# LTI system without constant coefficient differential equation

I have encountered a system where the output $y(t)$ and input $x(t)$ are related in the laplace domain as:

$$Y(s) = H(s)X(s) \tag1$$

which is typical. However, $H(s)$ is not a rational function of polynomials of $s$. Instead, $H(s)$ contains terms like $\log(s)$.

As far I knew, one can write equation $(1)$ only for an LTI system and LTI systems have rational polynomial transfer functions. Am I correct in this analysis?

If I have a logarithm in $H(s)$, that implies that I cannot write a linear differential equation with constant coefficients in the time domain. Is it possible to have an LTI system that is not representable by such differential equation?

Does a frequency domain analysis always imply an LTI system?

• Eq. $(1)$ is only correct for LTI systems. As far as I know, an LTI system's transfer function can often be written as a rational function, but not always. The same regarding differential equations: I believe that not every LTI system out there can be expressed as a differential equation with constant coefficients. Please correct me if I'm mistaken here. Apart from that, a frequency domain analysis is not only used for LTI systems. Nonlinear systems (or linear systems that are not time-invariant) are also analyzed in the frequency domain. – Tendero Mar 28 '16 at 4:41
• An LTI system that can be expressed as a linear constant coefficient difference / differential equation is easily realizable, provided the order of the equation is finite. Not all LTI systems are realizable; the ideal low pass filter is LTI, but it is not realizable nor is it expressive as a (finite order) difference equation. – Peter K. Mar 28 '16 at 20:58

If the transfer function is rational (and positive-real), the system can be realized by a circuit containing lumped elements, i.e., ideal R, L, and Cs. On the other hand, a simple example of a non-rational transfer function is an ideal delay line (e.g., approximated by a long cable), the transfer function of which is $H(s)=e^{-sT}$, where $T$ is the delay.
In sum, the exact expression for $H(s)$ is irrelevant for the LTI property of the corresponding system. However, $H(s)$ can only be realized by resistors, inductors, and capacitors if it is a rational function (and if it is positive-real).