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It is known, the relation between the transfer characteristic and the weighting function is:
$                  w(j \omega)=\mathfrak{F} \left { w(t) \right }

It is possible to see from the given figures
$                  K(\omega) = \begin{cases}
 K_0,\quad \text{for}\quad 0<\omega < \omega_0 
0 \quad \text{otherwise}
\end{cases} and $            \varphi(\omega)= \begin{cases} -\omega \tau,\quad \text{for} \quad 0<\omega < \omega_0  \\ 0 \quad \text{otherwise}

Consequently, the formula of the transfer characteristic is the following:
$                  w(j\omega)=K_0e^{-j\omega \tau} , for $                  0<\omega < \omega_0, otherwise $        0 .

The weighting function can be derived by the inverse Fourier transformation of the transfer characteristic:

$         w(t)=\mathfrak{F}^{-1} \left { w(j\omega) \right }= \frac{1}{2\pi} \int_{-\infty}^{\infty}w(j\omega)e^{j\omega t} d\omega = \frac{1}{2\pi} \int_{-\omega_0}^{\omega_0}K_0 e^{-j\omega \tau}e^{j\omega t} d\omega

$                  w(t)= \frac{1}{2\pi} \int_{-\omega_0}^{\omega_0}K_0 e^{j\omega( t-\tau )} d\omega = \frac{1}{2\pi} K_0 \left [ \frac{e^{j\omega( t-\tau )}}{j(t-\tau)} \right ] _{-\omega_0}^{\omega_0}=\frac{ K_0 }{2\pi} \left [\frac{e^{j\omega_0( t-\tau )}}{j(t-\tau)}- \frac{e^{-j\omega_0( t-\tau )}}{j(t-\tau)} \right ]

$                  w(t)= \frac{ K_0 }{\pi}\frac{\text{sin}(t-\tau)}{t-\tau}

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