Expectation Maximization Aided Modified Weighted Sequential Energy Detector for Distributed Cooperative Spectrum Sensing

A cognitive radio system is an intelligent wireless communication system that learns from its surrounding radio environment and adapts its operating parameters (e.g., carrier frequency, transmit power, and digital modulation scheme) in real-time to the spatiotemporal variations of the RF spectrum. The primary objective of the cognitive scheme is to enable the unlicensed (secondary) users to opportunistically utilize the spectrum owned by the licensed (primary) users, where the reconfigurability of the radio is accomplished using software-defined radio based platforms [1, 2, 3]. However, the opportunistic access of the wireless spectrum entails using the spectrum sensing algorithm at the secondary users (SUs) to detect the presence or absence of the primary user (PU) in the channel. For the detection of PU, several sensing algorithms have been presented over the past years including the energy detector (ED) [4], coherent ED [5], matched filter detector [6], cyclostationary feature detector [7, 8], information theoretic criterion based detector [9], and eigen-values based detectors [10, 11]. These detectors make different trade-offs between the spectrum sensing delay, computational complexity, and the amount of PU’s signal information needed for sensing. Among these detectors, ED stands out as a preferred choice because of its low computational complexity, ease of implementation, and due to the reason that it does not require any prior knowledge about the PU signal. As such, ED-based spectrum sensing has been exploited widely in the literature, e.g., [12, 13, 14, 15], including the present paper.

Spectrum sensing (or PU detection) can be done by the SUs either by using a non-cooperative scheme or a cooperative scheme. In a non-cooperative scheme, each SU performs PU detection individually without any direct communication with the other SUs or a fusion center (FC) [13, 11]. In contrast, in cooperative spectrum sensing, a group of SUs communicate with each other or with an FC to collaboratively perform the PU detection [15, 16, 17]. Consequently, in comparison, the cooperative sensing approach is resilient to the deep fading and shadowing at an SU level, aids in eliminating the hidden terminal problem, reduces the sensing duration per SU, and demonstrates a better detection performance across the SUs network[18, 19].

Cooperative spectrum sensing schemes can be further categorized into either a centralized scheme or a distributed scheme. In a centralized scheme, a fusion center collects the sensing information from the SUs, detects the unused band, and broadcasts the decision to the SUs [20, 16, 15, 21]. However, the centralized approach is not scalable to large networks as the available communication resources are usually limited at the FC. Furthermore, an FC involvement defines a single point of failure for the centralized network. In comparison, in a distributed scheme, the SUs share their sensing statistics with their neighboring users followed by using a consensus protocol to collaboratively decide on the presence or absence of PU in the channel [22, 23]. This approach not only eliminates the single point of failure from the network, but it is also scalable as the communication resources need to be shared only among the neighbors.

The distributed cooperative spectrum sensing (DCSS) scheme usually has three critical phases, namely the sensing phase, the consensus phase, and the channel use phase. In the sensing phase, a group of SUs observes the PU channel for a certain time duration to collect a sufficient number of samples for computing the summary statistics (e.g., energy statistics [24, 13]). Next, in the consensus phase, the SUs share their summary statistics with their neighbors, and use an average consensus protocol [25, 26] to iteratively compute a weighted average of the globally shared values across the network. Upon consensus in such an approach, the final value is compared against a threshold at each SU to locally detect the presence or absence of the PU in the channel. Finally, in the channel use phase, the detection outcome is used to decide on the opportunistic use of the PU channel before restarting the cycle. This DCSS scheme was proposed in [22] wherein the authors analyzed its convergence speed as well as the detection performance for varying false alarm rates. In [27, 28], DCSS was extended to protect against the eavesdropper attack by encrypting the summary statistics shared between the SUs, whereas in [17, 29], the authors considered the scenarios in which some malicious SUs (aka Byzantines) may share falsified data with the neighbors and thus proposed algorithms to mitigate the Byzantine attacks on the network.

