2,518 research outputs found
Signal Processing in Large Systems: a New Paradigm
For a long time, detection and parameter estimation methods for signal
processing have relied on asymptotic statistics as the number of
observations of a population grows large comparatively to the population size
, i.e. . Modern technological and societal advances now
demand the study of sometimes extremely large populations and simultaneously
require fast signal processing due to accelerated system dynamics. This results
in not-so-large practical ratios , sometimes even smaller than one. A
disruptive change in classical signal processing methods has therefore been
initiated in the past ten years, mostly spurred by the field of large
dimensional random matrix theory. The early works in random matrix theory for
signal processing applications are however scarce and highly technical. This
tutorial provides an accessible methodological introduction to the modern tools
of random matrix theory and to the signal processing methods derived from them,
with an emphasis on simple illustrative examples
On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval
We derive the probability that all eigenvalues of a random matrix lie
within an arbitrary interval ,
, when is a real or complex finite dimensional Wishart,
double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient
recursive formulas allowing the exact evaluation of for Wishart
matrices, even with large number of variates and degrees of freedom. We also
prove that the probability that all eigenvalues are within the limiting
spectral support (given by the Mar{\v{c}}enko-Pastur or the semicircle laws)
tends for large dimensions to the universal values and for
the real and complex cases, respectively. Applications include improved bounds
for the probability that a Gaussian measurement matrix has a given restricted
isometry constant in compressed sensing.Comment: IEEE Transactions on Information Theory, 201
Eigen-Inference for Energy Estimation of Multiple Sources
In this paper, a new method is introduced to blindly estimate the transmit
power of multiple signal sources in multi-antenna fading channels, when the
number of sensing devices and the number of available samples are sufficiently
large compared to the number of sources. Recent advances in the field of large
dimensional random matrix theory are used that result in a simple and
computationally efficient consistent estimator of the power of each source. A
criterion to determine the minimum number of sensors and the minimum number of
samples required to achieve source separation is then introduced. Simulations
are performed that corroborate the theoretical claims and show that the
proposed power estimator largely outperforms alternative power inference
techniques.Comment: to appear in IEEE Trans. on Information Theory, 17 pages, 13 figure
Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices
One of the key issues in the acquisition of sparse data by means of
compressed sensing (CS) is the design of the measurement matrix. Gaussian
matrices have been proven to be information-theoretically optimal in terms of
minimizing the required number of measurements for sparse recovery. In this
paper we provide a new approach for the analysis of the restricted isometry
constant (RIC) of finite dimensional Gaussian measurement matrices. The
proposed method relies on the exact distributions of the extreme eigenvalues
for Wishart matrices. First, we derive the probability that the restricted
isometry property is satisfied for a given sufficient recovery condition on the
RIC, and propose a probabilistic framework to study both the symmetric and
asymmetric RICs. Then, we analyze the recovery of compressible signals in noise
through the statistical characterization of stability and robustness. The
presented framework determines limits on various sparse recovery algorithms for
finite size problems. In particular, it provides a tight lower bound on the
maximum sparsity order of the acquired data allowing signal recovery with a
given target probability. Also, we derive simple approximations for the RICs
based on the Tracy-Widom distribution.Comment: 11 pages, 6 figures, accepted for publication in IEEE transactions on
information theor
- âŠ