Browsing by Subject "Wigner distribution"

Now showing 1 - 12 of 12

Open Access
Cross-term-free time-frequency distribution reconstruction via lifted projections
(Institute of Electrical and Electronics Engineers, 2015-01) Deprem, Z.; Çetin, A. Enis
A crucial aspect of time-frequency (TF) analysis is the identification of separate components in a multicomponent signal. The Wigner-Ville distribution is the classical tool for representing such signals, but it suffers from cross-terms. Other methods, which are members of Cohen's class of distributions, also aim to remove the cross-terms by masking the ambiguity function (AF), but they result in reduced resolution. Most practical time-varying signals are in the form of weighted trajectories on the TF plane, and many others are sparse in nature. Therefore, in recent studies the problem is cast as TF distribution reconstruction using a subset of AF domain coefficients and sparsity assumption. Sparsity can be achieved by constraining or minimizing the l(1) norm. In this article, an l(1) minimization approach based on projections onto convex sets is proposed to obtain a high-resolution, cross-term-free TF distribution for a given signal. The new method does not require any parameter adjustment to obtain a solution. Experimental results are presented.
Open Access
Digital computation of the fractional Fourier transform
(Institute of Electrical and Electronics Engineers, 1996-09) Özaktaş, Haldun M.; Arıkan, Orhan; Kutay, M. A.; Bozdağı, G.
An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.
Open Access
Efficient computation of the ambiguity function and the Wigner distribution on arbitrary line segments
(IEEE, Piscataway, NJ, United States, 1999) Özdemir, A. K.; Arıkan, Orhan
Efficient algorithms are proposed for the computation of Wigner distribution and ambiguity function samples on arbitrary line segments based on the relationship of Wigner distribution and ambiguity function with the fractional Fourier transformation. The proposed algorithms make use of an efficient computation algorithm of fractional Fourier transformation to compute N uniformly spaced samples O(N log N) flops. The ability of obtaining samples on arbitrary line segments provides significant freedom in the shape of the grids used in the Wigner distribution or in ambiguity function computations.
Open Access
Fractional Fourier domains
(Elsevier BV, 1995-09) Özaktaş, Haldun M.; Aytür, O.
It is customary to define the time-frequency plane such that time and frequency are mutually orthogonal coordinates. Representations of a signal in these domains are related by the Fourier transform. We consider a continuum of “fractional” domains making arbitrary angles with the time and frequency domains. Representations in these domains are related by the fractional Fourier transform. We derive transformation, commutation, and uncertainty relations among coordinate multiplication, differentiation, translation, and phase shift operators between domains making arbitrary angles with each other. These results have a simple geometric interpretation in time-frequency space.
Open Access
Fractional fourier transform
(Wolfram Research, 2003) Özaktaş, Haldun M.; Weisstein, E. W.
Open Access
High resolution time frequency representation with significantly reduced cross-terms
(IEEE, 2000-06) Özdemir, A. Kemal; Arıkan, Orhan
A novel algorithm is proposed for efficiently smoothing the slices of the Wigner distribution by exploiting the recently developed relation between the Radon transform of the ambiguity function and the fractional Fourier transformation. The main advantage of the new algorithm is its ability to suppress cross-term interference on chirp-like auto-components without any detrimental effect to the auto-components. For a signal with N samples, the computational complexity of the algorithm is O(N log N) flops for each smoothed slice of the Wigner distribution.
Open Access
High resolution time-frequency analysis by fractional domain warping
(IEEE, 2001-05) Özdemir, Ahmet Kemal; Durak, Lütfiye; Arıkan, Orhan
A new algorithm is proposed to obtain very high resolution time-frequency 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, chirp-like components is extended to signal components with curved time-frequency supports. The main advantage of the algorithm is its ability to suppress not only the cross-cross terms, but also the auto-cross terms in the Wigner distribution. For a signal with N samples duration, the computational complexity of the algorithm is O(N log N) flops for each computed slice of the new time-frequency distribution.
Open Access
Linear canonical domains and degrees of freedom of signals and systems
(Springer Verlag, 2016) Öktem, F. S.; Özaktaş, Haldun; Healy, J. J.; Kutay, Mehmet Alper; Özaktaş, Haldun; Sheridan, J. T.
