Discrete-time Fourier Transform Research Articles

Complex-Valued Noisy Sinewaves Frequency Estimator Based on DTFT Linearization and Interpolation

AbstractThis paper proposes an Interpolated Discrete-Time Fourier Transform (LIpDTFT) estimator for complex-valued noisy sinewave frequency based on the Linearization of the DTFT module behavior around the spectrum peak. It belongs to a class of LIpDTFT estimators that compensate the poor Discrete Fourier Transform (DFT) frequency resolution by interpolating two DTFT samples located very close to an initial frequency estimate. As compared with other DTFT-based frequency estimators recently proposed in the literature, the proposed LIpDTFT estimator ensures a smaller interpolation error and a lower processing effort. Moreover, if the rectangular window is adopted, it almost attains the unbiased Cramér-Rao Lower Bound (CRLB) even if a very small number of samples is analyzed. The accuracies ensured in different operating conditions by the proposed frequency estimator and other state-of-the-art DTFT-based algorithms are compared to each other through computer simulations.

Generalized multi-point interpolated DTFT frequency and damping factor estimators of real-valued damped sinusoids

In this paper a generalized multi-point interpolated Discrete-Time Fourier Transform (DTFT) is proposed for frequency and damping factor estimation of real-valued damped sinusoids. The acquired data are weighted by a Maximum Sidelobe Decay (MSD) window and the interpolation function is defined as the ratio between two linear combinations of DTFT samples located one bin apart, which is generalized exploiting a tuning parameter. Analytical expressions for the estimated parameter errors due to the interference of the fundamental image component are derived. Leveraging on these expressions an improved version of the algorithm is also proposed and optimized by selecting a suitable value for the tuning parameter. The developed theory is applied to both a three-point and a four-point interpolated Discrete Fourier Transform (IpDFT) estimators published in the literature and the achieved accuracy improvements are verified through simulations.

Marathi Speech Emotion recognition using Deep Learning techniques.

Marathi Speech Emotion recognition using Deep Learning techniques.

Doppler Parameters Estimation Using Digital PLL Based on SWIFT & αSWIFT Structures

A digital phase-locked loop (DPLL) based on sliding window infinite Fourier transform (SWIFT) and αSWIFT is proposed to extract the vibration and velocity parameters from the phase-modulated carrier signal. The phase of a carrier signal gets modulated when it is reflected from a vibration object due to the Doppler effect. Therefore, the phase-modulated carrier signal contains the Doppler signal that has the vibration information. The SWIFT or αSWIFT structure is designed from sliding discrete-time Fourier transform (DTFT) where the exponential window provides higher weightage to recent samples. It acts as a phase detector in the proposed DPLL, and its bandwidth can be varied based on the values of the time-constant. A controller unit, along with a bi-quad oscillator, is introduced in the proposed DPLL structure to remove any leakages in the system and to synchronize the system units in the parameter estimation. The proposed scheme is implemented in MATLAB/SIMULINK environment and validation for real-time applications is done using dSPACE 1202 MicrolabBox.

Co-occurrence of dental caries and periodontitis: multilevel modelling approach

BackgroundPrevious studies reported varyingly positive, negative, or no relationships between caries and periodontitis. Therefore, the aim was to assess the potential co-occurrence of caries experience and periodontal inflammation on the same teeth.MethodsThis cross-sectional study used data from the Lithuanian National Oral Health Survey. The study included a stratified random sample of 1405 individuals aged 34–78, recruited from 5 Lithuanian cities and 10 peri-urban/rural areas (response rate 52%). Information about sociodemographic (age, sex, education, residence), behavioral (sugar-containing diet, tooth brushing frequency, use of interdental care products, last dental visit, smoking) and biological (systemic disease, use of medication and xerostomia) determinants was collected using the World Health Organization (WHO) Oral Health Questionnaire for Adults supplemented with additional questions. Clinical data were recorded using the WHO criteria and collected by one trained and calibrated examiner. Dental caries status was recorded as sound, decayed, missing, filled surfaces. Subsequently for the analyses, status was recorded at a tooth-level as decayed- and filled-teeth (DT and FT) including proximal, buccal, and oral surfaces. Two measures were used for periodontal status. The probing pocket depth (PPD) was measured at six sites and recorded at a tooth level into the absence of PPD or presence of PPD ≥ 4 mm. Bleeding on probing (BOP) was measured at the same six sites and was recorded as either present or absent at a tooth-level. Univariable and multivariable 2-level random intercept binary logistic regression analyses were utilized.ResultsPositive associations were found between DT and BOP (OR 1.42, 95% CI 1.20–1.67), FT and BOP (OR 2.07, 95% CI 1.82–2.23), DT and PPD (OR 1.38, 95% CI 1.15–1.67) and FT and PPD (OR 2.01, 95% CI 1.83–2.20).ConclusionsOur findings add evidence for the co-occurrence of periodontal inflammation and caries on the same teeth. This suggests the need for increased emphasis on a transdisciplinary approach in designing oral health interventions that target dental caries and periodontal disease simultaneously. In addition, longitudinal studies exploring the co-occurrence of caries and periodontal disease at the same sites, taking into consideration the levels of both conditions and genetic variation, are warranted.

