#### Title

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

#### Date of Award

1995

#### Degree Type

Thesis

#### Degree Name

Master of Science in Electrical Engineering (MSEE)

#### Department

Electrical and Computer Engineering

#### First Advisor

G. Faye Boudreaux-Bartels

#### 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 P_{0} 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, the Hyperbolic class TFRs do remarkably well over the Cohen's class TFRs in terms of time frequency localization and cross term removal.

Finally, the proposed implementation technique for the TFDs of the Hyperbolic class using the Fast Mellin transform and numerical integration is compared with the Canfield/Jones warping technique regarding memory requirements and computational time. The new Altes Q-distribution algorithm, implemented using a Fast Mellin transform, is faster and requires less memory than using the Canfield/Jones algorithm to warp the Wigner distribution. The new smoothed Altes distribution algorithms, which require the Altes algorithm followed by numerical integration, require less memory, but takes more time to compute than the corresponding Canfield/Jones algorithm.

#### Recommended Citation

Praveenkumar, Sanigepalli, "IMPLEMENTATION OF THE HYPERBOLIC CLASS OF TIME-FREQUENCY DISTRIBUTIONS AND REMOVAL OF CROSS TERMS" (1995). *Open Access Master's Theses.* Paper 598.

https://digitalcommons.uri.edu/theses/598

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