4

4

8228

Lower Bound of Time Span Product for a General Class of Signals in Fractional Fourier Domain

Fractional Fourier Transform is a generalization of the classical Fourier Transform which is often symbolized as the rotation in time- frequency plane. Similar to the product of time and frequency span which provides the Uncertainty Principle for the classical Fourier domain, there has not been till date an Uncertainty Principle for the Fractional Fourier domain for a generalized class of finite energy signals. Though the lower bound for the product of time and Fractional Fourier span is derived for the real signals, a tighter lower bound for a general class of signals is of practical importance, especially for the analysis of signals containing chirps. We hence formulate a mathematical derivation that gives the lower bound of time and Fractional Fourier span product. The relation proves to be utmost importance in taking the Fractional Fourier Transform with adaptive time and Fractional span resolutions for a varied class of complex signals.

Fractional Fourier Transform, uncertainty principle, Fractional Fourier Span, amplitude, phase.

3

9355

Perturbation in the Fractional Fourier Span due to Erroneous Transform Order and Window Function

Fractional Fourier Transform is a generalization of the
classical Fourier Transform. The Fractional Fourier span in general
depends on the amplitude and phase functions of the signal and varies
with the transform order. However, with the development of the
Fractional Fourier filter banks, it is advantageous in some cases to
have different transform orders for different filter banks to achieve
better decorrelation of the windowed and overlapped time signal. We
present an expression that is useful for finding the perturbation in the
Fractional Fourier span due to the erroneous transform order and the
possible variation in the window shape and length. The expression is
based on the dependency of the time-Fractional Fourier span
Uncertainty on the amplitude and phase function of the signal. We
also show with the help of the developed expression that the
perturbation of span has a varying degree of sensitivity for varying
degree of transform order and the window coefficients.

Fractional Fourier Transform, Perturbation,Fractional Fourier span, amplitude, phase, transform order, filterbanks.

2

9485

Quality Factor Variation with Transform Order in Fractional Fourier Domain

Fractional Fourier Transform is a powerful tool,
which is a generalization of the classical Fourier Transform. This
paper provides a mathematical relation relating the span in Fractional
Fourier domain with the amplitude and phase functions of the signal,
which is further used to study the variation of quality factor with
different values of the transform order. It is seen that with the
increase in the number of transients in the signal, the deviation of
average Fractional Fourier span from the frequency bandwidth
increases. Also, with the increase in the transient nature of the signal,
the optimum value of transform order can be estimated based on the
quality factor variation, and this value is found to be very close to
that for which one can obtain the most compact representation. With
the entire mathematical analysis and experimentation, we consolidate
the fact that Fractional Fourier Transform gives more optimal
representations for a number of transform orders than Fourier
transform.

Fractional Fourier Transform, Quality Factor,
Fractional Fourier span, transient signals.

1

10293

An Efficient Hamiltonian for Discrete Fractional Fourier Transform

Fractional Fourier Transform, which is a
generalization of the classical Fourier Transform, is a powerful tool
for the analysis of transient signals. The discrete Fractional Fourier
Transform Hamiltonians have been proposed in the past with varying
degrees of correlation between their eigenvectors and Hermite
Gaussian functions. In this paper, we propose a new Hamiltonian for
the discrete Fractional Fourier Transform and show that the
eigenvectors of the proposed matrix has a higher degree of
correlation with the Hermite Gaussian functions. Also, the proposed
matrix is shown to give better Fractional Fourier responses with
various transform orders for different signals.

Fractional Fourier Transform, Hamiltonian, Eigen
Vectors, Discrete Hermite Gaussians.