We consider the secure transmission in ergodic fast Rayleigh fading multiple-input single-output single-antenna-eavesdropper (MISOSE) wiretap channels. We assume that the statistics of both the legitimate and eavesdropper channels are the only available channel state information at the transmitter (CSIT). By introducing a new secrecy capacity upper bound, we prove that the secrecy capacity is achieved by the Gaussian input without prefixing. To attain this result, we form another MISOSE channel for upper-bounding by relaxing the equivocation constraint, and tighten the bound by carefully selecting correlations between the legitimate and eavesdropper channel gains. The resulting upper bound is tighter than the others in the literature which are based on modifying the correlation between the noises at the legitimate receiver and eavesdropper. Next, we fully characterize the secrecy capacity by showing that the optimal channel input covariance matrix is a scaled identity matrix. The key to solve such a stochastic optimization problem is by exploiting the completely monotone property of the secrecy capacity. Finally, we prove that with only statistical CSIT of both channels, the capacity will neither scale with signal-to-noise ratio (SNR) nor the number of antenna. Our numerical results also match these observations and further confirm that having the legitimate CSIT (realizations) is very beneficial to increase the secrecy capacity.
We consider the secure transmission in ergodic fast Rayleigh fading multiple-input single-output single-antenna-eavesdropper (MISOSE) wiretap channels. We assume that the statistics of both the legitimate and eavesdropper channels are the only available channel state information at the transmitter (CSIT). By introducing a new secrecy capacity upper bound, we prove that the secrecy capacity is achieved by the Gaussian input without prefixing. To attain this result, we form another MISOSE channel for upper-bounding by relaxing the equivocation constraint, and tighten the bound by carefully selecting correlations between the legitimate and eavesdropper channel gains. The resulting upper bound is tighter than the others in the literature which are based on modifying the correlation between the noises at the legitimate receiver and eavesdropper. Next, we fully characterize the secrecy capacity by showing that the optimal channel input covariance matrix is a scaled identity matrix. The key to solve such a stochastic optimization problem is by exploiting the completely monotone property of the secrecy capacity. Finally, we prove that with only statistical CSIT of both channels, the capacity will neither scale with signal-to-noise ratio (SNR) nor the number of antenna. Our numerical results also match these observations and further confirm that having the legitimate CSIT (realizations) is very beneficial to increase the secrecy capacity.
In this paper, we consider the secure transmission in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When legitimate receiver has more (or equal) antennas than the eavesdropper, we prove the first MIMOME secrecy capacity result with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is forming an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of which the same as the eavesdropper's channel matrix. Next, subject to the total power constraint overall transmit antennas, we solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, with aids of the proposed degraded channel, we show that the stochastic MIMOME optimization problem can be transformed to be a Schur-concave problem to find its optimal solution. Finally, we find that the MIMOME secrecy capacities scale with the signal-to-noise ratios with large enough numbers of antennas at legitimate receiver. However, as shown in previous works, such a scaling does not exist for wiretap channels with single antenna at legitimate receiver and eavesdropper each.
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the antennas at the legitimate receiver are more than (or equal to) those at the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to be the same as the eavesdropper's channel matrix. Next, under the total power constraint over all transmit antennas, we analytically solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, thanks to the proposed degraded channel, we can transform the stochastic MIMOME optimization problem to be a Schur-concave one and then find its solution. Besides total power constraint, we also investigate the secrecy capacity when the transmitter is subject to the practical per-antenna power constraint. The corresponding optimization problem is even more difficult since it is not Schur-concave. Under the two power constraints considered, the corresponding MIMOME secrecy capacities can both scale with the signal-to-noise ratios (SNR) when the difference between numbers of antennas at legitimate receiver and eavesdropper are large enough. However, when the legitimate receiver and eavesdropper have a single antenna each, such SNR scalings do not exist for both cases.
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the antennas at the legitimate receiver are more than (or equal to) those at the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to be the same as the eavesdropper's channel matrix. Next, under the total power constraint over all transmit antennas, we analytically solve the channel-input covariance matrix optimization problem to fully characterize the MIMOME secrecy capacity. Typically, the MIMOME optimization problems are non-concave. However, thanks to the proposed degraded channel, we can transform the stochastic MIMOME optimization problem to be a Schur-concave one and then find its solution. Besides total power constraint, we also investigate the secrecy capacity when the transmitter is subject to the practical per-antenna power constraint. The corresponding optimization problem is even more difficult since it is not Schur-concave. Under the two power constraints considered, the corresponding MIMOME secrecy capacities can both scale with the signal-to-noise ratios (SNR) when the difference between numbers of antennas at legitimate receiver and eavesdropper are large enough. However, when the legitimate receiver and eavesdropper have a single antenna each, such SNR scalings do not exist for both cases.
In this paper, we consider the secure transmission in ergodic fast fading multiple-input single-output single-antenna-eavesdropper (MISOSE) wiretap channels with only the statistics of eavesdropper's channel state information at the transmitter (CSIT). Two kinds of the legitimate CSIT are assumed, that is, full and statistical legitimate CSIT. With full legitimate CSIT, we generalize and optimize the previously proposed artificial noise (AN) aided secure beamforming to improve its secrecy rate performance. The AN covariance matrix in our scheme is more flexible than previous scheme and the region of non-zero secrecy rate is enlarged significantly according to our simulations. For the case with statistical legitimate CSIT, we further prove that the secure beamforming is secrecy capacity achieving for the Rayleigh faded channels. In this case, the AN is not necessary. Extensions to cases where legitimate receiver and eavesdropper have multiple antennas will also be discussed.
A Gaussian multiple-input single-output (MISO) wiretap channel model is considered, where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper each equipped with a single antenna. We study the problem of finding the optimal input covariance that achieves ergodic secrecy capacity subject to a power constraint where only statistical information about the eavesdropper channel is available at the transmitter. This is a non-convex optimization problem that is in general difficult to solve. Existing results address the case in which the eavesdropper or/and legitimate channels have independent and identically distributed Gaussian entries with zero-mean and unit-variance, i.e., the channels have trivial covariances. This paper addresses the general case where eavesdropper and legitimate channels have nontrivial covariances. A set of equations describing the optimal input covariance matrix are proposed along with an algorithm to obtain the solution. Based on this framework, we show that when full information on the legitimate channel is available to the transmitter, the optimal input covariance has always rank one. We also show that when only statistical information on the legitimate channel is available to the transmitter, the legitimate channel has some general non-trivial covariance, and the eavesdropper channel has trivial covariance, the optimal input covariance has the same eigenvectors as the legitimate channel covariance. Numerical results are presented to illustrate the algorithm.
The characterization of secrecy capacities of Gaussian wiretap channels depends on the knowledge of channel state information at the transmitter (CSIT). When there is only statistical CSIT, the secrecy capacity is unknown in general. In this paper we investigate the relation between the usual multivariate stochastic order and the degradedness among the legitimate receiver's and eavesdropper's channels under fast fading with only statistical CSIT. The considered transmitter has multiple antennas and the legitimate receiver and the eavesdropper both have multiple antennas. Based on the technique of coupling we derive criteria to check the degradedness of the fast fading Gaussian wiretap channel including some commonly used channel models even there is only statistical CSIT. We illustrate one example of a Rician 2 × 2 × 2 channel where the ergodic secrecy capacity can be derived under different K-factors.