Abstract
The capacity-achieving input distribution for many channels like the additive white Gaussian noise (AWGN) channel and the free-space optical intensity (FSOI) channel under the peak-power constraint is discrete with a finite number of mass points. The number of mass points is itself a variable, and figuring it out is a part of the optimization problem. We wish to understand the behavior of the optimal input distribution at the transition points where the number of mass points changes. To this end, we give a new set of necessary and sufficient conditions at the transition points, which offer new insights into the transition and make the computation of the optimal distribution easier. For the real AWGN channel case, we show that for the zero-mean unit-variance Gaussian noise, the peak amplitude A of 1.671 and 2.786 mark the points where the binary and ternary signaling, respectively, are no longer optimal. For the FSOI channel, we give transition points where binary gives way to ternary, and in some cases where ternary gives way to quaternary, in the presence of the peak-power constraint and with or without the average-power constraint.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Smith, J.G., The Information Capacity of Amplitude- and Variance-Constrained Scalar Gaussian Channels, Inform. Control, 1971, vol. 18, pp. 203–219.
Shamai (Shitz), S. and Bar-David, I., The Capacity of Average and Peak-Power-Limited Quadrature Gaussian Channels, IEEE Trans. Inform. Theory, 1995, vol. 41, no. 4, pp. 1060–1071.
Kahn, J.M. and Barry, J.R., Wireless Infrared Communications, Proc. IEEE, 1997, vol. 85, no. 2, pp. 265–298.
Hranilovic, S. and Kschischang, F.R., Capacity Bounds for Power- and Band-Limited Optical Intensity Channels Corrupted by Gaussian Noise, IEEE Trans. Inform. Theory, 2004, vol. 50, no. 5, pp. 784–795.
Chan, T.H., Hranilovic, S., and Kschischang, F.R., Capacity-Achieving Probability Measure for Conditionally Gaussian Channels with Bounded Inputs, IEEE Trans. Inform. Theory, 2005, vol. 51, no. 6, pp. 2073–2088.
Lapidoth, A., Moser, S.M., and Wigger, M.A., On the Capacity of Free-Space Optical Intensity Channels, Proc. 2008 IEEE Int. Sympos. on Information Theory (ISIT’2008), Toronto, Canada, 2008, pp. 2419–2423.
Abou-Faycal, I.C., Trott, M.D., and Shamai (Shitz), S., The Capacity of Discrete-Time Memoryless Rayleigh-Fading Channels, IEEE Trans. Inform. Theory, 2001, vol. 47, no. 3, pp. 1290–1301.
Katz, M. and Shamai (Shitz), S., On the Capacity-Achieving Distribution of the Discrete-Time Noncoherent and Partially Coherent AWGN Channels, IEEE Trans. Inform. Theory, 2004, vol. 50, no. 10, pp. 2257–2270.
Goldsmith, A.J. and Varaiya, P.P., Capacity of Fading Channels with Channel Side Information, IEEE Trans. Inform. Theory, 1997, vol. 43, no. 6, pp. 1986–1992.
Gursoy, M.C., Poor, H.V., and Verdú, S., The Noncoherent Rician Fading Channel-Part I: Structure of the Capacity-Achieving Input, IEEE Trans. Wireless Commun., 2005, vol. 4, no. 5, pp. 2193–2206.
Tchamkerten, A., On the Discreteness of Capacity-Achieving Distributions, IEEE Trans. Inform. Theory, 2004, vol. 50, no. 11, pp. 2773–2778.
Huang, J. and Meyn, S.P., Characterization and Computation of Optimal Distributions for Channel Coding, IEEE Trans. Inform. Theory, 2005, vol. 51, no. 7, pp. 2336–2351.
Liang, Y. and Veeravalli, V.V., Capacity of Noncoherent Time-Selective Rayleigh-Fading Channels, IEEE Trans. Inform. Theory, 2004, vol. 50, no. 12, pp. 3095–3110.
Chen, J. and Veeravalli, V.V., Capacity Results for Block-Stationary Gaussian Fading Channels with a Peak Power Constraint, IEEE Trans. Inform. Theory, 2007, vol. 53, no. 12, pp. 4498–4520.
Lapidoth, A., On the Asymptotic Capacity of Stationary Gaussian Fading Channels, IEEE Trans. Inform. Theory, 2005, vol. 51, no. 2, pp. 437–446.
Luenberger, D.G., Optimization by Vector Space Methods, New York: Wiley, 1969.
Milnor, J., Morse Theory, Princeton: Princeton Univ. Press, 1963.
Arnold, V.I., Teoriya katastrof, Moscow: Nauka, 1990, 3rd ed. Translated under the title Catastrophe Theory, Berlin: Springer, 1992, 3rd ed.
Rajpurohit, P., Rawat, A., and Sharma, N., On the Capacity Achieving Distribution for a Peak Power Constrained Channel, in Proc. National Conf. on Communications (NCC’2007), Kanpur, India, 2007, pp. 79–83.
Sundaram, R.K., A First Course in Optimization Theory, Cambridge: Cambridge Univ. Press, 1996.
Bertsekas, D., Nonlinear Programming, Belmont: Athena Scientific, 2003.
Sharma, N. and Shamai (Shitz), S., Characterizing the Discrete Capacity Achieving Distribution with Peak Power Constraint at the Transition Points, in Proc. IEEE Int. Sympos. on Information Theory and Its Applications (ISITA’2008), Auckland, New Zealand, 2008.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N. Sharma, S. Shamai (Shitz), 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 4, pp. 14–32.
Supported in part by the Israel Science Foundation and the Technion Fund for Research.
The material in this paper was presented in part at the International Symposium on Information Theory and Applications (ISITA), Auckland, New Zealand, December 2008, and at the IEEE Information Theory Workshop (ITW), Cairo, Egypt, January 2010.
Rights and permissions
About this article
Cite this article
Sharma, N., Shamai (Shitz), S. Transition points in the capacity-achieving distribution for the peak-power limited AWGN and free-space optical intensity channels. Probl Inf Transm 46, 283–299 (2010). https://doi.org/10.1134/S0032946010040022
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0032946010040022