Let$\mathrm {W}\in \mathbb {C}^{n\times n}$be a single-spiked Wishart matrix in the class$\mathrm {W}\sim \mathcal {CW}_{n}(m,\mathrm {I}_{n}+ \theta \mathrm {v}\mathrm {v}^{\dagger}) $with$m\geq n$, where${\mathrm {I}}_{n}$is the$n\times n$identity matrix,$\mathrm {v}\in \mathbb {C}^{n\times 1}$is an arbitrary vector with unit Euclidean norm,$\theta \geq 0$is a non-random parameter, and$(\cdot)^{\dagger} $represents the conjugate-transpose operator. Let u1 and${\mathrm {u}}_{n}$denote the eigenvectors corresponding to the smallest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity$Z_{\ell }^{(n)}=\left |{\mathrm {v}^{\dagger} \mathrm {u}_\ell }\right |^{2}\in (0,1)$for$\ell =1,n$. In particular, we derive a finite dimensional closed-form p.d.f. for$Z_{1}^{(n)}$which is amenable to asymptotic analysis as$m,n$diverges with$m-n$fixed. It turns out that, in this asymptotic regime, the scaled random variable$nZ_{1}^{(n)}$converges in distribution to$\chi ^{2}_{2}/2(1+\theta)$, where$\chi _{2}^{2}$denotes a chi-squared random variable with two degrees of freedom. This reveals that u1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of$Z_{n}^{(n)}$is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension$(n-2)$. Although a simple solution to this double integral seems intractable, for special configurations of$n=2,3$, and 4, we obtain closed-form expressions.

Let$\mathbf {X}\in \mathbb {C}^{n\times m}$($m\geq n$) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean andsingle-spikedcovariance matrix$\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*}$, where$\mathbf {I}_{n}$is the$n\times n$identity matrix,$\mathbf {u}\in \mathbb {C}^{n\times 1}$is an arbitrary vector with unit Euclidean norm,$\eta \geq 0$is a non-random parameter, and$(\cdot)^{*}$represents the conjugate-transpose. This paper investigates the distribution of the random quantity$\kappa _{\text {SC}}^{2}(\mathbf {X})=\sum _{k=1}^{n} \lambda _{k}/\lambda _{1}$, where$0\le \lambda _{1}\le \lambda _{2}\le \ldots \leq \lambda _{n} < \infty $are the ordered eigenvalues of$\mathbf {X}\mathbf {X}^{*}$(i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so calledscaled condition numberor the Demmel condition number (i.e.,$\kappa _{\text {SC}}(\mathbf {X})$) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e.,$\kappa _{\text {SC}}^{-2}(\mathbf {X})$). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of$\kappa _{\text {SC}}^{2}(\mathbf {X})$which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as$m,n\to \infty $such that$m-n$is fixed and when$\eta $scales on the order of$1/n$,$\kappa _{\text {SC}}^{2}(\mathbf {X})$scales on the order of$n^{3}$. In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as$m,n\to \infty $such that$n/m\to c\in (0,1)$, properly centered$\kappa _{\text {SC}}^{2}(\mathbf {X})$fluctuates on the scale$m^{\frac {1}{3}}$.

We consider multiple-antenna signal detection of primary user transmission signals by secondary user receivers in cognitive radio networks. The optimal detector is analyzed for the scenario where the number of primary user signals is no less than the number of receive antennas at the secondary users. We first derive exact expressions for the moments of the generalized likelihood ratio (GLRT) statistic, yielding approximations for the false alarm and detection probabilities. We then show that the normalized GLRT statistic converges in distribution to a Gaussian random variable when the number of antennas and observations grow large at the same rate, which is used to obtain a simple design rule for the signal detection threshold.

This paper builds upon our recent work which computed the moment generating function of the multiple-input multiple-output mutual information exactly in terms of a Painlevé V differential equation. By exploiting this key analytical tool, we provide an in-depth characterization of the mutual information distribution for sufficiently large (but finite) antenna numbers. In particular, we derive systematic closed-form expansions for the high-order cumulants. These results yield considerable new insight, such as providing a technical explanation as to why the well-known Gaussian approximation is quite robust to large signal-to-noise ratio for the case of unequal antenna arrays, while it deviates strongly for equal antenna arrays. In addition, by drawing upon our high-order cumulant expansions, we employ the Edgeworth expansion technique to propose a refined Gaussian approximation which is shown to give a very accurate closed-form characterization of the mutual information distribution, both around the mean and for moderate deviations into the tails (where the Gaussian approximation fails remarkably). For stronger deviations where the Edgeworth expansion becomes unwieldy, we employ the saddle point method and asymptotic integration tools to establish new analytical characterizations which are shown to be very simple and accurate. Based on these results, we also recover key well-established properties of the tail distribution, including the diversity-multiplexing-tradeoff.

In this paper, we compute two important information-theoretic quantities which arise in the application of multiple-input multiple-output (MIMO) antenna wireless communication systems: the distribution of the mutual information of multiantenna Gaussian channels, and the Gallager random coding upper bound on the error probability achievable by finite-length channel codes. We show that the mathematical problem underpinning both quantities is the computation of certain Hankel determinants generated by deformed versions of classical weight functions. For single-user MIMO systems, it is a deformed Laguerre weight; for multiuser MIMO systems, it is a deformed Jacobi weight. We apply two different methods to characterize each of these Hankel determinants. First, we employ the ladder operators of the corresponding monic rthogonal polynomials to give an exact characterization of the Hankel determinants in terms of Painlevé differential equations. This turns out to be a Painlevé V for the single-user MIMO scenario and a Painlevé VI for the multiuser scenario. We then introduce Coulomb fluid linear statistics methods to derive closed-form approximations for the MIMO mutual information distribution and the error probability which, although formally valid for large matrix dimensions, are shown to give accurate results even when the matrix dimensions are small. Focusing on the single-user mutual information distribution, we then employ the exact Painlevé V representation with the help of the Coulomb fluid linear statistics approximation to yield deeper insights into the scaling behavior in terms of the number of antennas and signal-to-noise ratio (SNR). Among other things, these results allow us to study the asymptotic Gaussianity of the distribution as the number of antennas increase, and to investigate when and why such approximations break down as the SNR increases. Based on the Painlevé, we also derive recursive formulas for explicitly computing in closed form any desired number of correction terms to the asymptotic mean and variance, as well as closed-form asymptotic expressions for any desired number of higher order cumulants. Using these cumulants, we propose new closed-form approximations to the mutual information distribution which are shown to be very accurate, not only in the bulk but also in the tail region of interest for the outage probability.