# Optimal adaptive multiband spectrum sensing in cognitive radio networks.

1. IntroductionThe explosive increase in the wireless service demand has made radio spectrum scarcity a serious problem. Cognitive radio has been proposed to improve the spectrum utilization by allowing secondary users (SUs) to opportunistically access the vacant frequency bands [1]. Since a cognitive radio network is designed to be aware of its surroundings, it is necessary for SUs to sense whether primary users (PUs) are active or not. Spectrum sensing is a critical task for cognitive radio system mainly due to noise, channel fading and shadowing [2][3][4].

Recently, sequential spectrum sensing strategy has received growing attention [5][6], in which the SUs sequentially sense the channels according to a pre-defined order and stop to sense when specific criterions are met. Sequential sensing strategy only senses one channel at a time. However, when the bandwidth is wide and the channel occupancy rate is high, the sequential sensing strategy needs a large amount of sensing resources causing the sensing efficiency to be greatly reduced. An adaptive multiband sensing approach has been proposed for rapidly identifying multiple spectrum holes in wideband cognitive radio network. The sensing method consists of two phases, a exploration phase and a detection phase [7]. In the exploration phase, the size of candidate idle channel set is reduced by excluding channels that are likely to be occupied by the PUs. In the detection phase, the final detection is performed to determine the idle channels. The sensing samples are distributed to focus the limited sensing resources on the more promising channels. This adaptive sensing strategy provides a significant performance gain under the scenario that holes are sparsely scattered across the wideband spectrum. However, the exploration phase in the adaptive sensing approach lengthens the sensing time, which accordingly decreases the transmission time. Consequently, there exists a sensing-throughput tradeoff in sensing time setting and transmission time setting. In [8], the optimal tradeoff is investigated so as to optimally utilize the transmission opportunities in a single channel for a single SU. In [9]-[12], the optimal tradeoff of cooperative sensing with multiple SUs are studied in a single channel.

In this paper, we optimize both the overall sensing time and the sensing time for every stage in the exploration and detection phases for multiband spectrum sensing. An optimization problem is formulated to maximize the average achievable throughput by jointly designing the overall sensing time, and the sensing time for every stage in the exploration and detection phases, while keeping the miss detection probability for each channel under a pre-defined threshold. The initial non-convex optimization problem is further transformed into a convex bilevel optimization problem to make it mathematically tractable. Simulations show that compared with the existing studies, the proposed sensing time setting can provide a significant throughput performance gain.

The rest of this paper is organized as follows. In Section II, the system model is given. In Section III, we formulate the optimization problem and prove that the problem is a convex bilevel problem. In Section V, we provide simulation results to demonstrate the performance. Finally, conclusions are drawn in Section VI.

2. System Model

A cognitive radio system with N channels is considered. Each channel has a bandwidth of W. Time is divided into slots, each with a fixed length T. In each slot, the primary user in the channel is assumed to be either active or idle for the whole slot. The channels among primary and secondary users are assumed to remain unchanged within each slot.

For channel n, we have the following hypotheses:

[H.sup.0.sub.n] : [y.sub.n](m) = [w.sub.n](m),

[H.sup.1.sub.n] : [y.sub.n](m) = [s.sub.n](m) + [w.sub.n](m),

n = 1, 2, ..., N. (1)

where [H.sup.0.sub.n] and [H.sup.1.sub.n] mean that the primary user in channel n is idle and busy respectively, m is the sample index, y(*) is the received signal of channel n at the secondary user. [w.sub.n](*) is the additive white Gaussian noise with variance [[sigma].sup.2.sub.w] corresponding to the channel n. [s.sub.n](x) is PU's signal sample at the channel n and is assumed to be independent and identically distributed with variance [([[sigma].sup.n.sub.s]).sup.2]. Denote [[gamma].sub.n] = [([[sigma].sup.n.sub.s]).sup.2]/[[sigma].sup.2.sub.w] as the received signal-to-noise (SNR) in channel n.

In the adaptive multiband spectrum sensing method, each slot includes two phases: a multi-stage exploration phase and a detection phase [7]. The exploration phase is essentially a coarse sensing process, and it consists of K iteration stages. In each stage, the size of candidate idle channel set is reduced by excluding channels that are likely to be occupied by the PU. We denote [I.sub.k] as the set of surviving channels at the end of k-th stage, then after the k-th stage, the number of remaining channels is [N.sub.k] = [absolute value of [I.sub.k]] = N[[alpha].sup.k], where 0 < [alpha] < 1 is called the distillation ratio, which is the percentage of channels that survive at each stage. Fig. 1 shows the adaptive multiband spectrum sensing structure.

