On the effect of false alarm rate on the performance of cognitive radio networks

We assume that the primary channel occupancies are time varying alternating between idle and busy periods, and thus SUs must perform spectrum sensing continuously to detect the presence of returning PUs. PU connections arrive at the network according to a Poisson process at a rate of λ ₁. The PU service rate which is assumed to be exponentially distributed is μ ₁. We also assume that PUs can obtain primary channel occupancy information, for example, by accessing a core network that makes signaling or querying of the PUs’ base station [32], and thus it is further assumed that PUs do not collide with each other [28].

We assume that both PUs and SUs have some interference tolerance T _TOL of how many seconds of interference they will tolerate before withdrawing from the system. If the PU interference tolerance time T _TOL is 0, no SU transmission is allowed [33]. We assume equal interference tolerance for both systems leading to both colliding users withdrawing from the system simultaneously. A similar assumption has been considered previously in [28].

We assume that SU connections arrive at the network according to a Poisson process with λ ₂. The SU service rate is assumed to be exponentially distributed with μ ₂. During the absence of PUs, SUs can opportunistically access the free channels if they are not occupied by other SUs. We also assume that SUs are capable of broadcasting control messages on a common control channel (CCC) [34] to show their existence to neighboring SUs in the proximity. Therefore, SUs do not attempt accessing channels occupied by other SUs. Upon detection of the presence of a returning PU, a SU leaves its current channel and starts the spectrum handover process in order to find a new free channel. If the channel search process ends without finding a free channel, the SU terminates its call and leaves the network.

As illustrated in Fig. 1 and similar to the distributed (coordination function) interframe space (DIFS) operation in IEEE 802.11, the SU has to keep sensing the PU channel from the beginning of its transmission, since the PU can arrive at any time instant of a slot. This process forms a continuous sequence of sensing slots with length equals T ₂. For example, if the PU appears in the middle of a time slot, then the first slot will not get full PU energy, leading to a smaller detection probability than the later full slots. Since the first T ₂ may be wasted, we assume that the first partial slot sensing never leads to detection, i.e., the detection probability is close to zero. Hence, we should detect the PU arrival during T _TOL−T ₂ seconds which corresponds to \(\widehat {T_{\text {TOL}}}=\left \lfloor \frac {T_{\text {TOL}}-T_{2}}{T_{2}} \right \rfloor \) slots. Although partial slot sensing can enable the SU to perform sensing immediately after the arrival of the PU and hence have a prompt reaction to protect PUs, for the sake of simplifying the analysis, we consider only the full slot sensing by assuming that the detection process will begin from the first full time slot following the SU arrival.

Without loss of generality, we consider that initial and ongoing spectrum sensing are done using an energy detector [9] with an integrate and dump operation mode as described in [10, 11]. The analysis techniques presented in this paper are generic and not limited to any particular detector, provided that the used detector can be mapped to false alarm probabilities, probability of detections, and false alarm rates. Let y _q (t) denote the SU received signal process. We express the incoming SU received signal process in the form

and the ongoing SU received signal process can be formulated as

In Eqs. (1) and (2), \(s_{\!_{\text {PU}}}(t)\) is the PU transmitted signal, \(s_{\!_{\text {SU}}}(t)\) represents the leakage from the SU transmitted signal, n(t) is the additive white Gaussian noise (AWGN), \(h_{\!_{\text {PU}}}\) is the PU channel gain while \(h_{\!_{\text {SU}}}\) represents the SU leakage signal gain, and t is the time. In the above equations, H ₀ is the null hypothesis meaning that PU is not present in the sensed band, and H ₁ represents the alternative hypothesis referring to the presence of the PU signal.

