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# Convergence of FS II: Gibb's Phenomena

Module by: Richard Baraniuk. E-mail the author

Summary: An overview of the convergence of the FS in the case of Gibb's phenomena.

f ^ t= n c n ei w o nt f ^ t n c n w o n t
(1)
• In the late 1800's many machines, (mechanical) computers, were built for computing FS coefficients c n c n and resynthesizing f ^ t f ^ t .
• Albert Michelson's 1898 machine could compute c o c o up to c ± 79 c ± 79 and resynthesize: f ^ N t= n =NN c n ei w o nt f ^ N t n N N c n w o n t
• Clearly hoped that f N tft f N t f t as N N .
• It performed well in almost all test except those involving discontinuous functions

## Example 1

f N t f N t for various N N:
• Even as N N the height of the ripple never decreases below 9% of pulse height.
• But energy in ripple 0 0 since we know limit   N N f ^ N f=0 N N f ^ N f 0 . ie: (zoom in) f N f0 f N f 0 → area inside ripples 0 0 as N N , but area 0 0 by width 0 0 . In fact, height nonzero constant (9% of jump height).
• This is called Gibb's phenomena after the dude who first explained it in 1899. (read Lathi pp 205-206 for more history)
• More technically, Gibb's phenomena is a case of nonuniform convergence.

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