Ramanujan Subspaces Research Articles

Enhanced symplectic Ramanujan mode pursuit and its application in mechanical composite fault diagnosis

In practical engineering applications, traditional signal decomposition methods are often affected by various factors such as strong noise, alternating periods, etc. When signal mode analysis is conducted, it is often found that the decomposition results do not meet engineering requirements. To address these issues, Enhanced Symplectic Ramanujan Mode Pursuit (ESRMP) method is proposed in this paper, which aims to improve the accuracy and reliability of signal decomposition and period estimation. First, the cyclic symplectic geometry similarity transform is used to separate the components of different modes in the signal, and the anti-noise autocorrelation function is used to estimate the period of different components. Then, the rectangular length of the intercepted signal is determined based on the estimated period, and the periodic compensation is achieved through related detection and cubic spline interpolation. Finally, the reconstructed signal is projected onto the Ramanujan subspace to extract and enhance periodical pulses. The experimental results of multimodal composite fault signals show that the ESRMP method can accurately separate components of different modes, especially periodic pulse signals.

Periodic Detection Mode Decomposition and Its Application in Bearing Fault Diagnosis

In the complex background noise, it is difficult to extract the periodic pulse of the rolling bearing fault signal. In this paper, a new modal decomposition method supported by singular value ratio (SVR) spectrum is proposed to find the optimal period of bearing fault data, which can be named as periodic detection mode decomposition (PDMD). In order to detect the intervals of periods and eliminate the interference of useless periods, the initial measurement intervals in this method are divided according to the theoretical fault periods of different fault types of bearings. In each interval, the SVR spectrum is used to detect the appropriate period and suppress the influence of noise on the recognition process. This period is used to construct the optimal Ramanujan subspace (RS). Finally, the harmonic spectral kurtosis (HSK) is used to identify the extracted period as false information, interference or fault. Simulation and experimental data verify the effectiveness of the proposed method. This method can effectively extract and screen periodic pulses, and can successfully identify the outer and inner ring faults of bearings.

Cyclic Symplectic Ramanujan Component Pursuit: Algorithm and applications

In recent years, signal decomposition method has been widely used in the engineering field to solve the problem of multimodal segmentation, such as Empirical Mode Decomposition (EMD), Ramanujan Mode Decomposition (RMD) and Adaptive Periodic Mode Decomposition (APMD) methods. Although the above methods have excellent decomposition performance, they still face many challenges for strong noise, coupled modes and other complex signals. Based on this, this paper proposes a new Cyclic Symmetric Ramanujan Component Pursuit (CSRCP) method, which can effectively decompose multi-frequency signals while protecting the integrity of the original information. First, to prevent information leakage, the circular matrix is constructed to protect the state information of the signal. Then, symplectic geometry similarity transformation is used to constrain signal decomposition, and multiple component signals with single frequency are obtained while noise reduction is performed. Finally, by constructing the Ramanujan subspace, the projection components in the Ramanujan subspace can be obtained, which enhances the extraction of periodic pulse information. The proposed method is verified by simulation and experimental signals, and the verification results show that CSRCP has superior decomposition performance.

Optimization of Ramanujan Subspace Periodic and Its Application in Identifying Industrial Bearing Fault Features

Due to the complex working conditions, it is difficult to accurately extract the periodic pulse component of the rolling bearing vibration signal. The Ramanujan subspace (RS) theory has a good ability to extract periodic components (PCs). However, the theory divides signals by fixed period length, which means that the effect of extracting quasi-periodic signals is poor. Therefore, this article proposes an optimized periodic mode decomposition (OPMD) method to improve the defect and apply it to bearing fault diagnosis. First, the window length of the intercepted data segment is determined by periodicity measurement. Then, the pulse phase corresponding to each data segment is detected by correlation to divide the data segment, and cubic spline interpolation is used for phase compensation. Finally, the periodic pulse is extracted from the reconstructed signal. Simulation and experimental results show that compared with the existing RS methods, OPMD can extract better periodic pulses, reduce the impact of quasi-periodic signals on the extraction effect, and improve the adaptability of the algorithm.

