Mathematical And Physical Fundamentals Of Climate Change Pdf

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Chapter 1

Fourier Analysis

Abstract

Motivated by the study of heat diffusion, Joseph Fourier claimed that any periodic signals can be represented as a series of harmonically related sinusoids. Fourier's idea has a profound impact in geoscience. It took one and a half centuries to complete the theory of Fourier analysis. The richness of the theory makes it suitable for a wide range of applications such as climatic time series analysis, numerical atmospheric and ocean modeling, and climatic data mining.

keywords

Fourier series, Fourier transform

Parseval identity

Poisson summation formulas

Shannon sampling theorem

Heisenberg uncertainty principle

Arctic Oscillation index

1.1 Fourier series and fourier transform

Assume that a system of functions {ϕn (t)}n + in a closed interval [a, b . if

and there does not exist a nonzero function f such that

then this system is said to be an orthonormal basis in the interval [a, b].

For example, the trigonometric system

are both orthonormal bases in [−π, π].

Let f(t) be a periodic signal with period 2π and be integrable over [−π, π], write f ∈ L2π and

Then a0(f), an(f), bn(f)(n +) are said to be Fourier coefficients of f. The series

is said to be the Fourier series of f. The sum

is said to be the partial sum of the Fourier series of f. It can be rewritten in the form

where

are also called the Fourier coefficients of f.

It is clear that these Fourier coefficients satisfy

Let f ∈L2π. If f is a real signal, then its Fourier coefficients an (f) and bn (f) must be real. The identity

shows that the general term in the Fourier series of f is a sine wave with circle frequency n, amplitude An, and initial phase θn. Therefore, the Fourier series of a real periodic signal is composed of sine waves with different frequencies and different phases.

Fourier coefficients have the following well-known properties.

Property

Let f, g ∈ L2π and α, β be complex numbers.

(i) (Linearity). cn(αf + βg) = αcn(f) + βcn(g).

(ii) (Translation). Let F(t) =f(t + α). Then cn(F) = einαcn(f).

(iv) (Derivative). If f(t) is continuously differentiable, then cn(f′) = incn(f) (n ≠ 0).

. Then cn(f * g) = 2πcn(f)cn(g).

Proof

Here we prove only (v). It is clear that f * g ∈ L2π and

Interchanging the order of integrals, we get

Let v = t − u. Since f(v)e −inv is a periodic function with period 2π, the integral in brackets is

Therefore,

Throughout this book, the notation f ∈ L ) means that f and the notation f ∈ L[a, b] means that f (t) is integrable over a closed interval [a, b .

Riemann-LebesgueLemma. If f ∈ L ), then ∫ f(t)e−i ω t dt → 0 as ǀωǀ →∞. Especially,

(i)if f ∈ L[a, b ;

(ii)if f ∈ L2π, then cn(f) → 0(ǀnǀ → ∞) and an(f) → 0, bn(f) → 0(n → ∞).

The Riemann-Lebesgue lemma (ii) states that Fourier coeffcients off ∈ L2π tend to zero as n →∞.

Proof

If f is a simple step function and

where c is a constant, then

and so∫ f(t)e− i ωt dt → 0(ǀωǀ → ∞). Similarly, it is easy to prove that for any step function s(t),

If f , then, for ∈ > 0, there exists a step function s(t) such that

Since s(t) is a step function, for the above e, there exists an N such that

From this and ǀe − iωtǀ ≤ 1, it follows that

i.e., ∫ f(t)e− iωt dt → 0(ǀ ωǀ →∞).

Especially, if f ∈ L[a, b], take

Then F ∈ L ), and so ∫ F(t)e− iωt d t → 0(ǀ ωǀ →∞). From

Take a =−π, b =π, and ω = n as ǀnǀ →∞, i.e.,

Combining this with an(f) = c−n(f) + cn(f) and bn(f) = i(c− n(f) − cn(f)), we get

The partial sums of Fourier series can be written in an integral form as follows.

By the definition of Fourier coefficients,

Let v = t − u. then

(1.1)

and is called the Dirichlet kernel.

The Dirichlet kernel possesses the following properties:

(i)Dn(−v)= Dn(v), i.e., the Dirichlet kernel is an even function.

(ii)Dn(v+ 2π) = Dn(v), i.e., the Dirichlet kernel is a periodic function with period 2π.

. This is because

We will give the Jordan criterion for Fourier series. Its proof needs the following proposition.

Proposition 1.1

For any real numbers a and b, the following inequality holds:

Proof

When 1 ≤ a ≤ b, by the second mean-value theorem for integrals, there exists aξ(a≤ξ≤b) such that

When 0 ≤ a ≤ b ≤ 1, with use of the inequality ǀ sin uǀ ≤ ǀuǀ, it follows that

When 0 ≤ a ≤ 1 ≤ b,

is a even function, it can easily prove that for all cases of real numbers a and b,

If a signal is the difference of two monotone increasing signals in an interval, then this signal is called a signal of bounded variation in this interval. Almost all geophysical signals are signals of bounded variation.

