CN103354463A - Cooperative wave beam formation method capable of improving physical layer security based on null space - Google Patents

Abstract

The invention discloses a cooperative wave beam formation method capable of improving physical layer security based on null space. In the prior art, a safety transmission problem when there is a plurality of eavesdropping nodes in a relay network can not be guaranteed. By using the method of the invention, the above problem is mainly solved. The method comprises the steps that 1) a source node sends confidential information to all the relay nodes and a relay carries out weighting processing on a received signal and then forwards to a destination node and the eavesdropping nodes; 2) a received signal to noise ratio of the destination node and the eavesdropping nodes is calculated; 3) according to the received signal to noise ratio, a safe rate of the destination node is calculated; 4) according to the signal after the weighting processing which is performed by the relay, consumption of relay total power and the power is calculated so as to obtain a power constraint; 5) the null space of an equivalent channel matrix of the eavesdropping nodes is calculated so as to obtain a space constraint; 6) under the condition that the power constraint and the space constraint are satisfied, an optimal wave beam shaping weight vector is designed; 7) the optimal wave beam shaping weight vector is distributed to each relay node. By using the method of the invention, the safe rate of the relay network where the plurality of eavesdropping nodes exist can be effectively increased.

Description

Strengthen the cooperative beam formation method of physical layer safety based on kernel

Technical field

The invention belongs to wireless communication technology field, further relate to when existing a plurality of listener-ins and each via node to have self power constraint in the wireless relay network, the cooperative beam manufacturing process can be used for strengthening in the NGBW cooperation communication system physical layer safety.

Background technology

In order to satisfy the requirement of 3GPP Long Term Evolution LTE High Data Rate and high power system capacity, multiple-input and multiple-output MIMO technology is widely used in the up-downgoing system of LTE.The MIMO technology can take full advantage of the space diversity except time, frequency diversity of being brought by many antennas on the transceiver, effectively improves the quality that receives signal and improves message transmission rate.And for single antenna equipment, road has been paved in the application that is introduced as MIMO of cooperating relay technology, and the cooperating relay technology can be considered as a kind of virtual MIMO in essence.Therefore, the many technological expansion in the mimo system can be obtained space diversity and spatial multiplexing gain simultaneously in cooperative relay system.Wherein, cooperative beam is shaped and has obtained broad research as an important technology.Its basic thought is the restore one's right value by adjustment relay forwarding signal, and namely distributed beam forming weight vector forms a virtual beams of aiming at receiving terminal, and then improves signal to noise ratio, power system capacity or coverage.

In addition, along with progressively application and the development of LTE system, the syncretization of its communication system, the high speed of the rate of information throughput are so that traditional link layer security algorithm has run into very large challenge in LTE.And the physical layer safe practice does not rely on upper layer data with it and encrypts and safe and reliable communication is provided and receives much concern.The spatial degrees of freedom that tradition MIMO and distributed MIMO technology all can utilize many antennas to bring effectively weakens the listener-in to the interception of confidential information.Wherein, cooperative beam is shaped and has been proved to be to strengthen a kind of effective ways of physical layer safety, and it maximizes the security of system transmission rate by the weight vector of choose reasonable beam forming.

For the wireless relay network that adopts amplification forwarding AF agreement, the people such as L.Dong are at " Improving wireless physical layer security via cooperative relays " (IEEE Trans.Signal Process., vol.58, no.3, pp.1875-1888, a kind of cooperative beam shaping scheme that strengthens physical layer safety based on kernel has been proposed Mar.2010), but this scheme has only been considered the constraint of relaying gross power, and each relaying may have self power constraint in practice.For this reason, for the constrained situation of each relaying self power, the people such as J.Zhang are at " Relay beamforming strategies for physical-layer security(in Proc.44 ^ThConference on Information Sciences and Systems, Princeton, Mar.2010, pp.1-6) " in the design that a kind of iterative algorithm carries out cooperative beam shaping weight vector has been proposed.But this algorithm is only applicable to exist in the network single listener-in's situation, and is difficult to be applied to have in the NGBW cooperation communication system a plurality of listener-ins' situation.

Summary of the invention

The object of the invention is to for the deficiencies in the prior art, a kind of cooperative beam formation method that strengthens physical layer safety based on kernel is proposed, in junction network, to exist a plurality of eavesdropping nodes and each via node to satisfy under the condition of self power constraint, the effective safe speed of elevator system.

