Distribution of noncentral indefinite quadratic forms in complex normal variables

Abstract-This paper studies secrecy rate optimization in a wireless network with a single-antenna source, a multi-antenna destination and a multi-antenna eavesdropper. This is an unfavorable scenario for secrecy performance as the system is interference-limited. In the literature, assuming that the receiver operates in half duplex (HD) mode, the aforementioned problem has been addressed via use of cooperating nodes who act as jammers to confound the eavesdropper. This paper investigates an alternative solution, which assumes the availability of a full duplex (FD) receiver. In particular, while receiving data, the receiver transmits jamming noise to degrade the eavesdropper channel. The proposed self-protection scheme eliminates the need for external helpers and provides system robustness. For the case in which global channel state information is available, we aim to design the optimal jamming covariance matrix that maximizes the secrecy rate and mitigates loop interference associated with the FD operation. We consider both fixed and optimal linear receiver design at the destination, and show that the optimal jamming covariance matrix is rank-1, and can be found via an efficient 1-D search. For the case in which only statistical information on the eavesdropper channel is available, the optimal power allocation is studied in terms of ergodic and outage secrecy rates. Simulation results verify the analysis and demonstrate substantial performance gain over conventional HD operation at the destination.

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“…Notice that this approximation provides neither an upper nor a lower bound of the original problem (32). Its effect will be evaluated in Fig.…”

Section: A Expected Secrecy Ratementioning

confidence: 99%

“…seen that γ is an indefinite quadratic form in complex normal variables and the outage probability can be derived as [32] Prob(γ ≥ 1) = e…”

Section: A Expected Secrecy Ratementioning

confidence: 99%

Improving Physical Layer Secrecy Using Full-Duplex Jamming Receivers

Zheng

Krikidis

et al. 2013

IEEE Trans. Signal Process.

368

285

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“…3) We obtain the CDF of the quadratic form as a finite sum of expenentials (as opposed to the infinite summation expression obtained in [17]). The exponent is the eignevalue of the matrix of the resulting indefinite form.…”

Section: Babak Hassibi Department Of Electrical Engineeringmentioning

confidence: 99%

“…etc. ), or provide highly complex solutions using series expansion or approximations for indefinite quadratic forms [3], [8], [4], [6], [9], [10], [11], [12], [13], [2], [14], [15], [16], [17]. Numerical integration is used in [18], [19] to evaluate the distribution of an indefinite quadratic forms.…”

Section: Introduction and Problem Definitionmentioning

confidence: 99%

On the distribution of indefinite quadratic forms in Gaussian random variables

Al-Naffouri

Hassibi

2009

2009 IEEE International Symposium on Information Theory

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Abstract-In this work, we propose a transparent approach to evaluating the CDF of indefinite quadratic forms in Gaussian random variables and ratios of such forms. This quantity appears in the analysis of different receivers in communication systems and in various applications in signal processing. Instead of attem pting to find the pdf of this quantity as is the case in many papers in literature, we focus on finding the CDF. The basic trick that we implement is to replace inequalities that appear in the CDF calculations with the unit step function and replace the latter with its Fourier transform. This produces a multi-dimensional integral that can be evaluated using complex integration. We show how our approach extends to nonzero mean Gaussian real/com plex vectors and to the joint distribution of indefinite quadratic forms. 1

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“…In Fig. 1, we plot the optimal solution (obtained by graphic method) of p 1 as a function of ξ 12 based on (9). It is seen that p 1 can be well approximated by…”

Section: B Power Allocation For Two-transmit-antenna Systemmentioning

confidence: 99%

Capacity Analysis and Power Allocation over Non-Identical MISO Rayleigh Fading Channels

Cao

Tao

Kam

2008

2008 IEEE International Conference on Communications

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Abstract-We analyze the capacity of a multiple-input singleoutput system over Rayleigh fading channels. The channels are assumed to be independent and non-identically distributed. Simple, explicit and closed-form expressions of ergodic mutual information and outage probability are obtained. Moreover, two suboptimal but efficient analytical power allocation schemes for mutual information maximization and outage minimization are derived, respectively. In specific, for mutual information maximization, more power is assigned to those channels with higher channel variances, while for outage minimization the power allocation scheme follows the water-filling principle. I. INTRODUCTIONMultiple-input multiple-output (MIMO) technology has attracted attention in wireless communications, since it offers significant increases in data throughput and link reliability without additional bandwidth or transmit power. Tremendous research work has been done on co-located MIMO systems, where identical channel distribution is often assumed. Recently, MIMO systems over non-identical fading channels have also attracted great attention because of their application in cooperative communications and distributed antenna systems. In decode-and-forward (DF) cooperative communications systems [1], a transmission from a source to a destination is facilitated with the help of a set of relays. In the second stage when those relays have known the transmitted signal, the subsequent transmission can be regarded as a non-identical multi-input single-output (MISO) system. In distributed antenna systems [2], multiple antennas work with the help of coaxial cable to simulcast signals. A non-identical MISO system actually is formed, which enhances signal quality, increases system capacity and improves coverage. Unlike conventional MIMO systems, non-identical MIMO systems experience independent but not necessarily identically distributed (i.n.d) fading. Effects of i.n.d fading in MIMO systems have been studied on several aspects. The bit-error performances over i.n.d Rayleigh/Ricean and Nakagami fading channels using space-time block codes (STBC) are studied in [3] and [4], respectively. In [5], the biterror performance over semi-identical Rayleigh fading channels using differential STBC with noncoherent receivers is analyzed. On the other hand, authors in [6] show that, for ergodic capacity maximization, the eigenvalues of the covariance matrix of input signals are monotonically distributed with respect to the ordered singular values of non-identical MIMO channel matrix. Furthermore, the total power is only shared by those eigen-channels with large enough singular values.

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Distribution of noncentral indefinite quadratic forms in complex normal variables

Cited by 73 publications

References 28 publications

Improving Physical Layer Secrecy Using Full-Duplex Jamming Receivers

Improving Physical Layer Secrecy Using Full-Duplex Jamming Receivers

On the distribution of indefinite quadratic forms in Gaussian random variables

Capacity Analysis and Power Allocation over Non-Identical MISO Rayleigh Fading Channels

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