Signal Decomposition and Time-Frequency Representation Using Variable-Length Symmetric Filters

Milton José Porsani, Bjorn Ursin


We present a time-frequency decomposition method to represent a time signal into a 2D (time X frequency) image, which describes how the frequency content varies along the time. This is done in two steps: firstly, by filtering the signal to obtain time-components; and secondly, by computing the average instantaneous frequency (AIF), which is used for moving the data components to the time-frequency plane. For the filtering process, we present an algorithm to generate a suite of symmetric filters that are computed recursively, starting with the high-frequency content of the signal, going down in frequency and leaving the lowest frequencies in the last filter component. This can be further decomposed by continuing the procedure. The symmetric impulse responses are zero-phase with positive frequency response, and they add up to a spike at the origin with a unit frequency response. The filtering procedure gives an exact decomposition of the signal and the traveltimes are preserved. Next, the analytic signal of each component is used for computing the AIF in sliding time windows, so that for each time sample, we have an associated AIF value. The 2D time-frequency plane is obtained by distributing and adding the data components along the frequency variable. Finally, by using the time X frequency distribution, a time-frequency filtering may be performed by stacking data of sub-domains with similar features. The new technique has been applied to two synthetic signals which have previously been analyzed by many authors using a variety of algorithms. The new signal decomposition algorithm and the AIF computation are simple and produce effective results on the synthetic data.


time-series analysis; time-frequency representation; seismic noise

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