1
$\begingroup$

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

enter image description here

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

$\endgroup$
1

1 Answer 1

4
$\begingroup$

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.

$\endgroup$
9
  • $\begingroup$ Missed this, nice answer (deleted mine that assumed it was causal) $\endgroup$ Dec 3, 2021 at 13:01
  • $\begingroup$ "you imply that you're dealing with a stable system" Well, maybe. You can -- both theoretically and physically -- wrap an unstable subsystem with a control loop that stabilizes it, then excite that loop with sine waves. Then you can measure the subsystem's inputs and outputs, and calculate a Bode plot. It'll look just like the one calculated from the transfer function above. So I contend that having a Bode plot with negative group delay everywhere doesn't imply that you're looking at a stable, non-causal system. $\endgroup$
    – TimWescott
    Dec 3, 2021 at 18:37
  • $\begingroup$ @TimWescott: Not sure I can agree with that. Maybe you could write up an answer and explain in more detail ... $\endgroup$
    – Matt L.
    Dec 3, 2021 at 20:54
  • $\begingroup$ OK. This is what I don't get. Being able to take frequency response measurements from a physical unstable system and put them into a Bode plot isn't an opinion -- it's a fact, I've done it. One disagrees with opinions. What are you disagreeing with? $\endgroup$
    – TimWescott
    Dec 3, 2021 at 21:10
  • $\begingroup$ @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. $\endgroup$
    – Matt L.
    Dec 3, 2021 at 21:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.