CN104270210A - Soft-decision spectrum sensing method based on compression non-reconstruction - Google Patents

Abstract

The invention discloses a soft-decision spectrum sensing method based on compression non-reconstruction. The method mainly comprises the steps that firstly, a sensing model is established, secondary users establish a local spectrum sensing model, a binary presumptive model is adopted, and the energy detecting method is adopted under the non-cooperative mode; secondly, compressed sampling is carried out, and the secondary users utilize a two-dimensional Gaussian measurement matrix to carry out compressed sampling on a receiving signal; thirdly, under the situation that the compressed sampling data are not reconstructed, the secondary users directly utilize the compressed sampling data to establish the test statistics amount based on energy, whether a primary user exists or not is not judged, and the test statistics amount is transmitted to a fusion center; fourthly, the soft decision of the fusion center is carried out, and after the fusion center collects the test statistics amount sent by all the secondary users, the unified test statistics amount is established and compared with the detecting threshold value, and a final decision result is given.

Description

Based on the soft-decision frequency spectrum sensing method of the non-reconstruct of compression

Technical field

The present invention relates to a kind of frequency spectrum sensing method for distributed frequency spectrum monitoring system, particularly a kind of soft-decision frequency spectrum sensing method based on the non-reconstruct of compression.

Background technology

Frequency spectrum perception is the key technology in cognitive radio networks, is to determine whether primary user exists, finds the important means of idle channel.How fast and accurately in perception monitoring range, spectrum occupancy is the major issue that frequency spectrum perception faces.Current, the distributed frequency spectrum monitoring system based on wireless sense network utilizes cognitive radio technology to realize perception and the utilization of frequency spectrum.In order to overcome the impact of multipath fading and channel fading, usually adopt the mode of the cooperative detection such as soft-decision and hard decision, under identical circumstances, soft-decision performance is better than hard decision, but soft-decision needs sacrifice transmission bandwidth resource to be cost.During hard decision, secondary user's only transmits the court verdict information of 1 bit, and during soft-decision, secondary user's needs the sample value or the test statistics that transmit collection, and the transfer resource of needs is much larger than hard decision.But soft-decision has higher detection probability, in channel width not by limited time, be still used widely.

In cognitive radio networks, secondary user's is before use frequency spectrum resource, in order to not produce interference to primary user, efficient frequency spectrum perception must be carried out, in this locality, judgement adopts often as methods such as energy measuring, feature detection, matched filterings, and requires secondary user's to sample to the received signal with Nyquist rate, and sampling rate is at least the twice of signal highest frequency, in the undistorted reproduction primary signal of receiving terminal, and then signal analysis and processing could be carried out.But along with electronic technology development, when signal occupied bandwidth is very wide, the data volume that need process at receiving terminal is very large.In practice, to wish under the prerequisite ensured signal quality with minimum data from the sample survey to recover primary signal, thus reduce the cost of signal storage, process and transmission.And compression sampling is theoretical under hypothesis signal is sparse condition, original high dimensional signal is projected on a lower dimensional space with the incoherent observing matrix of transform-based by one, and then solve an optimization problem and just from these a small amount of projections, primary signal can be reconstructed with high probability.In compressive sensing theory, sampling and the compression of signal are carried out with low rate simultaneously, make sampling and assess the cost greatly to reduce, and signaling protein14-3-3 process are optimization computational processes.

Compression sampling and frequency spectrum perception combine by the present invention, propose a kind of soft-decision frequency spectrum sensing method based on the non-reconstruct of compression.In recent years, also document is had to carry out the theoretical application study in frequency spectrum detection of compression sampling, as the broader frequency spectrum detection algorithm based on compressed sensing, the compressed sensing signal detecting method based on sampled value numerical characteristic and broader frequency spectrum cognition technology of detecting based on sub-nyquist sampling and cycle specificity etc., what above-mentioned algorithm had need reconstruct primary signal from measured value, just be reduced in the pressure in transmission, do not give full play to the advantage of compression sampling theory, have do not provide compression sampling after perception algorithm performance close solution expression formula, just pass through simulating, verifying.The present invention proposes the soft-decision frequency spectrum sensing method based on the non-reconstruct of compression, utilize the observing matrix of compression sampling that original signal is carried out dimension-reduction treatment, when not reconstructing primary signal, the method of energy measuring is adopted in the judgement of this locality, simultaneously for reducing transmission pressure, and keep higher detection perform, respective test statistics is passed to fusion center by secondary user's, fusion center sets up unified test statistics, and makes final differentiation.In addition the present invention derive based on compression non-reconstruct soft-decision frequency spectrum sensing method performance close solution expression formula, invention achievement can be widely used in distributed frequency spectrum monitoring system.

