Sparse Fast Fourier Transform Research Articles

From FFT to sparse FFT: Innovations in efficient signal processing for large sparse data

This paper explores the advancements from the traditional Fast Fourier Transform (FFT) to the Sparse Fast Fourier Transform (sFFT) and their implications for efficient signal processing of large, sparse datasets. FFT has long been a fundamental component in digital signal processing, significantly lowering the runtime of the Discrete Fourier Transform. However, the ingress of big data has necessitated much more efficient algorithms. In contrast, the sFFT exploits the sparsity in the signals themselves to reduce computational demand, and it becomes very efficient. This paper will discuss the theoretical backing of these two developments, FFT and sFFT, and the algorithmic development in both. In addition, it will also discuss the practical applications of both with emphasis on how the latter outperforms the former in large, sparse data. Comparative analysis shows that sFFT has far greater efficiency and noise tolerance, which is of value for network traffic analysis, astrophysical data analysis, and real-time medical imaging. The purpose of this paper is to provide clarity regarding these transformations and their relationship to being paradigms in modern signal analysis.

The uniform sparse FFT with application to PDEs with random coefficients

We develop the uniform sparse Fast Fourier Transform (usFFT), an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The algorithm is an adaption of the sparse Fast Fourier Transform (sFFT), a dimension-incremental algorithm, which tries to detect the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. The usFFT does this w.r.t. the stochastic domain of the PDE simultaneously for multiple fixed spatial nodes, e.g., nodes of a finite element mesh. The key idea of joining the detected frequency sets in each dimension increment results in a Fourier approximation space, which fits uniformly for all these spatial nodes. This strategy allows for a faster and more efficient computation due to a significantly smaller amount of samples needed, than just using other algorithms, e.g., the sFFT for each spatial node separately. We test the usFFT for different examples using periodic, affine and lognormal random coefficients in the PDE problems.

Roll Eccentricity Signal Detection and Its Engineering Application Based on SFFT-IAA

The roll eccentricity signal is a weak and complex periodic signal that is difficult to be identified. To improve the detection accuracy of the roll eccentricity signal and to compensate effectively, this study proposed a roll eccentricity signal detection method by combining the sparse fast Fourier transform (SFFT) and the iterative adaptive approach (IAA). The proposed method can rapidly determine the frequency range of the roll eccentricity signal by using the SFFT. Then, it divides the frequency range into several small frequency bands. In each small frequency band, the frequency, amplitude, and phase angle of each harmonic component of the roll eccentricity signal were estimated by using IAA. The simulation results show that the proposed method can find all frequency components, and the frequency estimation accuracy is higher than 99.88%. Finally, the engineering application of this method and the eccentricity compensation control were investigated. In engineering applications, the proposed method can reduce the thickness fluctuation of the finished strip by 89.2%, and the product quality is improved significantly. The simulation results and engineering experiments show that the proposed method has an excellent effect on detecting and compensating roll eccentricity signals.

Two Efficient Sparse Fourier Algorithms Using the Matrix Pencil Method

The research on efficient computation of sparse signals by various Sparse Fast Fourier Transform (sFFT) algorithms has always been a hot topic in the direction of signal processing. The algorithms can decrease the sampling and running complexity by taking advantage of the signal’s inherent characteristics that a large number of signals are sparse in the frequency domain. The sFFT algorithm is generally divided into two stages: the first stage is bucketization. The process is to divide N frequencies into B buckets through the filter. The main filters used are the flat window filter and the aliasing filter. The second stage is the spectrum recovery. The process is to successfully locate the position of the large frequency in each bucket and successfully calculate the amplitude. Among these steps, the most difficult and time-consuming step is to successfully locate the position of the large value frequency. For the sFFT algorithm based on a flat window filter, the voting method used in sFFT1.0 and sFFT2.0 algorithms should use many rounds, so these two algorithms are time-consuming and indeterministic, while the phase estimation method used in sFFT3.0 and sFFT4.0 algorithms has medium robustness. For sFFT algorithms based on an aliasing filter, the Prony method used in sFFT-DT1.0 algorithm is only applicable to noiseless signals, while the enumeration method used in the sFFT-DT2.0 algorithm has high complexity and poor robustness. In view of the performance of the old methods, new and more efficient methods are needed to achieve spectrum recovery. The spectrum restoration can be converted to estimating the complex amplitudes and attenuation coefficient in the model of the sum of complex exponentials. The matrix pencil method is a standard technique for mode frequency identification. Therefore, we propose the sFFT5.0 algorithm and sFFT-DT3.0 algorithm using the matrix pencil method to do spectrum recovery. These two algorithms are low computational complexity and strong robustness and have achieved good results in the actual comparative test.