The received signal is filtered by a bandpass filter to remove the out-of-band and self-interference noise. The filtered signal is then squared by the squaring device and applied to the integrator. The integrator output Y _q (also denotes the decision variable of the energy detector) is sampled every T _q seconds. Then, the integrator is reset before integrating the next sample over the next T _q seconds. Finally, Y _q is compared with the decision threshold to decide about the presence of the PU. Let W _q denote the sensed bandwidth. Let γ _q denote the signal to noise ratio SNR and η _q denote the energy detection threshold. According to [9] where \(\chi _{2u_{q}}^{2}\) is a chi-square distribution with 2u _q degrees of freedom (i.e., the time-bandwidth product u _q =W _q T _q ) and \(\chi _{2u_{q}}^{2}(2\gamma _{q})\) is a non-central chi-square distribution with 2u _q degrees of freedom and a non-centrality parameter (2γ _q ). It has been shown in [40] that the probability of detection P _Dq and the false alarm probability P _FAq can be given as follows:

where Q _m(.,.) is the generalized mth Marcum Q-function [41].

Referring to Fig. 1, where W represents the PU channel bandwidth. To obtain the received signal energy, let P _PU denote the PU transmitted signal power. Let also N ₀ denote the one-sided power spectrum density (PSD). Let T ₁ denote the incoming SU initial sensing time. Assuming that the PU signal power is uniformly distributed over the PU channel, then the incoming SU signal energy within the initial sensed area can be obtained as

and the initial sensing SNR is then obtained with

Similarly, we can obtain the signal energy within the ongoing SU sensed area with a sensing time duration T ₂ as

Although we use guard band, bandpass filtering, and self-interference cancellation to eliminate the effect of SU leakage signal, we assume that the SU self-interference has not been fully removed. To handle the effect of any remaining self-interference, we follow the results presented in [42] to model the residual interference. Let assume that the SU operate with a single-antenna full-duplex transceiver. Let \(\alpha _{_{\text {SU}}}\) denote the SU’s residual interference distortion factor. By using Eq. (2), the effective ongoing sensing SNR can be expressed as [42]

In continuous spectrum sensing with full-duplex communication, the consideration of self-interference is particulary important since the self-interference can affect the sensing outcome and degrade SUs performance. Although the SU self-interference signal can have non-zero mean, it has been been assumed in the majority of related works to have a zero mean. For instance in [43], the authors mentioned that in practical full-duplex systems, the self-interference cannot be completely canceled, such that the signals received at each node is a combination of the signal transmitted by the other source, the residual self-interference (RSI), and the noise. They also assume that the RSI can be typically modeled as zero-mean additive white Gaussian noise (AWGN). The work reported in [42] assumed that the Gaussian distortion and noise follows central chi-square distribution in the absence of PU signals but potentially including RSI and noncentral chi-square distribution when PU signal is present.

Self-interference mitigation in full-duplex MIMO relays has been investigated in [23] where the authors focused on minimizing the residual loop interference so that it can be regarded as additional relay input noise. They assumed that all signal from the relay output to the relay input (including loop interference (LI) signal) and noise vectors have zero mean. Furthermore, the authors of [44‐46] assumed that the SU self-interfering signal before carrying out self-interference suppression (SIS) to be a zero-mean random signal with self-interference channel coefficient equal one. In [47], the residual self-transmitted signal is modelled with circular symmetric complex Gaussian variables. Following the common practice in existing models, the use of the assumption that SU’s leakage signal can be zero mean and follows central chi-square distributions is justified and can be hold in order to take into account the RSI signal and perform the analysis.

It should be noted that when we use a dedicated part of the bandwidth (subchannel C) for continuous sensing, the effect of the residual interference becomes much lower than when we use the full bandwidth for simultaneous sensing and transmission. Each incoming SU correctly detects channel occupancy with probability P _D1, and falsely classifies a free channel as occupied with P _FA1. Similarly, each SU with ongoing calls detects the arrival of a PU with probability P _D2 and falsely classifies a free channel as occupied with P _FA2. The corresponding misdetection probabilities for incoming and outgoing SUs are P _M1=1−P _D1 and P _M2=1−P _D2, respectively. The detection probability P _D2 refers to the probability of detecting incoming PU during the first \(\widehat {T_{\text {TOL}}}\) full slots of its arrival, instead of the per-slot detection probability. If the per-slot detection probability is denoted as z then \(P_{D2}=1-(1-z)^{\widehat {T_{\text {TOL}}}}\). This does not affect the FAR process since one per-slot false alarm event is enough to initiate the spectrum handoff and channel searching process. Modeling of partial slot sensing is left for future work.