Summary of the invention

The object of the present invention is to provide a kind of efficiently based on the soft-decision frequency spectrum sensing method of the non-reconstruct of compression.

Realizing technical scheme of the present invention is: the first, and sensor model is set up, and secondary user's sets up local frequency spectrum perception model, adopts dualism hypothesis model, considers that the present invention adopts energy detection method under non-approach to cooperation; The second, compression sampling, secondary user's utilizes dimensional Gaussian calculation matrix to carry out compression sampling to received signal; Whether the 3rd, not to the situation that compression sampling data are reconstructed, secondary user's directly utilizes compression sampling data to set up test statistics based on energy, but do not exist primary user and differentiate, but test statistics is passed to fusion center.4th, the soft-decision of fusion center, after collecting the test statistics of all secondary user's transmission, sets up unified test statistics, compares with detection threshold, provide final judging result at fusion center.

Compared with prior art, its remarkable advantage is in the present invention: 1, be incorporated in frequency spectrum perception by compression sampling, directly utilizes the measured value after compression sampling to carry out frequency spectrum perception, reduces the sample rate of signal, realize the detection to broadband primary user's signal; 2, compared with existing soft decision method, respective test statistics is passed to fusion center by secondary user's, instead of transmits the data of compression sampling, while reduction transmission pressure, keeps higher Detection accuracy; 3, give based on compression non-reconstruct soft-decision frequency spectrum sensing method detection perform close solution expression formula.

Accompanying drawing explanation

Fig. 1 soft-decision frequency spectrum sensing method flow chart based on the non-reconstruct of compression of the present invention;

Graph of a relation between Fig. 2 soft-decision frequency spectrum sensing method detection probability based on the non-reconstruct of compression of the present invention and secondary user's number

Graph of a relation between Fig. 3 soft-decision frequency spectrum sensing method detection probability based on the non-reconstruct of compression of the present invention and false alarm probability

Graph of a relation between Fig. 4 soft-decision frequency spectrum sensing method detection probability based on the non-reconstruct of compression of the present invention and signal to noise ratio

Embodiment

Below in conjunction with accompanying drawing, the present invention is further illustrated.

Fig. 1 illustrates the soft-decision frequency spectrum sensing method flow chart based on the non-reconstruct of compression, it comprises the following steps:

The first step, each secondary user's sets up binary hypothesis test model, H ₀represent that primary user does not exist, H ₁represent that primary user exists, that supposes that i-th secondary user's receive can not be expressed as through the signal model of compression sampling

H ₀：r _i(k)＝n _i(k)

H ₁：r _i(k)＝x _i(k)+n _i(k)

Wherein, r _ik () is the signal that i-th secondary user's receives in a kth sampling instant; x _ik () represents the primary user's signal arriving i-th secondary user's through fading channel, average is 0, and variance is n _ik () represents i-th secondary user's interchannel noise, be assumed to be additive white Gaussian noise, average is 0, and variance is

Supposing that i-th secondary user's receives number of samples is L, if represented in the form of vectors by Received signal strength, then binary hypothesis test model can be expressed as:

H ₀：r _i＝n _i

H ₁：r _i＝x _i+n _i

Wherein, r _i, x _i, n _irepresenting Received signal strength, primary user's signal, the vector that noise is corresponding respectively, is L × 1 dimensional vector, i.e. r _i=[r _i(1), r _i(2) ..., r _i(k) ... r _i(L)] ^t, x _i=[x _i(1), x _i(2) ..., x _i(L)] ^t, n _i=[n _i(1), n _i(2) ..., n _i(L)] ^t, subscript T represents transposed transform, lower same.

