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{\*\generator Msftedit 5.41.21.2507;}\viewkind4\uc1\pard\tx720\tx1440\tx2160\cf1\f0\fs28 Born the son of an Auxerre tailor in 1768, Jean Baptiste Joseph Fourier lived to become one of France's greatest administrators, historians, and mathematicians. He graduated with honors from the military school in Auxerre, joined the staff of the Ecole Normale and then the Polytechnique in Paris. He went to Egypt with Napoleon, who made him Governor of Lower Egypt after the 1798 Expedition. After returning to France he was made Prefect at Grenoble in 1802 and then Baron in 1809, finally ending up as secretary of the Academie des Sciences in 1816 and Fellow in 1817. He contributed heavily to Description de l'Egypte, which covered the cultural and scientific results of Napoleon's invasion of Egypt, issuing 21 volumes over the period from 1808 to 1825. This work led to establishing Egyptology as a recognized new and separate discipline [13].\par
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{\*\generator Msftedit 5.41.21.2507;}\viewkind4\uc1\pard\tx720\tx1440\tx2160\cf1\b\f0\fs28 Today many areas of study benefit from Fourier analysis, however the method did not gain acceptability in Fourier's time. The reason behind this was the distrust of the use of series. One influential mathematician in 1828 commented: "Divergent series are the invention of the Devil, and it is shameful to base on them any demonstration whatsoever"[13]. Since then work by Dirichlet, Friedrich, Riemann, and others have resolved any doubts about the validity of the Fourier series.\par
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{\*\generator Msftedit 5.41.21.2507;}\viewkind4\uc1\pard\tx720\tx1440\tx2160\cf1\b\f0\fs32 The major setback in using the transform is computational time. A major tool used in the analysis, the discrete Fourier transform (DFT) requires extensive computational time to evaluate.Techniques to reduce the computational time were welcomed in such minimization studies as that conducted by C. Runge in 1903 which essentially described the FFT [13]. Danielson and Lanczos in 1942 recognized certain symmetries and periodicities which reduced the number of operations [13].\par
\par
Even with the advent of the digital computer the techniques to reduce computational time were generally unknown until 1965 when James W. Cooley and John W. Tukey published their mathematical algorithm which has become known as the fast Fourier transform (FFT) [13]. \fs20\par
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{\*\generator Msftedit 5.41.21.2507;}\viewkind4\uc1\pard\tx720\tx1440\tx2160\cf1\b\f0\fs32 http://me.oregonstate.edu/classes/me452/winter95/ButlerKeithMurphy/insth.html#A1.22\par
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{\*\generator Msftedit 5.41.21.2507;}\viewkind4\uc1\pard\cf1\lang1033\b\f0\fs20 public\cf2\b0 \cf1\b static\cf2\b0 \cf1\b void\cf2\b0 computeFFT(\cf1\b int\cf2\b0 sign, \cf1\b int\cf2\b0 n, \cf1\b double\cf2\b0 ar[], \cf1\b double\cf2\b0 ai[])\cf0\par
\cf2 \{\cf0\par
\cf2 \cf1\b double\cf2\b0 scale = 2.0 / (\cf1\b double\cf2\b0 ) n;\cf0\par
\cf2 \cf1\b int\cf2\b0 i, j;\cf0\par
\cf2 \cf1\b for\cf2\b0 (i = j = 0; i < n; ++i)\cf0\par
\cf2 \{\cf0\par
\cf2 \cf1\b if\cf2\b0 (j >= i)\cf0\par
\cf2 \{\cf0\par
\cf2 \cf1\b double\cf2\b0 tempr = ar[j] * scale;\cf0\par
\cf2 \cf1\b double\cf2\b0 tempi = ai[j] * scale;\cf0\par
\cf2 ar[j] = ar[i] * scale;\cf0\par
\cf2 ai[j] = ai[i] * scale;\cf0\par
\cf2 ar[i] = tempr;\cf0\par
\cf2 ai[i] = tempi;\cf0\par
\cf2 \}\cf0\par
\cf2 \cf1\b int\cf2\b0 m = n / 2;\cf0\par
\cf2 \cf1\b while\cf2\b0 ((m >= 1) && (j >= m))\cf0\par
\cf2 \{\cf0\par
\cf2 j -= m;\cf0\par
\cf2 m /= 2;\cf0\par
\cf2 \}\cf0\par
\cf2 j += m;\cf0\par
\cf2 \}\cf0\par
\cf2 \cf1\b int\cf2\b0 mmax, istep;\cf0\par
\cf2 \cf1\b for\cf2\b0 (mmax = 1, istep = 2 * mmax; mmax < n; mmax = istep, istep = 2 * mmax)\cf0\par
\cf2 \{\cf0\par
\cf2 \cf1\b double\cf2\b0 delta = sign * Math.\cf3\i PI\cf2\i0 / (\cf1\b double\cf2\b0 ) mmax;\cf0\par
\cf2 \cf1\b for\cf2\b0 (\cf1\b int\cf2\b0 m = 0; m < mmax; ++m)\cf0\par
\cf2 \{\cf0\par
\cf2 \cf1\b double\cf2\b0 w = m * delta;\cf0\par
\cf2 \cf1\b double\cf2\b0 wr = Math.\i cos\i0 (w);\cf0\par
\cf2 \cf1\b double\cf2\b0 wi = Math.\i sin\i0 (w);\cf0\par
\cf2 \cf1\b for\cf2\b0 (i = m; i < n; i += istep)\cf0\par
\cf2 \{\cf0\par
\cf2 j = i + mmax;\cf0\par
\cf2 \cf1\b double\cf2\b0 tr = wr * ar[j] - wi * ai[j];\cf0\par
\cf2 \cf1\b double\cf2\b0 ti = wr * ai[j] + wi * ar[j];\cf0\par
\cf2 ar[j] = ar[i] - tr;\cf0\par
\cf2 ai[j] = ai[i] - ti;\cf0\par
\cf2 ar[i] += tr;\cf0\par
\cf2 ai[i] += ti;\cf0\par
\cf2 \}\cf0\par
\cf2 \}\cf0\par
\cf2 mmax = istep;\cf0\par
\cf2 \}\cf0\par
\pard\tx720\tx1440\tx2160\cf2 \}\par
}
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6330a3b2-e522-4b98-bda9-0dfa4d842fe4