Cagatay Candan

The discrete fractional Fourier transform

We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.

Digital Computation of Linear Canonical Transforms

We deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform (LCT) of a function, from the samples of the original function. Two approaches are presented and compared. The first is based on decomposition of the LCT into chirp multiplication, Fourier transformation, and scaling operations. The second is based on decomposition of the LCT into a fractional Fourier transform followed by scaling and chirp multiplication. Both algorithms take ~ N log N time, where N is the time-bandwidth product of the signals. The only essential deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus, the algorithms compute LCTs with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy.

The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

Certain solutions to Harpers equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.

On Higher Order Approximations for Hermite–Gaussian Functions and Discrete Fractional Fourier Transforms

Discrete equivalents of Hermite-Gaussian functions play a critical role in the definition of a discrete fractional Fourier transform. The discrete equivalents are typically calculated through the eigendecomposition of a commutator matrix. In this letter, we first characterize the space of DFT-commuting matrices and then construct matrices approximating the Hermite-Gaussian generating differential equation and use the matrices to accurately generate the discrete equivalents of Hermite-Gaussians.

Analysis and Further Improvement of Fine Resolution Frequency Estimation Method From Three DFT Samples

The bias and mean square error (MSE) analysis of the frequency estimator suggested in is given and an improved version of the estimator, with the removal of estimator bias, is suggested. The signal-to-noise ratio (SNR) threshold above which the bias removal is effective is also determined.

A unified framework for derivation and implementation of Savitzky-Golay filters

The Savitzky-Golay (SG) filter design problem is posed as the minimum norm solution of an underdetermined equation system. A unified SG filter design framework encompassing several important applications such as smoothing, differentiation, integration and fractional delay is developed. In addition to the generality and flexibility of the framework, an efficient SG filter implementation structure, naturally emerging from the framework, is proposed. The structure is shown to reduce the number of multipliers in the smoothing application. More specifically, the smoothing application, where an Lth degree polynomial to the frame of 2N+1 samples is fitted, can be implemented with N-L/2 multiplications per output sample instead of N+1 multiplications with the suggested structure.

Fine resolution frequency estimation from three DFT samples

An efficient and low complexity frequency estimation method based on the discrete Fourier transform (DFT) samples is described. The suggested method can operate with an arbitrary window function in the absence or presence of zero-padding. The frequency estimation performance of the suggested method is shown to follow the Cramer-Rao bound closely without any error floor due to estimator bias, even at exceptionally high signal-to-noise-ratio (SNR) values. HighlightsFrequency estimation.Windowing.Cramer-Rao bound.Interpolated DFT (IpDFT).

A fine-resolution frequency estimator using an arbitrary number of DFT coefficients

A method for the frequency estimation of complex exponential signals observed under additive white Gaussian noise is presented. Unlike competing methods based on relatively few Discrete Fourier Transform (DFT) samples, the presented technique can generate a frequency estimate by fusing the information from all DFT samples. The estimator is shown to follow the Cramer–Rao bound with a smaller signal-to-noise ratio (SNR) gap than the competing estimators at high SNR.

Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations

It is possible to obtain either exact realizations or useful approximations of linear systems or matrix-vector products arising in many different applications, by synthesizing them in the form of repeated or multi-channel filtering operations in fractional Fourier domains, resulting in much more efficient implementations with acceptable reduction in accuracy. By varying the number and configuration of the filter blocks, which may take the form of arbitrary flow graphs, it is possible to trade off between accuracy and efficiency in the desired manner. The proposed scheme constitutes a systematic way of exploiting the information inherent in the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent.

Digital Wideband Integrators With Matching Phase and Arbitrarily Accurate Magnitude Response

A new class of linear-phase infinite-impulse-response digital wideband integrators based on the numerical integration rules is presented. The proposed class of integrators exactly matches the desired phase response of the continuous-time integrator (after group delay compensation) and can approximate the magnitude response as closely as desired by increasing the number of system zeros, i.e., the order of the integrator. The low-order integrators (up to the fourth degree) generated by this technique can be immediately utilized in many applications such as strapdown inertial navigation systems, sampled data systems, and other applications that require accurate integration.