MATLAB Implementation of Third-Order All-Pass Systems

Third-order all-pass systems are commonly used phase equalization modules in digital signal processing. Their key characteristic is a constant magnitude response of 1, while only adjusting the phase of input signals. Compared to first-order all-pass systems, the third-order structure can handle more complex phase correction requirements.

The implementation approach can be divided into the following steps: Pole Configuration: The zeros and poles of all-pass systems must maintain conjugate reciprocal relationships. Third-order systems typically consist of one real pole and a pair of complex conjugate poles, with zero positions mirroring their corresponding poles. Transfer Function Construction: Derive the system function based on zero-pole positions, where numerator and denominator polynomials have the same order and symmetric coefficients. MATLAB Implementation: Use the `tf` (transfer function) or `zpk` (zero-pole-gain) functions to establish transfer function models, and analyze phase response characteristics through the `freqz` function. The implementation involves creating polynomial coefficients where denominator coefficients appear in reverse order in the numerator.

Differences from first-order systems include: First-order all-pass requires only a single pole/zero pair with limited phase adjustment range Third-order systems can generate nonlinear phase curves through complex pole pairs, suitable for higher-order phase compensation scenarios In group delay characteristics, third-order systems exhibit more complex fluctuation patterns with multiple extremum points

Typical applications include phase distortion correction and filter bank delay equalization. In MATLAB, phase response differences between all-pass systems of various orders can be observed by comparing `angle(h)` outputs from frequency response calculations. Code implementation typically involves specifying pole locations (e.g., using `poly` function for polynomial conversion) and verifying the all-pass property through magnitude response plots.