CN104079359A - Cooperative spectrum sensing threshold optimization method in cognitive wireless network - Google Patents

Abstract

The invention provides a cooperative spectrum sensing threshold optimization method in a cognitive wireless network. According to the method, a voting threshold and a detection threshold in dual-threshold cognitive spectrum sensing based on voting fusion are optimized respectively. The method comprises the steps of firstly, fixing a detection threshold value of dual-threshold energy detection, optimizing the voting threshold of voting fusion rules, and enabling the global error probability of the cooperative spectrum sensing under the condition of the energy detection threshold value to be the minimum; then, on the premise that the voting threshold of the voting fusion rules is the optimal value, optimizing the detection threshold value of the dual-threshold energy detection, and enabling the optimal detection threshold value to be dynamic under the condition of different receiving signal-to-noise ratios, and therefore the optimal detection threshold value is determined according the signal-to-noise ratio. According to the cooperative spectrum sensing threshold optimization method, the global error probability of the cooperative spectrum sensing under the condition of each signal-to-noise is made to be the minimum, and therefore the performance of the cooperative spectrum sensing is improved.

Description

Collaborative spectrum sensing thresholding optimization method in a kind of cognition wireless network

Technical field

The present invention relates to assist frequency spectrum perception thresholding optimization method in a kind of cognition wireless network, belong to communication technical field.

Background technology

Along with wireless communication technology constantly obtains great development, as bluetooth, Wifi, WSN, 3GLTE etc., increasing communication system need to be distributed wireless frequency spectrum, the characteristic of radio wave non-line-of-sight propagation has determined that wireless device is applicable to using the frequency range below 3GHZ, but traditional fixed allocation frequency spectrum mechanism each communication system all fixedly take a certain frequency range, causing so far can be very few for the frequency range below the 3GHZ distributing.Meanwhile, authorize the service efficiency of frequency spectrum but very low, only had 15%～85%.Cognitive radio can effectively solve frequency spectrum scarcity, and it allows time user carry out transfer of data in the mandate frequency range of free time.Frequency spectrum perception, as the key technology of cognitive radio, obtains research extensively and profoundly.Whether cognitive user need to periodically be carried out frequency spectrum perception, detect primary user and exist, if detect, primary user does not exist, and can utilize this mandate frequency range to carry out transfer of data.Cognitive user need to possess very high detection probability, once primary user reappears, primary user's appearance must very accurately be detected, and in official hour, exit rapidly this frequency range, avoids the interference to primary user as far as possible.

The method of frequency spectrum perception mainly comprises the methods such as matched filtering detection, energy measuring and cyclostationary characteristic detection, wherein the decision method of energy detection method is that a thresholding is first set, compare with the thresholding of setting by energy detector, exceed detection threshold, just think in this frequency range and have primary user to exist.Its advantage is that method is simple, computational complexity is low, and do not need primary user's prior information, it is the main method of local frequency spectrum perception, but detect poor performance under Low SNR, because signal is submerged in noise, energy measuring method can only be calculated the energy of signal, and can not distinguish interference is from signal or noise.Due to the uncertainty of noise, traditional simple gate limit energy measuring threshold value is difficult for setting, and near primary user's energy that cognitive user perceives is positioned at threshold value time, flase drop easily occurs, and employing double threshold energy measuring can reduce probability of false detection greatly.In addition, due to the impact of the problem such as hidden terminal, multipath fading, poor performance is detected at alone family, repeatedly user collaboration frequency spectrum perception can effectively improve detection performance, collaborative spectrum sensing has been utilized the advantage of identical data in many receptions of different terminals different transmission path, obtains space diversity gain.

The major parameter of weighing collaborative spectrum sensing performance is overall false alarm probability and overall false dismissal probability, and their sums are defined as global error probability.First, performance and the blending algorithm of collaborative spectrum sensing are closely related, the fusion criterion of collaborative spectrum sensing has AND criterion, OR criterion, Voting Fusion criterion etc., wherein OR criterion is equivalent to that the ballot thresholding of Voting Fusion criterion is 1, AND criterion is that the ballot thresholding that is equivalent to Voting Fusion criterion is total number of time user, so AND criterion and OR criterion are all the special cases of Voting Fusion criterion.Therefore ballot threshold value and the frequency spectrum perception performance of Voting Fusion criterion are closely connected.Secondly, while utilizing double threshold energy detection method to carry out frequency spectrum perception, detection threshold value choose the performance that affects to a great extent perception.Under each received signal to noise ratio condition, the detection threshold value of cognitive user how to confirm double threshold energy measuring and how to choose suitable Voting Fusion criterion ballot threshold value and become the problem that must research and solve.

