To preface, this is not a homework related question but purely for self-study purposes.

I'm try to do the analyse of z-transform of $a^nu[n+1]$. It is clearly a non-causal signal, I try to explain it by using the definition.

Based on the causality of z-transform definition: If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a. i.e. $|z| > |a|$.

I am not sure if my explanation is correct: $$\mathcal{Z}\big\{a^nu[n+1]\big\} = a^{-1}z+\sum_{n=0}^{\infty} (a/z)^{n}$$

The ROC of this function in z-plane is $|z| > |a|$ except at $z = ∞$.

It does not obey the causality definition because the boundary exists ( $|a|<|z|< ∞ $), i.e. the ROC pattern is a ring, it is a non-causal signal.

Am I correct? I am kinda confused with the causality definition.

  • $\begingroup$ Signals just are. Systems can be non-causal. That signal description just says that it starts before $n = 0$ and remains non-zero to $n = +\infty$. Because, in signal processing convention, $n = 0$ means "now" and not the moment of the big bang, it's perfectly acceptable (if mathematically awkward) to have a signal that's nonzero for $n < 0$. Generally, if it happens (and it's not an impulse response, which does indicate a non-causal system), then it's most convenient to just change your time index. $\endgroup$ – TimWescott Jan 18 at 18:13

You are correct. If the region of convergence of a right-sided signal (like the one you have) does not include infinity, then the signal is not causal.

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  • $\begingroup$ How can a signal be noncausal? Signals just are, whenever they are. They can be non-zero at any arbitrary spot in time. Systems can be noncausal, by predicting the future -- but signals are handed to us by the universe, not always on a platter. $\endgroup$ – TimWescott Jan 18 at 18:09
  • $\begingroup$ Technically, you are absolutely right. Signals just are and systems can be characterized as causal or not. However, it is very common to call signals as causal if $$x(t) = 0, t < 0$$ and anti-causal if $$x(t) = 0, t> 0$$ The term is obviously borrowed from systems and their impulse response. $\endgroup$ – GKH Jan 18 at 21:20
  • $\begingroup$ Hmm. I wonder when that came into common use? My intuition says "blech". $\endgroup$ – TimWescott Jan 19 at 0:24
  • $\begingroup$ At least 4 years ago :) : dsp.stackexchange.com/questions/27143/what-is-causal-signal $\endgroup$ – GKH Jan 19 at 13:56

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