For achieving the above object, concrete steps of the present invention comprise as follows:

1) cooperating relay was transmitted in two stages:

1a) in the phase I, source node is to all via node speech amplifier confidential informations, and the signal that via node receives is y _r

1b) in second stage, via node is y to the received signal _rBe weighted processing, obtain signal x to be transmitted _r(w)=D (y _r) w, wherein w is for needing the junction waves beam shaping weight vector of design, and D () expression becomes diagonal matrix with vector transformation;

1c) destination node and eavesdropping node receive the signal of relay forwarding, and the signal that the note destination node receives is y _d(w), remember that the signal that j eavesdropping node receives is y _{E, j}(w), E represents to contain the set of all eavesdropping nodes, and M represents to eavesdrop the number of node;

2) according to above-mentioned reception signal y _d(w) and y _{E, j}(w), calculate the received signal to noise ratio γ of destination node _d(w) and j the eavesdropping node received signal to noise ratio γ _{E, j}(w);

3) according to above-mentioned received signal to noise ratio γ _d(w) and γ _{E, j}(w), calculate the safe speed of destination node $R_s\left(w\right)=\frac{1}{2}\underset{j}{\min }\{\log (1+{\mathrm{\γ}}_d\left(w\right))-\log (1+{\mathrm{\γ}}_{e,j}\left(w\right))\};$

4) according to step 1b) in treat forward signal x _r(w), calculate the consumption of via node gross power || x _r(w) || ²Consumption with i via node self power | x _{R, i}(w) | ², and will || x _r(w) || ²≤ P _TotWith | x _{R, i}(w) | ²≤ P _iAs the power constraint condition of design junction waves beam shaping weight vector, wherein x _{R, i}(w) expression x _r(w) i element, P _TotThe total power thresholds of all via nodes, P _iThe power threshold of i via node self, this P _TotAnd P _iSet generation by actual cooperation communication system,

C represents to contain the set of all via nodes, and K represents to eavesdrop the number of node;

5) calculate the kernel U that all eavesdrop the node equivalent channel matrixes, with the space constraints of w=Uv as design junction waves beam shaping weight vector, utilize this space constraint w with above-mentioned R _s(w), || x _r(w) || ²With | x _{R, i}(w) | ²Be rewritten as R _s(v), || x _r(v) || ²With | x _{R, i}(v) | ², and will be converted into to the design of w design to v;

6) satisfy above-mentioned || x _r(v) || ²≤ P _TotWith | x _{R, i}(v) | ²≤ P _iConstraint under, design vector v is so that safe speed R _s(v) maximization obtains optimal design v ^*, and according to the space constraint of w=Uv in the step 5), obtaining optimal beam shaping weight vector is w ^*=Uv ^*

7) according to optimal beam shaping weight vector w ^*, with w ^*In i element

Distribute to i via node.

The present invention compared with prior art has the following advantages

The present invention is owing to design the beam forming weight vector in the kernel of a plurality of eavesdropping node equivalent channel matrixes, and the beam forming weight vector that obtains optimum under the prerequisite that satisfies each via node self power constraint designs, guaranteed that not only each via node can work under self power threshold of default, and Effective Raise the safe speed when having a plurality of eavesdropping node in the cooperative relay system.

Description of drawings

Fig. 1 is the system model figure that the present invention is suitable for;

Fig. 2 is workflow diagram of the present invention;

Fig. 3 is the present invention and existing methodical safe speed comparison diagram.

Embodiment

With reference to the accompanying drawings and in conjunction with example the present invention is described further.

With reference to Fig. 1, the system model that the present invention adopts is by a source node S, a destination node D, K via node and M eavesdrop node composition, wherein K〉M.Each node all is equipped with single antenna, and adopts the half-duplex mode of operation.Suppose that all channels all experience separate flat Rayleigh fading, do not have the link that direct transfers between source node and the destination node, and source node can obtain the global channel state information.For representing conveniently, source node is expressed as f to the channel vector of all via nodes, all via nodes are expressed as g to the channel vector of destination node, and all via nodes are expressed as h to m the channel vector of eavesdropping node _j, wherein,

E represents to contain the set of all eavesdropping nodes.The reception noise of supposing each Nodes is additive white Gaussian noise, and the obedience average is that zero variance is 1 multiple Gaussian Profile.Because via node is operated in semiduplex mode, whole cooperation transmission process is divided into two stages.Phase I, source node is given all via node speech amplifier confidential informations; Second stage, via node adopt based on the host-host protocol of AF and transmit the information of source node to destination node and eavesdropping node.

