CN104079359A

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

Info

Publication number: CN104079359A
Application number: CN201410256853.1A
Authority: CN
Inventors: 朱琦; 金燕君
Original assignee: Nanjing Post and Telecommunication University
Current assignee: Nanjing Post and Telecommunication University
Priority date: 2014-06-10
Filing date: 2014-06-10
Publication date: 2014-10-01
Anticipated expiration: 2034-06-10
Also published as: CN104079359B

Abstract

The invention provides a cooperative spectrum sensing threshold optimization method in a cognitive wireless network. The method optimizes respectively the voting threshold and the detection threshold in the dual-threshold cooperative spectrum sensing based on voting fusion. First, the detection threshold of dual-threshold energy detection is fixed, and the voting threshold of the voting fusion criterion is optimized, so that the global error probability of cooperative spectrum sensing is the smallest under the condition of the energy detection threshold; Under the premise of taking the optimal value of the threshold, the detection threshold value of the double-threshold energy detection is optimized. Under the condition of different receiving signal-to-noise ratio, the optimal detection threshold value is dynamic, so it should be determined according to the signal-to-noise ratio Optimal detection threshold. The invention can make the global error probability of the cooperative spectrum sensing reach the minimum value under each signal-to-noise ratio condition, thereby improving the performance of the cooperative spectrum sensing.

Description

A Cooperative Spectrum Sensing Threshold Optimization Method in Cognitive Wireless Networks

技术领域technical field

本发明涉及一种认知无线网络中协助频谱感知门限优化方法，属于通信技术领域。The invention relates to a threshold optimization method for assisting spectrum sensing in a cognitive wireless network, and belongs to the technical field of communication.

背景技术Background technique

随着无线通信技术不断取得巨大发展，如蓝牙、Wifi、WSN、3GLTE等，越来越多的通信系统需要分配无线频谱，无线电波非视距传播的特性决定了无线设备适合使用3GHZ以下的频段，但是传统的固定分配频谱机制保证了每一种通信系统都固定占用某一频段，导致目前为止可供分配使用的3GHZ以下的频段寥寥无几。与此同时，已授权频谱的使用效率却很低，只有15％～85％。认知无线电可以有效解决频谱匮乏，它允许次用户在空闲的已授权频段进行数据传输。频谱感知作为认知无线电的关键技术，得到广泛深入的研究。认知用户需要周期性地进行频谱感知，检测主用户是否存在，若检测到主用户不存在，则可以利用该授权频段进行数据传输。认知用户需要具备很高的检测概率，一旦主用户重新出现，必须很精确地检测到主用户的出现，并在规定的时间内迅速退出该频段，尽量避免对主用户的干扰。With the continuous great development of wireless communication technology, such as Bluetooth, Wifi, WSN, 3GLTE, etc., more and more communication systems need to allocate wireless spectrum. The characteristics of non-line-of-sight propagation of radio waves determine that wireless devices are suitable for using frequency bands below 3GHZ , but the traditional fixed spectrum allocation mechanism ensures that each communication system will occupy a certain frequency band, resulting in very few frequency bands below 3 GHz available for allocation so far. At the same time, the utilization efficiency of authorized spectrum is very low, only 15% to 85%. Cognitive radio can effectively solve spectrum scarcity by allowing secondary users to transmit data in vacant licensed frequency bands. As a key technology of cognitive radio, spectrum sensing has been extensively and deeply researched. Cognitive users need to perform spectrum sensing periodically to detect whether the primary user exists. If no primary user is detected, the authorized frequency band can be used for data transmission. Cognitive users need to have a high detection probability. Once the primary user reappears, the presence of the primary user must be detected accurately and quickly exit the frequency band within the specified time to avoid interference to the primary user.

频谱感知的方法主要包括匹配滤波检测、能量检测和循环平稳特征检测等方法，其中能量检测方法的判决方法是先设置一个门限，通过能量检测器与设定的门限相比较，超过检测门限，就认为该频段内有主用户存在。它的优点是方法简单，计算复杂性低，且不需要主用户的先验信息，是本地频谱感知的主要方法，但是在低信噪比条件下检测性能差，因为信号淹没在噪声中，能量检测法只能计算信号的能量，不能区分出干扰是来自信号还是噪声。由于噪声的不确定性，传统的单门限能量检测门限值不易设定，而且当认知用户感知到的主用户能量位于门限值附近时，容易发生误检，采用双门限能量检测可以大大降低误检概率。另外，由于隐蔽终端、多径衰落等问题的影响，单用户检测性能很差，多次用户协作频谱感知可以有效提高检测性能，协作频谱感知利用了相同数据在不同终端不同传输路径的多接收的优势，获得空间分集增益。Spectrum sensing methods mainly include matched filter detection, energy detection, and cyclostationary feature detection. The judgment method of the energy detection method is to set a threshold first, and compare it with the set threshold through the energy detector. It is considered that there is a primary user in this frequency band. Its advantage is that the method is simple, the computational complexity is low, and it does not need the prior information of the main user. It is the main method of local spectrum sensing, but the detection performance is poor under the condition of low signal-to-noise ratio, because the signal is submerged in the noise, and the energy The detection method can only calculate the energy of the signal, and cannot distinguish whether the interference comes from the signal or the noise. Due to the uncertainty of noise, the threshold value of traditional single-threshold energy detection is not easy to set, and when the primary user energy perceived by cognitive users is near the threshold value, false detection is prone to occur. Using double-threshold energy detection can greatly Reduce the probability of false detection. In addition, due to the influence of hidden terminals and multipath fading, single-user detection performance is very poor. Multi-user cooperative spectrum sensing can effectively improve detection performance. advantage, gaining space diversity gain.

衡量协作频谱感知性能的主要参数是全局虚警概率和全局漏检概率，它们之和被定义为全局错误概率。首先，协作频谱感知的性能与融合算法密切相关，协作频谱感知的融合准则有AND准则、OR准则、表决融合准则等，其中OR准则等价于表决融合准则的投票门限为1、AND准则是等价于表决融合准则的投票门限为次用户的总个数，所以AND准则和OR准则都是表决融合准则的特例。故表决融合准则的投票门限值与频谱感知性能有着紧密联系。其次，利用双门限能量检测方法进行频谱感知时，检测门限值的选取很大程度上影响着感知的性能。在各接收信噪比条件下，认知用户如何确定双门限能量检测的检测门限值以及怎样选取合适的表决融合准则投票门限值成为了必须研究解决的问题。The main parameters to measure the performance of cooperative spectrum sensing are the global false alarm probability and the global missed detection probability, and their sum is defined as the global error probability. First of all, the performance of cooperative spectrum sensing is closely related to the fusion algorithm. The fusion criteria of collaborative spectrum sensing include AND criterion, OR criterion, voting fusion criterion, etc., where the OR criterion is equivalent to the voting threshold of the voting fusion criterion is 1, and the AND criterion is Since the voting threshold of the voting fusion criterion is the total number of secondary users, both the AND criterion and the OR criterion are special cases of the voting fusion criterion. Therefore, the voting threshold of the voting fusion criterion is closely related to the spectrum sensing performance. Secondly, when using the dual-threshold energy detection method for spectrum sensing, the selection of the detection threshold greatly affects the sensing performance. How to determine the detection threshold of dual-threshold energy detection for cognitive users and how to select the appropriate voting threshold for voting fusion criteria has become a problem that must be studied and solved under the conditions of each receiving signal-to-noise ratio.

