Haldun M. Ozaktas

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.

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.

Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.

Optimal filtering in fractional Fourier domains

For time-invariant 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 mean-square-error estimate of the original undistorted signal. For time-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N/sup 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.

Limit to the Bit-Rate Capacity of Electrical Interconnects from the Aspect Ratio of the System Architecture

We show that there is a limit to the total number of bits per second,B, of information that can flow in a simple digital electrical interconnection that is set only by the ratio of the lengthlof the interconnection to the total cross-sectional dimensionAof the interconnect wiring?the “aspect ratio” of the interconnection. This limit is largely independent of the details of the design of the electrical lines. The limit is approximatelyB~BoA/l2bits/s, withBo~ 1015(bit/s) for high-performance strip lines and cables, ~1016for small on-chip lines, and ~1017?1018for equalized lines. Because the limit is scale-invariant, neither growing nor shrinking the system substantially changes the limit. Exceeding this limit requires techniques such as repeatering, coding, and multilevel modulation. Such a limit will become a problem as machines approach Tb/s information bandwidths. The limit will particularly affect architectures in which one processor must talk reasonably directly with many others. We argue that optical interconnects can solve this problem since they avoid the resistive loss physics that gives this limit.

Fractional Fourier optics

There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.

Fourier transforms of fractional order and their optical interpretation

Abstract Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a =1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Fractional Fourier transforms can extend the range of spatial filtering operations.

Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform

Two definitions of a fractional Fourier transform have been proposed previously. One is based on the propagation of a wave field through a graded-index medium, and the other is based on rotating a functions Wigner distribution. It is shown that both definitions are equivalent. An important result of this equivalency is that the Wigner distribution of a wave field rotates as the wave field propagates through a quadratic graded-index medium. The relation with ray-optics phase space is discussed.

Optimal filtering with linear canonical transformations

Optimal filtering with linear canonical transformations allows smaller mean-square errors in restoring signals degraded by linear time- or space-variant distortions and non-stationary noise. This reduction in error comes at no additional computational cost. This is made possible by the additional flexibility that comes with the three free parameters of linear canonical transformations, as opposed to the fractional Fourier transform which has only one free parameter, and the ordinary Fourier transform which has none. Application of the method to severely degraded images is shown to be significantly superior to filtering in fractional Fourier domains in certain cases.

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.