06-04-2011, 11:00 AM
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INTRODUCTION
function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. The weights, or coefficients, of the modes, are a one-to-one mapping of the original function. These modal coefficients are sometimes themselves confusingly referred to as "modes" for brevity, especially in physics literature. Fourier series serve many useful purposes, as manipulation and conceptualization of the modal coefficients are often easier than with the original function.
Areas of application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. Using the tools and techniques of spectroscopy, for example, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star's emitted light. Similarly, engineers can optimize the design of a telecommunications system using information about the spectral components of the data signal that the system will carry. See also spectrum analyzer.
Who is Fourier?
Fourier is one of the France’s greatest administrators, historians, and mathematicians.
He graduated with honors from the military school in Auxerre and became a teacher of math when he was 16 years old.
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27.
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition.
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817.
Lord Kelvin on Fourier’s theorem
Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.
Fourier’s basic idea
Trigonometric functions: sin(x) and cos(x) has the period 2π.sin(nx) and cos(nx) have period 2π/n.The linear combination of these functions or multiply each by a constant, the adding result still has a period 2π.
Fourier series
• Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine functions.A continuous periodic signal x(t) with a period T0 may be represented by:
• Dirichlet conditions must be placed on x(t) for the series to be valid: the integral of the magnitude of x(t) over a complete period must be finite, and the signal can only have a finite number of discontinuities in any finite interval
Trigonometric form for Fourier series:
• If the two fundamental components of a periodic signal are B1cosω0t and C1sinω0t, then their sum is expressed by trigonometric identities:
• If this procedure is applied to all the harmonic components of a Fourier series:
INTRODUCTION
function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. The weights, or coefficients, of the modes, are a one-to-one mapping of the original function. These modal coefficients are sometimes themselves confusingly referred to as "modes" for brevity, especially in physics literature. Fourier series serve many useful purposes, as manipulation and conceptualization of the modal coefficients are often easier than with the original function.
Areas of application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. Using the tools and techniques of spectroscopy, for example, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star's emitted light. Similarly, engineers can optimize the design of a telecommunications system using information about the spectral components of the data signal that the system will carry. See also spectrum analyzer.
Who is Fourier?
Fourier is one of the France’s greatest administrators, historians, and mathematicians.
He graduated with honors from the military school in Auxerre and became a teacher of math when he was 16 years old.
Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27.
He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition.
He was secretary of the Academy of Sciences in 1816 and Fellow in 1817.
Lord Kelvin on Fourier’s theorem
Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.
Fourier’s basic idea
Trigonometric functions: sin(x) and cos(x) has the period 2π.sin(nx) and cos(nx) have period 2π/n.The linear combination of these functions or multiply each by a constant, the adding result still has a period 2π.
Fourier series
• Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine functions.A continuous periodic signal x(t) with a period T0 may be represented by:
• Dirichlet conditions must be placed on x(t) for the series to be valid: the integral of the magnitude of x(t) over a complete period must be finite, and the signal can only have a finite number of discontinuities in any finite interval
Trigonometric form for Fourier series:
• If the two fundamental components of a periodic signal are B1cosω0t and C1sinω0t, then their sum is expressed by trigonometric identities:
• If this procedure is applied to all the harmonic components of a Fourier series: