Abstract

We address the problem of optical propagation in random lattices. This can be relevant in characterizing, among other phenomena, the urban shortwave channels such as those involved in cellular communications. We consider an ensemble of optical rays, generated by an isotropic source, that propagates in a two-dimensional disordered medium whose characteristic parameter is the density of inner square reflectors. The statistical characterization of the propagation mechanism is our aim. In a previous work [G. Franceschetti et al., IEEE Trans. Antennas Propag. 47(7) (1999)], a quite similar scenario has been considered, with a ray impinging on a semi-infinite layer of reflectors and with a Markov chain formulation. We report the extension of such an approach to the internal-source scenario and point out how the independence assumption of the ray characterization may not lead to particularly accurate results. Therefore we propose a different approach, based solely on the geometry of the random lattice. We exploit the intuition that the relevant geometry in such a propagation problem should be based on the city-block distance rather than on the usual Euclidean distance. This allows us to obtain a simple analytical solution in the form of a parametric family of distribution functions. This basic result is validated by means of computer simulations.

© 1999 Optical Society of America

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References

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  1. C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
    [Crossref]
  2. L. Bernstein, C. M. Yuas, “Managing the last mile,” IEEE Commun. Mag. 35, 72–76 (October1997).
    [Crossref]
  3. S. Dehghan, R. Steel, “Small cell city,” IEEE Commun. Mag.52–59 (August1997).
  4. E. Damosso, F. Tallone, “Coperture radio per ambienti microcellulari urbani: il caso DECT,” Not. Tec. Telecom Ital. 6, 67–78 (1997).
  5. H. Suzuki, “A statistical model of urban radio propagation,” IEEE Trans. Commun. C-25, 673–680 (1977).
    [Crossref]
  6. G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).
  7. A. J. Coulson, A. G. Williamson, R. G. Vaughan, “A statistical basis for lognormal shadowing effects in multipath fading channels,” IEEE Trans. Commun. 46, 494–502 (1998).
    [Crossref]
  8. H. Hashemi, “Impulse response modeling of indoor radio propagation channels,” IEEE J. Sel. Areas Commun. 11, 967–978 (1993).
    [Crossref]
  9. L. Talbi, G. Y. Delisle, “Experimental characterization of EHF multipath indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 431–439 (1996).
    [Crossref]
  10. L. Dossi, G. Tartara, F. Tallone, “Statistical analysis of measured impulse response functions of 2.0 GHz indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 405–410 (1996).
    [Crossref]
  11. G. E. Corazza, V. Degli Espositi, M. Frullone, G. Riva, “A characterization of indoor space and frequency diversity by ray-tracing modeling,” IEEE J. Sel. Areas Commun. 14, 411–419 (1996).
    [Crossref]
  12. M. F. Cátedra, J. Pérez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios,” IEEE Antennas Propag. Mag. 40(No. 2), 15–27 (1998).
    [Crossref]
  13. G. Liang, H. L. Bertoni, “A new approach to 3-D ray tracing for propagation prediction in cities,” IEEE Trans. Antennas Propag. 46, 853–863 (1998).
    [Crossref]
  14. C.-Fa Yang, B.-Cheng Wu, C.-Jyi Ko, “A ray-tracing method for modeling indoor wave propagation and penetration,” IEEE Trans. Antennas Propag. 46, 907–919 (1998).
    [Crossref]
  15. R. Mazar, A. Bronshtein, I.-Tai Lu, “Theoretical analysis of UHF propagation in a city street modeled as a random multislit waveguide,” IEEE Trans. Antennas Propag. 46, 864–871 (1998).
    [Crossref]
  16. C. C. Constantinou, L. C. Ong, “Urban radiowave propagation: a 3-D path-integral wave analysis,” IEEE Trans. Antennas Propag. 46, 211–217 (1998).
    [Crossref]
  17. G. Franceschetti, S. Marano, F. Palmieri, “Propagation without wave equation. Towards an urban area model,” IEEE Trans. Antennas Propag. (to be published).
  18. G. Franceschetti, S. Marano, F. Palmieri, “Studio della propagazione in un mezzo percolativo come nuovo modello di tessuto urbano” (Centro Studi e Laboratori Telecomunicazioni, Torino, Italy, 1998). Distributed in electronic form (CD) within Alta Freq. 10(4) (1998).
  19. D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).
  20. R. Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, San Diego, Calif., 1975).
  21. S. Marano, F. Palmieri, G. Franceschetti, “Optical propagation in a regular lattice: a Martingale based approach,” internal report (Department of Electrical Engineering and Communications, University of Naples, Naples, Italy, 1998).
  22. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).
  23. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  24. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  25. J. Spaner, K. B. Oldham, An Atlas of Functions (Springer-Verlag, Berlin, 1987).

