Impulse response of a system with a transfer function with a phase discontinuity

Question

I am puzzled about an impulse response of a system. The transfer function is: \begin{align} T(\omega) = A_0 e^{-j \Theta_0 \mathrm{sign}(\omega)} \end{align} where the sign() function is the sign of ω. There is a discontinuity at ω = 0.

The impulse response from the inverse Fourier transform of the transfer function:

\begin{equation} h(t) = A_0 cos(\Theta_0) \delta(t) + \frac{A_0 sin(\Theta_0)}{\pi t} \end{equation}

Since h(t) is not equal to zero for t < 0, the system is non-causal, is this correct? It doesn't look that way at first glance.

Answer 1

The system is not causal. The definition is that if the input $f(t)=0$ for $t<0$ then the output $g(t)=0$ for $t<0$. So it is not physically realizable. It is possible to make it causal by multiplying its impulse response by a rectangle or triangle function in order to have zero for $t<0$ (to see p 106-108["The Fourier Integral and Its Applications",1962, Papoulis, McGraw-Hill Professional])