S8.

Book for Physicists » Introduction to the mathematics of medical imaging » Mathematical Methods of Linear Model Based Image Processing Procedures » Appendix. » Solution of problems » S8.

Solution:

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} \end{cases}$

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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