Blind Detection Strategies

1.4.1.3 Other Feature Sensing Methods

Other feature spectrum sensing methods include matched filtering and multitaper spectral es- timation. Matched filtering is known as the optimum method for detection of PUs when the trans- mitted signal is known [23]. The main advantage of matched filtering is the short time to achieve a certain probability of false alarm or probability of a miss detection [24] as compared to other me- thods that are discussed in this section. However, matched filtering requires the CR to demodulate received signals. Hence, it requires knowledge of the PUs signaling features such as bandwidth, operating frequency, modulation type and order, pulse shaping, frame format, etc. Multitaper spec- tral estimation is proposed in [25]. The proposed algorithm is shown to be an approximation to maximum likelihood power spectral density (PSD) estimation. For wideband signals it is nearly optimal. Although the complexity of this method is less than the maximum likelihood estimator, it is still computationally demanding.

1.4.2.2 Model Selection Based Detection

Sub Space Analysis Based DetectionOne of the main contributions in this work is the inves- tigation of the sub space analysis in spectrum sensing. We propose in this context the dimension estimation detector (DED) which will be presented and analyzed in Chapter 3. This detector ex- ploits the sub space analysis of the PU received signal using AIC and MDL criteria as model selection tools [27] [11]. The same idea was applied in [28] and [29], published after our work, to develop two spectrum sensing algorithms exploiting the maximum or/and the minimum eigenva- lue as detection rule. However, in [28] and [29], the model selection has not been considered. This work will be presented in Subsection 1.4.2.3.

Distribution Analysis Based DetectionThe second contribution in this part is the distribu- tion analysis detector (DAD). To develop the DAD detector, we will compute the Kullback-Leibler distance between signal and noise distributions using AIC criteria and Akaike weight as model se- lection tools. Chapter 2 will describe this detector.

Kullback-Leibler Based Detection¹ The Kullback-Leibler detector (KLD) was developed for comparison with the DAD detector. Note that, this work in under progress and the simulation results which will be presented later are a preliminary step for this idea. We will give in this subsection the basic idea of this detector and the work done until now.

The Kullback-Leibler (KL) divergence, or relative entropy, is a measure of the distance bet- ween two probability distributions. The KL divergence between the two continuous probability density functionsf(x)andg(x)is defined as

D(f||g)=E

·

logf(x) g(x)

¸

(1.17)

where the expectation is taken with respect tof.D(f||g)is only finite if the support set off is contained in the support set ofg. Another important property of the KL divergence is that it is non negative and in general non-symmetric.

The literature surveys in the two papers [30] and [31] were good references for more elabo- rate KL divergence estimators. [30] suggests estimating the characteristic function of a normalized version of the input signal, composing a toeplitz matrix of the characteristic function, computing its eigenvalues and using these eigenvalues to estimate the KL divergence. The estimation pro- cedure is founded in the relationship between the sum of the eigenvalues of an autocorrelation matrix and the integral of the spectrum given by Szego’s theorem. [31] presents a completely different approach. The algorithm given suggests estimating the KL divergence between two dis- tributions through estimating their cumulative density functions. The analysis and ideas presented in the paper are thorough and consistent, and the author implies that the estimation variance of the algorithm only scales with the number of input samples.

The proposed algorithm depending on estimating the KL divergence is given on closed form as [31]

Υ_KLD(x) =D(f||g)=−κ−1 p

Xp i=1

ln (p∆G(x_i)) (1.18)

1. This work is a collaboration between our team in EURECOM and the Norwegian university of science and tech- nology (NTNU) team. Acknowledgements to Professor Tor Audun Ramstad at NTNU and Jorgen Berle Christiansen mastere student at NTNU for the collaboration we had.

whereκ= 0.577215is the Euler-Mascheroni constant,pis the number of input samples,∆G(xi) = G(x_i)−G(x_i−1)andGdenotes the CDF such thatg(x) =G⁰(x). The probability of false alarm for a given detection threshold is given as

P_{F A,KLD}=Q

µr p

π²/6−1γ_KLD

¶

(1.19) whereQ(.)denotes the cumulative distribution function [20] of aχ² distributed random variable with2pdegrees of freedom.

1.4.2.3 Maximum-Minimum Eigenvalue Based Detection

In [28] and [29], two sensing algorithms are suggested. One is based on the ratio of the maxi- mum eigenvalue to the minimum eigenvalue, the other is based on the ratio of the average eigenva- lue to the minimum eigenvalue. It is assumed that the signal to be detected is highly correlated. Let Rbe the covariance matrix of the received signal. Then, underH₀, all eigenvalues ofRare equal.

However, underH₁ some eigenvalues ofRwill be larger than others. A detector exploiting this property is called maximum-minimum eigenvalue detector (MMED) and was proposed in [28]. It will be described briefly in the context of this section. ConsideringN observationsx_nreceived in a sequence, the sample covariance matrix can be defined as

R=ˆ 1 N

XN n=1

x_nx^T_n (1.20)

Let λ_n|_n=1,...,N be the eigenvalues of R. There are two eigenvalue-based detectors proposed in [28]. The first detector uses the ratio of the largest eigenvalue to the smallest eigenvalue and compares it to a threshold. So the test statistic of the first proposal of [28] is based on a condition number

Υ_{M M ED}(x)=maxλ_n

minλ_n (1.21)

The probability of false alarm of the MMED is given by P_{F A,M M ED}=1−F₁





γ_{M M ED}

³√ N −√

p

´₂

−¡√

N−1−√ p¢₂

¡√N−1−√

p¢ ³ ₁

√N−1 +^√1_p

´¹

3



 (1.22)

whereF₁is the cumulative distribution function (CDF) of the Tracy-Widom distribution of order 1,N is the number of PU observations and pis the length of each observation. The distribution function is defined as

F₁=exp µ

−1 2

Z _∞

t

¡q(u) + (u−t)q²(u)¢ du

¶

(1.23) whereq(u)is the solution of the nonlinear Painleve II differential equation

q”(u)=uq(u) + 2q³(u) (1.24)

With the above expressions for the probability of false alarm, the expected detection performance can be evaluated. In [32] the authors propose a research perspective of the MMED considering a finite number of cooperative receivers and a finite number of samples. They calculate in this paper the exact decision threshold as a function of the desired probability of false alarm for the MMED detector.

Bandwidth 8MHz

Mode 2K

Guard interval 1/4

Channel models Rayleigh/Rician (K=1) Maximum Doppler shift 100Hz

Frequency-flat Single path

Sensing time 1.25ms

Location variability 10dB

TABLE1.1 – The transmitted DVB-T primary user signal parameters

1.4.2.4 Other Blind Sensing Methods

Another blind technique called multi resolution sensing was proposed in [33]. This technique produces a multi resolution PSD estimate using a tunable wavelet filter that can change its cen- ter frequency and its bandwidth [34]. In [35], wavelets are used for detecting PU signals in blind manner. The wavelet based approach is efficiently used for wideband spectrum sensing where a wi- deband signal spectrum is decomposed into elementary building blocks of sub-bands that are well characterized by local irregularities in frequency [35]. The wavelet transform is then employed in order to detect and to estimate the local spectral irregular structure that carries important informa- tion about the frequency location and power spectral densities of the sub-bands. Others methods that exploit a recorded form of the covariance matrix are also derived in the literature [36].