Single-cycle P Class phasor estimation based on harmonics whitening and off-nominal frequency offset adjustment

Phasor Measurement Units (PMUs) are essential for real-time power system monitoring and control. In this paper, a novel algorithm for protection (i.e., P Class) PMUs is presented. The proposed algorithm consists of four steps: preliminary narrowband disturbance whitening; estimation of the off-nominal fundamental frequency offset through the Interpolated Discrete Time Fourier Transform (IpDFT); data record length adjustment to refine narrowband disturbance whitening over an integer number of fundamental periods; time-domain estimation of the Taylor's series coefficients of the synchrophasor function through a weighted least-squares estimator (after correction of possible static off-nominal deviations). The key novel contribution of the proposed approach is its superior ability to smooth harmonic disturbances over shorter observation intervals than many existing algorithms (which typically require at least two-cycle-long data records to meet P Class accuracy requirements), while ensuring a good tracking capability of transient events with a reasonable execution time. Several simulation and experimental results confirm that, in the case of multi-harmonic distortion, the proposed approach returns more accurate results than both the basic Taylor-series-based Weighted Least-Squares (TWLS) estimator and other whitening-based techniques, while ensuring compliance with P Class specifications even over one-cycle-long data records.

Insights about aliasing and spectral leakage when analyzing discrete-time finite viscoelastic functions

Material property viscoelastic inversion studies often rely on the continuous -time framework for Fourier analysis, which may not accurately represent real experimentally collected data. In this paper, we address the discrete and finite nature of viscoelastic functions obtained from experiments and discuss the impact of these characteristics on the frequency spectrum analysis. We derive equations for the Discrete-Time Fourier Transform (DTFT) of a discrete-finite stress relaxation signal corresponding to the relaxation of a generalized Maxwell model. Our analysis highlights the limitations of the traditional continuous -time framework in capturing the inherent features of real signals, which are discrete and finite in nature. This results in two phenomena: aliasing and spectral leakage. We present equations that consider these phenomena, allowing experimentalists to anticipate and account for aliasing and leakage when performing model fitting. The proposed discrete-finite approach provides a more accurate representation of real viscoelastic data, enabling researchers to make better-informed decisions in the analysis and interpretation of sample viscoelastic functions.

Classification of Resampled Pediatric Epilepsy EEG Data Using Artificial Neural Networks with Discrete Fourier Transforms

Epilepsy is a neurological disorder commonly observed in children. Currently, electroencephalography (EEG) is widely used as the most important diagnostic method for epilepsy in medical practice. The diagnosis of epilepsy in pediatric patients is challenging due to their high level of activity and incomplete brain development. In this study, data sampled at 256 Hz were obtained from patients between the ages of 7–12, collected by Boston Children’s Hospital. First, the image intervals that contain seizure waves were identified in the datasets, and the discrete-time Fourier transform (DFT) was applied. The amplitude-frequency features of the frequency spectrum in seizure and nonseizure states were obtained, and patients were classified for seizure detection using a multilayer perceptron (MLP) based on an artificial neural network (ANN) architecture. In the next step, the EEG signals were resampled at low frequencies, and the same analyses were repeated to minimise the disadvantages of limiting factors such as storage space and processing power, resulting in reduced storage space usage and more efficient performance.

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step.

Sinusoidal Frequency Estimator by Using Interpolation of Four DTFT Spectral Lines

This paper proposes a sinusoid frequency estimator by interpolating four Discrete Time Fourier Transform (DTFT) samples. Firstly, Discrete Fourier Transform (DFT) is performed on the received sinusoid. Then we find the positional information of the maximal DFT bin and obtain the coarse estimation results. Next, the proposed method uses four symmetrical DTFT samples on both sides of the maximal DFT bin to interpolate the signal frequency and obtain accurate estimation results. Simulation experiment results illustrate that the proposed method’s accuracy is higher than that of the competitive methods. When the signal-to-noise ratio (SNR) increases from 0dB to 90dB, the presented estimator performs better than the competitive methods.

