Abstract
This article investigates the possibility and convenience of a filtering operation in the joint time/chirp-rate (TCR) domain, and proposes a novel linear TCR filter for decomposing multicomponent signals into their quadratic and/or cubic phase chirp components with monotonic instantaneous chirp-rate (ICR) laws only. The TCR domain mask of the filter is selected on a display of a TCR representation of an input signal to isolate the desired chirp component. Projecting the input signal onto the phase signal associated with the TCR mask and approximating the phase difference in this projection operation in terms of ICR values result in the proposed TCR filter that recovers the selected component. Simulations illustrate the proposed filtering in recovery of undersampled cubic phase signals and in resolving back-to-back objects from in-line holograms for which cases it is easier to design filter masks in the TCR domain than in the time-frequency domain.
Highlights
Multicomponent nonstationary signals are widely encountered in many applications including radar, sonar, communications and optics
There are various linear TF filter types; such as Zadeh [18], Weyl [19,20] and generalized Weyl filters [21,22] encompassing these two, TF projection filters [23,24,25], short-time Fourier transform (FT) filter by means of an analysis-masking-synthesis procedure [26,27], local polynomial FT filter [28], S-transform filters again based on analysis-masking-synthesis approaches [29,30,31], and a method for chirp signal reconstruction from ridges of Gabor and wavelet transforms of the analyzed signal [32,33,34], among others
Motivated by the above application, we propose a novel linear time-varying filter in the TCR domain, reminiscent to TF filtering, for decomposing multicomponent signals to reconstruct their chirp components of the form a(t) exp(j2π(t)), where (t) is a quadratic or cubic phase with a monotonic instantaneous chirp-rate (ICR) law
Summary
Multicomponent nonstationary signals are widely encountered in many applications including radar, sonar, communications and optics. Both approaches lead to time varying impulse responses with four product terms in them These alternative filters can successfully recover quadratic and cubic phase signals with monotonic ICR laws exhibiting single linear tracks, as the one proposed in Equation (8) does. Our simulations indicate that their performances in chirp signal recovery are worse than that of the proposed one, since their equivalent TF transfer functions exhibit more severe peaks near the origin of the TF plane Their discrete implementations require more than one discrete TCR mask functions to be prepared and used, each for a different product term in the filter impulse response. Our proposed filter in Equation (8) has the best separation performance and is easiest to implement, among them
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