The above-mentioned DCSS algorithms use the conventional approach in which each SU uses energy statistic only from the current sensing interval to make the PU detection. This usually requires a large number of samples per energy statistic, specially for lower SNRs, in order to achieve improved PU detection. As a result, the sensing duration increases which in turn increases the power consumption per SU as well as reduces the network throughput [30]. However, it has been shown that aggregating present and past ED samples at each SU, for lower SNRs, can result in an improved detection performance, even with using a fewer samples per energy statistics [14, 12, 16, 15]. As such, in [15, 16], a dynamic PU is modeled using a two-state Markov chain model and a weighted sequential energy detector (WSED) is proposed in which all the present and past ED samples over an observation window¹¹1An observation window is defined as a vector of length $D$ containing all the ED statistics from the $D-1$ past sensing periods as well as the ED statistic from the present sensing period. are aggregated to achieve improved detection capability. However, [15, 16] assume a centralized scheme for cooperative spectrum sensing which as discussed before is not a scalable approach. Using the average of past ED samples, a non-cooperative improved ED (IED) is also proposed in [14] in which the averaged value compared against the threshold is used to improve the PU detection probability at lower SNRs. Furthermore, a non-cooperative multi-slot ED (mED) is developed in [12] in which the ED samples from multiple sensing slots are compared against the threshold to detect the presence or absence of PU in the channel.

In this paper, we consider a dynamic PU that follows a two-state Markov chain model for switching between the active and idle states [16, 15]. However, for the PU detection, we develop a test statistic for SUs wherein the highly correlated present and past ED samples are aggregated to improve the PU detection at lower SNRs with a few samples per ED. The SUs deploy the DCSS scheme in which the sensed test statistics are shared between the neighboring users to reach a consensus on the observed value across the network. The contributions made in this paper are listed as follows:

• •

  A modified WSED (mWSED) is proposed in which only those past and present ED samples are aggregated in its test statistics that correspond to the PU’s state in the current sensing interval. Thus, in contrast to the detectors presented in [15, 12, 14], mWSED avoids including the outdated ED samples in its test statistics which improves the detection performance of SUs at lower SNRs (see the simulation results provided in Section III-A and Section V). The closed-form equations for the probability of detection and probability of false alarm are also derived for mWSED.

• •

  An underlying assumption in mWSED is that the actual present and past states visited by the PU are a priori known over the observation window. In practice, the states are unknown, and thus we develop a joint expectation-maximization (EM) and Viterbi based algorithm, referred to herein as EM-Viterbi, to estimate them using the ED samples collected over the window. Specifically, the EM algorithm provides an estimate of the model parameters of the joint probability distribution over the observation vector and the state vector. Next, using their estimate, the Viterbi algorithm [31] optimally estimates the latent state vector by the maximization and back tracing operations and using the properties of the two-state Markov chain model. The estimated state vector produced by the EM-Viterbi algorithm is then used in mWSED to aggregate only the highly correlated samples in its test statistic. The resulting algorithm is named here as the EM-mWSED algorithm.

• •

  Simulation results are included that demonstrate the estimation performance of EM-Viterbi and compare its detection performance with that of the EM-mWSED algorithm. Further, the detection performance of EM-mWSED is compared with the performances of other schemes in the literature. The results show that EM-mWSED performs better than the other methods and its detection performance improves by either increasing the average number of connections per SU in the network, or by increasing the SNR or the number of samples per energy statistics.

This paper is outlined as follows. Section II provides a brief review of the energy detection based spectrum sensing. Distributed cooperative spectrum sensing is discussed in Section III, also including a review of WSED and presentation of our proposed mWSED. Next, Section IV delivers an expectation-maximization and Viterbi algorithm based scheme for estimating the PU states over an observation window, including using it for mWSED. Simulation results are presented in Section V. Finally, Section VI summarizes this work.

