# A General Coding Scheme for Signaling Gaussian Processes Over Gaussian Decision Models

TL;DR: The n-finite transmission feedback information (FTFI) capacity of unstable Gaussian decision models with memory on past outputs is transformed into controllers-encoders-decoders that control the output process, encode a Gaussian process, reconstruct theGaussian process via a mean-square error (MSE) decoder, and achieve the n-FTFI capacity.

Abstract: In this paper, we transform the n-finite transmission feedback information (FTFI) capacity of unstable Gaussian decision models with memory on past outputs, subject to an average cost constraint of quadratic form derived in [1], into controllers-encoders-decoders that control the output process, encode a Gaussian process, reconstruct the Gaussian process via a mean-square error (MSE) decoder, and achieve the n-FTFI capacity. For a Gaussian RV message X N(0,σ2X) it is shown that the MSE decays according to E X-X' n n2= -2C 0, n (k)σ X 2, Kɞ(k min ,∞), where C 0, n (k) is the n-FTFI capacity, and k min is the threshold on the power to ensure convergence.

## Summary (1 min read)

### I. INTRODUCTION It has been recently shown

- That randomized strategies in decision systems are operational, in the sense that not only they stabilize the system but they also encode information, which can be decoded at the output of the control system with arbitrary small probability of decoding error.
- Another application is that of signaling digital messages available to the controller, such as, values associated with actuating devices, for failure detection and monitoring applications.

### B. Main Problem

- It should be mentioned that controllers-encoders have contradictory goals.
- The controller aims at stabilization, while the encoder aims at communicating new information.
- A lowcost control strategy would want the state process to be kept near the origin, with as little randomness as feasible injected by the coding part, while a communication strategy requires informative deviations.

### A. Optimal Controllers-Encoders

- In the class of linear controller-encoders that encode the Gaussian Markov process X n defined by (7) and operate at the n−FTFI capacity, the optimal controllerencoder exists, the conditional mean decoder minimizes the MSE, and these are given below, also known as Theorem 3.2.
- Then the optimal filter estimates satisfy the following recursions.
- EQUATION EQUATION EQUATION EQUATION 1 where the filter gains are difined by EQUATION (c) Conditional Mean Decoder.
- Below a certain value, then the constraint set is not feasible, i.e., it is empty.

### B. Asymptotic Properties

- The asymptotic properties of the controller-encoderdecoder are obtained by analyzing (6) , under the following assumptions (see [14] on Linear Quadratic stochastic optimal control theory with complete information).
- A constructive procedure is developed to synthesize {controller-encoder-decoder} strategies, that encode Gaussian Markov processes, communicate them over unstable Gaussian recursive models to the decoder.
- Examples illustrate the convergence of the MSE to zero, as the number of transmissions tends to infinity.

Did you find this useful? Give us your feedback

...read more

##### Citations

### Cites background or methods from "A General Coding Scheme for Signali..."

...Compared to [6], we wish to show global optimality of the tuple {controller-encoder, decoder} of strategies, according to the following definition....

[...]

...CONCLUSIONS The constructive procedure developed by the authors and collaborators [6] to synthesize {controller-encoder-decoder} strategies, that encode unstable Gaussian Markov processes, communicate them over unstable G-RMs to the decoder, is shown to be globally optimal and linear, among all strategies....

[...]

...Answer to the question: We consider the two-parameter coding scheme [6] of communicating X to the decoder...

[...]

##### References

1,610 citations

### "A General Coding Scheme for Signali..." refers background in this paper

...with or without feedback, the capacity is given by [3]...

[...]

1,037 citations

### "A General Coding Scheme for Signali..." refers methods in this paper

...Asymptotic Properties The asymptotic properties of the controller-encoderdecoder are obtained by analyzing (6), under the following assumptions (see [14] on Linear Quadratic stochastic optimal control theory with complete information)....

[...]

551 citations

### "A General Coding Scheme for Signali..." refers methods in this paper

...Schalkwijk and Kailath [5] showed that, when the Elias coding scheme is applied to a set of equiprobable messages { 0, 1, . . . ,Mn 4 = exp{(n+ 1)R} : n = 0, 1, . . . } , then the probability of ML decoding error at time n, decays doubly exponentially....

[...]

...Schalkwijk and Kailath [5] showed that, when the Elias coding scheme is applied to a set of equiprobable messages { 0, 1, ....

[...]

...[5] J. P. M. Schalkwijk and T. Kailath, “A coding scheme for additive noise channels with feedback-I: no bandwidth constraints,” IEEE Transactions on Information Theory, vol. 12, no. 2, pp. 172–182, April 1966....

[...]

...Variations of the Elias and Schalkwijk-Kailath schemes for network communication over memoryless AGN channels are extensive and given in [9]–[12]....

[...]

^{1}

349 citations

### "A General Coding Scheme for Signali..." refers background in this paper

...Variations of the Elias and Schalkwijk-Kailath schemes for network communication over memoryless AGN channels are extensive are given in [9]–[12]....

[...]

232 citations

### "A General Coding Scheme for Signali..." refers background or methods in this paper

...Kim [8] revisited the limited memory, stationary ergodic version of the Cover and Pombra [7] AGN channel, and applied frequency domain methods to conclude that Butman’s conjecture is true....

[...]

...[7] T. Cover and S. Pombra, “Gaussian feedback capacity,” IEEE Transactions on Information Theory, vol. 35, no. 1, pp. 37–43, Jan. 1989....

[...]

...Cover and Pombra [7] derived the characterization of feedback capacity for the AGN channel (8), when the noise V n is nonstationary, nonergodic, with distribution PV n ....

[...]