Frequency response of marginally stable LTI systems

The frequency response of a system is defined as: $$\int_0^\infty{h(t)e^{-j\omega t}dt}$$ where $h(t)$ is the impulse response. But in marginally stable systems, $h(t)$ does not decay so the integral doesn't converge. Does the frequency response "exist" in this case?

• that bottom limit of 0 on the integral is okay as long as everyone is on board with the stable LTI system also being causal. i think it would be better, for generality, to put the bottom limit at $-\infty$. – robert bristow-johnson Jun 11 '18 at 4:37

As long as the system is linear and time-invariant, it can be described using the frequency response / transfer function $H(j\omega)$. For marginally stable systems, $H(j\omega)$ does exist, but cannot be expressed in terms of the Fourier-transformed impulse response $h(t)$.
The Form $H(j\omega) = \frac{Y(j\omega)}{X(j\omega)}$ however is still valid for systems not stable in the more narrow sense simply because they are both linear and time-invariant.
• In this lecture: web.stanford.edu/~boyd/ee102/freq.pdf, the frequency response is used to solve the forced response of a system due to a sinusoidal excitation. You can see that $H(j\omega)$ was directly substituted for $\int_0^\infty{h(\tau)e^{-j\omega\tau}d\tau}$. What bugs me is that I can use this derivation to find the forced response of marginally stable systems and I get the correct answer, as if I solved it through undetermined coefficients if I instead formulated an ODE. – mjtsquared Jun 11 '18 at 1:42