CN103490835B - Super-geometry volume based frequency spectrum sensing method and system - Google Patents

Super-geometry volume based frequency spectrum sensing method and system Download PDF

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CN103490835B
CN103490835B CN201310478534.0A CN201310478534A CN103490835B CN 103490835 B CN103490835 B CN 103490835B CN 201310478534 A CN201310478534 A CN 201310478534A CN 103490835 B CN103490835 B CN 103490835B
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黄磊
李蓉娴
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Shenzhen Graduate School Harbin Institute of Technology
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Abstract

The invention is applied to the field of cognitive radio and provides a super-geometry volume based frequency spectrum sensing method. The method comprises the following steps: (A) constructing a sample covariance matrix S according to the inherent correlation of data y(k) received by a secondary user terminal, wherein the formula of S is shown in the specification; (B) constructing a super geometry according to the sample covariance matrix S, and defining detection statistics according to the volume of the super geometry, wherein the formula of the detection statistics is shown in the specification; (C) acquiring a detection threshold according to a false-alarm probability given by a system; (D) judging whether a main signal exists or not according to the size relationship of the detection statistics and the detection threshold. Due to the adoption of super-geometry volume detection, the noise power of a known background is not required, and the complexity of detection is not improved obviously when receiving antennas increase; the method is not affected by noise non-uniformity and has good applicability to actual cognitive radio networks.

Description

A kind of method and system based on hypergeometry body volume frequency spectrum perception
Technical field
The invention belongs to cognition wireless electrical domain, particularly relate to a kind of method and system based on hypergeometry body volume computing frequency spectrum perception.
Background technology
As everyone knows, the policy of existing frequency spectrum resource fixed allocation can not make limited frequency spectrum resource be fully used.The proposition of cognitive radio is intended to solve the problem.In cognitive radio networks, when primary user's (i.e. authorized user) does not work, the frequency band shared by it is in " free time " state; Now, secondary user (i.e. unauthorized user) perception effectively utilize idle frequency band, to reach the object improving band efficiency in a very wide band limits.The Wireless Telecom Equipment with cognitive function is operated according to the mode that certain " is waited for an opportunity " in the frequency range of having authorized, and namely " uses " frequency spectrum resource of authorizing, ensures that the communication of these wireless devices can not have influence on other authorized user work simultaneously.This " cognitive function " i.e. frequency spectrum perception technology.
Frequency spectrum perception technology can be modeled as binary hypothesis test problem---and main signal does not exist (namely primary user is idle) supposes that there is (i.e. primary user's work) supposes with main signal.Under the non-existent hypothesis of main signal, observation data only comprises noise; Under the hypothesis that main signal exists, observation data comprises signal and noise simultaneously.Whether main signal exists, and directly affects energy and the dependency structure of observation data; Frequency spectrum perception just applies energy and the dependency structure otherness under two kinds of hypothesis to detect the presence or absence of main signal.
Frequency spectrum perception technology mainly comprises the detection etc. of energy measuring, Likelihood ration test, feature detection and feature based value.During known noise power, energy detector detection perform optimum; And in the cognitive radio networks of reality, noise power is normally unknown, replace real noise power, the energy detector obtained by the noise power estimated performance has obvious decline.Utilize and receive the characteristic value of data covariance matrix and do the detection that the method detected is called feature based value.When main signal does not exist, receive data and only comprise noise, sample covariance matrix is similar to a unit matrix; When main signal exists, this covariance matrix loses unit structure due to the correlation of signal, and matrix is unfolded.The detector of feature based value proposes according to this dependency structure.Wherein, ratio (AGM) detector of algebraically-geometric average characteristic value reliably can detect coherent signal.Whether ratio (MME) detector of maximum-minimal eigenvalue corresponds to unit structure by detecting sample covariance matrix, carrys out the presence or absence of perception main signal.Owing to not using all characteristic values, the detection perform of MME detector declines under certain condition to some extent.If only have a primary user in cognitive radio networks, then the best performance of Generalized Likelihood Ratio (GLRT) detector, it is equivalent to signal-noise characteristic value (SNE) detector.In actual conditions, the internal structure of secondary each reception antenna of user side may be different, and cause noise power on each antenna unequal, namely noise is non-uniform noise.Adama (Hadamard) detector is not easy affected by noise, has robustness to non-uniform noise.
