On the condition number distribution of complex wishart matrices

Abstract

Abstract-This paper investigates the distribution of the condition number of complex Wishart matrices. Two closely related measures are considered: the standard condition number (SCN) and the Demmel condition number (DCN), both of which have important applications in the context of multiple-input multipleoutput (MIMO) communication systems, as well as in various branches of mathematics. We first present a novel generic framework for the SCN distribution which accounts for both central and non-central Wishart matrices of arbitrary dimension. This result is a simple unified expression which involves only a single scalar integral, and therefore allows for fast and efficient computation. For the case of dual Wishart matrices, we derive new exact polynomial expressions for both the SCN and DCN distributions. We also formulate a new closed-form expression for the tail SCN distribution which applies for correlated central Wishart matrices of arbitrary dimension and demonstrates an interesting connection to the maximum eigenvalue moments of Wishart matrices of smaller dimension.Based on our analytical results, we gain valuable insights into the statistical behavior of the channel conditioning for various MIMO fading scenarios, such as uncorrelated/semi-correlated Rayleigh fading and Ricean fading.

Power Allocation in Multi-Antenna Wireless Systems Subject to Simultaneous Power Constraints

“…The observation is that, as we increase the number of antennas at the transmitter, the value of α th decreases. In fact, we can conclude that for a well-conditioned Wishart matrix in which the number of transmit antennas is much larger than the number of receive antennas [21], the value of α th is smaller compared to the case of an ill-conditioned Wishart matrix. This suggests that as we increase the number of antennas at the transmitter, the short-term power constraint becomes less relevant, since the value of α th becomes smaller.…”

Maximum Eigenvalue detection for spectrum sensing under correlated noise

“…In [6], semi-asymptotic Maximum to Minimum Eigenvalue (MME) and Energy with Minimum Eigenvalue (EME) algorithms for SS have been proposed using the combination of the MP and Tracy-Widom (TW) distributions and a ratio based technique has been proposed in [12] using the TracyWidom Curtiss (TWC) distribution. The contribution in [17] provides a general framework for the Cumulative Distribution Function (CDF) of the SCN of different classes of Wishart matrices. Furthermore, in [18], analytic expressions for the Probability Density Function (PDF) and CDF of the ratio of the maximum eigenvalue to the trace of complex Wishart matrices with arbitrary dimensions have been derived.…”

The effect of noise correlation on fractional sampling based spectrum sensing

“…In this direction, firstly, we use the Standard Condition Number (SCN) of the noise covariance matrix to analyze the effect of noise correlation in eigenvalue based SS techniques and then the SCN of the received signal's covariance matrix for decision process. The SCN of a matrix is defined as the ratio of the maximum eigenvalue to the minimum eigenvalue [14] and can be used as a metric to characterize the support of the asymptotic eigenvalue probability distribution function (a.e.p.d.f.) of a random matrix.…”