Energy-Efficient Sensing and Transmission for Multi-Hop Relay Cognitive Radio Sensor Networks

Bin Han , Min Zeng, Qiumei Guo,Hong Jiang,＊, Qiuyun Zhang, Li Feng

1 School of Information Engineering, Southwest University of Science and Technology, Mianyang 621010, China

2 Engineering and Technology College, Sichuan Radio and TV University, Chengdu 610073, China

Abstract: How to achieve transmissions in an energy-efficient way in multi-hop decode and forward (DF) relay cognitive radio sensor networks (CRSNs) is important since sensor nodes in CRSNs are usually battery powered.This paper aims to maximize energy efficiency (EE) by joint optimizing sensing time and power allocation in multi-channels & multihops DF relay CRSNs under constraints on outage probability and sensing performance.First, we design a channel selection scheme for sensing according to the available probabilities of multi channels. Second, we analyze the expected throughput and energy consumption and formulate the EE problem as a concave/concave fractional program. Third, coordinate ascent and Charnes-Cooper Transformation(CCT) methods are used to transform the nonlinear fractional problem into an equivalent concave problem. Subsequently, the closed form of outage probability is derived and the convergence rate of the iterative algorithm is analyzed. Finally, simulation results show that the proposed scheme can achieve effective EE.

Keywords: energy efficiency; cognitive radio sensor network; outage probability; fractional program

I. INTRODUCTION

Energy efficiency (EE) is defined as the ratio of the average transmitted bits to the average consumed energy[1], which plays an important role in wireless communication and has been a focus in the literature[2][3]. Further, EE is a decisive factor for the lifetime of wireless sensor networks[4][5] since it may be impossible to recharge battery deployed in wireless sensor nodes.

Recently, cognitive radio networks(CRNs)receive considerable attentions to alleviate the contradiction between the scarcity and low utilization of spectrum resources. However,high energy consumption is one of the main challenges encountered by CRNs, which may limit their implementation especially in battery-powered secondary terminals. In order to increase the EE in CRNs, some research attention has been devoted to energy reduction in CRNs[6][7][8]. One example of this EE design is provided in [9], where the dynamically sensing and transmission are formulated as a partially observable Markov decision process and resolved by myopic policy. Another example is the joint multi-channel assignment and power control for EE in CRNs [10], where the EE is formulated as a nonlinear integer programming and is solved by a polynomial time heuristic algorithm. Similarly, [11] formulates a multi-objective optimization problem that jointly maximizes the ergodic capacity and minimizes the average transmission power of the secondary network, and the problem is solved by transferred into a single objective problem with ε-constraint method. Further,[12] proposes a general framework to evaluate the tradeoff between EE and SE in CRNs, and infers the EE-SE relation in the closed-form formulation.

However, the above EE optimizations in CRNs mainly are restricted to one-hop transmissions, which is unsuitable for many practical applications. As a consequence, EE optimization in multi-hop transmissions emerges as a very challenging issue due to multi-hop parameters’ control. For example, [13] describes the application of cooperative relays for both the spectrum sensing and secondary transmissions, and simulation results show that there exists a trade-off design of spectrum sensing with cooperative relays and secondary transmissions. While [14] proposes a distributed heuristic solution to address the EE broadcast in multihop CRNs and verifies the results based on MICAZ energy consumption model.Further, a two-level Stackelberg game model is presented in [15] and Nash equilibrium is reached by jointly considering the power allocation and spectrum price for multihop CRNs.

Due to the poor ability of sensor nodes,the EE optimization in CRSN scenario becomes even more important and challenging.In recent years, some research efforts have been made to improve the EE of CRSN. For example, the authors in [16] consider the EE in a cooperative sensing manner in CRSN, and an iterative algorithm is designed to obtain efficient sensing time and the length of the modulated symbol sequence. Ren et al. [17]investigate the dynamic channel access to improve EE for a clustered CRSN. By considering the energy consumption in channel sensing and switching, two dynamic channel accessing schemes are proposed for intra-cluster and inter-cluster transmission. In [18], an energy efficient handoff strategy is developed by using the partially observable Markov decision process to maximize throughput in each time slot. The authors in [19] propose a network cluster architecture, and the energy consumption and the end-to-end delay are reduced by dividing each cluster into disjoint subsets with overlapped sensing coverage of PU activity.In [20], a ε-optimal iterative algorithm is proposed to solve the nonlinear fractional problem in a form of concave/affine with affine constraints in underlay CRSN.

