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C entered D iscrete F ractional F ourier T ransform & L inear C hirp S ignals

C entered D iscrete F ractional F ourier T ransform & L inear C hirp S ignals. Balu Santhanam & Juan G. Vargas- Rubio SPCOM Laboratory, Department of E.C.E. University of New Mexico, Albuquerque Email: bsanthan ,jvargas @ece.unm.edu. ABSTRACT.

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C entered D iscrete F ractional F ourier T ransform & L inear C hirp S ignals

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  1. Centered Discrete Fractional Fourier Transform& Linear Chirp Signals Balu Santhanam & Juan G. Vargas- Rubio SPCOM Laboratory, Department of E.C.E. University of New Mexico, Albuquerque Email: bsanthan,jvargas@ece.unm.edu

  2. ABSTRACT • Centered discrete fractional Fourier transform (CDFRFT), based on the Grünbaum commutor produces a impulse-like transform for discrete linear chirp signals. • Relationship between chirp rate & angle of the transform approximated by a simple tangent function. • Multi-angle CDFRFT computed using the FFT & used to estimate the chirp rate of monocomponent & two-component chirps.

  3. Why Grunbaum eigenvectors? • Grünbaum tridiagonal commutor converges to Hermite--Gauss differential operator as N !1 • Grünbaum commutor furnishes a full orthogonal basis of eigenvectors independent of N. • Grünbaum eigenvectors are better approximations to Hermite-Gauss functions. • Corresponding CDFRFT is efficient in concentrating a discrete linear chirp signal.

  4. The Centered DFRFT • Define CDFRFT for parameter  as: • VT  matrix of Grünbaum eigenvectors. • L2a/p diagonal matrix with elements lk=e-jka. • CDFRFT in terms of the individual eigenvectors vk via single expression for N even or odd:

  5. Concentrating a linear chirp CDFRFT of the chirp a=102° DFT of the chirp

  6. Concentrating a linear chirp Number of coefficients that capture 50% of the energy for signals with average frequency zero (N=128) Number of coefficients that capture 50% of the energy for signals with average frequency /2 (N=128)

  7. Basis vectors of the CDFRFT (real part) Angle a Row index

  8. IF of CDFRFT basis vectors IF estimates of the rows of the CDFRFT for a=5° IF estimates of the rows of the CDFRFT for a=85°

  9. Chirp rate Vs. angle a The chirp rate can be approximated by:

  10. Better accuracy for cr • The empirical relation: • has an error slightly larger than 10%. • Restricting the angle to the range 45° to 135°: produces an error of less than 2% if we use the exact value of a at which the maximum occurs.

  11. Multi-angle CDFRFT • The CDFRFT of a signal x[n] can be written as: • For the set of equally spaced angles the CDFRFT can be rewritten using index r as • Multi-angle CDFRFT (MA-CDFRFT) is a DFT & can be computed using the FFT algorithm.

  12. MA-CDFRFT: chirp rate Vs. Frequency representation

  13. Chirp rate estimation: monocomponent chirp Peak occurs at r=36 that corresponds to a=1.7671. The value for the chirp rate obtained from the second empirical relation is 0.0053. Note that we are not obtaining the exact value of a. The actual chirp rate of the signal is 0.005.

  14. Chirp rate estimation: two component chirp Peaks occur at r=27 and r=36 that correspond to a=1.3254 and a=1.7671 respectively. The values for the chirp rate obtained from the second empirical relation are -0.0066 and 0.0053. The actual chirp rates of the signal are -0.007 and 0.005.

  15. Chirp rate estimation: three component chirp Peaks occur at r=24, r=30 and r=36 that correspond to a=1.1781, a=1.4726 and a=1.7671 respectively. The values for the chirp rate obtained from the second empirical relation are -0.0108, -0.0026 and -0.0053. The actual chirp rates of the signal are -0.011, -0.003 and -0.005.

  16. MA-CDFRFT: nonzero average frequency Zero average Nonzero average

  17. Chirp rate estimation: Noisy chirps Noiseless 3dB SNR

  18. Conclusions • CDFRFT based on the Grünbaum commuting matrix can concentrate a linear chirp signal in a few coefficients. • Multiangle version of the CDFRFT can be computedefficiently using the FFT algorithm. • Empirical relations that relate the chirp rate & the angle of the CDFRFT that produces an impulse-like transform were developed. • Multi-angle CDFRFT can be applied to chirp rate estimation of mono & multicomponent signals including noisy chirps.

  19. References • B. Santhanam and J. H. McClellan, “The DiscreteRotational Fourier Transform,”IEEE Trans. Sig.Process., Vol.44, No.4, pp.994-998, 1996. • S. Pei, M. Yeh, C. Tseng, “Discrete Fractional Fourier Transform Based on Orthogonal Projections,”IEEE Trans. Sig.Process., Vol. 47, No. 5, pp. 1335-1348, May 1999. • C. Candan, M. A. Kutay, H. M. Ozatkas, “The Discrete Fractional Fourier Transform,”IEEE Trans. Sig. Process., Vol. 48, No. 5, pp. 1329-1337, 2000. • D. H. Mugler and S. Clary, “Discrete Hermite Functions and The Fractional Fourier Transform,"in Proc. Int. Conf. Sampl. Theo. And Appl. Orlando Fl, pp. 303-308, 2001. • S. Clary and D. H. Mugler, "Shifted FourierMatrices and Their Tridiagonal Commutors,"SIAM Jour. Matr.Anal. & Appl., Vol.24, No.3, pp.809-821, 2003. • B. Santhanam and J. G. Vargas-Rubio, “On the Grünbaum Commutor Based Discrete Fractional Fourier Transform,”Proc. of ICASSP04, Vol. II, pp. 641-644, Montreal, 2004. • J. G. Vargas-Rubio and B. Santhanam, “Fast and Efficient Computation of a Class of Discrete Fractional Fourier Transforms,”Submitted Sig. Process. Lett., March 2004.

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