Periodicity-Aware Signal Denoising Using Capon-Optimized Ramanujan Filter Banks and Pruned Ramanujan Dictionaries

We propose a ‘periodicity-aware’ hybrid analysis-synthesis framework for denoising discrete-time periodic signals. Our method uses Ramanujan filter banks (RFB) for analysis and dictionaries for synthesis. The synthesis dictionary retains appropriate subspaces for signal reconstruction, by pruning the Ramanujan dictionary based on the outputs of the RFB. Unlike the other existing denoising methods, a unique advantage of the proposed method is that the denoised output is guaranteed to be composed of integer-periodic components with periods smaller than a pre-selected value. Our method works well even when the signal length is small, and has a high SNR gain over a wide range of input signal SNRs. Furthermore, we propose to adapt each filter in the analysis bank to the incoming data, by optimizing the filter coefficients through a multi-band Capon formulation. This helps in suppressing the spurious energy peaks generated from higher period filters in the analysis bank, further improving the denoising performance. Implementing multiband Capon filters requires inverses of several autocorrelation matrices. To reduce computations, a way to recursively compute these inverses based on Levinson's recursion is discussed. Next, we prove several multirate properties of Ramanujan subspace signals. An important property among these is that after decimation, a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -th Ramanujan subspace signal still remains in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -th Ramanujan space, if and only if the decimation rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> is coprime to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> . This is helpful to further reduce computations required in the analysis part of the denoising framework by downsampling the filter outputs. Extensive Monte-Carlo simulations comparing different variants of the proposed method and several existing denoising methods are also provided.

When Ramanujan sums meet affine Fourier transform

Ramanujan Fourier transform is one of the efficient multiresolution analysis tools, but it only works for stationary or periodically characterized signals. Affine Fourier transform, a general form of Fourier transform, is suitable for nonstationary signal analysis but with fixed resolution. In this study, the Ramanujan sums are combined with the affine Fourier transform to provide affine multiresolution analysis tools for nonstationary signals. First, affine Ramanujan sums are proposed and their main properties are introduced, based on which the affine Ramanujan Fourier transform (ARFT) for infinite sequences is proposed. Then, the first type ARFT (ARFT-I) for finite sequences is obtained by truncating the ARFT. Meanwhile, the transform matrix of ARFT-I is not orthogonal in general. Thus, the affine Ramanujan subspaces are constructed by affine Ramanujan sums and their canonical circular time shifts, among which different subspaces are orthogonal. Subsequently, the second type ARFT (ARFT-II) for finite sequences is developed, where the transform matrix is block orthogonal. Independent variables of ARFTs are chirp periods, which provides novel and direct methods to estimate (hidden) chirp periods. Different simulations, such as radar moving target detection and velocity estimation, are designed to show superior performance of ARFTs than affine Fourier transform based algorithms.

Operational modal analysis under harmonic excitation using Ramanujan subspace projection and stochastic subspace identification

The presence of deterministic harmonic excitation, such as that induced by rotating machinery, violates the classical operational modal analysis (OMA) assumption that the system output is strictly ergodic. This paper proves that the presence of harmonic excitation has no effect on the identification of structural modes except in the case where some of the harmonic excitation frequencies are close to the structural frequencies. Therefore, the harmonic component must be removed from the mixed random and harmonic system output before further processing. This paper proposes a Ramanujan subspace projection (RSP) method for harmonic removal, which is realized by projecting the raw system output onto the complex conjugate division (CCD) of the Ramanujan subspace. The novelty of this study is that it reveals the relationship between the frequencies defined in the period and frequency domains, allowing the RSP method to directly extract the harmonic component from the specific CCD without any frequency domain analysis. In addition, an energy indicator is proposed to select the underlying CCDs with the most robust harmonic feature. Using the indicator, one only needs to project the raw system output onto a subset of the CCDs rather than all of them, thereby significantly reducing the computational effort. After removing the harmonic components from the raw output, the remainder can be fed into the covariance-driven stochastic subspace identification (Cov-SSI) method for OMA. The numerical, experimental and field test results show that the proposed RSP method is not only resistant to random noise but also capable of precisely extracting the weak harmonic component from the raw output with less computation. Furthermore, the modal parameters of the structures subjected to mixed random and harmonic excitation can be accurately identified by combining the Cov-SSI and RSP methods.

