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__Definition:__ Interpretation of the transfer function of a linear shift invariant system among the {EQUATION(size="80")}$ s=j\omega {EQUATION} imaginary axis is called transfer characteristic function (or modulation transfer function MTF).
{EQUATION(size="80")}$ W(j\omega)=W(s)|_{s=j\omega}{EQUATION}
Consequence of the definition is, the transfer characteristic function can be derived by Fourier transformation due to the relation between the Laplace and Fourier transformations
{EQUATION(size="80")}$ \mathcal{L} \left \{ f(t) \right \}\big{|}_{s=j\omega}= \mathfrak{F} \left \{ f(t) \right \} \right {EQUATION}
Consequently the transfer characteristic function can be written by Fourier transformation as follow:
{EQUATION(size="80")}$W(j\omega) = \frac{\mathfrak{F}\{y(t)\}}{\mathfrak{F}\{f(t)\}} = \frac{Y(j\omega)}{F(j\omega)} {EQUATION}
Transfer characteristic function is also a system property function describing the system response in the frequency domain:
{EQUATION(size="80")}$ Y(j\omega) = W(j\omega)F(j\omega){EQUATION}
{EQUATION(size="80")}$ y(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} W(j\omega)F(j\omega)e^{j\omega t}d\omega{EQUATION}
Deriving {EQUATION(size="80")}$ y(t){EQUATION} by the above written improper integral do not seems to be a simple task. Therefore another tools and methods will be introduced in order to describe the system response in the complete frequency domain. Let’s consider the fundamental definition of the transfer function as was introduced by the Figure 31:
{EQUATION(size="80")}$ W(s) = \frac{Y(s)}{F(s)}{EQUATION}
{EQUATION(size="80")}$ Y(s) = W(s) F(s){EQUATION}
Let’s apply {EQUATION(size="80")}$ f(t) = \delta(t){EQUATION} as an input function:
Consequently, {EQUATION(size="80")}$ F(s) = \mathcal{L}\{\delta(t)\} = 1{EQUATION}, i.e. {EQUATION(size="80")}$ W(s) = Y(s){EQUATION}, where it is possible to see, the system response function is equivalent with the transfer function. All of these consequences in the real parameter space are as follow:
{EQUATION(size="80")}$ w(t) = \mathcal{L}^{-1}\{Y(s)\} \text{, if }f(t) = \delta(t){EQUATION}
If a linear invariant system is excited by the Dirac-delta input function, then the output response will be equivalent with the transfer function in the real parameter space -{EQUATION(size="80")}$ w(t){EQUATION}- calling impulse response function or weighting function. Other words: let’s suppose, an unknown structured – called black box- system satisfying the linear shift invariant conditions, is excited by a Dirac-delta input function, then the output response gives at the same time the system response, i.e. the weighting function “{EQUATION(size="80")}$ w(t){EQUATION}” in the real parameter space. Laplace transformation of the w(t) weighting function is the system transfer function, describing the system properties in the extended complex frequency domain.
Let’s it suppose, the weighting function of a linear invariant system is known and the input function is a
general step function as follow:
{EQUATION(size="80")}$ f(t) = \left\{\begin{matrix}
\varphi(t), &\text{ ha } &0\leq t \\
0, &\text{ ha } & t<0
\end{matrix}\right.{EQUATION}
The output response:
{EQUATION(size="80")}$ Y(s) = W(s)F(s){EQUATION}
{EQUATION(size="80")}$ y(t) = \mathcal{L}^{-1}\{W(s)F(s)\} = \int_{0}^{t}f(\tau)w(t-\tau)d\tau = f(t)\ast w(t),{EQUATION}
Where the consequence is: the output response can be determined by the convolution of f(t) input function and the system characteristic impulse response or weighting function w(t). Furthermore, if the weighting function of a linear invariant system is known and the input function is not a general step function, but {EQUATION(size="80")}$ \int_{-\infty}^{\infty}|f(t)|dt < \infty{EQUATION}, then the obtained system response is the following:
{EQUATION(size="80")}$ y(t) = \int_{-\infty}^{t}f(\tau)w(t-\tau)d\tau{EQUATION}
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If the weighting function of a linear shift invariant system is available then the transfer characteristic function can be determined by the Fourier transformation of the weighting function:
{EQUATION(size="80")}$ W(j\omega) = \mathfrak{F}\{w(t)\}{EQUATION}
Translating the above described principle and the obtained result into the imaging field plays very important role. In case of 2D imaging systems, as well as procedures, very often have to apply the above described method for the determination of contrast transfer function, resolution, and the noise elimination procedures in the frequency domain. In case of a particular 2D imaging system (where is possible to suppose: the system is linear and stationary i.e. shift invariant) can be characterized the system impulse response function (Point Spread Function i.e. PSF) by the real spatial point source acquisition or registration. An example from nuclear imaging is presented in the Figure 18. Fourier transformation of the measured PSF (which is the impulse response or weighting function of 2D imaging system) describes the property of a particular imaging system in the complete frequency domain. Most of the case {EQUATION(size="80")}$ MTF = \mathfrak{F}\{PSF\}{EQUATION} (Modulation Transfer Function) denoting is applied in the terminology of imaging systems.