Systems and methods for securing wireless communications

Abstract

Systems and methods of friendly jamming for securing wireless communications at the physical layer are presented. Under the assumption of exact knowledge of the eavesdropping channel, a resource-efficient distributed approach is used to improve the secrecy sum-rate of a multi-link network with one or more eavesdroppers while satisfying an information-rate constraint for all links. A method based on mixed strategic games can offer robust solutions to the distributed secrecy sum-rate maximization. In addition, a block fading broadcast channel with a multi-antenna transmitter, sending two or more independent confidential data streams to two or more respective users in the presence of a passive eavesdropper is considered. Lastly, a per-link strategy is considered and an optimization problem is formulated, which aims at jointly optimizing the power allocation and placement of the friendly jamming devices for a given link under secrecy constraints.

Claims

1. An integrated transmitter/receiver based friendly jamming system comprising: (a) a transmitting mobile device with three or more antennas, capable of transmitting at least two information signals and, simultaneously, an artificial noise signal; (b) two or more receiving mobile devices, each receiving mobile device having one or more antennas, wherein a set of channel conditions of each of the two or more receiving mobile devices is known to the transmitting mobile device, wherein each receiving mobile device achieves a secrecy rate, where the secrecy rate is a data rate at which an information signal from the transmitting device can be securely received by the receiving mobile device when one or more eavesdroppers of unknown locations are present, wherein the secrecy rate is zero if the signal-to-noise ratio of the information signal at the receiver drops below a threshold, wherein the signal-to-noise ratio is the ratio of the power of the information signal to the power of all noise sources combined; (c) a self-interference cancellation module embedded in each receiving mobile device, wherein the self-interference cancellation module allows each receiving mobile device to transmit a friendly jamming signal while simultaneously receiving an information signal from the transmitting mobile device, wherein the friendly jamming signal is a receiver-based artificial noise signal, wherein the self-interference cancellation module allows the receiver to cancel the receiver-based artificial noise signal, wherein a portion of self-interference is left after cancellation; (d) a transmitter precoder embedded in the transmitting mobile device wherein the transmitter precoder utilizes a set of signal processing algorithms for precoding one or more confidential information signals before transmission to the two or more receiving mobile devices, wherein the transmitter precoder also precodes the transmitter-based artificial noise signal; and (e) a receiver precoder embedded in each of the two or more receiving mobile devices for precoding the receiver-based artificial noise signal; wherein the three or more antennas of the transmitting mobile device are jointly used to simultaneously transmit two or more information signals along with the transmitter-based artificial noise signal to two or more receiving mobile devices, wherein the information signal transmitted by the transmitting mobile device to a receiving mobile device of the two or more receiving mobile devices is preceded by the transmitter precoder such that the information signal is cancelled out at unintended receiving mobile devices, and the signal strength of the information signal at each intended receiving mobile device is maximized, wherein the transmitter-based artificial noise signal is precoded by the transmitter precoder such that the transmitter-based artificial noise signal cancels out at each of the two or more receiving mobile devices, wherein the one or more antennas of each receiving mobile device receives the intended information signal, wherein the effect of the transmitter-based artificial noise signal is cancelled out at each receiving mobile device using zero-forcing beamforming technique implemented at the transmitting device, wherein the unintended information signals are cancelled out at each receiving mobile device using the zero-forcing beamforming technique implemented at the transmitting device, wherein the one or more antennas at each receiving device also transmit a receiver-based artificial noise signal, wherein the self-interference cancellation module cancels the receiver-based artificial noise signal, wherein a portion of self-interference of the artificial noise signal remains, wherein the power of the information signal and the receiver-based artificial noise signal at each receiver are set so that the signal-to-noise ratio of the information signal at each receiver exceeds a target threshold, wherein the power of the transmitter-based artificial noise signal is set at a level which optimizes a utility function, wherein the utility function is the sum of the secrecy rates of all receiving mobile devices in the system, wherein the optimal value of the utility function is achieved when the power of the transmitter-based artificial noise signal is equal to the total power budget of the transmitter minus the sum of the transmit powers of the information signals, wherein the codewords of the information signals are designed over multiple fading blocks to cope with fast-fading channel conditions and the lack of information about the eavesdroppers' locations, wherein the required amount of randomization in the codewords of the information signals is shared by all receiving mobile devices so as to confuse the eavesdropper.

2. The system of claim 1, wherein the receiver-based artificial noise signal is precoded by the receiver precoder such that the receiver-based artificial noise signal cancels out at every other receiving mobile device in the system.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows an exemplary system model according to an embodiment of the present invention.

(2) FIG. 2 shows a non-limiting example of achievable rate pairs for a two-user multiple access channel.

(3) FIG. 3 depicts an exemplary effect of Eve's location on secrecy sum-rate, wherein the configuration is

(4) $\left\{{\mathrm{Alice}}_1=\left(-5,8\right),{\mathrm{Bob}}_1=\left(5,8\right),{\mathrm{Alice}}_2=\left(-5,-8\right),{\mathrm{Bob}}_2=\left(5,-8\right),{\hat{y}}_e=3.5,{\hat{r}}_e=1.6,P_q=-32\mathrm{dBm},{}_q=\frac{1}{3},N_{T_q}=3\right\}.$

(5) FIG. 4 depicts an exemplary effect of Eve's location on secrecy sum-rate, wherein the configuration is

(6) $\left\{{\mathrm{Alice}}_1=\left(-50,10\right),{\mathrm{Bob}}_1=\left(5,10\right),{\mathrm{Alice}}_2=\left(-50,-10\right),{\mathrm{Bob}}_2=\left(50,10\right),{\hat{y}}_e=0,{\hat{r}}_e=10,P_q=0\mathrm{dBm},{}_q=\frac{1}{3},N_{T_q}=3\right\}.$

(7) FIGS. 5A and 5B each depict an exemplary effect of Eve's location on individual secrecy sum-rates. The configurations are the same as FIG. 4.

(8) FIG. 6 depicts an exemplary effect of Eve's location on secrecy sum-rate, wherein the configuration is {(.sub.q=, Alice.sub.1=(40,20), Bob.sub.1=(40,20), Alice.sub.2=(40,20), Bob.sub.2=(40,20), .sub.e=25, {circumflex over (r)}.sub.e=20, P.sub.q=10 dBm, N.sub.T.sub.q=3 q}.

(9) FIG. 7 depicts an exemplary effect of small number of sample data transmissions, wherein the configuration is {(.sub.q=, Alice.sub.1=(20,20), Bob.sub.1=(20,20), Alice.sub.2=(20,20), Bob.sub.2=(20,20), .sub.e=10, {circumflex over (r)}.sub.e=20, P.sub.q=10 dBm, N.sub.T.sub.q=4q}.

(10) FIG. 8 shows an exemplary system model with both TxFJ and RxFJ.