Summary of the invention
The object of the present invention is to provide a kind of method based on hypergeometry body volume frequency spectrum perception, be intended to solve noise power on each antenna unknown or noise power is unequal time the detection perform problem that reduces.The determinant of sample covariance matrix differs greatly under main signal exists and main signal does not exist these two kinds different hypothesis, and the present invention utilizes this otherness to do frequency spectrum perception.
The present invention is achieved in that a kind of method based on hypergeometry body volume frequency spectrum perception, said method comprising the steps of:
A, utilization time user side receive the inherent correlation of data y (k), structure sample covariance matrix S, its formula: S = 1 n Σ k = 1 n y ( k ) y T ( k ) ;
B, foundation sample covariance matrix S form hypergeometry body, according to hypergeometry body volume definition detection statistic, and its formula: ξ = Δ log det [ S ′ ] = log det [ D - 1 S ] ;
C, obtain detection threshold according to the false alarm probability that system is given;
D, to judge according to the magnitude relationship of detection statistic and detection threshold main signal with or without.
Further technical scheme of the present invention is: also comprise before described steps A:
The data that A0, basis time user side antenna receive set up signal model.
Further technical scheme of the present invention is: described step B comprises:
The length of each row vector of B1, compute matrix S, its formula: δ i=|| S (i) ||;
B2, length configuration diagonal matrix D according to matrix S row vector, its formula: D=diag (δ 1..., δ m);
B3, structural matrix S', its formula: S'=D -1the row vector of S, S' forms hypergeometry body, its volume equal matrix S ' determinant det [S'];
B4, according to hypergeometry body volume definition detection statistic, its formula:
ξ = Δ log det [ S ′ ] = log det [ D - 1 S ] .
Further technical scheme of the present invention is: described hypergeometry body volume definition detection statistic does not need the power of known background noise, does not affect, have robustness to heterogeneity noise by noise heterogeneity.
Further technical scheme of the present invention is: described hypergeometry body volume definition is less with the increase amplitude of variation of antenna number for the computing time needed for detection statistic, strong to the applicability of actual cognitive radio system.
Another object of the present invention is to provide a kind of system based on hypergeometry body volume frequency spectrum perception, this system comprises:
Sample covariance matrix module, for utilizing time user side to receive the inherent correlation of data y (k), structure sample covariance matrix S, its formula:
Detection statistic module, for forming hypergeometry body according to sample covariance matrix S, according to hypergeometry body volume definition detection statistic, its formula:
Detection threshold module, obtains detection threshold for the false alarm probability given according to system;
Judge module, for judge according to the magnitude relationship of detection statistic and detection threshold main signal with or without.
Further technical scheme of the present invention is: described system also comprises:
Signal model module, sets up signal model for the data received according to time user side antenna.
Further technical scheme of the present invention is: length computation unit, for the length of each row vector of compute matrix S, and its formula: δ i=|| S (i) ||;
Diagonal matrix unit, for the length configuration diagonal matrix D according to matrix S row vector, its formula: D=diag (δ 1..., δ m);
Cube unit, for structural matrix S', its formula: S'=D -1the row vector of S, S' forms hypergeometry body, its volume equal matrix S ' determinant det [S'];
Obtain detection statistic unit, for defining detection statistic according to hypergeometry body volume, its formula: ξ = Δ log det [ S ′ ] = log det [ D - 1 S ] .
Further technical scheme of the present invention is: described hypergeometry body volume definition detection statistic does not need the power of known background noise, does not affect, have robustness to heterogeneity noise by noise heterogeneity.
Further technical scheme of the present invention is: described hypergeometry body volume definition is less with the increase amplitude of variation of antenna number for the computing time needed for detection statistic, strong to the applicability of actual cognitive radio system.
The invention has the beneficial effects as follows: adopt hypergeometry body volume to detect, when Unknown Noise Environments power, detection perform is good, and when reception antenna number increases, computation complexity can not obviously increase, and does not affect by noise heterogeneity.