Empirical Evaluation of Typical Sparse Fast Fourier Transform Algorithms

Computing the Sparse Fast Fourier Transform(sFFT) has emerged as a critical topic for a long time. The sFFT algorithms decrease the runtime and sampling complexity by taking advantage of the signal's inherent characteristics that a large number of signals are sparse in the frequency domain. More than ten sFFT algorithms have been proposed, which can be classified into many types according to filter, framework, method of location, method of estimation. In this paper, the technology of these algorithms is completely analyzed in theory. The performance of them is thoroughly tested and verified in practice. The techniques involved in different sFFT algorithms include the following contents: five operations of signal, three methods of frequency bucketization, five methods of location, four methods of estimation, two problems caused by bucketization, three methods to solve these two problems, four algorithmic frameworks. All the above technologies and methods are introduced in detail and examples are given to illustrate the above research. After theoretical research, we make experiments for computing the signals of different SNR, length, sparsity by a standard testing platform and record the run time, percentage of the signal sampled and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> error with eight different typical sFFT algorithms. The result of experiments satisfies the inferences obtained in theory. It helps us to optimize these algorithms and use them selectively.

Cooperative wideband spectrum sensing in cognitive radio based on sparse real‐valued fast Fourier transform

The Cognitive Radio (CR) plays a key role in identifying free bandwidths in the Radio Frequency (RF) spectrum. High-speed Analog-to-Digital Converters are normally applied for spectrum sensing of sideband signals providing several raw data for digital signal processors resulted in energy-inefficient complex circuits and hardware resources in digital signal processing blocks. In some instances, the frequency spectrum is sparsely occupied by various users, i.e., only a few active frequency bands exist at the same time. This feature enables CR application to use sub-Nyquist sampling approaches for designing a system representing significantly reduced cost and power consumption, as well as improved processing speed. The current paper introduced a novel Cooperative Real-valued Sparse Spread Spectrum Sensing algorithm (CR4S) based on a sub-Nyquist sampling approach by employing the sparsity of the frequency spectrum and the real-valued properties of the RF signal to identify free bandwidth with minimum computational complexity. The CR4S algorithm aimed at improving CR spectrum sensing by utilizing techniques such as Real-valued FFT, Sparse Fast Fourier Transform, and collaborative spectrum sensing. The proposed algorithm has been approved by simulation to above 95% detection performance. The performance enhancement in the CR4S algorithm is an emerging advance being fascinating for portable CR devices.

A review on sparse fast fourier transform applications in image processing

Fast Fourier Transform has long been established as an essential tool in signal processing. To address the computational issues while helping the analysis work for multi-dimensional signals in image processing, sparse Fast Fourier Transform model is reviewed here when applied in different applications such as lithography optimization, cancer detection, evolutionary arts and wasterwater treatment. As the demand for higher dimensional signals in various applications especially multimedia appplications, the need for sparse Fast Fourier Transform grows higher.

On Performance of Sparse Fast Fourier Transform Algorithms Using the Flat Window Filter

The problem of computing the Sparse Fast Fourier Transform(sFFT) of a $K$ -sparse signal of size $N$ has received significant attention for a long time. The first stage of sFFT is hashing the frequency coefficients into $B(\approx {K})$ buckets named frequency bucketization. The process of frequency bucketization is achieved through the use of filters: Dirichlet kernel filter, aliasing filter, flat filter, etc. The frequency bucketization through these filters can decrease runtime and sampling complexity in low dimensions. It is a hot topic about sFFT algorithms using the flat filter because of its convenience and efficiency since its emergence and wide application. The next stage of sFFT is the spectrum reconstruction by identifying frequencies that are isolated in their buckets. Up to now, there are more than thirty different sFFT algorithms using the sFFT idea as mentioned above by their unique methods. An important question now is how to analyze and evaluate the performance of these sFFT algorithms in theory and practice. In this paper, it is mainly discussed about sFFT algorithms using the flat filter. In the first part, the paper introduces the techniques in detail, including two types of frameworks, five different methods to reconstruct spectrum and corresponding algorithms. We get the conclusion of the performance of these five algorithms, including runtime complexity, sampling complexity and robustness in theory. In the second part, we make three categories of experiments for computing the signals of different SNR, different $N$ , and different $K$ by a standard testing platform and record the run time, percentage of the signal sampled, and $L_{0},L_{1},L_{2}$ error both in the exactly sparse case and general sparse case. The result of experiments is consistent with the inferences obtained in theory. It can help us to optimize these algorithms and use them correctly in the right areas.

