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I'm trying to find the output where $h[n]$ is the impulse response of a real stable causal LTI system, and $s_0 \epsilon \mathbb{R}$, and where the input is $x[n] = e^{2\pi i n s_0}$ for $n \epsilon \mathbb{N}$. Supposedly my answer should be a compact formula with no infinite sums...

So since since my system is causal I obtain, $y[k] = e^{2\pi i ks_0}\sum_{n=0}^{\infty}h[n]e^{-2\pi i ns_0}$. From here I can conclude that since the system is stable this series converges, but without knowing more information about $h[n]$ I can't see any way to further simplify the expression and rid myself of this infinite sum. Of course I also know by assumption that $y[k],s_0 \epsilon \mathbb{R}$, but I can't see anyway to use this fact without, again, knowing more about $h[n]$.

I suppose my sum is looking a lot like the standard Fourier transform but I don't see how I can use that with out, once again, knowing more about $h[n]$. Hopefully someone can help me here, thanks.

Edit: Could it be that a real system implies that $h[n]$ is real? Although that's still not enough info to allow me to find a closed form for this series.

Edit 2: Or could "real" in this context be referring to a system which is actually implemented?

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  • $\begingroup$ By "real system" I would interpret that to mean that $h[n]$ is real. That does not mean that $y[k]$ is real: consider the simple case that $h[n] = \delta[n]$. Then, for the given input signal $x[n]$, $y[n]$ is complex. $\endgroup$ – Jason R Mar 9 '12 at 19:36
  • $\begingroup$ As Dilip pointed out, this should be a trivial problem if you're aware of LTI system properties relative to complex exponential inputs. Take a step back from the math and think about it from a higher level. You aren't going to get a single answer for a generic $h[n]$; the result must be a function of the system's impulse response (even if expressed in a different way). $\endgroup$ – Jason R Mar 9 '12 at 19:38
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    $\begingroup$ @Dilip: Nitpick alert: I would also term the signals discrete, not necessarily digital, since their numeric representation isn't discussed anywhere in the question. $\endgroup$ – Jason R Mar 9 '12 at 19:40
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So since since my system is causal I obtain, $y[k] = e^{2\pi i ks_0}\sum_{n=0}^{\infty}h[n]e^{-2\pi i ns_0}$. From here I can conclude that since the system is stable this series converges, but without knowing more information about $h[n]$ I can't see any way to further simplify the expression and rid myself of this infinite sum.

Hint: The function (of $z$) defined by $\sum_n h[n]z^{-n}$ is given a special name and is of some importance in signal-processing circles. Do you know about this function? What is the value of this function at $z = e^{2 \pi i s_0}$? which by the way lies on the unit circle in the complex plane, seeing as how we are talking of circles.

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