发明内容Contents of the invention

技术问题：本发明的目的是提供一种认知无线网络中优化协作频谱感知性能的方案，该方法可以使得协作频谱感知的全局错误概率在各接收信噪比条件下都达到最小值，从而达到提高协作频谱感知性能的目的。Technical problem: The purpose of the present invention is to provide a scheme for optimizing the performance of cooperative spectrum sensing in a cognitive wireless network. This method can make the global error probability of cooperative spectrum sensing reach the minimum value under each receiving signal-to-noise ratio condition, thereby achieving The purpose of improving the performance of cooperative spectrum sensing.

技术方案：该方法对基于表决融合的双门限协作频谱感知中的投票门限和检测门限分别进行了优化，首先，固定双门限能量检测的检测门限值，对表决融合准则的投票门限进行优化，使得在该能量检测门限值条件下，协作频谱感知的全局错误概率最小；然后在表决融合准则的投票门限取最优值的前提下，对双门限能量检测的检测门限值进行了优化，在不同接收信噪比条件下，最优的检测门限值是动态的，所以要根据信噪比确定最优的检测门限值。Technical solution: This method optimizes the voting threshold and detection threshold in the dual-threshold cooperative spectrum sensing based on voting fusion. First, the detection threshold of the dual-threshold energy detection is fixed, and the voting threshold of the voting fusion criterion is optimized. Under the condition of the energy detection threshold value, the global error probability of cooperative spectrum sensing is the smallest; then, on the premise that the voting threshold of the voting fusion criterion is the optimal value, the detection threshold value of the dual-threshold energy detection is optimized, Under the condition of different receiving signal-to-noise ratio, the optimal detection threshold is dynamic, so the optimal detection threshold should be determined according to the signal-to-noise ratio.

本发明所针对的认知无线网络中基于表决融合的集中式协作频谱感知系统结构如图1所示。包括一个主用户，多个次用户以及一个融合中心，认知用户通过各自的感知信道(次用户接收主用户信号经过的信道)接收主用户信号的能量信息，然后对接收到的主用户信号的能量信息进行双门限能量检测，做出本地判决结果，并将各自的判决结果通过报告信道(次用户将判决结果传输给融合中心经历的信道)发送至融合中心，融合中心对接收到的判决结果进行表决融合，得到最终判决结果。认知无线网络根据此最终检测结果来判断该授权频谱是否空闲。The structure of the centralized cooperative spectrum sensing system based on voting fusion in the cognitive wireless network targeted by the present invention is shown in FIG. 1 . Including a primary user, multiple secondary users and a fusion center, the cognitive users receive the energy information of the primary user signal through their respective sensing channels (the channel through which the secondary user receives the primary user signal), and then analyze the energy information of the received primary user signal Double-threshold energy detection is performed on the energy information to make local judgment results, and the respective judgment results are sent to the fusion center through the reporting channel (the channel through which the secondary user transmits the judgment results to the fusion center), and the fusion center evaluates the received judgment results Voting fusion is carried out to obtain the final judgment result. The cognitive wireless network judges whether the licensed spectrum is free according to the final detection result.

能量检测法是频谱检测最基本的方法。它测量信道中的无线频率能量或者接收到的信号强度指标(RSSI)来判断信道是否被占用。能量检测的性能决定了最终的感知结果和感知性能，而能量检测的性能主要与门限值、噪声平均功率、信号平均功率和采样数有关。若采用传统的传统的单门限能量检测法，选取λ作为单门限能量检测的门限值，则能量检测的虚警概率p_f和检测概率p_d为：Energy detection method is the most basic method of spectrum detection. It measures the radio frequency energy in the channel or the received signal strength indicator (RSSI) to determine whether the channel is occupied. The performance of energy detection determines the final perception result and performance, and the performance of energy detection is mainly related to threshold value, noise average power, signal average power and sampling number. If the traditional traditional single-threshold energy detection method is adopted and λ is selected as the threshold value of single-threshold energy detection, the false alarm probability p _f and detection probability p _d of energy detection are:

${p p}_{f f} = = P P ((E E. ((x x)) > > λ λ | | {H h}_{00})) = = Q Q ((((\frac{λ λ}{{σ σ}_{u u}^{22}} - - 11)) \sqrt{τ τ {f f}_{s the s}}))$

${p p}_{d d} = = P P ((E E. ((x x)) > > λ λ | | {H h}_{11})) = = Q Q ((((\frac{λ λ}{{σ σ}_{u u}^{22}} - - γ γ - - 11)) \sqrt{\frac{τ τ {f f}_{s the s}}{22 γ γ + + 11}}))$

其中：γ为次用户的接收信噪比；是噪声方差；τ是次用户的感知时间；f_s为采样频率； $Q (x) = \frac{1}{\sqrt{2 π}} {&Integral;}_{x}^{\infty} \exp (- \frac{t^{2}}{2}) dt .$ Where: γ is the receiving signal-to-noise ratio of the secondary user; is the noise variance; τ is the perception time of the secondary user; f _s is the sampling frequency; $Q (x) = \frac{1}{\sqrt{2 π}} {&Integral;}_{x}^{\infty} \exp (- \frac{t^{2}}{2}) dt .$

本发明所采用的双门限能量检测有两个检测门限λ₀、λ₁，如图2所示，则次用户对应的本地判决准则为：The dual-threshold energy detection adopted in the present invention has two detection thresholds λ ₀ and λ ₁ , as shown in Fig. 2, then the local decision criterion corresponding to the secondary user is:

$D D. = = \{\begin{matrix} 00,, E E. ((x x)) < < {λ λ}_{00} \\ 11,, E E. ((x x)) > > {λ λ}_{11} \\ U u,, {λ λ}_{00} < < E E. ((x x)) < < {λ λ}_{11} \end{matrix}$

其中：U表示能量值落入不定区间时，次用户不做出本地判决。Among them: U means that when the energy value falls into an indeterminate range, the secondary user does not make a local decision.

