Spectrum Sensing for Cognitive Radios with Transmission Statistics: Considering Linear Frequency Sweeping

In the recent years, Ultra Wideband (UWB) technology has emerged as one of the key candidates for CR based secondary user communications [7, 8]. When UWB technology is used as CRs for secondary communications, it is required to scan the entire spectrum from  GHz–  GHz (in many cases a significant portion of it) to detect the presence of any PUs in the network. In such situations, scanning a wide range of frequencies (  GHz) can be a time consuming process and hence a narrow band PU, which has a bandwidth much smaller than the UWB node, can be gone undetected when the UWB-CR node is scanning a large portion of the spectrum. It is obvious to state that such miss detection depends on the PU's transmission statistics, or in other words the Spectral occupancy Statistics (SoS) as well as the spectrum sniffing hardware unit of the UWB-CR node and the corresponding time required to sense the spectrum. Such a problem is considered to be a crucial one to be solved by many researchers and engineers working in this filed, such as in the European Union funded 20 M-Euros project on EUWB [8]. The problem defined here is not specific to UWB based CR systems only, but in general it applies to any CR systems with a bandwidth much larger than the bandwidth of the potential victim services (i.e., the PUs) in the network.

Spectrum sensing techniques have been heavily discussed and treated in the literature and one could refer to [3, 4, 9‐28] for further reading. The performance of different spectrum sensing techniques are measured in terms of the probability of false alarm and the probability of miss detection for noisy sensing. In our work however, in addition to noisy sensing, we also include the SoS (i.e., the temporal characteristics) of the PU and analyze the detection performance of the CR nodes for wideband sensing. In order to perform theoretical analysis some mathematical models should be followed for the temporal characteristics of the PU. The Poisson model for the arrival process is the most common model used in the research literature for theoretical derivations and performance analysis [29‐40], and also recommended in the International Telecommunication Union's (ITU) handbook on Teletraffic Engineering [32]. Moreover, Poisson models are also verified for several cases [34‐40], especially for the Internet traffic when the load is higher [34]. Therefore, we adopt such a model in our work. It should be noted that there also exist empirical models for various traffic sources [29, 30, 30, 31, 31‐44] and for various radio access technologies mostly derived from the Poisson arrival process.

Furthermore, the authors in [21] have used the Poisson traffic model to design an admission controller for the CR node to improve the Quality of Service (QoS) for secondary transmissions. In [22], for Poisson based PU traffic, the authors have studied by means of simulations the trade-off between sensing time and the achievable throughput by considering the probability of collision. The Poisson traffic model is also used in [23, 24] to derive a-priori probability based detection schemes for spectrum sensing and have presented some numerical results on the improvement over the traditional energy-based detection [45] and the classical Maximum Likelihood- (ML-) based detection techniques [46], respectively. A channel selection scheme for CR nodes for a multichannel multiuser environment is also proposed-based on Poisson traffic model in [25], and recently in [26, 27] we have studied the performance of shared spectrums sensing for UWB-based CR assuming the Poisson model for the PU. Furthermore, moving away from the Poisson based theoretical model, the authors in [47] have analyzed the performances of dynamic spectrum access techniques based on experimentally measured spectrum occupancy statistics.

In this paper, we study the performance of detecting the PU based on its time domain SoS, by classifying them as light, average or heavy users of the spectrum, together with noisy sensing at the CR nodes. We mainly consider the case where the PU bandwidth is much smaller than the CR's spectrum scanning frequency range. Though the references in [22‐26] have considered the Poisson model for the PU channel SoS they have presented mainly some simulation results to analyze the performances of the the CR node for spectrum sensing and channel occupancy efficiency. In our work presented here, we perform some detailed theoretical analysis of the performance of PU detection, based on an envelope-based energy detector, by initially considering the transmission statistics only case and then together with the sensing noise. The theoretical anaylses are also verified by simulations. We also study the minimum required sensing time for the CR node to reliably detect the narrow band PU given its temporal characteristics.

The rest of the paper is organized as follows. In Section 2, we provide the model for the CR network, and in Section 3, we derive the Poisson arrival model for the PU's transmission statistics from the fundamentals. In Section 4, we provide the signal envelope based spectrum sensing technique followed by some theoretical analysis on the detection performances for the noiseless case with a constant channel occupancy (hold) time in Section 5. In Section 6, we present a PU detection risk analysis based on the transmission statistics (Poisson process) of the PU. In Section 7, we present the detection performance considering noise, and in Section 8, we extend the analysis for a random channel occupancy (hold) time. In Section 9, we present the sensing time requirements for the CR based on the SoS of the PU, and finally we make some concluding remarks in Section 10.

In the noiseless case, the detection performance of the CR node is characterized by the SoS of the PU, the bandwidth of the PU, and the total time for the CR node to scan the entire bandwidth . As mentioned previously, we assume the CR nodes linearly scan the frequency in time over the desired (wideband) spectrum.

