An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the...

An uncertainty inequality for the Fourier–Dunkl series, introduced by the authors in [Ó. Ciaurri and J. L. Varona, A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007), 2939–2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.

A. Table of contents by sections: 1. Abstract (you’re reading this now) 2. Summary of the DFT (How do I do the homework?) 3. Review of continuous-time Fourier series 4. Bandlimited signals and finite Fourier series 5. Sampling theorem for periodic signals 6. Review of quirks of discrete-time frequency 7. Orthogonality and its significance 8. Discrete Fourier Transform (DFT) 9. Use of DFT to com...

In this paper, we focus on Fourier analysis and holographic transforms for signal representation. For instance, in the case of image processing, the holographic representation has the property that an arbitrary portion of the transformed image enables reconstruction of the whole image with details missing. We focus on holographic representation defined through the Fourier Transforms. Thus, We f...

We present a new method for time-frequency representation (TFR), which combines a Fourier Bessel (FB) series expansion and the Wigner-Ville distribution (WVD). The FB series decomposes a multi-component signal into a number of single component signals before the WVD is applied. and then WVD is applied on the single component signals to analyze its time-frequency distribution (TFD). The simulati...

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) approximation to N N commuting matrix is 2kþ1rN in Ca...

In this paper we introduce a modification of the real discrete Fourier transform and its inverse transform to filter noise and perform reduction on the data whilst preserving the trend of global moving of time series. The transformed data is still in the same time domain as the original data, and can therefore be directly used by any other mining algorithms. We also present a classification alg...

We extract the frequency content of a noisy signal by use of Discrete Fourier Transform. Our analysis overcomes the limitations imposed by incommensurate lattices. After computing the deterministic component, we show the relevance of the method in removing spurious autocorrelations from the signal residuals. Results are presented for a temperature time series.