Summary of the invention

Technical problem: the object of this invention is to provide the scheme of optimizing collaborative spectrum sensing performance in a kind of cognition wireless network, the method can make the global error probability of collaborative spectrum sensing all reach minimum value under each received signal to noise ratio condition, thereby reaches the object that improves collaborative spectrum sensing performance.

Technical scheme: the method is optimized respectively ballot thresholding and detection threshold in the double threshold collaborative spectrum sensing based on Voting Fusion, first, the detection threshold value of fixing double threshold energy measuring, ballot thresholding to voting fusion criterion is optimized, make under this energy measuring threshold value condition the global error probability minimum of collaborative spectrum sensing; Then get under the prerequisite of optimal value at the ballot thresholding of Voting Fusion criterion, detection threshold value to double threshold energy measuring is optimized, under different received signal to noise ratio conditions, optimum detection threshold value is dynamic, so will determine optimum detection threshold value according to signal to noise ratio.

The present invention for cognition wireless network in centralized collaborative spectrum sensing system configuration based on Voting Fusion as shown in Figure 1.Comprise a primary user, multiple users and a fusion center, cognitive user receives the energy information of primary user's signal by channel perception (inferior user receives the channel of primary user's signal process) separately, then the energy information of the primary user's signal receiving is carried out to double threshold energy measuring, make local court verdict, and court verdict is separately sent to fusion center by reporting channel (inferior user is transferred to court verdict the channel of fusion center experience), fusion center is to the court verdict the receiving fusion of putting to the vote, obtain final judging result.Cognition wireless network judges that according to this final detection result whether this mandate frequency spectrum is idle.

Energy measuring method is the most basic method of frequency spectrum detection.It measures the wireless frequency energy in channel or the signal strength indicator (RSSI) receiving judges that whether channel is occupied.The performance of energy measuring has determined final sensing results and perceptual performance, and the performance of energy measuring is main relevant with threshold value, noise average power, average power signal and hits.If adopt traditional traditional simple gate limit energy measuring method, choose the threshold value of λ as simple gate limit energy measuring, the false alarm probability p of energy measuring _fwith detection probability p _dfor:

$p_f=P(E\left(x\right)>\mathrm{\λ}|H_0)=Q\left((\frac{\mathrm{\λ}}{{\mathrm{\σ}}_u^2}-1)\sqrt{\mathrm{\τ}{f}_s}\right)$

$p_d=P(E\left(x\right)>\mathrm{\λ}|H_1)=Q\left((\frac{\mathrm{\λ}}{{\mathrm{\σ}}_u^2}-\mathrm{\γ}-1)\sqrt{\frac{\mathrm{\τ}{f}_s}{2\mathrm{\γ}+1}}\right)$

Wherein: γ is time user's received signal to noise ratio; it is noise variance; τ is time user's detecting period; f _sfor sample frequency; $Q\left(x\right)=\frac{1}{\sqrt{2\mathrm{\π}}}{\∫}_x^{\∞}\exp (-\frac{t^2}{2})\mathrm{dt}.$

Double threshold energy measuring of the present invention has two detection threshold λ ₀, λ ₁, as shown in Figure 2, time local decision rule corresponding to user is:

$D=\left\{\begin{array}{c}0,E\left(x\right)<{\mathrm{\&lambda;}}_0\\ 1,E\left(x\right)>{\mathrm{\&lambda;}}_1\\ U,{\mathrm{\&lambda;}}_0<E\left(x\right)<{\mathrm{\&lambda;}}_1\end{array}\right.$

Wherein: when U represents that energy value falls into interval of uncertainty, inferior user does not make local judgement.

Due to the impact of the problem such as hidden terminal, multipath fading, poor performance is detected at alone family, repeatedly user collaboration frequency spectrum perception can effectively improve detection performance, the fusion criterion of collaborative spectrum sensing has AND criterion, OR criterion, Voting Fusion criterion etc., wherein OR criterion is equivalent to that the ballot thresholding of Voting Fusion criterion is 1, AND criterion is that the ballot thresholding that is equivalent to Voting Fusion criterion is total number of time user, so AND criterion and OR criterion are all the special cases of Voting Fusion criterion.The detection performance of collaborative spectrum sensing and the ballot thresholding of Voting Fusion criterion, the detection threshold of double threshold energy measuring is closely related, so first the present invention is optimized the ballot thresholding of voting fusion criterion, then get under the prerequisite of optimal value at the ballot thresholding of Voting Fusion criterion, detection threshold value to double threshold energy measuring is optimized, under different received signal to noise ratio conditions, optimum detection threshold value is dynamic, so will determine optimum detection threshold value according to signal to noise ratio, make the global error probability of collaborative spectrum sensing all reach minimum value under each signal to noise ratio condition, thereby improve the performance of collaborative spectrum sensing.

