History

The Fast Fourier Transform (FFT) is a modern algorithm to compute the Fourier coefficients of a finite sequence. Fourier will forever be known by his assertion in 1807 that any function could be expressed as a linear combination of sines and cosines, its Fourier series. "Any" was a little ambitious, counter-examples coming to the fore in due time. A fair amount of mathematics from that time to this has been devoted to refining Fourier's insight and studying trigonometric series, a subject that led Georg Cantor to founding set theory. Piecewise smoothness is sufficient for pointwise convergence on \( [-\pi, \pi] \):

\[ f(x) = {a_0 \over 2} + \sum_{j=1}^\infty \left( a_j \cos jx + b_j \sin jx \right), \]

My original calendar program was on the Casio fx-3600p calculator in 1980 or so - my first programming venue and exercised partly in spare moments when driving truck out on the route; a precursor mobile device you could say. My buddy Dave got me started. I might have scarred myself permanently though. The transition from math to software engineering is always tricky, considering that there are many commonalities, but just as important differences to snag the self-taught and perhaps obstinate and all-too-confident mathematician (perish the thought). The 3600 had this bizarre little macro language providing for a trade-off between memory and program size. You could have (say) fifty memory locations and 400 instructions or twenty locations and 600 instructions. They're really variables of course, but the memory locations were designated K0 through K19 or something.

Two works by Americans are canonical references for anyone who really wants to understand this monumental, cataclysmic event:

• Red Star over China, by Edgar Snow
• Fanshen, by William Hinton

Edgar Snow worked as a journalist in China as a young man and traveled to Yenan, the remote base of the Chinese Communists, shortly after the Long March brought them there in 1935. This was a period where little to nothing was known of Mao Tse-Tung and his associates or the Communist movement in China, extensive as it was. There had been a number of Soviet areas in south China governing millions of people in the early thirties - it was because of their encirclement by the Chiang Kai-shek forces of the Kuomintang (KMT) in the fifth anti-Communist extermination campaign that the legendary Long March was necessary to escape the old bases and relocate in the northwest at Yenan in Shaanxi. Snow interviewed Mao and other top leaders fresh from this epic campaign; till that point and long after, his account was the only one known in the West. He somehow got Mao to open up about his own background, something almost irregular among these self-effacing revolutionaries.

Fondly do we hope -- fervently do we pray -- that this mighty scourge of war may speedily pass away. Yet, if God wills that it continue, until all the wealth piled by the bond-man's [slave's] two hundred and fifty years of unrequited toil shall be sunk, and until every drop of blood drawn with the lash, shall be paid by another drawn with the sword, as was said three thousand years ago, so still it must be said "the judgements of the Lord, are true and righteous altogether."

-- Abraham Lincoln (March 4, 1865 at his second Inauguration)

It is fitting to memorialize this epic battle today, the seventieth anniversary of its turning point. Throughout the summer and fall of 1942, the fascist hordes had thrown the Red Army back across to steppes, all the way to the banks of the mighty Volga. They had massacred their way through western Russia and the Ukraine, successfully continuing the blitzkrieg tactics resulting in the encirclement and near annihilation of multiple Soviet armies. Soviet propaganda then and subsequently put a stoic face on it all, but realistic accounts betray the sense of hopelessness and despair prevalent in the army as they saw their best and bravest cut down in unequal matches again and again, never ending it must have appeared at the time. The massed armor, air support, and superior organization and communications of the Wehrmacht seemed invincible.