The statistic of the received energy in channel n at the k-th stage is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [mu] is the sampling rate of the received signal, [[tau].sub.k] is the sampling time for per channel at the k-th stage. (2) is equivalent to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [T.sub.k] = [[summation].sup.k.sub.i=1] [[tau].sub.i]. Sort the [N.sub.k-1] = [absolute value of [I.sub.k-1]] energy values {[[GAMMA].sup.k.sub.n], n [member of] [I.sub.k-1]} in ascending order, to obtain [N.sub.k] channels with the smallest energy values as the new set of surviving channels [I.sub.k]. Although the new set is obtained by comparing the energy values between the [N.sub.k-1] channels, the decision process at the k-th stage can be seen as comparing the statistic in (2) with a virtual threshold [[lambda].sup.k.sub.n]. The probability distribution function (pdf) of [[GAMMA].sup.k.sub.n] in channel n at the k-th stage is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where N denotes Gaussian distribution. So, the false alarm probability (i.e., the probability that, under hypothesis [H.sup.0.sub.n], the SU falsely declares that the primary signal is active) of channel n at the k-th stage is given as

[P.sup.fa.sub.n,k] = Pr([[GAMMA].sup.k.sub.n] > [[lambda].sup.k.sub.n]|[H.sup.0.sub.n]) = Q(([[lambda].sup.k.sub.n]/[[sigma].sup.2.sub.w] - 1) [square root of ([mu][T.sub.k])]] (4)

and miss detection probability (i.e., the probability that, under hypothesis [H.sup.1.sub.n], the SU falsely declares that the primary signal is inactive) of channel n at the k-th stage is given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where Q(.) is the Q function, defined as

Q(x) = 1/[square root of (2[pi])] [[integral].sup.[infinity].sub.x] exp (-[z.sup.2]/2)dz.

At the end of the exploration phase, a small subset of the original N channel is obtained, for which the final detection to determine the identified idle channels is performed. In the detection phase, we update the energy of each channel in [I.sub.k] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [[tau].sub.D] is the sampling time for each channel in [I.sub.k] during the detection phase. (6) is equivalent to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [T.sub.D] = [[tau].sub.D] + [[summation].sup.K.sub.i=1][[tau].sub.i]. The candidate set of idle channels is [I.sub.D] = {n [member of] [I.sub.k] : [[GAMMA].sup.D.sub.n] [less than or equal to] [[lambda].sup.D.sub.n]}, where [[lambda].sup.D.sub.n] is the decision threshold. Similar to the k-th stage, we can get the false alarm probability [P.sup.fa.sub.n, d] and miss detection probability [P.sup.md.sub.n, D] of the detection phase as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The overall false alarm probability of channel n is seen as the probability that the SU falsely declares that the primary signal is active in every stage of the exploration phase and the detection phase under hypothesis [H.sup.0.sub.n]. The overall miss detection probability of channel n is seen as the probability that the SU falsely declares that the primary signal is inactive in every stage of the exploration phase and the detection phase under hypothesis [H.sup.1.sub.n]. The overall false alarm probability and the overall miss detection probability are calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

which are similar to the OR-Rule in the cooperative sensing scheme.

When channel n is indeed free and is detected to be free, the achievable transmission rate of channel n is [C.sup.0.sub.n] = [log.sub.2](1 + [P.sub.s]/[N.sub.0]), where [P.sub.s] is the received power of the secondary user and [N.sub.0] is the noise power [8]. When channel n is busy and is detected to be free, the achievable transmission rate of channel n is [C.sup.1.sub.n] = [log.sub.2](1 + [P.sub.s]/([P.sub.p] + [N.sub.0])), where [P.sub.p] is the interference power of primary user measured at the secondary receiver [8]. Then, the average throughput of secondary user is given as

