On the Binary Input Gaussian Wiretap Channel with/without Output Quantization
Abstract
:1. Introduction
- Both the legitimate receiver and the wiretapper have unquantized outputs;
- Both the legitimate receiver and the wiretapper have binary quantized outputs;
- The legitimate receiver has binary quantized outputs, while the wiretapper has unquantized outputs;
- The legitimate receiver has unquantized outputs, while the wiretapper has binary quantized outputs.
2. System Model
2.1. Gaussian Channel with Binary Inputs
- Unquantized outputs:If the channel output signal is directly processed by the receiver without binary quantization as shown in Figure 1, we call it the unquantized output. The channel output Y is a continuous signal, having the conditional probability density functionFor the sake of simplifying computation in what follows, the channel capacity can be approximated by keeping the first m terms of the summation as [11]
- Binary quantized outputs:Through a binary quantization, the binary quantized output isIn fact, this case can be modeled as the a binary symmetric channel with transition probability [10]. By means of the results for binary symmetric channel [12], we can obtain the channel capacity for the BI-GWC with binary quantized outputs as
2.2. Gaussian Wiretap Channel with Binary Inputs
3. BI-GWC with/without Output Quantization
3.1. Both the Legitimate Receiver and the Eavesdropper Have Unquantized Outputs or Both Have Binary Quantized Outputs
3.1.1. BI-GWC: Secrecy Capacity When the Legitimate Receiver and the Eavesdropper Have Unquantized Outputs
3.1.2. BI-GWC: Secrecy Capacity When the Legitimate Receiver and the Eavesdropper Have Binary Quantized Outputs
3.2. One of the Legitimate Receiver and the Eavesdropper Has Binary Quantized Outputs and the Other Has Unquantized Outputs
3.2.1. BI-GWC: Secrecy Capacity When the Legitimate Receiver Has Binary Quantized Outputs and the Eavesdropper Has Unquantized Outputs
3.2.2. BI-GWC: Secrecy Capacity When the Legitimate Receiver Has Unquantized Outputs and the Eavesdropper Has Binary Quantized Outputs
4. Numerical Results
4.1. Comparison of , , and
4.2. Comparison from the Perspectives of SNR and SNR/bit
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A. Proof of Theorem 1
Appendix B. Proof of Property 1
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Qi, C.; Chen, Y.; Vinck, A.J.H. On the Binary Input Gaussian Wiretap Channel with/without Output Quantization. Entropy 2017, 19, 59. https://doi.org/10.3390/e19020059
Qi C, Chen Y, Vinck AJH. On the Binary Input Gaussian Wiretap Channel with/without Output Quantization. Entropy. 2017; 19(2):59. https://doi.org/10.3390/e19020059
Chicago/Turabian StyleQi, Chao, Yanling Chen, and A. J. Han Vinck. 2017. "On the Binary Input Gaussian Wiretap Channel with/without Output Quantization" Entropy 19, no. 2: 59. https://doi.org/10.3390/e19020059
APA StyleQi, C., Chen, Y., & Vinck, A. J. H. (2017). On the Binary Input Gaussian Wiretap Channel with/without Output Quantization. Entropy, 19(2), 59. https://doi.org/10.3390/e19020059