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Fourier series
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Fourier series
Equation

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Every time function x(t) can be expressed as a Fourier series, that is, a sum of trigonometric functions of a base frequency and its harmonics:

$ x(t) = \displaystyle\sum_{k=-\infty}^{\infty}( a_k \cos 2 \pi \nu_k t + b_k \sin 2 \pi \nu_k t )$

ID:(14342, 0)


Base frequency
Equation

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The base frequency
u_k of the Fourier series is defined as a function of the time T of the time series x(t):

$ \nu_k = \displaystyle\frac{ k }{ T }$

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Definition coefficient $a_k$
Equation

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To calculate the coefficient a_k of the series

$ x(t) = \displaystyle\sum_{k=-\infty}^{\infty}( a_k \cos 2 \pi \nu_k t + b_k \sin 2 \pi \nu_k t )$



the function x(t) must be integrated weighted by the cosine of the corresponding frequency:

$ a_k = \displaystyle\frac{1}{ T } \displaystyle\int_0^T x(t) \cos(2 \pi \nu_k t ) dt$

ID:(14347, 0)


Definition coefficient $b_k$
Equation

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To calculate the coefficient b_k of the series

$ x(t) = \displaystyle\sum_{k=-\infty}^{\infty}( a_k \cos 2 \pi \nu_k t + b_k \sin 2 \pi \nu_k t )$



the function x(t) must be integrated weighted by the sine of the corresponding frequency:

$ b_k = \displaystyle\frac{1}{ T } \displaystyle\int_0^T x(t) \sin(2 \pi \nu_k t ) dt $

ID:(14348, 0)


Calculate coefficient $a_k$
Equation

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To estimate the integral

$ a_k = \displaystyle\frac{1}{ T } \displaystyle\int_0^T x(t) \cos(2 \pi \nu_k t ) dt$



you can discretize the function x(t) and replace the integral with a sum:

$ a_k = \displaystyle\frac{1}{ T } \displaystyle\sum_{n=0}^{ N -1} x_n \cos(2 \pi \nu_k n \Delta t )$

ID:(14349, 0)


Calculate coefficient $b_k$
Equation

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To estimate the integral

$ b_k = \displaystyle\frac{1}{ T } \displaystyle\int_0^T x(t) \sin(2 \pi \nu_k t ) dt $



you can discretize the function x(t) and replace the integral with a sum:

$ b_k = \displaystyle\frac{1}{ T } \displaystyle\sum_{n=0}^{ N -1} x_n \sin(2 \pi \nu_k n \Delta t )$

ID:(14350, 0)


Coefficient in complex form
Equation

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The coefficients of the Fourier transform

$ x(t) = \displaystyle\sum_{k=-\infty}^{\infty}( a_k \cos 2 \pi \nu_k t + b_k \sin 2 \pi \nu_k t )$



can be regrouped as a complex number by defining

$ X_k = a_k - i b_k $

ID:(14352, 0)


Complex version of the Fourier series
Equation

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Fourier transform

$ x(t) = \displaystyle\sum_{k=-\infty}^{\infty}( a_k \cos 2 \pi \nu_k t + b_k \sin 2 \pi \nu_k t )$



you can with Euler\'s relation



the definition

$ X_k = a_k - i b_k $



and the decreticization of time

$ t_n = n \Delta t $



redefine as the discrete transforms on the complex space of the time series x_n equal to

$ x_n = \displaystyle\sum_{k=0}^{N-1} X_k e^{ i 2 \pi \nu_k n \Delta t }$

ID:(14351, 0)


Definition coefficient $X_k$
Equation

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To calculate the coefficient X_k of the series

$ x_n = \displaystyle\sum_{k=0}^{N-1} X_k e^{ i 2 \pi \nu_k n \Delta t }$



the function x(t) must be integrated weighted by the cosine of the corresponding frequency:

$ X_k = \displaystyle\frac{1}{ T } \displaystyle\int_{0}^{ T } x(t) e^{ i 2 \pi \nu_k t } dt$

ID:(14353, 0)


Calculate coefficient $X_k$
Equation

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To estimate the integral

$ X_k = \displaystyle\frac{1}{ T } \displaystyle\int_{0}^{ T } x(t) e^{ i 2 \pi \nu_k t } dt$



you can discretize the function x(t) and replace the integral with a sum:

$ X_k = \displaystyle\frac{1}{ T } \displaystyle\sum_{ n =0}^{ N -1} x_n e^{ i 2 \pi \nu_k n \Delta t }$

ID:(14354, 0)


Magnitudes of the modes
Equation

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If the complex coefficient is

$ X_k = a_k - i b_k $



Its magnitude is defined as

$ r_k = \sqrt{ a_k ^2 + b_k ^2}$

ID:(14355, 0)


Modes phase
Equation

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If the complex coefficient is

$ X_k = a_k - i b_k $



the phase can be calculated from

$ \phi_k = \arctan\displaystyle\frac{ b_k }{ a_k }$

ID:(14356, 0)