(11) FIG. 9 depicts an exemplary effect of self-interference suppression on the secrecy sum-rate (bits/second/Hz).

(12) FIG. 10 depicts an exemplary effect of the minimum required SINR of the users, T, on the secrecy sum-rate (bits/second/Hz).

(13) FIG. 11 shows exemplary contours of secrecy sum-rate in (bits/second/Hz) for various locations of Eve (Rx FJ+Tx FJ).

(14) FIG. 12 shows exemplary contours of secrecy sum-rate in (bits/second/Hz) for various locations of Eve (Tx FJ).

(15) FIG. 13 shows exemplary contours of secrecy sum-rate in (bits/second/Hz) for various locations of Eve (Rx FJ).

(16) FIG. 14 depicts placements of single-antenna FJ devices (without nullification constraints.) Jamming power is a function of distances to and received information signals at each Eve.

(17) FIG. 15 shows an exemplary network topology for the case of five P2P links. Hollow circles, crossed circles, and diamonds represent receivers, transmitters, and potential eavesdropping locations.

(18) FIG. 16 shows an exemplary total jamming power vs. number of links for the P2P scenario (per-link).

(19) FIG. 17 depicts an exemplary outcome of the proposed per-link scheme for the one-link case. The hollow, crossed, and solid circles represent the locations of Bob, Alice, and the FJ nodes, respectively.

(20) FIG. 18 shows exemplary jamming power of each FJ node along with total jamming power vs. iteration for the example in FIG. 17.

(21) FIG. 19 depicts power consumption vs. number of potential eavesdropping locations for the case of two P2P links (per-link).

(22) FIG. 20 shows an example of a Network topology for routing simulations. Hollow circles represent the legitimate nodes and diamonds represent potential eavesdropping locations.

(23) FIG. 21 depicts examples of power consumption of the network-wide and per-link schemes vs. the number of potential eavesdropping locations for the case of two P2P link.

(24) FIG. 22 is a depiction of the resource-efficient, distributed friendly jamming system of Example 1.

(25) FIG. 23 is a depiction of the integrated transmitter/receiver-based friendly jamming system of Example 2.

(26) FIG. 24 is a depiction of the efficient friendly jamming system for multi-link wireless networks of Example 3.

DESCRIPTION OF PREFERRED EMBODIMENTS

(27) Referring now to FIGS. 1-7, the present invention features a price-based distributed friendly jamming system for wireless communications security. The system may comprise at least two transmitting mobile devices, each transmitting mobile device comprising at least two transmit antennas, and at least two receiving mobile devices. In some embodiments, each receiving mobile device communicates with the corresponding transmitting mobile device. The transmitting mobile device may know the location of the corresponding receiving mobile device. In other embodiments, each receiver has a secrecy rate, which is the data rate at which secure transmission can be received, assuming the presence of an unknown eavesdropper.

(28) In one embodiment, each transmitting mobile device can transmit an information signal intended for the corresponding receiving mobile device along with a signal of artificial noise. Preferably, the direction of the artificial noise is such that the artificial noise is zero at a location of the receiving mobile device, and the artificial noise is non-zero at locations of any other receiving device. In another embodiment, an interference at other receivers is increased when a power of the artificial noise signal increases. The secrecy rate at the corresponding receiver may be increased with the power of the artificial noise signal and the interference may reduce the secrecy rate at the other receivers.

(29) In some embodiments, a penalty (price) is charged for increasing the power of artificial noise. The transmitters may compete to optimize a utility function of the corresponding receiver's secrecy rate minus the price paid for generating artificial noise. At equilibrium, a reduction in noise power can increase the sum of all secrecy rates. The overall goal of the pricing approach is to regulate the amount of artificial noise power used by transmitting devices such that the sum of secrecy rates of all legitimate links in the network is maximized. Such maximization is strictly in the direction of reducing interference at legitimate receivers and increasing it at eavesdropper(s).

(30) Without any loss of generality, in the following, the case of two legitimate multi-antenna transmitters communicating with two respective receivers in the presence of a single-antenna eavesdropper is described. Each receiver is equipped with a single antenna. Extensions to more than two receivers, multiple antennas per receiver, and multiple eavesdroppers are straightforward. As shown in FIG. 1, multiple transmitters and receivers operate within the same geographic locality in the presence of an eavesdropper. Alice1 is attempting to transmit to Bob1 and Alice2 is attempting to transmit to Bob2, while Eve is eavesdropping. Both Alice1 and Alice2 have at least two antennas. The transmitted signal includes both information signal added by artificial noise. H.sub.11, H.sub.12, H.sub.21, and H.sub.22 are channel matrices that transform the signal between the transmitters and both receivers. These matrices are dependent on the location of the receivers and are known to the transmitters. Alice1 pre-codes the artificial noise signal such that the artificial noise is canceled out by the H.sub.11 channel matrix and becomes zero at Bob1's location. Alice2 pre-codes the artificial noise signal such that the artificial noise is canceled out by the H.sub.22 channel matrix and becomes zero at Bob2's location. Neither artificial noise signal is cancelled out at Eve's location. However, the artificial noise from Alice1 is not canceled out at Bob2's location and vice versa, which causes interference with the legitimate reception of Bob1 and Bob2. Increasing the signal strength of the artificial noise at one link may create more security for the Alice1-Bob1 link but degrades signal reception of the other link, which leads to suboptimal performance of the overall system. The present invention may comprise applying a price to adjust the artificial noise signal strength in Alice1 and Alice2 so as to maximize the sum of secrecy rates in the network. Applying the pricing scheme is done by modifying the secrecy-rate function at the respective receivers. Hence, each link attempts to maximize a function of the secrecy-rate at the respective receivers, minus the cost of the artificial noise power. Alice1 and Alice2 each are able to compute the secrecy-rate, which depends on Eve's location. Such a design helps nodes regulate their artificial noise powers in a way that interference at legitimate nodes is decreased, and yet the sum of secrecy rates of both links is maximized. Since Eve is closer to one transmitter then the other, the function will be optimized at different power levels for each receiver. For example, if Eve is closer to Alice2 than Alice1, as shown in FIG. 1, then Alice1 can reduce the power level of its artificial noise signal without degrading the secrecy rate at Bob1. This improves the secrecy-rate at Bob2, thus leading to a more optimal system-wide performance.

(31) Referring to FIGS. 8-13, another embodiment of the present invention features a receiver-based friendly jamming system. The system may comprise at least one transmitting mobile device comprising at least three transmit antennas, and at least two receiving mobile devices. In some embodiments, each receiving mobile device may comprise one or more antennas with the capability of full-duplexity. Note that FIG. 8 shows a sample scenario with two receiving mobile devices, each having a single antenna. Instead of labeling these devices Bob1 and Bob2, they are labeled Bob and Charlie to make the notation easier. Hereafter, the present invention will be analyzed for this sample scenario. The transmitting mobile device can communicate with the receiving mobile devices. The transmitting mobile device can also know the locations of the receiving mobile devices.