C = (1 - [tau]/T) [N.summation over (n=1)][Pr([H.sup.0.sub.n])(1 - [P.sup.fa.sub.n])[C.sup.0.sub.n] + Pr([H.sup.1.sub.n])[P.sup.md.sub.n][C.sup.1.sub.n],, (11)

where [tau] is the overall sensing time and [tau] = [[tau].sub.D]N[[alpha].sup.K] + [K.summation over (i=1)] ([[tau].sub.i]N[[alpha].sup.i-1]. Here we consider the scenario that all the candidate channels for the secondary user are licensed to a single primary user (e.g. a base station of cell networks), which means that the SNR of every candidate channel is approximately the same. For simplicity, we assume that the active probability is equal in all the channels, which means that [[gamma].sub.n] = [gamma], and Pr([H.sup.0.sub.n]) = Pr([H.sub.0]), Pr([H.sup.1.sub.n]) = Pr([H.sub.1]), [P.sup.fa.sub.n,k] = [P.sup.fa.sub.k], [P.sup.md.sub.n,k] = [P.sup.md.sub.k], [P.sup.fa.sub.n] = [P.sup.fa], [P.sup.md.sub.n] = [P.sup.md], [[lambda].sup.k.sub.n] = [[lambda].sub.k], [[lambda].sup.D.sub.n] = [[lambda].sub.D], [C.sup.0.sub.n] = [C.sub.0], [C.sup.1.sub.n] = [C.sub.1]. Therefore, the throughput in (11) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

3. Optimal Sensing Time Setting for Multi-stage Exploration Phase and Detection Phase

In this section, an optimal sensing time allocation problem for multi-stage exploration phase, the detection phase and overall sensing procedure is formulated and addressed resulting in maximizing the average throughput. The miss detection probability in each channel should be smaller than a threshold that is denoted [P.sub.th] so as to protect the activities of primary users. Then, the problem can be formulated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13b)

[T.sub.D] = [[tau].sub.D] + [[summation].sup.K.sub.i=1] [[tau].sub.i], [T.sub.k] = [[summation].sup.k.sub.i=1] [[tau].sub.i], k = 1, 2, ..., K (13c)

[tau] = [[tau].sub.D]N[[alpha].sup.K] + [K.summation over (i=1)] ([[tau].sub.i]N[[alpha].sup.i-1]) (13d)

[[tau].sub.D] [greater than or equal to] 0, [[tau].sub.i] [greater than or equal to] 0. i = 1, 2, ..., K (13e)

The problem above is not a convex problem. To solve it, we use the bilevel optimization [13][14], in which the lower level problem is to optimize {[[tau].sub.1], [[tau].sub.2], ..., [[tau].sub.K], [[tau].sub.D]} with a fix [tau], whereas the upper level problem is to optimize the overall sensing time [tau]. Specifically, the lower level problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

which is subject to the constraints (13b)-(13e).

Lemma 1: The objective function [U.sub.p] in problem PL1 achieves the maximal value when [P.sup.md] = [P.sub.th].

Proof: In the multi-stage exploration phase, at the k-th stage, we select [N.sub.k] = [alpha][N.sub.k-1] channels as the surviving channels from [N.sub.k-1] candidate channels. Because the idle channels are sparsely distributed among a large number of channels, the miss detection probability at the k-th stage is approximately calculated as [P.sup.md.sub.k]([T.sub.k], [[lambda].sub.k]) [approximately equal to] = [N.sub.k-1]/[N.sub.k] = [alpha]. Using (4) and (5), the false alarm probability at the k-th stage can be written as a function of the detection probability

[P.sup.fa.sub.k]([T.sub.k], [[lambda].sub.k]) = Q(([gamma] + 1)[Q.sup.-1](1 - [alpha]) + [gamma] [square root of ([mu][T.sub.k])]). (15)

It can be seen that the false alarm probability and the miss detection probability at the k-th stage are irrelevant to the threshold [[lambda].sub.k]. There are K stages in the exploration phase. From (9) and (10), we can obtain the overall miss detection probability and the overall false alarm probability as

[P.sup.md] = [[alpha].sup.K][P.sup.md.sub.D]([T.sub.D], [[lambda].sub.D]) (16)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

From Equation (7) and (8), it can be seen that (1 - [P.sup.fa.sub.D]([T.sub.D], [[lambda].sub.D])) and [P.sup.md.sub.D]([T.sub.D], [[lambda].sub.D]) grow with the increase of [[lambda].sub.D]. When the overall miss detection probability [P.sup.md] reaches the limit [P.sub.th], the overall false alarm probability [P.sup.fa] also reaches the maximum value. Therefore, The objective function [U.sub.p] achieves the maximum value when [P.sup.md] = [P.sub.th].

