In this paper, we study the aperiodic comb signal from the point of view of the Fourier Transform. The comb is very important in the theory of ideal sampling. The knowledge of its properties is crucial for the establishment of suitable interpolation schemes. Here, we present sufficient conditions so that the Fourier Transform of an aperiodic comb is an aperiodic comb. We use this result to prop...

The Lomb-Scargle periodogram is widely used for the estimation of the power spectral density of unevenly sampled data. A small extension of the algorithm of the Lomb-Scargle periodogram permits the estimation of the phases of the spectral components. The amplitude and phase information is sufficient for the construction of a complex Fourier spectrum. The inverse Fourier transform can be applied...

Time-varying spectra of non-stationary time-series commonly used are spectrograms from the Short-Time Fourier Transform (STFT). The most prominent limitation of the Fourier Transform is that of frequency resolution. To overcome the limitation the Wavelet Transform, Wigner-Ville Distribution and the Min-Norm subspace method have been applied for spectrum estimation of non-stationary signals caus...

The subsequence matching in a large time-series database has been an interesting problem. Many methods have been proposed that cope with this problem in an adequate extent. One of the good ideas is reducing properly the dimensionality of time-series data. In this paper, we propose a new method to reduce the dimensionality of high-dimensional time-series data. The method is simpler than existing...

Electroencephalograms (EEGs) are brain waves, which are recorded using scalp electrodes. Generally, signal attenuate on recording and amplitude of the evoked potentials (EPs) are low when merged with the base brain waves. Therefore, mathematical tools are needed to analysis the time series (EEGs) to discover the EPs in the base EEGs. This paper reviews spectral analysis based on periodic amplit...

We survey a number of applications of the wavelet transform in time series prediction. The Haar à trous wavelet transform is proposed as a means of handling time series data when future data is unknown. Results are exemplified on financial futures and S&P500 data. Nonlinear and linear multiresolution autoregression models are studied. Experimentally, we show that multiresolution approaches can ...

The theory of fuzzy transform (F-transform, for short) has been developed extensively in recent years. Based on the current successes of F-transform, we expect that in the future, F-transform will become as successful and as widely used as the well-known transforms such as Laplace, Fourier, and wavelet transforms. The most successful applications of F-transform are in the fields of image proces...

Although Fourier series approximation is ubiquitous in computational physics owing to the Fast Fourier Transform (FFT) algorithm, efficient techniques for the fast evaluation of a three-dimensional truncated Fourier series at a set of arbitrary points are quite rare, especially in MATLAB language. Here we employ the Nonequispaced Fast Fourier Transform (NFFT, by J. Keiner, S. Kunis, and D. Pott...

7 Periodic Series and Functions 128 7.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . 128 7.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . . 130 7.3 Basic Properties of Complex Numbers . . . . . . . . . . . . . 131 7.4 Complex Numbers and Trigonometrical Identities . . . . . . . 133 7.5 The Exponential of a Complex Number . . . . . . . . . . . . . 133 7.6...

Wavelets and Fourier analysis in digital signal processing are comparatively discussed. In data processing, the fundamental idea behind wavelets is to analyze according to scale, with the advantage over Fourier methods in terms of optimally processing signals that contain discontinuities and sharp spikes. Apart from the two major characteristics of wavelet analysis multiresolution and adaptivit...