With reference to Fig. 2, performing step of the present invention is as follows:

Step 1: cooperating relay was transmitted in two stages.

1a) in the phase I, source node is to all via node speech amplifier confidential informations, and the signal that via node receives is y _r, this receives signal y _rCan be expressed as vector form:

$y_r\stackrel{\mathrm{\Δ}}{=}{[y_{r,1},...,y_{r,i},...y_{r,K}]}^T=\sqrt{P_s}\mathrm{fs}+n_r,$

Wherein, s represents confidential information, P _sThe average transmit power of source node, y _{R, i}The reception signal that represents i via node place,

The expression source node is to the channel vector of all via nodes, f _iRepresent source node to the channel information of i via node,

The reception noise vector of expression via node, n _{R, i}The reception noise that represents i via node,

C represents to contain the set of all via nodes, and K represents to eavesdrop the number of node, () ^TThe operation that transposition is got in expression;

1b) in second stage, via node is y to the received signal _rBe weighted processing, obtain signal x to be transmitted _r(w), this signal x to be transmitted _r(w) can be expressed as vector form:

$x_r\left(w\right)\stackrel{\mathrm{\Δ}}{=}{[x_{r,1}\left(w\right),...,x_{r,i}\left(w\right),...,x_{r,K}\left(w\right)]}^T=D\left(y_r\right)w,$

Wherein, x _{R, i}(w) i via node place of expression treats forward signal, y _rBe the signal of via node reception,

Expression needs the junction waves beam shaping weight vector of design, w _iBe i the weights that relaying need to design,

C represents to contain the set of all via nodes, and K represents to eavesdrop the number of node, and D () expression becomes diagonal matrix with vector transformation;

1c) destination node and eavesdropping node receive the signal of relay forwarding, the signal y that the note destination node receives _d(w) and j the signal y that the eavesdropping node receives _{E, j}(w) be respectively:

$y_d\left(w\right)=g^H{x}_r\left(w\right)+n_d=\sqrt{P_s}{g}^{H}D\left(f\right)\mathrm{ws}+n_r^T{D}^H\left(g\right)w+n_d,$

$y_{e,j}\left(w\right)=h_j^H{x}_r\left(w\right)+n_{e,j}=\sqrt{P_s}{h_j}^{H}D\left(f\right)\mathrm{ws}+n_r^T{D}^H\left(h_j\right)w+n_{e,j},$

Wherein,

Represent that all via nodes are to the channel vector of destination node, g _iBe the channel information that i via node arrives destination node, Represent that all via nodes are to the channel vector of j eavesdropping node, h _{I, j}Be that i via node is to the channel information of j eavesdropping node, n _dThe reception noise of expression destination node, n _{E, i}The reception noise that represents j eavesdropping node,

E represents to contain the set of all eavesdropping nodes, and M represents to eavesdrop the number of node, () ^HThe conjugate transpose of matrix, () are got in expression ^TThe operation that transposition is got in expression.

Step 2: according to above-mentioned reception signal y _d(w) and y _{E, j}(w), calculate the received signal to noise ratio γ of destination node _d(w) and j the eavesdropping node received signal to noise ratio γ _{E, j}(w) be respectively:

${\mathrm{\γ}}_d\left(w\right)=\frac{P_s{w}^{H}D\left(f\right){\mathrm{gg}}^H{D}^H\left(f\right)w}{1+w^{H}D\left({\mathrm{gg}}^H\right)w},$

${\mathrm{\γ}}_{e,j}\left(w\right)=\frac{P_s{w}^{H}D\left(f\right)h_j{h}_j^H{D}^H\left(f\right)w}{1+w^{H}D\left(h_j{h}_j^H\right)w}.$

Step 3: according to above-mentioned received signal to noise ratio γ _d(w) and γ _{E, j}(w), calculate the safe speed R of destination node _s(w) be:

