Effect of BIBO-Instability on the frequency response of a ideal LPF

Question

I recently came across this post stating that the continuous ideal LPF is BIBO-unstable since the impulse response is not absolutely integrable, and this post stating some examples.

I have been trying to understand this using the frequency response. For example, take a system with bounded input $x(t) = \sin(2\pi B t)u(t)$ where $u(t)$ indicates the Heaviside unit step function and the impulse response of the ideal LPF $h(t) = \frac{\sin(2\pi B t)}{2\pi B t}$. If we find the convolution of these two functions $y(t) = x(t) * h(t)$ for $t = 0$, we would find that the integral diverges: the output is unbounded due to $h(t)$ not being absolutely summable. If we would use frequency-domain multiplication instead of time-domain convolution for this system, would we find the same result? And maybe more importantly: does the convolution property of the Fourier Transform always hold if the Fourier Transforms of the two signals exist?

Thanks in advance for any help, this has been bugging me for the last couple of days. I feel that there may be something larger to this problem which I don't quite grasp yet.

Answer 1

Note that BIBO-unstable systems can have well-defined output signals for a large class of input signals, i.e., there are many cases for which the convolution integral doesn't diverge. The example in your question is problematic because the frequency of the sinusoid at the input is exactly at the band edge of the ideal lowpass. That's why the result is ill-defined anyway, no matter if you're trying to solve it in the time domain or in the frequency domain. If instead you had a sinusoidal input signal $x(t)=\sin(\omega_0t)$ with $\omega_0$ not equal to the ideal lowpass filter's band edge, then the convolution integral would converge, and you would obtain the same result by multiplying the corresponding Fourier transforms.

The convolution theorem of the Fourier transform holds for a large class of functions, including distributions (e.g., the Dirac delta impulse). In (engineering) practice you won't usually come across functions for which the convolution theorem doesn't hold. For more mathematical details, take a look at the corresponding wikipedia page and the references therein.