This completes the proof.

Based on Lemma 1, the lower level problem PL1 is equivalent to the problem

PL2: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

which is subject to the constraints (13c)-(13e). The objective function in problem PL1 can be rewritten as [U.sub.p] = Pr([H.sub.0])[C.sub.0][S.sub.p] + Pr([H.sub.1])[C.sub.1][P.sub.th].

Lemma 2: Problem PL2 is a convex problem under condition C1 which includes that [[tau].sub.1] [greater than or equal to] [([gamma] + 1)[Q.sup.-1](1 - [alpha])/[gamma][square root of ([mu])]).sup.2] when [P.sub.th] [greater than or equal to] [[alpha].sup.K+1] at [alpha] [less than or equal to] 0.5 and [P.sub.th] > [[alpha].sup.K](1 - [alpha]) at [alpha] > 0.5 , [[tau].sub.1] [greater than or equal to] [([gamma] + 1)[Q.sup.-1](1 - [alpha])/[gamma][square root of ([mu])]).sup.2] and [[tau].sub.D] [greater than or equal to] [[tau].sub.D] [([gamma] + 1).sup.2]/[[gamma].sup.2][mu] [([Q.sup.-1](1 - [alpha]) - [Q.sup.-1](1 - [P.sub.th]/[[alpha].sup.K])).sup.2] when [P.sub.th] < [[alpha].sup.K+1] at [alpha] [less than or equal to] 0.5 and [P.sub.th] [greater than or equal to] [[alpha].sup.K](1 - [alpha]) at [alpha] > 0.5.

Proof: In the proof of Lemma 1, we have the miss detection probability [P.sup.md.sub.k]([T.sub.k]) [approximately equa; to] [alpha] and [P.sup.md.sub.D]([T.sub.D], [P.sub.th]) [approximately equal to] [P.sub.th]/[[alpha].sup.K]. Thus, the false alarm probability in the multi-stage exploration phase and detection phase are given as

[P.sup.fa.sub.k]([T.sub.k]) = Q(([gamma] + 1)[Q.sup.-1](1 - [alpha]) + [gamma] [square root of ([mu][T.sub.k])]) k = 1, 2, ... K (19)

and

[P.sup.fa.sub.k]([T.sub.D], [P.sub.th]) = Q(([gamma] + 1)[Q.sup.-1](1 - [P.sub.th]/[[al[ha].sup.K]) + [gamma] [square root of [mu][T.sub.D])]). (20)

Taking logarithm to the objective function [S.sub.p], we have

[S.sup.l.sub.p] = log [S.sub.p] = log(1 - [P.sup.fa.sub.D]([T.sub.D], [P.sub.th])) + [K.summation over (k=1)] log(1 - [P.sup.fa.sub.k]([T.sub.k])). (21)

Based on (19), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Further, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Since spectrum opportunity is sparsely distributed in the sensing band, we expect that the false alarm probability is no greater than 0.5 in every stage of the exploration phase. It is equivalent to the following two inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

which mean that

[T.sub.k] [greater than or equal to] [(([gamma] + 1)[Q.sup.-1](1 - [alpha])/[gamma][square root of ([mu])]).sup.2] k = 1, 2, ..., K (25)

and

[T.sub.D] [greater than or equal to] [(([gamma] + 1)[Q.sup.-1](1 - [P.sub.th]/[[alpha].sup.K])/[gamma] [square root of ([mu])]).sup.2]. (26)

Since [[tau].sub.k] [greater than or equal to] 0 for any k, (25) is equivalent to [[tau].sub.1] [greater than or equal to] [(([gamma] + 1)[Q.sup.-1](1 - [alpha])/[gamma][square root of ([mu])]).sup.2]. In the detection phase, we expect the miss detection probability is no larger than 0.5, and we can achieve this by properly setting parameter a and K to obtain [P.sub.th]/[[alpha].sup.K] < 0.5. And when [P.sub.th] < [[alpha].sup.K+1] at [alpha] [less than or equal to] 0.5 and [P.sub.th] [greater than or equal to] [[alpha].sup.K](1 - [alpha]) at [alpha] > 0.5, we have [([Q.sup.-1] (1 - [P.sub.th]/[[alpha].sup.K])).sup.2] > [([Q.sup.-1](1 - [alpha])).sup.2] which means [T.sub.D] > [T.sub.k]. So it should be satisfied that [[tau].sub.D] [greater than or equal to] [([gamma] + 1).sup.2]/[[gamma].sup.2][mu] [([Q.sup.-1](1 - [alpha]) - [Q.sup.-1](1 - [P.sub.th]/[[alpha].sup.K])).sup.2].

