Convergence of Fourier Series 2.2 2002/07/18 2003/07/09 14:05:19 GMT-5 Michael Haag mjhaag@rice.edu Justin Romberg jrom@rice.edu Justin Romberg jrom@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu converge convergence discontinuity fourier fourier coefficient fourier series pointwise pointwise convergence This module discusses the covergence of the Fourier Series to show that it can be a very good approximation for all signals.

Introduction Before looking at this module, hopefully you have become fully convinced of the fact that any periodic function, f t , can be represented as a sum of complex sinusoids. If you are not, then try looking back at eigen-stuff in a nutshell or eigenfunctions of LTI systems. We have shown that we can represent a signal as the sum of exponentials through the Fourier Series equations below: f t n c n ω 0 n t c n 1 T t T 0 f t ω 0 n t Joseph Fourier insisted that these equations were true, but could not prove it. Lagrange publicly ridiculed Fourier, and said that only continuous functions can be represented by (indeed he proved that holds for continuous-time functions). However, we know now that the real truth lies in between Fourier and Lagrange's positions.

Understanding the Truth Formulating our question mathematically, let f N t n N N c n ω 0 n t where c n equals the Fourier coefficients of f t (see ). f N t is a "partial reconstruction" of f t using the first 2 N 1 Fourier coefficients. f N t approximates f t , with the approximation getting better and better as N gets large. Therefore, we can think of the set N N 0 1 … f N t as a sequence of functions, each one approximating f t better than the one before. The question is, does this sequence converge to f t ? Does f N t f t as N ? We will try to answer this question by thinking about convergence in two different ways: Looking at the energy of the error signal: e N t f t f N t Looking at N f N t at each point and comparing to f t .

Approach #1 Let e N t be the difference (i.e. error) between the signal f t and its partial reconstruction f N t e N t f t f N t If f t L 2 0 T (finite energy), then the energy of e N t 0 as N is t T 0 e N t 2 t T 0 f t f N t 2 0 We can prove this equation using Parseval's relation: N t T 0 f t f N t 2 N N ℱ n f t ℱ n f N t 2 N n n N c n 2 0 where the last equation before zero is the tail sum of the Fourier Series, which approaches zero because f t L 2 0 T . Since physical systems respond to energy, the Fourier Series provides an adequate representation for all f t L 2 0 T equaling finite energy over one period.

Approach #2 The fact that e N 0 says nothing about f t and N f N t being equal at a given point. Take the two functions graphed below for example:

Given these two functions, f t and g t , then we can see that for all t , f t g t , but t T 0 f t g t 2 0 From this we can see the following relationships: energy convergence pointwise convergence pointwise convergence convergence in L 2 0 T However, the reverse of the above statement does not hold true. It turns out that if f t has a discontinuity (as can be seen in figure of g t above) at t 0 , then f t 0 N f N t 0 But as long as f t meets some other fairly mild conditions, then f t ′ N f N t ′ if f t is continuous at t t ′ .