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Convergence of Fourier Series

Module by: Michael Haag, Justin Romberg

Summary: 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, ft 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:
ft=∑n c n ⅇⅈ ω 0 nt f t n c n ω 0 n t (1)
c n =1T∫0Tftⅇ-ⅈ ω 0 ntdt c n 1 T t T 0 f t ω 0 n t (2)
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 Equation 1 (indeed he proved that Equation 1 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=-NN c n ⅇⅈ ω 0 nt f N t n N N c n ω 0 n t where c n c n equals the Fourier coefficients of ft f t (see Equation 2).
f N ′t f N t is a "partial reconstruction" of ft f t using the first 2N+1 2 N 1 Fourier coefficients. f N ′t f N t approximates ft f t , with the approximation getting better and better as NN gets large. Therefore, we can think of the set ∀N,N= 01… : f N ′t N N 0 1 … f N t as a sequence of functions, each one approximating ft f t better than the one before.
The question is, does this sequence converge to ft f t ? Does f N ′t→ft f N t f t as N→∞ N ? We will try to answer this question by thinking about convergence in two different ways:
  1. Looking at the energy of the error signal: e N t=ft- f N ′t e N t f t f N t
  2. Looking at limN→∞ f N ′t N f N t at each point and comparing to ft f t .

Approach #1

Let e N t e N t be the difference (i.e. error) between the signal ft f t and its partial reconstruction f N ′t f N t
e N t=ft- f N ′t e N t f t f N t (3)
If ft∈ L 2 0T f t L 2 0 T (finite energy), then the energy of e N t→0 e N t 0 as N→∞ N is
∫0T| e N t|2dt=∫0Tft- f N ′t2dt→0 t T 0 e N t 2 t T 0 f t f N t 2 0 (4)
We can prove this equation using Parseval's relation: limN→∞∫0T|ft- f N ′t|2dt=limN→∞∑N=-∞∞| ℱ n ft - ℱ n f N ′t |2=limN→∞∑|n|>N| c n |2=0 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 ft∈ L 2 0T f t L 2 0 T . Since physical systems respond to energy, the Fourier Series provides an adequate representation for all ft∈ L 2 0T f t L 2 0 T equaling finite energy over one period.

Approach #2

The fact that e N →0 e N 0 says nothing about ft f t and limN→∞ f N ′t N f N t being equal at a given point. Take the two functions graphed below for example:
cverg1.pngcverg2.png
Subfigure 1.1
Subfigure 1.2
Figure 1
Given these two functions, ft f t and gt g t , then we can see that for all t t, ft≠gt f t g t , but ∫0T|ft-gt|2dt=0 t T 0 f t g t 2 0 From this we can see the following relationships: energy convergence≠pointwise convergence energy convergence pointwise convergence pointwise convergence⇒ convergence in L 2 0T pointwise convergence convergence in L 2 0 T However, the reverse of the above statement does not hold true.
It turns out that if ft f t has a discontinuity (as can be seen in figure of gt g t above) at t 0 t 0 , then f t 0 ≠limN→∞ f N ′ t 0 f t 0 N f N t 0 But as long as ft f t meets some other fairly mild conditions, then f t ′ =limN→∞ f N ′ t ′ f t ′ N f N t ′ if ft f t is continuous at t= t ′ t t ′ .

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