1998 (7)

A. J. Coulson, A. G. Williamson, R. G. Vaughan, “A statistical basis for lognormal shadowing effects in multipath fading channels,” IEEE Trans. Commun. 46, 494–502 (1998).
[Crossref]

M. F. Cátedra, J. Pérez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios,” IEEE Antennas Propag. Mag. 40(No. 2), 15–27 (1998).
[Crossref]

G. Liang, H. L. Bertoni, “A new approach to 3-D ray tracing for propagation prediction in cities,” IEEE Trans. Antennas Propag. 46, 853–863 (1998).
[Crossref]

C.-Fa Yang, B.-Cheng Wu, C.-Jyi Ko, “A ray-tracing method for modeling indoor wave propagation and penetration,” IEEE Trans. Antennas Propag. 46, 907–919 (1998).
[Crossref]

R. Mazar, A. Bronshtein, I.-Tai Lu, “Theoretical analysis of UHF propagation in a city street modeled as a random multislit waveguide,” IEEE Trans. Antennas Propag. 46, 864–871 (1998).
[Crossref]

C. C. Constantinou, L. C. Ong, “Urban radiowave propagation: a 3-D path-integral wave analysis,” IEEE Trans. Antennas Propag. 46, 211–217 (1998).
[Crossref]

G. Franceschetti, S. Marano, F. Palmieri, “Studio della propagazione in un mezzo percolativo come nuovo modello di tessuto urbano” (Centro Studi e Laboratori Telecomunicazioni, Torino, Italy, 1998). Distributed in electronic form (CD) within Alta Freq. 10(4) (1998).

1997 (4)

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

L. Bernstein, C. M. Yuas, “Managing the last mile,” IEEE Commun. Mag. 35, 72–76 (October1997).
[Crossref]

S. Dehghan, R. Steel, “Small cell city,” IEEE Commun. Mag.52–59 (August1997).

E. Damosso, F. Tallone, “Coperture radio per ambienti microcellulari urbani: il caso DECT,” Not. Tec. Telecom Ital. 6, 67–78 (1997).

1996 (3)

L. Talbi, G. Y. Delisle, “Experimental characterization of EHF multipath indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 431–439 (1996).
[Crossref]

L. Dossi, G. Tartara, F. Tallone, “Statistical analysis of measured impulse response functions of 2.0 GHz indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 405–410 (1996).
[Crossref]

G. E. Corazza, V. Degli Espositi, M. Frullone, G. Riva, “A characterization of indoor space and frequency diversity by ray-tracing modeling,” IEEE J. Sel. Areas Commun. 14, 411–419 (1996).
[Crossref]

1993 (1)

H. Hashemi, “Impulse response modeling of indoor radio propagation channels,” IEEE J. Sel. Areas Commun. 11, 967–978 (1993).
[Crossref]

1977 (1)

H. Suzuki, “A statistical model of urban radio propagation,” IEEE Trans. Commun. C-25, 673–680 (1977).
[Crossref]

1972 (1)

G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Bernstein, L.

L. Bernstein, C. M. Yuas, “Managing the last mile,” IEEE Commun. Mag. 35, 72–76 (October1997).
[Crossref]

Bertoni, H. L.

G. Liang, H. L. Bertoni, “A new approach to 3-D ray tracing for propagation prediction in cities,” IEEE Trans. Antennas Propag. 46, 853–863 (1998).
[Crossref]

Bronshtein, A.

R. Mazar, A. Bronshtein, I.-Tai Lu, “Theoretical analysis of UHF propagation in a city street modeled as a random multislit waveguide,” IEEE Trans. Antennas Propag. 46, 864–871 (1998).
[Crossref]

Cátedra, M. F.

M. F. Cátedra, J. Pérez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios,” IEEE Antennas Propag. Mag. 40(No. 2), 15–27 (1998).
[Crossref]

Clapp, F. D.

G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).

