Druyd:Knowledge

Fourier series

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ID:(1921, 0)

Fourier series

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, then, select the variable:

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Equations

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

x_t = @SUM( a_k * cos(2 * pi * nu_k * t ) + b_k * sin(2 * pi * nu_k * t , -infty, infty)

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

nu_k = k / T

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

a_k = @INT( x( t ) *cos(2* pi * nu_k * t )/ T , t , 0 , T )

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

b_k = @INT( x(t) *sin(2* pi * nu_k * t )/ T , 0 , T )

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

a_k = @SUM( x_n *cos(2* pi * nu_k * n * Dt )/ T , n , 0 , N -1 )

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

b_k = @SUM( x_n *sin(2* pi * nu_k * n * Dt )/ T , n , 0 , N-1 )

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

x_n = @SUM( X_k * exp(2 * pi * nu_k * t ), k , 0, N -1)

$ X_k = a_k - i b_k $

X_k = a_k - i b_k

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

X_k = @INT( x( t ) *cos(2* pi * nu_k * t )/ T , t , 0 , T )

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

X_k = @SUM( x_n *exp( i *2* pi * nu_k * n * Dt / T ), n , 0 , N -1 )/ T

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

r_k = sqrt( a_k ^2 + b_k ^2)

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

phi_k = atan( b_k / a_k )

Examples

Fourier series

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:

equation

Base frequency

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

equation

Definition coefficient $a_k$

To calculate the coefficient a_k of the series

equation=14342

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

equation

Definition coefficient $b_k$

To calculate the coefficient b_k of the series

equation=14342

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

equation

Calculate coefficient $a_k$

To estimate the integral

equation=14347

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

equation

Calculate coefficient $b_k$

To estimate the integral

equation=14348

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

equation

Coefficient in complex form

The coefficients of the Fourier transform

equation=14342

can be regrouped as a complex number by defining

equation

Complex version of the Fourier series

Fourier transform

equation=14342

you can with Euler\'s relation

equation=9407

the definition

equation=14352

and the decreticization of time

equation=14344

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

equation

Definition coefficient $X_k$

To calculate the coefficient X_k of the series

equation=14351

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

equation

Calculate coefficient $X_k$

To estimate the integral

equation=14353

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

equation

Magnitudes of the modes

If the complex coefficient is

equation=14352

Its magnitude is defined as

equation

Modes phase

If the complex coefficient is

equation=14352

the phase can be calculated from

equation

>Model

ID:(1921, 0)