We handle the maximization problem of energy efficiency for overlay multi-hop relay CRSNs in this paper.

In summary, the previous works discussed above address important aspects of CRSNs.However, joint optimization of sensing time and energy allocation for maximization of EE has not been investigated thoroughly in overlay multi-hop CRSNs. In this paper, we formulate the energy-efficiency design in the form of throughput/energy as a concave/concave program. We propose joint sensing time and energy allocation scheme to maximize the EE for secondary network. The main contributions of this work are as follows.

1. We analyze the expected throughput and energy consumption and formulate the EE problem with respect to both sensing time and power allocation as a concave/concave fractional program by coordinate ascent method.

2. We transform the concave/concave problem into an equivalent concave problem by Charnes-Cooper Transformation (CCT).

3. We derive the closed form of outage probability, analyze the convergence rate of the proposed algorithm, and evaluate the effectiveness of the scheme by simulations.

Our proposed scheme seems to be similar to [20] with respect to transforming nonlinear fractional program by CCT. However, it is entirely different in two aspects. First, our scheme converts concave/concave nonlinear fractional program with convex constraints rather than concave/affine under affine constraints. Second, we maximize EE for multihop DF overlay CRSN scenario instead of underlay central one.

The rest of the paper is organized as follows. The system model is presented in Section 2. Sensing time and power allocation strategy and the algorithm are given in Section 3. Convergence analysis and simulation results discussion are shown in Section 4. Finally,conclusions are drawn in Section 5.

II. SYSTEM MODEL

Let Ht,1,Ht,0denote a channel being busy and idle, andbeing sensed as busy and idle at time slot t, respectively. For a spectrum sensing threshold ϕ, detection duration τ, and signal-to-noise ratio γ, probabilities of false alarm and detection for a channel spectrum sensing based on energy detection can be denoted as[21]

Fig. 1. The model of relay and time slot structure.

As seen in figure 1, we assume the multihop DF relay CRSN in question consists of a pair of source-destination and several relays in the same PU station coverage. Secondary transmission uses decoded and forwarded(DF) mode in a time slot manner through L hops, and we assume that the PU state remains unchanged during the total interval of T [22].Specifically, we divide each time slot into two portions respectively for sensing with length τ∈R+and transmitting comprising L subtime slots each with length (T −τ)/L, as shown in figure 1. If a channel is sensed idle and the received data from previous hop are decoded correctly, the ith hop forwards with power pi∈ R+, i ∈ [1,...,L] in corresponding sub-time slot. The communication mechanism is described as follows:

Step 1: Sensing phase: As shown in figure 1, the source secondary node S performs spectrum sensing to acquire the information of the vacant spectrum on licensed spectrum bands of PUs during the τ interval. If the channel is available, before transmission, the previous hop node transfers the free channel information through the out-band control channel to next hop node within the interval τ' of (τ' is transmission time of channel sensing results,because the amount of data is very small, the transmission time and corresponding power consumption can be ignored [8] [22] [33] .) . For convenience of illustration, we still assume that T −τ interval is used for L hop transmission if the channel is available.

Step 2: Transmission phase: After the received data is successfully decoded from previous node, relay node Ri, i∈[1,…,L] forwards the information to the next relay node by using the DF protocol during the (T −τ)/L subtime slot.

Without loss of generality, we assume there exist K Rayleigh block-fading channels, such that a channel condition remains unchanged during the transmission of one data block. In other words, the instantaneous channel power gain for the ith hop is independent identically distributed (i.i.d) exponential random variables Gi,i+1with mean λi,i+1.

