1 11th DSP Workshop Taos Ski Valley, NM 2004 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

2 11th DSP Workshop Taos Ski Valley, NM 2004 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 11th DSP Workshop Taos Ski Valley, NM 2004 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 11th DSP Workshop Taos Ski Valley, NM 2004 The Centered DFRFT Define CDFRFT for parameter  as:  V T  matrix of Grünbaum eigenvectors.     diagonal matrix with elements k =e -jk   CDFRFT in terms of the individual eigenvectors v k via single expression for N even or odd :

5 11th DSP Workshop Taos Ski Valley, NM 2004 Concentrating a linear chirp CDFRFT of the chirp  =102° DFT of the chirp

6 11th DSP Workshop Taos Ski Valley, NM 2004 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 11th DSP Workshop Taos Ski Valley, NM 2004 Basis vectors of the CDFRFT Row index Angle (real part)

8 11th DSP Workshop Taos Ski Valley, NM 2004 IF of CDFRFT basis vectors IF estimates of the rows of the CDFRFT for  =5° IF estimates of the rows of the CDFRFT for  =85°

9 11th DSP Workshop Taos Ski Valley, NM 2004 Chirp rate Vs. angle  The chirp rate can be approximated by:

10 11th DSP Workshop Taos Ski Valley, NM 2004 Better accuracy for c r has an error slightly larger than 10%. Restricting the angle to the range 45° to 135°: The empirical relation: produces an error of less than 2% if we use the exact value of a at which the maximum occurs.

11 11th DSP Workshop Taos Ski Valley, NM 2004 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 11th DSP Workshop Taos Ski Valley, NM 2004 MA-CDFRFT: chirp rate Vs. Frequency representation

13 11th DSP Workshop Taos Ski Valley, NM 2004 Chirp rate estimation: monocomponent chirp Peak occurs at r=36 that corresponds to  = The value for the chirp rate obtained from the second empirical relation is Note that we are not obtaining the exact value of . The actual chirp rate of the signal is

14 11th DSP Workshop Taos Ski Valley, NM 2004 Chirp rate estimation: two component chirp Peaks occur at r=27 and r=36 that correspond to  = and  = respectively. The values for the chirp rate obtained from the second empirical relation are and The actual chirp rates of the signal are and

15 11th DSP Workshop Taos Ski Valley, NM 2004 Chirp rate estimation: three component chirp Peaks occur at r=24, r=30 and r=36 that correspond to  =1.1781,  = and  = respectively. The values for the chirp rate obtained from the second empirical relation are , and The actual chirp rates of the signal are , and

16 11th DSP Workshop Taos Ski Valley, NM 2004 MA-CDFRFT: nonzero average frequency Zero averageNonzero average

17 11th DSP Workshop Taos Ski Valley, NM 2004 Chirp rate estimation: Noisy chirps Noiseless3dB SNR

18 11th DSP Workshop Taos Ski Valley, NM 2004 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 computed efficiently 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 11th DSP Workshop Taos Ski Valley, NM 2004 References B. Santhanam and J. H. McClellan, “The Discrete Rotational Fourier Transform,” IEEE Trans. Sig. Process., Vol. 44, No. 4, pp , S. Pei, M. Yeh, C. Tseng, “Discrete Fractional Fourier Transform Based on Orthogonal Projections,” IEEE Trans. Sig. Process., Vol. 47, No. 5, pp , May C. Candan, M. A. Kutay, H. M. Ozatkas, “The Discrete Fractional Fourier Transform,” IEEE Trans. Sig. Process., Vol. 48, No. 5, pp , 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 , S. Clary and D. H. Mugler, "Shifted Fourier Matrices and Their Tridiagonal Commutors," SIAM Jour. Matr. Anal. & Appl., Vol. 24, No. 3, pp , B. Santhanam and J. G. Vargas-Rubio, “On the Grünbaum Commutor Based Discrete Fractional Fourier Transform,” Proc. of ICASSP04, Vol. II, pp , Montreal, 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.