Design and implementation of the Walsh–Hadamard transform on a ternary optical computer
Zhehe Wang, Yunfu Shen
Abstract
Walsh–Hadamard is a commonly used mathematical transformation, a mathematical method for spectrum analysis in the real number domain, which has been widely used in many fields such as digital signal processing, spectral modulation, and image processing. Based on the characteristics of the ternary optical computer with a large number of processor bits, reconfigurable bit functions, and parallel computation, a ternary optical processor with all bits allocated is given, and the transformations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>W</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">′</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">′</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> of modified signed-digit (MSD) addition are reconfigured. On this basis, by configuring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>A</mml:mi> <mml:mi>d</mml:mi> <mml:mi>d</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> <mml:mi>N</mml:mi> </mml:math> MSD adders on the optical processor, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> -point Walsh–Hadamard transform fast parallel computing approach is given. It requires only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>3</mml:mn> <mml:mo>×</mml:mo> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mi>d</mml:mi> <mml:mi>d</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:math> clock cycles to compute the Walsh–Hadamard transform of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> points under parallel computing. Furthermore, complexity analysis shows that the clock cycles in an electronic computer are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:math> times that of in a ternary optical computer under the fully parallel optical computation. The experiment of the Walsh–Hadamard transform with eight points shows that parallel optical computing has more advantages than the conventional electronic computer in fast computing the problems with characteristics of parallel computing. At the same time, it highlights the advantages and potential of ternary optical computers in data-intensive computing.