由于隐蔽终端、多径衰落等问题的影响，单用户检测性能很差，多次用户协作频谱感知可以有效提高检测性能，协作频谱感知的融合准则有AND准则、OR准则、表决融合准则等，其中OR准则等价于表决融合准则的投票门限为1、AND准则是等价于表决融合准则的投票门限为次用户的总个数，所以AND准则和OR准则都是表决融合准则的特例。协作频谱感知的检测性能与表决融合准则的投票门限、双门限能量检测的检测门限密切相关，于是本发明首先对表决融合准则的投票门限进行优化，然后在表决融合准则的投票门限取最优值的前提下，对双门限能量检测的检测门限值进行了优化，在不同接收信噪比条件下，最优的检测门限值是动态的，所以要根据信噪比确定最优的检测门限值，使得协作频谱感知的全局错误概率在各信噪比条件下都达到最小值，从而提高了协作频谱感知的性能。Due to the influence of hidden terminals and multipath fading, single-user detection performance is very poor. Multi-user cooperative spectrum sensing can effectively improve detection performance. The fusion criteria of cooperative spectrum sensing include AND criterion, OR criterion, voting fusion criterion, etc., among which The OR criterion is equivalent to the voting threshold of the voting fusion criterion is 1, and the AND criterion is equivalent to the voting threshold of the voting fusion criterion being the total number of secondary users, so both the AND criterion and the OR criterion are special cases of the voting fusion criterion. The detection performance of cooperative spectrum sensing is closely related to the voting threshold of the voting fusion criterion and the detection threshold of the double-threshold energy detection, so the present invention first optimizes the voting threshold of the voting fusion criterion, and then takes the optimal value of the voting threshold of the voting fusion criterion Under the premise of , the detection threshold value of dual-threshold energy detection is optimized. Under the condition of different receiving signal-to-noise ratio, the optimal detection threshold value is dynamic, so the optimal detection threshold value should be determined according to the signal-to-noise ratio. The limit value makes the global error probability of cooperative spectrum sensing reach the minimum value under each SNR condition, thus improving the performance of cooperative spectrum sensing.

有益效果：双门限能量检测较之传统的单门限能量检测，可以大大降低频谱感知的误检概率，而且由于隐蔽终端、多径衰落等问题的影响，单用户检测性能很差，多次用户协作频谱感知可以有效提高检测性能。本发明基于双门限能量检测的协作频谱感知方法，在融合中心采用表决融合准则对各次用户的本地判决结果进行融合，将优化目标设为全局错误概率，基于该优化目标对表决融合准则的投票门限值和双门限能量检测的检测门限值进行选取，使全局错误概率最小，从而提高了协作频谱感知的性能。Beneficial effects: Compared with traditional single-threshold energy detection, double-threshold energy detection can greatly reduce the false detection probability of spectrum sensing, and due to the influence of hidden terminals, multipath fading and other problems, single-user detection performance is very poor, and multiple users cooperate Spectrum sensing can effectively improve detection performance. The cooperative spectrum sensing method based on double-threshold energy detection in the present invention adopts the voting fusion criterion in the fusion center to fuse the local judgment results of each user, sets the optimization goal as the global error probability, and votes for the voting fusion criterion based on the optimization goal The threshold value and the detection threshold value of double-threshold energy detection are selected to minimize the global error probability, thereby improving the performance of cooperative spectrum sensing.

附图说明Description of drawings

图1为本发明的系统模型。Fig. 1 is the system model of the present invention.

图2为本发明的双门限判决模型。Fig. 2 is the double-threshold decision model of the present invention.

具体实施方式Detailed ways

按照单门限能量检测的相关概率定义，定义双门限能量检测的相关概率如下：According to the definition of correlation probability of single-threshold energy detection, the correlation probability of dual-threshold energy detection is defined as follows:

${p p}_{f f} = = P P ((E E. ((x x)) > > {λ λ}_{11} | | {H h}_{00})) = = Q Q ((((\frac{{λ λ}_{11}}{{σ σ}_{u u}^{22}} - - 11)) \sqrt{τ τ {f f}_{s the s}})) - - - - - - ((11))$

${p p}_{a a} = = P P ((E E. ((x x)) < < {λ λ}_{00} | | {H h}_{00})) = = 11 - - Q Q ((((\frac{{λ λ}_{00}}{{σ σ}_{u u}^{22}} - - 11)) \sqrt{τ τ {f f}_{s the s}})) - - - - - - ((22))$

$\begin{matrix} {Δ Δ}_{00} = = P P (({λ λ}_{00} < < E E. ((x x)) < < {λ λ}_{11} | | {H h}_{00})) = = 11 - - {p p}_{f f} - - {p p}_{a a} \\ = = Q Q ((((\frac{{λ λ}_{00}}{{σ σ}_{u u}^{22}} - - 11)) \sqrt{τ τ {f f}_{s the s}})) - - Q Q ((((\frac{{λ λ}_{11}}{{σ σ}_{u u}^{22}} - - 11)) \sqrt{τ τ {f f}_{s the s}})) \end{matrix}- - - - - - - ((33))$

${p p}_{d d} = = P P ((E E. ((x x)) > > {λ λ}_{11} | | {H h}_{11})) = = Q Q ((((\frac{{λ λ}_{11}}{{σ σ}_{u u}^{22}} - - γ γ - - 11)) \sqrt{\frac{τ τ {f f}_{s the s}}{22 γ γ + + 11}})) - - - - - - ((44))$

${p p}_{m m} = = P P ((E E. ((x x)) < < {λ λ}_{00} | | {H h}_{11})) = = 11 - - Q Q ((((\frac{{λ λ}_{00}}{{σ σ}_{u u}^{22}} - - γ γ - - 11)) \sqrt{\frac{τ τ {f f}_{s the s}}{22 γ γ + + 11}})) - - - - - - ((55))$

$\begin{matrix} {Δ Δ}_{11} = = P P (({λ λ}_{00} < < E E. ((x x)) < < {λ λ}_{11} | | {H h}_{11})) = = 11 - - {p p}_{d d} - - {p p}_{m m} \\ = = Q Q ((((\frac{{λ λ}_{00}}{{σ σ}_{u u}^{22}} - - γ γ - - 11)) \sqrt{\frac{τ τ {f f}_{s the s}}{22 γ γ + + 11}})) - - Q Q ((((\frac{{λ λ}_{11}}{{σ σ}_{u u}^{22}} - - γ γ - - 11)) \sqrt{\frac{τ τ {f f}_{s the s}}{22 γ γ + + 11}})) \end{matrix}- - - - - - - ((66))$

其中：H₁和H₀分别表示主用户存在和不存在的情况，△₀和△₁分别表示在H₀、H₁条件下次用户接收到的能量值位于不定区间的概率，γ为次用户的接收信噪比，是噪声方差，τ是次用户的感知时间，f_s为采样频率，本发明所采用的双门限能量检测有两个检测门限λ₀、λ₁。Among them: H ₁ and H ₀ respectively represent the existence and non-existence of the primary user, △ ₀ and △ ₁ represent the probability that the energy value received by the user next time under the conditions of H ₀ and H ₁ is in the uncertain interval, and γ is the secondary user The receiving signal-to-noise ratio of is the noise variance, τ is the perception time of the secondary user, f _s is the sampling frequency, The double-threshold energy detection adopted in the present invention has two detection thresholds λ ₀ and λ ₁ .

