Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T18:19:21.978Z Has data issue: false hasContentIssue false
  • English
  • Français

Signalling over a Gaussian channel with feedback and autoregressive noise

Published online by Cambridge University Press:  14 July 2016

J. Wolfowitz
J. Wolfowitz*
Affiliation:
University of Illinois
Get access Rights & Permissions [Opens in a new window]

Abstract

We study in detail the case of first-order regression, but our results can be extended to the general regression in a straightforward manner. An average energy constraint ((1.2) below) is imposed on each signal. In Section 2 we give an optimal linear signalling scheme (definition and proof in Section 4) for this channel. We conjecture that this scheme is optimal among all signalling schemes. Then the capacity C of the channel is (see Section 5) – log b, where b is the unique positive root (in x) of the equation x2 = (1 + g2(1 + |α|x)2)–1. Here a is the regression coefficient, and g2 is the ratio of the average energy per signal to the variance of the noise. An equivalent expression is C = ½log(1 + g2(1 + |α| b)2).

Keywords

GAUSSIAN CHANNELFEEDBACKENERGY CONSTRAINT FOR EACH SIGNALOPTIMAL LINEAR FUNCTION OF SIGNALSESTIMATIONLINEAR CAPACITYREGRESSIONOPTIMAL LINEAR SIGNALLING SCHEME
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 4 , December 1975 , pp. 713 - 723
Copyright
Copyright © Applied Probability Trust 1975 

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.)

References

[1] Tiernan, J. C. and Schalkwijk, J. P. M. (1974) An upper bound to the capacity of the band-limited Gaussian autoregressive channel with noiseless feedback. IEEE Trans. Inf. Theory IT-20, 311–316.Google Scholar
[2] Tiernan, J. C. (1972) Autoregressive Gaussian channels with noiseless feedback. Ph.D. thesis, University of California at San Diego.Google Scholar
[3] Schalkwijk, J. P. M. (1966) A coding scheme for additive noise channels with feedback. IEEE Trans. Inf. Theory IT-12, 183–189.CrossRefGoogle Scholar
[4] Wolfowitz, J, (1968) Note on the Gaussian channel with feedback and a power constraint. Information and Control 12, 71–78.Google Scholar
[5] Shannon, C. E. (1948) A mathematical theory of communication. Bell System Tech. Journal 27, 379–424 and 623–657.CrossRefGoogle Scholar
[6] Wolfowitz, J. (1964) Coding Theorems of Information Theory , 2nd ed., Springer-Verlag, Heidelberg and New York.Google Scholar
[7] Butman, S. (1969) A general formulation of linear feedback communication systems with solutions. IEEE Trans. Inf. Theory IT–15, 392–400.CrossRefGoogle Scholar
[8] Tiernan, J. C. (1975) Analysis of the center-of-gravity processor for the autoregressive forward channel with noiseless feedback. To appear.Google Scholar
[9] Schalkwijk, J. P. M. (1966) Center-of-gravity information feedback. Project No. 502–3, May 1966, Applied Research Laboratory, Sylvania Electronic Systems, Waltham, Mass. Google Scholar