Physically realizable, linear time-invariant systems (LTI systems) can be described by a set of differential equations [Unb93]. The behaviour of a **LTI system**
then is completely characterized by its impulse response h(t).4If the impulse response of auch a system is known, the relation between the input x(t**) and the**
output y(t**) is given by the convolution integral**

$$y(t)~=~\int\limits_{-\infty}^{\infty}x(\tau)h(t-\tau)\mathrm{d}\tau.$$
$$\left(1\right)$$

Furthermore, a LTI system is stable, if and only if its impulse response h(t**) is** absolute integrable, that is there exists a constant K **such that**

$$\int\limits_{-\infty}^{\infty}\,|h(\tau)|\,\mathrm{d}\tau\ \leq\ K\ <\ \infty.$$
$$\left(2\right)$$

|h(τ)| dτ ≤ K < ∞. (2)
In network and filter analysis and design, however, one commonly employs the frequency domain rather than the time domain. To this end the system is described based on its transfer function H(s**). In case the Laplace transformation**
is used we have5

$$H(s)~=~\int\limits_{-\infty}^{\infty}h(t)e^{-s t}\mathrm{d}t.$$

where s = σ + jω is a complex variable with ℜ{s} = σ and ℑ{s} = ω.

![0_image_0.png](0_image_0.png)

$$(3)$$
H(s) h(t) System
The evaluation of H(s) for s = jω - in case of convergence6 **— enables**
the qualitative understanding of how the system handles and selects various frequencies ω**, so for example whether the system describes a high-pass filter,** low-pass filter, etc. Now the necessary condition to demonstrate **the stability of** LTI systems in the frequency domain reduces to show, that the jω**-axis lies in** the Laplace transformation's region of convergence (ROC).

4 h(t) is the output of the system, when the input is the Dirac delta **function** δ(t).

5 **Note that this is a generalization of the continuous-time Fourier transformation.**
6 In this case H(jω) equals the Fourier transform.