假设有N个次用户进行频谱感知，其中做出本地判决的次用户将判决结果发送给融合中心，融合中心对接收到的判决结果进行表决融合，表决融合的投票门限为n，则融合中心按照表决融合准则：Assuming that there are N secondary users performing spectrum sensing, the secondary users who make local decisions send the judgment results to the fusion center, and the fusion center performs voting fusion on the received judgment results. The voting threshold for voting fusion is n, then the fusion center follows Voting Convergence Guidelines:

$F f = = \{\begin{matrix} 00,, {Σ Σ}_{i i = = 11}^{N N} {D D.}_{i i} < < n no \\ 11,, {Σ Σ}_{i i = = 11}^{N N} {D D.}_{i i} &GreaterEqual; &Greater Equal; n no \end{matrix} - - - - - - ((77))$

判断主用户的存在与否，其中D_i表示第i个次用户做出的本地判决结果。则本地判决采用单门限能量检测时，全局虚警概率Q_f和全局检测概率Q_d分别为：Judging whether the primary user exists or not, where D _i represents the local judgment result made by the ith secondary user. Then when the local judgment adopts single-threshold energy detection, the global false alarm probability Q _f and the global detection probability Q _d are respectively:

${Q Q}_{f f} = = 11 - - {Σ Σ}_{l l = = n no}^{N N} {C C}_{N N}^{l l} {((11 - - {p p}_{f f}))}^{l l} {p p}_{f f}^{N N - - l l} - - - - - - ((88))$

${Q Q}_{d d} = = {Σ Σ}_{l l = = n no}^{N N} {p p}_{d d}^{l l} {((11 - - {p p}_{d d}))}^{N N - - l l} - - - - - - ((99))$

本地判决采用双门限能量检测时，N个进行频谱感知的次用户中，若有K个次用户接收到的能量值位于确定区间，则有(N-K)个次用户不做出本地判决。所以全局虚警概率Q_f、漏检概率Q_m和检测概率Q_d分别为：When double-threshold energy detection is used for local decision, among N secondary users performing spectrum sensing, if the energy values received by K secondary users are in a certain interval, then (NK) secondary users do not make local decision. Therefore, the global false alarm probability Q _f , missed detection probability Q _m and detection probability Q _d are respectively:

$\begin{matrix} {Q Q}_{f f} = = 11 - - P P {{F f = = 00 | | {H h}_{00}}} \\ = = 11 - - P P {{F f = = 00,, K K &NotEqual; &NotEqual; N N | | {H h}_{00}}} - - P P {{F f = = 00,, K K = = N N | | {H h}_{00}}} \\ = = 11 - - {Σ Σ}_{K K = = 00}^{N N - - 11} {C C}_{N N}^{K K} {Π Π}_{i i = = 11}^{K K} P P {{{O o}_{i i} \leq \leq {λ λ}_{00} \cup \cup {O o}_{i i} &GreaterEqual; &Greater Equal; {λ λ}_{11} | | {H h}_{00}}} {Π Π}_{i i = = K K + + 11}^{N N} P P {{{λ λ}_{00} \leq \leq {O o}_{i i} \leq \leq {λ λ}_{11} | | {H h}_{00}}} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} \\ - - {Π Π}_{i i = = 11}^{N N} P P {{{O o}_{i i} \leq \leq {λ λ}_{00} \cup \cup {O o}_{i i} &GreaterEqual; &Greater Equal; {λ λ}_{11} | | {H h}_{00}}} {Σ Σ}_{l l = = n no}^{N N} {C C}_{N N}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} \\ = = 11 - - {Σ Σ}_{K K = = 00}^{N N - - 11} {C C}_{N N}^{K K} {Π Π}_{i i = = 11}^{K K} ((11 - - {Δ Δ}_{00})) {Π Π}_{i i = = K K + + 11}^{N N} {Δ Δ}_{00} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} - - {Π Π}_{i i = = 11}^{N N} ((11 - - {Δ Δ}_{00})) {Σ Σ}_{l l = = n no}^{N N} {C C}_{N N}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{N N - - l l} \\ = = 11 - - {Σ Σ}_{K K = = 00}^{N N - - 11} {C C}_{N N}^{K K} {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} - - {((11 - - {Δ Δ}_{00}))}^{N N} {Σ Σ}_{l l = = n no}^{N N} {C C}_{N N}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{N N - - l l} \\ = = 11 - - {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} \end{matrix} - - - - - - ((1010))$

$\begin{matrix} {Q Q}_{m m} = = P P {{F f = = 00 | | {H h}_{11}}} \\ = = P P {{F f = = 00,, K K &NotEqual; &NotEqual; N N | | {H h}_{11}}} + + P P {{F f = = 00,, K K = = N N | | {H h}_{11}}} \\ = = {Σ Σ}_{K K = = 00}^{N N - - 11} {C C}_{N N}^{K K} {Π Π}_{i i = = 11}^{K K} P P {{{O o}_{i i} \leq \leq {λ λ}_{00} \cup \cup {O o}_{i i} &GreaterEqual; &Greater Equal; {λ λ}_{11} | | {H h}_{11}}} {Π Π}_{i i = = K K + + 11}^{N N} P P {{{λ λ}_{00} \leq \leq {O o}_{i i} \leq \leq {λ λ}_{11} | | {H h}_{11}}} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} \\ + + {Π Π}_{i i = = 11}^{N N} P P {{{O o}_{i i} \leq \leq {λ λ}_{00} \cup \cup {O o}_{i i} &GreaterEqual; &Greater Equal; {λ λ}_{11} | | {H h}_{11}}} {Σ Σ}_{l l = = n no}^{N N} {C C}_{N N}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} \\ = = {Σ Σ}_{K K = = 00}^{N N - - 11} {C C}_{N N}^{K K} {Π Π}_{i i = = 11}^{K K} ((11 - - {Δ Δ}_{11})) {Π Π}_{i i = = K K + + 11}^{N N} {Δ Δ}_{11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} + + {Π Π}_{i i = = 11}^{N N} ((11 - - {Δ Δ}_{11})) {Σ Σ}_{l l = = n no}^{N N} {C C}_{N N}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{N N - - l l} \\ = = {Σ Σ}_{K K = = 00}^{N N - - 11} {C C}_{N N}^{K K} {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} + + {((11 - - {Δ Δ}_{11}))}^{N N} {Σ Σ}_{l l = = n no}^{N N} {C C}_{N N}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{N N - - l l} \\ = = {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} \end{matrix} - - - - - - ((1111))$

Q_d＝1-Q_m (12)Q _d =1-Q _m (12)

a.表决融合准则的投票门限n的优化a. Optimization of the voting threshold n of the voting fusion criterion

对于协作频谱感知，融合准则的选取会影响频谱感知的性能，OR准则和AND准则分别是表决融合准则的投票门限n等于1和等于次用户的总个数N的特殊情况，由式(10)、(11)和(12)知，表决融合准则的投票门限n会影响协作频谱感知的性能，所以需要对它的值进行最优化，使得协作频谱感知的性能最优。For cooperative spectrum sensing, the selection of the fusion criterion will affect the performance of spectrum sensing. The OR criterion and the AND criterion are special cases where the voting threshold n of the voting fusion criterion is equal to 1 and equal to the total number of secondary users N. Formula (10) , (11) and (12), the voting threshold n of the voting fusion criterion will affect the performance of cooperative spectrum sensing, so its value needs to be optimized to make the performance of cooperative spectrum sensing optimal.

