Fast Sequential LS Estimation for Sinusoidal Modeling and Decomposition of Audio Signals

This work demonstrates a sequential Least Squares algorithm applied to the decomposition of sounds into sines-plus-residual models. For a given basis of r distinct frequency components, the algorithm derives recursively the Least Squares estimates of the associated amplitudes and phases. While a direct calculation achieves a O(nr
²
) complexity the main cost of our implementation is only of 4r multiplications per sample, whatever the length n of the analysis window. The technique is extended to basis of exponentially increasing or decreasing frequency components, which provides a fast and enhanced decomposition of rapidly varying segments of the sound. Finally, the proposed method is successfully applied to a real piano note.