Second step, sensing node introduces compression sampling technology, different from nyquist sampling, its each sampled value is obtained by the projection of signal on observation vector, and its general principle is projected by sparse signal obtain sampled value and use restructing algorithm to reconstruct primary signal by sampled value on observation vector.Suppose signal x ∈ R ^lfor L × 1 dimensional vector, suppose that its degree of rarefication is K (K < L), then the projection of signal on observing matrix Φ can be expressed as

y＝Φx

Wherein, Φ is that a M × L (M < < L) meeting restriction isometry ties up observing matrix.Y is the sample value obtained after compression sampling, is M × 1 dimensional vector.After compression sampling, under normal circumstances, the rarefaction representation reconstruct primary signal of signal on transform domain can be found, as by i ₀norm or its i with solution ₁norm optimization problem solving primary signal.But for frequency spectrum detection, in the y after compression sampling, maintain the structural information of primary signal, therefore, can not Accurate Reconstruction primary signal have been carried out, directly utilized compression sampling vector y, complete frequency spectrum detection task.

Therefore, for i-th secondary user's after employing compression sampling, its dualism hypothesis model becomes:

H ₀：r _c，i＝Φ _in _i

H ₁：r _c，i＝Φ _i(x _i+n _i)

Wherein, Φ _ibe the M × L dimension observing matrix of i-th secondary user's for compression sampling, in the present invention, the observing matrix of all secondary user's all adopts Gauss measurement matrix, and namely in observing matrix, each element is separate, obeying average is 0, and variance is the gaussian random distribution of 1/L.X _i, n _irepresenting Received signal strength, primary user's signal, the vector that noise is corresponding respectively, is L × 1 dimensional vector; Received signal vector r after compression sampling _{c, i}for M × 1 dimensional vector, i.e. r _{c, i}=[r _{c, i}(1), r _{c, i}(2) ..., r _{c, i}(M)] ^t, wherein r _{c, i}k () 1≤k≤M represents the kth sample value after compression sampling.

3rd step, i-th secondary user's sets up the detection statistic T after compression sampling _{c, i}

$T_{c,i}=\frac{1}{M}\underset{k=1}{\overset{M}{\mathrm{\&Sigma;}}}{\left|r_{c,i}\left(k\right)\right|}^2$

According to central-limit theorem, when M value is fully large, for hypothesis H ₀, T _{c, i}obeying average is 0, and variance is | Φ _iΦ _i ^t| ₀gaussian Profile; For hypothesis H ₁, T _{c, i}obeying average is 0, and variance is gaussian Profile, wherein | Φ _iΦ _i ^t| ₀representing matrix Φ _iΦ _i ^tdiagonal entry sum.Then T _{c, i}obey following Gaussian Profile, can be expressed as

$H_0:T_{c,i}~N(\frac{{\mathrm{\&sigma;}}_{n,i}^2{\mathrm{\&eta;}}_i}{M},\frac{2{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}{M^2})$

$H_1:T_{c,i}~N(\frac{({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2){\mathrm{\&eta;}}_i}{M},\frac{2{\mathrm{\&eta;}}_i^{*}{\left[({\mathrm{\&sigma;}}_{x,i}^2+{\mathrm{\&sigma;}}_{n,i}^2)\right]}^2}{M^2})$

Wherein, η _i=| Φ _iΦ _i ^t| ₀, representing matrix Φ _iΦ _i ^tdiagonal entry square sum.

Gaussian Profile is defined as:

If the probability density of continuous random variable X is

$f\left(x\right)=\frac{1}{\sqrt{2\mathrm{\&pi;}}\mathrm{\&sigma;}}{e}^{-\frac{{(x-\mathrm{\&mu;})}^2}{2{\mathrm{\&sigma;}}^2}},-\&infin;<x<\&infin;$

Wherein μ, σ (σ > 0) are constant, then claiming X to obey parameter is the Gaussian Profile of μ, σ, is designated as X ~ N (μ, σ ²).

