Consider a CR network consists of a pair of primary transmitters (PTs) and receivers u, v, and a pair of secondary communication nodes s, d, as depicted in Fig. 1.

Fig. 1.

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	Fig. 1. CR network model.

The CR network is assumed to operate in a time-slotted fashion and the SU slots are synchronized to the PU slots. Let T denote the slot length and h_{s, v}, h_{s, d}, h_{u, v}, h_{u, d} denote the path losses from s to v, from s to d, from u to v, and from u to d, which are given as

where d_{i, j} is the distance between nodes i and j. κ and μ denote the pass loss constant and path loss exponent, respectively. Each receiver is affected by an additive white Gaussian noise (AWGN) with zero mean and variance .

The spectrum sensing process is modeled as an HMM, as shown in Fig. 2. The status of the PU follows a discrete-time Markov process with state space X = {x₁, x₂}, where x₁ = 0, x₂ = 1. These states are hidden since they are not directly observable (due to the influence of the AWGN and the path losses). Hypotheses and denote the absence of a PU transmission in slot t and the existence of the PU, respectively. The prior probabilities of the two states are P(H₀) and P(H₁). Let A denote the state transition probability matrix: A = {a_ij}, a_ij = P(q_{t+ 1} = j| q_t = i) > 0, i, j ∈ X. For convenience, we assume P(H₀) and P(H₁) are equal to the steady-state probabilities of H₀ and H₁, which are given as

P_d = P(H₁| H₁) and P_f = P(H₁| H₀) denote the detection and false alarm probabilities, respectively. The SU senses the channel of the PU and obtains its observation sequence: , where o_t is the SU’ s local decision at slot t. The observation state space is Y = {y₁, y₂}, where y₁ = 0, y₂ = 1.

Fig. 2.

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	Fig. 2. Spectrum sensing process.

We assume that the ST is equipped with an energy harvester which harvests energy from ambient sources and stores the energy in a rechargeable battery with the maximum battery capacity C_max, as shown in Fig. 1. ST can be in one of the following two modes at any given time: Harvesting mode (harvests energy from ambient sources) and active mode (senses the channel of the PU and decides whether to transmit or not). We use the Bernoulli model to illustrate the energy arrival process, which is as follows: In each time slot, SU harvests energy E with probability p. Therefore, the average energy harvesting rate is E_h = pE. The Bernoulli arrival model is simple but it still can capture the random and sporadic nature of energy arrival.^[20] Similar to Ref. [22], we assume that the ST switches into the active mode only if it has E_sa (trigger energy) units energy in the battery, where E_sa ≤ C_max.

An HMM can be denoted by λ = (π , A, B), where π is the initial state distribution: π = {π _i}, π _i = P(q₁ = i), i ∈ X; A is the state transition probability matrix: A = {a_ij}, a_ij = P(q_{t+ 1} = j| q_t = i), i, j ∈ X; B is emission probability matrix: B = {b_jk}, b_jk = P(o_t = k| q_t = j), j ∈ X, k ∈ Y. In our sensing scenario, π _i = P(H_i), i ∈ X, b_jk = P(H_k| H_j), j ∈ X, k ∈ Y. The parameters of the HMM are determined by solving the following optimization problem, given as^{[25, 26]}

where p(O| λ ) is the probability of the observation sequence . In order to solve Eq. (4), a training phase is needed. At the beginning of the training, the SU senses the PU for M consecutive slots with energy detection to obtain the decision sequence (observation sequence or training sequence), where o_t ∈ {0, 1} is the decision made by SU at slot t. The training algorithm of HMM is an iterative algorithm, which is given as^[26]

where α _t (i) and β _t (i) are forward and backward partial likelihoods, respectively. The iteration is initialized by setting π _i = 1/2, i ∈ X, b_jk = 1/2, j ∈ X, k ∈ Y, α ₁ (i) = π _ib_io1, β _M (i) = 1, i ∈ X. The iteration process might be stopped if the increase of P(O| λ ) is not larger than a threshold or the number of iterations achieves the maximum number of iteration.^[26]

According to the above analysis, the complexity of the HMM training mainly concentrates on two phases: obtaining the training sequence with energy detection and the iteration process. Energy detection needs N_s multiplications and N_s − 1 additions, where N_s is the number of signal samples collected in one slot.^[14] Hence, the complexity of obtaining the training sequence is M(2N_s − 1). Since the number of iterations is generally not large, the complexity of the iteration process is O(M). Therefore, the complexity of HMM training is M(2N_s − 1) + O(M).

As previously mentioned, in order to switch to the active mode, the amount of energy in the battery of the ST must achieve E_sa. Since the average energy harvesting rate is E_h, the expected number of slots that the ST needs to harvest enough energy is given as

After E_sa units energy is achieved, the ST switches to the active mode. In active mode, the ST first uses E_s energy to sense the status of the PU and makes a decision θ _t ∈ {0, 1} during one slot, where θ _t = 0 denotes the PU being absent at slot t and θ _t = 1 denotes the existence of the PU’ s transmission.