(32) In one embodiment, the transmitting mobile device may transmit independent information signals to the first and the second receiving mobile devices. The information signals are pre-coded to cancel out at other receiving devices.

(33) In some embodiments, the transmitting mobile device may transmit an artificial noise signal. The artificial noise signal is pre-coded to cancel out at a location of each receiving mobile device. In other embodiments, the receiving antenna of each receiving mobile device receives the information signal. In still other embodiments, each receiving mobile device transmits an artificial noise signal. The receiving mobile device may use the same antenna to concurrently transmit artificial noise and receive the information message over the same channel.

(34) As shown in FIG. 8, Alice attempts to transmit information signals to Bob and Charlie in the presence of an eavesdropper Eve. Alice knows the channel states at Bob and Charlie, i.e., channel vectors h.sub.AB and h.sub.AC. The information signal to Bob is pre-coded to cancel out at Charlie's location and vice versa. In addition to the two information signals transmitted to Bob and Charlie, Alice transmits a pre-coded artificial noise signal, which is pre-coded to cancel out at the locations of both Bob and Charlie. As an additional layer of security, the present invention comprises the receivers emitting an additional artificial noise signal.

(35) Referring to FIGS. 14-21, a further embodiment of the present invention features an efficient, friendly jamming system for multi-link wireless networks. The system may comprise a plurality of wireless mobile devices, each having a mobile device location, and a plurality of wireless friendly-jamming (FJ) devices. Each friendly jamming device may transmit an artificial noise signal and may have an FJ device location and a power allocation. In some embodiments, the artificial noise signals of the FJ devices are jointly pre-coded to produce null spaces of the artificial noise signal at the locations of the wireless mobile devices. Preferably, the FJ device locations and joint power allocations of the FJ devices are jointly optimized to minimize power consumption under a secrecy constraint.

(36) As shown in FIG. 14, Alice (circle and x) attempts to transmit an information signal to Bob (marked by circle), in the presence of multiple eves droppers (Eve1 and Eve2). An independent friendly jamming device (square) is placed somewhere in the field, and emits an artificial noise signal. The position of the jamming device and the power allocation is optimized to minimize jamming power consumption under the constraint of a minimum requirement for the secrecy-rate at Bob, which is dependent on the FJ power seen by all eavesdroppers. (i.e. more noise at Eve obscures the information signal increasing Bob's secrecy rate).

(37) Referring to FIG. 17, a second example of an efficient friendly jamming system for multi-link mobile wireless network is shown. In this example, there are four eavesdroppers (1,2,3,4) and two FJ devices (FJ1, and FJ2). The FJ devices emit noise that is canceled out at Bob (circle), but not at the eavesdropper's locations.

(38) As used herein, the following notation is adopted. Vectors and matrices are denoted by bold lower-case and uppercase letters, respectively. Column and row vectors notations are used interchangeably. ().sup. and ().sup.T represent the complex conjugate transpose and the transpose of a vector or matrix, respectively. Frobenius norm and the absolute value of a real or complex number are denoted by and |.Math.|, respectively. [] indicates the expectation of a random variable. A.sup.MN means that A is an MN complex matrix. (, .sup.2) denotes a complex Gaussian random variable with mean and variance .sup.2. I.sub.N represents an NN identity matrix. [x]+=x for x>0, and [x]+=0 for x0.

Example 1

(39) The following is a non-limiting example of a price-based jamming control that can guarantee a local optimum for secrecy sum-rate.

(40) System Model

(41) A communication scenario is shown in FIG. 1, where two transmitters, Alice1 and Alice2 communicate with two respective receivers, Bob1 and Bob2. Each transmitter (TX) q, q=1, 2, has N.sub.Tq transmit antennas. A passive eavesdropper (i.e., Eve) with one antenna exists in the range of communication. The received signal by the qth receiver, y.sub.q, can be written as:
y.sub.q={tilde over (H)}.sub.qqu.sub.q+{tilde over (H)}.sub.rqu.sub.r+n.sub.q, r,q{1,2}, rq(1a)
where H.sub.qT denotes the 1N.sub.T.sub.q channel matrix between the qth transmitter and the rth receiver, u.sub.q is the transmitted signal from the qth transmitter, and n.sub.q is the complex Additive White Gaussian Noise (AWGN) with the power N.sub.0. We assume that H.sub.qT is a zero mean complex Gaussian matrix. Let G.sub.q denote the 1N.sub.T.sub.q complex channel matrix between the qth transmitter and Eve. Eve's received signal is
z={tilde over (G)}.sub.au.sub.a+{tilde over (G)}.sub.ru.sub.r+e(2a)
where e has the same characteristics of n.sub.q. We assume that channels remain stationary over the duration of each transmission. The term u.sub.q=s.sub.q+w.sub.q is the sum of an information bearing signal s.sub.q and artificial noise w.sub.q. For the AN, we write it as w.sub.e=Z.sub.qv.sub.q, where Z.sub.q is an orthonormal basis for the null space of H.sub.qq (H.sub.qqw.sub.q=0) and v.sub.q is a vector of i.i.d. complex Gaussian random variables with covariance matrix

(42) $E\left[v_q{v}_q^H\right]={}_q{I}_{N_{T_q}-1}$
where E[.] is the expectation operator. In here, the scalar value denotes the jamming power and I.sub.N is the NN identity matrix. For the information bearing signal, s.sub.q=T.sub.qx.sub.q where T.sub.q is the precoder and x.sub.q is the information signal. We assume that Gaussian codebook is used, Furthermore, Let X.sub.q=tr{E[x.sub.qx.sub.q.sup.H]} and assume that X.sub.qp.sub.q where tr[.] is the trace operator, (.).sup.H is the Hermitian of a matrix, and p.sub.q is a scalar representing a constraint on information signal's power. The power constraint for each transmitter q is written as
E[|u.sub.q|.sup.2]=tr{E[u.sub.qu.sub.q.sup.H]}P.sub.q,(3a)
Thus,
tr{E[v.sub.qv.sub.q.sup.H]}+X.sub.qP.sub.q
where P.sub.q=.sub.q(N.sub.T.sub.q1)+p.sub.q is a scalar value that represents the total power budget for the qth transmitter. We assume that qth receiver is able to estimate the channel H.sub.qq and feed it back to the qth transmitter, q=1, 2. Note that the process of acquiring channel state information (CSI) is assumed to be done securely, so that we can only focus on the secrecy of transmission phase. With the knowledge of H.sub.qq, the transmitter uses singular value decomposition of {tilde over (H)}.sub.qq, {tilde over (H)}.sub.qq=U.sub.q.sub.qV.sub.q.sup.H. Hence, the precoder T.sub.q is set to T.sub.q=V.sub.q.sup.1, where V.sub.q.sup.1 is the first column of V.sub.q and Z.sub.q=V.sub.q.sup.2, where V.sub.q.sup.2 is the matrix of N.sub.T.sub.q1 rightmost columns of V.sub.q.

