Secured Communication over Frequency-Selective Fading Channels: A Practical Vandermonde Precoding
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Research Article Secured Communication over Frequency-Selective Fading Channels: A Practical Vandermonde Precoding Mari Kobayashi,1 M´erouane Debbah,2 and Shlomo Shamai (Shitz)3 1 Department
of Telecommunications, SUPELEC, 3 Rue Joliot-Curie, Gif-sur-Yvette, 91192, France Chair on Flexible Radio, SUPELEC, 3 Rue Joliot-Curie, Gif-sur-Yvette, 91192, France 3 Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel 2 Alcatel-Lucent
Correspondence should be addressed to Mari Kobayashi, [email protected] Received 2 February 2009; Accepted 16 June 2009 Recommended by H. Vincent Poor We study the frequency-selective broadcast channel with confidential messages (BCC) where the transmitter sends a confidential message to receiver 1 and a common message to receivers 1 and 2. In the case of a block transmission of N symbols followed by a guard interval of L symbols, the frequency-selective channel can be modeled as a N × (N + L) Toeplitz matrix. For this special type of multiple-input multiple-output channels, we propose a practical Vandermonde precoding that projects the confidential messages in the null space of the channel seen by receiver 2 while superposing the common message. For this scheme, we provide the achievable rate region and characterize the optimal covariance for some special cases of interest. Interestingly, the proposed scheme can be applied to other multiuser scenarios such as the K + 1-user frequency-selective BCC with K confidential messages and the two-user frequency-selective BCC with two confidential messages. For each scenario, we provide the secrecy degree of freedom (s.d.o.f.) region of the corresponding channel and prove the optimality of the Vandermonde precoding. One of the appealing features of the proposed scheme is that it does not require any specific secrecy encoding technique but can be applied on top of any existing powerful encoding schemes. Copyright © 2009 Mari Kobayashi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction We consider a secured medium such that the transmitter wishes to send a confidential message to its receiver while keeping the eavesdropper, tapping the channel, ignorant of the message. Wyner [1] introduced this model named the wiretap channel to model the degraded broadcast channel where the eavesdropper observes a degraded version of the receiver’s signal. In this model, the confidentiality is measured by the equivocation rate, that is, the mutual information between the confidential message and the eavesdropper’s observation. For the discrete memoryless degraded wiretap channel, Wyner characterized the capacityequivocation region and showed that a nonzero secrecy rate can be achieved [1]. The most important operating point on the capacity-equivocation region is the secrecy capacity, that is, the largest reliable communication rate
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