Skip to main content

Linear Transformations in Signal and Optical Systems

  • Reference work entry
  • First Online:
Operator Theory
  • 4460 Accesses

Abstract

In this survey article some linear transformations that play a fundamental role in signal processing and optical systems are reviewed. After a brief discussion of the general theory of linear systems, specific linear transformations are introduced. An important class of signals to which most of these linear transformations are applied is the class of bandlimited signals and some of its generalizations. The article begins by an introduction to this class of signals and some of its properties, in particular, the property that a bandlimited signal can be perfectly reconstructed from its samples on a discrete set of points. The main tool for the reconstruction is known as the sampling theorem. Some of the transformations presented, such as the windowed Fourier transform, the continuous wavelet transform, the Wigner distribution function, the radar ambiguity function, and the ambiguity transformation, fall into the category of time–frequency, scale-translation, or phase-space representations. Such transformations make it possible to study physical systems from two different perspectives simultaneously. Another group of transformations presented is closely related to the Fourier transform, such as the fractional Fourier transform. Generalizations of the fractional Fourier transform, including the special affine Fourier transformation, and their applications in optical systems are introduced, together with sampling theorems for signals bandlimited in the domains of the aforementioned transformations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 999.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abe, S., Sheridan, J.: Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation. Opt. Lett. 19(22), 1801–1803 (1994)

    Article  Google Scholar 

  2. Almeida, L.B.: The fractional Fourier transform and time–frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)

    Article  Google Scholar 

  3. Auslander, L., Tolimieri, R.: Radar ambiguity functions and group theory. SIAM J. Math. Anal. 16, 577–601 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bastiaans, M.J., van Leest, A.J.: From the rectangular to the quincunx Gabor lattice via fractional Fourier transformation. IEEE Signal Process. Lett. 5, 203–205 (1998)

    Article  Google Scholar 

  5. Birman, M., Solomyak, M.: Spectral Theory of Selfadjoint Operators in Hilbert Space. D. Reidel Publishing, Dordrecht (1987)

    Google Scholar 

  6. Boas, R.: Entire Functions. Academic, New York (1954)

    MATH  Google Scholar 

  7. Boashash, B. (ed.): Time–Frequency Signal Analysis-Method and Applications. Halsted Press, New York (1992)

    Google Scholar 

  8. Candan, C., Kutay, M.A., Ozakdas, H.M.: The discrete fractional Fourier transform. IEEE Trans. Signal Process. 48(5), 1329–1337 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cariolaro, G., Erseghe, T., Kraniauskas, P., Laurenti, N.: Multiplicity of fractional Fourier transforms and their relationships. IEEE Trans. Signal Process. 48(1), 227–241 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Claasen, T.A.C.M., Mecklenbrauker, W.F.G.: The Wigner distribution, part 2. Philips Res. J. 35, 276–300 (1980)

    MathSciNet  MATH  Google Scholar 

  11. Cohen, L.: Time–Frequency Analysis. Prentice Hall, Endlewood Cliffes (1995)

    Google Scholar 

  12. Daubechies, I.: Ten Lectures on Wavelets. SIAM Publications, Philadelphia (1992)

    Book  MATH  Google Scholar 

  13. de Bruijn, N.G.: A theory of generalized functions with applications to Wigner distribution and Weyl correspondence. Nieuv. Archief voor Wiskunde 21, 205–280 (1973)

    MATH  Google Scholar 

  14. Erseghe, T., Kraniauskas, P., Carioraro, G.: Unified fractional Fourier transform and sampling theorem. IEEE Trans. Signal Process. 47(12), 3419–3423 (1999)

    Article  MATH  Google Scholar 

  15. Gröchenig, K.: Foundations of Time–Frequency Analysis. Birkhauser, Boston (2001)

    Book  MATH  Google Scholar 

  16. Hlawatsch, F., Boudreaux-Bartels, G.F.: Linear and quadratic time-frequency signal representations. IEEE Signal Process. Mag. 9(2), 21–67 (1992)

    Article  Google Scholar 

  17. Jerri, A.J.: The Shannon sampling theorem-its various extensions and applications. A tutorial review. Proc. IEEE 11, 1565–1596 (1977)

    Article  Google Scholar 

  18. Kotel’nikov, V.: On the carrying capacity of the ether and wire in telecommunications. In: Material for the First All-Union Conference on Questions of Communications, Izd. Red. Upr. Svyazi RKKA, Moscow, Russia (1933)

    Google Scholar 

  19. Kramer, H.: A generalized sampling theorem. J. Math. Phys. 38, 68–72 (1959)

    Article  MATH  Google Scholar 

  20. Kutay, M.A., Ozaktas, H.M., Arikan, O., Onural, L.: Optimal filtering in fractional Fourier domains. IEEE Trans. Signal Process. 45 1129–1143 (1997)

    Article  Google Scholar 

  21. Lee, A.J.: Characterization of bandlimited functions and processes. Inform. Control 31, 258–271 (1976)

    Article  MATH  Google Scholar 

  22. Lohmann, A.W.: Image rotation, Wigner rotation and the fractional Fourier transform. J. Opt. Soc. Am. A. 10, 2181–2186 (1993)