Accompanying drawing explanation
Fig. 1 is the flow chart of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides;
The shape figure of hypergeometry body when Fig. 2 is the observation data independence of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides;
Fig. 3 is the shape figure of the observation data of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides hypergeometry body when being correlated with;
Fig. 4 is the shape figure of the observation data of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides hypergeometry body when being concerned with;
Fig. 5 is the tendency chart that the detection probability of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides changes with signal to noise ratio, and main signal is real gaussian signal, main signal number d=1;
Fig. 6 is the tendency chart that the detection probability of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides changes with signal to noise ratio, and main signal is real gaussian signal, main signal number d=3;
Fig. 7 is the tendency chart that the detection probability of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides changes with signal to noise ratio, and main signal is multiple QPSK signal, main signal number d=1;
Fig. 8 is the tendency chart that the detection probability of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides changes with signal to noise ratio, and main signal is multiple QPSK signal, main signal number d=3;
Fig. 9 is receiver performance characteristics (ROC) curve chart of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides, and its noise is Uniform noise;
Figure 10 is receiver performance characteristics (ROC) curve chart of the method based on hypergeometry body volume frequency spectrum perception that the embodiment of the present invention provides, and its noise is non-uniform noise;
Figure 11 is the tendency chart that the computation complexity of each detector changes with secondary user side antenna number m.
Embodiment
As shown in Figure 1, the method flow diagram based on hypergeometry body volume frequency spectrum perception provided by the invention, details are as follows:
In step sl, in a cognitive radio networks, suppose have m time user's (unauthorized user) to receive the signal launched from d primary user's (authorized user), primary user, secondary user side are all equipped with antenna to launch or Received signal strength, m time user's perceived spectral to cooperatively.Use H respectively 1and H 0represent that main signal existence or not two kinds with main signal and supposes, then the reception data of a secondary user side m antenna constitute the observation vector that m × 1 is tieed up, and can be expressed as:
y ( k ) = w ( k ) H 0 Hs ( k ) + w ( k ) H 1 , Wherein k=1 ..., n represents the observation moment, and n, for always to observe duration, is called fast umber of beats, H ∈ R m × dit is the multidiameter fading channel matrix between primary user and secondary user;
Y (k)=[y 1(k) ..., y m(k)] trepresent the vector that time user side antenna receives, subscript T represents transposition;
S (k)=[s 1(k) ..., s d(k)] tfor signal vector, w (k)=[w 1(k) ..., w m(k)] tfor noise vector, known signal and noise separate, noise is white noise, obeys that average is 0, variance is gaussian Profile, be denoted as into: (i=1 ..., m), the data received according to time user side antenna thus establish signal model.
In step s 2, secondary user side antenna receives the signal that primary user is launched, and utilize and receive data letter y (k) inherent correlation, the covariance matrix R of structure time user side observation vector, its formula is: R=E [y (k) y t(k)], wherein E [] is mathematic expectaion operator.At formula: R=E [y (k) y t(k)] middle covariance matrix R is the result asking mathematic expectaion, but R can not obtain in practice, and with the maximal possibility estimation of R, namely the sample covariance matrix S of observation vector replaces covariance matrix R, and its sample covariance matrix formula is:
Matrix S has the row vector of m 1*m dimension, and they are tieed up in Euclidean space at m and form a hypergeometry body, and the m being public point with certain summit of hypergeometry body is faced m the row vector that limit is S respectively.For m=3, namely secondary user side has 3 antenna cooperative perception main signals.Now, covariance matrix R is the matrix of 3 × 3.Suppose to receive data y ik the average of () is 0, the stochastic variable that variance is equal.According to the relation of covariance matrix R and observation vector y (k), i.e. R=E [y (k) y t(k)], if receive data independently, then R is diagonal matrix, and 3 row vector R (1), the R (2) and R (3) of R form a cube in three dimensions; If have correlation between reception data, then matrix R is full rank non-diagonal battle array, and the row vector of R forms a parallelepiped; If it is completely relevant to receive data, then matrix R to be order be 1 Arbitrary Matrix, the row vector of R is parallel to each other, and the solid of formation is the end to end line segment of three vectors, and above-mentioned three kinds of situations are respectively as in Figure 2-4.For ease of volume ratio comparatively, be 1 entirely in the length of side of this hypothesis solid, namely the European norm of each row of R is 1, is denoted as || R (i) || and=1, i=1,2,3.Solid is cube corresponding no signal hypothesis H 0; Solid is that parallelepiped or straightway are to there being signal hypothesis H 1.Due to the maximal possibility estimation of R, namely the length of the row vector of S differs and is decided to be 1, therefore needs step S3-S5 to the row vector normalization of S.