The source secondary node senses the channels according to the descending order of their available probabilities, ϑ(k), k∈[1,…,K]. The state transition probability of a channel k from BUSY (IDLE) to IDLE (BUSY) is denoted by βk( αk), which are updated from a previous period of time measurements. Then, the average idle and busy probabilities are as follows:

Let, in a unit time, psebe the energy consumption of sensing, and pcbe the circuit energy consumption of node in the active mode. In addition, transmission power vector is denoted by P∈, and optimization variable vector is expressed by X=(τ,pi)∈ RL+1. The sensing-transmitting structure makes the SUs relay secondary data in a multi-hop manner in the current timeslot. Specifically, taking an action will lead to one of the following four possible consequences[23]:

(1) Presence detection with probability: the sensed channel is busy while the sensing result is correct. As a consequence, the transmission rate is zero, and the energy consumption is τpse+(L +1)T pc.

(2) Miss detection with probability(1 − ϑt(k ))(1 −Pd,k(τ)): the sensed channel is busy while the sensing result is incorrect. Although transmission occurs in the first sub-transmission-slot, collisions will result in zero rate and energy consumption τpse+ (L + 1)T pc+ p1( T −τ) /ςL , where p1is transmission power allocated to first hop node,and ς is the power amplifier efficiency which is assumed to be identical for all nodes without loss of generality[24].

(3) False alarm with probability, the sensed channel is idle while the sensing result is incorrect. There is no transmission with energy consumption τpse+(L +1)T pc.

(4) Absence detection with probability ϑt(k )(1 −Pf,k(τ)): the sensed channel is idle while the sensing result is correct. Transmission will be successful with throughputand energy consumption, where E[R] is expected data rate for L-hop transmission.

Therefore, based on above four cases, expected energy consumption can be expressed as

Combining (5) and (6), we can express the source-to-destination energy efficiency in information bits per Joule as

where Pout(X) denotes the outage probability when the expected throughput of secondary source-destination pair falls below a specified threshold Rth, Rminis the upper threshold probability of outage, anddenotes detection probability is above a given threshold.Similar to [25], we can obtain the optimal τ and the optimal piby optimizing OP1.

III. SENSING AND TRANSMISSION STRATEGY IN A SLOT

It is dif ficult to obtain a closed form for OP1with low complex algorithm due to the complexity of OP1. This paper uses coordinate ascent to solve OP1in (7) and (8). Specifically,OP1is decomposed into two sub-problems,namely, power allocation OP2with given τ and sensing time selection OP3for a given power allocation.

Before solving OP2and OP3, we provide the following lemmas.

Lemma 1. When energy efficiency in OP2on P is maximized, power allocation should satisfy the following equation:

Proof. When the energy efficiency reaches its maximization in OP2, we assume there still exists pk, and satisfies. Then, pkcan be decreased to getCk=( P). Since, when numerator is unchanged, reducing pkwill reduce, which results in increasing ξ（2P).This contradicts the maximization result.

Lemma 2. The constraint Pout(X)≤ Rminmeans that there exist Xs satisfying ψ (τ, P) =ws1+(ws2(τ)−1)ws3(P) ≤0, where ψ(P) and ψ τ() are convex with respect to P and τ, respectively.

Proof. According to Rayleigh block-fading channel assumption and Lemma 1, outage probability constraint in (8) is derived by

where ws1=ln(1− Rmin),. Rthand Rmindenote a minimum rate threshold and an outage probability requirement, respectively.

For a given τ, ψ(P) is obviously convex for P≻=0.

For a given p, since 1/(T−)τ is an increasing convex function and 1/(1 −Pf,k(τ))is a decreasing convex function on τ, thenis convex[26],which leads to ws2()τ, and then ψ τ(), being convex on τ[28].

