Power-bandwidth-distortion scaling laws for sensor networks

ABSTRACT

The goal of a class of sensor networks is to monitor an underlying physical reality at the highest possible fidelity. Sensors acquire noisy measurements and have to communicate them over a power- and possibly bandwidth-constrained interference channel to a set of base stations. The goal of this paper is to analyze, as a function of the number of sensors, the trade-offs between the degrees of freedom of the underlying physical reality, the communication resources (power, temporal and spatial bandwidth), and the resulting distortion at which the physical reality can be estimated by the base stations. The distortion can be expressed as the sum of two fundamentally different terms. The first term reflects the fact that the measurements are noisy. It depends on the number of sensors and on their locations, but it cannot be influenced by the communication resources. The second contribution to the distortion can be controlled by the communication resources, and the key question becomes: What resources are necessary to make it decay at least as fast as the first distortion term, as a function of the number of sensors? This question is answered threefold: First, a lower bound to the power-bandwidth trade-o is derived, showing that at least a constant to linearly increasing total power is required for typical cases (as a function of M). But is this also sufficient? In the second answer, communication strategies are considered where each sensor applies the best possible distributed compression algorithm, followed by capacity-achieving channel codes. For such a separation strategy, it is shown for typical cases that the power must increase exponentially as a function of the number of sensors, suggesting that the lower bound derived in this paper is far too optimistic. However, in the third answer, it is shown that this is not the case: For some example scenarios, the power requirements of the lower bound are indeed achievable, but joint source-channel coding is required. Finally, the problem of sensor synchronization is considered, and it is shown that the scaling laws derived in this paper continue to hold under a Rician fading model.

REFERENCES

• 1: T. Berger. Rate Distortion Theory: A Mathematical Basis For Data Compression. Prentice-Hall, Englewood Cliffs, NJ, 1971.
• 2: T. Berger, Z. Zhang, and H. Viswanathan. The CEO problem. IEEE Transactions on Information Theory, IT--42:887--902, May 1996.
• 3: W. L. Briggs and V. E. Henson. The DFT: An owner's manual for the Discrete Fourier Transform. SIAM, Philadelphia, 1995.
• 4: Thomas M. Cover , Joy A. Thomas, Elements of information theory, Wiley-Interscience, New York, NY, 1991
• 5: I. Csiszár and J. K.orner. Information Theory: Coding Theory for Discrete Memoryless Systems. Academic Press, New York, 1981.
• 6: M. Gastpar. On the role of feedback in large sensor networks. In Proc International Zurich Seminar, Zurich, Switzerland, February 2004. Invited.
• 7: M. Gastpar, B. Rimoldi, and M. Vetterli. To code, or not to code: Lossy source-channel communication revisited. IEEE Transactions on Information Theory, 49(5):1147--1158, May 2003.
• 8: M. Gastpar and M. Vetterli. Scaling laws for homogeneous sensor networks. In Proc 2003 Allerton Conference, October 2003. Invited paper.
• 9: M. Gastpar and M. Vetterli. Source-channel communication in sensor networks. In Leonidas J. Guibas and Feng Zhao, editors, 2nd International Workshop on Information Processing in Sensor Networks (IPSN'03), pages 162--177. Lecture Notes in Computer Science, Springer, New York, NY, April 2003.
• 10: M. Gastpar and M. Vetterli. Scaling laws for sensor networks. to be submitted to IEEE Transactions on Information Theory, 2004.
• 11: R. M. Gray. On the asymptotic eigenvalue distribution of Toeplitz matrices. IEEE Transactions on Information Theory, IT--18:725--730, November 1972.
• 12: Y. Oohama. The rate-distortion function for the quadratic Gaussian CEO problem. IEEE Transactions on Information Theory, IT--44(3):1057--1070, May 1998.
• 13: C. E. Shannon. A mathematical theory of communication. Bell Sys. Tech. Journal, 27:379--423, 623--656, 1948.
• 14: I. E. Telatar. Capacity of multi-antenna Gaussian channels. Bell Labs Technical Memorandum, June 1995. Also published in European Transactions on Telecommunications, 10(6):585--596, Nov.-Dec. 1999.
• 15: H. Viswanathan and T. Berger. The quadratic Gaussian CEO problem. IEEE Transactions on Information Theory, IT--43(5):1549--1559, September 1997.

INDEX TERMS

Primary Classification:
  H. Information Systems
  H.1 MODELS AND PRINCIPLES
      H.1.1 Systems and Information Theory
          Subjects: Information theory

Additional Classification:
  E. Data
  E.4 CODING AND INFORMATION THEORY
      Subjects: Error control codes; Data compaction and compression

General Terms:
Performance, Theory

Keywords:
OPTA (optimum performance theoretically attainable), joint source-channel coding, sensor networks, separation theorem