LOG#184. Absement, Fourier and transforms.

In this short post, I am going to relate the relatively unknown notion of absement, the quantity

$A=\int X(t)dt$

with some common transforms, in particular, with Fourier series and transforms more naturally.

The Fourier series for a periodic signal is

$x(t)=a_0+\sum_{n=1}^\infty a_n \cos (n\omega_0 t)+ b_n \sin (n\omega_0 t)$

The complex forms reads

$x(t)=\sum_{n=-\infty}^\infty c_n\exp (j n \omega_0 t)$

The coefficients for the real case read

$a_0=\int x(t)dt/T_0$

$a_n=\dfrac{2}{T_0}\int x(t) \cos (n\omega_0 t)dt$

$b_n=\dfrac{2}{T_0}\int x(t) \sin (n\omega_0 t) dt$

and in the complex case

$c_n=\dfrac{1}{T_0}\int x(t) \exp (-jn\omega_0t) dt$

Here, the Fourier transforms reads

$X(f(t))=\int_{-\infty}^\infty x(t) \exp (-2\pi j f t) dt$

Key remark: the $a_0$ coefficient in the Fourier transforms is the average absement of $x(t)$ .Indeed, the remaining coefficients are also the absement weighted with sinusoidal functions.

Exercise: given the Laplace transform

$F(s)=L(f(t))=\int_0^\infty f(t) \exp (-st) dt$

the discrete Fourier transformation

$X(\exp (j\omega))=\sum_{n=-\infty}^\infty x(n) \exp (-nj\omega)$

and the zeta transform

$X(z)=\sum_{n=-\infty}^\infty x(n)z^{-n}$

What is the relationship with the absement of the transformation coefficients? Interpret your results.

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The Spectrum Of Riemannium

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LOG#184. Absement, Fourier and transforms.

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