A joint optimization of energy harvesting and spectrum sensing for wireless sensor networks under Middleton Class A noise

Abstract

With the growing popularity of wireless sensor networks, the environment in which the network is located becomes more undesirable. In addition, the problems of spectrum scarcity and the short sensor lifetime have become increasingly prominent. In this article, we incorporate the two technologies of cognitive radio and energy harvesting to solve the above problems of wireless sensor networks under impulsive noise. First, we use a Middleton Class A noise model to imitate the practical environment and the fractional lower order moments detector is employed to perform spectrum sensing for the sensors of wireless sensor networks, which are performing as the second users. Second, a new time-slots structure is proposed for the self-powered second user and the analytical expression of the second user’s average throughput is derived. Finally, we maximize the second user’s average throughput by a joint optimization of the sensing duration and data transmission duration while giving the primary user sufficient protection. Simulation shows that a much better performance can be achieved by fractional lower order moment detector than the traditional energy detector. Moreover, our optimization of the time-slots allocation is feasible and the maximum second user’s average throughput can be obtained.

Keywords

Joint optimization energy harvesting spectrum sensing wireless sensor networks Middleton Class A noise

Introduction

With the increasing popularity of wireless services and applications, such as smart cities and e-healthcare systems, the wireless sensor networks (WSNs) play an indispensable role as a popular solution of information collection.¹ Taking into account the application scenario, WSNs should ensure that the user can be provided with reliable data for a relatively long period. However, traditional WSNs operate in the industrial, scientific, and medical (ISM) band where sensors may suffer spectrum shortage and severe interference due to a large number of potential applications in this band.² To deal with the issue of spectrum scarcity, cognitive radio (CR) can be integrated into WSNs for allowing sensors to utilize licensed bands opportunistically when they are not heavily occupied by licensed primary users (PUs).^3,4

For establishing cognitive WSNs, reliable spectrum sensing is essential because the operation of CR sensors as the second users (SUs) starts with detecting the absence of PUs. Compared with traditional sensors, spectrum sensing will increase the energy consumption of CR sensors which are typically powered by batteries. For guaranteeing longer lifetime of CR sensors, promising technologies for improving energy efficiency has been proposed.⁵ As one of the effective approaches, energy-harvesting (EH) technology is the most widely studied one for improving the energy efficiency of WSNs.^6–11 CR sensors equipped with EH circuits can harvest energy from either PU signals or ambient energy sources, which support the sensor to work sustainably without replacing the battery.⁷

After EH technology being employed in cognitive WSNs, the frame structure of the network will be composed of EH duration, spectrum-sensing duration, and data transmission duration. How to maximize the network throughput by allocating time-slots for each duration is a question worthy of study. Extensive research efforts have been made in the literature. An EH SU model, investigated by Hoang et al.,⁸ states that the SU can harvest the radio frequency (RF) energy of the PU signal after detecting that the channel is occupied by a PU. For supplying the energy consumption of spectrum sensing, Liu et al.⁹ proposed a multislot simultaneous spectrum-sensing and EH model and maximized the throughput of the SU under the constraints of sensing performance and EH. Li et al.¹⁰ used the Markov decision process framework to determine the optimal spectrum-sensing energy, the transmit energy, and spectrum-sensing interval to optimize the long-term average throughput of the SU and the interference caused to the PU. For most of the existing contributions, there are several assumptions in common. First, EH is always operating after spectrum sensing, which makes the SU not to be self-powered; second, spectrum sensing is usually considered being performed under the additive white Gaussian noise (AWGN) channel; and third, energy detector (ED)-based sensing method is employed for simplicity. However, WSNs are usually corrupted by low-probability, high-amplitude impulsive interference due to the environments where the networks are implemented. This kind of interference can be natural or man-made and has been found to be significant in many environments.^12–14 Examples include atmospheric noise, power lines, automobile ignitions, and wireless systems in the vicinity of the cognitive WSNs.

Being different from AWGN hypothesis, ED-based sensing in a channel contaminated by strong impulsive noise suffers a significant performance loss. Therefore, the analysis of network performance needs to be reconsidered under impulsive noise assumption. In this article, we investigate the EH-aided self-powered cognitive WSNs performing under impulsive noise. Fractional lower order moments (FLOM)-based detector is utilized in our scheme, and we maximize the SU’s average throughput through the joint optimization of spectrum-sensing duration, data transmission duration, and sensing parameters. The main contributions of this article are summarized as follows:

We study the problem of spectrum sensing under Middleton Class A noise adopting FLOM-based detector and derive the analytical expressions of the probability of false alarm $P_{f a}$ and the probability of detection $P_{d}$ . Then we show the performance enhancement that FLOM-based detector can achieve.

We propose a frame structure of the self-powered SU with three durations of EH, spectrum sensing, and data transmission. Using the proposed structure, SU can perform EH in the data transmission time-slots when the PU is detected as present.

We make the optimization of SU’s average throughput without interfering the PU’s signal by satisfying the tolerant detection probability constraint.

