## Abstract

Many real-time experiments involve the measurement of signals which fall exponentially with time. The task is then to determine, from such measurements, the number of terms n and the value of all the parameters in the exponentially damped model
φ(t) = sum_{j=1}^n α_j exp(φ_j t), α_j =β_j +i γ_j, φ_j =ψ_j +i ω_j
Here φ_j, ω_j, β_j and γ_j are respectively called the damping, frequency, amplitude and phase of each exponential term. Exponential models appear, for instance, in power system transient detection, motor fault diagnosis, electrophysiology, drug clearance monitoring and glucose tol- erance testing, magnetic resonance and infrared spectroscopy, vibration analysis, seismic data analysis, music signal processing, odour recognition and the electronic nose, typed keystroke recognition from a keyboard sound recording, nuclear science,...From this list, music is arguably the most highly structured signal. Decomposing a complex music signal into different components is an important and powerful preprocessing step for many applications. Much current research focuses on individual aspects of music (such as rhythm, or chords, or instruments) instead of the overall model, although these are anything but independent. Music is also one of the most popular types of online information and there are hundreds of music streaming and download services in operation. The general technique of multiexponential modelling is closely related to what is commonly known in the applied sciences as the Padé-Laplace method and the technique of sparse interpolation in the field of symbolic computation. We emphasize that it is generally believed that none of the listed limitations can be overcome!
• The problem of multiexponential modelling is an inverse problem and may be ill-posed. Even the recent algorithms do not yield reliable output when applied to our problem statement.
• The analysis of band-limited signals has given rise to a well-developed theory of resolution, associated with the names of Shannon and Nyquist. It states that, if Ω/2 is the highest frequency present in its spectrum, the frequency content of a signal is completely determined by its values at equidistantly spaced time points, 2π/Ω apart. A coarser time grid causes aliasing, identifying higher frequencies with lower frequencies without being able to distinguish between them. As a consequence, the exponential analysis as it is used today, suffers a similar frequency resolution limitation.
• In addition, in the presence of noise, different exponential decays ψi cannot be resolved if the ratio of the damping factors is less than some threshold, in other words when the damping constants are too much alike.
Our purpose is essentially to propose a regularization of the ill-posed problem above, by making use of the connections of the subject with sparse interpolation and Padé approximation. The trick is to exploit aliasing rather than avoid it! So the objective of the project can be summarized in the following keywords:
• fast → because we make use of much smaller-sized structured numerical linear algebra problems than theoretically required in Padé-Laplace,
• high-resolution → because we overcome the resolution limitations imposed by the Shannon-Nyquist bound on the one hand and the decay-grid on the other hand,
• well-conditioned → because we use the aliasing effect as a way to control the condition number of all the involved structured matrices.
Hence the regularization of a popular and widely present ill-posed multiexponential analysis problem!

## Researcher(s)

## Research team(s)