We consider the spectrum sensing problem in cognitive radio networks. We offer a framework for optimal joint detection and parameter estimation when the secondary users have only a small number of signal samples. We discuss the finite-sample optimality of the generalized likelihood ratio test (GLRT) and derive the corresponding GLRT spectrum sensing algorithms by exploiting the statistics of the received signal and the prior information on the channel, noise, as well as the data signal. An iterative GLRT sensing algorithm, and a simple non-iterative GLRT sensing algorithm are developed for slow and fast-fading channels, respectively, with the latter also serving as an approximate sensing method for slow-fading channels. The proposed techniques are also extended for spectrum sensing in orthogonal frequency-division multiple-access (OFDMA) systems and in multiple-input multiple-output (MIMO) systems. It is seen that the proposed simple non-iterative fast-fading GLRT sensing algorithm offers the best performance in all systems under considerations, including slow fading channels, fast fading channels, OFDMA systems, and MIMO systems, and it significantly outperforms several state-of-the-art spectrum sensing methods in these systems when there is noise uncertainty.

Saddlepoint approximations of the meta-converse and random-coding union bounds are derived. These bounds accurately characterize the channel coding minimum error probability for symmetric memoryless channels in a wide range of system parameters. The proposed approximations are simple to compute and yield a unified analysis of both hypothesis-testing lower bounds and random-coding upper bounds.

Cyclostationary processes exhibit a form of frequency diversity. Based on that, we show that a digital waveform with symbol period T can be asymptotically represented as a rank-1 frequency-domain vector process which exhibits uncorrelation at different frequencies inside the Nyquist spectral support of 1/T. By resorting to the fast Fourier transform (FFT), this formulation obviates the need of estimating a cumbersome covariance matrix to characterize the likelihood function. We then derive the generalized likelihood ratio test (GLRT) for the detection of a cyclostationary signal in unknown white noise without the need of a assuming a synchronized receiver. This provides a sound theoretical basis for the exploitation of the cyclostationary feature and highlights an explicit link with classical square timing recovery schemes, which appear implicitly in the core of the GLRT. Moreover, to avoid the well-known sensitivity of cyclostationary-based detection schemes to frequency-selective fading channels, a parametric channel model based on a lower bound on the coherence bandwidth is adopted and incorporated into the GLRT. By exploiting the rank-1 structure of small spectral covariance matrices, the obtained detector outperforms the classical spectral correlation magnitude detector.

The basic concepts of exterior calculus for space–time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space–time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms.

This paper shows that, for discrete memoryless channels, the error exponent of a randomly generated code with independent codewords converges in probability to its expectation—the typical error exponent. For high rates, the result follows from the fact that the random-coding error exponent and the sphere-packing error exponent coincide. For low rates, instead, the convergence is based on the fact that the union bound accurately characterizes the error probability. The paper also zooms into the behavior at asymptotically low rates, and shows that the normalized error exponent converges in distribution to the standard Gaussian or a Gaussian-like distribution. We also state several results on the convergence of the error probability and error exponent for generic ensembles and channels.