Efficient Timing/Frequency Synchronization Based on Sparse Fast Fourier Transform

Efficient synchronization is of great importance for fiber optical coherent communication system. Here, we propose an accurate and low-complexity timing/frequency synchronization scheme based on sparse fast Fourier transform (S-FFT). The proposed scheme consists of coarse timing/frequency synchronization, fine timing synchronization, and fine frequency synchronization. Inspired by the idea of S-FFT, we take full advantages of the sparse nature of training symbols (TSs) and realize the synchronization in the frequency domain. The proposed scheme enables accurate timing synchronization at low signal-to-noise ratio (SNR) region. A complete frequency offset estimation (FOE) range of [−symbol rate/2, + symbol rate/2] can be realized with high resolution. Finally, the proposed scheme is experimentally verified through 10 Gbaud dual-polarization (DP) 16/32-quadrature amplitude modulation (QAM) transmission, without the BER performance penalty. In particular, the total computation complexity is reduced by nearly 200 times in comparison with that using conventional sliding window correlation scheme.

Modulation format identification assisted by sparse-fast-Fourier-transform for hitless flexible coherent transceivers.

For hitless flexible coherent transceivers based next-generation agile optical network, efficient modulation format identification (MFI) is an essential element in digital signal processing (DSP) flow at the receiver-side (Rx). In this paper, we propose a blind and fast MFI scheme with high identification accuracy at low optical signal-to-noise ratio (OSNR) regime. This is achieved by first raising the signal to the 4th power and calculate the peak-to-average power ratio (PAPR) of the corresponding spectra to distinguish 32 quadrature amplitude modulation (QAM) from quadrature phase shift keying (QPSK), 16 and 64QAM signals. Then, followed by iterative partition schemes to remove signals with phase ±π4,±3π4 (or QPSK-like phases) based on the signal amplitudes, the PAPR of the remaining signals is calculated to distinguish the other three formats. Additionally, by frequency offset (FO) pre-compensation, the spectrum can be obtained using sparse-fast-Fourier-transform (S-FFT), which greatly reduces the total complexity. The MFI performance is numerically and experimentally investigated by 28 Gbaud dual-polarization (DP) coherent optical back-to-back (B2B) and up to 1500 km standard single mode fiber (SSMF) transmission system using QPSK, 16QAM, 32QAM, and 64QAM. Results show that high identification accuracy can be achieved, even when OSNR is lower than that required for the 20% forward error correction (FEC) threshold of BER=2×10-2 for each format. Furthermore, fast format switching between 64QAM-32QAM and 32QAM-16QAM are demonstrated experimentally for B2B scenario and 900 km SSMF with the proposed MFI technique, respectively.

On Performance of Sparse Fast Fourier Transform and Enhancement Algorithm

Sparse fast Fourier transform (FFT) is a promising technique that can significantly reduce computational complexity. However, only a handful of research has been conducted on precisely analyzing the performance of this new scheme. Accurate theoretical results are important for new techniques to avoid numerous simulations when applying them in various applications. In this study, we analyze several performance metrics and derive the corresponding closed-form expressions for the sparse FFT including 1) inter sparse interference due to nonideal windowing effects, 2) the probability of sparse elements overlapping, and 3) the recovering rate performance. From the analytical results, we gain insights and propose a novel mode-mean estimation algorithm for improving the performance. Simulation results are provided to show the accuracy of the derived results as well as the performance enhancement. We also show how to determine parameters to achieve the lowest computational complexity using these theoretical results.

Face recognition using a new compressive sensing-based feature extraction method

This paper proposes a novel face recognition algorithm that utilizes a sparse Fast Fourier Transform (FFT)-based feature extraction method. In our algorithm, we use Compressive Sampling (CS) theory two times. First, in the feature extraction process for extracting the feature vectors from a face images, and second, in the classification process where the CS reconstruction is used for selecting true classes. As a result, a significant reduction in the dimensionality of the signals is achieved. Extensive and comparative experiments have been conducted to evaluate the performance of the proposed scheme. The experiment results show that the combined Compressive Sensing and Sparse Representation Classification (SRC) achieves a high recognition accuracy, while maintaining a reasonable computational complexity.