A New Method of Wheelset Bearing Fault Diagnosis

During the movement of rail trains, trains are often subjected to harsh operating conditions such as variable speed and heavy loads. It is therefore vital to find a solution for the issue of rolling bearing malfunction diagnostics in such circumstances. This study proposes an adaptive technique for defect identification based on multipoint optimal minimum entropy deconvolution adjusted (MOMEDA) and Ramanujan subspace decomposition. MOMEDA optimally filters the signal and enhances the shock component corresponding to the defect, after which the signal is automatically decomposed into a sequence of signal components using Ramanujan subspace decomposition. The method’s benefit stems from the flawless integration of the two methods and the addition of the adaptable module. It addresses the issues that the conventional signal decomposition and subspace decomposition methods have with redundant parts and significant inaccuracies in fault feature extraction for the vibration signals under loud noise. Finally, it is evaluated through simulation and experimentation in comparison to the current widely used signal decomposition techniques. According to the findings of the envelope spectrum analysis, the novel technique can precisely extract the composite flaws that are present in the bearing, even when there is significant noise interference. Additionally, the signal-to-noise ratio (SNR) and fault defect index were introduced to quantitatively demonstrate the novel method’s denoising and potent fault extraction capabilities, respectively. The approach works well for identifying bearing faults in train wheelsets.

Symplectic Ramanujan Mode Decomposition and its application to compound fault diagnosis of bearings

The existing compound-fault diagnosis methods of rolling bearings have their own defects, which makes their accuracy of fault diagnosis impossible to be guaranteed. Therefore, this paper attempts to combine symplectic similarity transformation with Ramanujan subspace theory, and then a periodic impulse extraction method called symplectic Ramanujan mode decomposition (SRMD) method is proposed. SRMD separates the components with different fault features through symplectic similarity transformation and hierarchical clustering method to obtain symplectic clustering components (SCCs). At the same time, SRMD uses the Ramanujan subspace theory to extract the major periodic impulse components of each component to be extracted, and then obtains symplectic Ramanujan components (SRCs). The results show that SRMD is a resultful method in compound-fault diagnosis of bearings with excellent periodic impulse extraction ability.

Adaptive periodic mode decomposition and its application in rolling bearing fault diagnosis

As is known to all, rolling bearing fault will induce periodic impulses. Although the existing fault diagnosis methods, such as wavelet transform (WT) and ensemble empirical mode decomposition (EEMD), can effectively diagnose rolling bearing fault, their weak period recognition abilities limit their application in rolling bearing fault diagnosis. The Ramanujan subspace is introduced from the perspective of period recognition and extraction, and an adaptive periodic mode decomposition (APMD) method that can extract periodic components (PCs) without setting any parameters is proposed in this paper. APMD determines the major periods (MPs) in the signal by periodicity measurement, and projects the signal into respective Ramanujan subspaces of MPs to form the corresponding PCs. The analysis results of rolling bearing signals show that APMD has excellent ability to identify and extract PCs and is a valid method for rolling bearing fault diagnosis.

Signal periodicity detection using Ramanujan subspace projection

Abstract Signal periodic decomposition and periodic estimation are two crucial problems in the signal processing domain. Due to its significance, the applications have been extended to fields like periodic sequence analysis of biomolecules, stock market predictions, speech signal processing, and musical pitch analysis. The recently proposed Ramanujan sums (RS) based transforms are very useful in analysing the periodicity of signals. This paper proposes a method for periodicity detection of signals with multiple periods based on autocorrelation and Ramanujan subspace projection with low computational complexity. The proposed method is compared with other signal periodicity detection methods and the results show that the proposed method detects the signal period correctly in less time.

Stagger period estimation algorithm for multiple sets of radar pulses

To apply the Ramanujan filter bank based on the compressed sensing theory to the period estimation of multiple sets of radar pulses, alternatives to period data and the Ramanujan Subspace are proposed in this study. First, the rationality of alternatives to the TOA (time of arrival) model is illustrated by comparing the advantages and disadvantages of the time-point model and the TOA model. Next, by clarifying the zero-sum energy property of the Ramanujan Subspace, the possibility of finding an alternative to the discrete Fourier transform sequence with smaller data is proved. On this basis, a new algorithm for estimating the stagger period of mixed pulses is proposed, which introduces the initial phase of the stagger pulses. The results of the experiments show that the algorithm can accurately estimate the hidden stagger periods of mixed pulses, and it is adept at dealing with false and missing mixed pulses.