For OP2,due to the concavity of min(log)function,(P) is concave, namely, Th2(P)is concave on P. In addition,is affine on P from (5). Based on Charnes-Cooper Transformation (CCT) [26][27], let, then (9) can be transformed into

Since Th2(P) is concave, OP4is an equivalent concave program of problem OP2[26][27]. Furthermore, If (y*, v*) is an optimal solution of OP4, then y*/v* is an optimal solution of OP2[26][27]. In addition, Lemma 2 shows when τ is fixed, ψ(P) in (11) is a decreasing convex function on P(≻=0), which meansfor ψ(P)≤ 0. In conclusion,we obtain the optimal power allocation in OP4as follows:

where ζ4(y*, v*) denotes optimal solution for the concave problem (14).

For OP3, by the convexity of ψ (τ)from lemma 2, we know that,i s a i n c r e a s i n g f u n c t i o n, w h e r e. Obviously,when; and τ→T−,, then there exists a pointfor τ＜τout,for τ=τout, andfor τ＞τout.From (13), ψ (τout)＜ 0, there is a point τ+, ψ(τ+) = 0, for τout＜ τ＜T−. Similarly,there exists a point τ−during 0＜τ＜τout,ψ (τ−) = 0, for ψ(0)＞0, and τ−=0 for ψ(0)＜0.

Next, we discuss the concavity and convexity of ξ（3τ）. The first derivative of Th3(τ)with respect to τ is given byand the second-order derivative of Th3()τ can be further derived byObviously,, thenwhich means Th3(τ) is convex. For purposes of easy computation we let Pd(τ)=,then, the first and second derivative of q(X) in terms of τ are respectively derived by

In addition, e1inis affine. Sois concave, which means that ξ（3τ） in (11) is a fractional programming in the form of concave/concave.

?

where ζ5(x*, t*) denotes optimal solution for the concave problem (17), and ψ−1the inverse function of ψ.

In summary, the pseudo-code for this scheme is presented in Algorithm 1.

In algorithm 1, after choosing a channel for sensing, the secondary source iteratively finds the optimal policy given a component in X.Specifically, an optimal p＊ is found for a given τ and then an optimal τ*is obtained for a given p. If the |ξ（ Xi）-ξ（ Xi+!）|≤ ε holds,the algorithm ends since the convergence is reached.

IV. CONVERGENCE ANALYSIS AND SIMULATION DISCUSSION

4.1 convergence analysis of coordinate ascent

Since the original problem OP1is partitioned into two equivalent concave sub-problems by coordinate ascent and CCT methods in Section 3, we concentrate on analyzing the convergence of coordinate ascent.

For OP2,

obviously, under the constraint of (10),subjects to an upper bound Lp,then ζ(P) has component Lipschitz constant with respect to p, i.e,

For OP3,

We d e f i n e t h e “g l o b a l” L i ps c h i t z c o n s t a n t b y Lτp. T h a t i s, where Lτp≤ Lτ+ LPdenotes Lipschitz constant[30][32].

Let Lmax=max(LP,Lτ) and

By the Lemma 1, we know that all vector components of P other than p1can be expressed by p1, so only two components,namely τ and p1, need to be computed in the convergence analysis. Furthermore, we use cyclic variants [31] to perform iterations, and at the kth iteration, we have the following [32]

4.2 simulation results and discussions

In this sub-section, we present simulation results to indicate the effective performance of the proposed scheme. The simulation parameters are summarized in table 1. Further, the distance between the ith hop and (i+1)th hop(i=1, 2, 3) is approximately equals to 1000/L(m). Figure 2 and fig.3 show the relationship between system EE of SUs , transmission power p and the sensing time τ, respectively.Figure 4 to figure 6 show the performance metrics when choosing maximum available probability ϑ(k)=0.75. In addition, in order to illustrate the effectiveness of the proposed scheme, we assume that there exists an ideal scheme in which the channel to be sensed isalways idle. Figure 7 shows the EE of the ideal scheme, while figure 8 illustrates the EE by randomly choosing a channel to be sensed.

Table I. Simulation parameters.

Fig. 2. System EE of SUs versus τ.

Fig. 3. Transmission power p versus τ.

Fig. 4. Expected throughput versus Rth.

Fig. 5. Expected energy consumption versus Rth.