We consider a distributed spectrum sensing system in which a network of $N$ SUs are spatially distributed and cooperating with each other to detect the PU in the channel. As discussed in the previous section, we assume that the SUs deploy an energy based statistic to sense the channel. Thus, the energy computed by an $i$-th SU under the null hypothesis ($H_{0}$) and the alternate hypothesis ($H_{1}$) can be written as follows.

in which $L$ is the total number of samples collected over the sensing interval, $h_{i}$ is the channel gain for SU $i$, $s_{l}$ represents the PU signal at time index $l$, and finally, $n_{i,l}$ denotes the noise in the sensing interval which is assumed to be normally distributed with zero mean and variance $\sigma^{2}_{n}$. The signal to noise ratio (SNR) at the SU is defined by $\eta_{i}=\frac{\sum^{L}_{l=1}|h_{i}s_{l}|^{2}}{\sigma^{2}_{n}}$ which is $L$ times the SNR at the output of the energy detector. The ED statistics in (1) follow a central chi-squared distribution with $L$ degrees of freedom under $H_{0}$, and a non-central chi-squared distribution with $L$ degrees of freedom and non-centrality parameter $\eta_{i}\sigma^{2}_{n}$ under $H_{1}$ [32]. However, for the case of low SNR, we usually need a sufficient large number of samples ($L$) per energy statistics for improved PU detection²²2This can be observed from Eqn. (II) that for smaller $\eta_{i}$ values we need larger $L$ values to realize non-overlapping distributions of ED statistics under both hypotheses and thereby achieve improved PU detection. However, note that larger $L$ implies larger sensing duration which increases the power consumption per SU as well as reduces the network throughput [30]. In this paper, we show that an improved detection capability can be achieved for lower SNRs and $L$ values by averaging the highly correlated present and past ED statistics.. Thus, we can invoke the central limit theorem (CLT) [33] to assume that $x_{i}$ is Gaussian distributed under the hypothesis $H_{h}$ with mean $\mu_{h}$ and variance $\sigma^{2}_{h}$, for $h\in\{0,1\}$ [12, 14]. These means and variances can be computed as

As such, the probability of false alarm ($P_{f}$) and probability of detection ($P_{d}$) can be computed as

where $\lambda$ is the threshold for energy detection, and $Q(.)$ is the complementary cumulative distribution function of Gaussian distribution [33]. Note that for a selected value of false alarm probability, the threshold $\lambda$ can be computed from (II) by using the inverse of the $Q(.)$ function. Furthermore, given $\lambda$, the detection probability can be computed from (II) as well.

We consider a scenario wherein the PU follows a two-state Markov chain model for switching between the $H_{0}$ and $H_{1}$ as shown in Fig. 1 [16, 15]. Specifically, the next state visited by the PU depends only on its immediate previous state. Accordingly, in this figure, the parameter $\alpha$ denotes the transition probability of switching to an idle state ($H_{0}$) given that previously the PU was in the active state ($H_{1}$), whereas $\beta$ represents the transition probability of switching to an active state ($H_{1}$) if previously the PU was in the idle state ($H_{0}$). Thus, higher values of $\alpha$ and $\beta$ imply a highly dynamic PU, whereas their smaller values represent its slowly time-varying behavior. In the following, we first describe a scalable distributed cooperative spectrum sensing scheme as considered here, followed with a brief review of the WSED algorithm and our modified WSED algorithm.

Consider a network of SUs represented by an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ in which $\mathcal{V}=\{1,2,\ldots,N\}$ is the set of $N$ number of SUs in the network, and $\mathcal{E}=\{(i,j),\forall i,j\in V\}$ represents the set of all possible bidirectional communication links between them. The connectivity of the network in any realization is denoted by $c$ which is defined as a ratio of the number of active connections in the network ($N_{a}$) to the number of all possible connections among the SUs ($N(N-1)/2$). We consider a strongly connected network in which the neighboring users that are one hop away from each other share their information to reach a consensus. Therefore, wireless and computing resources such as bandwidth, processing power, and data storage capabilities, need to be only locally managed at the SUs, and scale proportionally to the average number of connections per SU in a network.