(43) Problem Formulation

(44) Given that {tilde over (H)}.sub.qqV.sub.q.sup.2=0, we set H.sub.qq={tilde over (H)}.sub.qqV.sub.q.sup.1, H.sub.qr={tilde over (H)}.sub.qrV.sub.q.sup.1, H.sub.jq={tilde over (H)}.sub.qrV.sub.q.sup.2, G.sub.q=GqVq1, Gjg=GqVq2. For q=1,2, the terms G.sub.q and G.sub.iq are called the eavesdropping channel components. Hence,
y.sub.q=H.sub.qqx.sub.q+H.sub.rqx.sub.r+H.sub.jrv.sub.r+n.sub.q(4a)
z.sub.q=G.sub.qx.sub.q+G.sub.jqv.sub.q+G.sub.rx.sub.r+G.sub.j.sub.rv.sub.r+e.sub.q.(5a)
The information rate for the qth user can be written as

(45) $\begin{array}{cc}{C}_q=\log \left(1+\frac{{.Math.H_{\mathrm{qq}}{x}_q.Math.}^2}{{.Math.H_{\mathrm{rq}}{x}_r.Math.}^2+{.Math.H_{j_r}.Math.}^2{}_r+N_0}\right).& \left(6a\right)\end{array}$

(46) The channel between the two Alices and Eve can be modeled as a multiple-access channel, since Eve is simultaneously receiving signals from both Alices. If Eve is capable of using successive interference cancellation (SIC) technique, it can simultaneously decode the signals from both Alices. The achievable-rate region of Eve's multi-access channel is shown in FIG. 2. C.sub.eq denotes the achievable rate at Eve while decoding the qth user's signal, where q=1,2. The points B.sub.q and F.sub.q are defined in the following. FIG. 2 suggests that to prevent Eve from using SIC, we must have C.sub.q>B.sub.q [12a], i.e.,

(47) $\begin{array}{cc}\log \left(1+\frac{{.Math.H_{\mathrm{qq}}{x}_q.Math.}^2}{{.Math.H_{\mathrm{rq}}{x}_r.Math.}^2+{.Math.H_{j_r}.Math.}^2{}_r+N_0}\right)>{}_q\mathrm{where}{}_q=\log \left(|1+\frac{{.Math.G_q{x}_q.Math.}^2}{{.Math.G_{j_q}.Math.}^2{}_q+{.Math.G_r{x}_r.Math.}^2+{.Math.G_{j_r}.Math.}^2{}_r+N_0}\right),q.& \left(7a\right)\end{array}$

(48) If inequality (7a) is satisfied, then Eve has to decode the signal of the rth user by considering the signal of the qth user as interference. Thus, the achievable rate received at Eve is C.sub.er=B.sub.r and the secrecy rate of the rth user would be

(49) $\begin{array}{cc}{C}_r^{\mathrm{se}c}=\max \left\{C_r-C_{\mathrm{cr}},0\right\}=\log \left(1+\frac{{.Math.H_{\mathrm{rr}}{x}_r.Math.}^2}{{.Math.H_{\mathrm{qx}}{x}_q.Math.}^2+{.Math.H_{j_q}.Math.}^2{}_q+N_0}\right)-\log \left(1+\frac{{.Math.G_r{x}_r.Math.}^2}{{.Math.G_{j_r}.Math.}^2{}_r+{.Math.G_q{x}_q.Math.}^2+{.Math.G_{j_q}.Math.}^2{}_q+N_0}\right),rq.& \left(8a\right)\end{array}$

(50) If (7a) is not satisfied, Eve considers the signal of the rth user as interference and decodes the signal of the qth user. Knowledge of the qth user's signal allows Eve to deduct the qth user's signal from the received signal and obtain an interference-free signal from the rth user. Hence, C.sub.er=F.sub.r and the secrecy rate of the rth user would be

(51) $\begin{array}{cc}{C}_r^{\mathrm{se}c}=\max \left\{\log \left(1+\frac{{.Math.H_{\mathrm{rr}}{x}_r.Math.}^2}{{.Math.H_{\mathrm{qr}}{x}_q.Math.}^2+{.Math.H_{j_q}.Math.}^2{}_q+N_0}\right)-{}_r,0\right\}& \left(9a\right)\\ {}_r=\log \left(1+\frac{{.Math.G_r{x}_r.Math.}^2}{{.Math.G_{j_q}.Math.}^2{}_q+{.Math.G_{j_r}.Math.}^2{}_r+N_0}\right).& \left(10a\right)\end{array}$

(52) It is obvious from (7a) and (9a) that in order to achieve the maximum secrecy, the two users have to choose a transmission rate higher than Eve's decodable rate. As can be seen in (8a), the interference caused by the friendly jamming signal can degrade the received signal to interference and noise ratio (SINR) at unintended receivers. On the contrary, the sum of interference accumulated at Eve can ensure low SINR at Eve. The degradation of both information rate and Eve's rate creates a conflicting situation. The performance of the network is comprised of the performance of both users. Hence, we define the term secrecy sum-rate to be
C.sup.sec=C.sub.1.sup.sec+C.sub.2.sup.sec.(11a)
We want to maximize C.sup.sec while ensuring a minimum QoS for both users. This problem can be formally written as:

(53) $\begin{array}{cc}\underset{\left\{X_1,X_2,{}_1,{}_2\right\}}{\mathrm{maximize}}{C}^{\mathrm{se}c}s.t.\{\begin{array}{c}{X}_q+{}_q\left(N_{T_q}-1\right)P_q\\ C_q{c}_q\end{array},q\left\{1,2\right\}& \left(12a\right)\end{array}$
where the second constraint ensures a minimum QoS (i.e., c.sub.q) for each user. The optimization in (12a) is non-convex. Thus, finding its optimum is prohibitively expensive. One relaxation to this problem is to eliminate the dependency of the problem on X.sub.1 and X.sub.2. We assume that the minimum QoS constraint, is satisfied with equality. Hence, C.sub.q=c.sub.q which can be reduced to a relation in the form of X.sub.q=.sub.q where .sub.qp.sub.q, q=1,2. By doing so, the second constraint can be embedded into the objective function and the first constraint. Hence, we have

(54) $\begin{array}{cc}\begin{array}{cc}\underset{\left\{{}_1,{}_2\right\}}{\mathrm{maxmize}}& C^{\mathrm{se}c}\\ s.t.& {}_q\frac{P_q-{}_q}{N_{T_{}}-1},q\left\{1,2\right\}.\end{array}& \left(13a\right)\end{array}$

