right) $$

(100)

where $W = {e^{{ - j\frac{{2\pi }}{N}}}}$ In symbolic form, (100) can be written as

$$ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}} = {A_N}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}} $$

(101)

and the inverse transform as

$$ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}} = \frac{1}{N}A_N^{*}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}} $$

(102)

where ‘*’ denotes conjugation, and a typical entry, a _ik , in A _N is given by

$$ {a_{{ik}}} = {W^{{(i - 1)(k - 1)}}},1 \leqslant i,k \leqslant N $$

(103)