Constantinou, C. C.

C. C. Constantinou, L. C. Ong, “Urban radiowave propagation: a 3-D path-integral wave analysis,” IEEE Trans. Antennas Propag. 46, 211–217 (1998).
[Crossref]

Corazza, G. E.

G. E. Corazza, V. Degli Espositi, M. Frullone, G. Riva, “A characterization of indoor space and frequency diversity by ray-tracing modeling,” IEEE J. Sel. Areas Commun. 14, 411–419 (1996).
[Crossref]

Coulson, A. J.

A. J. Coulson, A. G. Williamson, R. G. Vaughan, “A statistical basis for lognormal shadowing effects in multipath fading channels,” IEEE Trans. Commun. 46, 494–502 (1998).
[Crossref]

Damosso, E.

E. Damosso, F. Tallone, “Coperture radio per ambienti microcellulari urbani: il caso DECT,” Not. Tec. Telecom Ital. 6, 67–78 (1997).

Degli Espositi, V.

G. E. Corazza, V. Degli Espositi, M. Frullone, G. Riva, “A characterization of indoor space and frequency diversity by ray-tracing modeling,” IEEE J. Sel. Areas Commun. 14, 411–419 (1996).
[Crossref]

Dehghan, S.

S. Dehghan, R. Steel, “Small cell city,” IEEE Commun. Mag.52–59 (August1997).

Delisle, G. Y.

L. Talbi, G. Y. Delisle, “Experimental characterization of EHF multipath indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 431–439 (1996).
[Crossref]

Dossi, L.

L. Dossi, G. Tartara, F. Tallone, “Statistical analysis of measured impulse response functions of 2.0 GHz indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 405–410 (1996).
[Crossref]

Fine, S. B.

G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).

Franceschetti, G.

G. Franceschetti, S. Marano, F. Palmieri, “Studio della propagazione in un mezzo percolativo come nuovo modello di tessuto urbano” (Centro Studi e Laboratori Telecomunicazioni, Torino, Italy, 1998). Distributed in electronic form (CD) within Alta Freq. 10(4) (1998).

G. Franceschetti, S. Marano, F. Palmieri, “Propagation without wave equation. Towards an urban area model,” IEEE Trans. Antennas Propag. (to be published).

S. Marano, F. Palmieri, G. Franceschetti, “Optical propagation in a regular lattice: a Martingale based approach,” internal report (Department of Electrical Engineering and Communications, University of Naples, Naples, Italy, 1998).

Frullone, M.

G. E. Corazza, V. Degli Espositi, M. Frullone, G. Riva, “A characterization of indoor space and frequency diversity by ray-tracing modeling,” IEEE J. Sel. Areas Commun. 14, 411–419 (1996).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Gutierrez, O.

M. F. Cátedra, J. Pérez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios,” IEEE Antennas Propag. Mag. 40(No. 2), 15–27 (1998).
[Crossref]

Hashemi, H.

H. Hashemi, “Impulse response modeling of indoor radio propagation channels,” IEEE J. Sel. Areas Commun. 11, 967–978 (1993).
[Crossref]

Johnston, T. L.

G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).

Karlin, R.

R. Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, San Diego, Calif., 1975).

Ko, C.-Jyi

C.-Fa Yang, B.-Cheng Wu, C.-Jyi Ko, “A ray-tracing method for modeling indoor wave propagation and penetration,” IEEE Trans. Antennas Propag. 46, 907–919 (1998).
[Crossref]

Lavry, D.

G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).

Liang, G.

G. Liang, H. L. Bertoni, “A new approach to 3-D ray tracing for propagation prediction in cities,” IEEE Trans. Antennas Propag. 46, 853–863 (1998).
[Crossref]

Lu, I.-Tai

R. Mazar, A. Bronshtein, I.-Tai Lu, “Theoretical analysis of UHF propagation in a city street modeled as a random multislit waveguide,” IEEE Trans. Antennas Propag. 46, 864–871 (1998).
[Crossref]

Marano, S.

G. Franceschetti, S. Marano, F. Palmieri, “Studio della propagazione in un mezzo percolativo come nuovo modello di tessuto urbano” (Centro Studi e Laboratori Telecomunicazioni, Torino, Italy, 1998). Distributed in electronic form (CD) within Alta Freq. 10(4) (1998).