我们定义频谱感知全局错误概率为(Q_f+Q_m)，假设做出本地判决的次用户的个数K已知，则优化问题可以写为：We define the global error probability of spectrum sensing as (Q _f +Q _m ), assuming that the number K of secondary users making local decisions is known, then the optimization problem can be written as:

min(Q_f+Q_m) (13)min(Q _f +Q _m ) (13)

s.t.n≤Ks.t.n≤K

求表决融合准则的最优投票门限n_opt，使得全局错误概率为(Q_f+Q_m)达到最小值。Find the optimal voting threshold n _opt of the voting fusion criterion, so that the global error probability (Q _f +Q _m ) reaches the minimum value.

由式(10)、(11)，得全局错误概率：From equations (10) and (11), the global error probability is obtained:

$\begin{matrix} {Q Q}_{f f} + + {Q Q}_{m m} \\ = = 11 + + {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} (({((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} - - {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l})) \end{matrix} - - - - - - ((1414))$

设：set up:

$G G ((n no)) = = {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} - - {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} - - - - - - ((1515))$

则由式(14)、(15)可得：Then from equations (14) and (15), we can get:

${Q Q}_{f f} + + {Q Q}_{m m} = = 11 + + {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} G G ((n no)) - - - - - - ((1616))$

因为：because:

$\frac{&PartialD; &PartialD; (({Q Q}_{f f} + + {Q Q}_{m m}))}{&PartialD; &PartialD; n no} {| |}_{{n no}_{opt opt}} = = {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; n no} {| |}_{{n no}_{opt opt}} = = 00 - - - - - - ((1717))$

$\begin{matrix} \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; n no} \approx \approx \frac{G G ((n no + + 11)) - - G G ((n no))}{((n no + + 11)) - - n no} = = G G ((n no + + 11)) - - G G ((n no)) \\ = = {C C}_{K K}^{n no} (({((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {p p}_{a a}^{n no} {p p}_{f f}^{K K - - n no} - - {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {p p}_{m m}^{n no} {p p}_{d d}^{K K - - n no})) \end{matrix}- - - - - - - ((1818))$

所以由式(17)、(18)可得：So from equations (17) and (18), we can get:

$\begin{matrix} \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; n no} {| |}_{{n no}_{opt opt}} = = 00 &DoubleRightArrow; &DoubleRightArrow; {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {p p}_{a a}^{{n no}_{opt opt}} {p p}_{f f}^{{K K - - n no}_{opt opt}} = = {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {p p}_{m m}^{{n no}_{opt opt}} {p p}_{d d}^{{K K - - n no}_{opt opt}} \\ &DoubleRightArrow; &DoubleRightArrow; {((\frac{{p p}_{a a}}{{p p}_{m m}}))}^{{n no}_{opt opt}} = = {((\frac{{p p}_{d d}}{{p p}_{f f}}))}^{{K K - - n no}_{opt opt}} {((\frac{11 - - {Δ Δ}_{11}}{11 - - {Δ Δ}_{00}}))}^{K K} {((\frac{{Δ Δ}_{11}}{{Δ Δ}_{00}}))}^{N N - - K K} \end{matrix} - - - - - - ((1919))$

对上式两边取对数得：Take the logarithm on both sides of the above formula to get:

${n no}_{opt opt} ln ln \frac{{p p}_{a a}}{{p p}_{m m}} = = ((K K - - {n no}_{opt opt})) ln ln \frac{{p p}_{d d}}{{p p}_{f f}} + + ln ln {((\frac{11 - - {Δ Δ}_{11}}{11 - - {Δ Δ}_{00}}))}^{K K} {((\frac{{Δ Δ}_{11}}{{Δ Δ}_{00}}))}^{N N - - K K} - - - - - - ((2020))$

解得：Solutions have to:

${n no}_{opt opt} = = \frac{K K ln ln \frac{{p p}_{d d}}{{p p}_{f f}} + + ln ln {((\frac{11 - - {Δ Δ}_{11}}{11 - - {Δ Δ}_{00}}))}^{K K} {((\frac{{Δ Δ}_{11}}{{Δ Δ}_{00}}))}^{N N - - K K}}{ln ln \frac{{p p}_{a a} {p p}_{d d}}{{p p}_{m m} {p p}_{f f}}} - - - - - - ((21 twenty one))$

所以当 $n = n_{opt} = \frac{K \ln \frac{p_{d}}{p_{f}} + \ln {(\frac{1 - Δ_{1}}{1 - Δ_{0}})}^{K} {(\frac{Δ_{1}}{Δ_{0}})}^{N - K}}{\ln \frac{p_{a} p_{d}}{p_{m} p_{f}}}$ 时， $\frac{&PartialD; G (n)}{&PartialD; n} |_{n_{opt}} = 0, (Q_{f} + Q_{m})$ 取得极值。又因为：so when $no = {no}_{opt} = \frac{K \ln \frac{p_{d}}{p_{f}} + \ln {(\frac{1 - Δ_{1}}{1 - Δ_{0}})}^{K} {(\frac{Δ_{1}}{Δ_{0}})}^{N - K}}{\ln \frac{p_{a} p_{d}}{p_{m} p_{f}}}$ hour, $\frac{&PartialD; G (no)}{&PartialD; no} |_{{no}_{opt}} = 0, (Q_{f} + Q_{m})$ Get the extreme value. also because:

$\frac{{&PartialD; &PartialD;}^{22} G G ((n no))}{{&PartialD; &PartialD; n no}^{22}} = = \frac{\frac{&PartialD; &PartialD; G G ((n no + + 11))}{&PartialD; &PartialD; n no} - - \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; n no}}{((n no + + 11)) - - n no} = = \frac{&PartialD; &PartialD; G G ((n no + + 11))}{&PartialD; &PartialD; n no} - - \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; n no} - - - - - - ((22 twenty two))$

$\begin{matrix} \frac{{&PartialD; &PartialD;}^{22} G G ((n no))}{&PartialD; &PartialD; {n no}^{22}} {| |}_{{n no = = n no}_{opt opt}} = = \frac{&PartialD; &PartialD; G G ((n no + + 11))}{&PartialD; &PartialD; n no} {| |}_{{n no = = n no}_{opt opt}} - - \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; n no} {| |}_{{n no = = n no}_{opt opt}} = = \frac{&PartialD; &PartialD; G G ((n no + + 11))}{&PartialD; &PartialD; n no} {| |}_{n no = = {n no}_{opt opt}} - - 00 \\ = = {C C}_{K K}^{{n no}_{opt opt} + + 11} (({((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {p p}_{a a}^{{n no}_{opt opt} + + 11} {p p}_{f f}^{{K K - - n no}_{opt opt} - - 11} - - {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {p p}_{m m}^{{n no}_{opt opt} + + 11} {p p}_{d d}^{K K - - {n no}_{opt opt} - - 11})) \\ = = {C C}_{K K}^{{n no}_{opt opt} + + 11} (({((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} \frac{{p p}_{a a}}{{p p}_{f f}} {p p}_{a a}^{{n no}_{opt opt}} {p p}_{f f}^{{K K - - n no}_{opt opt}} - - {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} \frac{{p p}_{m m}}{{p p}_{d d}} {p p}_{m m}^{{n no}_{opt opt}} {p p}_{d d}^{{K K - - n no}_{opt opt}})) \end{matrix} - - - - - - ((23 twenty three))$

