This work addresses the problem of determining whether two multivariate random time series have the same power spectral density

(PSD), which has applications, for instance, in physical-layer security and cognitive radio. Remarkably, existing detectors for this

problem do not usually provide any kind of optimality. Thus, we study here the existence under the Gaussian assumption of optimal

invariant detectors for this problem, proving that the uniformly most powerful invariant test (UMPIT) does not exist. Thus, focusing on

close hypotheses, we show that the locally most powerful invariant test (LMPIT) only exists for univariate time series. In the multivariate

case, we prove that the LMPIT does not exist. However, this proof suggests two LMPIT-inspired detectors, one of which outperforms

previously proposed approaches, as computer simulations show.

},
keywords = {cognitive radio, winter},
author = {David Ram{\'\i}rez and Daniel Romero and Javier V{\'\i}a and R. L{\'o}pez-Valcarce and Ignacio Santamar{\'\i}a}
}
@article {941,
title = {Testing equality of multiple power spectral density matrices},
journal = {IEEE Trans. Signal Processing},
volume = {66},
year = {2018},
month = {12/2018},
pages = {6268-6280},
keywords = {cognitive radio, winter},
doi = { 10.1109/TSP.2018.2875884},
author = {David Ram{\'\i}rez and Daniel Romero and Javier V{\'\i}a and R. L{\'o}pez-Valcarce and Ignacio Santamar{\'\i}a}
}
@conference {654,
title = {Detection of Gaussian signals in unknown time-varying channels},
booktitle = {IEEE Statistical Signal Processing Workshop (SSP 2012)},
year = {2012},
month = {08/2012},
publisher = {IEEE},
organization = {IEEE},
address = {Ann Arbor, MI},
abstract = {Detecting the presence of a white Gaussian signal distorted by a\ noisy time-varying channel is addressed by means of three different\ detectors. First, the generalized likelihood ratio test (GLRT) is\ found for the case where the channel has no temporal structure, resulting\ in the well-known Bartlett{\textquoteright}s test. Then it is shown that, under\ the transformation group given by scaling factors, a locally most\ powerful invariant test (LMPIT) does not exist. Two alternative approaches\ are explored in the low signal-to-noise ratio (SNR) regime:\ the first assigns a prior probability density function (pdf) to the channel\ (hence modeled as random), whereas the second assumes an underlying\ basis expansion model (BEM) for the (now deterministic)\ channel and obtains the maximum likelihood (ML) estimates of the\ parameters relevant for the detection problem. The performance of\ these detectors is evaluated via Monte Carlo simulation.

},
keywords = {cognitive radio, dynacs, spectrum sensing},
doi = {10.1109/SSP.2012.6319858},
author = {Daniel Romero and Javier V{\'\i}a and R. L{\'o}pez-Valcarce and Ignacio Santamar{\'\i}a}
}
@article {RamirezVazquezTSP11,
title = {Detection of Rank-{P} Signals in Cognitive Radio Networks With Uncalibrated Multiple Antennas},
journal = {IEEE Trans. on Signal Processing},
volume = {59},
number = {8},
year = {2011},
month = {08/2012},
pages = {3764-3774},
abstract = {Spectrum sensing is a key component of the Cognitive\ Radio paradigm. Typically, primary signals have to be\ detected with uncalibrated receivers at signal-to-noise ratios\ (SNRs) well below decodability levels. Multiantenna detectors\ exploit spatial independence of receiver thermal noise to boost\ detection performance and robustness. We study the problem

of detecting a Gaussian signal with rank-P unknown spatial

covariance matrix in spatially uncorrelated Gaussian noise with

unknown covariance using multiple antennas. The generalized

likelihood ratio test (GLRT) is derived for two scenarios. In the

first one, the noises at all antennas are assumed to have the same\ (unknown) variance, whereas in the second, a generic diagonal\ noise covariance matrix is allowed in order to accommodate\ calibration uncertainties in the different antenna frontends. In\ the latter case, the GLRT statistic must be obtained numerically,\ for which an efficient method is presented. Furthermore, for\ asymptotically low SNR, it is shown that the GLRT does\ admit a closed form, and the resulting detector performs well\ in practice. Extensions are presented in order to account for\ unknown temporal correlation in both signal and noise, as well\ as frequency-selective channels.

},
keywords = {cognitive radio, dynacs, spectrum sensing},
doi = {10.1109/TSP.2011.2146779},
author = {David Ram{\'\i}rez and Gonzalo V{\'a}zquez-Vilar and R. L{\'o}pez-Valcarce and Javier V{\'\i}a and Ignacio Santamar{\'\i}a}
}
@conference {RamirezVazquezICASSP11,
title = {Multiantenna detection under noise uncertainty and primary user{\textquoteright}s spatial structure},
booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
year = {2011},
month = {May},
pages = {2948-2951},
publisher = {IEEE},
organization = {IEEE},
address = {Prage, Czech Republic},
keywords = {cognitive radio, generalized likelihood ratio test (GLRT), spectrum sensing},
author = {David Ram{\'\i}rez and Gonzalo V{\'a}zquez-Vilar and R. L{\'o}pez-Valcarce and Javier V{\'\i}a and Ignacio Santamar{\'\i}a}
}
@conference {RamirezValcarceICASSP10,
title = {Multiantenna spectrum sensing: detection of spatial correlation among time-series with unknown spectra},
booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
year = {2010},
address = {Dallas, TX},
keywords = {cognitive radio, coherence spectrum, generalized likelihood ratio test, Hadamard ratio, multiple-channel signal detection},
author = {David Ram{\'\i}rez and Javier V{\'\i}a and Ignacio Santamar{\'\i}a and R. L{\'o}pez-Valcarce and L. L. Scharf}
}