We model the occurrence of the false alarm at each sensing decision with the Bernoulli process. The energy detector makes only one sensing decision in each slot which results into a binary variable (0 or 1). Since the sensing decisions with only white Gaussian noise present are independent, the resulting binary output of the sensing clearly follows the Bernoulli process (i.e., independent and identically distributed process generating 1 and 0 s), and the Bernoulli parameter corresponds to the probability of FAR occurrence (binary output 1) in each spectrum sensing decision.

At each spectrum sensing decision epoch T ₂, a false alarm occurs with probability P _FA2 and does not occur with probability 1−P _FA2, independently of the decision outcome of the last sensing period. The λ _FAR parameter is the product of the decision rate and the false alarm probability [10, 11, 48]. Therefore, λ _FAR is given by P _FA2/T ₂. Let us assume that the sensing interval T ₂ is short and therefore we assume that the decision rate given by 1/T ₂ is large, and that the false alarm P _FA2 is small as otherwise there would be too many false alarms for successful SU operation. Then, the arrival process of false alarms can be approximated by a Poisson process as a limit of a shrinking Bernoulli process [13] with parameter λ _FAR.

The goal of this section is to extend the results presented previously to describe a CRN with an arbitrary number of channels N. When N is large, constructing a state transition diagram and finding a solution to the corresponding balance equations is complicated and time consuming. Similar to [30], we use a recursive method to calculate the state transition rates to and from all different states of the CTMC representing the CRN network. The state transition rates are used to get all the possible balance equations. Recalling that the number of PUs is denoted by i and the number of SUs is denoted by j. Let us also assume that the number of free channels is denoted by k which is given by k=N−i−j.

It is important to notice that the state transition rates presented in this paper are different from those defined in [30] for several reasons: (1) the inclusion of the effect of the FAR in the CTMC. (2) In our previous work [30] we assumed that ongoing SUs perfectly detect the arrival of PUs and that there are no false alarms during ongoing data transmission. However, in this paper, we assume that SUs with ongoing connections do not perfectly detect the arrival of PUs and also that the false alarm probability during ongoing calls is not negligible. With this assumption, ongoing SUs detect PU arrivals with P _D2 where P _D2 is an arbitrary value between 0 and 1 and that the false alarm probability during ongoing calls equal P _FA2 which is also an arbitrary value. Note that this is a more realistic assumption for practical CRNs and represents a significant improvement over our previous work [30] that leads to obtaining accurate state transition rates and state probabilities. (3) Because of the false alarms during ongoing sensing, we require a new state transition that defines the transition from state (i,j) to state (i−1,j−1), with i>0 and j>0. In addition to state transitions because of the FAR events, we refer the reader to [30] for details concerning the other different events that trigger state transitions. All possible state transition types are described as follows:

Let Q denote the state transition rate matrix (also known as infinitesimal generator) of the CTMC. Let π denote the steady state probability vector with π _(i,j) denoting the probability that the system is in the steady state (i,j). When the system is in the steady or equilibrium state, the normalization condition is given by \(\sum \limits _{i = 0}^{N} {\sum \limits _{j = 0}^{N} {{\boldsymbol {\pi _{(\textit {i,j})}} = 1} }}\) [49] with the condition that 0≤i≤N, 0≤j≤N, and 0≤i+j≤N. Let D equals the total number of states in CTMC. We map the elements of the steady state probability vector π _(i,j) from state to index by assigning a unique integer index to identify each state. Therefore, the steady state probability vector can be represented as π=(π ₁,π ₂,...,π _D ) and the normalization condition is given by \(\sum \limits _{d} {{\boldsymbol {\pi }_{d}}} = 1\). The steady state probabilities of the CTMC can be found by applying the following procedure:

Since the number of states of the CTMC grows exponentially with the number of the channels in the network, it would be impossible to derive the CTMC transition rates by hand for large number of states. In this sense, the utilized recursive approach solves one part of this problem. However, the number of states is still exponential, which lead to higher memory and processing time requirements when the number of channels increases since the full-state transition matrix is used to obtain exact results. With very large number of channels, approximation solutions with reduced number of channel states would be beneficial. In the literature, some approximation methods have been presented for CTMCs with a large number of states [51, 52]. The results presented in this paper have been obtained using the exact full CTMC. However, when the number of channels is very large which brings some inefficiency, we can apply approximate solutions of large CTMCs to overcome this problem [51, 52].

