Signal decomposition and reconstruction using complex exponential models

• We develop a signal decomposition method theoretically identical to Prony's method.
• Unlike Prony's method the new method is robust to sampling rate and round-off error.
• The new method can reject signal noise via truncated singular value decomposition.
• The new method is applicable to intermittent signals.
• The new method outperforms discrete Fourier transform in harmonic analysis.

The theme of this paper is signal decomposition and reconstruction, not specific for or limited to system identification. In dealing with aperiodic damped signals, Prony-based techniques — which decompose a signal into real- and/or complex-valued exponential components — are often utilized. In essence, the derivation of Prony's method has been based on a high order homogeneous difference equation. In this paper, an alternative approach that uses a first-order matrix homogenous difference equation (state-space model) to replace the high order homogenous difference equation is advocated. Although the proposed method and Prony's method appear to be theoretically identical, this paper shows that they are drastically different over crucial numerical issues, including conditioning and stability. While Prony's method is very sensitive to sampling rate and round-off error, the proposed method is not. While Prony's method has trouble to deal with noise embedded in the signal, the proposed method can handle noisy signals properly because it has a build-in noise rejection mechanism via the usage of truncated singular value decomposition. While root-finding of a high order polynomial — a classical ill-conditioned problem — is a required step in Prony's method, the proposed method completely avoids it. The proposed method is also applicable to intermittent signals, and can recover the missing parts of intermittent signals nicely through reconstruction.