On Continuous-Time Gaussian Channels †
Abstract
:1. Introduction
Moments of reflection, however, reveals that the sampling approach for the channel capacity (with bandwidth limit or not) is heuristic in nature: For one thing, a bandwidth-limited signal cannot be time-limited, which renders it infeasible to define the data transmission rate if assuming a channel has bandwidth limit. In this regard, rigorous treatments coping with this issue and other technicalities can be found in [5,6]; see also [7] for a relevant in-depth discussion. Another issue is that, even disregarding the above technical nuisance arising from the bandwidth limit assumption, the sampling approach only gives a lower bound for the capacity of (1): it shows that is achievable via a class of special coding schemes, but it is not clear that why transmission rate higher than cannot be achieved by other coding schemes. The capacity of (1) was rigorously studied in [8,9], and a complete proof establishing as its de facto capacity can be found in [10,11].a continuous-time infinite-bandwidth Gaussian channel without feedback or memory is “equivalent” to a discrete-time Gaussian channel without feedback or memory at low signal-to-noise ratio (SNR).(A)
- (1)
- either as a feedback channel, where can be rewritten as M, interpreted as the message to be transmitted through the channel, and can be rewritten as , interpreted as the channel input, which depends on M and , the channel output up to time s that is fed back to the sender,
- (2)
- or as a memory channel, where can rewritten as , interpreted as the channel input, g is “part” of the channel, and , the channel output at time t, depends on and , the channel input and output up to time t that are present in the channel as memory, respectively.
2. Sampling Theorems
- (a)
- The solution to the stochastic differential Equation (6) uniquely exists;
- (b)
- (c)
- (d)
- Uniform Lipschitz condition: There exists a constant such that for any , any and ,
- (e)
- Uniform linear growth condition: There exists a constant such that for any and any ,
- (f)
- Regularity conditions on W: There exists such that
3. Approximation Theorems
4. The Approximation Approach
We remark however that for the purpose of deriving the capacity of (25) though, the approximation approach, like the conventional sampling approach, is heuristic in nature: Theorem 3 does require Conditions (d)–(f), which are much stronger than the power constraint (26). Nevertheless, this approach is of fundamental importance to our treatment of continuous-time Gaussian channels: as elaborated in Section 5, not only can it channel the ideas and techniques in discrete time to rigorously establish new results in continuous time, more importantly, it can also provide insights and intuition for our rigorous treatments where we will employ established tools and develop new tools in stochastic calculus.a continuous-time infinite-bandwidth Gaussian channel with feedback is “equivalent” to a discrete-time Gaussian channel with feedback at low SNR.(B)
5. Continuous-Time Multi-User Gaussian Channels
5.1. Gaussian MACs
5.2. Gaussian ICs
5.3. Gaussian BCs
6. Conclusions and Future Work
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Proof of Theorem 1
Appendix B. Proof of Lemma 1
Appendix C. Proof of Theorem 2
Appendix D. Proof of Theorem 3
Appendix E. Proof of Theorem 7
Appendix F. Proof of Theorem 8
Appendix G. Proof of Theorem 10
- when encoding, only carries the message meant for receiver i;
- when decoding, receiver i treats , , as noise,
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Liu, X.; Han, G. On Continuous-Time Gaussian Channels. Entropy 2019, 21, 67. https://doi.org/10.3390/e21010067
Liu X, Han G. On Continuous-Time Gaussian Channels. Entropy. 2019; 21(1):67. https://doi.org/10.3390/e21010067
Chicago/Turabian StyleLiu, Xianming, and Guangyue Han. 2019. "On Continuous-Time Gaussian Channels" Entropy 21, no. 1: 67. https://doi.org/10.3390/e21010067
APA StyleLiu, X., & Han, G. (2019). On Continuous-Time Gaussian Channels. Entropy, 21(1), 67. https://doi.org/10.3390/e21010067