Estimation of PRI Stagger in Case of Missing Observations

For a moving target indicating (MTI) radar system to eliminate “blind speeds” or synthetic aperture radar (SAR) to eliminate “blind ranges,” two or more pulse repetition intervals (PRIs) are often used sequentially. This article proposes a new correlation-matching algorithm to estimate the PRIs, despite the radar receiver possibly missing some time samples. This proposed algorithm operates in the orthonormal Ramanujan subspace, leading to fewer nonzero elements therein, thus reducing the computational load. This proposed algorithm employs the ${l}_{p}$ -norm (instead of the ${l}_{2}$ -norm) in computing the correlation coefficients, for better robustness against outliers. Simultaneously, a new divisor rule, instead of the least common multiple (LCM) rule, and an algorithm for computing the threshold are proposed to identify the frame period of the staggered pulse train in the presence of spurious peaks. This new algorithm improves estimation accuracy and is robustness to missing pulse and PRI jitter.

Period Determination in Cyclo-Stationary Signals by Autocorrelation and Ramanujan Subspaces

Period determination in a periodic-like signal is a challenging process, if the signal is contaminated with a noise or noise-like interference signals. In this work, multiple period determination was considered in aforementioned signals. Recently, cyclostationary properties of the periodic-like signals were utilized to determine the time-varying autocorrelation function (TVAC). First we proved that TVAC can be expressed in terms of Ramanujan sums, then we used TVAC in the periodicity metric to identify the periods. Periodicity metric provides energy distribution of each periodic component as a function of block folding index and reaches a maximum at the points representing the hidden periods in the signal. Proposed method was verified by artificial signals and mean estimation errors versus signal to noise ratio were illustrated. Finally, hidden periods in the noisy respiration signals were estimated by the proposed method successfully.

Ramanujan periodic subspace division multiplexing

In this study, a new modulation method defined as Ramanujan periodic subspace division multiplexing (RPSDM) is proposed using Ramanujan subspaces. Each subspace contains an integer-valued Ramanujan sum and its circular downshifts as a basis. The proposed RPSDM decomposes the linear time-invariant wireless channels into a Toeplitz stair block diagonal matrices, whereas orthogonal frequency division multiplexing (OFDM) decompose the same into diagonal. Advantages of such structured subspaces representation are studied and compared with an OFDM representation in terms of a peak-average power ratio and bit error rate. Zero-forcing and minimum mean square error detectors are applied to evaluate the performance of OFDM and RPSDM techniques. Finally, the simulation results show that the proposed design (with an additional receiver complexity) outperforms OFDM under both detectors.

IMUSIC: A Family of MUSIC-Like Algorithms for Integer Period Estimation

The MUSIC algorithm is one of the most popular techniques today for line spectral estimation. If the line spectrum is that of a periodic signal, can we adapt MUSIC to exploit the additional harmonicity in the spectrum? Important prior work in this direction includes the Harmonic MUSIC algorithm and its variations. For applications where the period of the discrete signal is an integer (or can be well approximated by an integer), this paper introduces a new and simpler class of alternatives to MUSIC. This new family, called iMUSIC, also includes techniques where simple integer valued vectors are used in place of complex exponentials for both representing the signal subspace, and for computing the pseudo-spectrum. It will be shown that the proposed methods not only make the computations much simpler than prior periodicity-adaptations of MUSIC, but also offer significantly better estimation accuracies for applications with integer periods. These advantages are demonstrated on examples that include repeats in protein and DNA sequences. The iMUSIC algorithms are based on the recently proposed Ramanujan subspaces and nested periodic subspaces. The resulting signal space bases are non-Vandermonde in structure. Consequently, many aspects of classical MUSIC that were based on the Vandermonde structure of complex-exponentials, such as guarantees for identifiability of the frequencies (periods in our case), are addressed in new ways in this paper.

Monaural Source Separation Using Ramanujan Subspace Dictionaries

Most source separation algorithms are implemented as spectrogram decomposition. In contrast, time-domain source separation is less investigated since there is a lack of an efficient signal representation that facilitates decomposing oscillatory components of a signal directly in the time domain. In this letter, we utilize the Ramanujan subspace and the nested periodic subspace to address this issue, by constructing a parametric dictionary that emphasizes period information with less redundancy. Methods including iterative subspace projection and convolutional sparse coding can decompose a mixture into signals with distinct oscillation periods according to the dictionary. Experiments on score-informed source separation show that the proposed method is competitive to the state-of-the-art, frequency-domain approaches when the provided pitch information and the signal parameters are the same.

