La terminología b ásica asociada a una DLT

The material in this section will only be used to determine the Fourier transform of the so-called comb distribution in section 9.1.3 and also to prove the sampling theorem in chapter 15. Sections 7.3 and 9.1.3 and the proof of the sampling theorem can be omitted without any consequences for the remainder of the material.

With the conclusion of section 7.2 one could state that we have finished the the-ory of the Fourier integral for non-periodic functions. In the next two chapters we extend the Fourier analysis to objects which are no longer functions, but so-called distributions. Before we start with distribution theory, the present section will first examine Poisson’s summation formula. It provides an elegant connection between the Fourier series and the Fourier integral. Moreover, we will use Poisson’s summa-tion formula in chapter 9 to determine the Fourier transform of the so-called comb distribution, and in chapter 15 to prove the sampling theorem. We note, by the way, that in the proof of Poisson’s summation formula we will not use the fundamental theorem of the Fourier integral.

In order to make a connection between the Fourier series and the Fourier integral, we will try to associate a periodic function with period T with an absolutely inte-grable function f(t). We will do this in two separate ways. First of all we define the periodic function fp(t) in the following obvious way:

f_p(t) = ^∞ n=−∞

f(t + nT ). (7.20)

Replacing t by t+ T in (7.20), it follows from a renumbering of the sum that f_p(t + T ) = ^∞

n=−∞

f(t + (n + 1)T ) = ^∞ n=−∞

f(t + nT ) = fp(t).

Hence, the function fp(t) is indeed periodic with period T . There is, however, yet another way to associate a periodic function with f(t). First take the Fourier transform F(ω) of f (t) and form a sort of Fourier series associated with f (note again that f is non-periodic):

1 T

∞

n=−∞F(2πn/T )e^2πint/T. (7.21)

(We will see that this is in fact the Fourier series of fp(t).) If we replace t by t+ T , then (7.21) remains unchanged and (7.21) is thus, as a function of t, also pe-riodic with period T . (We have taken F(2πn/T )/T instead of F(n) since a similar connection between Fourier coefficients and the Fourier integral has already been derived in (6.9).) Poisson’s summation formula now states that the two methods to obtain a periodic function from a non-periodic function f(t) lead to the same result.

Of course we have to require that the resulting series converge, preferably abso-lutely. In order to give a correct statement of the theorem, we also need to impose some extra conditions on the function f(t).

Let f(t) be an absolutely integrable and continuous function on R with spectrum THEOREM 7.8

Poisson’s summation formula F(ω). Let T > 0 be a constant. Assume furthermore that there exist constants p> 1, A > 0 and M > 0 such that | f (t) | < M | t |^−pfor| t | > A. Also assume

that_∞

(with absolutely convergent series). In particular

∞

Define f_p(t) as in (7.20). Without proof we mention that, with the conditions on the function f(t), the function fp(t) exists for every t ∈ R and that it is a continuous function. Furthermore, we have already seen that f_p(t) is a periodic function with period T . The proof now consists of the determination of the Fourier series of fp(t) and subsequently applying some of the results from the theory of Fourier series.

For the nth Fourier coefficient cnof fp(t) one has

c_n= 1

From the conditions on the function f(t) it follows that this termwise integration is allowed, but again this will not be proven. Changing to the variableτ = t + kT in the integral, it then follows that

cn= 1

This determines the Fourier coefficients c_nof f_p(t) and because of the assumption on the convergence of the series_∞

n=−∞| F(2πn/T ) | we now have that

∞

n=−∞| cn| converges.

Since cne^2πint/T = | cn|, it then also follows that the Fourier series of fp(t) converges absolutely (see theorem 4.5). For the moment we call the sum of this series g(t), then g(t) is a continuous function with Fourier coefficients cn. The two continuous functions fp(t) and g(t) thus have the same Fourier coefficients and according to the uniqueness theorem 4.4 it then follows that f_p(t) = g(t). Hence,

∞

which proves (7.22). For t= 0 we obtain (7.23).
In the proof of theorem 7.8 a number of results were used without proof. All of these results rely on a property of series – the so-called uniform convergence – which is not assumed as a prerequisite in this book. The reader familiar with the properties of uniform convergent series can find a more elaborate proof of Poisson’s summation formula in, for example, The theory of Fourier series and integral by P.L. Walker, Theorem 5.30.