Fig. 6. EE sensing channel with the idlest probability.

In figure 2, the energy efficiency versus the sensing time is compared among different Rminand Pmax. Firstly, we observed that at the small values of sensing time, the value of EE is almost fixed and relatively low, which is due to the fact that there is no data transmission at the small sensing times. Second, under the same Rmin, the EE of the SUs increases with the increasing of the sensing time, and the greater the Pmax, the longer the sensing time is allowed. From figure 2, we can also see that the larger the Rminis, the longer the sensing time can be allowed under the same Pmax. This is because the accuracy of sensing is decreasing with the increase of Rmin.

Figure 3 shows the relationship between sensing time and transmission power when different sensing times are given and the maximum value of the EE is taken. We observe a relatively low power consumption at small given sensing times. As the given τ increases,the total transmission power also gradually increases until it reaches or exceeds a given threshold, then, becomes zero. In addition,under the same Rmin, the transmission power of the SUs increases with the increase of Pmax; the greater the Rminis, the longer the sensing time is allowed under the same Pmax.

Figure 4 illustrates the expected throughput versus the minimum rate requirement Rthfrom three aspects: First, we observe that each expected throughput curve consists of two parts. The first one is almost fixed, while the second one increases with the minimum rate requirement. This is due to the fact that when the rate requirement is low, very low energy is needed to relay little data, which results in almost fixed throughput. On the other hand,when the rate requirement becomes high,more energy is needed to transmit data, which results in high expected throughput. Second,each curve indicates that there exists an upper Rthbeyond which the throughput becomes zero due to lack of available solution. Finally, it is observed that the higher Rmin, the lower the expected throughput. This can be explained by the fact that a larger Rminimplies that the network can suffer from more outage, which leads to lower energy and thus lower throughput. In addition, we also observe that increasing the Pmaxwill increase the throughput under a given upper value.

Figure 5 shows the effect of Rthon the expected energy consumption. In general, the energy consumption increases with increase in Rthand dramatically decreases to near zero when Rthgoes beyond some upper value. This is due to the fact that increase in the Rthincreases throughput which leads to more energy consumption. Obviously, when Rthexceeds an upper value, no available solutions exist which means near-zero energy consumption.In addition, we observe the higher in Rmin, the lower energy consumption. The reason is that higher Rmincan tolerate more outage which is equivalent to lower energy consumption.

From figure 6, we can observe that Rthand Rminhave direct impacts on maximization of EE. When Rthincrease, EE will decrease. The reason for this decrease is that although higher Rthwill lead to increases in both throughput and energy consumption, the consumption increases more than throughput, which is also supported by figure 4 and figure 5. Additionally, when Rminis low, more energy is needed to ensure the throughput, which results in a more serious decrease in EE.

Under different Rminand Pmax, Figure 6 and figure 7 show a small gap in EE between the proposed scheme and the ideal one. We observe that the proposed scheme reduces its average EE by around 10% compared to the ideal scheme.

In figure 8, the maximization EE results are computed by averaging the results of 20 simulation runs, and each run randomly chooses a channel from the available channel list. The EE values in figure 8, compared to figure 6 become lower for all Rthand decrease obviously with the increasing of Rth. The reason is straight forward; that is, the available probability of a randomly chosen channel will be less than or equal to the maximum available probability, which results in a smaller throughput for a given energy consumption.

Fig. 7. EE of the ideal scheme.

Fig. 8. Average EE sensing randomly chosen channel.

V. CONCLUSIONS

We handle the maximization problem of energy efficiency for overlay multi-hop relay CRSNs in this paper. We formulate the problem as a concave/concave fractional one by analyzing the expected throughput and energy consumption, and convert the problem into an equivalent concave one by using coordinate ascent and CCT methods. We derive the closed form of outage probability for the scenario and analyze the convergence performance of the iterative algorithm. The performance of the scheme is verified by simulation results.

ACKNOWLEDGEMENTS

This work was mainly supported by the National Nature Science Foundation of China.(Grant No. 61771410)