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

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.

• Matt's answer is spot on. Also see dsp.stackexchange.com/questions/66574/… as to why group delay can be positive or negative without violating causality. Dec 3, 2021 at 13:04

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.
• @TimWescott: The question is what exactly it is that you've done. What is the frequency response of an unstable system? E.g., a causal system with transfer function $H(s)=1/(s-1)$ is unstable and doesn't have a frequency response. And that's not an opinion either. Dec 3, 2021 at 21:25