The secondary forced termination probability, denoted by S U _FTP , is the probability of terminating SU calls because of SU’s failure to find a new free channel after moving from its current channel. The S U _FTP is calculated and defined by Eq. (15). It reflects the ratio of terminated SUs’ call to total SU call arrivals λ ₂.

The primary forced termination probability, denoted by P U _FTP and given by Eq. (16), is calculated as the ratio of terminated PU calls because of collisions with SUs to the total primary call arrivals λ ₁.

Here, we introduce a new performance metric that measures the secondary self-termination probability. The motivation of proposing this new metric starts with the fact that in CRNs, the occurrence of false alarms is of critical importance since significant amount of SUs’ calls could be terminated because of FARs. The new metric helps in accurately determining the percentage of SUs’ connection terminated due to SUs’ own errors. By identifying this metric, it allows for measuring the SUs’ ability of utilizing spectrum opportunities and helps in designing CRNs by setting correct spectrum sensing parameters. The metric determines the ratio of terminated SU calls because of FAR occurrence to the total secondary call arrivals λ ₂. As mentioned earlier, upon λ _FAR arrival, an SU has to terminate its own active call if it finishes the channel searching process without finding a new idle channel. The S U _STP can be calculated as:

where \({{\overline {T}}_{(i,j - 1)}^{(\textit {i,j})}}\) represents the portion of the transition rate from state (i,j) to state (i,j−1) that occur because of the λ _FAR.

The receiver operating characteristic (ROC) for both initial and continuous spectrum sensing is shown in Fig. 3. For continuous spectrum sensing, it can be clearly seen that better detection performance is achieved when large values of continuous sensing durations are used. However, for initial sensing, small initial sensing time T ₁ is enough for good detection level. This improved initial detection performance can be attributed to the fact that incoming SUs perform spectrum sensing over the whole PU bandwidth and hence their time bandwidth product is improved.

The impact of the residual interference distortion factor \(\alpha _{_{\text {SU}}}\) on the detection and false alarm probabilities is demonstrated by the ROC curves shown in Fig. 3. The SNR value for the initial sensing is 19 dB. However, the SNR values for the continuous sensing vary depending on the spectrum sensing time duration T ₂ that affects the time bandwidth product. We assume that the PU signal power is uniformly distributed over the PU channel. It can be seen that the residual interference affects the ROC curves, as the ROC performance drops significantly with increases in \(\alpha _{_{\text {SU}}}\) values. We can also see from the figure that the curve for the initial sensing with spectrum sensing time duration T ₁=10 μs is identical with the curve for continuous sensing with spectrum sensing time duration T ₂=100 μs and \(\alpha _{_{\text {SU}}}=0\). This is due to the fact that their time bandwidth products are the same and equal to 200. Figure 3 also shows the effect of the residual interference distortion factor \(\alpha _{_{\text {SU}}}\) on the false alarm and detection probabilities.

