# Z-transform of difference equations and stability of a process

According to this paper:

$y(t)$ is stationary if all of the roots (of characteristic equation) lie outside the unit circle

Here, $y(t)$ is causal.

To me it seems the case is exactly the opposite, the roots should be inside the unit circle for causal process to be stable.

Example, consider the difference equation:

$$y_t-5y_{t-1}+6y_{t-2}=0$$

The characteristic equation is then: $$r^2-5r+6=0$$

Solution for characteristic roots: $$r=2, r=3$$

Solution for the difference equation (assuming 1 as initial conditions): $$y(t)=2^t+3^t$$

Clearly the roots are outside the unit circle (as 1<2,3), and the process unstationary (as it grows with t). To me it seems that the 3rd equation in the paper should have negative exponent for zs (in which case the solution would be the same as mine, even if arrived through the z transform).

Yet, the claim appears in many places so it can't be wrong (I doubt the paper is wrong either). However, for LTI system to be stable the poles must be inside the circle, a other claim that is repeated everywhere. Is the statement just plain wrong or are these things somehow not equivalent?

The paper deals in the lag operator $L$, rather than $z^{-1}$ as DSP people tend to use.
• A causal system expressed in terms of $L$ will have its impulse response with only non-negative exponents (i.e. zero or positive). $L^n$ terms in the system impulse response with $n<0$ will be anti-causal. – Peter K. Apr 20 '16 at 15:34
• How is it more limiting than saying they should be inside the unit circle? The substitution is $L = z^{-1}$. The two ways of looking at it are precisely equivalent. There is nothing more complex than that. Statisticians / econometrists sometimes use the lag operator instead of what DSP people consider the standard $z^{-1}$ operator. In some ways, they are right: $z$ is really an "advance" operator, which is somewhat nonsensical given the way equations are usually written in terms of past values of the input and output (for causal systems). – Peter K. Apr 20 '16 at 16:22