A distributed cooperative spectrum sensing algorithm is a scalable and a fully distributed approach which deploys a consensus protocol at an SU. The protocol iteratively updates the sensing information at the user, by using the locally shared information, to reach consensus with the other users in the network. To elaborate, let $y_{i}(0)=x_{i}$ represents the initial energy statistic for an $i$-th SU, then in iteration $k$ of the DCSS algorithm, the SU updates its estimate by using a weighted average method as follows.

in which $N_{i}$ is the set of neighboring users of the $i$-th SU. The weight $w_{ij}$ can be selected as $w_{ij}=1/\text{max}(d_{i},d_{j})$ where $d_{i}$ and $d_{j}$ represent the number of neighbors of SU $i$ and SU $j$, respectively. This selection of weights results in a doubly-stochastic Metropolis-Hasting weighting matrix which guarantees convergence of the consensus algorithm [26]. Thus, starting with a set of initial values $\{y_{i}(0),\text{ for }i=1,2,\ldots,N\}$, the algorithm running locally at each SU iteratively updates the values using (4) until it converges to an average of the globally shared values across the network. The average value is defined by $y^{*}=\frac{\sum_{i=1}^{N}x_{i}}{N}$. Upon convergence, the decision can be made locally at the $i$-th SU by using the following rule

The mWSED algorithm described in the previous section assumes that the actual states visited by the PU over the observation window are a priori known to the SUs. In practice, this may not be a valid assumption, and thus in this section we aim to compute an estimate of the states locally at each SU from the samples collected over the observation window.

To begin, let an SU $i$ collect $D$ ED samples over consecutive sensing intervals using (1), denoted by $\mathbf{x}_{i}=[x_{i,1},x_{i,2},\ldots,x_{i,D}]^{T}$ with $T$ representing the transpose operation. Using the CLT assumption [33, 17], we assume that $x_{id}$ follows a Gaussian distribution represented by $\mathcal{N}\left(x_{i,d}|\mu_{h},\sigma^{2}_{h}\right)$ with mean $\mu_{h}$ and variance $\sigma^{2}_{h}$, and with $h=0$ when the PU is idle, and $h=1$ when the PU is active. These means and variances are defined in (II). Further, if for the $i$-th SU the state of the PU at the sensing interval $d$ is denoted by $s_{i,d}\in\{0,1\}$, then for $\bm{\theta}_{0}\triangleq\{\mu_{0},\sigma^{2}_{0}\}$ and $\bm{\theta}_{1}\triangleq\{\mu_{1},\sigma^{2}_{1}\}$, the conditional probability distribution of $\mathbf{x}_{i}$ can be written as

where $\mathbf{s}_{i}=[s_{i,1},s_{i,2},\ldots,s_{i,D}]^{T}$ denotes the PU state vector, and $\mathds{1}(A)$ is an indicator function which is one if $A$ is true, and is zero otherwise [33]. Next, as discussed before and shown in Fig. 1, we assume that the state vector $\mathbf{s}_{i}$ follows a two-state Markov chain model with the transition probabilities $\alpha$ and $\beta$. Thus, the probability distribution of $\mathbf{s}_{i}$ is written as

where considering the steady-state distribution for the Markov process, we assume $s_{i,1}$ is Bernoulli distributed with mean $\frac{\beta}{\alpha+\beta}$.

Now if the model parameters of the above probability distributions are defined by $\bm{\Theta}=\{\bm{\theta}_{0},\bm{\theta}_{1},\alpha,\beta\}$, an optimal scheme for estimating both $\bm{\Theta}$ and $\mathbf{s}_{i}$ for SU $i$ involves solving the following optimization problem

where for the sake of simplicity, we assumed a uniform prior distribution on $\bm{\Theta}$. Note that due to the large dimensionality of the search space, jointly optimizing for $\mathbf{s}_{i}$ and $\bm{\Theta}$ is computationally difficult. Alternatively, we can aim to sequentially optimize for $\mathbf{s}_{i}$ and $\bm{\Theta}$ which involves solving the following two optimization problems:

where (IV) maximizes the likelihood function of $\bm{\Theta}$. Then using $\hat{\bm{\Theta}}$ we can solve,

However, note that due to the log-sum in (IV), directly optimizing for the elements of $\bm{\Theta}$, e.g., using the derivative trick, does not result in the closed-form update equations, whereas using the numerical methods for optimization have the inherent complexity with the tuning of the step-size parameter [34]. Furthermore, the optimization problem in (IV) is still a complex combinatorial search problem where the dimensionality of the search space increases exponentially with $D$. In the following, we develop a joint expectation maximization and Viterbi (EM-Viterbi) algorithm to estimate $\mathbf{s}_{i}$ and $\bm{\Theta}$ in a computationally efficient and optimal way using the closed-form update equations.

