## 4 Frequency Response Function (FRF)
### Frequency Response Functions  
The response of a linear system to a unit impulse, defined by the Dirac function \(\delta(t)\), is called the weight function \(h(t)\), by means of which it is possible to characterize the linear system by the so-called Duhamel integral. Said integral divides the input signal into individual pulses, and then determines the output signal by summarizing the system responses to each of these pulses:  
\[r(t)=\int\limits_{0}^{t}h(t-\tau)\cdot F(\tau)\cdot\mathrm{d}\tau=\int \limits_{0}^{t}h(\tau)\cdot F(t-\tau)\cdot\mathrm{d}\tau, \tag{11}\]  
where \(F(t)\) is the input value and \(r(t)\) is the output value. It can be proved that it is true:  
\[\begin{split} r(t)=\int\limits_{0}^{t}h(t-\tau)\cdot F(\tau) \cdot\mathrm{d}\tau=\int\limits_{0}^{t}h(\tau)\cdot F(t-\tau)\cdot\mathrm{d }\tau=\int\limits_{-\infty}^{t}h(t-\tau)\cdot F(\tau)\cdot\mathrm{d}\tau=\\ =\int\limits_{0}^{\infty}h(\tau)\cdot F(t-\tau)\cdot\mathrm{d} \tau=\int\limits_{-\infty}^{\infty}h(\tau)\cdot F(t-\tau)\cdot\mathrm{d}\tau \endsubstitution:  
\(t\)\(-\tau=z\)  
-d\(\tau=\) dz  
d\(\tau=\) -d\(z\) \begin{tabular}{c c c} \hline  & \(t\)\(-\tau\) & \(z\) \\ \hline L.L. & \(\tau=-\infty\) & \(z=+\infty\) \\ \hline U.L. & \(\tau=t\) & \(z=0\) \\ \hline \end{tabular}   
The transition from the integration limits from \(0\) to \(+\infty\) to the integration limits from \(-\infty\) to \(+\infty\) in the last expression of the relation (12) is possible on the basis of this consideration. For \(t<0\), the system is at a point of rest. It follows from the above definition of the weight function \(h(t)\) that \(h(t)=0\) for \(t<0\). In Equation (12), it is then possible to extend the integration limits, and the value of the integral does not change. The Dirac function \(\delta(t)\) is a function that is zero everywhere except the point \(t=0\). In practice, the Dirac function \(\delta_{n}(t)\) is used:  
\[0 \qquad t<0\] \[\delta_{n}(t)= n \quad\text{pre }0\leq t\leq 1/n, \tag{13}\] \[0 \qquad t> 1/n\]  
for which:  
\[\int\limits_{-\infty}^{\infty}\delta_{n}(t)\cdot\text{d}t=1. \tag{14}\]  
The Laplace transform of the Dirac function is:  
\[L\{\delta(t)\}=\lim_{n\to\infty}\int\limits_{0}^{\infty}\delta_{n}(t)\cdot \text{e}^{-p\cdot t}\cdot\text{d}t=\lim_{n\to\infty}\int\limits_{0}^{1/n}n \cdot\text{e}^{-p\cdot t}\cdot\text{d}t=1. \tag{15}\]  
The weight function \(h(t)\) and the Dirac function \(\delta_{n}(t)\) are shown in Figure 4.  
The transition function \(H(t)\) is defined as the response to the excitation in the form of a unit jump (Figure 5):  
\[H(t)=\int\limits_{0}^{t}h(t-\tau)\cdot\text{d}\tau=\int\limits_{0}^{t}h(\tau) \cdot\text{d}\tau. \tag{16}\]  
Figure 4: Dirac function \(\delta_{n}(t)\) and weight function \(h(t)\) for the SDF system. By this the linear system is mathematically described in the time domain. A mathematical description of a linear system in the frequency domain can also be conducted by means of the frequency response function \(H(p)\), which is defined as a Laplace transform of the weight function \(h(t)\):  
\[H(p)=L\{h(t)\}=\int\limits_{0}^{\infty}h(t)\cdot\mathrm{e}^{-p\cdot t}\cdot \mathrm{d}t, \tag{17}\]  
where \(p\) is complex variable and \(L\) is operator of Laplace transform.