Pole-Zero Analysis and Stability

System Function $H(z)$ and Pole-Zero Analysis

For an LTI system with impulse response $h[n]$:

\[H(z) = \sum_{n=-\infty}^{\infty} h[n] z^{-n} = \frac{Y(z)}{X(z)}\]

For systems described by linear constant-coefficient difference equations:

\[\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]\]

Taking Z-transform:

\[H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} = \frac{B(z)}{A(z)}\]

Poles and Zeros

• Zeros: Roots of numerator $B(z)$ → $H(z) = 0$
• Poles: Roots of denominator $A(z)$ → $H(z) \to \infty$

Stability from Pole Locations

Pole Location	System
All poles inside unit circle ($	z	< 1$)	Stable (causal)
Any pole outside unit circle ($	z	> 1$)	Unstable
Poles on unit circle ($	z	= 1$)	Marginally stable (oscillatory)

Zero Locations → Frequency Response Shaping

• Zeros on unit circle → notches (complete rejection at that frequency)
• Zeros near unit circle → dips in response
• Poles near unit circle → peaks (resonance)

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