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2002 | Buch

Fourier Transforms of Stable Signals Kapitel lesen Erstes Kapitel lesen

Mathematical Principles of Signal Processing

Fourier and Wavelet Analysis

verfasst von: Pierre Brémaud

Verlag: Springer New York

Enthalten in: Professional Book Archive

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Über dieses Buch

Fourier analysis is one of the most useful tools in many applied sciences. The recent developments of wavelet analysis indicates that in spite of its long history and well-established applications, the field is still one of active research.
This text bridges the gap between engineering and mathematics, providing a rigorously mathematical introduction of Fourier analysis, wavelet analysis and related mathematical methods, while emphasizing their uses in signal processing and other applications in communications engineering. The interplay between Fourier series and Fourier transforms is at the heart of signal processing, which is couched most naturally in terms of the Dirac delta function and Lebesgue integrals.
The exposition is organized into four parts. The first is a discussion of one-dimensional Fourier theory, including the classical results on convergence and the Poisson sum formula. The second part is devoted to the mathematical foundations of signal processing - sampling, filtering, digital signal processing. Fourier analysis in Hilbert spaces is the focus of the third part, and the last part provides an introduction to wavelet analysis, time-frequency issues, and multiresolution analysis. An appendix provides the necessary background on Lebesgue integrals.

Inhaltsverzeichnis

Frontmatter

Fourier Analysis in L 1

A1. Fourier Transforms of Stable Signals
Abstract
This first chapter gives the definition and elementary properties of the Fourier transform of integrable functions, which constitute the specific calculus mentioned in the introduction. Besides linearity, the toolbox of this calculus contains the differentiation rule and the convolution—multiplication rule. The general problem of recovering a function from its Fourier transform then receives a partial answer that will be completed by the results on pointwise convergence of Chapter A3.
Pierre Brémaud
A2. Fourier Series of Locally Stable Periodic Signals
Abstract
A periodic signal is neither stable nor of finite energy unless it is almost everywhere null, and therefore, the theory of the preceding Chapter is not applicable. The relevant notion is that of Fourier series. (Note that Fourier series were introduced before Fourier transforms, in contrast with the order of appearance chosen in this text.) The elementary theory of Fourier series of this section is parallel to the elementary theory of Fourier transforms of the previous section. The connection between Fourier transforms and Fourier series is made by the Poisson sum formula, of which we present a weak (yet useful) version in this chapter.
Pierre Brémaud
A3. Pointwise Convergence of Fourier Series
Abstract
The inversion formula for Fourier series obtained in Chapter A2 requires a rather strong condition of summability of the Fourier coefficients series. Moreover, this condition implies that the function itself is almost everywhere equal to a continuous function. In this section, the class of functions for which the inversion formula holds is extended.
Pierre Brémaud

Signal Processing

Frontmatter
B1. Filtering
Abstract
We introduce a particular and very important class of filters.
Pierre Brémaud
B2. Sampling
Abstract
In a digital communication system, an analog signal {s(t)}t∈ℝ must be transformed into a sequence of binary symbols, 0 and 1. This binary sequence is generated by first sampling the analog signal, that is, extracting a sequence of samples {s(nΔ){n∈ℤ, and then quantizing which means converting each sample into a block of 0 and 1.
Pierre Brémaud
B3. Digital Signal Processing
Abstract
Suppose we need to compute numerically the FT of a stable signal s(t). In practice only a finite vector of samples is available,
$$s = \left( {{s_0},...,{s_{N - 1}}} \right)$$
where s n = s(nΔ). The Fourier sum of this vector evaluated at pulsations ω k = 2kπ/N is the discrete Fourier transform (DFT).
Pierre Brémaud
B4. Subband Coding
Abstract
Let x(t) be a stable base-band (B) real signal that we seek to analyze in the following sense. For fixed N = 2 k we wish to obtain for all 1 ≤ i ≤2 k the signals x i (t) with Fourier transforms
$${\hat x_i}\left( v \right) = {1_{{B_i}}}\left( v \right)\hat x\left( v \right)$$
, where B i is the frequency band
$${B_i}\left[ {\frac{{i - 1}} {{{2^k}}}B,\frac{i} {{{2^k}}}B} \right]$$
.
Pierre Brémaud

