Fourier, was a 19th Century French
mathematician. He made a remarkable discovery. He found - or more
correctly he came to believe, because he couldn't prove
it - that any repetitive wave function could be expressed as an
infinite series of sine functions. Each member of the infinite
sequence has a different amplitude and frequency, and can have
a different phase. Fourier's work quickly became a corner-stone
of mathematical physics, but it was 150 years before mathematicians
finally proved that Fourier's series do in fact converge.
I remember being impressed as a freshman undergraduate at the length of the shelves in the Canterbury University Physics Library that were devoted to books on Fourier Analysis. Floor to ceiling and as far as I could see. We are fortunate now to have computing facilities which Fourier could never have imagined. Fourier analysis has become routine. SignalScope performs the millions of calculations required for a Fourier Transform - the calculation of the members of the Fourier Series from the shape of the wave form.
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Paul Falstad has kindly allowed us to use his Fourier Series Java Applet |
A square wave is built with the first 60 terms of the sequence....
A saw-tooth wave is built with the first 120 terms of the sequence....
Fourier had lots of trouble convincing his peers that a sequence could always be found that would converge to the right shape.
"Ye of little faith." watch this!
"Ye of very little faith" read this.
A Fourier transform is the reverse process. A given complex wave is resolved into a series of sine waves. Exact calculations are difficult and time consuming even with a fast computer. Engineers get the result with an approximation. The wave is sampled at set frequency intervals (say every 25 Hz) and the Fourier sequence is compiled as a series of amplitudes for sine waves at 25 Hz intervals. SoundVision for Macintosh calculates a Fast Fourier Transform (FFT) in this way and displays the resulting approximate frequency spectrum in real time. MacCRO allows the sampling interval to be reduced to as little as 1 Hz. The analysis takes longer with smaller intervals.
Sound vision.... and a square wave
A square wave signal generator is connected to a speaker. The sound is not pleasant. The FFT spectrum plotted in SoundVision shows the square wave resolved into the first few terms of the Fourier series. (SoundVision was an FFT program for Mac OS9 and below.)
> Clip 1
Increasing the switching rate of the square wave raises the apparent pitch of the sound and changes the FFT spectrum to components of higher frequency.
> Clip 2
