Identiication of Chirps with Continuous Wavelet Transform

Chirps are signals (or sums of signals) that may be characterized by a local (i.e. time-dependent) amplitude and a local frequency. Time-frequency representations such as wavelet representations are well adapted to the characterization problem of such chirps. Ridges in the modulus of the transform determine regions in the transform domain with a high concentration of energy, and are regarded as natural candidates for the characterization and the reconstruction of the original signal. A couple of algorithmic procedures for the estimation of ridges from the modulus of the (continuous) wavelet transform of one-dimensional signals are described, together with a new reconstruction procedure, using only information of the restriction of the wavelet transform to a sample of points from the ridge. This provides with a very eecient way to code the information contained in the signal. 1 Generalities There exists a large class of signals that may be modeled as sums of amplitude and frequency modulated components, i.e. in the form f(x) = X k A k (x) cos k (x) (1) where the relative variations of the amplitudes are assumed to be small compared with the oscillations, and the local frequencies k (x) = 1 2 0 k (x) (2) are assumed to be slowly varying. The characterization of such signals and the separation of their components (in the presence of noise) is a classical problem of signal analysis and signal processing, that goes back to pioneering work of J. Ville 16]. Applications can be found in many situations, such as for instance radar/sonar detection and speech processing 13]. Clearly, time-frequency methods (linear methods such as wavelet or short time Fourier transforms, or bilinear methods such as Wigner distributions) can provide satisfactory answers, at least in some situations and for large values of the signal to noise ratio.