In step s3, according to the length of European each row vector of norm calculation matrix S, its formula: δ i=|| S (i) ||, wherein i=1 ..., m, S (i) represent i-th row vector of S, operator || || represent the Euclidean Norm of vector.
In step s 4 which, structure diagonal matrix D, i-th diagonal element δ of D ifor the length of matrix S i-th row vector, its formula: D=diag (δ 1..., δ m), wherein diag represents and makes δ 1..., δ mfor the diagonal element of matrix.
In step s 5, structural matrix S', its formula: S'=D -1s, the diagonal element due to diagonal matrix D is the mould of each row vector of S, inverse to matrix S premultiplication matrix D, the matrix S obtained ' the length of row vector is 1, namely S' has carried out normalization to the row vector of S.Tieing up in Euclidean space at m utilizes the row vector of S' to form hypergeometry body, wherein the volume of hypergeometry body equal matrix S ' determinant det [S'].Such as, as m=3, the hypergeometry body of the row vector formation of S' is the solid in three dimensions, and its volume equals the determinant of S'.
As described in step S2, at H 0and H 1suppose that each row vector length of lower matrix S is unequal.If the hypergeometry body volume directly formed by the row vector of S is as detection statistic, the presence or absence of detection signal just cannot be carried out liberally according to the corresponding different volumes of the difformity of hypergeometry body.Therefore be necessary to the normalization of S row vector.
In step s 6, according to the volume definition detection statistic of hypergeometry body in step S5, its formula: wherein det [] is for asking determinant operator, to det [D -1s] to get common logarithm log ' be to make detection statistic ξ at H 0and H 1difference under two kinds of hypothesis is more obvious.
In the step s 7, be P according to the false alarm probability that system is given fa, obtain detection threshold γ with Monte Carlo simulation experimental technique vD.
In step s 8, the presence or absence of main signal is judged according to the magnitude relationship of the detection statistic of trying to achieve and detection threshold.When wherein main signal does not exist, D -1s is close to unit matrix, and the volume of hypergeometry body is approximately 1; When main signal exists, D -1correlation between each row of S makes hypergeometry body be stretched, and volume has obvious reduction, and the volume differences opposite sex according to this hypergeometry body does input, obtains judging criterion with presence or absence of main signal: wherein γ vDit is the detection threshold experimentally obtained according to false alarm probability.Work as ξ vD< γ vDtime, then main signal exists; Work as ξ vD> γ vDtime, then main signal does not exist.
Following table gives energy detector with detector AGM and MME of feature based value, the detection criteria of the detector based on hypergeometry body volume computing (VD) that signal-noise characteristic value (SNE) detector, Adama (Hadamard) detector and the present invention propose.Wherein represent the energy detector that ideally noise variance is known, represent that noise variance is the energy detector of estimated value, be applicable to actual conditions.
As viewed in figures 5-8, compare respectively main signal be real gaussian signal and multiple QPSK signal time, the trend that the detection perform of each detector changes with signal to noise ratio, wherein the detection criteria of each detector is as shown above.Signal to noise ratio snr is defined as unit is decibel (dB), represent the power of signal s (k); False alarm probability is set to P fa=10 -2, detection threshold γ vDaccording to false alarm probability by 10 6secondary Monte Carlo simulation obtains; Antenna number m=6, fast umber of beats n=12; Noise is Uniform noise.In figure, curve VD represents the detector based on hypergeometry body volume computing that the present invention proposes.The situation of to be main signal the be real gaussian signal that Fig. 5,6 shows.As main signal number d=1, as shown in Figure 5, VD detector performance be better than MME, Hadamard and detector, lower than AGM and SNE detector; As main signal number d=3, as shown in Figure 6, VD detector performance is still lower than AGM detector, but be better than other detectors, wherein SNE detector performance has obvious decline when main signal number increases, and illustrates that the detector (VD) based on hypergeometry body volume computing also has good detection perform when main signal number increases.The situation of to be main signal the be multiple QPSK signal that Fig. 7,8 shows.Main signal number d=1(is as shown in Figure 7) and d=3(is as shown in Figure 8) in two kinds of situations, the performance of VD detector is all obviously better than AGM detector, a little less than SNE detector.