The following parts of this article are organized as follows. The noise model are defined in section “Middleton Class A noise model.” The sensing performance of the ED and FLOM detector are derived, and the frame structure with EH are proposed in section “Spectrum sensing and energy harvesting under Middleton Class A noise.” In section “Throughput analysis and optimization,” the average throughput of SU is analyzed and optimized. Section “Simulation and results” includes the simulation and the results, and the conclusion is drawn in section “Conclusion.”

Middleton Class A noise model

There are several models approximating practical impulsive noise such as Middleton Class A model, symmetric $α$ -stable (S $α$ S) model, and Gaussian mixture models. Among these models, Middleton Class A noise is one of the widely investigated statistical distributions that can well match physical phenomena and is used to model the narrow band impulsive noise in different systems.¹⁵ So in this article, we assume that the sensors are operating over a channel impaired by Middleton Class A noise $z$ . The probability density function $(p d f)$ of the instantaneous amplitude of the noise is defined as

f_{Z} (z) = \sum_{m = 0}^{\infty} β_{m} G (z; σ_{m}^{2}) = \sum_{m = 0}^{\infty} \frac{β_{m}}{\sqrt{2 π} σ_{m}} e^{- \frac{z^{2}}{2 σ_{m}^{2}}}

(1)

where $β_{m} = (e^{- A} A^{m}) / m!$ indicates that $m$ noise sources contribute to the impulsive event simultaneously, and $A = E {m} = \sum_{m = 0}^{\infty} m β_{m}$ is the corresponding overlap index denoting the average number of impulse noise sources active at any given time. Note that we use $E {\cdot}$ to denote expectation operation in this article. $A$ is usually within the range of $[0.01, 1]$ in practice and larger values of $A$ make the characteristic of the noise closer to Gaussian noise.

In addition, $σ_{m}^{2} = (((m / A) + T) / (1 + T)) σ_{Z}^{2} = σ_{G}^{2} + σ_{I}^{2} (m / A)$ , where $σ_{Z}^{2} = E {z^{2}} = σ_{G}^{2} + σ_{I}^{2}$ is the variance of the noise, denoting the noise power, and $σ_{G}^{2}$ is the power of Gaussian portion, $σ_{I}^{2}$ is the power of impulsive portion. $T = σ_{G}^{2} / σ_{I}^{2}$ represents the average power ratio of the two components. Thus, the Middleton Class A noise is totally characterized by the parameters $A$ , $T$ and $σ_{Z}^{2}$ .

Spectrum sensing and EH under Middleton Class A noise

Spectrum sensing under Middleton Class A noise

A CR-based sensor wants to access licensed bands opportunistically, it must detect the presence and absence of the PU signal in advance, which is known as spectrum sensing. Due to the two opposite states of the PU signal, the problem of spectrum sensing can be considered as the following binary hypothesis testing problem¹⁶

\begin{matrix} H_{0} : y (n) = z (n) P U a b s e n t \\ H_{1} : y (n) = x (n) + z (n) P U p r e s e n t \end{matrix}

(2)

in which, $n = 1, 2, 3, \dots, N$ , $N$ is the number of observed samples; $y (n)$ is the signal observed by sensing receiver with $x (n)$ and $z (n)$ denoting the PU signal and the additive impulsive noise, respectively. Obviously, hypothesis $H_{0}$ denotes the PU signal being absent and $y (n)$ consists only of noise $z (n)$ . On the contrary, hypothesis $H_{1}$ denotes the PU signal being present along with noise $z (n)$ .

First, the ED-based spectrum sensing is considered as most of the existing literature. The corresponding test statistic can be expressed as

Λ_{E D} (y) = \frac{1}{N} \sum_{n = 1}^{N} {| y (n) |}^{2}

(3)

Based on the Bayesian model defined in Kay,¹⁷ we assume that the PU signal $X ~ N (0, σ_{X}^{2})$ is a Gaussian random process and is independent with the noise $z (n)$ , then we can obtain that

\begin{matrix} {f_{Y} (y (n)) |}_{H_{0}} = f_{Z} (y (n)) = \sum_{m = 0}^{\infty} \frac{β_{m}}{\sqrt{2 π} σ_{m}} e^{- \frac{y {(n)}^{2}}{2 σ_{m}^{2}}} \\ {f_{Y} (y (n)) |}_{H_{1}} = f_{X} (y (n)) * f_{Z} (y (n)) \\ = \sum_{m = 0}^{\infty} \frac{β_{m}}{\sqrt{2 π (σ_{m}^{2} + σ_{X}^{2})}} e^{- \frac{y {(n)}^{2}}{2 (σ_{m}^{2} + σ_{X}^{2})}} \end{matrix}

(4)

According to the central limit theorem (CLT),¹⁸ when the number of observed samples $N$ is large enough, the metric $Λ_{E D} (y)$ can be approximated as a Gaussian random variable with

\begin{matrix} {Λ_{E D} (y) |}_{H_{0}} ~ N (μ_{0}, \frac{σ_{0}^{2}}{N}) \\ {Λ_{E D} (y) |}_{H_{1}} ~ N (μ_{1}, \frac{σ_{1}^{2}}{N}) \end{matrix}