Beneficial effect: double threshold energy measuring is than traditional simple gate limit energy measuring, can greatly reduce the probability of false detection of frequency spectrum perception, and due to the impact of the problem such as hidden terminal, multipath fading, poor performance is detected at alone family, and repeatedly user collaboration frequency spectrum perception can effectively improve detection performance.The present invention is based on the cooperative frequency spectrum sensing method of double threshold energy measuring, adopt Voting Fusion criterion to merge each user's local court verdict at fusion center, optimization aim is made as to global error probability, based on this optimization aim, the voting ballot threshold value of fusion criterion and the detection threshold value of double threshold energy measuring are chosen, make global error probability minimum, thereby improved the performance of collaborative spectrum sensing.

Brief description of the drawings

Fig. 1 is system model of the present invention.

Fig. 2 is dual-threshold judgement model of the present invention.

Embodiment

According to the dependent probability definition of simple gate limit energy measuring, the dependent probability of definition double threshold energy measuring is as follows:

$p_f=P(E\left(x\right)>{\mathrm{\&lambda;}}_1|H_0)=Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right)---\left(1\right)$

$p_a=P(E\left(x\right)<{\mathrm{\&lambda;}}_0|H_0)=1-Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right)---\left(2\right)$

$\begin{array}{c}{\mathrm{\&Delta;}}_0=P({\mathrm{\&lambda;}}_0<E\left(x\right)<{\mathrm{\&lambda;}}_1|H_0)=1-p_f-p_a\\ =Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right)-Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right)\end{array}----\left(3\right)$

$p_d=P(E\left(x\right)>{\mathrm{\&lambda;}}_1|H_1)=Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right)---\left(4\right)$

$p_m=P(E\left(x\right)<{\mathrm{\&lambda;}}_0|H_1)=1-Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right)---\left(5\right)$

$\begin{array}{c}{\mathrm{\&Delta;}}_1=P({\mathrm{\&lambda;}}_0<E\left(x\right)<{\mathrm{\&lambda;}}_1|H_1)=1-p_d-p_m\\ =Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right)-Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right)\end{array}----\left(6\right)$

Wherein: H ₁and H ₀represent that respectively primary user exists and non-existent situation, △ ₀and △ ₁be illustrated respectively in H ₀, H ₁the energy value that the next user of condition receives is positioned at the probability of interval of uncertainty, and γ is time user's received signal to noise ratio, be noise variance, τ is time user's detecting period, f _sfor sample frequency, double threshold energy measuring of the present invention has two detection threshold λ ₀, λ ₁.

Suppose to have N time user to carry out frequency spectrum perception, court verdict is sent to fusion center by the inferior user who wherein makes local judgement, fusion center is to the court verdict the receiving fusion of putting to the vote, and the ballot thresholding of Voting Fusion is n, and fusion center is according to Voting Fusion criterion:

$F=\left\{\begin{array}{c}0,\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{D}_i<n\\ 1,\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{D}_i\&GreaterEqual;n\end{array}\right.---\left(7\right)$

Whether judge primary user's existence, wherein D _irepresent the local court verdict that i time user makes.When local judgement adopts simple gate limit energy measuring, overall false alarm probability Q _fwith global detection probability Q _dbe respectively:

$Q_f=1-\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^l{(1-p_f)}^l{p}_f^{N-l}---\left(8\right)$

$Q_d=\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{p}_d^l{(1-p_d)}^{N-l}---\left(9\right)$

When local judgement employing double threshold energy measuring, N is carried out in the inferior user of frequency spectrum perception, determines interval if the energy value that has K time user to receive is positioned at, and has (N-K) individual user not make local judgement.So overall false alarm probability Q _f, false dismissal probability Q _mwith detection probability Q _dbe respectively:

$\begin{array}{c}{Q}_f=1-P\{F=0|H_0\}\\ =1-P\{F=0,K\&NotEqual;N|H_0\}-P\{F=0,K=N\left|H_0\right\}\\ =1-\underset{K=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{C}_N^K\underset{i=1}{\overset{K}{\mathrm{\&Pi;}}}P\{O_i\&le;{\mathrm{\&lambda;}}_0\&cup;O_i\&GreaterEqual;{\mathrm{\&lambda;}}_1|H_0\left\}\underset{i=K+1}{\overset{N}{\mathrm{\&Pi;}}}P\right\{{\mathrm{\&lambda;}}_0\&le;O_i\&le;{\mathrm{\&lambda;}}_1\left|H_0\right\}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}\\ -\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}P\{O_i\&le;{\mathrm{\&lambda;}}_0\&cup;O_i\&GreaterEqual;{\mathrm{\&lambda;}}_1|H_0\}\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^l{p}_a^l{p}_f^{K-l}\\ =1-\underset{K=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{C}_N^K\underset{i=1}{\overset{K}{\mathrm{\&Pi;}}}(1-{\mathrm{\&Delta;}}_0)\underset{i=K+1}{\overset{N}{\mathrm{\&Pi;}}}{\mathrm{\&Delta;}}_0\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}-\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}(1-{\mathrm{\&Delta;}}_0)\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^l{p}_a^l{p}_f^{N-l}\\ =1-\underset{K=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{C}_N^K{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}-{(1-{\mathrm{\&Delta;}}_0)}^N\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^l{p}_a^l{p}_f^{N-l}\\ =1-\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}\end{array}---\left(10\right)$

$\begin{array}{c}{Q}_m=P\{F=0|H_1\}\\ =P\{F=0,K\&NotEqual;N|H_1\}+P\{F=0,K=N\left|H_1\right\}\\ =\underset{K=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{C}_N^K\underset{i=1}{\overset{K}{\mathrm{\&Pi;}}}P\{O_i\&le;{\mathrm{\&lambda;}}_0\&cup;O_i\&GreaterEqual;{\mathrm{\&lambda;}}_1|H_1\left\}\underset{i=K+1}{\overset{N}{\mathrm{\&Pi;}}}P\right\{{\mathrm{\&lambda;}}_0\&le;O_i\&le;{\mathrm{\&lambda;}}_1\left|H_1\right\}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}\\ +\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}P\{O_i\&le;{\mathrm{\&lambda;}}_0\&cup;O_i\&GreaterEqual;{\mathrm{\&lambda;}}_1|H_1\}\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^l{p}_m^l{p}_d^{K-l}\\ =\underset{K=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{C}_N^K\underset{i=1}{\overset{K}{\mathrm{\&Pi;}}}(1-{\mathrm{\&Delta;}}_1)\underset{i=K+1}{\overset{N}{\mathrm{\&Pi;}}}{\mathrm{\&Delta;}}_1\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}+\underset{i=1}{\overset{N}{\mathrm{\&Pi;}}}(1-{\mathrm{\&Delta;}}_1)\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^l{p}_m^l{p}_d^{N-l}\\ =\underset{K=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{C}_N^K{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}+{(1-{\mathrm{\&Delta;}}_1)}^N\underset{l=n}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^l{p}_m^l{p}_d^{N-l}\\ =\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}\end{array}---\left(11\right)$

Q _d＝1-Q _m (12)

A. the optimization of the ballot thresholding n of Voting Fusion criterion

For collaborative spectrum sensing, the performance that affect frequency spectrum perception is understood in choosing of fusion criterion, OR criterion and AND criterion are respectively that the ballot thresholding n of Voting Fusion criterion equals 1 and equal the special circumstances of total number N of time user, known by formula (10), (11) and (12), the ballot thresholding n of Voting Fusion criterion can affect the performance of collaborative spectrum sensing, so need to carry out optimization to its value, make the best performance of collaborative spectrum sensing.

We define frequency spectrum perception global error probability is (Q _f+ Q _m), suppose that the inferior user's who makes local judgement number K is known, optimization problem can be written as:

min(Q _f+Q _m) (13)

s.t.n≤K

Ask the optimum ballot thresholding n of Voting Fusion criterion _opt, making global error probability is (Q _f+ Q _m) reach minimum value.

By formula (10), (11), obtain global error probability:

$\begin{array}{c}{Q}_f+Q_m\\ =1+\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K({(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}-{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l})\end{array}---\left(14\right)$

If:

$G\left(n\right)={(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}-{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}---\left(15\right)$

Can be obtained by formula (14), (15):

$Q_f+Q_m=1+\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^{K}G\left(n\right)---\left(16\right)$

Because:

$\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;n}{|}_{n_{\mathrm{opt}}}=\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}{|}_{n_{\mathrm{opt}}}=0---\left(17\right)$

$\begin{array}{c}\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}\&ap;\frac{G(n+1)-G\left(n\right)}{(n+1)-n}=G(n+1)-G\left(n\right)\\ =C_K^n({(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}{p_a}^n{p_f}^{K-n}-{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}{p_m}^n{p_d}^{K-n})\end{array}----\left(18\right)$

So can be obtained by formula (17), (18):