Under the above conditions, we have [[partial derivative].sup.2][P.sup.fa.sub.k]([T.sub.k])/[partial derivative][T.sup.2.sub.k] > 0. Then it can be obtained that [[partial derivative].sup.2][S.sup.l.sub.p]/[partial derivative][T.sup.2.sub.k] < 0 and [[partial derivative].sup.2][S.sup.l.sub.p]/[partial derivative][T.sup.2.sub.D]. Therefore, [S.sup.l.sub.p] is a concave function for {[T.sub.1], [T.sub.2], ..., [T.sub.K], [T.sub.D]}. Because the objective function [S.sub.p] is the exponent function of [S.sup.l.sub.p] and exponent function is concave and nondecreasing, [S.sub.p] is also a concave function for {[T.sub.1], [T.sub.2], ..., [T.sub.K], [T.sub.D]} [14, page 84].

Denote T = [[[T.sub.1], [T.sub.2], ..., [T.sub.K], [T.sub.D]].sup.T], [tau] = [[[[tau].sub.1], [[tau].sub.2], ... [[tau].sub.K], [[tau].sub.D]].sup.R]. From the constraint (13c), we have T = A[tau], where A is a (K + 1)? (K 1) lower triangular matrix with the nonzero elements equal to 1. Since [S.sub.p] is a concave function for {[T.sub.1], [T.sub.2], ..., [T.sub.K], [T.sub.D]}, after the operation of composition with an affine mapping [14, page 79], [S.sub.p] is also a concave function for {[[tau].sub.1], [[tau].sub.2], ..., [[tau].sub.K], [[tau].sub.D]}.

This completes the proof.

Since the problem PL2 is convex, the lower level problem PL1 is also convex, and it can be solved by convex optimization methods. So, the optimal solution of {[[tau].sub.1], [[tau].sub.2], ... [[tau].sub.K], [[tau].sub.D]} for a given [tau] can be obtained. By denoting [U.sup.*.sub.p]([tau]) as the optimal objective value of the lower level problem PL1 with a specific [tau], the upper level problem is given as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Lemma 3: Problem PU1 is a convex problem under condition C1.

Proof: Define two variables [[tau].sup.(1)] and [[tau].sup.(2)], assume that [[tau](1)] < [[tau].sup.(2)]. For [tau] = [[tau].sup.(1)], the optimal solution to problem PL1 is {[[tau].sup.(1).sub.1], [[tau].sup.(1).sub.2], ... [[tau].sup.(1).sub.K], [[tau].sup.(1).sub.D]}, i.e. [[tau].sup.(1)]/N = [[tau].sup.(1).sub.D][[alpha].sup.K] + [K.summation over (i=1)] ([[tau].sup.(1).sub.i] [[alpha].sup.i-1], and the optimal objective value is [U.sup.*.sub.p] ([[tau].sup.(1)]). For [tau] = [[tau].sup.(2)], the optimal solution to problem PL1 is {[[tau].sup.(2).sub.1], [[tau].sup.(2).sub.2], ... [[tau].sup.(2).sub.K], [[tau].sup.(2).sub.D]}, i.e. [[tau].sup.(2)]/N = [[tau].sup.(2).sub.D][[alpha].sup.K] + [K.summation over (i=1)] ([[tau].sup.(2).sub.i] [[alpha].sup.i-1]), and the optimal objective value is [U.sup.*.sub.p]([[tau].sup.(2)]).