G. Franceschetti, S. Marano, F. Palmieri, “Propagation without wave equation. Towards an urban area model,” IEEE Trans. Antennas Propag. (to be published).

S. Marano, F. Palmieri, G. Franceschetti, “Optical propagation in a regular lattice: a Martingale based approach,” internal report (Department of Electrical Engineering and Communications, University of Naples, Naples, Italy, 1998).

Mazar, R.

R. Mazar, A. Bronshtein, I.-Tai Lu, “Theoretical analysis of UHF propagation in a city street modeled as a random multislit waveguide,” IEEE Trans. Antennas Propag. 46, 864–871 (1998).
[Crossref]

Morton, D.

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

Oldham, K. B.

J. Spaner, K. B. Oldham, An Atlas of Functions (Springer-Verlag, Berlin, 1987).

Ong, L. C.

C. C. Constantinou, L. C. Ong, “Urban radiowave propagation: a 3-D path-integral wave analysis,” IEEE Trans. Antennas Propag. 46, 211–217 (1998).
[Crossref]

Palmieri, F.

G. Franceschetti, S. Marano, F. Palmieri, “Studio della propagazione in un mezzo percolativo come nuovo modello di tessuto urbano” (Centro Studi e Laboratori Telecomunicazioni, Torino, Italy, 1998). Distributed in electronic form (CD) within Alta Freq. 10(4) (1998).

G. Franceschetti, S. Marano, F. Palmieri, “Propagation without wave equation. Towards an urban area model,” IEEE Trans. Antennas Propag. (to be published).

S. Marano, F. Palmieri, G. Franceschetti, “Optical propagation in a regular lattice: a Martingale based approach,” internal report (Department of Electrical Engineering and Communications, University of Naples, Naples, Italy, 1998).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

Pérez, J.

M. F. Cátedra, J. Pérez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios,” IEEE Antennas Propag. Mag. 40(No. 2), 15–27 (1998).
[Crossref]

Riva, G.

G. E. Corazza, V. Degli Espositi, M. Frullone, G. Riva, “A characterization of indoor space and frequency diversity by ray-tracing modeling,” IEEE J. Sel. Areas Commun. 14, 411–419 (1996).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Saez de Adana, F.

M. F. Cátedra, J. Pérez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios,” IEEE Antennas Propag. Mag. 40(No. 2), 15–27 (1998).
[Crossref]

Spaner, J.

J. Spaner, K. B. Oldham, An Atlas of Functions (Springer-Verlag, Berlin, 1987).

Stauffer, D.

D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).

Steel, R.

S. Dehghan, R. Steel, “Small cell city,” IEEE Commun. Mag.52–59 (August1997).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Stumpf, C.

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

Suzuki, H.

H. Suzuki, “A statistical model of urban radio propagation,” IEEE Trans. Commun. C-25, 673–680 (1977).
[Crossref]

Talbi, L.

L. Talbi, G. Y. Delisle, “Experimental characterization of EHF multipath indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 431–439 (1996).
[Crossref]

Tallone, F.

E. Damosso, F. Tallone, “Coperture radio per ambienti microcellulari urbani: il caso DECT,” Not. Tec. Telecom Ital. 6, 67–78 (1997).

L. Dossi, G. Tartara, F. Tallone, “Statistical analysis of measured impulse response functions of 2.0 GHz indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 405–410 (1996).
[Crossref]

Tartara, G.

L. Dossi, G. Tartara, F. Tallone, “Statistical analysis of measured impulse response functions of 2.0 GHz indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 405–410 (1996).
[Crossref]

Taylor, H. M.

R. Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, San Diego, Calif., 1975).

Turin, G. L.

G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).

Ulema, M.

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

Vaughan, R. G.

A. J. Coulson, A. G. Williamson, R. G. Vaughan, “A statistical basis for lognormal shadowing effects in multipath fading channels,” IEEE Trans. Commun. 46, 494–502 (1998).
[Crossref]

White, R. G.

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

Wilkes, J. E.

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

Williamson, A. G.

