MEMORIA CONSTRUCTIVA
3. SISTEMA ENVOLVENTE
3.1. SUBSISTEMA FACHADAS
In this Chapter, the existence of the noise enhancement phenomenon has been shown by addressing the spectral analysis problem under both empirical and theoretical frame- works.
On the one hand, the problem of spectral analysis employing the periodogram and Capon estimates from nonuniformly sampled data has been addressed. The nonuniform sampling has been modeled as a linear projection onto a lower dimensional space. The correlation-matching metric has been proposed in this setting as a rank-1 fitting. This for-
mulation has allowed the derivation of the nonuniform periodogram and Capon estimates as particular cases. It is concluded that the nonuniform Capon, which performs denoising and adaptive pattern, provides good resolution for spectral analysis.
On the other hand, an equivalence between the noise enhancement produced by Ber- noulli nonuniform sampling and SNR has been introduced. The equivalence is firstly es- tablished in terms of signal correlation and signal spectral density, and is further extended to matrix formulation and spectrum sensing detection. Numerical results on spectral anal- ysis and spectrum sensing show that the degradation incurred by nonuniform sampling can be easily translated to a penalty in SNR.
Chapter
4
Multi-Frequency Primary Signal Detection
4.1
Introduction
Statistical signal detection is one of the most fundamental signal processing problems for decision making in radar, communications, biomedicine, and so on [Kay98a]. In this Chap- ter, the problem of detecting the presence of a primary signal over a set of frequency re- sources is addressed. As illustrated in Figure 4.1, the objective is the design of test statistics Tq(X) that, based on the nonuniformly sampled observations X, provide optimal detec-
tion probability at each of the q-th frequency resource.
4.1.1 Generalized Likelihood Ratio Test
This thesis addresses the optimal spectrum sensing detection based on the complete or par- tial side information on the signal and noise statistics. The primary function of interweave cognitive radios [GJMS09] is to reliably identify the available spectrum resources tempo- rally unused by primary users. This awareness can be obtained through a database, using beacons, or by local spectrum sensing [ALLP12]. This thesis focuses on spectrum sensing performed at the cognitive radio receivers as it constitutes a broader solution and has less infrastructure requirements. The energy detector, cyclostationarity feature detection, and match-filtering are the most commonly employed techniques for spectrum sensing. How-
PQ
q=1Sq+W −→ Nonuniform Sampling −→ X −→ Signal Detection −→ Tq(X)
ever, the performance of such detectors is severely degraded when the side information on the signal and noise features is incomplete, e.g., the cyclic frequencies. Spectrum sensing detectors based on the generalized likelihood ratio test (GLRT) have received recent atten- tion as the GLRT statistic is optimal in the Neyman-Pearson sense [Kay12] and natively incorporates joint parameter maximum likelihood (ML) estimation for inaccurate model parameters [MJTW12]. The effect of side information on the signal and noise statistics has been reported in [FSW10] in realistic scenarios.
Wideband spectrum sensing has gained recent attention [SNWC13]. It is recognized that ML estimation in wideband cognitive radios is especially challenging because wide- band regimes are characterized by close to zero spectral efficiency and low signal-to-noise ratio (low SNR) [Ver02]. Furthermore, the method of ML, despite its theoretical appeal, is often difficult to implement, and analytical solutions are not available in many circum- stances [Por08]. However, the tractability of the ML formulation is shown in asymptoti- cally low SNR regimes, and it is identified that the second-order statistics of the observa- tions are sufficient statistics for the spectrum detection problem when the noise variance is high. In this thesis, the optimal ML estimates for GLRT spectrum sensing detection are derived in the single-frequency scenario, and show through low SNR approximations that any GLRT spectrum sensing detector exclusively depends on a kernel operator and the sample covariance matrix of the observations, asymptotically as the SNR tends to zero. Further, it is extended to cognitive radio networks operating over primary systems that employ multi-frequency communications, such as the terrestrial digital video broadcast- ing (DVB-T). Simulation results assess the performance comparison of the derived algo- rithms.