We discuss the relationships between linear canonical transform (LCT) domains, fractional Fourier transform (FRT) domains, and the space-frequency plane. In particular, we show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and monotonically ordered by the corresponding fractional order parameter and provides a more transparent view of the evolution of light through an optical system modeled by LCTs. We then study the number of degrees of freedom of optical systems and signals based on these concepts. We first discuss the bicanonical width product (BWP), which is the number of degrees of freedom of LCT-limited signals. The BWP generalizes the space-bandwidth product and often provides a tighter measure of the actual number of degrees of freedom of signals. We illustrate the usefulness of the notion of BWP in two applications: efficient signal representation and efficient system simulation. In the first application we provide a sub-Nyquist sampling approach to represent and reconstruct signals with arbitrary space-frequency support. In the second application we provide a fast discrete LCT (DLCT) computation method which can accurately compute a (continuous) LCT with the minimum number of samples given by the BWP. Finally, we focus on the degrees of freedom of first-order optical systems with multiple apertures. We show how to explicitly quantify the degrees of freedom of such systems, state conditions for lossless transfer through the system and analyze the effects of lossy transfer.
Open Access
Optimal filtering in fractional Fourier domains
(IEEE, 1995) Kutay, M. Alper; Onural, Levent; Özaktaş Haldun M.; Arıkan, Orhan
The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum-mean-square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.
Embargo
Relationships between two definitions of the discrete Wigner distribution and the continuous Wigner distribution
(Elsevier, 2025-03) Korkmaz, Sayit; Özaktaş, Haldun M.
We present a very simple relationship between two widely used discrete-time discrete-frequency Wigner distributions. The first one is obtained through sampling and the second one is obtained from the representation theory of the finite Heisenberg group. This relation shows that the values of one can simply be obtained by permuting the values of the other along the frequency axis, which in turn implies a relationship of the second definition to the samples of the continuous Wigner distribution, and the first definition to group representation theory. In the process, we derive a simplified form for the second definition which is completely analogous to the continuous Wigner distribution, and develop a set of relationships relating this definition to a discrete ambiguity function and auxiliary functions.
Open Access
Time-frequency component analyzer
(2003) Özdemir, Ahmet Kemal
In this thesis, a new algorithm, time–frequency component analyzer (TFCA), is proposed to analyze composite signals, whose components have compact time–frequency supports. Examples of this type of signals include biological, acoustic, seismic, speech, radar and sonar signals. By conducting its time–frequency analysis in an adaptively chosen warped fractional domain the method provides time–frequency distributions which are as sharp as the Wigner distribution, while suppressing the undesirable interference terms present in the Wigner distribution. Being almost fully automated, TFCA does not require any a priori information on the analyzed signal. By making use of recently developed fast Wigner slice computation algorithm, directionally smoothed Wigner distribution algorithm and fractional domain incision algorithm in the warped fractional domain, the method provides an overall time-frequency representation of the composite signals. It also provides time–frequency representations corresponding to the individual signal components constituting the composite signal. Since, TFCA based analysis enables the extraction of the identified components from the composite signals, it allows detailed post processing of the extracted signal components and their corresponding time–frequency distributions, as well.
Open Access
Use of time-frequency representations in the analysis of stock market data
(Springer, 2002) Sayan, G. T.; Sayan, Serdar; Kontoghiorghes, E. J.; Rustem, B.; Siokos, S.
The analysis of economic/financial time series in the frequency domain is a relatively underexplored area of the literature, particularly when the statistical properties of a time series are time-variant (evolutionary). In this case, the spectral content of the series varies as time progresses, rendering the conventional Fourier theory inadequate in describing the cyclical characteristics of the series fully. The joint Time-Frequency Representation (TFR) techniques overcome this problem, as they are capable of analyzing a given (continuous or discrete) function of time in time and frequency domains simultaneously. To illustrate the potential of some of the TFR techniques widely used in various fields of science and engineering for use in the analysis of stock market data, the behavior of ISE-100 index of the Istanbul Stock Exchange is analyzed first, using two linear (the Gabor Transformation and the Short Time Fourier Transform) and two quadratic (the Wigner Distribution and the Page Distribution) TFRs. The performance of each TFR in detecting and decoding cycles that may be present in the original ISE data is evaluated by utilizing a specially synthesized time series whose trend and/or cycle components can be analytically specified and computed. This series is constructed in such a way to roughly mimic the pattern of a stock index series such as the original ISE series and is used as a benchmark for comparative performance analysis. The results indicate that the performance of the Page distribution, used for the first time in economics/finance literature, is significantly superior to the other TFRs considered. The analysis is then repeated using NASDAQ-100 index data recorded over the last 15 years so as to see if the results are robust to a change in the source of stock data from an emerging to a well-established market. The results point to a superior performance by the Page distribution once again, demonstrating the robustness of our previous results.