(55) Note that p.sub.q=.sub.q and .sub.q (N.sub.T.sub.q1)=(1)P.sub.q where .sub.q is a scalar variable within the interval [0,1] representing the proportion of power allocated to the information signal and to the AN. We also note that the jamming power has to be chosen such that inequality (7a) is satisfied. Reducing (7a), we have

(56) 0$\begin{array}{cc}{}_q>\frac{A_q}{B_q},\mathrm{where}& \left(14a\right)\\ A_q={.Math.G_q.Math.}^2{}_q\left({.Math.H_{\mathrm{rq}}.Math.}^2{}_r+{.Math.H_{j_r}.Math.}^2{}_r+N_0\right)-{.Math.H_{\mathrm{qq}}.Math.}^2{}_q\left({.Math.G_r.Math.}^2{}_r+{.Math.G_{j_r}.Math.}^2{}_r+N_0\right),B_q={.Math.G_{jq}.Math.}^2{.Math.H_{\mathrm{qq}}.Math.}^2{}_q.& \left(15a\right)\end{array}$
Simplifying (14a), we have the following constraint for .sub.q:

(57) $\begin{array}{cc}{}_q=\frac{P_q-p_q}{N_{T_q}-1}\mathrm{if}\frac{A_q}{B_q}>\frac{P_q-p_q}{N_{T_q}-1}& \left(16a\right)\\ {}_q>\frac{A_q}{B_q}\mathrm{if}{A}_q>0\&\frac{A_q}{B_q}<\frac{P_q-p_q}{N_{T_q}-1},& \left(17a\right)\\ {}_q>0\mathrm{if}{A}_q<0.& \left(18a\right)\end{array}$
For the case in (16a), no power can prevent Eve from using SIC to decode two information messages, and the solution to (13a) would be infeasible. Hence, we assume that if (16a) is true for any of the users, they will not start any communications. Considering that we always have either of (17a) or (18a), the optimization in (13a) becomes

(58) $\begin{array}{cc}\begin{array}{cc}\underset{\left\{{}_1,{}_2\right\}}{\mathrm{maximize}}& C^{\mathrm{se}c}\\ s.t.& {}_q{D}_q,q\left\{1,2\right\}\end{array}& \left(19a\right)\end{array}$
where

(59) $D_q\stackrel{}{=}\left[\max \left\{\frac{A_q}{B_q},{}_q\right\},\frac{P_q-p_q}{N_{T_q}-1}\right]\mathrm{and}\left[a,b\right]$
represents an interval between a and b.

(60) Since the inequality in (18a) strictly suggests that the jamming power has to be positive, we set .sub.q to be the smallest positive value in the D.sub.q. The optimization in (13a) tries to find the best trade-off between the jamming powers of the two users, In other words, the Pareto-optimal jamming powers can be found by solving (13a). Unfortunately, the optimization in (13a) is still non-convex. Furthermore, it requires the exact knowledge of the eavesdropping channel components.

(61) Game Formulation

(62) 1. Greedy Friendly Jamming:

(63) One solution to reduce the complexity of (13a) is to let each user maximize its own secrecy rate and ignore the effect of its friendly jamming on the other user. This locally optimized jamming control leads to a game theoretic interpretation of this network. Assuming that each user myopically chooses the best strategy for itself, we formulate this scenario as a non-cooperative game, in which the best strategy of each user q, q=1,2 is

(64) $\begin{array}{cc}\begin{array}{cc}\underset{{}_q}{\mathrm{maximize}}& C_q^{\mathrm{se}c}\\ s.t.& {}_q{D}_q.\end{array}& \left(20a\right)\end{array}$
In this game, the utility function of each player (user) is its secrecy rate and his strategy is to choose the best jamming power subject to the jamming power constraint (i.e., strategy set) to maximize its utility. The existence of Nash equilibrium (NE) in this game can be proven by showing that the strategy set of each user is a non-empty, compact, and convex subset of , and the utility function of each user is a continuous and quasi-concave function of the jamming power. Verifying these properties in the game is straightforward and is skipped for brevity. Since the objective function in (20a) is strictly concave in .sub.q, the best strategy that maximizes the secrecy rate of the qth user is to select the maximum available jamming power

(65) $\left(i.e.,{}_q=P_q^{\mathrm{jam}}=\frac{P_q-p_q}{N_{T_q}-1},q=1,2\right).$
Given the strategy of the user in the network, when .sub.q=P.sub.q.sup.jam, q, no user is willing to unilaterally change its own strategy because choosing any jamming power less than that can degrade the individual secrecy rate of that user. Therefore the point .sub.q=P.sub.q.sup.jam, q is the NE. This result is in line with [4a] for the single-user case.

(66) This NE point, however, might not always be efficient, because the selfish maximization of secrecy rate by each user is not always guaranteed to be Pareto-optimal. As an intuitive explanation, consider the case where the interference from Alice2 is large enough to affect Bob1, but Eve is much closer to Alice1 than to Alice2. Considering almost equal jamming power constraints, if both users select the maximum jamming power, Alice1 can make its transmission more secure by applying maximum jamming power. However, although Alice2 is not that much in the risk of being eavesdropped, it chooses maximum jamming power which has little impact on its own secrecy rate, but can degrade the received SINR at Bob1. Degradation of C.sub.1 makes the transmission of Alice1 less secure, so the secrecy sum-rate of the network will be reduced.

(67) 2. Price-Based Friendly Jamming:

(68) The efficiency of the NE in greedy friendly jamming can be improved by using appropriate pricing policies. Hence, for q{1,2}, the objective function of player q in (20a) is modified into:

(69) $\begin{array}{cc}\begin{array}{cc}\underset{{}_q}{\mathrm{maximize}}& C_q^{\mathrm{se}c}-{}_q{}_q\\ s.t.& {}_q{D}_q\end{array}& \left(21a\right)\end{array}$
where .sub.q is the pricing factor for the qth user, defined in (22a).

