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I've been struggling with the following exercise in Ljung's "System Identification: Theory for the User" (Problem 3G.1):

Given a rational transfer function $G(z)$ such that its poles are all contained in $|z| \leq \mu$ where $\mu < 1$, show that $$ |g(k)| \leq c \cdot \mu^k. $$ Here, $g(k)$ is defined by $$ G(z) = \sum_{k=1}^\infty g(k) z^{-k}. $$

Since the poles are all contained in the unit circle, the inverse of the z-transform should just be the inverse Fourier transform, but I don't quite see how to bound that either. I've read in several places that the impulse response decays whenever the poles are contained in the unit circle, but I haven't been able to find a proof. Any help would be appreciated.

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The transfer function $G(z)$ is usually expressed as a ratio of $\mathcal{z}$-transforms in the following way: $$G(z) = \frac{Y(z)}{X(z)} = \frac{b_0 + b_1z^{-1} + b_2z^{-2} +\,\cdots + b_Mz^{-M}}{1+ a_1z^{-1} + a_2z^{-2} +\,\cdots + a_Nz^{-N}}$$ If $G(z)$ is strictly proper ($M < N$), it can also be expressed using partial fraction decomposition/expansion: $$G(z) = \sum_{i=1}^{N}\frac{r_i}{1-p_iz^{-1}}$$ where $r_i$ are the residues of the poles $p_i$. Notice that each term in that sum is a first order system.


For $M \geq N$, other steps are needed but we'll leave that out here, see the end of this answer


The inverse $\mathcal{z}$-transform of each of these terms is a geometric series: $$\mathcal{Z}^{-1}\bigg\{\frac{r_i}{1-p_iz^{-1}}\bigg\} = r_ip_i^k$$ So the inverse $\mathcal{z}$-transform of $G(z)$ yields: $$g(k) = \sum_{i=1}^{N}r_ip_i^k$$

$k = 0,1,2,\cdots$

Given that the poles $p_i$ are bounded by $\mu < 1$, i.e. $|p_i| < \mu$, each term $|p_i|^k$ is also bounded $|p_i^k| < \mu^k$. Can you take it from there? Try yourself before clicking on the spoiler.

Therefore, the magnitude of each component of the impulse response decays exponentially with $k$, and we can write: $$|g(k)| \leq \sum_{i=1}^{N}|r_i||p_i^k| \leq \left(\sum_{i=1}^{N}|r_i|\right)\mu^k$$ Setting $c = \left(\sum_{i=1}^{N}|r_i|\right)$, which is a finite constant since $N$ is finite, concludes the proof.

For more details, I suggest you read through this chapter from INTRODUCTION TO DIGITAL FILTERS by Julius O. Smith III

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  • $\begingroup$ Thanks for the detailed answer! One quick question, are the residues automatically bounded? $\endgroup$
    – LSK21
    Commented May 15 at 6:59
  • $\begingroup$ for stable systems (poles inside the unit circle), yes. $\endgroup$
    – Jdip
    Commented May 15 at 22:35
  • $\begingroup$ Ok. I don't quite see immediately why this is the case... is there a simple explanation? $\endgroup$
    – LSK21
    Commented May 16 at 6:36
  • $\begingroup$ Yes! start with the expression for residues: $$r_i = \lim_{z\to p_i}G(z)$$ do you see why necessarily $r_i$ is bounded if $|p_i| < 1$? $\endgroup$
    – Jdip
    Commented May 16 at 15:42
  • $\begingroup$ Ah so since the stability guarantees that the transfer function is bounded for bounded input we get that the limit is always bounded as well? $\endgroup$
    – LSK21
    Commented May 17 at 6:26

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