A. J. Coulson, A. G. Williamson, R. G. Vaughan, “A statistical basis for lognormal shadowing effects in multipath fading channels,” IEEE Trans. Commun. 46, 494–502 (1998).
[Crossref]

Wu, B.-Cheng

C.-Fa Yang, B.-Cheng Wu, C.-Jyi Ko, “A ray-tracing method for modeling indoor wave propagation and penetration,” IEEE Trans. Antennas Propag. 46, 907–919 (1998).
[Crossref]

Yang, C.-Fa

C.-Fa Yang, B.-Cheng Wu, C.-Jyi Ko, “A ray-tracing method for modeling indoor wave propagation and penetration,” IEEE Trans. Antennas Propag. 46, 907–919 (1998).
[Crossref]

Yu, C. C.

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

Yuas, C. M.

L. Bernstein, C. M. Yuas, “Managing the last mile,” IEEE Commun. Mag. 35, 72–76 (October1997).
[Crossref]

Alta Freq. (1)

G. Franceschetti, S. Marano, F. Palmieri, “Studio della propagazione in un mezzo percolativo come nuovo modello di tessuto urbano” (Centro Studi e Laboratori Telecomunicazioni, Torino, Italy, 1998). Distributed in electronic form (CD) within Alta Freq. 10(4) (1998).

IEEE Antennas Propag. Mag. (1)

M. F. Cátedra, J. Pérez, F. Saez de Adana, O. Gutierrez, “Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: application to picocell and microcell scenarios,” IEEE Antennas Propag. Mag. 40(No. 2), 15–27 (1998).
[Crossref]

IEEE Commun. Mag. (3)

C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, M. Ulema, “Low-tier wireless local loop radio systems—Part 1: Introduction,” IEEE Commun. Mag. 35, 84–92 (March1997);“Part 2: Comparison of systems,” IEEE Commun. Mag. 35, 94–98 (March1997).
[Crossref]

L. Bernstein, C. M. Yuas, “Managing the last mile,” IEEE Commun. Mag. 35, 72–76 (October1997).
[Crossref]

S. Dehghan, R. Steel, “Small cell city,” IEEE Commun. Mag.52–59 (August1997).

IEEE J. Sel. Areas Commun. (4)

H. Hashemi, “Impulse response modeling of indoor radio propagation channels,” IEEE J. Sel. Areas Commun. 11, 967–978 (1993).
[Crossref]

L. Talbi, G. Y. Delisle, “Experimental characterization of EHF multipath indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 431–439 (1996).
[Crossref]

L. Dossi, G. Tartara, F. Tallone, “Statistical analysis of measured impulse response functions of 2.0 GHz indoor radio channels,” IEEE J. Sel. Areas Commun. 14, 405–410 (1996).
[Crossref]

G. E. Corazza, V. Degli Espositi, M. Frullone, G. Riva, “A characterization of indoor space and frequency diversity by ray-tracing modeling,” IEEE J. Sel. Areas Commun. 14, 411–419 (1996).
[Crossref]

IEEE Trans. Antennas Propag. (4)

G. Liang, H. L. Bertoni, “A new approach to 3-D ray tracing for propagation prediction in cities,” IEEE Trans. Antennas Propag. 46, 853–863 (1998).
[Crossref]

C.-Fa Yang, B.-Cheng Wu, C.-Jyi Ko, “A ray-tracing method for modeling indoor wave propagation and penetration,” IEEE Trans. Antennas Propag. 46, 907–919 (1998).
[Crossref]

R. Mazar, A. Bronshtein, I.-Tai Lu, “Theoretical analysis of UHF propagation in a city street modeled as a random multislit waveguide,” IEEE Trans. Antennas Propag. 46, 864–871 (1998).
[Crossref]

C. C. Constantinou, L. C. Ong, “Urban radiowave propagation: a 3-D path-integral wave analysis,” IEEE Trans. Antennas Propag. 46, 211–217 (1998).
[Crossref]

IEEE Trans. Commun. (2)

H. Suzuki, “A statistical model of urban radio propagation,” IEEE Trans. Commun. C-25, 673–680 (1977).
[Crossref]

A. J. Coulson, A. G. Williamson, R. G. Vaughan, “A statistical basis for lognormal shadowing effects in multipath fading channels,” IEEE Trans. Commun. 46, 494–502 (1998).
[Crossref]

IEEE Trans. Veh. Technol. (1)

G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, D. Lavry, “A statistical model of urban multipath propagation,” IEEE Trans. Veh. Technol. VT-21, 1–9 (1972).

Not. Tec. Telecom Ital. (1)

E. Damosso, F. Tallone, “Coperture radio per ambienti microcellulari urbani: il caso DECT,” Not. Tec. Telecom Ital. 6, 67–78 (1997).