Fast interpolated DTFT estimators of frequency and damping factor of real-valued damped sinusoids

This paper deals with the estimation of real-valued damped sinusoid frequency and damping factor. To that aim an Interpolated Discrete Time Fourier Transform (IpDTFT) algorithm is proposed which extends to real-valued signal an algorithm proposed in the literature for complex-valued damped sinusoids. The proposed algorithm provides fast estimates of the unknown parameters of either undistorted or distorted real-valued noisy damped sinusoids with accuracy generally better than the one ensured by state-of-the-art estimators, especially when few sinusoid cycles are analysed. Accuracy improvement is obtained by both weighting the acquired signal using a Maximum Sidelobe Decay (MSD) window and compensating the contribution of the spectral image component on the estimated parameters. Analytical expressions for the estimator variances due to wideband noise are also derived.

High-precision frequency estimator by using discrete Fourier transform and asymmetric discrete time Fourier transform samples

Fast and accurate frequency estimation is significant for instrumentation and measurement. A sinusoid frequency estimator using discrete Fourier transform (DFT) is presented. DFT is implemented on the sinusoid and the maximum DFT bin is sought out to obtain the coarse estimate. Different from all the existing methods, two asymmetric discrete-time Fourier transform (DTFT) samples situated at arbitrary positions on the same side of the maximum DFT bin are employed to get the fine estimate. The theoretical mean square error is analyzed. To evaluate the estimation performance, the presented estimator is compared with the Cramer-Rao lower bound (CRLB) and state-of-art estimators through computer simulations. Simulation results demonstrate that the presented algorithm is closer to the CRLB compared with the competing methods when the signal-to-noise ratio (SNR) varies in a large range and is unbiased at high SNR.

Design and Analysis of Convolutional Neural Layers: A Signal Processing Perspective

Convolutional layers (CLs) are ubiquitous in contemporary deep neural network (DNN) models, commonly used for automatic feature extraction. A CL performs cross-correlation between the input to the layer and a set of learnable kernels to produce the layer output. Typically, kernel weights are randomly initialized and automatically learned during model training using the backpropagation and gradient descent algorithms to minimize a specific loss function. Modern DNN models comprise deep hierarchical stacks of CLs and pooling layers. Despite their prevalence, CLs are perceived as a magical tool for feature extraction without solid interpretations of their underlying working principles. In this work, we advance a method for designing and analyzing CLs by providing novel signal processing interpretations of the CL by exploiting the correlation and equivalent convolution functions of the layer. The proposed interpretations enable the employment of CLs to develop finite impulse response (FIR) filters, matched filters (MFs), short-time Fourier transform (STFT), discrete-time Fourier transform (DTFT), and continuous wavelet transform (CWT) algorithms. The main idea is to pre-assign the CL kernel weights to implement a specific convolution- or correlation-based DSP algorithm. Such an approach enables building self-contained DNN models in which CLs are utilized for various preprocessing and feature extractions tasks, enhancing the model portability, and cutting down the preprocessing computational cost. The proposed DSP interpretations provide an effective means to analyze and explain the operation of automatically trained CLs in the time and frequency domains by reversing the design procedures. The presented interpretations are mathematically established and experimentally validated with a comprehensive machinery fault diagnosis application example illustrating the potential of the proposed methodology.

Accurate Frequency Estimator for Real Sinusoid Based on DFT

An accurate frequency estimator for real sinusoid based on Discrete Fourier Transform (DFT) is proposed. The proposed estimator is based on the interpolation of the maximum DFT spectral line and two Discrete-Time Fourier Transform (DTFT) spectral lines and can operate with both the rectangular window and the maximum sidelobe decay (MSD) window. In the coarse estimation step, the proposed estimator with the MSD window is used. According to the value of the coarse estimate, the negative frequency spectral component is removed. In the fine estimation step, the proposed estimator with the rectangular window is utilized to achieve the Cramer–Rao lower bound (CRLB). Simulation results show that the performance of this algorithm is better than that of the AM algorithm, Candan algorithm, Djukanovic algorithm, and FDIAM algorithm.

Comparative Study of AC Signal Analysis Methods for Impedance Spectroscopy Implementation in Embedded Systems

For embedded impedance spectroscopy, a suitable method for analyzing AC signals needs to be carefully chosen to overcome limited processing capability and memory availability. This paper compares various methods, including the fast Fourier transform (FFT), the FFT with barycenter correction, the FFT with windowing, the Goertzel filter, the discrete-time Fourier transform (DTFT), and sine fitting using linear or nonlinear least squares, and cross-correlation, for analyzing AC signals in terms of speed, memory requirements, amplitude measurement accuracy, and phase measurement accuracy. These methods are implemented in reference systems with and without hardware acceleration for validation. The investigation results show that the Goertzel algorithm has the best overall performance when hardware acceleration is excluded or in the case of memory constraints. In implementations with hardware acceleration, the FFT with barycentre correction stands out. The linear sine fitting method provides the most accurate amplitude and phase determinations at the expense of speed and memory requirements.