In this section, we demonstrate the PU detection and states estimation performances of our proposed EM-Viterbi algorithm and the detection performance of our proposed EM-mWSED scheme. For comparison purposes, we compare the performances of these algorithms to the conventional energy detector (ED), the weighted sequential energy detector (WSED) of [16, 15], multi-slot ED (msED) of [12], and the improved ED (IED) of [14] under different scenarios, when these detectors are used with the DCSS scheme and with the consensus happening on the present and past ED samples as proposed herein. As suggested in [16, 15] for WSED, we use a total of $3$ past ED samples in its test statistics for a highly dynamic PU, whereas ED uses only the present energy sample in its test statistics. For the msED, we consider only the present and past ED samples for a fair comparison and plot its performance when $2$ slots are used in making the PU detection [12]. Notably increasing the number of slots for msED results in a degraded performance for the considered time-varying PU. We consider a network of $N$ secondary users randomly generated with a connectivity $c$ and the weighting matrix as defined in (4). The average number of connections per SU in the network is given by $R=c(N-1)$. The primary user follows a two-state Markov chain model to switch between the active and idle states with the transition probabilities $\alpha$ and $\beta$. The SUs collect $L$ samples individually to compute the energy statistic, and the length of the observation window is set as $D=150$. As described in Algorithm 1, we assume that the SUs reach consensus on all the present and past ED samples over the observation window, which improves the SNR proportionally to ${R}$ and aids in improving the detection and estimation performances. For the initialization of the EM algorithm, we determine the initial estimate of the means and variances by using the K-means clustering algorithm [36] over the window with $K=2$, whereas the initial estimate of the transition probabilities can be computed by performing a coarse grid search over the likelihood function in (IV) in the $(0,1)$ interval with a grid resolution of $0.1$.

In Fig. 6, we demonstrate the PU’s states estimation performance of the EM-Viterbi algorithm as a function of SNR (dB) and the number of samples per energy statistics ($L$). The states estimation error for SU $i$ is defined here as $\text{Estimation Error }(\mathbf{s}_{i})=\frac{1}{D}\sum^{D}_{d=1}\mathbb{E}\left[\mathds{1}\left(\hat{s}_{i,d}\neq s_{i,d}\right)\right]$ where the expectation is computed over several Monte Carlo trials. Further, we assume that the consensus is reached on the ED samples in Step $1$ of Algorithm 1 prior to estimation, and thus the error plots in this figure are observed at all the SUs in the network. We consider here that the secondary users network has $N=10$, $20$, and $60$ users with connectivity $c=0.2$. The PU displays a highly dynamic nature with transition probabilities $\alpha=\beta=0.1$. Firstly, it is observed that for a fewer number of SUs ($N=10$) in the network, the estimation error is higher at the lower SNRs and the lower $L$ values, but when the number of SUs in the network increases from $N=10$ to $20$ and then to $60$, the estimation error decreases significantly even for the lower SNR and $L$ values. This is because the SNR upon consensus in Step 1 of Algorithm 1 improves proportionally to ${R}$, as each SU exploits about $R$ independent ED samples of the PU’s channel. This implies that better performance can be achiever at lower PU’s SNR environments by deploying DCSS-based larger and highly connected networks of SUs, i.e., networks with larger $N$ and $c$ values., This explains our motivation behind using DCSS scheme prior to the estimation process in Algorithm 1. Secondly, as discussed in Section I, it is observed that the estimation error is higher at lower SNR and $L$ values for all the considered cases in Fig. 6. As discussed in Section II, this is because the distribution of the ED samples under the two hypotheses highly overlap at those values which degrades the estimation and detection performances, but increasing $L$ at lower SNRs separates the means of the two distributions that in turn improves the performance\footrefED_footnote. However, larger $L$ implies larger sensing durations which reduces the network throughput and increases the energy consumption [30].

