A data driven compressive sensing approach for time-frequency signal enhancement
Introduction
In many real-life applications signal frequency content is a key information which needs to be extracted. In order to do this, one can simply use the Fourier transform, but in doing so, signal time attributes are lost. However, by using time-frequency distributions (TFD) one can analyze the evolution of signal energy as a function of both time and frequency, providing additional information about the signal nature. However, if signal has more than one linear frequency modulated (LFM) component, or a non-LFM component, its TFD gets corrupted by highly oscillatory artifacts. Quadratic TFDs (QTFD), most commonly used TFDs in practice, utilise filtering with 2D low-pass filters, which inherently affects time-frequency (TF) localization properties [1]. The need for a trade-off between interference suppression and TF localization has led to a number of TF localization improvement methods, one of which is described in the sequel.
Over the last few years, compressive sensing (CS) has been an important research topic [2], [3], [4], [5], with applications in medicine [6], [7], geophysics [8], [9], communication [10], [11], etc. Traditionally, compressive sensing (CS) implies signal sampling with sub-Nyquist frequencies, with samples randomly picked [2], [3], [4], [6]. However, the samples can be picked to favour specific signal features, while discarding the others [12], [13], [14]. This is followed by a signal reconstruction algorithm for solving unconstrained optimization problems (i.e. basis pursuit (BP) [15], [16], [17], [18], [19], [20], [21], [22], matching pursuit (MP) [23], orthogonal matching pursuit (OMP) [24], [25], etc.). However, the reconstruction algorithm leads to a meaningful result only if the signal is sparse, which means that the signal can be represented in a certain domain with K non-zero coefficients, where K ≪ Nt (Nt being the number of signal samples in the time domain). Most signals are non-sparse in the domain of interest, but can become sparse (or approximately sparse) by applying a domain transformation. For example, a sinusoidal signal can be represented with only one sample in the frequency domain. Ideal TFDs are sparse since they are composed of components instantaneous frequency (IF) trajectories, hence CS can be utilized in such a way to include only the signal components samples, while discarding the interference samples; by applying a reconstruction algorithm, resolution loss is minimised [7], [8], [13], [14], [25], [26], [27], [28], [29], [30].
Current TF signal processing approaches have focused on various optimization algorithms for signal reconstruction, leaving CS area size and shape selection underutilized. The CS area is usually a rectangle, containing approximately Nt samples inside of it [13], [26], [27]. In this paper, we propose a method for data driven automatic CS area selection. Our goal is to select CS area as large as possible without artifact inclusion, which then increases the signal reconstruction algorithm input data amount, hence decreasing its computational requirements.
The paper is organized as follows. Section 2 gives a short introduction to TFDs, while Section 3 describes TFDs as a sparsity inducing signal representation, and introduces the CS area selection algorithm. In Section 4 we compare different optimization algorithms performances based on the measure of reconstructed TFD concentration [31] for the case when the CS area is selected both manually and when it is selected automatically by the proposed method.
Section snippets
Quadratic time-frequency distributions
Let us consider an LFM signal z(t) with a time-varying phase φ(t), and a slowly varying amplitude A(t) of the form: Its ideal TFD, is a set of Dirac functions, with a perfect energy localization around the signals instantaneous frequency, ω0(t): with . The ideal TFD in most cases is impossible to accomplish, since practical TFDs are not perfectly localized, and furthermore, they are corrupted by the cross-terms. The Wigner-Ville
Compressed sensed ambiguity function
As mention earlier, ideal TFDs are inherently sparse, since they are composed of the components IFs, thus requiring only NcNt < <NωNt samples, where Nc, Nt, and Nω are the number of components, the number of time instances, and the number of frequency bins, respectively. The CS-AF, is formulated as: where the operator ⊙ denotes element-by-element matrix multiplication, and ϕ(ν, τ) is the sensing matrix which defines area Ω around the AF plane origin:
Experimental results
Example 1 A multi-component signal with samples is composed of LFMs:
The first component, z1(t), has been modeled in such a way that it is almost completely masked by the cross-terms between the other two components, making it almost unidentifiable in the WVD, as shown in Fig. 1(a). Fig. 1(b) shows RGK of the considered signal with the kernel volume parameter . The current CS area selection methodology gives
Conclusion
We have presented a method for an automatic data-driven selection of the CS area for enhancement of the signal TFD. The CS area is generally selected experimentally, with the area size being approximately Nt samples in the ambiguity domain. The method proposed in this paper automates this procedure, resulting in CS areas that contain more useful samples. Furthermore, the proposed method also allows a less strict optimization criterion of the sparse signal reconstruction algorithm, ultimately
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2018, Signal ProcessingCitation Excerpt :For example, the work in [21] using a reduced interference TFD, first suppresses the cross-terms of the multi-component signal and then by implementing two image processing tools, i.e. local peak detection and component linking, determines the TF representation of non-stationary signal in high noise environments. Another category of references estimates the signal parameters or represents a localized TFD by defining an optimization problem [22–26]. For instance, the work in [26] considers the non-stationary signals contaminated by additive white Gaussian noise at low SNRs and using the ant colony optimization algorithm, specifies the auto-terms of TFD.