Unified commutation-pruning technique for efficient computation of composite DFTs

An efficient computation of a composite length discrete Fourier transform (DFT), as well as a fast Fourier transform (FFT) of both time and space data sequences in uncertain (non-sparse or sparse) computational scenarios, requires specific processing algorithms. Traditional algorithms typically employ some pruning methods without any commutations, which prevents them from attaining the potential computational efficiency. In this paper, we propose an alternative unified approach with automatic commutations between three computational modalities aimed at efficient computations of the pruned DFTs adapted for variable composite lengths of the non-sparse input-output data. The first modality is an implementation of the direct computation of a composite length DFT, the second one employs the second-order recursive filtering method, and the third one performs the new pruned decomposed transform. The pruned decomposed transform algorithm performs the decimation in time or space (DIT) data acquisition domain and, then, decimation in frequency (DIF). The unified combination of these three algorithms is addressed as the DFTCOMM technique. Based on the treatment of the combinational-type hypotheses testing optimization problem of preferable allocations between all feasible commuting-pruning modalities, we have found the global optimal solution to the pruning problem that always requires a fewer or, at most, the same number of arithmetic operations than other feasible modalities. The DFTCOMM method outperforms the existing competing pruning techniques in the sense of attainable savings in the number of required arithmetic operations. It requires fewer or at most the same number of arithmetic operations for its execution than any other of the competing pruning methods reported in the literature. Finally, we provide the comparison of the DFTCOMM with the recently developed sparse fast Fourier transform (SFFT) algorithmic family. We feature that, in the sensing scenarios with sparse/non-sparse data Fourier spectrum, the DFTCOMM technique manifests robustness against such model uncertainties in the sense of insensitivity for sparsity/non-sparsity restrictions and the variability of the operating parameters.

Efficient Software Implementation of the Nearly Optimal Sparse Fast Fourier Transform for the Noisy Case

In this paper we present an optimized software implementation (sFFT-4.0) of the recently developed Nearly Optimal Sparse Fast Fourier Transform (sFFT) algorithm for the noisy case. First, we developed a modified version of the Nearly Optimal sFFT algorithm for the noisy case, this modified algorithm solves the accuracy issues of the original version by modifying the flat window and the procedures; and second, we implemented the modified algorithm on a multicore platform composed of eight cores. The experimental results on the cluster indicate that the developed implementation is faster than direct calculation using FFTW library under certain conditions of sparseness and signal size, and it improves the execution times of previous implementations like sFFT-2.0. To the best knowledge of the authors, the developed implementation is the first one of the Nearly Optimal sFFT algorithm for the noisy case.

Efficient Software Implementation of the Nearly Optimal Sparse Fast Fourier Transform for the Noisy Case

In this paper we present an optimized software implementation (sFFT-4.0)of the recently developed Nearly Optimal Sparse Fast Fourier Transform (sFFT) algorithm for the noisy case. First, we developed a modified versionof the Nearly Optimal sFFT algorithm for the noisy case, this modified algorithm solves the accuracy issues of the original version by modifying theflat window and the procedures; and second, we implemented the modifiedalgorithm on a multicore platform composed of eight cores. The experi-mental results on the cluster indicate that the developed implementation isfaster than direct calculation using FFTW library under certain conditions of sparseness and signal size, and it improves the execution times of previous implementations like sFFT-2.0. To the best knowledge ofthe authors,the developed implementation is the first one of the Nearly Optimal sFFT algorithm for the noisy case.

Sound source localization using compressive sensing-based feature extraction and spatial sparsity

In this paper, we propose a source localization algorithm based on a sparse Fast Fourier Transform (FFT)-based feature extraction method and spatial sparsity. We represent the sound source positions as a sparse vector by discretely segmenting the space with a circular grid. The location vector is related to microphone measurements through a linear equation, which can be estimated at each microphone. For this linear dimensionality reduction, we have utilized a Compressive Sensing (CS) and two-level FFT-based feature extraction method which combines two sets of audio signal features and covers both short-time and long-time properties of the signal. The proposed feature extraction method leads to a sparse representation of audio signals. As a result, a significant reduction in the dimensionality of the signals is achieved. In comparison to the state-of-the-art methods, the proposed method improves the accuracy while the complexity is reduced in some cases.

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a common feature: edges are identified through separation of scales. To this end, we advocate here the use of concentration kernels (Tadmor, Acta Numer. 16:305–378, 2007), to convert the global spectral data into an approximate jump function which is localized in the immediate neighborhoods of the edges. Building on these concentration kernels, we show that the sIFT method, the TV-based compressed sensing and the zero crossing yield effective edge detectors, where finitely many jump discontinuities are accurately recovered. One- and two-dimensional numerical results are presented.