In general, a poor estimation error for PU’s states do effect the detection performances of EM-Viterbi⁴⁴4Since the last element $\hat{s}_{i,D}$ in the state vector estimate, $\hat{\mathbf{s}}_{i}$ at SU $i$, represents an estimate of the PU’s state in the present sensing period $D$, EM-Viterbi can also be used for PU detection. and EM-mWSED schemes, but for an appropriate choice of $L$ value and by aggregating present and past ED samples, we can improve the detection performance at lower SNRs. To illustrate this fact, in Fig. 7, we compare the detection performances of EM-Viterbi and EM-mWSED by varying SNR and $L$ values, when $N=20$, $c=0.5$, and $\alpha=\beta=0.1$. Note that the state vector estimated by EM-Viterbi is used in EM-mWSED to average the highly correlated present and past ED samples. The weights on the aggregated ED samples in EM-mWSED are chosen as exponentially distributed as it results in better detection performance than equal weighting at lower SNR or $L$ values, due to the rise in the estimation error. Further, the threshold at each SNR and $L$ values is selected based on the false alarm rate of EM-Viterbi. However, it is observed that the detection performance of EM-mWSED is better than that of EM-Viterbi at lower SNRs, and that the performance of EM-mWSED improves significantly by a slight increase in the $L$ value. While the increase in $L$ aids in improving the EM’s estimation performance, the better performance of mWSED is due to the fact that it averages the highly correlated present and past ED samples in its test statistics which reduces the variances under the two hypotheses as illustrated in Section III-D. This improves the detection performance at lower SNRs and smaller $L$ values. Finally, as expected, we also observe that the performances of both algorithms achieve that of mWSED with an increase in SNR or $L$ values.

In Fig. 8, we illustrate the convergence performance of the EM algorithm in estimating the model parameters $\bm{\Theta}$ when $N=10$ SUs are considered in the network with network connectivity $c=0.2$. The mean-squared error (MSE) of $\bm{\Theta}$ is defined as $\text{MSE}\left(\bm{\Theta}\right)=\mathbb{E}\left[\mid\mid\bm{\Theta}-\hat{\bm{\Theta}}\mid\mid_{2}\right]$. It is observed that for $L=12$ samples per energy statistics, as the SNR increase from $-5$ dB, to $-3$ dB, and then to $0$ dB, the EM algorithm converges faster in fewer iterations. Similar observation is also made when for a lower SNR value of $-5$ dB, we increase $L$ from $12$ to $36$. This is due to the fact that the initial estimates for EM are improved at the larger SNR and $L$ values which results in its faster convergence response and improved estimation performance.

While EM-Viterbi provides a single operating point for SUs due to the estimation of the present state, in terms of detection probability and false alarm probability; in contrast, by using the estimated state vector in mWSED, the EM-mWSED algorithm can provide a wide range of operating points for SUs. Further, as observed in Fig. 7, EM-mWSED outperforms EM-Viterbi at lower SNRs which makes it a preferable choice. As such, in Figs. 9 and 10, we show the receiver operating characteristic (ROC) curves of the proposed EM-mWSED algorithm and compare it with those of the ED, WSED, IED, and msED schemes. In Fig. 9, we consider $N=20$ SUs in the network with connectivity $c=0.2$. The PU states transitioning probabilities are selected as $\alpha=\beta=0.1$. The SNR is assumed to be $-5$ dB, whereas the number of samples per energy statistic $L$ is assumed to be either $8$ or $12$. As expected, it is observed that EM-mWSED outperforms all other methods at increasing the detection probability and reducing the false alarm probability, and thereby provides a wide range of operating points for SU. Furthermore, the detection performance of EM-mWSED improves with the increase in the $L$ values, due to decrease in the states estimation error.

