Is the filter $1/(1-s)$ anti-causal?

Question

The filter with the response function

$$ H(s) = \frac{1}{1 - s} $$ Produces a positive phase shift and a negative group delay for all frequencies

Is it anti-causal? Is there a way to deduce such information from the frequency response of the system.

Answer 1

From the transfer function alone it is generally impossible to say whether a system is causal or not. Only in combination with a given region of convergence (ROC), or, equivalently, with an assumption about stability, can we know for sure if a given system is causal or not.

The given transfer function has a pole at $s=1$. There are two possible time domain functions (impulse responses) that correspond to this transfer function. For the ROC to the right of the pole, i.e., $|s|>1$, the system is causal but unstable. It is unstable because the ROC does not include the imaginary axis ($\omega$-axis). The other system that is described by the same transfer function is obtained by assuming that the ROC is to the left of the pole, i.e., $|s|<1$. Now the imaginary axis is inside the ROC, so the corresponding filter is stable. However, it is anti-causal because of the ROC being a left half-plane.

By evaluating the transfer function on the imaginary axis (i.e., by plotting magnitude and phase), you imply that you're dealing with a stable system, i.e., you choose the ROC that includes the $\omega$-axis ($|s|<1$), which means that the system you're looking at is indeed anti-causal.