$\begin{array}{c}\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}{|}_{n_{\mathrm{opt}}}=0\&DoubleRightArrow;{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}{p_a}^{n_{\mathrm{opt}}}{p_f}^{{K-n}_{\mathrm{opt}}}={(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}{p_m}^{n_{\mathrm{opt}}}{p_d}^{{K-n}_{\mathrm{opt}}}\\ \&DoubleRightArrow;{\left(\frac{p_a}{p_m}\right)}^{n_{\mathrm{opt}}}={\left(\frac{p_d}{p_f}\right)}^{{K-n}_{\mathrm{opt}}}{\left(\frac{1-{\mathrm{\&Delta;}}_1}{1-{\mathrm{\&Delta;}}_0}\right)}^K{\left(\frac{{\mathrm{\&Delta;}}_1}{{\mathrm{\&Delta;}}_0}\right)}^{N-K}\end{array}---\left(19\right)$

Taken the logarithm in above formula both sides:

$n_{\mathrm{opt}}\ln \frac{p_a}{p_m}=(K-n_{\mathrm{opt}})\ln \frac{p_d}{p_f}+\ln {\left(\frac{1-{\mathrm{\&Delta;}}_1}{1-{\mathrm{\&Delta;}}_0}\right)}^K{\left(\frac{{\mathrm{\&Delta;}}_1}{{\mathrm{\&Delta;}}_0}\right)}^{N-K}---\left(20\right)$

Solve:

$n_{\mathrm{opt}}=\frac{K\ln \frac{p_d}{p_f}+\ln {\left(\frac{1-{\mathrm{\&Delta;}}_1}{1-{\mathrm{\&Delta;}}_0}\right)}^K{\left(\frac{{\mathrm{\&Delta;}}_1}{{\mathrm{\&Delta;}}_0}\right)}^{N-K}}{\ln \frac{p_a{p}_d}{p_m{p}_f}}---\left(21\right)$

So work as $n=n_{\mathrm{opt}}=\frac{K\ln \frac{p_d}{p_f}+\ln {\left(\frac{1-{\mathrm{\&Delta;}}_1}{1-{\mathrm{\&Delta;}}_0}\right)}^K{\left(\frac{{\mathrm{\&Delta;}}_1}{{\mathrm{\&Delta;}}_0}\right)}^{N-K}}{\ln \frac{p_a{p}_d}{p_m{p}_f}}$ Time, $\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}{|}_{n_{\mathrm{opt}}}=0,(Q_f+Q_m)$ Obtain extreme value.Again because:

$\frac{{\&PartialD;}^{2}G\left(n\right)}{{\&PartialD;n}^2}=\frac{\frac{\&PartialD;G(n+1)}{\&PartialD;n}-\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}}{(n+1)-n}=\frac{\&PartialD;G(n+1)}{\&PartialD;n}-\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}---\left(22\right)$

$\begin{array}{c}\frac{{\&PartialD;}^{2}G\left(n\right)}{\&PartialD;n^2}{|}_{{n=n}_{\mathrm{opt}}}=\frac{\&PartialD;G(n+1)}{\&PartialD;n}{|}_{{n=n}_{\mathrm{opt}}}-\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}{|}_{{n=n}_{\mathrm{opt}}}=\frac{\&PartialD;G(n+1)}{\&PartialD;n}{|}_{n=n_{\mathrm{opt}}}-0\\ =C_K^{n_{\mathrm{opt}}+1}({(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}{p_a}^{n_{\mathrm{opt}}+1}{p}_f^{{K-n}_{\mathrm{opt}}-1}-{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}{p_m}^{n_{\mathrm{opt}}+1}{p}_d^{K-n_{\mathrm{opt}}-1})\\ =C_K^{n_{\mathrm{opt}}+1}({(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\frac{p_a}{p_f}{p_a}^{n_{\mathrm{opt}}}{p}_f^{{K-n}_{\mathrm{opt}}}-{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\frac{p_m}{p_d}{p_m}^{n_{\mathrm{opt}}}{p}_d^{{K-n}_{\mathrm{opt}}})\end{array}---\left(23\right)$

Know p by formula (1), (2), (4) and (5) _a>p _m, p _d>p _ftherefore:

$\frac{p_a}{p_f}>\frac{p_m}{p_d}---\left(24\right)$

So obtained by formula (19), (23) and (24):

${(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\frac{p_a}{p_f}{p_a}^{n_{\mathrm{opt}}}{p}_f^{{K-n}_{\mathrm{opt}}}>{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\frac{p_m}{p_d}{p_m}^{n_{\mathrm{opt}}}{p}_d^{{K-n}_{\mathrm{opt}}}---\left(25\right)$

Substitution formula (23): $\frac{{\&PartialD;}^{2}G\left(n\right)}{\&PartialD;n^2}{|}_{n=n_{\mathrm{opt}}}>0,$

