Ahmet Serbes

The discrete fractional Fourier transform based on the DFT matrix

We introduce a new discrete fractional Fourier transform (DFrFT) based on only the DFT matrix and its powers. Eigenvectors of the DFT matrix are obtained in a simple-yet-elegant and straightforward manner. We show that this DFrFT definition based on the eigentransforms of the DFT matrix mimics the properties of continuous fractional Fourier transform (FrFT) by approximating the samples of the continuous FrFT. By appropriately combining existing commuting matrices we obtain a new commuting matrix which performs better. We show the validity of the proposed algorithms by computer simulations comparing DFrFT points and continuous FrFT samples for various signals.

Efficient computation of DFT commuting matrices by a closed-form infinite order approximation to the second differentiation matrix

In order to define the discrete fractional Fourier transform, Hermite Gauss-like eigenvectors are needed and one way of extracting these eigenvectors is to employ DFT commuting matrices. Recently, Pei et al. exploited the idea of obtaining higher order DFT-commuting matrices, which was introduced by Candan previously. The upper bound of O(h^2^k) approximation to NxN commuting matrix is 2k+1@?N in Candans work and Pei et al. improved the proximity by removing this upper bound at the expense of higher computational cost. In this paper, we derive an exact closed form expression of infinite-order Taylor series approximation to discrete second derivative operator and employ it in the definition of excellent DFT commuting matrices. We show that in the limit this Taylor series expansion converges to a trigonometric function of second-order differentiating matrix. The commuting matrices possess eigenvectors that are closer to the samples of Hermite-Gaussian eigenfunctions of DFT better than any other methods in the literature with no additional computational cost.

Spectrally efficient OFDMA lattice structure via toroidal waveforms on the time-frequency plane

We investigate the performance of frequency division multiplexed (FDM) signals, where multiple orthogonal Hermite-Gaussian carriers are used to increase the bandwidth efficiency. Multiple Hermite-Gaussian functions are modulated by a data set as a multicarrier modulation scheme in a single time-frequency region constituting toroidal waveform in a rectangular OFDMA system. The proposed work outperforms in the sense of bandwidth efficiency compared to the transmission scheme where only single Gaussian pulses are used as the transmission base. We investigate theoretical and simulation results of the proposed methods.

On the Estimation of LFM Signal Parameters: Analytical Formulation

We propose analytical formulations, approximations, upper and lower bounds for the angle sweep of maximum magnitude of fractional Fourier transform of mono- and multicomponent linear frequency modulated (LFM) signals. We employ a successive coarse-to-fine grid-search algorithm to estimate the chirp rates of multicomponent nonseparable LFM signals. Extensive numerical simulations show the validity of analytical formulations and performance of the proposed estimator. Obtained analytical results may also find themselves other application areas, where nonstationary signals are of interest.

A fast and accurate chirp rate estimation algorithm based on the fractional Fourier transform

In this work, a fast and accurate chirp-rate estimation algorithm is presented. The algorithm is based on the fractional Fourier transform. It is shown that utilization of the golden section search algorithm to find the maximum magnitude of the fractional Fourier transform domains not only accelerates the process, but also increases the accuracy in a noisy environment. Simulation results validate the proposed algorithm and show that the accuracy of parameter estimation nearly achieves the Cramer-Rao lower bound for SNR values as low as −7dB.

Estimation of LFM signal parameters by using the fractional fourier transform

A detector for estimating the chirp rate of linear frequency modulated signals is proposed. The proposed detector is formulated by calculating the supremum of the 4th-power modulus of the fractional Fourier transform of the signal. It has similar performance to the modulus square detector of the Radon transform yet its computational complexity is much lower. The proposed detector has been generalized for the nth-power modulus of the fractional Fourier transform (n > 2) and computer simulations have been performed to confirm its efficiency amongst existing detectors.

Eigenvectors of the DFT and discrete fractional fourier transform based on the bilinear transform

Orthonormal eigenvectors of the DFT matrix, which is closer to the samples of Hermite-Gaussian functions, are crucial to define the discrete fractional Fourier transform. In this work we determine the eigenvectors of the DFT matrix inspired by the bilinear transform. The bilinear transform maps the analog space to the discrete sample and it maps jw in the analog s-domain to the unit circle in the discrete z-domain one-to-one without aliasing, it is appropriate to use in the discretization of the eigenfunctions of the Fourier transform. We obtain Hermite-Gaussian like eigenvectors of the DFT matrix and confirm the results with extensive simulations.

Eigenvectors of the discrete Fourier transform based on the bilinear transform

Determining orthonormal eigenvectors of the DFT matrix, which is closer to the samples of Hermite-Gaussian functions, is crucial in the definition of the discrete fractional Fourier transform. In this work, we disclose eigenvectors of the DFT matrix inspired by the ideas behind bilinear transform. The bilinear transform maps the analog space to the discrete sample space. As j in the analog s-domain is mapped to the unit circle one-to-one without aliasing in the discrete z-domain, it is appropriate to use it in the discretization of the eigenfunctions of the Fourier transform. We obtain Hermite-Gaussian-like eigenvectors of the DFT matrix. For this purpose we propose three different methods and analyze their stability conditions. These methods include better conditioned commuting matrices and higher order methods. We confirm the results with extensive simulations.

Fast Frequency Estimation of a Complex Sinusoidal When the Phase Is Already Known

This paper describes a fast and precise algorithm for the frequency estimation of a single complex sinusoidal in noisy environments. The method interpolates on the shifted samples of its DFT. Computer simulations show that the proposed method outperforms existing popular DFT-interpolation based frequency estimation algorithms. The proposed method can be successfully employed in noisy environments, where the phase of the complex sinusoidal is already known.

Performance analysis of feature extraction methods in indoor sound classification

In this paper, by using a novel database of home environment warning sounds, the classification and recognition performances of these sounds are compared over feature extraction algorithms. Following the sample reduction of the feature vectors by Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA), k-Nearest Neighbour (k-NN) algorithm is employed for classification. Besides, a modified version of the algorithm for MF coefficients is proposed and we observe that the classification performance is better than MFCC and LPC even at low SNR values.