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Research in fields such as communication, speech, oceanography, seismic exploration, economics, and biomedical data processing is often directed toward the analysis of nonstationary or transient data. Complex demodulation is shown to be a valuable method that can be used to decompose a composite signal composed of differing transient wavelets and to estimate spectra. For the latter application it is shown that several other techniques recently advanced in the literature are special cases of complex demodulation. The algorithm discussed can achieve the decomposition of a certain class of noisy composite signals composed of nonidentical unknown multiple wavelets overlapping in time, namely those signals with reasonably well-defined independent resonances in the spectrum. The decomposition estimates the arrival time, peak, envelope, and frequency of the damped oscillatory transient wavelet. The procedure has been tested extensively and several selected experimental results are tendered. It has been found that for wavelets of the type , , that an uncertainty relationship for the product of the 3-dB bandwidth and the time duration of the wavelet must be satisfied. An error analysis has established a relationship between envelope-and phase-estimation errors to wavelet and filter parameters. The results obtained via complex demodulation are discussed relative to those obtained via inverse filtering, the complex cepstrum, and the chirp z transform.