  We present a mathematically justifiable, computationally simple, sample
eigenvalue based procedure for estimating the number of high-dimensional
signals in white noise using relatively few samples. The main motivation for
considering a sample eigenvalue based scheme is the computational simplicity
and the robustness to eigenvector modelling errors which are can adversely
impact the performance of estimators that exploit information in the sample
eigenvectors.
  There is, however, a price we pay by discarding the information in the sample
eigenvectors; we highlight a fundamental asymptotic limit of sample eigenvalue
based detection of weak/closely spaced high-dimensional signals from a limited
sample size. This motivates our heuristic definition of the effective number of
identifiable signals which is equal to the number of "signal" eigenvalues of
the population covariance matrix which exceed the noise variance by a factor
strictly greater than 1+sqrt(Dimensionality of the system/Sample size). The
fundamental asymptotic limit brings into sharp focus why, when there are too
few samples available so that the effective number of signals is less than the
actual number of signals, underestimation of the model order is unavoidable (in
an asymptotic sense) when using any sample eigenvalue based detection scheme,
including the one proposed herein. The analysis reveals why adding more sensors
can only exacerbate the situation. Numerical simulations are used to
demonstrate that the proposed estimator consistently estimates the true number
of signals in the dimension fixed, large sample size limit and the effective
number of identifiable signals in the large dimension, large sample size limit.