We start the analysis by investigating the impact of the PU interference tolerance on the performance of the CRN. We tested the cases of \(\widehat {T_{\text {TOL}}}=0, 1, 2, 3, 4, 5\) slots. By recalling the fact that PU detection starts from the first full \(\widehat {T_{\text {TOL}}}\) slots, if \(\widehat {T_{\text {TOL}}}=0\), PUs tolerate only very little interference from an SU, and strict interference constraints should be satisfied. This means that the SU should leave the channel immediately upon the arrival of a PU. However, in practice, this is not possible since the SU user needs some sensing time to detect the presence of the PU. In Fig. 4, we compare the PU successful probability P U _SP for different values of T _TOL and against the PU arrival rate λ ₁. We can observe from the figure the improvement in PU performance as we relax the interference constraint by increasing \(\widehat {T_{\text {TOL}}}\). We can also observe that the P U _SP drops significantly when \(\widehat {T_{\text {TOL}}}=0\). However, when \(\widehat {T_{\text {TOL}}}\) is sufficiently large, PU performance starts to improve, indicating that SUs are getting enough sensing time to complete the process of detecting incoming PUs. This implies that the \(\widehat {T_{\textit {TOL}}}\) has to be chosen in a way that meets the PUs interference constraint, and at the same time also maximizes the CRN performance. Figure 4 also presents the impact of the primary arrival rate λ ₁ on the CRN’s performance in term of P U _SP . It can be easily seen that there are peak values in P U _SP curves. As λ ₁ increases, the P U _SP improves. However, as λ ₁ become large, most of the channels will be occupied by PUs, thus increasing the possibility of collisions with SUs who move away from their channels because of the λ _FAR.

Figure 5 confirms what we asserted above regarding the impact of \(\widehat {T_{\text {TOL}}}\) on the PU performance. While varying λ ₁ from 1–10, the figure compares primary forced termination probability P U _FTP for each T _TOL value. It can be observed that P U _FTP monotonically decreases with increasing λ ₁. It is also evident that the P U _FTP decreases with the increase of the \(\widehat {T_{\text {TOL}}}\). The reason behind this trend can be attributed to the fact that employing small \(\widehat {T_{\text {TOL}}}\) reduces the SUs capability of correctly detecting incoming PUs. In such a case, SUs collisions with incoming PUs increases forcing PUs to leave the network and hence increasing the P U _FTP .

In Fig. 6, we plot the secondary self termination probability S U _STP against λ ₁ for different values of \(\widehat {T_{\textit {TOL}}}\). The figure shows that increasing \(\widehat {T_{\textit {TOL}}}\) can result in a significant SU performance degradation. For example, when \(\widehat {T_{\textit {TOL}}}=0\), SUs collide with incoming PUs, making life easier for other secondary users since they could find free channels more easily. On the one hand, when λ ₁ is small, PUs will have a smaller network resource share, leaving more opportunities for SUs. For example, SUs who leave their channels due to λ _FAR could find free channel and therefore reduce the S U _STP . On the other hand, when λ ₁ is large, most of the channels will be occupied by PUs, leaving smaller network resources for SUs opportunistic access. In this case, the effect of S U _STP is more noticeable since with FAR occurrence (which trigger SUs to leave their channels), SUs either correctly detect the presence of PUs or collide with them. In both causes this leads to an increase in the S U _STP . Intuitively, PUs are better protected by employing large \(\widehat {T_{\text {TOL}}}\) as it provides SUs with enough sensing time to complete the detection process.

Figure 7 shows the effect of λ _FAR on the primary forced termination probability P U _FTP for different number of channels N. It can be seen from Fig. 7 that P U _FTP curves for different N have unique minimums at different λ _FARs. The minimum points represent the optimum sensing parameters for SUs that would strongly protect PUs against forced connection termination. According to Fig. 7, the P U _FTP decreases with the increase in λ _FAR until it reaches the minimum point after which it starts to monotonically increase. On the one hand, too low λ _FAR reduces the PU’s performance. The degradation in PUs performance is due to the fact that small values of λ _FAR leads to small P _D2, and therefore existing SUs frequently collide with incoming PUs and thereby increase P U _FTP . On the other hand, with high values of λ _FAR (and consequently high P _D2), it would be easy for SUs to detect incoming PUs and therefore they can avoid collision with them. However, too high λ _FAR is not good since SUs initiate unnecessary spectrum handoffs, leading to sharp increases in P U _FTP . As shown in the figure, the gap between P U _FTP ’s curves shrinks as N increases. This reflects the fact that when N is sufficiently large, most of the channels are occupied by PUs because λ ₁>λ ₂, and the combined effect of λ _FAR, P _M2 and P _M1 forces some PUs to terminate their calls. With further increases in N, the effect of λ _FAR on P U _FTP is flat, as there would be enough radio resources to meet SUs demands. After SUs avoid incoming PUs and initiate a handoff and channel switching process, SUs will most likely find new free channels and hence will not harm existing PUs as there are more channels to accommodate them.

