# Determine the stability of a system without using the $\mathcal Z$-transform (described by a difference equation)

For example, let's say a causal LTI System is described by the following equation:

$$y[n] - ay[n-1] = x[n] - bx[n-1],\quad n \in Z$$

Is there a way to determine (in this case) the stability of the system without using the $\mathcal Z$-transform?

(Solving the problem using the $\mathcal Z$-transform would be a classical easy problem - verify that ROC of $H(z)$ contains the unit circle, I know how to do that)

Yes you can. Just find the roots of the charactheristic polynomial associated with the LCCDE. Then for the causal system to be stable, all of those roots should be inside the unit circle; i.e., $|r_k| < 1$ where $r_k$ is the k-th root.

The characteristic polynomial for the above causal LCCDE is: $$1 - az^{-1}$$ and its single root is $r = a$, hence for this LCCDE to be stable you need $|a|<1$.

• So... LCCDE means using Laplace to my equation and getting H? But laplace for discrete systems isn't the same with Z-transform? – S'en dou Feb 2 '18 at 11:11
• In this context LCCDE means Linear Constant Coefficient Difference Equation and Z-transform applies only. Laplace transform is for continuous time systems where LCDDE means linear constant coefficient differential equations. – Fat32 Feb 2 '18 at 11:12
• So how did you determine the characteristic polynomial? – S'en dou Feb 2 '18 at 11:17
• @S'endou characteristic equation associated with $y[n]+a y[n-1] +b y[n-2] + c y[n-3]$ is $1 + a z^{-1} + b z^{-2} + c z^{-3}$... – Fat32 Feb 2 '18 at 12:23
• I get it now. So you care only about the left hand, you ignore $x[n] - bx[n-1]$. Thanks – S'en dou Feb 2 '18 at 19:43

It boils down to the same thing, with or without $\mathcal{Z}$-transform. You determine the roots of the characteristic equation, which in your case is

$$\lambda=a\tag{1}$$

For stability, the magnitude of the roots of the characteristic equation must be smaller than $1$. So for your example you get the condition $|a|<1$ for stability.

• happened again... :-) – Fat32 Feb 2 '18 at 11:10
• I am not sure I understand. What is the characteristic equation in this case and why? – S'en dou Feb 2 '18 at 11:16
• @S'endou: Did you check the link in my answer? That should make it clear. – Matt L. Feb 2 '18 at 11:42