Tracking the frequencies of superimposed time-varying harmonics

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Abstract
The problem of tracking the frequencies of slowly time-varying superimposed sinusoids in the presence of noise is addressed. An algorithm based on singular value decomposition (SVD) of the data matrix is developed for the problem, and its performance is evaluated by numerical experiments on computer-synthesized data. The algorithm is based on the discovery that even when the frequencies are changing with time, as long as they change slowly locally, a Hankel matrix constructed directly from the noise-free signal is close to a matrix of rank equal to twice the number of real-valued sinusoids superimposed in the signal. Thus, in the presence of additive noise, instead of using the SVD of many small matrices constructed from local blocks of data, all the available data can be included in one large matrix, which can then be approximated by its principal singular vectors and singular values, to achieve greater noise suppression. The number of sinusoidal components and the instantaneous frequency tracks are directly estimated from the principal singular vectors of the large Hankel matrix.

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