Orthogonal Mode Decomposition for Finite Discrete Signals

TL;DR

This work addresses the challenge of decomposing finite-length, non-stationary signals into meaningful components by proposing Orthogonal Mode Decomposition (OMD), a local, projection-based framework operating in an interpolation-function space $oldsymbol{IFS_{(n,\Delta)}}$. By defining the intrinsic mode set via narrow-band bands and monotone intrinsic phase, the method yields a unique, orthogonal collection of modes, unlike traditional EMD and VMD approaches that suffer from mode-mixing and non-uniqueness. The authors introduce parity-based intrinsic phase and instantaneous frequency calculations, enabling precise bandwidth determination and selective mode extraction through orthogonal projection onto real and imaginary part spectra. The approach offers computational efficiency, explicit bandwidth definitions, and robust handling of boundary effects, making it well-suited for real-time applications such as signal filtering and fault diagnosis. Overall, OMD provides a rigorous, targetable alternative for time-frequency analysis of finite discrete signals with strong theoretical and practical implications.

Abstract

In this paper, an orthogonal mode decomposition method is proposed to decompose ffnite length real signals on both the real and imaginary axes of the complex plane. The interpolation function space of ffnite length discrete signal is constructed, and the relationship between the dimensionality of the interpolation function space and its subspaces and the band width of the interpolation function is analyzed. It is proved that the intrinsic mode is actually the narrow band signal whose intrinsic instantaneous frequency is always positive (or always negative). Thus, the eigenmode decomposition problem is transformed into the orthogonal projection problem of interpolation function space to its low frequency subspace or narrow band subspace. Different from the existing mode decomposition methods, the orthogonal modal decomposition is a local time-frequency domain algorithm. Each operation extracts a speciffc mode. The global decomposition results obtained under the precise deffnition of eigenmodes have uniqueness and orthogonality. The computational complexity of the orthogonal mode decomposition method is also much smaller than that of the existing mode decomposition methods.

Figures (14)

• Figure 1: The interpolation function for the finite length discrete real signals
• Figure 2: $U_u (\omega)$ and $U_{\Omega_\Delta} (\omega)$
• Figure 3: Example 1, $\omega(t)$ is always positive, and $\Psi_{u}(t)$ is a mode
• Figure 4: Example 2, both $Fre(\omega)$ and $Fim(\omega)$ have only one principal lobe,$\Psi_{u}(t)$ is a mode
• Figure 5: Example 3, $\omega(t)$ is located at the geometric center of the Fourier spectrum