Jordan Criterion. Suppose that a signal f ∈ L2π is of bounded variation in (t − η,t + η), η > 0. Then the partial sums of the Fourier series of f

Proof

The assumption that f(t) is of bounded variation in (t − η, t + η) shows that f(t + 0) and f(t − 0) exist. By (1.1) and the properties of Dirichlet kernel, it follows that

where ψt(v) =f(t + v) +f(t − v) −f(t + 0) −f(t − 0). It is clear that

Therefore,

(1.2)

, and ψt(v) ∈ L[0, π]. By Riemann-Lebesgue Lemma, it follows that

Combining this with (1.2), we get

(1.3)

whereψt (v) = f(t + v) +f(t − v) −f(t + 0) −f(t − 0).

Since ψt (v) is of bounded variation in (− η, η) and ψt(0 + 0) = 0, there exist two monotone increasing functions h1 (v) and h1 (v) satisfying h1 (0 + 0) = h2(0 + 0) = 0 such that

Since h1(0 + 0) = h 2(0 + 0) = 0, for any given ∈ > 0, there is a δ(0 < δ <π) such that

For the fixed δ, by (1.3), there exists an N such that

and so

However, using the second mean-value theorem, there exist ζ i(0 < ζ i < δ) such that

and by Proposition 1.1,

Therefore,

at t.

be a periodic function with period T. Then its Fourier series is

where the Fourier coeffcients are

and

An orthogonal basis and an orthogonal series on [−1,1] used often are stated as follows.

Denote Legendre polynomials by Xn(t)(n = 0,1,…):

Especially, X0(t) = 1, X1(t) = t .

By use of Leibnitz's formula, the Legendre polynomials are

. Let t = 1 and t = − 1. Then

Legendre polynomials possess the property:

So Legendre polynomials conform to an orthogonal basis on the interval [−1,1]. In terms of this orthogonal basis, any signal f , where

The coeffcients ln are called Legendre coefficients.

Now we turn to introduce the concept of the Fourier transform.

Suppose that f ∈ L ). The integral

is called the Fourier transform of f . The integral

is called the inverse Fourier transform. Suppose that f ∈ L . It can be proved easily that

Theorem 1.1

Let f ∈ L ). then

is continuous uniformly on .

Proof

The first conclusion is just the Riemann-Lebesgue lemma. It follows from the definition that

Since

with use of the dominated convergence theorem, it follows that for any ω ,

Fourier transforms have the following properties.

Property

Let f, g ∈ L ). then

, where α, β be constants.

, where Daf =f(a t) is the dilation operator.

, where Taf =f(t −α) is the translation operator.

(vii) (Convolution in time). Let the convolution (f*g)(t f(t − u)g(u) du. then

i.e., the Fourier transform of the convolution of two signals equals the product of their Fourier transforms.

Proof

These seven properties are derived easily by the defnition. We prove only (ii), (iii), and (vii).

The Fourier transform of Da (f) is

If a > 0, then ǀaǀ = a and

If a < 0, then ǀaǀ = −a and

We get (ii).

The Fourier transform of Tαf is

Let u = t −α. then

We get (iii).

By the defnition of the Fourier transform,

Interchanging the order of integrals, and then letting v = t − u, we get

So we get (vii).

The notationf ∈ L ) means that f . The defnition of the Fourier transform of f ∈ L ) is based on the Schwartz space.

A space consists of the signals f satisfying the following two conditions:

(i)f ;

(ii) for any non-negative integers p, q,

This space is called the Schwartz space. Denote it by f ∈ S.

From the definition of the Schwartz space, it follows that if f ∈ S, then f ∈ L ) and f ∈ L ). It can be proved easily that if f ∈ S .

On the basis of the Schwartz space, the Fourier transform of f ∈ L ) is defined as follows.

Definition 1.1

Let f ∈ L ). Take arbitrarily fn(t) ∈ S such that fn(t) → f(t)(L in L ) is said to be the Fourier transform of f(t .

Remark

fn(t) → f(t)(L²) means that f (fn(t) −f(t)) dt → 0(n → ∞).

Similarly, on the basis of Definition 1.1, Fourier transforms for L ) have the following properties.

Property

Let f, g ∈ L ) and α, β be constants. Then

, where Daf =f(at) and a ≠ 0 is a constant.

, where Tαf =f(t − α).

(v)

A linear continuous functional F, which is defined as a linear map from the Schwartz space to the real axis, is called a generalized distribution on the Schwartz space. Denote it by F ∈ S′. For any g ∈ S, denote F(g F, g . For each f ∈ L ), we can define a linear continuous functional on the Schwartz space as follows:

which implies that L ) ⊂ S′.

