Lecture Notes on Fourier Transforms (III)

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In this note, we will extend our model of periodic functions from finite series to infinite series. ... sible to represent a non-differentiable function as a finite sum of.
Lecture Notes on Fourier Transforms (III) Christian Bauckhage B-IT, University of Bonn In this note, we will extend our model of periodic functions from finite series to infinite series. Avoiding mathematical rigor, we qualitatively discuss several motivations for this step and briefly address issues of convergence that are raised by this extension.

Fourier Series In this note, we address a crucial open question as to the generality of the linear model of periodic functions which we introduced in a previous note. That is, we ask: for any periodic function f ( x ), can we really write n

f (x) =



1

Z

k =−n

0

f (x) e

−i2πkx

 dx

ei2πkx =

n



fˆ(k) ei2πkx

(1)

k =−n

for some finite n? In a nutshell, the answer is: no! In general, we cannot restrict ourselves to using finite sums! To better understand this, we observe the following: 1. Recall that ei2πkx (sin and cos) are continuous functions; the rectangular function Π( x ) is not continuous (see Fig. 1); it is impossible to represent a discontinuous function as a finite sum of continuous functions!

Figure 1: The rectangle function Π( x ).

2. Recall that ei2πkx (sin and cos) are differentiable functions; the triangle function Λ( x ) is not differentiable (see Fig. 2); it is impossible to represent a non-differentiable function as a finite sum of differentiable functions! In fact, if there is some discontinuity in some higher derivative of a function f ( x ), i.e. if there is a lack of smoothness, the argument applies again. In other words, any discontinuity in any derivative of f ( x ) prevents us from representing f ( x ) by means of (1) for some finite n! From these observations, we can deduce the following general principle: it takes (very) high frequencies to make sharp corners or edges! Figure 3 illustrates this principle by means of a few examples. The functions f ( x ) and g( x ) shown in the figure are continuous but not differentiable. That is to say, they are discontinuous in their first derivative. Nevertheless, both can be reconstructed fairly well using a series where k ranges from −9 to 9. However, as the ramps in g( x ) are slightly steeper than those in f ( x ), the representation of g( x ) by means of a finite series of 19 terms is slightly worse than that of f ( x ).

Figure 2: The triangle function Λ( x ).

lecture notes on fourier transforms (iii)

f (x)

f (x)

1

0

0

1

0

0

1

g(x)

0

0

g(x)

1

0

0

0

0

(i) n = 1

g(x)

1

h(x)

0

0

1

h(x)

1

(j) n = 3

0

(h) n = 9 h(x)

1

0

1

1

(g) n = 5

1

1

0

g(x)

1

h(x)

1

0

(d) n = 9

1

(f) n = 3

0

1

(c) n = 5

1

(e) n = 1

0

0

1

(b) n = 3

1

f (x)

1

(a) n = 1

0

f (x)

1

0

2

1

0

1

(k) n = 5

The function h( x ) in Fig. 3 is neither continuous nor differentiable. Its ramps are maximally steep. Similar to the other two functions, we see that including more terms into the linear combination of elementary functions e−i2πkx , k = −n, . . . , n improves the result. Yet, in contrast to the other two examples, we observe significant overand undershooting even for the most accurate approximation shown in the figure.

0

0

1

(l) n = 9

Figure 3: It takes high frequencies to make sharp edges! Three simple functions f ( x ), g( x ), and h( x ) and their Fourier series approximations where n = 1, 3, 5, 9. The steeper the step, the less accurate are the approximations, if only a few low frequencies 2πk are considered.

At this point, we could set out for a more rigorous mathematical analysis as to the consequences of the above observations. However, we will refrain from a principled treatment and trust our intuition to deduce that, in order to represent any periodic signal f ( x ), we must consider infinite Fourier series of the form ∞

f (x) =



fˆ(k ) ei2πkx .

(2)

k =−∞

Issues of Convergence One of the drawbacks of resorting to intuition is that we cannot really be sure that (2) is indeed a valid model of general periodic functions. After all, whenever we are dealing with a series that contains infinitely many terms, we do need to make sure that we can in fact compute it. In other words, we have to consider the convergence behavior of the series. Again avoiding mathematical rigor and proofs, we will confine ourselves to a case that is of considerable practical interest and simply summarize what is known about this situation.

The question of whether or not the Fourier series of a periodic function indeed converges to that function is a topic of classic harmonic analysis, a branch of pure mathematics. In the general albeit rather theoretical case, convergence is not guaranteed and there is a rich and deep theory about criteria which need to be met in order for convergence to occur.

lecture notes on fourier transforms (iii)

If we suppose a periodic function f ( x ), again w.lo.g. of period 1, and if we furthermore suppose the hypothesis of finite energy Z 1 0

| f ( x )|2 dx < ∞

(3)

is valid, then it is fairly easy to prove that

Z 1 n 0

∑ k=−n

2 ˆf (k) ei2πkx − f ( x ) dx −−−→ 0. n→∞

(4)

In other words, if f ( x ) is a square integrable function, the result in (4) states that the difference between f ( x ) and its approximation by means of a series of elementary wave functions becomes infinitesimally small if the number of different elementary wave functions becomes infinitely large. From the point of view of signal- or image processing this is indeed comforting. Functions (or signals) which we are processing by means of a computer are necessarily square integrable (i.e. of finite energy). If this was not the case, we would have a hard time storing them on a hard-drive or in computer memory.

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