Controlled-coverage discrete S-transform (CC-DST): Theory and applications

Abstract

The S-transform (ST) is a popular linear time-frequency (TF) transform with hybrid characteristics from the short-time Fourier transform (STFT) and the wavelet transform. It enables a multi-resolution TF analysis and returns globally referenced local phase information, but its expensive computational requirements often overshadow its other desirable features. In this paper, we develop a fully discrete ST (DST) with a controllable TF sampling scheme based on a filter-bank interpretation. The presented DST splits the analyzed signal into subband channels whose bandwidths increase progressively in a fully controllable manner, providing a frequency resolution that can be varied and made as high as required, which is a desirable property for processing oscillatory signals lacked by previously presented DSTs. Thanks to its flexible sampling scheme, the behavior of the developed transform in the TF domain can be adjusted easily; with specific parameter settings, for example, it samples the TF domain dyadically, while by choosing different settings, it may act as a STFT. The spectral partitioning is performed through asymmetric raised-cosine windows whose collective amplitude is unitary over the signal spectrum to ensure that the transform is easily and exactly invertible. The proposed DST retains all the appealing properties of the original ST, representing a local image of the Fourier transform; it requires low computational complexity and returns a modest number of TF coefficients. To confirm its effectiveness, the developed transform is utilized for different applications using real-world and synthetic signals.

Introduction

The spectral analysis of signals is a crucial task in countless scientific applications. As well known, the Fourier transform (FT) is a powerful tool for decomposing a stationary signal into individual frequency components, but it is unsuitable to represent dynamic spectral contents. The need for properly analyzing nonstationary signals, whose spectra evolve with time, has motivated the development of various tools to enable efficient time-frequency (TF) analysis [1], [2], [3], [4]. Among these tools, the S-transform (ST) [5], [6] uniquely combines appealing characteristics of two conventional TF transforms: Like the classical short-time Fourier transform (STFT), the ST provides a TF representation by projecting the analyzed signal onto local TF functions derived from the Fourier basis by a simple windowing operation. Therefore, the ST independently localizes the real and imaginary parts of the spectrum, providing globally referenced local phase information, thereby preserving the intuitive meaning of phase given by the FT. Unlike the STFT, however, and similarly to the wavelet transform (WT) [7], the ST enables a multi-resolution analysis by using windows the temporal width of which reduces with increasing frequency. In doing so, the ST captures local variations in the global spectrum with a progressive trade-off between time and frequency resolution in accordance with Heisenberg's uncertainty principle. Furthermore, the analytic Fourier-similar basis functions of the ST makes certain tasks relevant to oscillatory signal processing, such as instantaneous frequency (IF) estimation, easier to handle compared with the WT (we refer the reader to [8], [9] for a detailed comparison between the two transforms). In addition, the progressive resolution renders a TF representation that is more consistent than that presented by a constant-bandwidth transform, like the STFT. Although the previous desirable features have made the ST a popular TF tool in various fields of study, such as medicine [10], seismology [11], and power engineering [12], just to name a few, the computational complexity required to realize this transform has restrained its fields of application to signal processing of relatively short signals, hindering more widespread usage. This excessive complexity is because, in a departure from the uncertainty principle, the original discrete ST (DST), as presented by Stockwell et al. in [5], samples the TF domain with a uniform lattice, which results in over-redundant TF information. Most of the research on the ST focused on designing new observation windows based on energy concentration (EC) criteria to improve localization of the signal components in the TF domain [13], [14], [15]. However, unlike the case with STFT and the WT, developing efficient sampling schemes for the ST is a topic that has not been explored enough and needs more research. In this context, Stockwell in [6] and Brown et al. in [16] presented dyadically sampled one-to-one DSTs. The core idea is based on partitioning the signal spectrum into non-overlapping subbands the sizes of which double from one subband to the next one. The time-domain signals carrying these subbands are sampled at the Nyquist rate, meaning that the rate is also doubled from one subband signal to the next one. To divide the spectrum this way, non-overlapping boxcar functions are used in [6], whereas in [16], truncated non-overlapping Gaussian windows are employed for their improved time localization. Over-redundancy of the TF information provided by the ST is also recognized in [17]; the authors therein proposed a harmonic sampling scheme tailored for power applications. Because of the rigid manner in which they partition the frequency domain, all the previous transforms are characterized by having low spectral resolution (i.e., they have a low Q-factor, as explained later), which limits their effectiveness in processing signals of clear oscillatory behavior, such as audio, frequency-modulated, and some biomedical signals, as is the case with the dyadic WT. To process such signals, we are particularly interested in a transform that allows for a finer frequency analysis, where the spectral resolution increases more gradually. To the best of our knowledge, a DST with the previous characteristics has not been developed yet.

Our main contribution in this paper is the construction of a DST with a controllable covering of the TF domain. By means of the proposed transform, the frequency domain is divided into subbands through asymmetric raised-cosine windows whose bandwidths grow progressively but in a fully controllable manner. That is, two parameters are introduced in the developed transform: one defines the size of the narrowest frequency window, while the second controls the relative frequency spacing between two successive windows. The time domain is sampled non-uniformly in accordance with the Nyquist Theorem, meaning that each subband channel has its own data rate, which is equivalent to the double-sided bandwidth of the respective partitioning window. With this level of control on its sampling scheme, the presented DST can be made a dyadic transform, and by choosing particular parameter settings, it can provide spectral resolution as high as desired. This transform requires low computational resources and produces a modest number of TF coefficients, comparable to the number of signal samples. Moreover, the windows by which the frequency domain is partitioned have a unitary composite amplitude, making the proposed transform easily and exactly invertible through a simple method, which resembles the overlap addition (OLA) algorithm used for inverting the STFT.