# Decay of the impulse response for poles contained in the unit circle

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.

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

• Thanks for the detailed answer! One quick question, are the residues automatically bounded? Commented May 15 at 6:59
• for stable systems (poles inside the unit circle), yes.
– Jdip
Commented May 15 at 22:35
• Ok. I don't quite see immediately why this is the case... is there a simple explanation? Commented May 16 at 6:36
• 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 $|p_i| < 1$?
– Jdip
Commented May 16 at 15:42
• 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? Commented May 17 at 6:26