Denote a new set {[[tau].sup.(1).sub.1](1), [[tau].sup.(1).sub.2], ... [[tau].sup.(1).sub.K], [[tau].sub.D]}, which satisfies

[[tau].sup.(2)]/N = [[tau].sub.D][[alpha].sup.K] + [K.summation over (i=1)]([[tau].sup.(1).sub.i] [[alpha].sup.i-1]). (28)

It can be seen that the new set is a feasible solution to problem PL1 with [tau] = [[tau].sup.(2)]. Because [U.sup.*.sub.p]([[tau].sup.(2)]) is the optimal objective value to problem PL1 with [tau] = [[tau].sup.(2)], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

where [T.sup.(1).sub.k] = [[summation].sup.k.sub.i=1] [[tau].sup.(1).sub.i], [T'.sub.D] = [[tau]'.sub.D] + [[summation].sup.K.sub.i=1] [[tau].sup.(1).sub.i]. Since [[tau].sup.(1)] < [[tau].sup.(2)], we have [[tau].sub.D] > [[tau].sup.(1).sub.D], and further we have (1 - [P.sup.fa.sub.D]([T.sub.D], [P.sub.th])) > (1 - [P.sup.fa.sub.D]([T.sup.(1).sub.D], [P.sub.th])). Therefore, it can be obtained that [U.sup.*.sub.p] ([[tau].sup.(2)]) > [U.sup.*.sub.p]([[tau].sup.(1)]). So [U.sup.*.sub.p]([tau]) is an increasing function with respect to [tau]. Then, we have d[U.sup.*.sub.p]([tau])/d[tau] > 0.

We rewrite the optimal objective value of the lower level problem PL1 as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where A is the feasible domain defined in problem PL1. It has been proved that [U.sub.p]([tau], [tau]) is the concave function with respect to [tau]. Since [tau] is the linear combination of the elements in [tau], with the operation of composition with an affine mapping, [U.sub.p]([tau], [tau]) is also the concave function for [tau]. Then, according to the pointwise supremum property [14, page 79], it can be obtained that [U.sup.*.sub.p]([tau]) is the concave function with respect to [tau]. So, we have [d.sup.2][U.sup.*.sub.p]([tau])/d[[tau].sup.2] < 0.

The second order derivative of [C.sub.p]([tau]) is given as

[d.sup.2][C.sub.p]([tau])/d[[tau].sup.2] = (1 - [tau]/T) [d.sup.2][U.sup.*.sub.p]([tau])/d[[tau].sup.2] - 2/T d[U.sup.*.sub.p]([tau])/d[tau]. (30)

Since [d.sup.2][U.sup.*.sub.p]([tau])/d[[tau].sup.2] < 0 and d[U.sup.*.sub.p]([tau])/d[tau] > 0, from Equation (30) we have [d.sup.2][C.sub.p]([tau])/d[[tau].sup.2] < 0. Therefore, problem PU1 is a convex problem under condition C1.

This completes the proof.

By Lemma 2 and Lemma 3, we prove that the problem is a convex bilevel problem, which can be solved by existing methods.

4. Simulation Results

In this section, simulations are provided to illustrate the performance of the proposed algorithm. Similar to [7], we consider a system consisting of N = 100 channels. The sampling frequency is set to be [mu] = 600 kHz. The overall miss detection probability constraint is set as [P.sub.th] = 0.0625. The distillation ratio of the adaptive method is set to be [alpha] = 0.5 and the number of stages in the exploration phase is set as K = 3.

Fig. 2 shows the maximum achievable normalized throughput of the proposed algorithm and compares the proposed algorithm with the exhaustive search algorithm, the no optimization scheme [7] with the equal sampling budget allocation and fixed overall sensing time, the adaptive partial optimization scheme [8] with the total sensing time as the optimization variable. The SNR is -10dB. It is obvious that the concave-shaped curve of proposed algorithm is consistent with the conclusion in Lemma 3. Compared with the exhaustive search curve, the proposed algorithm is close to the optimal. Compared with the no optimization scheme [7] and partial optimization scheme [8], the proposed algorithm provides a significant performance gain.

Fig. 3 compares the throughput performance between the proposed algorithm, the exhaustive search algorithm, the no optimization scheme [7] and the adaptive partial optimization scheme [8] in different SNR. For every SNR, there exists an optimal overall sensing time for a fix frame duration T [8]. When SNR is low, it needs more sensing time to maintain the given target probability of detection, thus, the optimal overall sensing time is relatively long. For example, when SNR = -10dB the optimal sensing time is 19ms, which is shown in figure 2. So the performance of no optimization [tau] = 10ms and 15ms are better than that of no optimization [tau] = 5ms. When SNR is 0dB, the case is opposite. The curve of partial optimization in figure 3 shows the performance with the optimal overall sensing time. The optimal overall sensing time at SNR=0dB is close to 5ms. So the curves of no optimization [tau] = 10ms and [tau] = 15ms are worse than that of no optimization [tau] = 5ms when SNR is 0dB. The curve of proposed algorithm is almost coincide to the exhaustive search algorithm. Comparing with the scheme without optimization [7] under different fixed overall sensing time, the throughput of the proposed algorithm shows significant performance improvement. The performance of the proposed algorithm is also better than that of the partial optimization algorithm [8]. The gain is obtained from the optimization of the sensing time for each stage of the exploration phase.

