Applying the Spectral Method for Modeling Linear Filters: Butterworth, Linkwitz-Riley, and Chebyshev filters

TL;DR

The paper introduces a spectral form of mathematical description for continuous-time linear filters, representing input and output signals via orthonormal expansions and describing filters with a two-dimensional non-stationary transfer function $W$. It derives explicit spectral representations for Butterworth, Linkwitz–Riley, and Chebyshev families (Type I and II) and demonstrates how to compute the output signal analytically from the input by algebraic transformations of expansion coefficients, without time discretization. Through deterministic and random-noise tests, it shows that the spectral-method error is negligible compared to intrinsic filter error, with performance improving as filter order increases and truncation order $L$ is chosen appropriately. The approach is extensible to other filter families (e.g., elliptic, Gaussian, Legendre) and provides a natural framework for continuous-time modeling and comparison in signal-processing tasks, offering potential advantages in avoiding aliasing and frequency-warping associated with discrete-time methods.

Abstract

This paper proposes a new technique for computer modeling linear filters based on the spectral form of mathematical description of linear systems. It assumes the representation of input and output signals of the filter as orthogonal expansions, while filters themselves are described by two-dimensional non-stationary transfer functions. This technique allows one to model the output signal in continuous time, and it is successfully tested on the Butterworth, Linkwitz-Riley, and Chebyshev filters with different orders.

Figures (9)

• Figure 1: The characteristic polynomial of the Butterworth filter ($n = 3$)
• Figure 2: The characteristic polynomial of the Linkwitz--Riley filter ($n = 4$)
• Figure 3: The characteristic polynomial of the Chebyshev Type I filter ($n = 5$)
• Figure 4: The characteristic polynomial of the Chebyshev Type II filter ($n = 2$)
• Figure 5: The graphs of the useful signal and the sum of useful signal and deterministic noise