    Article  Google Scholar 

  23. McBride, A., Kerr, F.: On Namias’s fractional Fourier Transforms. IMA J. Appl. Math. 39, 159–175 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mendlovic, D., Ozaktas, H.M.: Fractional Fourier transforms and their optical implementation 1. J. Opt. Soc. Am. A 10, 1875–1881 (1993)

    Article  Google Scholar 

  25. Mendlovic, D., Ozaktas, H.M., Lohmaann, A.: Graded-index fibers, Wigner-distribution functions, and the Fractional Fourier transform. J. Appl. Opt. 33(26), 6188–6193 (1994)

    Article  Google Scholar 

  26. Mendlovic, D., Zalevsky, Z., Ozakdas, H.M.: The applications of the fractional Fourier transform to optical pattern recognition. In: Optical Pattern Recognition, Ch. 3. Academic, New York (1998)

    Google Scholar 

  27. Mustard, D.: The fractional Fourier transform and the Wigner distribution. J. Aust. Math. Soc. B-Appl. Math. 38, 209–219 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Namias, V.: The fractional order Fourier transforms and its application to quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nikol’skii, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, New York (1975)

    Book  Google Scholar 

  30. Ozaktas, H.M., Barshan, B., Mendlovic, D., Onural, L.: Convolution filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms. J. Opt. Soc. Am. A 11, 547–559 (1994)

    Article  MathSciNet  Google Scholar 

  31. Ozaktas, H.M., Kutay, M.A., Mendlovic, D.: Introduction to the fractional Fourier transform and its applications. In: Advances in Imaging Electronics and Physics, Ch. 4. Academic, New York (1999)

    Google Scholar 

  32. Ozaktas, H., Zalevsky, Z., Kutay, M.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2001)

    Google Scholar 

  33. Paley, R., Wiener, N.: Fourier Transforms in the Complex Domain. Am. Math. Soc. Colloquium Publ. Ser., vol. 19. American Mathematical Society, Providence (1934)

    Google Scholar 

  34. Parzen, E.: A simple proof and some extensions of the sampling theorem. Technical report N. 7. Stanford University, Stanford, CA (1956)

    Google Scholar 

  35. Pei, S.-C., Yeh, M.-H., Luo, T.-L.: Fractional Fourier series expansion for finite signal and dual extension to discrete-time fractional Fourier transform. IEEE Trans. Signal Process. 47(10), 2883–2888 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pesenson, I.: Sampling of Band limited vectors. J. Fourier Anal. Appl. 7(1), 93–100 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pesenson, I., Zayed, A.: Paley–Wiener subspace of vectors in a Hilbert space with applications to integral transforms. J. Math. Anal. Appl. 353, 566–582 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Shakhmurov, V.B., Zayed, A.I.: Fractional Wigner distribution and ambiguity functions. J. Frac. Calc. Appl. Anal. 6(4), 473–490 (2003)

    MathSciNet  MATH  Google Scholar 

  39. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  40. Whittaker, E.T.: On the functions which are represented by the expansion of interpolation theory. Proc. R. Soc. Edinb. Sect. A 35, 181–194 (1915)

    Article  Google Scholar 

  41. Wiener, N.: Hermitian polynomials and Foureir analysis. J. Math. Phys. MIT 8, 70–73 (1929)

    Article  MATH  Google Scholar 

  42. Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)

    Article  Google Scholar 

  43. Wilcox, C.: The synthesis problem for radar ambiguity functions. MRC Technical Report, 157, Math. Research Center, U.S. Army, University of Wisconsin, Madison (1960)

    Google Scholar 

  44. Wolf, K.B.: Integral Transforms in Science and Engineering. Plenum Press, New York (1979)

    Book  MATH  Google Scholar 

  45. Woodward, P.: Probability and Information Theory with Applications to Radar. McGraw-Hill, New York (1953)

    MATH  Google Scholar 

  46. Zakai, M.: Bandlimited functions and the sampling theorem. Inform. Control 8, 143–158 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zalevsky, Z., Mendlovic, D.: Fractional Wiener filter. Appl. Opt. 35, 3930–3936 (1996)

    Article  Google Scholar 

  48. Zayed, A.I.: Kramer’s sampling theorem for multidimensional signals and its relationship with Lagrange-type interpolation. J. Multidimen. Syst. Signal Process. 3, 323–340 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  50. Zayed, A.I.: Function and Generalized Function Transformations. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  51. Zayed, A.I.: On the relationship between the Fourier and fractional Fourier transforms. IEEE Signal Process. Lett. 3, 310–311 (1996)

    Article  Google Scholar 

  52. Zayed, A.I.: Convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 4, 15–17 (1997)

    Article  MathSciNet  Google Scholar 

  53. Zayed, A.I.: Fractional Fourier transform of generalized functions. J. Integr. Trans. Special Funct. 7(4), 299–312 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zayed, A.I.: A class of fractional integral transforms: a generalization of the fractional Fourier transform. IEEE Trans. Signal Process. 50, 619–627 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ahmed I. Zayed .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Basel

About this entry

Cite this entry

Zayed, A.I. (2015). Linear Transformations in Signal and Optical Systems. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_48

Download citation

Publish with us

Policies and ethics