A numerous amount of spectrum sensing detectors based on the GLRT have been re- ported in the last years. It is a well-known signal processing result that in the low SNR regime, the second-order statistics of the signal and noise involved in the detection are sufficient statistics for detection [Kay98a]. More precisely, the cross-correlation between signal and noise can be exploited in the frequency, spatial, or temporal domains. Firstly, frequency correlation detectors are intended to uncover the spectral coherence of cyclo- stationary processes (see [GNP06] and the references thereby). However, these detectors are very sensitive to the signal and noise features, and require in general high computa- tional complexity. A spectral feature based detector has been reported in [QZSS11]. Sec- ondly, a bunch of signal detectors that exploit spatial correlation have been proposed in the recent literature [ZLLZ10, WFHL10, TNKG10, RVS10, VVLVS11, RVVLV+11, SWZZ12, SAVVLV12, STM13, RVSS13, ZQ13, ZKLZ13]. Addressing the GLRT problem in the mul- tiple antenna framework has allowed the formulation of well-known Gaussian detectors such as the arithmetic-to-geometric mean (AGM) detector [ZLLZ10], rank-1 detectors [RVS10, SAVVLV12, ZQ13], the rank-P detector [RVVLV+11], and the locally most powerful in- variant test (LMPIT)[RVSS13]. In the case of fading, the algorithms reported in [WFHL10,
TNKG10, SWZZ12] address blind detection with unknown channel parameters, whereas the works by [VVLVS11, STM13, ZKLZ13] investigate the use of prior information in the detection process. The common frawmework of [ZLLZ10, WFHL10, TNKG10, RVS10, VVLVS11, RVVLV+11, SWZZ12, SAVVLV12, STM13, RVSS13, ZQ13, ZKLZ13] is that the detectors are based on the eigenvalue decomposition (EVD) of the spatial correlation ma- trix. Therefore, these algorithms are only valid when the secondary users are equipped with multiple antennas. Thirdly, the exploitation of temporal correlation has gained lit- tle attention, even though temporal correlation matrices exhibit good properties of sta- bility and computational complexity. Related work involving GLRT detection include [NPI10, YLPC11, BFPRI12]. In [NPI10], the sensitivity of oversampled temporal corre- lation based detection to frequency offsets is investigated. In order to further improve the efficiency, [YLPC11] reports a detector based on the Cholesky factorization of the sam- ple correlation matrix, whereas the work by [BFPRI12] is concerned in performing signal detection while communicating at the same time. These works does not consider side information on the primary signal correlation.
In this thesis, the temporal correlation of the primary users is exploited given the Toeplitz structure of the correlation matrices when one single antenna is employed at the cognitive receiver. An N × N Hermitian Toeplitz matrix has only N degrees of freedom. Therefore, the algorithms derived in this Chapter will have good properties of stability and computational complexity. It is noted that the detection of a stationary process with a single antenna constitutes a well-defined fundamental problem by itself, which takes advantage of the second-order statistics distinctness between hypotheses. Theoretical in- terpretations and numerical results show the tradeoff between detection performance and the degree of side information on the most informative statistics for detection, i.e., the modulation format and spectrum distribution of the primary users.
4.1.2 Nonuniform Correlation Matching
Due to the low occupancy of many communication systems, it is recognized that primary signals are sparse in the spectrum domain, which allows the application of the foreseen theory of compressed sensing [CW08] to further reduce the sampling rate required for spectrum sensing detection. A comparison addressed by [SLH09] shows that nonuniform sampling does not suffer from some drawbacks present in traditional uniform Nyquist rate sampling.
As stated above, ML estimation is an important method used in a wide range of sta- tistical signal processing problems [Kay98b], which is often difficult to implement, even more in the nonuniform sampling scenario. As an alternative, this thesis proposes the ap- plication of the correlation-matching technique to spectrum sensing of wideband signals. Correlation-matching is a least-squares fitting of second-order statistics, and behaves as an approximation to ML at the low SNR regime with asymptotic large data records [Por08].
More specifically, the nonuniform sampling version of the correlation-matching approach for spectrum reconstruction of sparse wideband signals is discussed. The potentials of correlation-matching as a matrix-level fitting has been recently addressed by [PNLRS09], and a reformulation is carried out under the nonuniform sampling scenario for asymptot- ically sparse wideband regimes. A general closed-form estimate of the signal power level based on nonuniformly sampled observations. This unified formulation allows physical interpretation of the problem, and straightforward application to the problem of GLRT spectrum sensing in multi-frequency cognitive radios.
4.1.3 Chapter Organization
This Chapter is organized as follows. The single and multi-frequency GLRT detection based on ML estimation with side information is addressed in Section 4.2. The extension to nonuniform sampling with nonuniform correlation-matching estimation is provided in Section 4.3. Numerical performance evaluation of the detection performance of the GLRT detectors derived in this Chapter is reported in Section 4.4. Section 4.5 concludes the Chapter.