Other (8)

D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).

R. Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, San Diego, Calif., 1975).

S. Marano, F. Palmieri, G. Franceschetti, “Optical propagation in a regular lattice: a Martingale based approach,” internal report (Department of Electrical Engineering and Communications, University of Naples, Naples, Italy, 1998).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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Figures (4)

Fig. 1
Fig. 1

Example of an optical ray impinging on a random lattice. An interesting problem is to characterize the penetration capability of the ray in such a disordered medium, which can model some dense urban areas.

Fig. 2
Fig. 2

Optical propagation of a ray generated inside a random lattice. The actual path of the ray is a function of the reflector positions. We are interested in the average behavior of rays on the ensemble of possible lattice realizations for a given density q and on the ray angle of emission from the source located at (0, 0). With the formalism introduced (see Section 3 and Appendix A), the successive reflections take place at (i1, j1)=(-1, -1), (i2, j2)=(-1, 0), (i3, j3)=(-1, 1), (i4, j4)=(0, -2), and so on.

Fig. 3
Fig. 3

Results of a computer simulation procedure (circles) compared with the theoretical pmf Pin(m), described by Eq. (7) (solid curves), with the values {Pn(0)} obtained from simulations. In (a) the dotted–dashed curve is drawn according to the Markov chain approach described in Section 3. The discrepancy of this latter method with simulated data is evident. Plot (a) refers to Pin(m) with n=10, namely, to the pmf at the time of the tenth reflection. The lattice parameter is p=0.8. Plot (b) refers to Pin(m) with n=4 for lattices with p=0.7.

Fig. 4
Fig. 4

Simulated values of the distribution parameter Pn(0), n=1, 2,, 100, for three values of p. As expected, greater values of p imply a lower Pn(0) and hence a lower probability of finding the ray in the neighborhood of the source. The rays tend to escape far from the origin in less-dense lattices.

Equations (25)

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rn=r0+m=1nxm,n=1, 2,.
Pr{rnkbeforern0, n0}=pqek (1-pek).
qe=1-pe,pe=ptan|θ|+1.
Pxm(i)  Pr{xm=i}=qei=012qe pe|i|i0,  ,
Pn(m, k)=Pin(m)Pjn(k),
Pn(m, k)=Q(d),
Q(m+k+1)Q(m+k)=Pin(m+1)Pjn(k)Pin(m)Pjn(k)=Pin(m)Pjn(k+1)Pin(m)Pjn(k)=Pin(m+1)Pin(m)=Pjn(k+1)Pjn(k)=λ=const.
Pin(m)=Pjn(m)=Pn(0)1-Pn(0)1+Pn(0)|m|,
Pn(m, k)=Pn2(0)1-Pn(0)1+Pn(0)|m|+|k|.
in=m=1nxm,n=1, 2,,
jn=m=1nym,n=1, 2,.
fpe(ξ)=2πξ ln(1/p)ln2(1/p)+ln2(ξ/p) u(ξ)u(p-ξ),
Fpe(ξ)=1+2π arctanln(ξ/p)ln(1/p)u(p-ξ)u(ξ)
E(pem)=0p 2πξ ξm ln(1/p)ln2(1/p)+ln2(ξ/p) dξ=2π pm ln(1/p)0 exp(-mξ)ln2(1/p)+ξ2 dξ.
0 exp(-mξ)α2+ξ2 dξ
=1α 0π/2 exp(-mα tan ξ)dξ=1α J(mα)
J(ξ)=ci(ξ)sin(ξ)-si(ξ)cos(ξ),
Pxm(i)=1π [ p|i|I(p|i|)-p|i|+1I(p|i|+1)],
Pxm(0)=1-2π pI(p).
Pxm(i)=1π [ p|i|I(p|i|)-p|i|+1I(p|i|+1)],i01-2π pI(p),i=0,
I(x)=J (-ln x)=0π/2 exp(ln x tan ξ)dξ,
0<x1.
J(ξ)1ξ ξ4+a1ξ2+a2ξ4+b1ξ2+b2,
J(ξ)γ+ln ξ-ξ24 exp(-ξ2/14)sin ξ-ξ exp(-ξ2/18)-π2cos ξ,
Pin(m)=mthentryofδMn,

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