So: $\frac{{\&PartialD;}^2(Q_f+Q_m)}{\&PartialD;n^2}{|}_{n_{\mathrm{opt}}}=\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K\frac{{\&PartialD;}^{2}G\left(n\right)}{{\&PartialD;n}^2}{|}_{n_{\mathrm{opt}}}>0.$

In sum, when $n=n_{\mathrm{opt}}=\frac{K\ln \frac{p_d}{p_f}+\ln {\left(\frac{1-{\mathrm{\&Delta;}}_1}{1-{\mathrm{\&Delta;}}_0}\right)}^K{\left(\frac{{\mathrm{\&Delta;}}_1}{{\mathrm{\&Delta;}}_0}\right)}^{N-K}}{\ln \frac{p_a{p}_d}{p_m{p}_f}}$ Time, $\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;n}{|}_{n_{\mathrm{opt}}}=\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K\frac{\&PartialD;G\left(n\right)}{\&PartialD;n}{|}_{n_{\mathrm{opt}}}=0,\frac{{\&PartialD;}^2(Q_f+Q_m)}{\&PartialD;n^2}{|}_{n_{\mathrm{opt}}}=\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K\frac{{\&PartialD;}^{2}G\left(n\right)}{{\&PartialD;n}^2}{|}_{n_{\mathrm{opt}}}>0,$ So (Q _f+ Q _m) obtain minimum.In practical application, ballot thresholding n is positive integer, so:

represent to be not less than the smallest positive integral of x.As the threshold value λ of double threshold energy measuring ₀, λ ₁while determining, can be by λ ₀, λ ₁value substitution (1)～(6) formula, try to achieve p _d, p _f, p _a, p _mand △ ₀, △ ₁, substitution above formula can be obtained and make (Q _f+ Q _m) obtain minimizing ballot thresholding n _opt.

B. the threshold value λ of double threshold energy measuring ₀, λ ₁optimization

For double threshold energy measuring, threshold value λ ₀, λ ₁value and the relation between them all can exert an influence to the performance of frequency spectrum perception, wherein λ ₀≤ λ ₁.From (1)～(6) formula, work as λ ₀value while reducing, false dismissal probability can reduce works as λ ₁value while increasing, false alarm probability can reduce, but simultaneously detection probability also can reduce; In the time that the distance between two thresholdings widens, the probability that the energy value that inferior user receives is positioned at interval of uncertainty strengthens, otherwise reduces, and works as λ ₀=λ ₁time, being equivalent to simple gate limit energy measuring, the probability that the energy value that time user receives is positioned at interval of uncertainty is 0.

We define frequency spectrum perception global error probability is (Q _f+ Q _m), suppose that the inferior user's who makes local judgement number K is known, optimization problem can be written as:

min(Q _f+Q _m) (27)

s.t.0<λ ₀<λ ₁<+∞

Ask the optimum gate limit value of double threshold energy measuring making global error probability is (Q _f+ Q _m) reach minimum value, meet:

$\left\{\begin{array}{c}\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=0\\ \frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=0\end{array}\right.---\left(28\right)$

Wherein:

$\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=\frac{\&PartialD;(1+\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^{K}G\left(n\right))}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K\frac{\&PartialD;G\left(n\right)}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=0---\left(29\right)$

$\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=\frac{\&PartialD;(1+\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^{K}G\left(n\right))}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K\frac{\&PartialD;G\left(n\right)}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=0---\left(30\right)$

$\frac{\&PartialD;G\left(n\right)}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=0,\frac{\&PartialD;G\left(n\right)}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=0.$

Formula (15) substitution above formula is obtained:

$\begin{array}{c}\frac{\&PartialD;G\left(n\right)}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=\frac{\&PartialD;p_m}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}(K{(1-{\mathrm{\&Delta;}}_1)}^{K-1}{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}\\ -{(1-{\mathrm{\&Delta;}}_1)}^K(N-K){{\mathrm{\&Delta;}}_1}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}+{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^{l}l{p}_m^{l-1}{p}_d^{K-l}\\ -\frac{\&PartialD;p_a}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}(K{(1-{\mathrm{\&Delta;}}_0)}^{K-1}{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p_f}^{K-l}\\ +{(1-{\mathrm{\&Delta;}}_0)}^K(N-K){{\mathrm{\&Delta;}}_0}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}-{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^{l}l{p}_a^{l-1}{p}_f^{K-l}\\ =0\end{array}---\left(31\right)$

Wherein:

$\frac{\&PartialD;p_m}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=-\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}(2\mathrm{\&gamma;}+1)}}\exp \left(\frac{{({\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}-\mathrm{\&gamma;}-1)}^2\mathrm{\&tau;}{f}_s}{2(2\mathrm{\&gamma;}+1)}\right)---\left(32\right)$

