Tutela administrativa y judicial

In this Section, numerical results are provided to show the existence of the noise enhance- ment phenomenon in nonuniform sampling, to illustrate the performance of the proposed nonuniform periodogram and Capon spectrum estimates, as well as to assess the SNR equivalences established in the Bernoulli nonuniform sampling framework. In the sequel, s(t) is modeled as the equivalent baseband primary signal based on the terrestrial digital video broadcasting (DVB-T) standard [ETS04], and w(t) is modeled as a zero-mean Gaus- sian noise.

3.4.1 Nonuniform Periodogram and Nonuniform Capon

The performance of the spectral analysis methods presented in Section 3.2, i.e., (3.18), (3.20), (3.23) and (3.24), is presented by means of simulation results. The size of the uni- formly sampled observations is N = 64. In order to strictly focus on the performance behavior due to sampling rate reduction and remove the effect of insufficient data records,

0 0.2 0.4 0.6 0.8 1 10−6 10−4 10−2 100 102 κ

Normalized estimation error

Periodogram Periodogram Noisy Capon

Capon Noisy

Figure 3.4: Normalized estimation error of the periodogram and Capon estimates versus the

sampling density of a primary signal with occupancy of 1/8 at average SNR of 15 dB.

the size of the observations X is fixed to 2N by setting M (κ) = 2N κ−1. The normalized estimation error is defined as

2[φs(ω), ˆφ(ω)]=. Z B E " |φs(ω)− ˆφ(ω)|2 |φs(ω)|2 # dω, (3.52)

where ˆφ(ω) is the spectrum estimate, and φs(ω) is the spectrum of the uniformly sampled

s(t) at Nyquist rate.

An example of spectral analysis with occupancy κ0 = 1/8 (e.g., two active channels out

of eight in this example), an SNR of 15 dB, and sampling density κ = 1/4 is depicted in Figure 3.3. The resulting spectrum estimates are averaged according to 1,000 Monte Carlo runs. Firstly, it is observed that the noisy periodogram and noisy Capon, even in the high SNR regime, introduce noise enhancement because of the projection implicit in nonuni- form sampling. The denoising process in (3.18) reduces the noise level by approximately 60 dB, whereas the Capon estimate is able to completely subtract the noise floor.

The behavior of the normalized estimation error (3.52) versus the sampling density is illustrated in Figure 3.4, when the occupancy is κ0 = 1/8. As it can be appreciated, both

Capon and noisy Capon estimates provide better performance when compared to the peri- odogram counterparts for a wide range of κ. The gain introduced by the denoising process is outlined when comparing the nonuniform spectral estimates to their noisy counterparts. In the case of the Capon estimate, a change of performance behavior is observed around a sampling density of κ = 1/4.

Figure 3.5 plots the normalized estimation error versus average SNR. In the low SNR regimes, the noisy periodogram and noisy Capon are not able to provide good spectral res-

−20 −15 −10 −5 0 5 10 15 20 10−5 10−4 10−3 10−2 10−1 100 SNR (dB)

Normalized estimation error

Periodogram Periodogram Noisy Capon

Capon Noisy

Figure 3.5: Normalized estimation error of the periodogram and Capon estimates versus average

SNR of a primary signal with occupancy of 1/8 and sampling density of 1/4.

olution because of the noise floor; whereas the nonuniform periodogram and Capon are capable of distinguishing the primary signal. In the high SNR regimes, the two versions of the periodogram saturate because the spectral pattern is not adaptive. Hence, it is iden- tified that the nonuniform Capon estimate (3.23) might provide outstanding performance for detection purposes for a wide range of SNRs.

Finally, Figure 3.6 illustrates the performance of the spectrum estimates when, for a fixed sampling density κ = 1/4, the occupancy of the primary signal κ0 increases from

κ0 = 0(no occupancy) to κ0 = 1(100% occupied). It is observed that the normalized es-

timation error increases with the occupancy, as the dimensionality of the lower space, i.e., K, is not sufficient for high resolution spectral analysis when the sparsity of the spectrum is low. However, it is concluded that the Capon estimate (3.23) behaves as a high resolution spectral method, e.g., 2 ≤ 0.01, as far as κ < κ0.

3.4.2 SNR Equivalence

The periodogram of two DVB-T signals immersed in Gaussian noise is depicted in Figure 3.7, for N = 512. The solid black curve depicts φx(ω) for an SNR of 10 dB. After Bernoulli

nonuniform sampling with a sampling density of κ = 1/4, the resulting spectrum φx(ω),

which is represented by the dashed red line, suffers from noise enhancement and signal attenuation, according to the expression obtained in (3.31). By employing the SNR equiva- lence established in (3.32), it is confirmed that φx(ω) with an SNR of 10 dB and φx(ω) with

an equivalent SNR of

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 10−5 10−4 10−3 10−2 10−1 100 κ

Normalized estimation error

Periodogram Periodogram Noisy Capon

Capon Noisy

Figure 3.6: Normalized estimation error of the periodogram and Capon estimates versus primary

signal occupancy level at average SNR of 15 dB and sampling density of 1/4.

0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 Frequency Spectrum (dB) Uniform sampling Nonuniform sampling Equivalent uniform sampling

6 dB

6 dB

Figure 3.7: Spectrum of uniform sampling, nonuniform sampling with sampling density of 1/4, and

10−3 10−2 10−1 100 10−4 10−3 10−2 10−1 100 False−alarm probability Missed−detection probability κ = 1 /4 κ = 1 /8 κ = 1 /2 Uniform sampling Nonuniform sampling Equivalent uniform sampling

Figure 3.8: ROC of uniform sampling, nonuniform sampling, and equivalent uniform sampling at

SNR of_{−12.5 dB.}

show the same spectral SNR. Therefore, the penalty of nonuniform sampling with κ = 1/4 equals to an SNR penalty of 11 dB.

Finally, the receiver operating characteristics (ROC) of the estimator-correlator (3.49) shown in Figure 3.8 is evaluated. The performance curves are obtained in several sam- pling conditions for an average SNR of−12.5 dB when detecting a DVB signal immersed in Gaussian noise with N = 32 samples and an occupancy of κ0 = 1/8. The degrada-

tion incurred by nonuniform sampling is observed (dashed lines). The dotted lines depict the performance of the equivalent uniform estimator-detector associated to the given sam- pling densities, i.e., T1(x)is evaluated under the SNR matrix conditions (3.51). From Figure

3.8 it follows that the equivalence between SNR and noise enhancement introduced in this thesis accurately models the effect of nonuniform sampling. As a result, the performance of any spectrum sensing detector based on nonuniformly sampled data can be forecasted beforehand by applying an SNR shift according to (3.51).