$R_s\left(w\right)=\frac{1}{2}\underset{j}{\min }(\log (1+\frac{P_s{w}^{H}D\left(f\right){\mathrm{gg}}^H{D}^H\left(f\right)\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="HDA00003376108000031.png"\; state="\{\{state\}\}">w</figure-callout>}}{1+w^{H}D\left({\mathrm{gg}}^H\right)w})-\log (1+\frac{P_s{w}^{H}D\left(f\right)h_j{h}_j^H{D}^H\left(f\right)\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="HDA00003376108000031.png"\; state="\{\{state\}\}">w</figure-callout>}}{1+w^{H}D\left(h_j{h}_j^H\right)w}))$

Wherein, log () the expression end is 2 logarithmic function.

Step 4: according to step 1b) treats forward signal x in _r(w), calculate the total power consumption at via node place || x _r(w) || ²With self power consumption | x _{R, i}(w) | ²Be respectively:

||x _r(w)|| ²=Tr(D ^H(y _r)D(y _r)ww ^H)=Tr((P _sD(f)D ^H(f)+I _K)ww ^H)，

${\left|x_{r,i}\left(w\right)\right|}^2=e_i^T\left(D^H\left(y_r\right)D\left(y_r\right){\mathrm{ww}}^H\right)e_i=e_i^T\left((P_{s}D\left(f\right)D^H\left(f\right)+I_K){\mathrm{ww}}^H\right)e_i,$

Wherein, incite somebody to action || x _r(w) || ²≤ P _TotWith | x _{R, i}(w) | ²≤ P _iAs total power constraint and self power constraint of design junction waves beam shaping weight vector, P _TotThe total power thresholds of all via nodes, P _kThe power threshold of k via node self, this P _TotAnd P _kSet generation by actual cooperation communication system, matrix trace, e are got in Tr () expression _iRepresent that i element is 1 unit column vector,

I _KThe unit matrix of expression K * K, K is the number of via node, and satisfies K〉constraint of M, wherein M is the number of eavesdropping node.

Step 5: the kernel U that calculates all eavesdropping node equivalent channel matrixes:

5a) calculating j the equivalent channel vector of eavesdropping node is

With all forms of eavesdropping the equivalent channel vector matrix of being write as of nodes be $\left[\begin{array}{c}{h}_1^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_j^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_M^{H}D\left(f\right)\end{array}\right],$

Wherein, f represents that source node arrives the channel vector of all via nodes, h _jRepresent all via nodes to the channel vector of j eavesdropping node,

E represents to contain the set of all eavesdropping nodes, and M is the number of eavesdropping node, () ^HThe conjugate transpose of matrix is got in expression, and D () expression becomes diagonal matrix with vector transformation;

5b) utilize the encapsulation function null () of MATLAB software, calculate the kernel U of above-mentioned eavesdropping node equivalent channel matrix:

$U=\mathrm{null}\left(\left[\begin{array}{c}{h}_1^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_j^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_M^{H}D\left(f\right)\end{array}\right]\right).$

Step 6: with the space constraints of w=Uv as design junction waves beam shaping weight vector, utilize this space constraint w with above-mentioned R _s(w), || x _r(w) || ²With | x _{R, i}(w) | ²Be rewritten as R _s(v), || x _r(v) || ²With | x _{R, i}(v) | ², namely $R_s\left(v\right)=\frac{1}{2}\underset{j}{\min }(\log (1+\frac{P_s{v}^H{U}^{H}D\left(f\right){\mathrm{gg}}^H{D}^H\left(f\right)\mathrm{<figure-callout\; id="1"\; label="Uv"\; filenames="HDA00003376108000031.png"\; state="\{\{state\}\}">Uv</figure-callout>}}{1+v^H{U}^{H}D\left({\mathrm{gg}}^H\right)\mathrm{Uv}})-\log (1+\frac{P_s{v}^H{U}^{H}D\left(f\right)h_j{h}_j^H{D}^H\left(f\right)\mathrm{<figure-callout\; id="1"\; label="Uv"\; filenames="HDA00003376108000031.png"\; state="\{\{state\}\}">Uv</figure-callout>}}{1+v^H{U}^{H}D\left(h_j{h}_j^H\right)\mathrm{Uv}})),$

||x _r(v)|| ²=Tr(D ^H(y _r)D(y _r)Uvv ^HU ^H)=Tr((P _sD(f)D ^H(f)+I _K)Uvv ^HU ^H)，

