Fourier History

Páginas: 13 (3063 palabras) Publicado: 6 de diciembre de 2012
EUROCON 2007 The International Conference on “Computer as a Tool”

Warsaw, September 9-12

The History of Applications of Analytic Signals in Electrical and Radio Engineering
Stefan L. Hahn
Institute of Radioelectronics,Warsaw University of Technology, Warsaw, Poland, e-mail: hahn@ire.pw.edu.pl
Abstract— Analytic signals have been originally described in 1946 by Dennis Gabor. The paperdescribes briefly the mathematical background created in the past by prominent mathematicians, physisists and engineers, especially the complex notation of functions, the theory of analytic functions and the Fourier spectral analysis and the theory of distributions. Due to the personal experience of the author, the extension of the Gabor’s notion of the analytic signal for N-dimensional signals isbriefly described. As well, problems of analytic spectra of causal signals with extension for N dimensions and 2N-dimensional Wigner distributions are shortly presented. Keywords—analytic functions, analytic signals, historical data.
Fig. 1. Leonhard Euler, 1707-1783 (Wikipedia)

I. INTRODUCTION The author received the invitation from Prof. Peter Hill to present a historical review atEUROCON2007. The choice of the subject is based on personal experience of the author and not on profound historical studies. These would be impossible, since the author is an engineer and not a historian. As well, the paper should have been written in two weeks. Certainly, the paper is biased by the personal experience of the author. II. MATHEMATICAL BACKGROUND CREATED BY PROMINENT MATHEMATICIANS A. Complexvariables A complex variable z = x + jy uses the imaginary unit j = −1 introduced in XVI century to solve the “casus irredu-cibilis” in cubic equations. B. Leonhard Euler, 1707-1783 In the work “Introductio in analysis infinutorum” (1748), Euler (Fig. 1) derived the formula e jx = cos ( x ) + j sin ( x ) . (1) C. William Rowan Hamilton, 1805-1865 The English mathematician W. R. Hamilton extendedthe notion of complex numbers and complex functions and defined so-called quaternions q = 1a + ib + jc + kd (2) where a, b, c, d ∈ ℜ, and the products of imaginary units obey non-commutative quaternion rules., i.e., ij ≠ ji.

D. Augustus Louis Cauchy, 1789 – 1857 Analytic signals can be defined as boundary distributions of analytic functions. A prominent cofounder of the theory of analyticfunctions is A. L. Cauchy (Fig. 2). Let us present the one-dimensional Cauchy integral

ψ (ς ) 1 ∫ C ς − z dς 2π j where z = t +jτ is a complex variable. ψ ( z) =
E. Bernhard Riemann, 1826 – 1866 We may write

(3)

ψ ( z ) = ψ ( t , τ ) = u ( t ,τ ) + jv ( t ,τ ) . This function is analytic, if

(4)

∂u ∂v ∂u ∂v ; (5) = =− . ∂t ∂τ ∂τ ∂t Conditions (5) of analyticity are called Cauchy-Riemannequations.

Fig. 2. Augustus Louis Cauchy, 1789 – 1857 (Wikipedia)

1-4244-0813-X/07/$20.00 2007 IEEE.

2627

F. Jean Baptiste Joseph de Fourier, 1768-1830 Today the Fourier transform is a fundamental tool in engineering and many other disciplines. Fourier developed Fourier series and Fourier transforms (FT) using real notations. The real notation of the FT is
f (t ) = ∫
∞ 0four-pole is defined. Note that Euler’s formula (1) found its way to engineering after 150 years.

{ A (ω ) cos (ω t ) + B (ω ) sin (ω t )} dω ,
A (ω ) = B (ω ) =

(6) (7) (8)

π∫ π∫
1

1



−∞ ∞

f ( t ) cos (ω t ) dt , f ( t ) sin (ω t ) dt .

−∞

Actually in most applications we use the complex notation of the direct and inverse Fourier transforms:
F (ω ) = ∫ f (t ) =
∞ −∞

f(t ) e

− jω t

dt ,

(9)

Fig. 4. Charles Proteus Steimetz, 1865-1923, born in Wrocław (at that time Breslau).

1 ∞ F (ω ) e jω t d ω . (10) 2π ∫−∞ The convenience of notation is obtained at the cost of the redundant part of the spectrum F( ) at negative frequencies.

A. Oliver Heaviside, 1850-1925 The Heaviside (Fig. 5) operational calculus is nowadays classified as a precursor of...
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