(5)

in which

\begin{matrix} μ_{0} = E {{| y (n) |}^{2} | H_{0}} = \sum_{m = 0}^{\infty} β_{m} σ_{m}^{2} = σ_{Z}^{2} \\ μ_{1} = E {{| y (n) |}^{2} | H_{1}} \\ = \sum_{m = 0}^{\infty} β_{m} (σ_{m}^{2} + σ_{X}^{2}) = σ_{Z}^{2} + σ_{X}^{2} \\ σ_{0}^{2} = E {{({| y (n) |}^{2} - μ_{0})}^{2} | H_{0}} \\ = E {{| y (n) |}^{4} | H_{0}} - μ_{0}^{2} = \sum_{m = 0}^{\infty} 3 β_{m} σ_{m}^{4} - σ_{Z}^{4} \\ σ_{1}^{2} = E {{({| y (n) |}^{2} - μ_{1})}^{2} | H_{1}} \\ = E {{| y (n) |}^{4} | H_{1}} - μ_{1}^{2} \\ = \sum_{m = 0}^{\infty} 3 β_{m} σ_{m}^{4} - σ_{Z}^{4} + 2 σ_{X}^{4} + 4 σ_{X}^{2} σ_{Z}^{2} \end{matrix}

(6)

Thus, the probability of false alarm $P_{f a_E D}$ and probability of detection $P_{d_E D}$ can be given by

\begin{matrix} P_{f a_E D} = P_{r} (Λ_{E D} (y) > γ_{E D} | H_{0}) = Q (\frac{γ_{E D} - μ_{0}}{\sqrt{σ_{0}^{2} / N}}) \\ P_{d_E D} = P_{r} (Λ_{E D} (y) > γ_{E D} | H_{1}) = Q (\frac{γ_{E D} - μ_{1}}{\sqrt{σ_{1}^{2} / N}}) \end{matrix}

(7)

where $Q (\cdot)$ is the complementary cumulative distribution function (Q-function) defined by $Q (x) = 1 / \sqrt{2 π} \int_{x}^{\infty} e x p - (t^{2} / 2) d t$ , and $γ_{E D}$ is the threshold chosen for judging the presence or absence of the PU signal.

The structure of ED is simple and easy to be established. However, the performance of ED-based sensing is not satisfactory, especially when the noise has high impulsiveness. FLOM-based detector showed its capability of sensing under Middleton Class A noise in a study by Xu et al.¹⁹ By means of fractional power operations, the large amplitudes of impulsive noise will be significantly mitigated, as a result the sensing performance will be improved.

So, second, FLOM-based sensing is considered in this article. The corresponding test statistic of the FLOM-based detector is given in equation (8)

Λ_{F} (y) = \frac{1}{N} \sum_{n = 1}^{N} {| y (n) |}^{p}

(8)

where $0 < p < 2$ is the fractional lower order.

The structure of FLOM-based detector is shown in Figure 1. Similarly, when the number of observed samples $N$ is large enough, the metric $Λ_{F} (y)$ in equation (9) can be approximated as a Gaussian random variable with

\begin{matrix} {Λ_{F} (y) |}_{H_{0}} ~ N (μ_{0, p}, \frac{σ_{0, p}^{2}}{N}) \\ {Λ_{F} (y) |}_{H_{1}} ~ N (μ_{1, p}, \frac{σ_{1, p}^{2}}{N}) \end{matrix}

(9)

in which

\begin{matrix} μ_{0, p} = E {{| y (n) |}^{p} | H_{0}} \\ μ_{1, p} = E {{| y (n) |}^{p} | H_{1}} \\ σ_{0, p}^{2} = E {{({| y (n) |}^{p} - μ_{0, p})}^{2} | H_{0}} = E {{| y (n) |}^{2 p} | H_{0}} - μ_{0, p}^{2} \\ σ_{1, p}^{2} = E {{({| y (n) |}^{p} - μ_{1, p})}^{2} | H_{1}} = E {{| y (n) |}^{2 p} | H_{1}} - μ_{1, p}^{2} \end{matrix}

(10)

Figure 1.

Structure of FLOM-based detector.

Similarly, with the assumption that the noise $z (n)$ and the PU signal $x (n)$ are independent of each other, the probability density function of $y (n)$ is the sum of infinite terms of the Gaussian probability density function, as shown in equation (4). The $p th$ order central absolute moment of $y (n)$ is equal to the sum of $p th$ order central absolute moment of each summation item. Take $μ_{0, p}$ as an example

\begin{matrix} μ_{0, p} = \int_{- \infty}^{\infty} {| y (n) |}^{p} {f_{Y} (y (n)) |}_{H_{0}} d y (n) \\ = \int_{- \infty}^{\infty} {| y (n) |}^{p} \sum_{m = 0}^{\infty} β_{m} G (y (n); σ_{m}^{2}) d y (n) \\ = \sum_{m = 0}^{\infty} β_{m} \int_{- \infty}^{\infty} {| y (n) |}^{p} G (y (n); σ_{m}^{2}) d y (n) \end{matrix}

(11)

Since the $p th$ order central absolute moments of a Gaussian random variable $S ~ N (0, σ^{2})$ can be expressed as

E [{| S |}^{p}] = \frac{2^{p / 2}}{\sqrt{π}} Γ (\frac{p + 1}{2}) σ^{p}

(12)

where $Γ (\cdot)$ is the usual gamma function defined by $Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t$ .