$\frac{\&PartialD;p_a}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=-\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}}}\exp (-\frac{{({\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}-1)}^2\mathrm{\&tau;}{f}_s}{2})---\left(33\right)$

Formula (31), (32) and (33) substitution formulas (33) are obtained:

$\begin{array}{c}\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;{\mathrm{\&lambda;}}_0}{|}_{{\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}}=\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K(-\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}(2\mathrm{\&gamma;}+1)}}\exp (-\frac{{({\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}-\mathrm{\&gamma;}-1)}^2\mathrm{\&tau;}{f}_s}{2(2\mathrm{\&gamma;}+1)}))\\ (K{(1-{\mathrm{\&Delta;}}_1)}^{K-1}{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}-{(1-{\mathrm{\&Delta;}}_1)}^K(N-K){{\mathrm{\&Delta;}}_1}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-1}\\ +{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^{l}l{p}_m^{l-1}{p}_d^{K-l})\\ +\left(\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}}}\exp (-\frac{{({\mathrm{\&lambda;}}_{0_{\mathrm{opt}}}-1)}^2\mathrm{\&tau;}{f}_s}{2})\right)(K{(1-{\mathrm{\&Delta;}}_0)}^{K-1}{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}\\ +{(1-{\mathrm{\&Delta;}}_0)}^K(N-K){{\mathrm{\&Delta;}}_0}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}-{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^{l}l{p}_a^{l-1}{p}_f^{K-l})\\ =0\end{array}---\left(34\right)$

Can solve optimum gate limit value by formula (34)

$\begin{array}{c}\frac{\&PartialD;G\left(n\right)}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=\frac{\&PartialD;p_d}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}(K{(1-{\mathrm{\&Delta;}}_1)}^{K-1}{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}\\ -{(1-{\mathrm{\&Delta;}}_1)}^K(N-K){{\mathrm{\&Delta;}}_1}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}+{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l(K-l)p_d^{K-l-1})\\ -\frac{\&PartialD;p_f}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}(K{(1-{\mathrm{\&Delta;}}_0)}^{K-1}{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p_f}^{K-l}+{(1-{\mathrm{\&Delta;}}_0)}^K(N-K){{\mathrm{\&Delta;}}_0}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}\\ -{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l(K-l)p_f^{K-l-1})\\ \end{array}---\left(35\right)$

Wherein:

$\frac{\&PartialD;p_d}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}(2\mathrm{\&gamma;}+1)}}\exp (-\frac{{({\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}-\mathrm{\&gamma;}-1)}^2\mathrm{\&tau;}{f}_s}{2(2\mathrm{\&gamma;}+1)})---\left(36\right)$

$\frac{\&PartialD;p_f}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}}}\exp (-\frac{{({\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}-1)}^2\mathrm{\&tau;}{f}_s}{2})---\left(37\right)$

Formula (35), (36) and (37) substitutions (28) are obtained:

$\begin{array}{c}\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}(2\mathrm{\&gamma;}+1)}}\exp (-\frac{{({\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}-\mathrm{\&gamma;}-1)}^2\mathrm{\&tau;}{f}_s}{2(2\mathrm{\&gamma;}+1)})\\ (K{(1-{\mathrm{\&Delta;}}_1)}^{K-1}{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}-{(1-{\mathrm{\&Delta;}}_1)}^K(N-K){{\mathrm{\&Delta;}}_1}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}\\ +{(1-{\mathrm{\&Delta;}}_1)}^K{{\mathrm{\&Delta;}}_1}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l(K-l)p_d^{K-l-1})-\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;}}}\exp (-\frac{{({\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}-1)}^2\mathrm{\&tau;}{f}_{\text{s}}}{2})\\ (K{(1-{\mathrm{\&Delta;}}_0)}^{K-1}{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}+{(1-{\mathrm{\&Delta;}}_0)}^K(N-K){{\mathrm{\&Delta;}}_0}^{N-K-1}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{K-l}\\ -{(1-{\mathrm{\&Delta;}}_0)}^K{{\mathrm{\&Delta;}}_0}^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l(K-l)p_f^{K-l-1})\\ =0\end{array}---\left(38\right)$

Can solve optimum gate limit value by formula (38)

From formula (34) (38), under different signal to noise ratio conditions, the thresholding optimal value of double threshold energy measuring value different.Get the ballot thresholding n=n of Voting Fusion criterion _opt, Voting Fusion criterion reaches under optimum prerequisite, can be calculated under each signal to noise ratio condition by formula (34) (38) value.