The contrast of Fig. 9 and Figure 10 is intended to illustrate that detector VD based on volume computing is for the robustness of non-uniform noise.The heterogeneity of noise refers to that the power of noise on a receiving terminal m antenna is unequal.Two figure have optimum configurations identical as follows: antenna number m=6, fast umber of beats n=10; A multiple QPSK main signal is had to exist, and signal to noise ratio snr=-3dB.In Fig. 9, noise is set to Uniform noise (namely the noise power of each antenna is equal), as seen from Figure 9, the detection perform of VD detector a little less than SNE detector, with Fig. 7,8 result consistent; When on each antenna, noise power is unequal, as shown in Figure 10, the performance of VD detector is better than the other kinds detector comprising Hadamard detector, and Hadamard detector is acknowledged as and has robustness to non-uniform noise, has the function of good opposing non-uniform noise.The performance which illustrating VD detector does not affect by noise is heteropical, has robustness to non-uniform noise.
Notice, in above-mentioned experimentation, noise power information do not use; And energy measuring must known noise power just can carry out; When noise power is estimated value time, based on the energy measuring of noise power estimation performance has obvious decline.This illustrates, the detector (VD) based on hypergeometry body volume does not need known noise power, therefore has advantage compared to energy measuring.
Figure 11 illustrates the computation complexity of various detector, and the time required for calculating with algorithm is weighed.As shown in Figure 11, along with antenna number m increases, each algorithm computation complexity difference is more and more significant.Based on volume computing detector VD needed for computing time less with the increase amplitude of variation of antenna number m, i.e. perception complexity be not vulnerable to antenna number increase impact, the requirement of the actual cognitive radio networks that is content with very little.Integrated comparative detection perform and computation complexity, the detector VD based on volume computing is optimum selection.
The present invention also provides a kind of system based on hypergeometry body volume frequency spectrum perception, and this system comprises:
Sample covariance matrix module, for utilizing time user side to receive the inherent correlation of data y (k), structure sample covariance matrix S, its formula:
Detection statistic module, for forming hypergeometry body according to sample covariance matrix S, according to hypergeometry body volume definition detection statistic, its formula:
Detection threshold module, obtains detection threshold for the false alarm probability given according to system;
Judge module, for judge according to the magnitude relationship of detection statistic and detection threshold main signal with or without.
Described system also comprises:
Signal model module, sets up signal model for the data received according to time user side antenna.
Described detection statistic module comprises:
Length computation unit, for the length of each row vector of compute matrix S, its formula: δ i=|| S (i) ||;
Diagonal matrix unit, for the length configuration diagonal matrix D according to matrix S row vector, its formula: D=diag (δ 1..., δ m);
Cube unit, for structural matrix S', its formula: S'=D -1the row vector of S, S' forms hypergeometry body, its volume equal matrix S ' determinant det [S'];
Obtain detection statistic unit, for defining detection statistic according to hypergeometry body volume, its formula: &xi; = &Delta; log det [ S &prime; ] = log det [ D - 1 S ] .
Described hypergeometry body volume definition detection statistic does not need the power of known background noise, does not affect, have robustness to heterogeneity noise by noise heterogeneity.
Described hypergeometry body volume definition is less with the increase amplitude of variation of antenna number for the computing time needed for detection statistic.Namely computation complexity is not subject to the impact that reception antenna number increases, and has good applicability for real system.
The foregoing is only preferred embodiment of the present invention, not in order to limit the present invention, all any amendments done within the spirit and principles in the present invention, equivalent replacement and improvement etc., all should be included within protection scope of the present invention.

