All-Pole Filter
What it Does
An all-pole filter shapes the frequency response using only poles and no zeros. It can act as a low-pass, high-pass, or resonator depending on the pole locations. All-pole filters are commonly used to model resonances in physical systems or speech formants.
Mathematical Model (Continuous-Time)
H(s) = G / ((s - p₁)(s - p₂) ... (s - pโ))
Where:
- G = gain constant
- pแตข = poles of the system
- n = filter order
All poles → numerator is just a constant → no frequency is completely canceled
Frequency Response
- Magnitude depends entirely on pole locations
- Poles closer to the imaginary axis → sharper resonance
- Poles closer to the origin → smoother response
Digital (Discrete-Time) Version
H(z) = G / (1 - a₁ z⁻¹ - a₂ z⁻² - ... - aโ z⁻โฟ)
aแตข = filter coefficients; poles are solutions of the denominator polynomial. Magnitude and phase are controlled entirely by the poles.
Applications
- Speech processing: Linear Predictive Coding (LPC) models vocal tract
- Audio: Resonators, formant filters
- Control systems: Feedback-based filter designs
Core Idea
Remember: All-pole filter = only poles, no zeros; peaks in magnitude occur near poles, phase changes smoothly.