Claims (1)

1. an optimization method for collaborative spectrum sensing thresholding in cognition wireless network, is characterized in that the method comprises the following steps:

A, definition λ ₀and λ ₁be two threshold values of double threshold energy measuring, and require threshold value λ ₀≤ λ ₁, [0, λ ₀] ∪ [λ ₁,+∞) and interval for determining, (λ ₀, λ ₁) be interval of uncertainty, the global error probability (Q of collaborative spectrum sensing _f+ Q _m) be:

$Q_f+Q_m=1+\underset{K=0}{\overset{N}{\mathrm{\&Sigma;}}}{C}_N^K({(1-{\mathrm{\&Delta;}}_1)}^K{\mathrm{\&Delta;}}_1^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_m^l{p}_d^{K-l}-{(1-{\mathrm{\&Delta;}}_0)}^K{\mathrm{\&Delta;}}_0^{N-K}\underset{l=n}{\overset{K}{\mathrm{\&Sigma;}}}{C}_K^l{p}_a^l{p}_f^{k-l})$

The signal energy that wherein has K cognitive user to receive in N cognitive user falls into determines interval, determines in interval have so the signal energy that specifically which K cognitive user receives drops on the possible situation of planting; H ₁and H ₀represent that respectively primary user exists and non-existent situation, △ ₀and △ ₁represent respectively H ₀, H ₁the energy value that the next user of condition receives is positioned at the probability of interval of uncertainty:

${\mathrm{\&Delta;}}_0=P({\mathrm{\&lambda;}}_0<E\left(x\right)<{\mathrm{\&lambda;}}_1|H_0)=1-p_f-p_a=Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right)-Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right)$

${\mathrm{\&Delta;}}_1=P({\mathrm{\&lambda;}}_0<E\left(x\right)<{\mathrm{\&lambda;}}_1|H_1)=1-p_d-p_m=Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right)-Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right);$

P _d, p _fand p _mrepresent respectively detection probability, false alarm probability and the false dismissal probability of local frequency spectrum perception:

$p_d=Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right),p_f=Q\left((\frac{{\mathrm{\&lambda;}}_1}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right),p_m=1-Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-\mathrm{\&gamma;}-1)\sqrt{\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&gamma;}+1}}\right),$ $p_a=1-Q\left((\frac{{\mathrm{\&lambda;}}_0}{{\mathrm{\&sigma;}}_u^2}-1)\sqrt{\mathrm{\&tau;}{f}_s}\right),$ Wherein $Q\left(x\right)=\frac{1}{\sqrt{2\mathrm{\&pi;}}}{\&Integral;}_x^{\&infin;}\exp (-\frac{t^2}{2})\mathrm{dt},$ γ is time user's received signal to noise ratio, be noise variance, τ is time user's detecting period, f _sfor sample frequency;

B, to voting fusion criterion ballot thresholding n be optimized, make the global error probability (Q of collaborative spectrum sensing _f+ Q _m) minimum, therefore requirement $\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;n}{|}_{n=n_{\mathrm{opt}}}=0,\frac{{\&PartialD;}^2(Q_f+Q_m)}{\&PartialD;n^2}{|}_{n=n_{\mathrm{opt}}}>0,$ Obtain optimum ballot threshold value $n_{\mathrm{opt}}=\frac{K\ln \frac{p_d}{p_f}+\ln {\left(\frac{1-{\mathrm{\&Delta;}}_1}{1-{\mathrm{\&Delta;}}_0}\right)}^K{\left(\frac{{\mathrm{\&Delta;}}_1}{{\mathrm{\&Delta;}}_0}\right)}^{N-K}}{\ln \frac{p_a{p}_d}{p_m{p}_f}},$ In practical application, ballot thresholding n is positive integer, so represent to be not less than the smallest positive integral of x;

C, detection threshold value λ to double threshold energy measuring ₀, λ ₁be optimized, make the global error probability (Q of collaborative spectrum sensing _f+ Q _m) reach minimum value, therefore order $\frac{\&PartialD;(Q_f+Q_m)}{\&PartialD;{\mathrm{\&lambda;}}_1}{|}_{{\mathrm{\&lambda;}}_1={\mathrm{\&lambda;}}_{1_{\mathrm{opt}}}}=0,$ Can obtain following equation group:

When the received signal to noise ratio γ of cognitive user, the number N of cognitive user, the signal energy receiving fall into definite interval cognitive user number K, detecting period τ and sample frequency f _swhile determining, get under the prerequisite of its optimal value i.e. n=n at the ballot thresholding of Voting Fusion criterion _opt, just can be obtained by above-mentioned equation group the optimal detection threshold value of double threshold energy measuring