Then the equation (11) can be further simplified as

\begin{matrix} μ_{0, p} = \sum_{m = 0}^{\infty} β_{m} \frac{2^{p / 2}}{\sqrt{π}} Γ (\frac{p + 1}{2}) σ_{m}^{p} \end{matrix}

(13)

So we can obtain that

\begin{matrix} μ_{0, p} = \frac{2^{p / 2}}{\sqrt{π}} Γ (\frac{p + 1}{2}) \sum_{m = 0}^{\infty} β_{m} σ_{m}^{p} \\ μ_{1, p} = \frac{2^{p / 2}}{\sqrt{π}} Γ (\frac{p + 1}{2}) \sum_{m = 0}^{\infty} β_{m} {\sqrt{σ_{m}^{2} + σ_{X}^{2}}}^{p} \\ σ_{0, p}^{2} = \frac{2^{p}}{\sqrt{π}} Γ (\frac{2 p + 1}{2}) \sum_{m = 0}^{\infty} β_{m} σ_{m}^{2 p} - μ_{0, p}^{2} \\ σ_{1, p}^{2} = \frac{2^{p}}{\sqrt{π}} Γ (\frac{2 p + 1}{2}) \sum_{m = 0}^{\infty} β_{m} {(σ_{m}^{2} + σ_{X}^{2})}^{p} - μ_{1, p}^{2} \end{matrix}

(14)

Thereafter, the probability of false alarm $P_{f a_F}$ and probability of detection $P_{d_F}$ can be expressed in terms of the $Q$ function by

\begin{matrix} P_{f a_F} = P_{r} (Λ_{F} (y) > γ_{F} | H_{0}) = Q (\frac{γ_{F} - μ_{0, p}}{\sqrt{σ_{0, p}^{2} / N}}) \\ P_{d_F} = P_{r} (Λ_{F} (y) > γ_{F} | H_{1}) = Q (\frac{γ_{F} - μ_{1, p}}{\sqrt{σ_{1, p}^{2} / N}}) \end{matrix}

(15)

where $γ_{F}$ is the corresponding threshold for judging the presence or absence of the PU signal.

EH

For a practical implemented WSN, the lifetime of the sensor is the longer the better. Spectrum sensing solves the problem of spectrum access for the SU, but it increases the energy consumption compared to the traditional sensors. So we investigate a self-powered SU that has no fixed power and has to extract energy by harvesting the ambient RF energy, then converts the harvested energy to the electrical power to supply itself. Moreover, the SU is assumed to be equipped with only one antenna, due to the hardware limitation. Therefore, at any certain instant, the SU can only perform one of the following three operations: EH, spectrum sensing, or data transmission.²⁰ The time-slots structure of the SU proposed in this article is shown in Figure 2. In each frame with duration $T_{f}$ , a fraction of time $T_{h}$ is devoted exclusively to EH (yellow colored). Another following fraction of time $T_{s}$ is dedicated to spectrum sensing (blue colored). The rest time of the frame $T_{t}$ is used for data transmission (red colored) or EH depending on the result of the spectrum sensing and the remaining transmission energy. Obviously, the equation $T_{f} = T_{h} + T_{s} + T_{t}$ should always hold.

Figure 2.

The time-slots structure of the SU.

As shown in Figure 2, the SU will perform the following 3 steps:

1. EH: During time-slots (0, $T_{h}$ ), an EH circuit gathers energy from ambient radio signals and deposits the energy into a rechargeable battery;

2. Spectrum sensing: During time-slots ( $T_{h}$ , $T_{h} + T_{s}$ ), the EH circuit shuts down and a spectrum sensor is powered by the battery and performs spectrum sensing for searching access opportunities;

3a. EH: If the result of spectrum sensing is that the PU signal is present, the SU still performs EH during time-slots ( $T_{h} + T_{s}$ , $T_{f}$ );

3b. Data transmission: If the result of spectrum sensing is that the PU signal is absent, the transmitter of the SU is powered on for data transmission during time-slots ( $T_{h} + T_{s}$ , $T_{f}$ ).

During EH time-slots, energy is harvested and deposited at a fixed rate $P_{h}$ . So the amount of harvested energy is $P_{h} T_{h}$ . During spectrum-sensing time-slots, the SU operates spectrum sensing to determine the channel status. Usually, the power consumption $P_{s}$ is a fixed value because it is decided by the sensing method. As mentioned above, if the PU signal is sampled at frequency $f_{s}$ , then the number of observed samples $N = f_{s} T_{s}$ . The sensing result is determined by the comparison of the test statistic against detection threshold $γ$ .