由式(1)、(2)、(4)和(5)知p_a>p_m，p_d>p_f，故：It is known from formulas (1), (2), (4) and (5) that p _a >p _m , p _d >p _f , so:

$\frac{{p p}_{a a}}{{p p}_{f f}} > > \frac{{p p}_{m m}}{{p p}_{d d}} - - - - - - ((24 twenty four))$

所以由式(19)、(23)和(24)得：So from equations (19), (23) and (24):

${((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} \frac{{p p}_{a a}}{{p p}_{f f}} {p p}_{a a}^{{n no}_{opt opt}} {p p}_{f f}^{{K K - - n no}_{opt opt}} > > {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} \frac{{p p}_{m m}}{{p p}_{d d}} {p p}_{m m}^{{n no}_{opt opt}} {p p}_{d d}^{{K K - - n no}_{opt opt}} - - - - - - ((2525))$

代入式(23)得： $\frac{{&PartialD;}^{2} G (n)}{&PartialD; n^{2}} |_{n = n_{opt}} > 0,$ Substitute into formula (23) to get: $\frac{{&PartialD;}^{2} G (no)}{&PartialD; {no}^{2}} |_{no = {no}_{opt}} > 0,$

所以： $\frac{{&PartialD;}^{2} (Q_{f} + Q_{m})}{&PartialD; n^{2}} |_{n_{opt}} = Σ_{K = 0}^{N} C_{N}^{K} \frac{{&PartialD;}^{2} G (n)}{{&PartialD; n}^{2}} |_{n_{opt}} > 0 .$ so: $\frac{{&PartialD;}^{2} (Q_{f} + Q_{m})}{&PartialD; {no}^{2}} |_{{no}_{opt}} = Σ_{K = 0}^{N} C_{N}^{K} \frac{{&PartialD;}^{2} G (no)}{{&PartialD; no}^{2}} |_{{no}_{opt}} > 0 .$

综上所述，当 $n = n_{opt} = \frac{K \ln \frac{p_{d}}{p_{f}} + \ln {(\frac{1 - Δ_{1}}{1 - Δ_{0}})}^{K} {(\frac{Δ_{1}}{Δ_{0}})}^{N - K}}{\ln \frac{p_{a} p_{d}}{p_{m} p_{f}}}$ 时， $\frac{&PartialD; (Q_{f} + Q_{m})}{&PartialD; n} |_{n_{opt}} = Σ_{K = 0}^{N} C_{N}^{K} \frac{&PartialD; G (n)}{&PartialD; n} |_{n_{opt}} = 0, \frac{{&PartialD;}^{2} (Q_{f} + Q_{m})}{&PartialD; n^{2}} |_{n_{opt}} = Σ_{K = 0}^{N} C_{N}^{K} \frac{{&PartialD;}^{2} G (n)}{{&PartialD; n}^{2}} |_{n_{opt}} > 0,$ 所以(Q_f+Q_m)取得极小值。实际应用中，投票门限n为正整数，所以：In summary, when $no = {no}_{opt} = \frac{K \ln \frac{p_{d}}{p_{f}} + \ln {(\frac{1 - Δ_{1}}{1 - Δ_{0}})}^{K} {(\frac{Δ_{1}}{Δ_{0}})}^{N - K}}{\ln \frac{p_{a} p_{d}}{p_{m} p_{f}}}$ hour, $\frac{&PartialD; (Q_{f} + Q_{m})}{&PartialD; no} |_{{no}_{opt}} = Σ_{K = 0}^{N} C_{N}^{K} \frac{&PartialD; G (no)}{&PartialD; no} |_{{no}_{opt}} = 0, \frac{{&PartialD;}^{2} (Q_{f} + Q_{m})}{&PartialD; {no}^{2}} |_{{no}_{opt}} = Σ_{K = 0}^{N} C_{N}^{K} \frac{{&PartialD;}^{2} G (no)}{{&PartialD; no}^{2}} |_{{no}_{opt}} > 0,$ Therefore (Q _f +Q _m ) takes a minimum value. In practical applications, the voting threshold n is a positive integer, so:

表示不小于x的最小整数。当双门限能量检测的门限值λ₀、λ₁确定时，可以将λ₀、λ₁的值代入(1)～(6)式，求得p_d、p_f、p_a、p_m以及△₀、△₁，代入上式即可求出使得(Q_f+Q_m)取得极小值的投票门限n_opt。 Indicates the smallest integer not less than x. When the threshold values λ ₀ and λ ₁ of the dual-threshold energy detection are determined, the values of λ ₀ and λ ₁ can be substituted into formulas (1) to (6) to obtain p _d , p _f , p _a , p _m and △ ₀ and △ ₁ are substituted into the above formula to obtain the voting threshold n _opt that makes (Q _f +Q _m ) obtain a minimum value.

b.双门限能量检测的门限值λ₀、λ₁的优化b. Optimization of threshold values λ ₀ and λ ₁ for dual-threshold energy detection

对于双门限能量检测，门限值λ₀、λ₁的取值以及它们之间的关系都会对频谱感知的性能产生影响，其中λ₀≤λ₁。由(1)～(6)式可知，当λ₀的取值减小时，漏检概率会降低当λ₁的取值增大时，虚警概率会降低，但是同时检测概率也会降低；当两个门限之间的距离拉大时，次用户接收到的能量值位于不定区间的概率加大，反之则减小，当λ₀＝λ₁时，等价于单门限能量检测，则次用户接收到的能量值位于不定区间的概率为0。For dual-threshold energy detection, the threshold values λ ₀ , λ ₁ and the relationship between them will affect the performance of spectrum sensing, where λ ₀ ≤ λ ₁ . It can be seen from (1)~(6) that when the value of λ ₀ decreases, the probability of missed detection will decrease; when the value of λ ₁ increases, the probability of false alarm will decrease, but at the same time the detection probability will also decrease; when When the distance between the two thresholds increases, the probability that the energy value received by the secondary user is in the indeterminate range increases, and vice versa, it decreases. When λ ₀ =λ ₁ , it is equivalent to single-threshold energy detection, and the secondary user The probability that the received energy value is in the indeterminate interval is 0.

min(Q_f+Q_m) (27)min(Q _f +Q _m ) (27)

s.t.0<λ₀<λ₁<+∞st0<λ ₀ <λ ₁ <+∞

求双门限能量检测的最优门限值使得全局错误概率为(Q_f+Q_m)达到最小值，即满足：Calculating the Optimal Threshold Value of Dual-Threshold Energy Detection Make the global error probability (Q _f +Q _m ) reach the minimum value, namely satisfy:

$\{\begin{matrix} \frac{&PartialD; &PartialD; (({Q Q}_{f f} + + {Q Q}_{m m}))}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = 00 \\ \frac{&PartialD; &PartialD; (({Q Q}_{f f} + + {Q Q}_{m m}))}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = 00 \end{matrix} - - - - - - ((2828))$

