Análisis de componentes principales

Blind detection techniques based on PCA can be used to detect communication signal activity with improved performance compared with other techniques, such as cyclic feature analysis and ED [68, 69].

PCA is a technique where the essential or principal components of a correlated data set are isolated to form a reduced and uncorrelated data set, with the aim of retaining most of the information [70]. PCA has found application in several fields including pattern recognition, image compression [71], and more recently in spectrum sensing used especially in cognitive radio [72].

2.3.5.1 Isolating the principal components

The first step in finding the principal components of a real data set, consisting of N variables (with M observations each) is to remove the mean from each variable [70]. The M observations of each zero-mean variable are organised as column vectors to form the M _×N data matrix X. The N _×N sample covariance matrix (SCM) ofX is then [73]

C= 1 MX

T_{X (2.56)}

withXT the transpose of X. TheN eigenvalues and associated eigenvectors ofCare then calculated and sorted in decreasing order such thatλ1 > λ2 > . . . > λN. The eigenvectors

associated with the largest eigenvalues of C are the principal components of X. The number of eigenvectors used depends on the application and level of reduction required.

2.3.5.2 Spectrum sensing application

The eigenvalues of Cin (2.56) can be used to perform detection ifX contains the samples of the received signal. A number of cognitive radio spectrum sensing algorithms, that use these eigenvalues in their detection test statistics are listed below [68, 72].

• The largest-eigenvalue (LE) method uses the largest eigenvalue directly with [74] TLE=λ1(C). (2.57)

• The maximum-minimum-eigenvalue (MME) method uses the ratio between the largest and smallest eigenvalue with [69]

TMME =

λ1(C) λN(C)

• The energy-with-minimum-eigenvalue (EME) method uses the ratio between the received signal power Ps and smallest eigenvalue with [69]

TEME = Ps

λN(C)

. (2.59)

• The scaled-largest-eigenvalue (SLE) method normalises the largest eigenvalue with the mean of all the eigenvalues with [68]

TSLE = λ1(C) 1 N N X n=1 λn(C) = N λ1(C) tr(C) (2.60)

since the trace tr(_·) of a matrix (sum of its diagonal elements) equals the sum of its eigenvalues [75].

In addition to outperforming classical detection algorithms, the PCA algorithms asso- ciated with (2.57) to (2.60) do not need prior information of the signal to be detected. If accurate noise estimates are available, (2.57) will outperform ED and the other PCA algorithms. If accurate noise estimates are however unavailable, (2.58)-(2.60) can be used instead, as they are insensitive to noise estimation error [68].

2.3.5.3 DSSS detection

Similar to the detection of primary users in cognitive radio applications [74], detection of DSSS signals can be performed using the largest eigenvalue of the SCM of the data matrix

X containing the intercepted signal [16]. The technique is semi-blind since it depends on knowledge of the spreading code length. Knowledge of the noise statistics is also required to determine the threshold [76]. More details of this technique are given in Chapter 3.

2.3.5.4 Critical evaluation of principal component analysis

The PCA techniques considered here are popular in especially cognitive radio spectrum sensing applications [68]. Although these techniques are processor intensive (they require the calculation of eigenvalues), they show promising performance for DSSS detection purposes. A number of publications with mathematical proof and simulated performance results indicate the superiority of PCA techniques to perform both DSSS detection and sequence estimation (compared with ED and autocorrelation techniques) [14, 16, 68, 77].

2.3.6

Chaos theory

Chaos theory involves nonlinear dynamical systems that exhibit apparent disordered be- haviour [78, 79]. An important characteristic of chaotic systems is their sensitive depen- dence on initial conditions or system parameters [80]. This sensitivity is also referred to as the “butterfly effect” which expresses the idea that a seemingly insignificant event (the flap of a butterfly’s wings) may possibly have dramatic consequences (setting off a distant tornado) [81].

Chaotic systems (e.g. Duffing oscillators [79, 82]) can potentially be made very sensitive to the presence of sinusoidal signals of a given frequency [83, 84]. The presence of a very weak signal (the butterfly’s flap) will then have a dramatic effect (the tornado) on the behaviour of the chaotic oscillator, such that signal detection at low SNR is made possible.

2.3.6.1 DSSS detection

The Duffing oscillator may be used to detect the presence of a DSSS transmission, after performing a nonlinear operation (such as squaring) on the intercepted signal [85]. The nonlinear operation causes distinct spectral lines to reappear, that are not present in the intercepted digital transmission. These spectral lines indicate the presence of sinusoidal signals (which will be weak in the case of DSSS) which can be used to perform detection.

2.3.6.2 Critical evaluation of chaos theory

It was claimed that chaotic techniques can perform detection at much lower SNR levels compared with classical approaches [83], although no simulation (or measured) results to support this claim have yet been published. Also, several important details (e.g. the number of samples used and the false alarm rate) were not considered in [83].

Furthermore, two important issues still need to be resolved to realise detection using chaotic systems. Firstly, the sensitive-dependence principle not only implies the detector will be sensitive to the signal of interest, but also to many other factors (including noise). Methods to limit the false alarm rate should therefore be developed. Secondly, solving nonlinear dynamical systems involves highly complex and computationally expensive nu- merical methods, which may limit the practical application of these methods.