(70) $\begin{array}{cc}{}_q=\frac{{.Math.H_{j_q}{H}_{\mathrm{rr}}.Math.}^2{}_r}{\begin{array}{c}\left({.Math.H_{\mathrm{qr}}.Math.}^2{}_q+{.Math.H_{j_q}.Math.}^2{}_q+N_0\right)\\ \left({.Math.H_{\mathrm{qr}}.Math.}^2{}_q+{.Math.H_{j_q}.Math.}^2{}_q+{.Math.H_{\mathrm{rr}}.Math.}^2{}_r+N_0\right)\end{array}}-\frac{{.Math.G_{j_q}{G}_r.Math.}^2{}_r}{\begin{array}{c}\left({.Math.G_q.Math.}^2{}_q+{.Math.G_{j_q}.Math.}^2{}_q+{.Math.G_{j_c}.Math.}^2{}_r+N_0\right)\\ \left({.Math.G_q.Math.}^2{}_q+{.Math.G_{j_q}.Math.}^2{}_q+{.Math.G_r.Math.}^2{}_r+{.Math.G_{j_r}.Math.}^2+{}_r+N_0\right)\end{array}},rq,\left(r,q\right){\left\{1,2\right\}}^2.& \left(22a\right)\\ \begin{array}{c}{}_q^{*}=[\frac{1}{{.Math.G_{j_{}}.Math.}^2}(\sqrt{{.Math.G_q{G}_{j_q}.Math.}^2\frac{{}_q}{{}_q}+\frac{{.Math.G_q.Math.}^4{}_q^2}{4}}-\\ {\frac{{.Math.G_q.Math.}^2{}_q}{2}-\left({.Math.G_r.Math.}^2{}_r+{.Math.G_{j_r}.Math.}^2{}_r+N_0\right))]}_{{}_q}^{\frac{P_q-p_q}{N_{T_q}-1}}\end{array}& \left(23a\right)\end{array}$
The rationale behind pricing has been discussed in many works (e.g., [6a], [19a], [20a]). In principle, pricing is a mechanism that incentivizes players to spend their jamming power more wisely by charging each user a price per unit of jamming power, thus discouraging users from acting selfishly. In the present invention, a linear pricing will be used to improve the efficiency of jamming control. The optimal jamming power can be found by writing the K.K.T conditions for (21a). Hence, close-form representation of the optimal jamming power for the qth user can be written as (23a),

(71) ${\mathrm{where}\left[\mathrm{.Math.}\right]}_b^a\stackrel{}{=}\min \left\{\max \left\{\mathrm{.Math.},b\right\},a\right\}\mathrm{and}{}_q\stackrel{}{=}\min \left\{\max \left\{\frac{A_q}{B_q},{}_q\right\},P_q^{\mathrm{jam}}\right\}.$
It is easy to verify that by setting .sub.q=0, we end up with the previously mentioned greedy friendly jamming.

(72) Using (23a) iteratively in setting the jamming power for both users leads to a convergence point (i.e., NE) from which neither user is willing to deviate. In what follows, we further explain the feasibility of converging to a NE using pricing. The following theorem clarifies the reason for setting the pricing factor as in (22a).

(73) Theorem 1.

(74) The NE of the game wherein the users use (22a) as the pricing factor to solve the optimization in (21a) equals to that of a locally optimal solution of (19a).

(75) Next, we introduce two properties of pricing mechanism.

(76) Proposition 1.

(77) The greedy friendly jamming can increase both users' secrecy rates (i.e., it is optimal) if .sub.q0, q{1,2}.

(78) Proposition 2.

(79) If .sub.q>0, the NE tuple of jamming powers (.sub.1, .sub.2) will be one of the following forms:

(80) $\begin{array}{cc}\left({}_1,{}_2\right)=\left({}_{\mathrm{int}},{}_2\right)\mathrm{or}\left({}_{\mathrm{int}},P_2^{\mathrm{jam}}\right)\mathrm{or}\left({}_1,{}_{\mathrm{int}}\right)\mathrm{or}\left(P_1^{\mathrm{jam}},{}_{\mathrm{int}}\right)\mathrm{or}\left({}_1,{}_2\right)\mathrm{or}\left(P_1^{\mathrm{jam}},P_2^{\mathrm{jam}}\right),\mathrm{where}{P}_q^{\mathrm{jam}}=\frac{P_q-p_q}{N_{T_q}-1}\mathrm{and}{}_q<{}_{\mathrm{int}}<P_q^{\mathrm{jam}}.& \left(24a\right)\end{array}$

(81) Proof of Theorem 1: For *.sub.q to be a locally optimal solution of (19a), K.K.T. (Karush-Kuhn-Tucker) conditions of both (19a) and (21a) must be equivalent. Hence, the optimal jamming power in (23a) is locally optimal if

(82) 0$\begin{array}{cc}{}_q=-\frac{C_r^{\sec }}{{}_q},& \left(33a\right)\end{array}$
which leads to setting .sub.q as in (22a). The local optimality of NE requires to prove that using (23a) converges to the NE. In proposition 2, convergence to NE is proved. Assuming that iteratively using (23a) reaches to a convergence point (i.e., NE), still the K.K.T. conditions of (21a) equals to that of (19a) which corresponds to a locally optimal solution of (19a).

(83) Proof of Proposition 1: According to (33a), if

(84) ${}_q>0\mathrm{then}\frac{C_r^{\sec }}{{}_q}<0.$
Hence, the positive price is effective as long as the increase in one user's jamming power reduces the secrecy rate in the other. Hence, the positive price .sub.q can make a user reduce its jamming power if this reduction is beneficial for the other user. Now, considering .sub.q0, the increase in one user's jamming power results in either no change (i.e., .sub.q=0) or increase (i.e., .sub.q<0) in the other user's secrecy rate. Therefore, whenever .sub.q0 the right decision would be using maximum jamming power (i.e., setting .sub.q=0) because each user can increase its own secrecy rate without reducing the secrecy sum-rate.

(85) Proof of Proposition 2: Assume that w.l.o.g. .sub.q defined in (23a), has the values

(86) ${}_q<{}_q<\frac{P_q-p_q}{N_{T_q}-1}.$
Furthermore, assume that the iterative use of (23a) is done sequentially, meaning that only one user is updating its jamming power at each iteration. Let the initial jamming power for the qth user be *.sub.q.sup.(1), q, where the superscript .sup.(1) represents the first iteration. In the next iteration, .sub.r gets updated using (23a) and *.sub.q.sup.(1)=*.sub.q.sup.(2). In the third iteration, *.sub.r.sup.(2)=*.sub.r.sup.(3), and .sub.q gets updated. It can be seen from (23) that *.sub.q is a decreasing function of .sub.r. Therefore, if *.sub.q.sup.(1)<*.sub.q.sup.(3) the rth user will select a smaller jamming power in the fourth iteration comparing to the second iteration (i.e. *.sub.r.sup.(2)>*.sub.r.sup.(4)). Consequently, in the fifth iteration, the qth user selects a higher jamming power comparing to the third iteration. This trend continues until either the qth user reaches to P.sub.q.sup.jam or the rth user reaches to .sub.r. Depending on which user reaches to either of the extreme points faster than the other, the first four forms in right hand side of (24a) are expected to be achieved. For the case of (.sub.1,.sub.2) and (P.sub.1.sup.jam, P.sub.2.sup.jam) we first derive the price above which we always have *.sub.q=.sub.q. Let this price be .sub.q.sup.1. Reducing the inequality *.sub.q.sub.q, we end up with an inequality in the form of .sub.q.sub.q.sup.1. Next, we find a price below which we have *.sub.q=P.sub.q.sup.jam. Let this price be .sub.q.sup.2. Reducing the inequality *.sub.qP.sub.q.sup.jam, we end up with an inequality in the form of .sub.q.sup.2.sub.q (note that when 0<.sub.q.sub.q.sup.2, using greedy friendly jamming is optimal in terms of secrecy sum-rate, but it might not always be beneficial for both of the users unless we have .sub.q0. The bound .sub.q0,q found in proposition 1 can also guarantee the optimality of greedy friendly jamming in terms of individual secrecy rates). Since .sub.q is a decreasing function of .sub.q, if P.sub.q.sup.jam>.sub.q then .sub.q.sup.1<.sub.q.sup.2. Therefore, the tuple (.sub.1,.sub.2) or (P.sub.1.sup.jam, P.sub.2.sup.jam) happen when .sub.q>.sub.q.sup.1,q or .sub.q<.sub.q.sup.2,q, respectively. It was assumed w.l.o.g. that