Figure 8 presents the effect of λ _FAR on the secondary self termination probability S U _STP for different number of channels N. It can be seen that there are peak values in the S U _STP curves. The following explanation is related to this behavior: on the one hand, when λ _FAR is relatively low, S U _STP increases with the increase in λ _FAR, until it reaches the peak value and then starts to decline. The degradation in SU performance can be explained by the fact that as λ _FAR increases, a growing number of SUs leave their current channels. If they cannot find new free channels elsewhere they terminate their calls, and thereby increase the S U _STP . On the other hand, when λ _FAR is relatively large, the detection probability P _D2 improves, which enables SUs to detect incoming PUs and move away from their channels. Hence, the proportion of SUs in the system is reduced, leading to a decrease in the S U _STP . It came as no surprise that increasing N greatly influences the S U _STP . When N is small, it would not be easier for SUs who initiate spectrum handoff to find new free channels to move to and the SUs have to leave the network. This results in a sharp increase in S U _STP .

In Fig. 9, for several values of N, we investigate the effect of λ _FAR on the primary successful probability P U _SP . The plot shows when λ _FAR is low; increasing λ _FAR slightly increases P U _SP until it reaches some (flat) peak points. The increase is a reflection of the improved detection P _D2. As can be observed from Fig. 10, increasing λ _FAR has an obvious negative impact on the secondary successful probability S U _SP . The reason for the reduction in S U _SP is that when λ _FAR is high, SUs increasingly initiate unnecessary spectrum handoff processes. SUs also initiate spectrum handoff processes if they detect the arrival of returning PUs. If SUs cannot find new channels, even though there are some free channel(s), they are forced to terminate their own connections and hence reducing S U _SP . The results shown in Fig. 10 indicate that SU optimal performance is when λ _FAR is close to zero. However, in practice, this value means SUs do not perform spectrum sensing and therefore cannot be used since too low λ _FAR leads to poor PU performance. Results shown in Fig. 7 confirm that low λ _FARs are not the best for PUs performance as they do not satisfy their QoS/interference constraint. Note that in order to improve network performance, there is a critical sensing tradeoff to be made. This result is a good motivation and illustrates the significance for optimizing detection parameters such that the effects of the λ _FAR can be kept within limits that would not harm PUs.

Figure 11 shows the influence of λ _FAR on the system resource utilization for different number of channels. When λ _FAR is small, SUs do not frequently initiate handoff processes and the effect of λ _FAR remains flat. However, as λ _FAR increases, a growing number of SUs terminate their calls if they cannot find other free channels. This explains the reduction in the system resource utilization. As illustrated in Fig. 11, PUs’ own resource utilization is lower than the overall system resource utilization. However, at high λ _FAR, PUs’ and system resource utilization get closer. This indicates that SUs do not complete their service when λ _FAR is too high and the network resource is mainly utilized by PUs.

The effect of perfect sensing on system resource utilization is shown in Fig. 12 where we plot a set of curves showing network resource utilization versus the secondary arrival rate λ ₂ for different number of channels N. The dash-dotted curves depict the CRN performance when SUs operate without sensing errors. It can be observed that, due to the absence of false alarms, network resources are better utilized. In this case, all unoccupied PUs’ channels can be opportunistically utilized by SUs. This effect is more noticeable when λ ₂ is high. On the other hand, sensing errors can severely reduce the network resource utilization. The impact of the resource underutilization can be reflected in the huge gap between the resource utilization curves. As λ ₂ increases, a growing number of SUs gain network access. However, λ _FAR prevents them from using network resources because of the increasing spectrum handoff initiation, which may lead to premature connection termination. With a small N, the system resources will be highly utilized since PUs and SUs will occupy most of the channels. However, when N is large, the radio resources are not fully utilized and therefore decreases the overall system resource utilization.