Cosine transforf

Abstract

Abstract COSE TRANSFORMS DEFINED, THE FOURIER TRANSFORM AT ANY FREQUENCY MAY BE COMPUTED. THE FOURIER TRANSFORM OF A NON-PERIODIC WAVE IS CALCULATED BY PROGRAMMED COMPUTER USING MATRIX MULTIPLICATION. THE NON-PERIODIC WAVE TO BE TRANSFORMED IS CONVERTED TO A STAIRCASE WAVEFORM WHERE THE AMPLITUDE OF EACH STEP IN THE STAIRCASE WAVEFORM IS THE AVERAGE AMPLITUDE OF THE ORIGINAL WAVE. WALSH COEFFICIENTS OF THE STAIRCASE WAVEFORM ARE THEN COMPUTED BY MATRIX MULTIPLICATION OF THE WALSH MATRIX TIMES SAMPLE VALUES FOR THE WAVE WHERE THE SAMPLE VALUES ARE THE VARIOUS AMPLITUDES OF THE STAIRCASE WAVE DURING SEPARATE STEP OR SAMPLE INTERVALS. THEREAFTER, THE CONTINUOUS FOURIER TRANSFORM OF A NON-PERIODIC WAVE IS CALCULATED BY COMPUTING THE FOURIER TRANSFORM OF THE WALSH FUNCTIONS. THE LATTER COMPUTATION IS SIMPLIFIED BY MULTIPLYING THE WALSH COEFFICIENTS BY A SINCE ARRAY AND A COSINC (DEFINED HEREINAFTER) ARRAY. A FURTHER AND ULTIMATE SIMPLIFICATION OF THE ENTIER PROCESS IS ACCOMPLISHED BY MULTIPLYING THE WALSH MATRIX BY AN INTERMINGLING OF THE SINC AND COSINC ARRAYS. THE RESULT IS A MATRIX CONSISTING MOSTLY OF ZEROES AND HEREINAFTER DEFINED AS THE SINC MATRIX. MULTIPLICATION OF THE INPUT WAVE VALUES BY THE SINC MATRIX WILL YIELD DIRECTLY SINE/COSINE COEFFICIENTS OF THE FOURIER SINE/COSINE TRANSFORMS FOR THE STAIRCASE WAVEFORM THAT SIMULATES THE INPUT WAVE WITH THE COEFFICIENTS OF THE FOURIER SINE/

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