## Frame-theory-based approach for determining the time–frequency distribution characterizing a dense group of reflecting objects

JOSA A, Vol. 17, Issue 6, pp. 1077-1085 (2000)

http://dx.doi.org/10.1364/JOSAA.17.001077

Enhanced HTML Acrobat PDF (191 KB)

### Abstract

An inverse-scattering problem concerning the determination of a time–frequency spreading function is addressed. Such a function characterizes a dense group of reflecting objects at different ranges and moving with different velocities. The problem, arising in radar and other remote-sensing techniques, is a classical inverse problem. The aim is to reconstruct a function of two variables by means of signals (of one variable) reflected from the environment being observed. The proposed approach is developed by recourse to the frame theory in order to provide a reconstruction formula that asymptotically converges to a unique spreading function. The realistic situation with respect to the transmission of a finite number of signals is further considered. In this case the reconstruction formula is shown to yield the orthogonal projection of the spreading function onto a subspace generated by the outgoing signals.

© 2000 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(100.3190) Image processing : Inverse problems

(280.0280) Remote sensing and sensors : Remote sensing and sensors

(280.5600) Remote sensing and sensors : Radar

(290.3200) Scattering : Inverse scattering

**History**

Original Manuscript: July 29, 1999

Revised Manuscript: February 1, 2000

Manuscript Accepted: February 1, 2000

Published: June 1, 2000

**Citation**

Laura Rebollo-Neira and A. Plastino, "Frame-theory-based approach for determining the time–frequency distribution characterizing a dense group of reflecting objects," J. Opt. Soc. Am. A **17**, 1077-1085 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-6-1077

Sort: Year | Journal | Reset

### References

- E. J. Kelly, R. P. Wishner, “Matched-filter theory for high-velocity targets,” IEEE Trans. Mil. Electron. ME-9, 56–59 (1965). [CrossRef]
- G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Berlin, 1994).
- G. Kaiser, “Physical wavelets and radar—a variational approach to remote-sensing,” IEEE Antennas Propag. Mag. 38(1), 15–24 (1996). [CrossRef]
- C. H. Wilcox, “The synthesis problem for radar ambiguity functions,” (U.S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisc., 1960).
- C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).
- H. Naparst, “Radar signal choice and processing for dense target environment,” Ph.D. dissertation (University of California, Berkeley, Calif., 1988).
- L. Rebollo-Neira, J. Fernandez-Rubio, “On the windowed Fourier transform,” IEEE Trans. Inf. Theory 45, 2668–2671 (1999). [CrossRef]
- M. Bernfeld, “Chirp Doppler radar,” Proc. IEEE 72, 540–541 (1984). [CrossRef]
- M. Bernfeld, “On the alternatives for imaging rotational targets,” in Radar and Sonar, Part II, F. A. Grünbaum, M. Bernfeld, R. E. Bluhat, eds. (Springer-Verlag, New York, 1992), pp. 37–44.
- H. Naparst, “Dense target signal processing,” IEEE Trans. Inf. Theory 37, 317–327 (1991). [CrossRef]
- R. J. Duffin, A. C. Shaffer, “A class of nonharmonic Fourier series,” Trans. Am. Math. Soc. 72, 341–366 (1952). [CrossRef]
- R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).
- J. R. Klauder, S. K. Skagerstam, Coherent States (World Scientific, Singapore, 1985).
- I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. (N.Y.) 27, 1271–1283 (1986). [CrossRef]
- I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
- C. Heil, D. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989). [CrossRef]
- S. T. Ali, J. P. Antoine, J. P. Gazeau, “Square integrability of group representation on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincaré group,” Ann. Inst. Henri Poincaré 55, 857–890 (1991).
- S. T. Ali, J. P. Antoine, J. P. Gazeau, “Continuous frames in Hilbert space,” Ann. Phys. (N.Y.) 222, 1–37 (1993). [CrossRef]
- I. Daubechies, “The Wavelets transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1 |

« Previous Article | Next Article »

OSA is a member of CrossRef.