From Figure 2, it is obvious that SU has two different operations after EH and spectrum sensing. Depending on whether EH or data transmission is performed after spectrum sensing, we can classify SU’s successive operations into two cases: Case A, SU performs EH, spectrum sensing and data transmmission sequentially; Case B, SU goes through EH, spectrum sensing, and data transmission. The probability of occurrence of different cases is different due to the randomness of the channel status and sensing results. So the probability of each case needs to be further calculated.

Case A: It happens when the result of spectrum sensing is PU signal present. No matter the result is correct or not, the SU will operate EH after spectrum sensing. The probability of Case A can be written as

P_{A} (T_{s}, γ) = P (H_{0}) P_{f a} (T_{s}, γ) + P (H_{1}) P_{d} (T_{s}, γ)

(16)

where $P (H_{0})$ and $P (H_{1})$ denote the probability of PU being really absent and really present, respectively.

Case B: It happens when the result of spectrum-sensing PU signal is absent. No matter whether the result is correct or not, the SU will operate data transmission after spectrum sensing. The probability of Case B can be written as

P_{B} (T_{s}, γ) = P (H_{0}) (1 - P_{f a} (T_{s}, γ)) + P (H_{1}) (1 - P_{d} (T_{s}, γ))

(17)

Obviously, $P_{A} (T_{s}, γ) + P_{B} (T_{s}, γ) = 1$ . However, in order to facilitate the discussion of the average throughput of the SU, Case B should be further divided into two sub-cases.

Case B1: SU correctly detects the absence of PU. The probability can be written as

P_{B 1} (T_{s}, γ) = P (H_{0}) (1 - P_{f a} (T_{s}, γ))

(18)

Case B2: On the other hand, SU wrongly detects the absence of PU. The probability can be written as

P_{B 2} (T_{s}, γ) = P (H_{1}) (1 - P_{d} (T_{s}, γ))

(19)

Throughput analysis and optimization

Throughput analysis

The SU’s throughput may vary with time due to the randomness of channel status. So it is reasonable to analyze the average throughput of SU over a long period of time. In other words, the throughput should be a probability-weighted statistic. Consider that the channel is correctly detected as PU absent at an instant, before which the channel may be detected as PU present for $k$ consecutive times. It means that Case A happens $k$ times before SU goes into Case B1. The probability of this condition can be expressed as

P_{B 1} (k, T_{s}, γ) = P_{A} (T_{s}, γ)^{k} P_{B 1} (T_{s}, γ)

(20)

Similarly, the SU may detect the PU’s presence for $k$ successive times before it wrongly detects the PU’s absence. It means that Case A happens $k$ times before SU goes into Case B2. The probability of this condition can be derived as

P_{B 2} (k, T_{s}, γ) = P_{A} (T_{s}, γ)^{k} P_{B 2} (T_{s}, γ)

(21)

Before data transmission, the energy gathered by harvesting under both conditions is $E_{h} (k, T_{s}, T_{t}) = (k + 1) [P_{h} (T_{f} - T_{s}) - P_{s} T_{s}] - P_{h} T_{t}$ , where $T_{h} = T_{f} - T_{s} - T_{t}$ is substituted. Let $C_{1}$ and $C_{2}$ denote the maximum data rate of the SU when it operates in the absence and presence of the PU, respectively. Then $C_{1} = B l o g_{2} (1 + (E_{h} / T_{t} σ_{n}^{2})), C_{2} = B l o g_{2} (1 + (E_{h} / T_{t} σ_{n}^{2} (1 + γ_{P U})))$ , where $B$ is the bandwidth, $σ_{n}^{2}$ denotes the noise power and $γ_{P U}$ represents the signal-to-noise power ratio (SNR) of the PU signal. Obviously, we have $C_{1} > C_{2}$ . Then, the instantaneous throughput of the SU can be obtained as

\begin{matrix} T h_{1} (k, T_{s}, T_{t}) = \frac{T_{t}}{(k + 1) T_{f}} C_{1} (k, T_{s}, T_{t}) \\ T h_{2} (k, T_{s}, T_{t}) = \frac{T_{t}}{(k + 1) T_{f}} C_{2} (k, T_{s}, T_{t}) \end{matrix}

(22)

By combining equations (20)–(22), the average throughput of the SU for a relatively long time can be expressed as a probability-weighted summation of the instantaneous throughput as follows

\begin{matrix} T h (T_{s}, T_{t}, γ) = \sum_{k = 0}^{\infty} P_{B 1} (k, T_{s}, γ) T h_{1} (k, T_{s}, T_{t}) \\ + P_{B 2} (k, T_{s}, γ) T h_{2} (k, T_{s}, T_{t}) \end{matrix}

(23)

