IMPLEMENTATION OF THE HYPERBOLIC CLASS OF TIME-FREQUENCY DISTRIBUTIONS AND REMOVAL OF CROSS TERMS

Abstract

This thesis proposes new, efficient techniques for implementing some members of the Hyperbolic class of time-frequency distributions (TFD). A new, efficient algorithm is proposed for the Altes-Marinovic Q-distribution. The algorithm is a generalization of the Fast Mellin transform implementation of the Bertrand P0 distribution. Other TFDs of the Hyperbolic class, which are the smoothed versions of the Altes-Marinovic distribution, are implemented using Fast Mellin transform to compute the Altes distribution of the signal, which is then appropriately weighted and smoothed using numerical integration and first order linear interpolation techniques. Also, an extensive analysis of the warping technique proposed by Canfield and Jones is done in this thesis. Canfield and Jone have implemented the data adaptive, radially Gaussian kernel hyperbolic class time-frequency representation. We have used their warping technique but the different kernels to implement various members of the Hyperbolic class like the Pseudo Altes-Marinovic distribution, Smoothed Pseudo Altes-Marinovic distribution, etc. Our implementation has the following approach. The Woodward Ambiguity function of the logarithmically frequency warped signal is calculated to compute the Hyperbolic ambiguity function. The Hyperbolic ambiguity function (HAF) is examined to determine the mapping of the various signal auto and cross components. Many complicated signals map to simple regions in the HAF plane. A suitable kernel is chosen so as to remove the cross terms without distorting the auto terms and is multiplied with the Hyperbolic Ambiguity function. Since HC class kernels can be formulated to be equivalent to (up to a proportionality factor) Cohen's class kernels, all of the useful kernel design strategies this far developed for Cohen's class TFDs can now be applied straight forwardly to HAF plane kernel design. Then, a two dimensional Fourier transform of the product is taken. The time frequency localization of the result is corrected to compute the corresponding hyperbolic class time frequency distribution. This approach is intuitive and often makes kernel design easier for cross term removal. The logarithmic signal spectrum warping and the time frequency localization are implemented using the warpings techniques proposed by Canfield and Jones. Also, a new Hyperbolic class member is proposed which uses the Multiform tiltable exponential kernel of Costa in the Hyperbolic Ambiguity function domain. This kernel does as wells as or better than other Hyperbolic class TFDs in time-frequency scenarios that were considered in this thesis. Various nonstationary signals like Hyperbolic impulses, linear FM chirps, etc. were analyzed. In the case of Hyperbolic impulses,