(87) ${}_q<\frac{P_q-p_q}{N_{T_q}-1}$
because if

(88) ${}_q\frac{P_q-p_q}{N_{T_q}-1}$
the forms of NE tuples overlap with each other (note that the effect of .sub.q is negligible on the convergence behavior of (23a) because .sub.q is a sublinear function of jamming powers.

(89) Robust Optimization Procedure

(90) So far, we proved that price-based jamming control results in locally optimum jamming powers. The next challenging question is how to determine the best price without having exact knowledge of the eavesdropping channel.

(91) When exact knowledge of the eavesdropping channel is available, iterative computation of *.sub.q*in (23a) can lead to the NE. Also, the price that results in locally optimal jamming powers of the secrecy sum-rate can be computed using (22a). However, the unknown eavesdropping channel imposes uncertainties in computing *.sub.q and .sub.q. In the following, we propose a method to overcome the issue of uncertainty in the eavesdropping channel.

(92) Let R.sub.q(s.sub.1s.sub.2) be the utility of the qth user where s.sub.1 and s.sub.2 represents the strategy taken by user1 and user2, respectively. The strategy space of each user is a continuous interval that can be written as .sub.q [.sub.q,P.sub.q.sup.jam]. The strategy set of the players has infinite real numbers. If we are to analyze this game using strategy tables, then the strategy set of each player should be countable and finite. In order to create a finite strategy set, we introduce discrete levels of the jamming power. Assuming that we have n bits to convey M power levels, the power level increment is

(93) ${}_q=\frac{P_q^{\mathrm{jam}}}{M}=\frac{P_q^{\mathrm{jam}}}{2^n}.$
The discrete power level is small enough that it constrains the continuous set of utility values to a relatively small discrete set with minimal quantization error. Hence, the strategy set would be S.sub.q={.sub.q,2.sub.q, . . . , (M1).sub.q, P.sup.jam}, q. Considering that s.sub.q S.sub.q, q, a utility matrix R.sub.q, q=1,2 can be obtained such that its (i,j) entry is [R.sub.q].sub.ij={R.sub.q(i.sub.q,j.sub.q)|(i,j){1, . . . , M}.sup.2}.

(94) Since the problem in (13a) is non-convex w.r.t the jamming power, Pareto-optimal points can be found via exhaustive search in Table I. Considering a finite jamming game, the complexity of this optimization is in the order of (n.sup.2). Proposition 2 reduces the complexity order to (4n4) because only a small set of jamming power tuples comprise the NE points of price-based friendly jamming, meaning that the locally optimal points of the secrecy sum-rate can be found by searching a small portion of Table I. To get more intuition into the order reduction, we mention a special case of proposition 2. If (18a) is always satisfied for both users, then only the rows corresponding to .sub.1 and P.sub.1.sup.jam, and the columns corresponding to .sub.2 and P.sub.2.sup.jam need to be searched. In the following, we introduce the concept of mixed strategic games.

(95) Table I: Strategy table for the two user finite jamming game with pricing.

(96) TABLE-US-00001 s.sub.1\s.sub.2 .sub.2 2.sub.2 . . . P.sub.2.sup.jam .sub.1 R.sub.1(.sub.1, .sub.2), R.sub.1(.sub.1, 2.sub.2), . . . R.sub.1(.sub.1, P.sub.2.sup.jam), R.sub.2(.sub.1, .sub.2) R.sub.2(.sub.1, 2.sub.2) R.sub.2(.sub.1, P.sub.2.sup.jam) 2.sub.1 R.sub.1(2.sub.1, .sub.2), R.sub.1(2.sub.1, 2.sub.2), . . . R.sub.1(2.sub.1, P.sub.2.sup.jam), R.sub.2(2.sub.1, .sub.2) R.sub.2(2.sub.1, 2.sub.2) R.sub.2(2.sub.1, P.sub.2.sup.jam) . . . . . . . . . . . . . . . P.sub.1.sup.jam R.sub.1(P.sub.1.sup.jam, .sub.2), R.sub.1(P.sub.1.sup.jam, 2.sub.2), . . . R.sub.1(P.sub.1.sup.jam, P.sub.2.sup.jam), R.sub.2(P.sub.1.sup.jam, .sub.2) R.sub.2(P.sub.1.sup.jam, 2.sub.2) R.sub.2(P.sub.1.sup.jam, P.sub.2.sup.jam)

(97) Definition 1.

(98) A mixed strategy vector for the qth user .sub.q={[.sub.i.sup.q].sub.i=1.sup.M|0.sub.i.sup.q1, iiq=1, q denotes a probability distribution of the strategy set's elements. That is to say the qth player chooses the power level i.sub.q with probability .sub.i.sup.q.

(99) In the mixed strategic jamming game, both users choose their jamming power level based on a probability distribution. Hence, the best response of each user is to maximize the expected value of its own utility. We should note that some games can be limited to only pure strategies. In particular, if the utility function of a player is concave w.r.t. its strategy then using Jensen's inequality, we deduce that
E.sub.s1[E.sub.s2[R.sub.1(s.sub.1,s.sub.2)]]E.sub.s2[R.sub.1(E.sub.s.sub.1[s.sub.1],s.sub.2)],
(s.sub.1,s.sub.2)S.sub.1S.sub.2,(25a)
The inequality in (25a) is satisfied with equality if and only if s.sub.1 reduces to pure strategies. Hence, whatever the strategies of other players are, every NE of the game is achieved using pure strategies [18a]. Sufficiency of pure strategies cannot be guaranteed if the utility function of a player is not concave w.r.t. its action. Hence, mixed strategies should also be investigated for non-concave utilities.