Throughput optimization

In this article, we aim to maximize the average throughput of the SU under some essential constraints. First, the SU should provide sufficient protection for PU. It means that the detection probability of spectrum sensing should be no less than a tolerant threshold. In addition, for the SU is self-powered, the energy harvested should be sufficient to perform spectrum sensing and data transmission. This implies that, the harvested energy should be no less than the energy consumption of spectrum sensing. Considering the worst condition where Case B first happens, that is $k = 0$ , the harvested energy $P_{h} T_{h}$ should be no less than the energy used for spectrum sensing. This can be denoted as follows

\begin{matrix} \underset{T_{s}, T_{t}, γ}{m a x i m i z e} & T h (T_{s}, T_{t}, γ) \\ subject to & P_{d} (T_{s}, γ) ⩾ {\bar{P}}_{d} \\ P_{h} (T_{f} - T_{s} - T_{t}) ⩾ P_{s} T_{s} \end{matrix}

(24)

where ${\bar{P}}_{d}$ is the tolerant detection probability, and $T_{h} = T_{f} - T_{s} - T_{t}$ is substituted.

Assume that ${T_{s}^{*}, T_{t}^{*}, γ^{*}}$ are the optimal values maximizing the average throughput $T h$ with $P_{d} (T_{s}^{*}, γ^{*}) > {\bar{P}}_{d}$ holding. Then there must exist $γ' > γ^{*}$ letting $P_{d} (T_{s}^{*}, γ') = {\bar{P}}_{d}$ , because $P_{d}$ is a monotonously decreasing function with respect to $γ$ . At the same time, $P_{f a}$ will also decrease with $γ$ increasing, that is $P_{f a} (T_{s}^{*}, γ^{*}) > P_{f a} (T_{s}^{*}, γ')$ . It can be concluded that the SU will obtain higher probability to go into Case B from equations (16) and (17). In other words, this means that the SU will have more chances to operate data transmission which results in an increasing average throughput. This conclusion indicates that ${T_{s}^{*}, T_{t}^{*}, γ'}$ is a better set of values than ${T_{s}^{*}, T_{t}^{*}, γ^{*}}$ , which is contrary to the optimal values assumption. Therefore, the maximum average throughput will be achieved only if $P_{d} (T_{s}^{*}, γ^{*}) = {\bar{P}}_{d}$ . Then the optimization problem can be reformed as

\begin{matrix} \underset{T_{s}, T_{t}}{m a x i m i z e} & T h (T_{s}, T_{t}, γ) \\ subject to & P_{d} (T_{s}, γ) = {\bar{P}}_{d} \\ P_{h} (T_{f} - T_{s} - T_{t}) ⩾ P_{s} T_{s} \end{matrix}

(25)

Now consider a certain value of spectrum-sensing duration $T'_{s}$ . A corresponding $γ'$ which makes $P_{d} (T'_{s}, γ') = {\bar{P}}_{d}$ can be obtained. Substituting ${T'_{s}, γ'}$ into $T h (T_{s}, T_{t}, γ)$ , $T h (T'_{s}, T_{t}, γ')$ becomes a function of $T_{t}$ . By taking the first derivation of $T h (T'_{s}, T_{t}, γ')$ with respect to $T_{t}$ , we found that $T h (T_{s'}, T_{t}, γ')$ is a convex function of $T_{t}$ . The certain value $T_{t}^{p}$ can be easily obtained, which makes $T h (T_{s'}, T_{t}^{p}, γ')$ reach its peak value. Moreover, considering the second constraint of equation (25), $T_{t}$ has its maximum value of $T_{t}^{m a x} = T_{f} - T_{s'} - (P_{s} / P_{h}) T_{s'}$ . Hence, for the certain spectrum-sensing duration $T'_{s}$ , the optimal duration of data transmission $T_{t'}$ can be expressed as

T'_{t} = {\begin{matrix} T_{t}^{p}, & i f & T_{t}^{p} < T_{t}^{m a x} \\ T_{t}^{m a x}, & i f & T_{t}^{p} ⩾ T_{t}^{m a x} \end{matrix}

(26)

Then the maximum average throughput with ${T'_{s}, γ'}$ can be obtained as

T h^{*} (T'_{s}, T_{t}, γ') = {\begin{matrix} T h (T'_{s}, T_{t}^{p}, γ'), & i f & T_{t}^{p} < T_{t}^{m a x} \\ T h (T'_{s}, T_{t}^{m a x}, γ'), & i f & T_{t}^{p} ⩾ T_{t}^{m a x} \end{matrix}

(27)

The same as $T_{t}$ , $T_{s}$ also has its maximum value according to the second constraint of equation (25) and $T_{t} ⩾ 0$ . So we can obtain that $T_{s}^{m a x} = (P_{h} / (P_{h} + P_{s})) T_{f}$ . Finally, through varying $T'_{s}$ within $(0, T_{s}^{m a x})$ and computing the optimal ${T'_{t}, γ'}$ correspondingly, we can eventually find the optimal values ${T_{s}^{*}, T_{t}^{*}, γ^{*}}$ maximizing the average throughput $T h^{*}$ .

It is worth mentioning that, the SU’s average throughput $T h (T_{s}, T_{t}, γ)$ is also a convex function of $T_{s}$ which will be validated in the following section. So searching for the optimal value of $T_{s}^{*}$ and the corresponding $T_{t}^{*}, γ^{*}$ can be obtained by a linear searching algorithm as follows.