We will call both (7.22) and (7.23) Poisson’s summation formula. From the proof we see that the right-hand side of (7.22) is the Fourier series of fp(t), that is, of the function f(t) made periodic according to (7.20). The occurring Fourier coefficients are obtained from the spectrum F(ω) using (7.24). In this manner we have linked the Fourier series to the Fourier integral. It is even possible to derive the fundamental theorem of the Fourier integral from the fundamental theorem of Fourier series using Poisson’s summation formula. This gives a new proof of the fundamental theorem of the Fourier integral. We will not go into this any further. In conclusion we present the following two examples.

Take f(t) = a/(a²+ t²) with a > 0, then F(ω) = πe^{−a| ω |}(see table 3). We want EXAMPLE 7.8

to apply (7.23) with T = 1 and so we have to check the conditions. The assumption about the convergence of the series_∞

n=−∞| F(2πn) | is easy since

which is a geometric series with ratio r = e^−2πa. Since a > 0 it follows that

| r | < 1, and so the geometric series converges (see section 2.4.1). In this case we even know the sum: arbitrary. Poisson’s summation formula can thus be applied. The right-hand side of (7.23) has just been calculated and hence we obtain

∞

By rewriting (7.25) somewhat, it then follows for any a> 0 that

∞ (apply, for example, De l’Hˆopital’s rule three times). This gives a new proof of the famous identity (also see exercise 4.8)

∞

Every rapidly decreasing function f(t) (see section 6.6) satisfies the conditions of EXAMPLE 7.9

theorem 7.8. First of all, it follows straight from the definition of the notion ‘rapidly decreasing’ that the condition on the function f(t) is met, since for f (t) one has, for example, that there exists a constant M > 0 such that | f (t) | < M/t²for all t = 0 (so we can choose p = 2 and A arbitrary positive). It only remains to be shown that the series_∞

n=−∞| F(2πn/T ) | converges. Now it has been proven in theorem 6.12 that the spectrum F(ω) of f (t) is again a rapidly decreasing function.

In particular, there again exists a constant M> 0 such that | F(ω) | < M/ω²for all ω = 0. Hence,

n=1n⁻²converges. All the conditions of theorem 7.8 are thus satisfied and so Poisson’s summation formula can be applied to any f ∈ S. This result will be used in section 9.1.3 to determine the Fourier transform of the so-called comb

distribution. 

EXERCISES

Show that (see example 7.8) 7.25^∗

Indicate why Poisson’s summation formula may be applied to the function f(t) = 7.26^∗

Prove the following generalization of (7.25) (here sinh x = (e^x − e^−x)/2 and 7.27^∗

For an absolutely integrable and piecewise continuous function f(t) on R with spec-trum F(ω) one has lim_ω→±∞F(ω) = 0 (Riemann–Lebesgue lemma). Using this important property, the fundamental theorem of the Fourier integral was proven:

1 2π

_∞

−∞F(ω)e^iωtdω =1

2( f (t+) + f (t−))

for any absolutely integrable and piecewise smooth function f(t) on R (from now on, all functions in the time domain will be assumed to be absolutely integrable and piecewise smooth onR). Here the Fourier integral in the left-hand side converges as a Cauchy principal value.

From the fundamental theorem the uniqueness of the Fourier transform onR immediately follows: if F(ω) = G(ω) on R, then f (t) = g(t) at all points t where

f and g are continuous.

In many cases the Fourier integral will exist as an improper integral as well, resulting in the duality or reciprocity property:

F(−t) ↔ 2π f (ω).

Because of this, Fourier transforms almost always occur in pairs. The duality prop-erty certainly holds for rapidly decreasing functions and this was used to show that the Fourier transform is a one-to-one mapping onto the spaceS of rapidly decreas-ing functions.

For square integrable functions f and g onR, the fundamental theorem was used to prove the convolution theorem in the frequency domain:

f(t)g(t) ↔ (F ∗ G)(ω)/2π.

