Optimized sparse fractional Fourier transform: Principle and performance analysis
Introduction
Fractional Fourier transform (FrFT) is a powerful mathematical tool to analyze nonstationary signals, which can be interpreted as a projection of the original signal onto a rotated time axis in the time frequency domain [1], with the rotation extent controlled by an adjustable rotation angle α. Although the real world is analog, computations are digital. Hence, instead of explicit numerical integration of input signals, the discrete counterpart is essential for all transforms to process the point-sampled signals in the digital world using computers or dedicated hardware. In this context, many types of definitions for discrete FrFT (DFrFT) have been proposed since the past decades (See, e.g., [2], [3], [4], [5]). Among these definitions, the algorithm in Ozaktas et al. [4] and Pei and Ding [5], which are both derived by sampling the continuous FrFT, can achieve low computational cost and good approximation to the continuous FrFT. Hence, they have been widely applied in practical scenarios. This work is developed based on the latter, which is referred to as Pei’s algorithm in this paper. Using Pei’s algorithm, an order-α DFrFT, which is denoted as of an input time sequence si with a sampling interval Δt and data length N is computed aswhere and represent fast Fourier transform (FFT) and inverse FFT (IFFT), respectively. The Λ and the sgn( · ) respectively denote an arbitrary integer and a signum function, and . The output are discrete points in the fractional Fourier domain (FrFD) with a sampling interval Δu. Variable m represents the fractional Fourier index in the FrFD, and Eqs. (1a) and (1c) holds for all . Note that, N should be smaller than or equal to M, and we only consider the case in this paper. If α ≠ Λπ, the computational complexity of Pei’s algorithm is . When N is large, Pei’s algorithm becomes time demanding, which is unaffordable for real-time processing.
Fast numerical algorithms analogous to the well-known FFT algorithm is indispensable to the practical application of DFrFT. Recent attempts in designing such algorithms include the work in Neto and Lima [6], where eigenvector decomposition type DFrFT is considered and orthonormal bases of Hermite-Gaussian-like eigenvectors are used to fractionalize the DFrFT. This algorithm is further optimized by introducing a rounding strategy [7], which is able to reduce up to 65% of multiplications. In [8], real DFrFT is constructed using a real random discrete Fourier transform (DFT)-commuting matrix, and its eigenvalues are either or 1. Nevertheless, the above eigendecompostion-based approach cannot be expressed in a closed form, and the runtime is O(N2) for an N-point dataset.
Leveraging recent advances in sparse Fourier transform (SFT) [9], we proposed a sparse discrete fractional Fourier transform (SDFrFT) based on Pei’s sampling algorithm to facilitate the fast computation of large-scale DFrFT [10]. SDFrFT is able to resolve multiple signal components and it has achieved some initial successes in practical applications such as wideband spectrum sensing [11] and radar target detection [12]. SDFrFT is composed of three stages. In the first stage, a rotation is performed with a proper rotation angle α, such that the fractional Fourier spectrum is sparse in the corresponding FrFD. In the second stage, the large Fourier coefficients is mapped into a down-sampled spectrum. The resulting spectrum is then utilized to recover the large coefficients. In the third stage, the signal is multiplied by another quadratic phase to obtain the output. The complexity of the second stage for SDFrFT is reduced to where k denotes the number of large coefficients in the spectrum. Later in Chen et al. [13], the closed-form expressions of several performance metrics of SFT were derived, and a mode-mean estimation algorithm was proposed to improve the estimation performance. Collectively, these performance metrics can serve as a guideline to design and evaluate the second stage of SDFrFT. In [14], the exact and fast DFT computation was considered in polar and spherical coordinates, and the corresponding inverse transform was also presented. Fast Fourier aliasing-based sparse transform (FFAST) [15] was also designed for sparse signals. Nevertheless, FFAST framework is quite different from SDFrFT. Concretely, the sparse-graph alias code was induced in the spectrum, and Chinese-remainder-theorem-guided down-sampling was performed in time domain. Then the induced code is utilized by a decoder to achieve k-sparse DFT. Subsequently, FFAST was generalized to the noise-corrupted signals and robust FFAST (R-FFAST) [16] was proposed. R-FFAST can approach perfect recovery accuracy, but its computational complexity is higher than that of SFT. Additionally, to use R-FFAST, the values of k and N should satisfy certain requirements, while SDFrFT works for more generic input signals. In [17], the arithmetic discrete fractional Fourier transform (ADFrFT) is proposed, and the computational complexity is reduced to . Nevertheless, ADFrFT requires nonuniform sampling of the time domain signal, and the sampling points are specially designed. As such, its applicability is narrowed and its computational complexity is incommensurable with generic uniform-sampling algorithms such as Pei’s algorithm.