4th step, the soft-decision of fusion center, when all secondary user's are by test statistics T _{c, i}pass to fusion center, fusion center sets up unified test statistics, and carries out conclusive judgement, supposes total J secondary user's, then the test statistics T at fusion center place _{c, Soft}can be expressed as

$T_{c,\mathrm{Soft}}=\frac{1}{J}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{T}_{c,i}=\frac{1}{J}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left(\frac{1}{M}\underset{k=1}{\overset{M}{\mathrm{\&Sigma;}}}{\left|r_{c,i}\left(k\right)\right|}^2\right)$

According to central-limit theorem, when secondary user's number is abundant, T _{c, Soft}meet following Gaussian Profile:

$H_0:T_{c,\mathrm{Soft}}~N(\frac{\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i{\mathrm{\&sigma;}}_{n,i}^2}{\mathrm{JM}},\frac{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}{J^2{M}^2})$

$H_1:T_{c,\mathrm{Soft}}~N(\frac{\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i({\mathrm{\&sigma;}}_{n.i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}{\mathrm{JM}},\frac{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}^2}{J^2{M}^2})$

The then detection probability at fusion center place and false alarm probability can be expressed as:

$P_{\mathrm{cD}}^S=P\{T_{c,\mathrm{Soft}}\&GreaterEqual;{\mathrm{\&lambda;}}_{c,S}/H_1\}=Q\left(\frac{{\mathrm{\&lambda;}}_{c,S}-\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)/\left(\mathrm{JM}\right)}{\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}^2}/\left(\mathrm{JM}\right)}\right)$

$P_{\mathrm{cF}}^S=P\{T_{c,\mathrm{Soft}}\&GreaterEqual;{\mathrm{\&lambda;}}_{c,S}/H_0\}=Q\left(\frac{{\mathrm{\&lambda;}}_{c,S}-\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{{\mathrm{\&eta;}}_i\mathrm{\&sigma;}}_{n,i}^2/\left(\mathrm{JM}\right)}{\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}/\left(\mathrm{JM}\right)}\right)$

Wherein, Q function expression is

Detection threshold λ _{c, S}can be determined by following formula:

${\mathrm{\&lambda;}}_{c,S}=Q^{-1}\left(P_{\mathrm{cF}}^S\right)\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}/\left(\mathrm{JM}\right)+\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i{\mathrm{\&sigma;}}_{n,i}^2/\left(\mathrm{JM}\right)$

Q ^-1() is the inverse function of Q function, then detection probability can be expressed as

$\begin{array}{c}{P}_{\mathrm{cD}}^S=P\{T_{c,\mathrm{Soft}}\&GreaterEqual;{\mathrm{\&lambda;}}_{c,S}/H_1\}\\ =Q\left(\frac{Q^{-1}\left(P_{\mathrm{cF}}^S\right)\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}+\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i{\mathrm{\&sigma;}}_{n,i}^2-\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}{\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}^2}}\right)\end{array}$

Conveniently simulation analysis, supposes that each secondary user's has identical channel condition, and has identical with then make signal to noise ratio then detection probability can be reduced to:

$P_{\mathrm{cD}}^S=Q\left(\frac{Q^{-1}\left(P_{\mathrm{cF}}^S\right)\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}}+\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i-\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i(1+r_{\mathrm{sn}})}{\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{(1+r_{\mathrm{sn}})}^2}}\right)$

Fusion center sets up decision rule:

$\left\{\begin{array}{c}{T}_{c,\mathrm{Soft}}>{\mathrm{\&lambda;}}_{c,S}\&DoubleRightArrow;1\\ T_{c,\mathrm{Soft}}<{\mathrm{\&lambda;}}_{c,S}\&DoubleRightArrow;0\end{array}\right.$

Wherein, 1 represents that primary user's signal exists, and 0 represents that primary user's signal does not exist.Thus the soft-decision frequency spectrum sensing method completed based on the non-reconstruct of compression.Accompanying drawing 2, accompanying drawing 3, accompanying drawing 4 sets forth based on graph of a relation between the compression soft-decision frequency spectrum sensing method detection probability of non-reconstruct and secondary user's number, false alarm probability, signal to noise ratio, can be found out by emulation, under identical circumstances, along with the increase of compression ratio M/L, the performance of soft-decision frequency spectrum perception algorithm improves gradually, in addition, the detection perform that the present invention derives closes that to separate expression formula and actual emulation performance comparatively identical, describes the validity that detection perform that the present invention derives closes solution expression formula.