From this, Parseval’s identities immediately follow:

_∞

Both the fundamental theorem and Parseval’s identities can be used to determine definite integrals.

(where T > 0 is a constant) provided a link between Fourier series and the Fourier integral. This formula can, for example, be applied to any f ∈ S.

S E L F T E S T

The function f(t) is defined by 7.28

f(t) =

sin t for 0≤ t ≤ π, 0 elsewhere.

(see exercise 6.9 for a similar function).

a Determine the spectrum F(ω) of f (t).

b Show that for each t∈ R one has

Let the function f(t) = sin at/(t(1 + t²)) (a > 0) with spectrum F(ω) be given.

7.29

Find F(ω) explicitly as follows.

a Show that g(t) = (p2a(v) ∗ e^{−| v |})(t) has as spectrum G(ω) = 4 f (ω).

b Determine g(t) explicitly by calculating the convolution from part a.

c Verify that the duality property can be applied and then give F(ω).

Let q_a(t) be the triangle function and pa(t) the block function (a > 0), then it is 7.30

known that (see table 3) q_a(t) ↔4 sin²(aω/2)

aω² and p_a(t) ↔ 2 sin(aω/2)

ω .

a For which values of t ∈ R does the Fourier integral corresponding to qa(t) converge to qa(t)? Does this Fourier integral converge only as Cauchy principal value or also as improper integral? Justify your answers.

b Use Parseval’s identity to show that

_∞ 0

sin³x

x³ d x= 3π 8 .

Distributions Introduction 188

8.1 The problem of the delta function 189 8.2 Definition and examples of distributions 192 8.2.1 Definition of distributions 192

8.2.2 The delta function 193 8.2.3 Functions as distributions 194 8.3 Derivatives of distributions 197

8.4 Multiplication and scaling of distributions 203 Summary 206

Selftest 206

Distributions

I N T R O D U C T I O N

Many new concepts and theories in mathematics arise from the fact that one is con-fronted with problems that existing theories cannot solve. These problems may originate from mathematics itself, but often they arise elsewhere, such as in physics.

Especially fundamental problems, sometimes remaining unsolved for years, decades or even centuries, have a very stimulating effect on the development of mathemat-ics (and science in general). The Greeks, for example, tried to find a construction of a square having the same area as the unit circle. This problem is known as the

‘quadrature of the circle’ and remained unsolved for some two thousand years. Not until 1882 it was found that such a construction was impossible, and it was discov-ered that the area of the unit circle, hence the numberπ, was indeed a very special real number.

Many of the concepts which one day solved a very fundamental problem are now considered obvious. Even the concept of ‘function’ has one day been heavily de-bated, in particular relating to questions on the convergence of Fourier series. Prob-lems arising in the context of the solutions of quadratic and cubic equations were solved by introducing the now so familiar complex numbers. As is well-known, the complex numbers form an extension of the set of real numbers.

In this chapter we will introduce new objects, the so-called ‘distributions’, which form an extension of the concept of function. For twenty years, these distributions were used successfully in physics, prior to the development, in 1946, of a mathe-matical theory which could handle these problematic objects. It will turn out that these distributions are an important tool, just as the complex numbers. They are indispensable when describing, for example, linear systems in chapter 10.

In section 8.1 we will show how distributions arise in the Fourier analysis of non-periodic functions. We will first concentrate on the so-called delta function – a misleading term by the way, since it is by no means a function. In section 8.2 we then present a mathematically rigorous introduction of distributions, and we treat our first important examples. We will show, among other things, that most functions can be considered as distributions; hence, distributions form an extension of functions (although not every function can be considered as a distribution).

It is remarkable that distributions can always be differentiated, as will be estab-lished in section 8.3. In this way, one can obtain new distributions by differentiation.

In particular one can start with an ordinary function, consider it as a distribution, and then differentiate it (as a distribution). In this manner one can obtain distributions from ordinary functions, which themselves can then no longer be considered as functions. For example, the delta function mentioned above arises as the derivative of the unit step function. In the final section of this chapter two more properties will be developed, which will be useful later on: multiplication and scaling. Fourier analysis will not return until chapter 9.

188

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