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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?

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  • $\begingroup$ 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. $\endgroup$ – Tendero Mar 28 '16 at 4:41
  • $\begingroup$ 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. $\endgroup$ – Peter K. Mar 28 '16 at 20:58
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As long as a system can be characterized by its transfer function, or, equivalently, by its impulse response, the system is LTI. For the LTI property it's irrelevant if the transfer function is rational or not.

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).

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