Spectral_Analysis_Fourier_series - Spectral Analysis...

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Spectral Analysis (Fourier Series) Author: John M. Cimbala, Penn State University Latest revision: 21 February 2007 Introduction There are many applications of spectral analysis , in which we determine the frequency content of a signal . For analog signals, we use Fourier series , which we discuss in this learning module. For digital signals, we use discrete Fourier transforms , which we discuss in a later learning module. Even function Odd function Even and odd functions An even function is one in which f ( t ) = f ( t ); an even function is symmetric about t = 0 . o Consider a cosine wave as sketched to the right. o Notice that the function on the left is the mirror image of that on the right around t = 0. o Even functions are also called symmetric functions . An odd function is one in which f ( t ) = f ( t ); an odd function is antisymmetric about t = 0 . o Consider a sine wave as sketched to the right. o Notice that the function on the left is the negative of the mirror image of that on the right around t = 0. o Odd functions are also called antisymmetric functions . Most functions encountered in t laboratory are a combination of both even and odd functions. he he Fourier series analysis Consider an arbitrary periodic function or signal, f ( t ). Suppose t signal repeats itself over and over again at some period T , as sketched to the right. The fundamental frequency of the signal is defined as 0 1/ f T = ; f 0 is also called the first harmonic frequency . f 0 is the frequency at which the signal repeats itself . We also define the fundamental angular frequency , 00 2 f ω π = . f 0 has units of Hz (cycles/s), but 0 has units of radians/s since there are 2 radians per cycle. Other names for fundamental angular frequency are fundamental radial frequency and fundamental radian frequency . Period, T In a signal like that shown above, there may be more frequency components than just the fundamental one. In fact, there may be many other frequency components, called harmonics , present in the signal. It turns out that any periodic function f(t) can be expressed as the sum of a constant plus a series of sine and cosine terms representing the contribution of each harmonic . Such a series is called a Fourier series , ( ) () 11 sin cos nn 0 f tc a n t b n t ∞∞ == =+ + ∑∑ .
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The first term on the right is a constant, which is simply the average of the function over the entire period T .
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This note was uploaded on 07/23/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

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Spectral_Analysis_Fourier_series - Spectral Analysis...

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