Fourier Analysis in L 2

C1. Hilbert Spaces
Abstract
Hilbert space theory is the fundamental tool in Fourier analysis of finite-energy signals. It is a huge chapter of functional analysis, but we shall only give the definitions and prove the basic facts used in this book, in particular, the projection theorem and the theorem of extension of isometries.
Pierre Brémaud
C2. Complete Orthonormal Systems
Abstract
The result of this section is the pillar of the L 2-theory of Fourier series and wavelet expansions. It concerns the possibility of decomposing a vector of a Hilbert space along an orthonormal base.
Pierre Brémaud
C3. Fourier Transforms of Finite Energy Signals
Abstract
A stable signal as simple as the rectangular pulse has a Fourier transform that is not integrable, and therefore one cannot use the Fourier inversion theorem for stable signals as it is. However, there is a version of this inversion formula that applies to all finite-energy functions (for instance, the rectangular pulse). The analysis becomes slightly more involved, and we will have to use the framework of Hilbert spaces. This is largely compensated by the formal beauty of the results, due to the fact that a square-integrable function and its FT play symmetrical roles.
Pierre Brémaud
C4. Fourier Series of Finite Power Periodic Signals
Abstract
Let us consider the Hilbert space ℓ ℂ 2 of complex sequences a = {a n }, n ∈ ℤ, such that \({\sum\nolimits_{n \in \mathbb{Z}} {|{a_n}|} ^2} \prec \infty \) with the Hermitian product
$${\left\langle {a,b} \right\rangle _{l_\mathbb{C}^2}} = \sum\limits_{n \in \mathbb{Z}} {{a_n}b_n^*} $$
(43)
and the Hilbert space L ℂ 2 ([0, T], dt/T) of complex signals x = {x(t)}, t ∈ ℝ, such that \(\int_0^T {{{\left| {x(t)} \right|}^2}dt} < \infty \) , with the Hermitian product
$${\left\langle {x,y} \right\rangle _{L_\mathbb{C}^2\left( {[0,T],\frac{{dt}}{T}} \right)}} = \int_0^T {x(t)y{{(t)}^*}\frac{{dt}}{T}} $$
(44)
Pierre Brémaud

Wavelet Analysis

Frontmatter
D1. The Windowed Fourier Transform
Abstract
Fourier analysis, well as wavelet analysis have an intrinsic limitation, which is contained in the uncertainty principle. In order to state this result, we need a definition of the “width” of a function. Here is the one that suits our purpose.
Pierre Brémaud
D2. The Wavelet Transform
Abstract
We mentioned in the introduction to Part D the shortcomings of the windowed Fourier transform. This chapter gives another approach to the time-frequency issue of Fourier analysis. The role played in the windowed Fourier transform by the family of functions
$${\omega _{v,b}}(t) = \omega (t - b){e^{ + 2i\pi vt}},\quad b,v \in \mathbb{R}$$
is played in the wavelet transform (WT) by a family
$${\psi _{a,b}}(t) = {\left| a \right|^{ - 1/2}}\psi (\frac{{t - b}}{a}),\quad a,b \in \mathbb{R},\;a \ne 0$$
(16)
where Ψ(t) is called the mother wavelet. The function Ψ a, b is obtained from the mother wavelet Ψ by successively applying a change of time scale (accompanied by a change of amplitude scale in order to keep the energy constant) and a time shift (see Fig. D2.1).
Pierre Brémaud
D3. Wavelet Orthonormal Expansions
Abstract
The wavelet analysis of Chapter D2 is continuous, in that the original function of L 2 is reconstructed as an integral, not as a sum. One would rather store the original function not as a function of two arguments, but as the doubly indexed sequence of coefficients of a decomposition along an orthonormal base of L 2. Multiresolution analysis is one particular way of obtaining such orthonormal bases.
Pierre Brémaud
D4. Construction of an MRA
Abstract
The Fourier structure of an MRA is now elucidated, and we know how to obtain a wavelet basis when an MRA is given. This chapter gives two methods for obtaining an MRA.
Pierre Brémaud
D5. Smooth Multiresolution Analysis
Abstract
The axiomatic framework of multiresolution analysis is Fourier analysis in L 2, and the convergence of the wavelet expansion is therefore in the L 2-norm. The smoothness properties of the scaling function and of the mother wavelet are, however, of great interest to obtain fast L 2-convergence of the wavelet expansion, or to obtain pointwise convergence of this expansion.
Pierre Brémaud
Backmatter
Metadaten
Titel
Mathematical Principles of Signal Processing
verfasst von
Pierre Brémaud
Copyright-Jahr
2002
Verlag
Springer New York
Electronic ISBN
978-1-4757-3669-4
Print ISBN
978-1-4419-2956-3
DOI
https://doi.org/10.1007/978-1-4757-3669-4