${\left|x_{r,i}\left(v\right)\right|}^2=e_i^T\left(D^H\left(y_r\right)D\left(y_r\right){\mathrm{Uvv}}^H{U}^H\right)e_i=e_i^T\left((P_{s}D\left(f\right)D^H\left(f\right)+I_K){\mathrm{Uvv}}^H{U}^H\right)e_i,$

Wherein, v is for needing the vector of design, and the design of w namely is converted into design to v.

Step 7: satisfying || x _r(v) || ²≤ P _TotWith | x _{R, i}(v) | ²≤ P _iConstraint under, maximize safe speed R _s(v), obtain optimal design v ^*, and according to the space constraint of above-mentioned w=Uv, obtaining optimal beam shaping weight vector is w ^*=Uv ^*

Step 8: according to optimal beam shaping weight vector w ^*, with w ^*In i element

Distribute to i via node.

Effect of the present invention can further specify by following emulation:

1, simulation parameter is set:

All channels all independently produce and obey average is 0, and variance is 1 multiple Gaussian Profile.Via node number K=10, eavesdropping node number M=5.Suppose that when the numbering i of relaying node is odd number this relaying self power threshold is P _i=0.5P _TotK, when the numbering i of relaying node was even number, this relaying self power threshold was P _i=2P _TotK, wherein P _TotRelaying gross power threshold value for actual cooperation communication system.

2, simulation result:

The present invention and the existing cooperative beam method of formationing that strengthens physical layer safety based on kernel have been carried out the emulation comparison to safe speed under above-mentioned simulated conditions, simulation result as shown in Figure 3.

As can be seen from Figure 3, the via node gross power threshold value P to setting _Tot, the present invention compares existing cooperative beam formation method based on kernel enhancing physical layer safety, and energy is the safe speed of elevator system obviously, and the effect that promotes is along with source node transmitting power P _sIncrease and increase.

Claims (4)

1. one kind strengthens the cooperative beam formation method of physical layer safety based on kernel, may further comprise the steps:

1) cooperating relay was transmitted in two stages:

1a) in the phase I, source node is to all via node speech amplifier confidential informations, and the signal that via node receives is y _r

1b) in second stage, via node is y to the received signal _rBe weighted processing, obtain signal x to be transmitted _r(w)=D (y _r) w, wherein w is for needing the junction waves beam shaping weight vector of design, and D () expression becomes diagonal matrix with vector transformation;

1c) destination node and eavesdropping node receive the signal of relay forwarding, and the signal that the note destination node receives is y _d(w), remember that the signal that j eavesdropping node receives is y _{E, j}(w),

E represents to contain the set of all eavesdropping nodes, and M represents to eavesdrop the number of node;

2) according to above-mentioned reception signal y _d(w) and y _{E, j}(w), calculate the received signal to noise ratio γ of destination node _d(w) and j the eavesdropping node received signal to noise ratio γ _{E, j}(w);

3) according to above-mentioned received signal to noise ratio γ _d(w) and γ _{E, j}(w), calculate the safe speed of destination node $R_s\left(w\right)=\frac{1}{2}\underset{j}{\min }\{\log (1+{\mathrm{\&gamma;}}_d\left(w\right))-\log (1+{\mathrm{\&gamma;}}_{e,j}\left(w\right))\};$

4) according to step 1b) in treat forward signal x _r(w), calculate the consumption of via node gross power || x _r(w) || ²Consumption with i via node self power | x _{R, i}(w) | ², and will || x _r(w) || ²≤ P _TotWith | x _{R, i}(w) | ²≤ P _iAs the power constraint condition of design junction waves beam shaping weight vector, wherein x _{R, i}(w) expression x _r(w) i element, P _TotThe total power thresholds of all via nodes, P _iThe power threshold of i via node self, this P _TotAnd P _iSet generation by actual cooperation communication system,

C represents to contain the set of all via nodes, and K represents to eavesdrop the number of node;

