15.5 : 極とシステムの安定性

In analyzing audio systems, understanding transfer functions is crucial.

The ratio of two polynomials represents the system's transfer function. Zeros and poles characterize this function, influencing system stability.

Simple poles are unique roots in the denominator, representing distinct solutions to the characteristic equation.

Repeated poles occur more than once, indicating oscillatory behavior or slower decay rates.

Complex poles with real and imaginary parts suggest oscillatory components in the system response.

These pole combinations apply to proper rational functions, where the numerator's degree is less than the denominator's.

Distinct poles undergo partial fraction expansion and inverse Laplace transform.

Pole locations on the s-plane determine the stability of an LTI system.

For BIBO stability, all poles must be in the left half-plane, leading to decay over time. Repeated poles contribute to BIBO stability, yielding an integrable impulse response.

Poles in the right half-plane indicate instability due to exponential growth.

Proper rational functions follow strictly proper stability rules while improper functions are BIBO unstable.