Claims (8)

1., based on a method for hypergeometry body volume frequency spectrum perception, it is characterized in that, said method comprising the steps of:
A, utilization time user side receive the inherent correlation of data y (k), structure sample covariance matrix S, its formula: S = 1 n &Sigma; k = 1 n y ( k ) y T ( k ) ;
B, foundation sample covariance matrix S form hypergeometry body, according to hypergeometry body volume definition detection statistic, and its formula: ξ log det [S ']=log det [D -1s];
C, obtain detection threshold according to the false alarm probability that system is given;
D, to judge according to the magnitude relationship of detection statistic and detection threshold main signal with or without; When main signal does not exist, D -1s is close to unit matrix, and the volume of hypergeometry body is approximately 1; When main signal exists, D -1correlation between each row of S makes hypergeometry body be stretched, and volume has obvious reduction, and the volume differences opposite sex according to this hypergeometry body does input, obtains judging criterion with presence or absence of main signal:
wherein γ vDbe the detection threshold experimentally obtained according to false alarm probability, work as ξ vD< γ vDtime, then main signal exists; Work as ξ vD> γ vDtime, then main signal does not exist;
Described step B comprises:
The length of each row vector of B1, compute matrix S, its formula: δ i=|| S (i) ||;
B2, length configuration diagonal matrix D according to matrix S row vector, its formula: D=diag (δ 1..., δ m);
B3, structural matrix S ', its formula: S '=D -1s, S ' row vector form hypergeometry body, its volume equal matrix S ' determinant det [S '];
B4, according to hypergeometry body volume definition detection statistic, its formula:
ξ□log det[S′]＝log det[D -1S]。
2. method according to claim 1, is characterized in that, also comprises before described steps A:
The data that A0, basis time user side antenna receive set up signal model.
3. method according to claim 1 and 2, is characterized in that, described hypergeometry body volume definition detection statistic does not need the power of known background noise, does not affect, have robustness to heterogeneity noise by noise heterogeneity.
4. method according to claim 1 and 2, is characterized in that, described hypergeometry body volume definition is less with the increase amplitude of variation of antenna number for the computing time needed for detection statistic, strong to the applicability of actual cognitive radio system.
5. based on a system for hypergeometry body volume frequency spectrum perception, it is characterized in that, this system comprises:
Sample covariance matrix module, for utilizing time user side to receive the inherent correlation of data y (k), structure sample covariance matrix S, its formula:
Detection statistic module, for forming hypergeometry body according to sample covariance matrix S, according to hypergeometry body volume definition detection statistic, its formula: ξ log det [S ']=log det [D -1s];
Detection threshold module, obtains detection threshold for the false alarm probability given according to system;
Judge module, for judge according to the magnitude relationship of detection statistic and detection threshold main signal with or without; When main signal does not exist, D -1s is close to unit matrix, and the volume of hypergeometry body is approximately 1; When main signal exists, D -1correlation between each row of S makes hypergeometry body be stretched, and volume has obvious reduction, and the volume differences opposite sex according to this hypergeometry body does input, obtains judging criterion with presence or absence of main signal: wherein γ vDbe the detection threshold experimentally obtained according to false alarm probability, work as ξ vD< γ vDtime, then main signal exists; Work as ξ vD> γ vDtime, then main signal does not exist;
Described detection statistic module comprises:
Length computation unit, for the length of each row vector of compute matrix S, its formula: δ i=|| S (i) ||;
Diagonal matrix unit, for the length configuration diagonal matrix D according to matrix S row vector, its formula: D=diag (δ 1..., δ m);
Cube unit, for structural matrix S ', its formula: S '=D -1s, S ' row vector form hypergeometry body, its volume equal matrix S ' determinant det [S '];
Obtain detection statistic unit, for defining detection statistic according to hypergeometry body volume, its formula: ξ log det [S ']=log det [D -1s].
6. system according to claim 5, is characterized in that, described system also comprises:
Signal model module, sets up signal model for the data received according to time user side antenna.
7. system according to claim 6, is characterized in that, described hypergeometry body volume definition detection statistic does not need the power of known background noise, does not affect, have robustness to heterogeneity noise by noise heterogeneity.
8. the system according to claim 6 or 7, is characterized in that, described hypergeometry body volume definition is less with the increase amplitude of variation of antenna number for the computing time needed for detection statistic, strong to the applicability of actual cognitive radio system.
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