The operation rules for generalized distributions on the Schwartz space are as follows:

(i) (Limit). Let Fn ∈ S′(n = 1,2,…) and F ∈ S′. For any g ∈ S, define Fn → F(S′)(n→ ∞) as

(ii) (Multiplier). Let F ∈ S′ and α be a constant. For any g ∈ S, define αF as

(iii) (Derivative). Let F ∈ S′. For any g ∈ S, define the derivative F′ ∈ S′ as

(iv) (Dilation). Let F ∈ S′. For any g ∈ S, define DaF = F(at) as

where a ≠ 0 is a constant.

(v) (Translation). Let F ∈ S′. For any g ∈ S, define TaF = F(t − a) as

where a is a constant.

(vi) (Antiderivative). Let F ∈ S′. For any g ∈ S, define the antiderivative

Definition 1.2

Let F ∈ S′.

(i) The Fourier series of F is defined as ∑n Cnei nt, where the Fourier coefficients are

where T2π is the translation operator and (Fe− i nt)−1 is the antiderivative of Fe− i nt.

(ii) The Fourier transform of F for any g ∈ S.

Fourier transforms of generalized distributions on the Schwartz space have the following properties.

Property Let F ∈ S′. then

, where a is a constant and TaF = F(t − a).

, where a ≠ 0 and DaF = F(at).

The Dirac function and the Dirac comb are both important tools in geophysical signal processing. Define the Dirac function δ as a generalized distribution on the Schwartz space which satisfies for any g ∈ S,

as a generalized distribution on the Schwartz space which satisfies for any g ∈ S,

Clearly, δ0 = δ is the generalization of the Dirac function δ.

By operation rule (iv) of generalized distributions on a Schwartz space, it is easy to prove that for any g ∈ S, the first-order generalized derivative of the Dirac function is

and the second-order generalized derivative of the Dirac function is

In general, the n-order generalized derivative of the Dirac function is

. By satisfies

Since g ∈ S ⊂ L ), by the definition of the Fourier transform, we have

. Especially, noticing that δ0 = δ, we find that the Fourier transform of the Dirac function is equal to 1.

On the other hand, by Definition 1.2(ii), for any g ∈ S,

Since g ∈ L holds. So

, it follows that

. Noticing that δ0 = δ, we obtain that the Fourier transform of 1 is equal to 2π δ.

Summarizing all the results, we have the following.

Formula 1.1

Remark

In engineering and geoscience, instead of the rigid definition, one often uses the following alternative definition for the Dirac function δ:

δ(t) dt = 1,

(iii) ∫R δ(t)g(t) dt = g(0) for any g(t).

The series ∑nδ2n π is called the Dirac comb which is closely related to sampling theory. In order to show that it is well defined, we need to prove that the series ∑nδ2n π is convergent.

Let Sn . Clearly, Sn are generalized distributions on the Schwartz space, i.e., Sn ∈ S′ and for any g ∈ S,

δ2kπ = g(2kπ), we get

Since g ∈ S, the series ∑n g(2nπ) converges. So there exists a δ* ∈ S′ such that

i.e., the series ∑n δ2nπ converges to δ δ*, g = ∑n g(2nπ) for any g ∈ S.

Secondly, we prove that δ* is a 2π-periodic generalized distribution.

By operation rule (v) of generalized distributions on a Schwartz space, for any g ∈ S,

This means that δ* is a periodic generalized distribution with period 2π.

Third, by Definition 1.2(i), we will find the Fourier series of δ*. We only need to find its Fourier coefficients.

Denote the Fourier coefficients of δ* by Cn. Since δ* ∈ S′, by Definition 1.2(i), for any g ∈ S,

Using operation rule (v) of generalized distributions on a Schwartz space, we get

. Therefore

Using operation rule (vi) of generalized distributions on a Schwartz space, we get

where

Since

, we get

and so

Using operation rule (ii) of generalized distributions on a Schwartz space, we get

and so

δ*, g = ∑kg(2kπ) for any g∈S. Noticing that e−in²kπ = 1, we find the right-hand side is

and so

. By Definition 1.2(i), the Fourier series of δ .

converges to δ .

. This is the Dirichlet kernel Dn(t). Using property (ii) of the Dirichlet kernel, we get

By the Jordan criterion for Fourier series, we have

Sn, g → ∑kg(2kπ)(n δ*, g = ∑kg(2kπ), it follows that

This means that Sn → δ*(S′)(n → ∞). From this and δ* = ∑n δ 2 nπ, we get

Taking the Fourier transform on both sides and using Formula 1.1, we get

Formula 1.2

The Fourier transform of a Dirac comb is still a Dirac comb, i.e.,

The Laplace transform is a generalization of the Fourier transform. Since it can convert differential or integral equations into algebraic equations, the Laplace transform can be used to solve differential/integral equations with initial