Claims (3)

1. based on the soft-decision frequency spectrum sensing method of the non-reconstruct of compression, it comprises the following steps: the first, and sensor model is set up, secondary user's sets up local frequency spectrum perception model, adopt dualism hypothesis model, consider that the present invention adopts energy detection method under non-approach to cooperation; The second, compression sampling, secondary user's utilizes dimensional Gaussian calculation matrix to carry out compression sampling to received signal; Whether the 3rd, not to the situation that compression sampling data are reconstructed, secondary user's directly utilizes compression sampling data to set up test statistics based on energy, but do not exist primary user and differentiate, but test statistics is passed to fusion center.4th, the soft-decision of fusion center, after collecting the test statistics of all secondary user's transmission, sets up unified test statistics, compares with detection threshold, provide final judging result at fusion center.

2. the soft-decision frequency spectrum sensing method based on the non-reconstruct of compression according to claim 1, it is characterized in that: after fusion center collects the test statistics of all secondary user's transmission, set up unified test statistics, and provide the gauss of distribution function of test statistics obedience.

Test statistics T _{c, Soft}for:

$T_{c,\mathrm{Soft}}=\frac{1}{J}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{T}_{c,i}=\frac{1}{J}\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}\left(\frac{1}{M}\underset{k=1}{\overset{M}{\mathrm{\&Sigma;}}}{\left|r_{c,i}\left(k\right)\right|}^2\right)$

According to central-limit theorem, when secondary user's number is abundant, T _{c, Soft}meet following Gaussian Profile:

$H_0:T_{c,\mathrm{Soft}}~N(\frac{\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i{\mathrm{\&sigma;}}_{n,i}^2}{\mathrm{JM}},\frac{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}{J^2{M}^2})$

$H_1:T_{c,\mathrm{Soft}}~N(\frac{\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}{\mathrm{JM}},\frac{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}^2}{J^2{M}^2})$

Wherein, J is the quantity of secondary user's; r _{c, i}k () represents the kth sample value after i-th secondary user's compression sampling; M is the number of samples after compression sampling; η _i=| Φ _iΦ _i ^t| ₀representing matrix Φ _iΦ _i ^tdiagonal entry sum, representing matrix Φ _iΦ _i ^tdiagonal entry square sum, Φ _iit is the dimensional Gaussian calculation matrix of the i-th secondary user's; be respectively the variance of i-th secondary user's noise and primary user's signal, subscript T represents transposition.

3., according to claim 1 and the soft-decision frequency spectrum sensing method based on the non-reconstruct of compression according to claim 2, it is characterized in that: the detection probability giving fusion center place and false alarm probability close solution expression formula, namely

$P_{\mathrm{cD}}^S=P\{T_{c,\mathrm{Soft}}\&GreaterEqual;{\mathrm{\&lambda;}}_{c,S}/H_1\}=Q\left(\frac{{\mathrm{\&lambda;}}_{c,S}-\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)/\left(\mathrm{JM}\right)}{\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}^2}/\left(\mathrm{JM}\right)}\right)$

$P_{\mathrm{cF}}^S=P\{T_{c,\mathrm{Soft}}\&GreaterEqual;{\mathrm{\&lambda;}}_{c,S}/H_0\}=Q\left(\frac{{\mathrm{\&lambda;}}_{c,S}-\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i{\mathrm{\&sigma;}}_{n,i}^2/\left(\mathrm{JM}\right)}{\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}/\left(\mathrm{JM}\right)}\right)$

Wherein, Q function expression is detection threshold λ _{c, S}can be determined by following formula:

${\mathrm{\&lambda;}}_{c,S}=Q^{-1}\left(P_{\mathrm{cF}}^S\right)\sqrt{2\underset{j=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}/\left(\mathrm{JM}\right)+\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i{\mathrm{\&sigma;}}_{n,i}^2/\left(\mathrm{JM}\right)$

Q ^-1() is the inverse function of Q function, then detection probability can be expressed as

$\begin{array}{c}{P}_{\mathrm{cD}}^S=P\{T_{c,\mathrm{Soft}}\&GreaterEqual;{\mathrm{\&lambda;}}_{c,S}/H_1\}\\ =Q\left(\frac{Q^{-1}\left(P_{\mathrm{cF}}^S\right)\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{\mathrm{\&sigma;}}_{n,i}^4}+\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i{\mathrm{\&sigma;}}_{n,i}^2-\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}{\sqrt{2\underset{i=1}{\overset{J}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i^{*}{({\mathrm{\&sigma;}}_{n,i}^2+{\mathrm{\&sigma;}}_{x,i}^2)}^2}}\right)\end{array}$