We present the simulation results in this section using Matlab to verify the theoretical analysis performed in Section 5.1 Simulations were performed using Monte-Carlo techniques to generate the random (Poisson based) PU's transmissions. In our simulations, the Poissonian arrivals are generated using the Binomial counting process by generating a binary random event with a probability of within a small time duration of , as described in [46, Sectio 21.3], which converges to a Poissonian process as with . Simulations were performed in Matlab to generate the random arrival process with a deterministic hold time. Figure 3 presents the occupancy probability for various time durations . From the figure, we observe that increases with the arrival rate , as expected, and also marginally improves with the time duration . The figure shows how the the spectral occupancy probability of the PU (defined over a period of ) increases when the scanning duration increases for a given arrival rate . This observation is quite important, especially when we study the minimum time requirement (minimum ) for sensing the PU, which we present in the later sections. Figure 4 depicts the miss detection probability for various holding times and arrival rate . From the figures, we see that the detection performance at the CR node improves when both and increase, we further observe that the improvement in the detection performance is greater when increases than when increases, relatively. From the figure we also see a very close match between the theoretical expressions derived in the previous section and the simulated results further verifying our analysis.

First, we define the four probabilities , , , and for the noiseless case as, , , and . From the analysis performed in Section 5, we identify that , , and . Then, the overall probability of detection and the probability of false alarm for the noisy case are given by,

From (16), by using (7), we come up with two closed form expressions for and as

where, is the Marcum -Function defined by,

From the expressions we, observe that the probability of false alarm does not depend on the transmission statistics of the PU but only depends on the sensing noise. In the following section, we perform some simulations to study the detection performances and also verify the theoretical analysis performed in this section.

The Complementary Receiver Operating Characteristic Curves (C-ROCs) for the envelope-based detector is presented here considering the PU transmission statistics. The C-ROC curve is the plot between and by varying the detection threshold . Together with the theoretical C-ROC curves, derived from the previous section, we also present some simulation results to verify our theoretical analysis. Monte-Carlo simulations were performed to generate the PU's (Poisson) transmission process and the noisy sensing process with Gaussian noise. Figure 7 shows the C-ROC curves for various SNR values for and . As expected, we see that the C-ROC curves improve when the SNR is increased, the figure also shows the probability of miss detection ( ) for the noiseless case as well. Figure 8 shows the C-ROC curves for different values of arrival rate with  dB and . From the figure, we observe how the detection performance improves when the arrival rate (spectral occupancy rate) of the PU is increased. Moreover, we see that when is increased further the detection performance predominantly depends on the sensing noise. Figure 9 shows the C-ROC curves for various values of spectral hold time for and  dB. Improvements in the detection performances are observed when is increased but not as much as when is increased as in Figure 8. Furthermore, we clearly see that the simulation results very closely match with the theoretical results on all the figures, verifying our analysis in the previous section.

Using the exponential and Pareto hold time models with Poisson arrival for the PU transmission we rederive the expression for the probability of detection here for random . The probability of false alarm stays unchanged, as in (17), because is independent of the transmission statistics of the PU (i.e., is independent of ). From (17), we derive the new probability of detection for the random hold time by averaging over all possible values of . Therefore, the probability of detection is given by

(Note that for the Pareto model the integration range in the above is from to .) By solving the integral in (19) for both exponential and Pareto models, we come up with a closed form expression for as

where is the probability of miss detection for the random in the absence of the sensing noise (i.e., considering only the transmission statistics of the PU), given by

Furthermore,the probability of detection for random in the absence of noise is given by . The curves and defining the risk regions (Section 6) can also be redefined for the random holding time case. From Section 8.1 we have two new curves for the exponential hold time model given by

The new curves in (22) differ mainly for low values of but has the same limit values of and defining the risk regions, as in (15), when .

We compare the differences in the detection performance between the constant and the random cases. The random case obviously relates to reality more than the former. Figure 10 depicts the theoretical curves for the probability of miss detection with respect to the arrival rate for the noiseless case. The figure shows both the constant and the random cases for the exponential and the Pareto models. As we observe from the figure, the differences between the two arise for higher values of mean hold time and . This is due to the fact that for larger values of and , the probability of having smaller hold times is greater in the random case, and hence we observe poorer performances compared to the constant case. Figure 11 on the other hand shows the complementary ROC curves for the random and constant cases for noisy sensing. In Figure 10, we observed that the differences between the random and the constant detection performances arise for higher values of (both) mean hold time and , and in Figure 11 we observe that the difference between the two arise only when the SNR is higher, provided that the mean hold time and are high. For lower values of we see that the difference is negligible. Furthermore, we observe that Pareto model shows better detection performance compared to the exponential model based on the value of for the same mean hold time. Note that when increases the, Pareto model approaches the constant hold time case because the density function gets more peakier.