The computational complexity of the proposed algorithm is summarized as follows. When optimizing the overall sensing time, we divide the overall sensing time into M parts for searching. For each sensing time in the exploration and detection phases, it needs K operations. There is K+1 sensing time in the proposed algorithm, so the computational complexity is O(M[K.sup.2]). The comparison of computational complexity in our computer between the proposed algorithm, the no optimization scheme [7] and the adaptive partial optimization scheme [8] is given in Table 1. It is observed that the proposed algorithm obtains throughput improvement at the price of higher complexity. Considering the fact that in practice, the number of stages K in the exploration stage is very small (e.g., K < 10), the complexity of the proposed algorithm is allowable for general systems.

5. Conclusions

In this paper, we have studied the problem of throughput tradeoff in adaptive multiband spectrum sensing procedures. This has been achieved by optimizing not only the overall sensing time but also the sensing time for the exploration and detection phases. We have transformed the initial non-convex optimization problem to a convex bilevel optimization problem. Comparing the adaptive method with equal sampling budget allocation and fixed overall sensing time, the proposed scheme provides a significant improvement on the throughput of the secondary user.

In this research, the problem is formulated when the distillation ratio of the adaptive method a and the number of stages in exploration phase K are fixed. How to implement the joint optimization of the distillation ratio a, the number of stages in the exploration phase K, overall sensing time t and the sensing time for the exploration and detection phases are interesting research topics for further investigation.

http://dx.doi.org/10.3837/tiis.2014.03.015

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This work was supported by the National Science Foundation of China under Grant No. 61172062, and in part by Jiangsu Province Natural Science Foundation of China under Grant No. BK2011116.

Long Yu (1), Qihui Wu (1) and Jinlong Wang (1)

(1) College of Communications Engineering, PLA University of Science and Technology. Nanjing, 210007, China

[Email: elong025@163.com, wqhqhw@163.com, wj1543@sina.com]

* Corresponding author: Long Yu

Received September 11, 2013; revised December 23, 2013; revised January 28, 2014; accepted February 27, 2014; published March 31, 2014

Long Yu received his B.S. degree in mobile communications, M.S. degree in communications and information system from Institute of Communications Engineering, Nanjing, China, in 2003 and 2006, respectively. He is currently pursuing the Ph.D. degree in communications and information system in College of Communications Engineering, PLA University of Science and Technology. His research interests focus on spectrum sensing, opportunistic spectrum access and optimization techniques.

Qihui Wu received his B.S. degree in communications engineering, M.S. degree and Ph.D. degree in communications and information system from Institute of Communications Engineering, Nanjing, China, in 1994, 1997 and 2000, respectively. He is currently professor at the PLA University of Science and Technology, China. His current research interests are algorithms and optimization for cognitive wireless networks, soft-defined radio and wireless communication.

Jinlong Wang received the B.S. degree in mobile communications, M.S. degree and Ph.D. degree in communications engineering and information systems from Institute of Communications Engineering, Nanjing, China, in 1983, 1986 and 1992, respectively. Since 1979, Dr. Wang has been with the College of Communications Engineering, PLA University of Science and Technology, where he is currently a Full Professor. He has published over 100 papers in refereed mainstream journals and reputed international conferences and has been granted over 20 patents in his research areas. His current research interests are the broad area of digital communications systems with emphasis on cooperative communication, adaptive modulation, multiple-input-multiple-output systems, soft defined radio, cognitive radio, green wireless communications, and game theory.

Table 1. Computational complexity Scheme Computational complexity No optimization scheme [7] O (K) Adaptive partial optimization scheme [8] O (MK) Proposed algorithm O (M[K.sup.2])

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Author: | Yu, Long; Wu, Qihui; Wang, Jinlong |
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Publication: | KSII Transactions on Internet and Information Systems |

Article Type: | Report |

Date: | Mar 1, 2014 |

Words: | 5676 |

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