Multiscale Chirplets and Near-Optimal Recovery of Chirps

This paper considers the model problem of recovering a signal f(t) from noisy sampled measurements. The objects we wish to recover are chirps which are neither smoothly varying nor stationary but rather, which exhibit rapid oscillations and rapid changes in their frequency content. We introduce a mathematical model to describe classes of chirps of the general form f(t) = A(t) cos(λφ(t)) where λ is a (large) base frequency, φ(t) is time-varying and A(t) is slowly varying, by imposing some smoothness conditions on the amplitude A(t) and the “instantaneous frequency” φ′(t). For example, our models allow the unknown object to oscillate at nearly the sampling/Nyquist rate. Building on recent advances in computational harmonic analysis, we construct libraries of tight frames of multiscale chirplets which are rapidly searchable and with fast algorithms for analysis and synthesis. We show that it is possible to invoke lowcomplexity algorithms which select a best tight-frame from our library in which simple thresholding achieves nearly minimax mean-squared errors over our classes of chirps. Our methodology is adaptive in the sense that it does not require a-priori knowledge of the degree of smoothness of the amplitude and the instantaneous frequency, and nearly attains the minimax risk over a meaningful range of chirp classes.