Numerical Methods for Chemical Engineering: Applications in MATLAB
Written in a pedagogic style, this textbook unites the applications of numerical mathematics and scientific computing to the practice of chemical engineering.
TABLE OF CONTENTS Numerical Methods for Chemical Engineering: Applications in MATLAB
Chapter 9: Fourier Analysis
Fourier analysis treats the representation of periodic functions as linear combinations of sine and cosine basis functions. In chemical engineering, Fourier analysis is applied to study time-dependent signals in spectroscopy and to analyze the spatial structure of materials from scattering experiments. Here, the basic foundation of Fourier analysis is presented, with an emphasis upon implementation in MATLAB.
Fourier Series and Transforms in One Dimension
We begin our discussion of Fourier analysis by considering the representation of a periodic function f( t) with a period of 2 P, f( t + 2 P) = f( t). If f( t) has a finite number of local extrema and a finite number of times t j ? [0, 2 P] at which it is discontinuous, Dirichlet s theorem states that it may be represented as the Fourier series
such that at all t ? where f( t) is continuous, 
, and at all points t j where f( t) is discontinuous, 
( t j) is the average of the right-and left-hand limits:
a 0, { a 1, a 2, }, and { b 1, b 2, } are calculated using the orthogonality properties of sine and cosine functions. First, to compute a 0, we integrate f( t) over the period [0, 2 P], and do the same for ( t):
As the...
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