Orhan Arikan

Digital computation of the fractional Fourier transform

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.

Short-time Fourier transform: two fundamental properties and an optimal implementation

Shift and rotation invariance properties of linear time-frequency representations are investigated. It is shown that among all linear time-frequency representations, only the short-time Fourier transform (STFT) family with the Hermite-Gaussian kernels satisfies both the shift invariance and rotation invariance properties that are satisfied by the Wigner distribution (WD). By extending the time-bandwidth product (TBP) concept to fractional Fourier domains, a generalized time-bandwidth product (GTBP) is defined. For mono-component signals, it is shown that GTBP provides a rotation independent measure of compactness. Similar to the TBP optimal STFT, the GTBP optimal STFT that causes the least amount of increase in the GTBP of the signal is obtained. Finally, a linear canonical decomposition of the obtained GTBP optimal STFT analysis is presented to identify its relation to the rotationally invariant STFT.

Optimal filtering in fractional Fourier domains

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.

Adaptive filtering for non-Gaussian stable processes

A large class of physical phenomena observed in practice exhibit non-Gaussian behavior. In the letter /spl alpha/-stable distributions, which have heavier tails than Gaussian distributions, are considered to model non-Gaussian signals. Adaptive signal processing in the presence of such a noise is a requirement of many practical problems. Since direct application of commonly used adaptation techniques fail in these applications, new algorithms for adaptive filtering for /spl alpha/-stable random processes are introduced.<<ETX>>

Fast computation of the ambiguity function and the Wigner distribution on arbitrary line segments

By using the fractional Fourier transformation of the time-domain signals, closed-form expressions for the projections of their auto or cross ambiguity functions are derived. Based on a similar formulation for the projections of the auto and cross Wigner distributions and the well known two-dimensional (2-D) Fourier transformation relationship between the ambiguity and Wigner domains, closed-form expressions are obtained for the slices of both the Wigner distribution and the ambiguity function. By using discretization of the obtained analytical expressions, efficient algorithms are proposed to compute uniformly spaced samples of the Wigner distribution and the ambiguity function located on arbitrary line segments. With repeated use of the proposed algorithms, samples in the Wigner or ambiguity domains can be computed on non-Cartesian sampling grids, such as polar grids.

Optimal Stochastic Signaling for Power-Constrained Binary Communications Systems

Optimal stochastic signaling is studied under second and fourth moment constraints for the detection of scalar-valued binary signals in additive noise channels. Sufficient conditions are obtained to specify when the use of stochastic signals instead of deterministic ones can or cannot improve the error performance of a given binary communications system. Also, statistical characterization of optimal signals is presented, and it is shown that an optimal stochastic signal can be represented by a randomization of at most three different signal levels. In addition, the power constraints achieved by optimal stochastic signals are specified under various conditions. Furthermore, two approaches for solving the optimal stochastic signaling problem are proposed; one based on particle swarm optimization (PSO) and the other based on convex relaxation of the original optimization problem. Finally, simulations are performed to investigate the theoretical results, and extensions of the results to M-ary communications systems and to other criteria than the average probability of error are discussed.

Optimal signaling and detector design for power-constrained binary communications systems over non-gaussian channels

In this letter, joint optimization of signal structures and detectors is studied for binary communications systems under average power constraints in the presence of additive non-Gaussian noise. First, it is observed that the optimal signal for each symbol can be characterized by a discrete random variable with at most two mass points. Then, optimization over all possible two mass point signals and corresponding maximum a posteriori probability (MAP) decision rules are considered. It is shown that the optimization problem can be simplified into an optimization over a number of signal parameters instead of functions, which can be solved via global optimization techniques, such as particle swarm optimization. Finally, the improvements that can be obtained via the joint design of the signaling and the detector are illustrated via an example.

Perturbed Orthogonal Matching Pursuit

Compressive Sensing theory details how a sparsely represented signal in a known basis can be reconstructed with an underdetermined linear measurement model. However, in reality there is a mismatch between the assumed and the actual bases due to factors such as discretization of the parameter space defining basis components, sampling jitter in A/D conversion, and model errors. Due to this mismatch, a signal may not be sparse in the assumed basis, which causes significant performance degradation in sparse reconstruction algorithms. To eliminate the mismatch problem, this paper presents a novel perturbed orthogonal matching pursuit (POMP) algorithm that performs controlled perturbation of selected support vectors to decrease the orthogonal residual at each iteration. Based on detailed mathematical analysis, conditions for successful reconstruction are derived. Simulations show that robust results with much smaller reconstruction errors in the case of perturbed bases can be obtained as compared to standard sparse reconstruction techniques.

Space–bandwidth-efficient realizations of linear systems

One can obtain either exact realizations or useful approximations of linear systems or matrix-vector products that arise in many different applications by implementing them in the form of multistage or multichannel fractional Fourier-domain filters, resulting in space-bandwidth-efficient systems with acceptable decreases in accuracy. Varying the number and the configuration of filters enables one to trade off between accuracy and efficiency in a flexible manner. The proposed scheme constitutes a systematic way of exploiting the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent.

Adaptive filtering approaches for non-Gaussian stable processes

A large class of physical phenomenon observed in practice exhibit non-Gaussian behavior. In this paper, /spl alpha/-stable distributions, which have heavier tails than Gaussian distribution, are considered to model non-Gaussian signals. Adaptive signal processing in the presence of such kind of noise is a requirement of many practical problems. Since, direct application of commonly used adaptation techniques fail in these applications, new approaches for adaptive filtering for /spl alpha/-stable random processes are introduced.