Algorithm 1:

Searching algorithm for optimal $T_{s}^{*}$

Initialize:

N_{m i n} = 1

and

N_{m a x} = T_{s}^{m a x} f_{s}

;
1: while

N_{m a x} - N_{m i n} > 1

do
2:

N = f l o o r ((N_{m i n} + N_{m a x}) / 2)

N^{n e x t} = N + 1

;
3:

T_{s} = N / f_{s}

T_{s}^{n e x t} = N^{n e x t} / f_{s}

;
4: Compute the SU’s maximum average throughput

T h^{*} (T_{s})

and

T h^{*} (T_{s}^{n e x t})

based on equation (27);
5: if

T h^{*} (T_{s}) ⩽ T h^{*} (T_{s}^{n e x t})

then
6: set

N_{m i n} = N

N^{*} = N^{n e x t}

;
7: else
8: set

N_{m a x} = N^{n e x t}

N^{*} = N

;
9: end if
10: end while
Output: The optimal

T_{s}^{*} = N^{*} / f_{s}

Simulation and results

Spectrum-sensing performance analysis of FLOM-based detector

In this section, we give the simulations of the sensing performance based on FLOM detector and show how it outperforms ED with different order $p$ . Set ${\bar{P}}_{f a} = 0.1$ as the target probability of false alarm; Figure 3 shows how $P_{d}$ changes with $p$ varying from 0.1 to 2 at $γ_{P U} = - 15 dB$ and $N = 1000$ .

Figure 3.

Probability of detection $P_{d}$ versus $p$ for different $A$ and $T$ with target probability of false alarm ${\bar{P}}_{f a} = 0.1$ , $N = 1000$ at $γ_{P U} = - 15 dB$ .

From Figure 3, it can be seen that traditional ED-based sensing $(p = 2)$ has very poor performance $(P_{d} < 0.2)$ for all the values of $A$ and $T$ at $γ_{P U} = - 15 dB$ . However, with the order $p$ getting smaller, the sensing performance shows large improvement. Taking $p = 0.4$ as an example, a satisfactory performance can be achieved when the noise is highly impulsive $(T ⩽ 0.1)$ . Nevertheless, the performance improvement is not considerable for the noise close to Gaussian noise $(T = 1)$ . In addition, the parameter $A$ has only slight effect on the sensing performance compared to $T$ . Therefore, the results indicate that the improvement of sensing performance depends on the noise impulsiveness besides the order $p$ , and the FLOM-based detector is much more suitable than ED for the circumstances where the noise has high impulsiveness.

Average throughput of SU versus time-slots allocation

For evaluating the impact of the time-slots allocation on the SU’s average throughput and verifying the feasibility of the optimization method, the SU’s average throughput varying with sensing duration and transmission duration is simulated. To simulate the typical impulsive noise environment, the noise parameters are set as $A = 0.1, T = 0.01, σ_{n}^{2} = - 20 dBm$ . The PU signal detected by the SU has low SNR, that is, $γ_{P U} = - 20 dB$ . As for the parameter of FLOM-based detector, we set $p = 1$ to reduce the complexity of the detector, thereby to reduce the energy consumption for spectrum sensing. The tolerant detection probability is set as ${\bar{P}}_{d} = 0.9$ according to 802.22 standard.²¹ The sampling frequency and bandwidth are both set as $1 MHz$ . Assume that the PU signal has high probability of absence that $P (H_{0}) = 0.7$ . On the other hand, the presence probability of PU signal $P (H_{1}) = 0.3$ . The EH rate $P_{h} = - 20 dBm$ and the power consumption of spectrum sensing is also $P_{s} = - 20 dBm$ . The frame length is set as $5 ms$ . All the simulation parameters are shown in Table 1.

Table 1.

Simulation parameters and values.

Description	Parameters	Values
Overlap index	$A$	$0.1$
Power ratio	$T$	$0.01$
Noise power	$σ_{n}^{2}$	$- 20 dBm$
SNR of PU signal	$γ_{P U}$	$- 20 dB$
Fractional lower order	$p$	1
Tolerant detection probability	${\bar{P}}_{d}$	$0.9$
Sampling frequency	$f_{s}$	$1 MHz$
Bandwidth	$B$	$1 MHz$
PU absent probability	$P (H_{0})$	$0.7$
PU present probability	$P (H_{1})$	$0.3$
Energy-harvesting rate	$P_{h}$	$- 20 dBm$
Spectrum-sensing power	$P_{s}$	$- 20 dBm$
Frame length	$T_{f}$	$5 ms$

SNR: signal-to-noise power ratio; PU: primary user.