5) calculate the kernel U that all eavesdrop the node equivalent channel matrixes, with the space constraints of w=Uv as design junction waves beam shaping weight vector, utilize this space constraint w with above-mentioned R _s(w), || x _r(w) || ²With | x _{R, i}(w) | ²Be rewritten as R _s(v), || x _r(v) || ²With | x _{R, i}(v) | ², and will be converted into to the design of w design to v;

6) satisfy above-mentioned || x _r(v) || ²≤ P _TotWith | x _{R, i}(v) | ²≤ P _iConstraint under, design vector v is so that safe speed R _s(v) maximization obtains optimal design v ^*, and according to the space constraint of w=Uv in the step 5), obtaining optimal beam shaping weight vector is w ^*=Uv ^*

7) according to optimal beam shaping weight vector w ^*, with w ^*In i element

Distribute to i via node.

2. according to claim 1ly strengthen the cooperative beam formation method of physical layer safety, wherein step 2 based on kernel) the received signal to noise ratio γ of described calculating destination node _d(w) and j the eavesdropping node received signal to noise ratio γ _{E, j}(w), undertaken by following formula

${\mathrm{\&gamma;}}_d\left(w\right)=\frac{P_s{w}^{H}D\left(f\right){\mathrm{gg}}^H{D}^H\left(f\right)w}{1+w^{H}D\left({\mathrm{gg}}^H\right)w}$

${\mathrm{\&gamma;}}_{e,j}\left(w\right)=\frac{P_s{w}^{H}D\left(f\right)h_j{h}_j^H{D}^H\left(f\right)w}{1+w^{H}D\left(h_j{h}_j^H\right)w}$

Wherein, P _sThe transmitting power of expression source node, f represents that source node arrives the channel vector of all via nodes, g represents that all via nodes are to the channel vector of destination node, h _jRepresent that all via nodes are to the channel vector of j eavesdropping node, () ^TTranspose of a matrix, () are got in expression ^HThe conjugate transpose of matrix is got in expression.

3. according to claim 1ly strengthen the cooperative beam formation method of physical layer safety, the wherein consumption of the described calculating via node of step 4) gross power based on kernel || x _r(w) || ²Consumption with k via node self power | x _{R, i}(w) | ², undertaken by following formula

||x _r(w)|| ²＝Tr(D ^H(y _r)D(y _r)ww ^H)＝Tr((P _sD(f)D ^H(f)+I _K)ww ^H)

${\left|x_{r,i}\left(w\right)\right|}^2=e_i^T\left(D^H\left(y_r\right)D\left(y_r\right){\mathrm{ww}}^H\right)e_i=e_i^T\left((P_{s}D\left(f\right)D^H\left(f\right)+I_K){\mathrm{ww}}^H\right)e_i$

Wherein, P _sThe transmitting power of expression source node, f represents that source node arrives the channel vector of all via nodes, matrix trace, () are got in Tr () expression ^HThe conjugate transpose of matrix is got in expression, and D () expression becomes diagonal matrix with vector transformation, e _iRepresent that i element is 1 unit column vector, I _KThe unit matrix of expression K * K, K is the number of via node, and satisfies the constraint of K＞M, wherein M is the number of eavesdropping node.

4. according to claim 1ly strengthen the cooperative beam method of formationing of physical layer safety based on kernel, wherein the described calculating of step 5) all eavesdrop the kernel U of node equivalent channel matrixes, carry out as follows:

5a) calculate j the equivalent channel vector of eavesdropping node

And with forms of the equivalent channel vector matrix of being write as of all eavesdropping nodes be $\left[\begin{array}{c}{h}_1^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_j^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_M^{H}D\left(f\right)\end{array}\right],$

Wherein, f represents that source node arrives the channel vector of all via nodes, h _jRepresent all via nodes to the channel vector of j eavesdropping node,

E represents to contain the set of all eavesdropping nodes, and M is the number of eavesdropping node, () ^HThe conjugate transpose of matrix is got in expression, and D () expression becomes diagonal matrix with vector transformation;

5b) utilize the encapsulation function null () of MATLAB software, calculate the kernel U of above-mentioned eavesdropping node equivalent channel matrix:

$U=\mathrm{null}\left(\left[\begin{array}{c}{h}_1^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_j^{H}D\left(f\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ h_M^{H}D\left(f\right)\end{array}\right]\right).$

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