Ramanujan subspace pursuit for signal periodic decomposition

The period estimation and periodic decomposition of a signal represent long-standing problems in the field of signal processing and biomolecular sequence analysis. To address such problems, we introduce the Ramanujan subspace pursuit (RSP) based on the Ramanujan subspace. As a greedy iterative algorithm, the RSP can uniquely decompose any signal into a sum of exactly periodic components by selecting and removing the most dominant periodic component from the residual signal in each iteration. In the RSP, a novel periodicity metric is derived based on the energy of the exactly periodic component obtained by orthogonally projecting the residual signal into the Ramanujan subspace. The metric is then used to select the most dominant periodic component in each iteration. To reduce the computational cost of the RSP, we also propose the fast RSP (FRSP) based on the relationship between the periodic subspace and the Ramanujan subspace and based on the maximum likelihood estimation of the energy of the periodic component in the periodic subspace. The fast RSP has a lower computational cost and can decompose a signal of length N into a sum of K exactly periodic components in O(KNlogN). In short, the main contributions of this paper are threefold: First, we present the RSP algorithm for decomposing a signal into its periodic components and theoretically prove the convergence of the algorithm based on the Ramanujan subspaces. Second, we present the FRSP algorithm, which is used to reduce the computational cost. Finally, we derive a periodic metric to measure the periodicity of the hidden periodic components of a signal. In addition, our results show that the RSP outperforms current algorithms for period estimation.

Signal Periodic Decomposition With Conjugate Subspaces

In this paper, we focus on hidden period identification and the periodic decomposition of signals. Based on recent results on the Ramanujan subspace, we reveal the conjugate symmetry of the Ramanujan subspace with a set of complex exponential basis functions and represent the subspace as the union of a series of conjugate subspaces. With these conjugate subspaces, the signal periodic model is introduced to characterize the periodic structure of a signal. To achieve the decomposition of the proposed model, the conjugate subspace matching pursuit (CSMP) algorithm is proposed based on two different greedy strategies. The CSMP is performed iteratively in two stages. In the first stage, the dominant hidden period is chosen with the periodicity strategy. Then, the dominant conjugate subspace is chosen with the energy strategy in the second stage. Compared with the current state-of-the-art methods for hidden period identification, the main advantages provided by the CSMP are the following: (i) the capability of identifying all the hidden periods in the range from $1$ to the maximum hidden period $Q$ of a signal of any length, without truncating the signal; (ii) the ability to identify the time-varying hidden period with its shifted version; and (iii) the low computational cost, without generating and using a large over-complete dictionary. Moreover, we provide examples and applications to demonstrate the abilities of the proposed two-stage CSMP algorithm, which include hidden period identification, signal approximation, time-varying period detection, and pitch detection of speech.

Ramanujan Sums in the Context of Signal Processing—Part II: FIR Representations and Applications

The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). In a companion paper (Part I), properties of Ramanujan sums were reviewed, and Ramanujan subspaces S_q introduced, of which the Ramanujan sum is a member. In this paper, the problem of representing finite duration (FIR) signals based on Ramanujan sums and spaces is considered. First, it is shown that the traditional way to solve for the expansion coefficients in the Ramanujan-sum expansion does not work in the FIR case. Two solutions are then developed. The first one is based on a linear combination of the first N Ramanujan-sums (with N being the length of the signal). The second solution is based on Ramanujan subspaces. With q_1, q_2,..., q_K denoting the divisors of N; it is shown that x(n) can be written as a sum of K signals x_(qi) (n) ∈ S_(qi). Furthermore, the i_(th) signal x_(qi) (n) has period q_i, and any pair of these periodic components is orthogonal. The components x_(qi) (n) can be calculated as orthogonal projections of x(n) onto Ramanujan spaces S_(qi). Then, the Ramanujan Periodic Transform (RPT) is defined based on this, and is useful to identify hidden periodicities. It is shown that the projection matrices (which compute x_(qi) (n) from x(n)) are integer matrices except for an overall scale factor. The calculation of projections is therefore rendered easy. To estimate internal periods N_∞ <; N of x(n), one only needs to know which projection energies are nonzero.