On the multiangle centered discrete fractional Fourier transform

Vargas-Rubio, J.G.; Santhanam, Balu

doi:10.1109/lsp.2005.843762

Cited by 67 publications

(34 citation statements)

References 18 publications

(35 reference statements)

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“…Therefore, for these elliptical rotations an angle modification and a post-phase compensation in the DFRFT are required to obtain results similar to the continuous FRFT [60]. This approach has been extended to the socalled multiple-parameter discrete fractional Fourier transform (MPDFRFT) [65], [66]. In fact, the MPDFRFT maintains all of the desired properties and reduces to the DFRFT when all of its order parameters are the same.…”

Section: Dfrft Based On Eigenvectorsmentioning

confidence: 99%

Fractional Fourier transform as a signal processing tool: An overview of recent developments

2011

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-Fractional Fourier transform (FRFT) is a generalization of the Fourier transform, rediscovered many times over the past hundred years. In this paper, we provide an overview of recent contributions pertaining to the FRFT. Specifically, the paper is geared toward signal processing practitioners by emphasizing the practical digital realizations and applications of the FRFT. It discusses three major topics. First, the manuscripts relates the FRFT to other mathematical transforms. Second, it discusses various approaches for practical realizations of the FRFT. Third, we overview the practical applications of the FRFT. From these discussions, we can clearly state the FRFT is closely related to other mathematical transforms, such as time-frequency and linear canonical transforms. Nevertheless, we still feel that major contributions are expected in the field of the its digital realizations and applications, especially, since many digital realizations of the FRFT still lack properties of the continuous FRFT. Overall, the FRFT is a valuable signal processing tool. Its practical applications are expected to grow significantly in years to come, given that the FRFT offers many advantages over the traditional Fourier analysis. I. IIn very simple terms, the fractional Fourier transform (FRFT) is a generalization of the ordinary Fourier transform [1]. Specifically, the FRFT implements the so-called order parameter α which acts on the ordinary Fourier transform operator. In other words, the αth order fractional Fourier transform represents the αth power of the ordinary Fourier transform operator. When α = π/2, we obtain the Fourier transform, while for α = 0, we obtain the signal itself. Any intermediate value of α (0 < α < π/2) produces a signal representation that can be considered as a rotated time-frequency representation of the signal [2], Signal Processing, Vol. 91 No. 6, June, 2011 [3].Interestingly enough, the idea of the fractional powers of the Fourier operator has been "discovered" several times in the literature. Initially, the idea appeared in the mathematical literature between the two world wars (e.g., [4], [5]). More publications relating to this idea appeared after the second world war, however they were sporadic (e.g. [6]). The idea of fractional Fourier operator re-gains a momentum in 1980's with publications by Namias (e.g. [7]). Following Namias' contributions, a large number of papers appeared in the literature during 1990's tying the concept of the fractional Fourier operators to many other fields (e.g., time-frequency analysis as described in [2]). We have also witnessed a number of recent contributions attempting to understand the practical applications of the FRFT beyond optics.The main goal of this publication is to provide an overview of recent developments regarding the FRFT and its applications. Although a number of publications reviewing the FRFT has also appeared in recent years (e.g., [1], [8], [9]), some of these publications are geared towards explaining the mathematical eloquence b...

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Section: Dfrft Based On Eigenvectorsmentioning

confidence: 99%

Fractional Fourier transform as a signal processing tool: An overview of recent developments

2011

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“…The authors had derived another type of discrete fractional Fourier transform as reported in [13,[18][19][20][21][22][23][24][25] by searching the eigenvectors and eigenvalues of the DFT matrix followed by computing the fractional power of the DFT matrix based on Hermite function. This type of DFRFT worked very similarly to the continuous FRFT, and it fulfills the properties of orthogonality, additivity, and reversibility.…”

Section: Eigenvector Decomposition Type Dfrftmentioning

confidence: 99%

“…Therefore, for these elliptical rotations an angle modification and a postphase compensation in the DFRFT have been required to obtain results similar to the continuous FRFT [24]. This approach has been extended to the so-called multiple-parameter discrete fractional Fourier transform (MPDFRFT) [20,25]. In fact, the MPDFRFT maintains all desired properties and reduces to the DFRFT when all of its order parameters are the same.…”

Section: Eigenvector Decomposition Type Dfrftmentioning

confidence: 99%

DFRFT: A Classified Review of Recent Methods with Its Application

Singh¹,

Saxena²

2013

Journal of Engineering

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In the literature, there are various algorithms available for computing the discrete fractional Fourier transform (DFRFT). In this paper, all the existing methods are reviewed, classified into four categories, and subsequently compared to find out the best alternative from the view point of minimal computational error, computational complexity, transform features, and additional features like security. Subsequently, the correlation theorem of FRFT has been utilized to remove significantly the Doppler shift caused due to motion of receiver in the DSB-SC AM signal. Finally, the role of DFRFT has been investigated in the area of steganography.

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“…The target vibration can be estimated through successive chirp-rate estimations using the discrete fractional Fourier transform (DFrFT), which is inherently geared toward chirprate estimation [3], [4], [5], [6], [7]. Later on, a subspace method was incorporated into the DFrFT-based method and it provides better performance than the direct DFrFT-based method in noise [8], [9].…”

Section: Introductionmentioning

confidence: 99%

Performance analysis on synthetic aperture radar-based vibration estimation in clutter

Wang

Santhanam

Pepin

et al. 2012

2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)

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Recently, a time-frequency method based on the discrete fractional Fourier transform (DFrFT) was proposed for estimating target vibrations using synthetic aperture radar (SAR). Later on, a subspace method was incorporated into the DFrFT-based method. It is shown that the subspace method provides better performance than the direct DFrFT-based method in noise. However, the performance of these two methods has not been studied in clutter that cause strong interference with signals from vibrating targets in real-world applications. In this paper, the performance of the two vibration estimation methods in clutter is characterized and compared via simulations. Simulation results demonstrate that the DFrFT-based method, that yielded reliable results when signal-to-clutter ratios (SCR) exceeds 18 dB, now yields reliable results when SCR exceeds 8 dB with the incorporation of the subspace method. Experimental results show that the subspace method correctly estimates the vibration frequency of a 7 Hz vibration from actual SAR data at an estimated SCR of 14 dB.

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On the multiangle centered discrete fractional Fourier transform

Cited by 67 publications

References 18 publications

Fractional Fourier transform as a signal processing tool: An overview of recent developments

Fractional Fourier transform as a signal processing tool: An overview of recent developments

DFRFT: A Classified Review of Recent Methods with Its Application

Performance analysis on synthetic aperture radar-based vibration estimation in clutter

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