scholarly journals Feedback-capacity of degraded Gaussian Vector BC using directed information and concave envelopes

Author(s):  
Viswanathan Ramachandran ◽  
Sibi Raj B Pillai
2017 ◽  
Author(s):  
Viswanathan Ramachandran

It is known that the capacity region of a two user physically degraded discrete memoryless (DM) broadcastchannel (BC) is not enlarged by feedback. An identical result holds true for a physically degraded Gaussian BC,established later using a variant of the Entropy Power Inequality (EPI). In this paper, we extend the latter resultto a physically degraded Gaussian Vector BC (PD-GVBC). However, the extension is not EPI based, but employs arecent result on the factorization of concave envelopes. While the existing concave envelope factorization results donot hold in the presence of feedback, we show that factorizing the corresponding directed information quantitiessuffice to attain the feedback capacity region of a PD-GVBC. Our work demonstrates that factorizing concaveenvelopes of directed information can handle situations involving feedback. We further show that the capacityregion of a discrete memoryless reversely physically degraded BC is not enlarged by feedback.


Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 533
Author(s):  
Milan S. Derpich ◽  
Jan Østergaard

We present novel data-processing inequalities relating the mutual information and the directed information in systems with feedback. The internal deterministic blocks within such systems are restricted only to be causal mappings, but are allowed to be non-linear and time varying, and randomized by their own external random input, can yield any stochastic mapping. These randomized blocks can for example represent source encoders, decoders, or even communication channels. Moreover, the involved signals can be arbitrarily distributed. Our first main result relates mutual and directed information and can be interpreted as a law of conservation of information flow. Our second main result is a pair of data-processing inequalities (one the conditional version of the other) between nested pairs of random sequences entirely within the closed loop. Our third main result introduces and characterizes the notion of in-the-loop (ITL) transmission rate for channel coding scenarios in which the messages are internal to the loop. Interestingly, in this case the conventional notions of transmission rate associated with the entropy of the messages and of channel capacity based on maximizing the mutual information between the messages and the output turn out to be inadequate. Instead, as we show, the ITL transmission rate is the unique notion of rate for which a channel code attains zero error probability if and only if such an ITL rate does not exceed the corresponding directed information rate from messages to decoded messages. We apply our data-processing inequalities to show that the supremum of achievable (in the usual channel coding sense) ITL transmission rates is upper bounded by the supremum of the directed information rate across the communication channel. Moreover, we present an example in which this upper bound is attained. Finally, we further illustrate the applicability of our results by discussing how they make possible the generalization of two fundamental inequalities known in networked control literature.


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