Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T08:15:45.187Z Has data issue: false hasContentIssue false
  • English
  • Français

Electromagnetic wave propagation and inequalities for moments of chord lengths

Part of: General convexity Global differential geometry Geometric probability and stochastic geometry Optics, electromagnetic theory - General

Published online by Cambridge University Press:  01 July 2016

Jan Hansen  and
Matthias Reitzner
Jan Hansen*
Affiliation:
Stanford University
Matthias Reitzner*
Affiliation:
Technische Universität Wien
*
∗ Postal address: Communication Technology Laboratory, ETH Zürich, Sternwartstrasse 7, CH-8092 Zürich, Switzerland.
∗∗ Postal address: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In a convex domain K in ℝd, a transmitter and a receiver are placed at random according to the uniform distribution. The statistics of the power received by the receiver is an important quantity for the design of wireless communication systems. Bounds for the moments of the received power are given, which depend only on the volume and the surface area of the convex domain.

Keywords

Convex bodyintegral geometrychord lengthinterpoint distancedual quermassintegral

MSC classification

Primary: 52A22: Random convex sets and integral geometry 60D05: Geometric probability and stochastic geometry 78A40: Waves and radiation
Secondary: 52A40: Inequalities and extremum problems 53C65: Integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 987 - 995
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by the German Research Foundation HA3499/1-1.

Supported by Austrian Science Foundation J1940 MAT and J2193 MAT.

References

[1] Alagar, V. S. (1976). The distribution of the distance between random points. J. Appl. Prob. 13, 558–566.CrossRefGoogle Scholar
[2] Bertoni, H. (2000). Radio Propagation for Modern Wireless Systems. Prentice-Hall PTR, Upper Saddle River, NJ.Google Scholar
[3] Brannen, N. S. (1997). The Wills conjecture. Trans. Amer. Math. Soc. 349, 3977–3987.CrossRefGoogle Scholar
[4] Chakerian, G. D. (1967). Inequalities for the difference body of a convex body. Proc. Amer. Math. Soc. 18, 879–884.CrossRefGoogle Scholar
[5] Coleman, R. (1969). Random paths through convex bodies. J. Appl. Prob. 6, 430–441.CrossRefGoogle Scholar
[6] Davy, P. (1984). Inequalities for moments of secant length. Z. Wahrscheinlichkeitsth. 68, 243–246.CrossRefGoogle Scholar
[7] Gates, D. J. (1985). Asymptotics of two integrals from optimization theory and geometric probability. Adv. Appl. Prob. 17, 908–910.CrossRefGoogle Scholar
[8] Gates, J. (1987). Some properties of chord length distributions. J. Appl. Prob. 24, 863–874.CrossRefGoogle Scholar
[9] Hadwiger, H. (1957). Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin.CrossRefGoogle Scholar
[10] Hammersley, J. M. (1950). The distribution of distance in a hypersphere. Ann. Math. Statist. 21, 447–452.CrossRefGoogle Scholar
[11] Hammersley, J. M. (1952). Lagrangian integration coefficients for distance functions taken over right circular cylinders. J. Math. Phys. 31, 139–150.CrossRefGoogle Scholar
[12] Hansen, J. and Reitzner, M. (2004). Efficient indoor radio channel modeling based on integral geometry. IEEE Trans. Antennas Propagation 52, 2456–2463.CrossRefGoogle Scholar
[13] Hashemi, H. (1993). The indoor radio propagation channel. Proc. IEEE 81, 943–968.CrossRefGoogle Scholar
[14] Hassan-Ali, M. and Pahlavan, J. (2002). A new statistical model for site-specific indoor radio propagation prediction based on geometrical optics and geometric probability. IEEE Trans. Wireless Commun. 1, 112–124.CrossRefGoogle Scholar
[15] Kingman, J. F. C. (1969). Random secants of a convex body. J. Appl. Prob. 6, 660–672.CrossRefGoogle Scholar
[16] Lord, R. D. (1954). The distribution of distance in a hypersphere. Ann. Math. Statist. 25, 794–798.CrossRefGoogle Scholar
[17] Lutwak, E. (1975). Dual mixed volumes. Pacific J. Math. 58, 531–538.CrossRefGoogle Scholar
[18] Miles, R. E. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
[19] Piefke, F. (1978). Beziehungen zwischen der Sehnenlängenverteilung und der Verteilung des Abstandes zweier zufälliger Punkte im Eikörper. Z. Wahrscheinlichkeitsth. 43, 129–134.CrossRefGoogle Scholar
[20] Rappaport, T. S. (2002). Wireless Communications. Principles and Practice, 2nd edn. Prentice-Hall PTR, Upper Saddle River, NJ.Google Scholar
[21] Santaló, L. A. (1986). On the measure of line segments entirely contained in a convex body. In Aspects of Mathematics and Its Applications (North-Holland Math. Library 34), North-Holland, Amsterdam, pp. 677–687.CrossRefGoogle Scholar
[22] Schneider, R. (1985). Inequalities for random flats meeting a convex body. J. Appl. Prob. 22, 710–716.CrossRefGoogle Scholar
[23] Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.CrossRefGoogle Scholar
[24] Schneider, R. and Wieacker, J. A. (1993). Integral geometry. In Handbook of Convex Geometry, eds Gruber, P. M. and Wills, J. M., Vol. B, North-Holland, Amsterdam, pp. 1349–1390.CrossRefGoogle Scholar
[25] Sheng, T. K. (1985). The distance between two random points in plane regions. Adv. Appl. Prob. 17, 748–773.CrossRefGoogle Scholar
[26] Sulanke, R. (1961). Die Verteilung der Sehnenlängen an ebenen und räumlichen Figuren. Math. Nachr. 23, 51–74.CrossRefGoogle Scholar
[27] Voss, K. (1982). Powers of chords for convex sets. Biometrical J. 24, 513–516.CrossRefGoogle Scholar
[28] Zhang, G. Y. (1991). Restricted chord projection and affine inequalities. Geom. Dedicata 39, 213–222.CrossRefGoogle Scholar