Interpolated DFT Algorithm for Frequency Estimation by Using Maximum Sidelobe Decay Windows

A sinusoidal frequency estimator based on interpolated Discrete Fourier Transform (DFT) algorithm by using Maximum Sidelobe Decay (MSD) windows is proposed in this paper. Firstly, the received sinusoid is weighted by an appropriate MSD window. Then DFT is carried out on the weighted sinusoid and the coarse estimation is acquired by finding the position of the maximum DFT sample. Different from all the existing algorithms, the presented estimator adopts the maximum DFT sample and two Discrete Time Fourier Transform (DTFT) spectral lines which are on the same side of the maximum DFT sample in the fine estimation step. MSE formulas of the presented estimator in additive white noise are derived. Simulation results indicate that the presented estimator outperforms the competing estimators.

DFT-Based Frequency Estimation of Multiple Sinusoids

An accurate frequency estimation method of multi-component sinusoidal signal in additive white Gaussian noise (AWGN) is proposed. The algorithm is implemented in the frequency domain and based on discrete Fourier transform (DFT). The maximum DFT spectral line and two Discrete-Time Fourier Transform (DTFT) samples located on the same side of the maximum DFT spectral line are used to estimate the frequency of a sinusoidal component. And this algorithm is utilized in both the coarse estimation and the fine estimation. Simulation results show that compared with the competing estimators, the presented method is closer to the Cramer-Rao lower bound (CRLB). And it is almost independent of the frequency displacement. The numerical complexity of the presented method is similar with the competing DFT-based algorithms.

Efficient Frequency Estimation Algorithm Based on Chirp-Z Transform

The Discrete Fourier Transform (DFT), which can be calculated efficiently by the Fast Fourier Transform (FFT), is one of the most commonly used tools for frequency estimation of a multi-frequency exponential signal. However, the performance of the FFT is inherently affected by spectral leakage when non-coherent sampling occurs. The Chirp-Z Transform (CZT), which can refine the spectrum to the narrowband range of the source signal accurately, can be generalized to a special form of DFT. This paper proposes a CZT based algorithm which can estimate frequencies of the multi-frequency exponential signal accurately. In our method, the CZT bin, as a nonlinear equation, is firstly used to describe the correspondence among the frequencies, amplitudes, phases and the observed spectrum. Then, the above nonlinear equation is rewritten as a linear equation with unknown frequencies. We further extend the proposed method to the Discrete-time Fourier Transform (DTFT), zero-padding FFT and DFT, which are the special cases of the CZT. We also give the derivation procedure of theoretical variance of our method. In our method, the spectrum leakage is fully considered as useful information to establish the linear equation set. Therefore, it can estimate the frequencies of multi-frequency exponential signal with high accuracy. Simulation results verify the performance of the proposed method.

Algorithms for Real-Valued Noisy Damped Sinusoid Parameter Estimation

In this paper an improved version of three state-of-the-art Interpolated Discrete-Time Fourier Transform (IpDTFT) algorithms for the estimation of the frequency and the damping factor of noisy damped sinusoids is proposed. Sinusoid amplitude and phase are also estimated. Improvement is obtained by both compensating the effect on the estimated parameters of the spectral leakage from the fundamental image component and minimizing the estimator variance through a suitable selection of the interpolation points. The obtained estimates are then further improved by applying a linear signal fit (LSF) algorithm. The accuracies of the proposed algorithms are compared with each other and with the related Cramer-Rao Lower bound through computer simulations. The required computational times are also briefly analyzed in order to assess the algorithms usefulness in real-time applications.

Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences

The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal sequences in ℓ2(Z) which are strictly bandlimited to a frequency band [−W,W] and maximally concentrated in a time interval {0,…,N−1}. The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in CN whose discrete time Fourier transform (DTFT) is maximally concentrated in a frequency band [−W,W]. Due to these properties, DPSSs have a wide variety of signal processing applications. The DPSSs are the eigensequences of a timelimit-then-bandlimit operator and the Slepian basis vectors are the eigenvectors of the so-called prolate matrix. The eigenvalues in both cases are the same, and they exhibit a particular clustering behavior – slightly fewer than 2NW eigenvalues are very close to 1, slightly fewer than N−2NW eigenvalues are very close to 0, and very few eigenvalues are not near 1 or 0. This eigenvalue behavior is critical in many of the applications in which DPSSs are used. There are many asymptotic characterizations of the number of eigenvalues not near 0 or 1. In contrast, there are very few non-asymptotic results, and these don't fully characterize the clustering behavior of the DPSS eigenvalues. In this work, we establish two novel non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and 1−ϵ. Also, we obtain bounds detailing how close the first ≈2NW eigenvalues are to 1 and how close the last ≈N−2NW eigenvalues are to 0. Furthermore, we extend these results to the eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the continuous-time version of the DPSSs. Finally, we present numerical experiments demonstrating the quality of our non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and 1−ϵ.