The simulation results with the parameters aforementioned are shown in Figure 4. Several conclusions can be drawn from observing Figure 4. First, the SU’s maximum average throughput will be obtained with an optimal pair of values of spectrum-sensing duration and data transmission duration. In this scenario, the optimal sensing duration and data transmission duration are $0.27 ms$ and $2.95 ms$ , respectively. Therefore, the corresponding EH duration is $1.78 ms$ , and the SU’s maximum average throughput is about $321.2 kbps$ . Second, the SU’s average throughput $T h (T_{s}, T_{t}, γ)$ is also a convex function of $T_{s}$ . So the optimal values of $T_{s}^{*}$ and $T_{t}^{*}$ can be easily found by aforementioned searching method in section “Throughput analysis and optimization.”

Figure 4.

The SU’s average throughput versus sensing duration and data transmission duration.

For comparing the performance of FLOM-based sensing and ED-based sensing, Figure 5 shows the corresponding SU’s maximum average throughput with the sensing duration $T_{s}$ varying within $(0, 0.5 ms)$ at (a) $γ_{P U} = - 20 dB$ and (b) $γ_{P U} = - 10 dB$ . It can be concluded that, the FLOM-based sensing outperforms ED-based sensing under both conditions. When $γ_{P U} = - 10 dB$ , the SU’s maximum average throughput of both sensing methods are convex functions of $T_{s}$ , shown in Figure 5(b). However, the SU’s maximum average throughput of ED-based sensing is monotonically decreasing over $T_{s}$ when $γ_{P U} = - 20 dB$ , shown in Figure 5(a). This indicates that the sensing result of ED is not reliable when the SNR of PU signal is sufficiently low and ED-based sensing can not be used as a robust method any more.

Figure 5.

The SU’s maximum average throughput versus sensing duration of FLOM-based sensing and ED-based sensing at (a) $γ_{P U} = - 20 dB$ and (b) $γ_{P U} = - 10 dB$ , respectively.

Impact of the EH rate and the PU signal’s SNR

Now we consider the impact of the EH rate on the time-slots allocation and the SU’s maximum average throughput for both sensing methods at $γ_{P U} = - 20 dB$ , which is shown in Figure 6. It can be seen that, with the EH rate $P_{h}$ getting higher, the optimal sensing duration and data transmission duration are both increasing. As a result, the EH duration decreases. This indicates that more time-slots within the frame are allocated to improve sensing accuracy and data transmission time. Intuitively, the SU’s maximum average throughput will increase with $P_{h}$ getting high. As shown in Figure 6(b), the SU’s maximum average throughput increases almost three times with $P_{h}$ changing from $- 20 dBm$ to $- 10 dBm$ . Furthermore, although more time-slots are allocated for data transmission in ED than that of FLOM detector, the SU’s maximum average throughput of ED is still lower than that of FLOM detector.

Figure 6.

(a) The optimal sensing duration $T_{s_o p t}$ and data transmission duration $T_{t_o p t}$ versus the energy-harvesting rate $P_{h}$ and (b) the SU’s maximum average throughput versus the energy-harvesting rate $P_{h}$ .

Next, the effect of PU signal’s SNR on the SU’s maximum average throughput and time-slots allocation is investigated. At a fixed EH rate $P_{h} = - 20 dBm$ , Figure 7 shows the variation of the SU’s maximum average throughput, the optimal sensing duration and data transmission duration with the PU signal’s SNR varying from $- 20$ to $10 dB$ . As shown in Figure 7(a), the optimal data transmission duration is decreasing with PU signal’s SNR getting higher. However, the trend of the optimal sensing duration is different which is caused by the following reasons. First, the sensing result will be no longer reliable with $γ_{P U}$ getting too low. Second, the sensing accuracy will be improved with $γ_{P U}$ getting high. So for both conditions, the sensing duration should decrease which results in the nonmonotonous of the optimal sensing duration. From Figure 7(b), it can be seen that, the SU’s maximum average throughput will increase even though the data transmission duration is getting shorter with the increase of $γ_{P U}$ . It is not difficult to analyze the reason of this result. The SU will obtain more opportunities to perform data transmission with the false alarm reducing, which is caused by the increasing sensing accuracy at higher PU signal’s SNR. In addition, all the parameters of both sensing method trend to be the same and be stable at $γ_{P U} = 10 dB$ . This indicates that the sensing performance of both methods trend to be the same at high PU signal’s SNR.

Figure 7.

(a) The optimal sensing duration $T_{s_o p t}$ and data transmission duration $T_{t_o p t}$ versus the PU signal’s SNR and (b) the SU’s maximum average throughput versus the PU signal’s SNR.

Conclusion

Spectrum scarcity and the short sensor lifetime are two urgent problems needed to be overcome for establishing WSNs. The two technologies of CR and EH can help WSNs solve these problems. By dividing the frame of SU into three segments, naming EH duration, spectrum-sensing duration, and data transmission duration, the SU can be self-powered and access unlicensed spectrum bands. A maximum SU’s average throughput can be obtained by a joint optimization of the three durations under the constraint of protecting PU’s communication. In addition, by employing the FLOM detector, spectrum sensing can work reliably under Middleton Class A noise, and the maximum SU’s average throughput will be much higher than using ED-based sensing at low PU signal’s SNR.

Footnotes

Handling Editor: Ning Zhang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Enwei Xu

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