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Optimal filtering in fractional Fourier domains
 IN PROC. IEEE INT. CONF. ACOUST., SPEECH, SIGNAL PROCESSING
, 1997
"... For timeinvariant degradation models and stationary signals and noise, the classical Fourier domain Wiener filter, which can be implemented in O(N log N) time, gives the minimum meansquareerror estimate of the original undistorted signal. For timevarying degradations and nonstationary processes, ..."
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Cited by 37 (12 self)
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For timeinvariant degradation models and stationary signals and noise, the classical Fourier domain Wiener filter, which can be implemented in O(N log N) time, gives the minimum meansquareerror estimate of the original undistorted signal. For timevarying degradations and nonstationary processes, however, the optimal linear estimate requires O(N 2) time for implementation. We consider filtering in fractional Fourier domains, which enables significant reduction of the error compared with ordinary Fourier domain filtering for certain types of degradation and noise (especially of chirped nature), while requiring only O(N log N) implementation time. Thus, improved performance is achieved at no additional cost. Expressions for the optimal filter functions in fractional domains are derived, and several illustrative examples are given in which significant reduction of the error (by a factor of 50) is obtained.
An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain
 IEEE Transactions on Signal Processing
, 2001
"... Abstract—The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle in the time–frequency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signal ..."
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Cited by 11 (0 self)
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Abstract—The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle in the time–frequency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signals is obtained, and it is shown that a Gaussian signal achieves the lower bound. The effect of shifting and scaling the signal on the uncertainty relation is discussed. An example is given in which the uncertainty relation for a real signal is obtained, and it is shown that this relation matches with that given by the uncertainty relation derived. Index Terms—Fractional Fourier transform, time–frequency analysis, uncertainty principle. I.
Interpolating Between Periodicity and Discreteness Through the Fractional Fourier Transform
"... Abstract—Periodicity and discreteness are Fourier duals in the same sense as operators such as coordinate multiplication and differentiation, and translation and phase shift. The fractional Fourier transform allows interpolation between such operators which gradually evolve from one member of the du ..."
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Cited by 6 (2 self)
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Abstract—Periodicity and discreteness are Fourier duals in the same sense as operators such as coordinate multiplication and differentiation, and translation and phase shift. The fractional Fourier transform allows interpolation between such operators which gradually evolve from one member of the dual pair to the other as the fractional order goes from zero to one. Here, we similarly discuss the interpolation between the dual properties of periodicity and discreteness, showing how one evolves into the other as the order goes from zero to one. We also discuss the concepts of partial discreteness and partial periodicity and relate them to fractional discreteness and periodicity. Index Terms—Chirp functions, discrete functions, fractional Fourier transform, periodic functions, sampling. I.
P.G Student
"... The Fractional Fourier Transform is a generalized form of Fourier Transform, which can be interpreted as a rotation by angle α in timefrequency plane or decomposition of signals in terms of chirps. However it fails in locating Fractional Fourier Domain Frequency contents. Shorttime FRFT variants a ..."
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The Fractional Fourier Transform is a generalized form of Fourier Transform, which can be interpreted as a rotation by angle α in timefrequency plane or decomposition of signals in terms of chirps. However it fails in locating Fractional Fourier Domain Frequency contents. Shorttime FRFT variants are suitable for analysis of multicomponent and nonlinear chirp signals with improved timefrequency resolution. ShortTime FRFT is the simultaneous representation of, combination of the time and FRFDfrequency information. Filtering in the fractional domain separates the noise and the highly concentrated signal. Filtering results depict that the results in fractional domain give better results. The timefrequency representation in fractional domain is useful tool for various applications like filtering of chirp signals. Simulations are performed on MATLAB platform.
SIGNAL RECOVERY FROM PARTIAL FRACTIONAL FOURIER DOMAIN INFORMATION AND PULSE SHAPE DESIGN USING ITERATIVE PROJECTIONS
, 2005
"... that I have read this thesis and that in my opinion it is fully adequate, ..."
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that I have read this thesis and that in my opinion it is fully adequate,
High Resolution TimeFrequency Analysis By Fractional Domain
"... A new algorithm is proposed to obtain very high resolution timefrequency analysis of signal components with curved time frequency supports. The proposed algorithm is based on fractional Fourier domain warping concept introduced in this work. By integrating this warping concept to the recently de ..."
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Cited by 1 (0 self)
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A new algorithm is proposed to obtain very high resolution timefrequency analysis of signal components with curved time frequency supports. The proposed algorithm is based on fractional Fourier domain warping concept introduced in this work. By integrating this warping concept to the recently developed directionally smoothed Wigner distribution algorithm [1], the high performance of that algorithm on linear, chirplike components is extended to signal components with curved timefrequency supports. The main advantage of the algorithm is its ability to suppress not only the crosscross terms, but also the autocross terms in the Wigner distribution. For a signal with # samples duration, the computational complexity of the algorithm is ### ### ## flops for each computed slice of the new timefrequency distribution.
Fractional power spectrum
 IEEE Trans. Signal Process
, 2008
"... Abstract—In this paper, by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process, we consider the case associated with the random process in an explicit manner. The fractional power spectral relations for the fractional Fourier do ..."
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Abstract—In this paper, by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process, we consider the case associated with the random process in an explicit manner. The fractional power spectral relations for the fractional Fourier domain filter are derived, and the expression for the fractional power spectrum in terms of the fractional correlation is obtained. In addition, the definitions and the properties of the fractional white noise and the chirpstationary process are presented. Simulation results verify the theoretical derivations and demonstrate the potential applications, such as detection and parameter estimation of chirp signals, fractional power spectral estimation and system identification in the fractional Fourier domain. Index Terms—Fractional correlation function, fractional Fourier transform, fractional power spectrum, fractional white noise. I.
時頻分 析 term paper(Filter Design)
"... Filtering in a time domain or a frequency domain is a normal method we knew before. Recently, filtering in a single fractional Fourier domain has been generalized. In this correspondence, we will generalize this to repeated filtering in consecutive fractional Fourier domain and discuss its applicati ..."
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Filtering in a time domain or a frequency domain is a normal method we knew before. Recently, filtering in a single fractional Fourier domain has been generalized. In this correspondence, we will generalize this to repeated filtering in consecutive fractional Fourier domain and discuss its application to signal restoration through an illustrative example. For timeinvariant degradation model and stationary signals and noise, the Fourier domain of Wiener filter which can be implemented in logN N time, and the meansquareerror of the original undistorted signal are minimum. Others, for timevarying degradations and nonstationary processes, the optimal linear estimate requires time for implementation is 2N. Here, we consider filtering in fractional Fourier domains which enables significant reduction of the error compared with ordinary Fourier domain filtering for certain types of degradation and noise. However, require time of fractional Fourier domain is 2N. Thus, improved the optimal filter functions in fractional domains are derived, and several illustrative examples are given in which significant reduction of the error is obtained.