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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