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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PFA ε PD 1- δ N=5 × 10 4 N = 10 5 N=2 × 10 5 Figure 3: ROC of an energy detector for several values of N , SNR=-21 dB. Energy detection approach relies only on the energy present in the channel. Since the energy of a signal is defined as integraltext | f ( t ) | 2 dt , no phase information is required. The underlying assumption is that with the presence of a signal in the channel, there would be significantly more energy than if there was no signal present. Within this premise lie two major disadvantages of energy detectors. First is that by definition, no signal will be detected that is below the noise floor. Second, the detector is susceptible to two types of errors, as shown in Figure 4. 1.3.2 Cyclostationary Feature Detector Cyclostationary feature detection relies upon periodic redundancy introduced into a signal by sampling and modulation. It uses the non-random periodic statistics of these signals to detect and possibly even classify a signal of interest. Cyclic detection is a robust spectrum sensing technique since it relies on what are called cyclostationary processes. 6
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Figure 4: Possible Type I and Type II errors of 4-FSK detection using an energy detector. A random process is cyclostationary if its mean and autocorrelation vary periodically in time. Modulated information is a cyclostationary process, while noise is not. As a result, cyclic detectors can successfully operate in extremely low SNR environments. A cyclic detector operates by calculating the cyclic autocorrelation function given by: R α x ( τ ) = lim T →∞ 1 T integraldisplay T x parenleftBig t + τ 2 parenrightBig x parenleftBig t - τ 2 parenrightBig e j 2 παt dt. (8) From this equation, it extracts the cyclic features of the signal. Different modulation schemes have different features. In fact, the difference between some modulation schemes is so distinct that these profiles can be used to accurately classify the PHY parameters of the signal. An example of these cyclic features is given in Figure 5. Cyclostationary feature detector is easily implemented via FFTs. Knowledge of the noise variance is not required to set the detection threshold. Hence, the detector does not suffer from the “snr wall” problem of the energy detector. However, the performance of the detector degrades in the presence of timing and frequency jitters (which smear out the spectral lines), and RF non-linearities (which induce spurious peaks). Representative papers that consider the approach are [4, 9, 10]. 7
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Figure 5: An example of 4-FSK cyclic features. 1.4 Problems 1. Calculate and plot the PSD of a 100MHz sinusoidal tone. How would you expect this to look on a spectrum analyzer? 2. Consider the problem of detecting between two real Gaussian random variables with means μ i and variance σ 2 i , i = 0 , 1. Show that the P F A is given by P( Z > τ |H 0) = Q ( τ μ 0 σ 0 ), where Q ( τ ) := 1 2 π integraltext τ e t 2 / 2 dt is the standard Gaussian tail function. Hence the ROC can be expressed as P D = 1 - δ = Q parenleftbigg σ 0 Q
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