其中：in:

$\frac{&PartialD; &PartialD; (({Q Q}_{f f} + + {Q Q}_{m m}))}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = \frac{&PartialD; &PartialD; ((11 + + {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} G G ((n no))))}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = 00 - - - - - - ((2929))$

$\frac{&PartialD; &PartialD; (({Q Q}_{f f} + + {Q Q}_{m m}))}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = \frac{&PartialD; &PartialD; ((11 + + {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} G G ((n no))))}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = 00 - - - - - - ((3030))$

则 $\frac{&PartialD; G (n)}{&PartialD; λ_{0}} |_{λ_{0_{opt}}} = 0, \frac{&PartialD; G (n)}{&PartialD; λ_{1}} |_{λ_{1_{opt}}} = 0 .$ but $\frac{&PartialD; G (no)}{&PartialD; λ_{0}} |_{λ_{0_{opt}}} = 0, \frac{&PartialD; G (no)}{&PartialD; λ_{1}} |_{λ_{1_{opt}}} = 0 .$

将式(15)代入上式得：Substitute (15) into the above formula to get:

$\begin{matrix} \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = \frac{&PartialD; &PartialD; {p p}_{m m}}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} ((K K {((11 - - {Δ Δ}_{11}))}^{K K - - 11} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} \\ - - {((11 - - {Δ Δ}_{11}))}^{K K} ((N N - - K K)) {Δ Δ}_{11}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} + + {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} l l {p p}_{m m}^{l l - - 11} {p p}_{d d}^{K K - - l l} \\ - - \frac{&PartialD; &PartialD; {p p}_{a a}}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} ((K K {((11 - - {Δ Δ}_{00}))}^{K K - - 11} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} \\ + + {((11 - - {Δ Δ}_{00}))}^{K K} ((N N - - K K)) {Δ Δ}_{00}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} - - {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} l l {p p}_{a a}^{l l - - 11} {p p}_{f f}^{K K - - l l} \\ = = 00 \end{matrix} - - - - - - ((3131))$

其中：in:

$\frac{&PartialD; &PartialD; {p p}_{m m}}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = - - \sqrt{\frac{τ τ {f f}_{s the s}}{22 π π ((22 γ γ + + 11))}} exp exp ((\frac{{(({λ λ}_{00_{opt opt}} - - γ γ - - 11))}^{22} τ τ {f f}_{s the s}}{22 ((22 γ γ + + 11))})) - - - - - - ((3232))$

$\frac{&PartialD; &PartialD; {p p}_{a a}}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = - - \sqrt{\frac{τ τ {f f}_{s the s}}{22 π π}} exp exp ((- - \frac{{(({λ λ}_{00_{opt opt}} - - 11))}^{22} τ τ {f f}_{s the s}}{22})) - - - - - - ((3333))$

将式(31)、(32)和(33)代入式(33)得：Substituting equations (31), (32) and (33) into equation (33), we get:

$\begin{matrix} \frac{&PartialD; &PartialD; (({Q Q}_{f f} + + {Q Q}_{m m}))}{&PartialD; &PartialD; {λ λ}_{00}} {| |}_{{λ λ}_{00_{opt opt}}} = = {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} ((- - \sqrt{\frac{τ τ {f f}_{s the s}}{22 π π ((22 γ γ + + 11))}} exp exp ((- - \frac{{(({λ λ}_{00_{opt opt}} - - γ γ - - 11))}^{22} τ τ {f f}_{s the s}}{22 ((22 γ γ + + 11))})))) \\ ((K K {((11 - - {Δ Δ}_{11}))}^{K K - - 11} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} - - {((11 - - {Δ Δ}_{11}))}^{K K} ((N N - - K K)) {Δ Δ}_{11}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - 11} \\ + + {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} l l {p p}_{m m}^{l l - - 11} {p p}_{d d}^{K K - - l l})) \\ + + ((\sqrt{\frac{τ τ {f f}_{s the s}}{22 π π}} exp exp ((- - \frac{{(({λ λ}_{00_{opt opt}} - - 11))}^{22} τ τ {f f}_{s the s}}{22})))) ((K K {((11 - - {Δ Δ}_{00}))}^{K K - - 11} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} \\ + + {((11 - - {Δ Δ}_{00}))}^{K K} ((N N - - K K)) {Δ Δ}_{00}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} - - {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} l l {p p}_{a a}^{l l - - 11} {p p}_{f f}^{K K - - l l})) \\ = = 00 \end{matrix} - - - - - - ((3434))$

由式(34)即可解得最优门限值 The optimal threshold value can be obtained by formula (34)

$\begin{matrix} \frac{&PartialD; &PartialD; G G ((n no))}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = \frac{&PartialD; &PartialD; {p p}_{d d}}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} ((K K {((11 - - {Δ Δ}_{11}))}^{K K - - 11} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} \\ - - {((11 - - {Δ Δ}_{11}))}^{K K} ((N N - - K K)) {Δ Δ}_{11}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} + + {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} ((K K - - l l)) {p p}_{d d}^{K K - - l l - - 11})) \\ - - \frac{&PartialD; &PartialD; {p p}_{f f}}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} ((K K {((11 - - {Δ Δ}_{00}))}^{K K - - 11} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} + + {((11 - - {Δ Δ}_{00}))}^{K K} ((N N - - K K)) {Δ Δ}_{00}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} \\ - - {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} ((K K - - l l)) {p p}_{f f}^{K K - - l l - - 11})) \end{matrix} - - - - - - ((3535))$

其中：in:

$\frac{&PartialD; &PartialD; {p p}_{d d}}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = \sqrt{\frac{τ τ {f f}_{s the s}}{22 π π ((22 γ γ + + 11))}} exp exp ((- - \frac{{(({λ λ}_{11_{opt opt}} - - γ γ - - 11))}^{22} τ τ {f f}_{s the s}}{22 ((22 γ γ + + 11))})) - - - - - - ((3636))$

$\frac{&PartialD; &PartialD; {p p}_{f f}}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = \sqrt{\frac{τ τ {f f}_{s the s}}{22 π π}} exp exp ((- - \frac{{(({λ λ}_{11_{opt opt}} - - 11))}^{22} τ τ {f f}_{s the s}}{22})) - - - - - - ((3737))$

将式(35)、(36)和(37)代入(28)得：Substitute (35), (36) and (37) into (28) to get:

$\begin{matrix} \frac{&PartialD; &PartialD; (({Q Q}_{f f} + + {Q Q}_{m m}))}{&PartialD; &PartialD; {λ λ}_{11}} {| |}_{{λ λ}_{11_{opt opt}}} = = \sqrt{\frac{τ τ {f f}_{s the s}}{22 π π ((22 γ γ + + 11))}} exp exp ((- - \frac{{(({λ λ}_{11_{opt opt}} - - γ γ - - 11))}^{22} τ τ {f f}_{s the s}}{22 ((22 γ γ + + 11))})) \\ ((K K {((11 - - {Δ Δ}_{11}))}^{K K - - 11} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} - - {((11 - - {Δ Δ}_{11}))}^{K K} ((N N - - K K)) {Δ Δ}_{11}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} \\ + + {((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} ((K K - - l l)) {p p}_{d d}^{K K - - l l - - 11})) - - \sqrt{\frac{τ τ {f f}_{s the s}}{22 π π}} exp exp ((- - \frac{{(({λ λ}_{11_{opt opt}} - - 11))}^{22} τ τ {f f}_{s the s}}{22})) \\ ((K K {((11 - - {Δ Δ}_{00}))}^{K K - - 11} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} + + {((11 - - {Δ Δ}_{00}))}^{K K} ((N N - - K K)) {Δ Δ}_{00}^{N N - - K K - - 11} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{K K - - l l} \\ - - {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} ((K K - - l l)) {p p}_{f f}^{K K - - l l - - 11})) \\ = = 00 \end{matrix} - - - - - - ((3838))$

由式(38)即可解得最优门限值 The optimal threshold value can be obtained by formula (38)

由公式(34)(38)可知，不同的信噪比条件下，双门限能量检测的门限最优值的取值各异。取表决融合准则的投票门限n＝n_opt，即表决融合准则达到最优的前提下，由公式(34)(38)可计算得在各信噪比条件下的值。From formulas (34)(38), it can be seen that under different SNR conditions, the optimal value of the threshold for dual-threshold energy detection The value of is different. Take the voting threshold n=n _opt of the voting fusion criterion, that is, under the premise that the voting fusion criterion is optimal, it can be calculated by formulas (34)(38) that under each SNR condition value.

Claims

1. an optimization method for cooperative spectrum sensing threshold in a cognitive wireless network, characterized in that the method comprises the following steps:

a. Define λ ₀ and λ ₁ as the two threshold values of double-threshold energy detection, and require the threshold value λ ₀ ≤ λ ₁ , [0,λ ₀ ]∪[λ ₁ ,+∞) is the definite interval, ( λ ₀ , λ ₁ ) are uncertain intervals, and the global error probability (Q _f +Q _m ) of collaborative spectrum sensing is:

{Q Q}_{f f} + + {Q Q}_{m m} = = 11 + + {Σ Σ}_{K K = = 00}^{N N} {C C}_{N N}^{K K} (({((11 - - {Δ Δ}_{11}))}^{K K} {Δ Δ}_{11}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{m m}^{l l} {p p}_{d d}^{K K - - l l} - - {((11 - - {Δ Δ}_{00}))}^{K K} {Δ Δ}_{00}^{N N - - K K} {Σ Σ}_{l l = = n no}^{K K} {C C}_{K K}^{l l} {p p}_{a a}^{l l} {p p}_{f f}^{k k - - l l}))

Among the N cognitive users, the signal energy received by K cognitive users falls within the definite interval, so specifically which K cognitive users receive the signal energy within the definite interval, there is H ₁ and H ₀ respectively represent the existence and non-existence of the primary user, △ ₀ and △ ₁ respectively represent the probability that the energy value received by the user next time under the conditions of H ₀ and H ₁ is in an uncertain interval:

{Δ Δ}_{00} = = P P (({λ λ}_{00} < < E E. ((x x)) < < {λ λ}_{11} | | {H h}_{00})) = = 11 - - {p p}_{f f} - - {p p}_{a a} = = Q Q ((((\frac{{λ λ}_{00}}{{σ σ}_{u u}^{22}} - - 11)) \sqrt{τ τ {f f}_{s the s}})) - - Q Q ((((\frac{{λ λ}_{11}}{{σ σ}_{u u}^{22}} - - 11)) \sqrt{τ τ {f f}_{s the s}}))

{Δ Δ}_{11} = = P P (({λ λ}_{00} < < E E. ((x x)) < < {λ λ}_{11} | | {H h}_{11})) = = 11 - - {p p}_{d d} - - {p p}_{m m} = = Q Q ((((\frac{{λ λ}_{00}}{{σ σ}_{u u}^{22}} - - γ γ - - 11)) \sqrt{\frac{τ τ {f f}_{s the s}}{22 γ γ + + 11}})) - - Q Q ((((\frac{{λ λ}_{11}}{{σ σ}_{u u}^{22}} - - γ γ - - 11)) \sqrt{\frac{τ τ {f f}_{s the s}}{22 γ γ + + 11}}));;

p _d , p _f and _pm represent the detection probability, false alarm probability and missed detection probability of local spectrum sensing respectively:

p_{d} = Q ((\frac{λ_{1}}{σ_{u}^{2}} - γ - 1) \sqrt{\frac{τ f_{the s}}{2 γ + 1}}), p_{f} = Q ((\frac{λ_{1}}{σ_{u}^{2}} - 1) \sqrt{τ f_{the s}}), p_{m} = 1 - Q ((\frac{λ_{0}}{σ_{u}^{2}} - γ - 1) \sqrt{\frac{τ f_{the s}}{2 γ + 1}}),

p_{a} = 1 - Q ((\frac{λ_{0}}{σ_{u}^{2}} - 1) \sqrt{τ f_{the s}}),

in

Q (x) = \frac{1}{\sqrt{2 π}} {&Integral;}_{x}^{\infty} \exp (- \frac{t^{2}}{2}) dt,

γ is the receiving signal-to-noise ratio of the secondary user, is the noise variance, τ is the perception time of the secondary user, and f _s is the sampling frequency;

b. Optimize the voting threshold n of the voting fusion criterion, so that the global error probability (Q _f +Q _m ) of cooperative spectrum sensing is the smallest, so it is required

\frac{&PartialD; (Q_{f} + Q_{m})}{&PartialD; no} |_{no = {no}_{opt}} = 0, \frac{{&PartialD;}^{2} (Q_{f} + Q_{m})}{&PartialD; {no}^{2}} |_{no = {no}_{opt}} > 0,

get the optimal voting threshold

{no}_{opt} = \frac{K \ln \frac{p_{d}}{p_{f}} + \ln {(\frac{1 - Δ_{1}}{1 - Δ_{0}})}^{K} {(\frac{Δ_{1}}{Δ_{0}})}^{N - K}}{\ln \frac{p_{a} p_{d}}{p_{m} p_{f}}},

In practical applications, the voting threshold n is a positive integer, so Indicates the smallest integer not less than x;

c. Optimize the detection thresholds λ ₀ and λ ₁ of dual-threshold energy detection, so that the global error probability (Q _f +Q _m ) of cooperative spectrum sensing reaches the minimum value, so that

\frac{&PartialD; (Q_{f} + Q_{m})}{&PartialD; λ_{1}} |_{λ_{1} = λ_{1_{opt}}} = 0,

The following equations can be obtained:

When the receiving signal-to-noise ratio γ of cognitive users, the number N of cognitive users, the number K of cognitive users whose received signal energy falls into a certain interval, the perception time τ and the sampling frequency f _s are determined, in the voting fusion Under the premise that the voting threshold of the criterion takes its optimal value, that is, n=n _opt , the optimal detection threshold value of the double-threshold energy detection can be obtained from the above equations