IIR Filters: Difference Equations, Stability, and Analog-to-Digital Design

All flashcards

Infinite impulse response (IIR) system

A system whose impulse response does not become exactly zero after any finite time, even if it decays.

Pole of a transfer function

A value of $z$ that makes the transfer function's denominator equal to zero.

BIBO stability condition for a causal IIR filter

Every pole must lie strictly inside the unit circle: $|p_k|<1$.

IIR and FIR design trade-off

IIR filters often meet specifications with lower order; FIR filters can be designed for exact linear phase.

Limit cycle

A small persistent oscillation that can arise from quantization in a feedback filter, even with no external input.

Quiz questions

1. For the zero-initial-condition difference equation $y[n]=sum_{i=0}^{P} b_i x[n-i]+sum_{i=1}^{Q} a_i y[n-i]$, which expression gives $H(z)=Y(z)/X(z)$?Answer: $\frac{\sum_{i=0}^{P} b_i z^{-i}}{1-\sum_{i=1}^{Q} a_i z^{-i}}$
2. For a causal IIR system, which pole condition ensures BIBO stability?Answer: Every pole lies strictly inside the unit circle.
3. What is a key property of the bilinear transform in analog-to-digital filter design?Answer: It avoids frequency aliasing but warps the frequency scale.
4. Which property can an FIR filter be designed to have exactly?Answer: Linear phase across frequency.
5. In a quantized feedback filter, what are limit cycles?Answer: Small persistent oscillations that can remain when the external input is idle.