Without loss of generality, we use energy detection (ED) to execute spectrum sensing. The detection performance of SU is determined by P_d and P_f. Correct detection can protect the quality of service (Qos) of the PU. In order to guarantee a certain Qos of the PU, P_d is set to be a constant, which depends on the requirement of the PU. The P_f of ED is given by^[27]

where Q(· ) is the complementary Gaussian distribution

f_s is the sampling frequency of the SU. τ is the length of the sensing time and γ is the received signal-to-noise ratio (SNR) of the PU signal measured at the ST when the PU is busy, as the energy consumption is roughly linear in the number of samples, which is linear in τ . Similar to Ref. [20], we assume that τ is proportional to E_s with a proportionality c. Then, equation (14) becomes

If the sensing decision made by the ST is θ _t = 1, the ST will not transmit and switch to the harvesting mode at the next slot to harvest E_s energy. Therefore, the expected number of slots needed for ST to harvest E_s is given as

It is seen that the expected number of slots mainly comes from two parts. One is required when the PU is busy and the SU correctly detects the presence of the PU. The other is required when the PU is idle but the SU detects the presence of the PU by mistake.

If θ _t = 0, the ST will transmit its data. In this paper, we adopt a myopic strategy: The ST will exhaust all its residual energy to transmit during N consecutive slots, where N ≥ 1. As the status of the PU follows a Markov chain with 2 states, the states of the PU may maintain several consecutive slots. Let N_i denote a variable that the probability of the state x_i lasting for more than N_i slots is less than 1 − ρ , where ρ is a predefined threshold. According to Ref. [25], N_i is given as

The maximum of N_i is given as

Therefore, we have 1 ≤ N ≤ N_max. The transmit power of the ST is given as

where P_max is the maximum transmit power of the ST. If there is sill some residual energy in the battery after N slots, the ST must release all of them to guarantee that the battery is empty at each harvesting-sensing-transmitting period, and we assume that the time for releasing the energy is negligible. Let γ _p and γ _s denote the target SNR for the primary user receiver and secondary user receiver, respectively.^[22] Hence, the minimum transmit power of the ST is given as

The minimum E_sa should afford the ST to transmit at least one slot, which means that E_sa ≥ P_minT. When the true state of the PU is idle, according to Eqs. (19) and (20), the data rate of the ST is given as

Note that the SU’ s data rate in Eq. (21) is normalized with W, ^[15] where W is the bandwidth of the channel.

Since a₀₁ > 0, the PU may change its state several times during the N consecutive transmission slots. The PU and SU will cause interference to each other when a transmission collision appears between them. In the collision scenario, the data rate of the SU is given as

where P_P is the transmit power of the PT. Meanwhile, we define the lost data rate of the PU as follows:

Let X₀ denote the number of idle slots during N consecutive slots. As shown in Fig. 3, ^[25] when the current true state of the PU is busy, the expectation of X₀ is given by

where m is the number of state changes during N slots and the probability p(X₀ = k) is given as follows:

where

Fig. 3.

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	Fig. 3. Illustration of the states changes of the PU during N consecutive slots.

When the current true state of the PU is idle, the expectation of X₀ is given by

where

where

Therefore, the expected number of idle slots in the N transmission slots is given as

The throughput of the SU in N transmission slots is given as

The ST needs to spend several slots to harvest enough energy before transmission, which results in loss of opportunities for data transmission. The performance of the SU’ s transmission cannot be evaluated by only using the throughput achieved in N transmission slots, i.e., the energy harvesting process must be considered. Hence, taking into account the energy harvesting process, we define the average throughput of the SU as

It can be seen that equation (30) is very different from that in the fixed powered scenario. Meanwhile, the interference caused to the PU is defined as

Our goal is to maximize f_SU and minimize f_in at the same time. From the perspective of PU, the smaller the interference caused by SU is, the better the performance of PU is. Alternatively, from the perspective of SU, the goal of SU is to achieve a transmit rate as high as possible. Thus, it is not appropriate to just simply construct an objective function as f_SU − f_in before all the terms are normalized. Motivated by Ref. [25], we use the sigmoid function to normalize the terms f_SU and f_in as

and are called the satisfaction degrees of SU and PU, respectively. Obviously, is an increasing function of f_SU while is a negative value decreasing function of f_in. It means that the satisfaction degree of SU increases with f_SU and the satisfaction degree of PU decreases with f_in, which matches our intuition.

We use the weighted sum of Eqs. (32) and (33) to denote the whole satisfaction degree (WSD) of the CR network, which is given as follows:

where ω ₁ and ω ₂ are the weight coefficients, which reflect the requirement of the CR network. For example, if the SU’ s average throughput is more important than Qos of PU, ω ₁ can be set as a large value, and vice versa.

The optimization problem to find the optimal set (E_sa, E_s, N) that maximizes the WSD under the energy causality and Qos constraints can be formulated as

Obviously, E_sa and E_s are continuous variables while N is an integer variable. Therefore, the problem in Eq. (35) is a mixed-integer nonlinear programming (MINLP) problem, which is a hot topic that attracts tremendous interest. In this paper, the well-known differential evolution (DE) algorithm^{[28, 29]} is applied to solve the optimization problem given in Eq. (35). Before using the DE algorithm, we first define and replace (E_sa, E_s, N) with (x, y, z). Furthermore, we use (x_low, y_low, z_low) and (x_up, y_up, z_up) denote the lower and upper limit of (x, y, z). The pseudocode of the DE algorithm is given in Algorithm 1.

The operator ⌊ (x, y, z)⌋ in Algorithm 1 is defined as (x, y, INT[z]), where INT[z] refers to the nearest integer to z.^[28] The mutation factor F is set as^[29]

where F₀ is a constant in the ranges from 0 to 1.