For large-scale input signals, SDFrFT can significantly reduce the computational complexity of DFrFT. However, when the length of input signal is relatively small, SDFrFT has no obvious advantage compared to DFrFT in terms of computational complexity. In this regard, we aim to optimize SDFrFT to further reduce the computational complexity, as well as to achieve sufficient anti-noise performance and estimation accuracy. For noiseless exactly k-sparse signals in the frequency domain, phase information has been exploited to locate the large coefficients in very few loops [18]. To realize fractional frequency estimation for noise-corrupted signals, Neyman-Pearson (NP) detection [19] can be applied. Nevertheless, the impact of noise on the phase difference retrieval must be analyzed and mitigated beforehand. Otherwise, the resultant location estimation error can be very large. Thereby, the novel contributions are summarized as follows. (1) The noise level in different stages of the k-sparse algorithm in Hassanieh et al. [18] are analyzed, and NP detection is used to deal with noise-corrupted signal; (2) The noise effect on the location step is analyzed, and the distribution of phase error is obtained via Parzen-Rosenblatt window method [20], [21]; (3) Location error correction method is proposed. (4) Important properties of OSFrFT are investigated via extensive simulations. (5) We apply OSFrFT to processing the real data collected from a continuous wave (CW) radar, and the velocity of a free falling target is estimated.
The rest of the paper is organized as follows. An overview of the core modules of SDFrFT algorithm is provided in Section 2. Then, in Section 3, we propose the NP detection in OSFrFT. Analysis of noise effect on the location step of OSFrFT and the location error correction method are elaborated in Section 4. Algorithm flow and performance analysis of OSFrFT are given in Section 5. Numerical results are shown in Section 6. The application of OSFrFT on CW radar for falling target detection is presented in Section 7. This paper is concluded in Section 8.
Notations: We use {N} to denote the set of indices and the indices are interpreted modulo N. xi represents the ith element of vector . represents the maximum value in and represents the spectrum of . The permutation operator Pσ,a,b is defined as where . The modular inversion of σ in the sense of modulus N is defined as where . The symbol * represents convolution between two signals. ϕ( · ) represents the phase of a complex variable. ⌊ · ⌋, ⌈ · ⌉, and round( · ) respectively represent round towards negative infinity, positive infinity, and nearest integer.
Section snippets
Preliminaries
First of all, we provide a brief review of the two core modules of SDFrFT in Liu et al. [10], i.e., hash-to-bins process and location and estimation loops.
NP Detection of OSFrFT algorithm
In this section, we assume that the input signal is a linear frequency modulated (LFM) signal, whose amplitude is A. The signal is corrupted by additive complex Gaussian noise satisfying . Assume α is matched with the chirp rate. After the first stage of SDFrFT, is composed of noise spectrum and line spectrum of the complex exponential signal. The where . The (i ∈ {N}) are N samples of the noise spectrum. The spectrum of the exponential signal is
Noise effect on the location step
The effect of noise on the location step is analyzed in the following two perspectives. The first perspective is to analyze the relationship between the phase error and the noise level. The second perspective is to analyze the effect of phase error on the estimation result.
Algorithm flow and performance analysis of OSFrFT algorithm
Based on the optimized spectrum permutation, NP detection and location error correction, OSFrFT is proposed. The pseudo code of OSFrFT is summarized in Algorithm 2.
Analysis of phase error
f(ϕerr) is influenced by the number of bins B and the input SNR. The effect of these parameters is shown in the following numerical simulations. Parameters are set as 4096, 256 or 512, and 0. The input SNRs are set as -8.1648 dB and −2.1442 dB, respectively. The result is illustrated in Fig. 1. The probability histograms represent the simulation results and the red lines are the PDF estimated by KDE.
Fig. 1 indicates that the estimated PDF fits well with the distribution
Application to falling target detection
We demonstrate in this section a successful application of OSFrFT to the CW radar signal processing. The experimental radar platform is SDR-KIT 580B from Ancortek, which operates at CW mode with a center frequency . We previously have also used this radar platform to test our human fall detection scheme [25]. In this experiment, a metal cubic portable power pack is released at a hight of around 1.25 m, and it starts to free fall with a zero initial velocity. The transmitting and
Conclusion
The growing demands for realtime, robust and accurate time-frequency analysis of nonstationary signals in recent years have spurred efforts to develop advanced algorithms to solve a plethora of grand challenges. In this article, we have presented an OSFrFT algorithm on the basis of state-of-the-art sparsity-based approaches to further optimize the numerical computation of DFrFT. A comprehensive optimization scheme has been proposed. Concretely, NP detection is used to cope with the scenarios
CRediT authorship contribution statement
Hongchi Zhang: Methodology, Software, Investigation, Data curation, Writing - original draft. Tao Shan: Conceptualization, Methodology, Supervision, Resources, Validation, Project administration, Funding acquisition. Shengheng Liu: Conceptualization, Writing - original draft, Writing - review & editing, Supervision, Visualization, Funding acquisition. Ran Tao: Conceptualization, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was supported in part by the National Natural Science Foundation of China under grant nos.61731023, 61671060, 61421001, U1833203. The work of S. Liu was also supported in part by the Basic Research Program of Jiangsu Province under grant no. SBK2019042353.
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