Fig. 10 compares the ROC curves of EM-mWSED with that of ED, WSED, IED, and msED schemes. The PU is either considered to be slowly time-varying with $\alpha=\beta=0.05$ or highly dynamic with $\alpha=\beta=0.1$ as considered earlier. There are $N=10$ SUs in the network with connectivity $c=0.2$, and the SNR is considered to be $-3$ dB with the number of samples per energy statistics as either $L=12$ or $36$. Firstly, by comparing Figs. 9 and 10 for $L=12$, SNR$=-3$ dB, and $\alpha=\beta=0.1$, we observe a decay in the detection performance of EM-mWSED due to decrease in the value of $R$ in Fig. 10, which reduces the SNR upon consensus as discussed above. Secondly, it is observed in Fig. 10 that for both slowly and highly dynamic natures of the PU, our EM-mWSED algorithm performs better than the other detectors as expected. However, when $L=12$, its detection performance appears to be dependent on the time-varying nature of the PU, and it is seen to be better in case of slowly time-varying PU than a highly dynamic PU. This is because at the lower SNR or $L$ values, EM-Viterbi can easily characterize the ED samples, corresponding to the two states of PU, when the PU is slowly time-varying than when it is highly dynamic, and thus results in a lower estimation error in the former case. However, when $L$ increases to $36$, then the estimation error of EM-Viterbi decreases for a highly dynamic PU as well, which in turn results in the similar performance of EM-mWSED for both kinds of PUs, as shown in this figure.

Finally, while the focus herein is on investigating the detection vs. false alarm probabilities, it is worth noting that, on the one hand, where the throughput performance of an SU under $H_{1}$ is proportional to ${(1-p_{d})}$, on the other hand, the throughput under $H_{0}$ is proportional to $(1-p_{f})$ [40]. Thus, the higher detection probability of EM-mWSED as compared with that of ED, WSED, IED, and msED implies a higher throughput of SUs with reduced interference to the primary user, whereas its capability of simultaneously decreasing the false alarm probability with the increase in the average connections per SUs in the network, SNR, or $L$, implies a higher throughput during the idle state of the PU.

We considered the problem of DCSS at lower SNRs when the present and past ED samples are aggregated in a test statistic for improved PU detection with fewer samples per energy statistics. Using a few samples per energy aids in reducing the sensing duration and power consumption per SU. Furthermore, it increases the throughput of the network. A modified weighted sequential energy detector is proposed which utilizes the PU states information over an observation window to combine only the highly correlated ED samples in its test statistics. In practice, the states information is unknown, and thus we developed an EM-Viterbi algorithm to iteratively estimate them using the ED samples collected over the window. The estimated states are then used in mWSED to compute its test statistics, and the resulting algorithm is named here as the EM-mWSED algorithm. Simulation results are included to demonstrate the estimation performance of EM-Viterbi and compare the detection performance of both EM-Viterbi and EM-mWSED schemes. Furthermore, the detection performance of EM-mWSED is compared with that of the conventional ED, WSED, IED, and msED methods. The results demonstrate that our proposed EM-mWSED performs better than the other schemes, and its performance improves by either increasing the average number of connections per SU in the network, or by increasing the SNR or the number of samples per energy statistics, for both slowly varying and highly dynamic PU.

Herein, we present the derivation of the probabilities $\gamma(s_{i,d}=h)=p\left(s_{i,d}=h|\mathbf{x}_{i},\bm{\Theta}^{(l-1)}\right)$ and $\xi(s_{i,d}=h,s_{i,d-1}=g)=p\left(s_{i,d}=h,s_{i,d-1}=g|\mathbf{x}_{i},\bm{\Theta}^{(l-1)}\right)$ for $g,h\in\{0,1\}$ and the $l$-th iteration of EM. To begin, the posterior distribution of $s_{i,d}$ given $\mathbf{x}_{i}$ and $\bm{\Theta}^{(l-1)}$ can be written as

where the distributions $\pi_{d}(s_{i,d})$ and $\nu_{d}(s_{i,d})$ are computed later herein. The notation $\mathbf{x}_{i,m:n}\triangleq[x_{i,m},x_{i,m+1},\ldots,x_{i,n}]^{T}$ which is a shorthand to represent the elements in $\mathbf{x}_{i}$ from index $m$ to $n$ where $m,n\in\{1,2,\ldots,D\}$. Similarly, we can write the joint distribution of $s_{i,d}$ and $s_{i,d-1}$ as

where the conditional distribution of $x_{i,d}$ and the conditional prior distribution of $s_{i,d}$ that are used above are both defined in (10) and (IV). Next we follow the forward-backward recursion approach in [37] to compute the distributions $\nu_{d}(s_{i,d})$ and $\pi_{d}(s_{i,d})$. First, to compute $\nu_{d}(s_{i,d})$, we write