(100) In price-based friendly jamming, the utility function to each player changes at every iteration. Furthermore, the strategy table looks at the outcome of the game when iterative usage of (23a) has converged. Hence, it is not possible to use the objective function in (21a) as the utility functions in the strategy table. In order to establish the strategy table, we look at (19a) again. As theorem 1 suggests that the K.K.T. conditions of (19a) are met at the NE point, the utility of each user at the NE point is R.sub.q(s.sub.1,s.sub.2)=C.sup.sec(.sub.q),q{1,2} which is a non-concave function w.r.t. .sub.q. By setting C.sup.sec as a function of .sub.q, we want to emphasize that each user locally computes its own jamming power and checks its effect on secrecy sum-rate. Recalling proposition 2, at a locally optimal point, only one tuple of jamming power at the corner entries of Table I happen. Hence, the objective of 1st user is,

(101) $\begin{array}{cc}\underset{{}^1}{\mathrm{maximize}}\underset{i=1}{\overset{M}{.Math.}}{}_i^1{R}_1\left(i{}_1,s_2\right),s.t.\underset{i=1}{\overset{M}{.Math.}}{}_i^1=1,0<{}_i^1<1,i,s_2\left\{.Math.\frac{{}_2}{M}.Math.{}_2,P_2^{\mathrm{jam}}\right\}& \left(26a\right)\end{array}$
where .sup.q=[.sub.1.sup.1, . . . , .sub.M.sup.1].sup.T is the probability set, and [ . . . ] rounds to positive infinity. Problem (26a) is a linear programming which can be solved efficiently using Simplex method. The 2nd user's strategy can be found accordingly.

(102) So far, all of the derivations were based on complete knowledge of the eavesdropping channel. However, the existence of a passive Eve in the network contradicts with our assumption. For the qth user, the computation of the secrecy rate defined in (8a) depends on knowing C.sub.q and C.sub.eq. Since we assumed that Bob can measure his received interference and Alice is aware of the channel between her and her corresponding Bob, the computation of C.sub.q can be done locally. However, the eavesdropping channel is unknown. The eavesdropping channel components can be equivalently shown as the product of some large-scale and small-scale fading, so G.sub.q=G.sub.q(d.sub.q.sup.) and G.sub.j.sub.q=G.sub.j.sub.q (d.sub.j.sub.q.sup.), where G.sub.q and G.sub.j.sub.q are scalar and 1(N.sub.T.sub.q1) matrix, respectively, and the entries of both are i.i.d. standard complex Gaussian random variables with unit variance (The same representation can be done for H.sub.qr and H.sub.j.sub.q). The terms d.sub.q.sup. and d.sub.j.sub.q.sup. are the equivalent distances where d.sub.q.sup. is a scalar and d.sub.j.sub.q.sup. is a diagonal (N.sub.T.sub.q1)(N.sub.T.sub.q1) matrix, and is the equivalent path-loss exponent. The secrecy rate in such settings is,

(103) $\begin{array}{cc}{C}_q^{\sec }=C_q-\underset{\left[\begin{array}{c}{d}_q{\stackrel{\_}{G}}_q\\ d_r{\stackrel{\_}{G}}_r\\ {\stackrel{\_}{G}}_{j_r}\end{array}\right]}{E}\left[C_{\mathrm{eq}}\right]=C_q-& \left(27a\right)\\ E\left[\log \left(1+\frac{{.Math.G_q.Math.}^2{}_q}{{.Math.G_{j_q}.Math.}^2{}_q+{.Math.G_r.Math.}^2{}_r+{.Math.G_{j_r}.Math.}^2{}_r+N_0}\right)\right].& \left(28a\right)\end{array}$
We derive (28a) as

(104) $\begin{array}{cc}\underset{\left[\begin{array}{c}{d}_q{\stackrel{\_}{G}}_q\\ d_r{\stackrel{\_}{G}}_r\\ {\stackrel{\_}{G}}_{j_r}\end{array}\right]}{E}\left[C_{\mathrm{eq}}\right]=\underset{\left[\begin{array}{c}{d}_q{W}_q\\ d_r{Y}_q\end{array}\right]}{E}\left[\log .Math.\frac{W_q{}_{1q}{W}_q^H}{Y_q{}_{2q}{Y}_q^H}.Math.\right]& \left(29a\right)\end{array}$
where W.sub.q=[G.sub.q, G.sub.j.sub.q, G.sub.r, G.sub.j.sub.r, e.sub.q], and Y.sub.q=[G.sub.j.sub.q, G.sub.r, G.sub.j.sub.r, e.sub.q]. Let the expression diag{E,F} represent a 22 diagonal matrix whose diagonal entries are E and F. Then,

(105) $\begin{array}{cc}{}_{1q}=\mathrm{diag}\left\{{}_q,{}_q\underset{\underset{N_{T_q}-1}{}}{\left[1,\mathrm{.Math.},1\right]}{\left(\frac{d_{j_q}}{d_q}\right)}^{-2},{\left(\frac{d_r}{d_q}\right)}^{-2}{}_r,{}_r\underset{\underset{N_{T_r}-1}{}}{\left[1,\mathrm{.Math.},1\right]}{\left(\frac{d_{j_r}}{d_q}\right)}^{-2},{\left(d_q\right)}^2{N}_0\right\},& \left(30a\right)\\ {}_{2q}=\mathrm{diag}\left\{{}_q\underset{\underset{N_{T_q}-1}{}}{\left[1,\mathrm{.Math.},1\right]}{\left(\frac{d_{j_q}}{d_r}\right)}^{\left(-2\right)},{}_r,{}_r\underset{\underset{N_{T_r}-1}{}}{\left[1,\mathrm{.Math.},1\right]}{\left(\frac{d_{j_r}}{d_r}\right)}^{-2},{\left(d_r\right)}^2{N}_0\right\}.& \left(31a\right)\end{array}$

(106) The expectation in (29a) w.r.t. Wq and Yq can be efficiently computed using the random matrix result in [21a, Appendix A, Lemma 2]. However, C.sub.eq is still an expectation over the distances d.sub.q and d.sub.r, which corresponds to geometry distribution of Eve. Since we were not able to analytically formulate this distribution, we numerically sample these parameters to approximate the expectation of C.sub.eq w.r.t. the distances. In simulations, we assumed that Eve is uniformly distributed within a circle with a given radius, and the center of this circle is determined depending on our simulation scenario. Similar idea can be found in [17a] where the authors assume that Eve is uniformly distributed around the transmitter. Another example is [22a] where the authors assumed that the location of Eve follows a Poisson process. With the same technique used in (29a), the expectation on the right hand side of equation (14a) that yields positive secrecy is as follows:

(107) 0$\begin{array}{cc}{}_q>\frac{\left({.Math.H_{\mathrm{rq}}.Math.}^2{}_r+{.Math.H_{j_r}.Math.}^2{}_r+N_0\right)}{{.Math.H_{\mathrm{qq}}.Math.}^2}\underset{\left[\begin{array}{c}{G}_q\\ G_{j_q}\end{array}\right]}{E}\left[\frac{{.Math.G_q.Math.}^2}{{.Math.G_{j_q}.Math.}^2}\right]-\underset{\left[\begin{array}{c}{G}_r{d}_q{d}_r\\ G_{j_r}{G}_{j_q}\end{array}\right]}{E}\left[\frac{\left({.Math.G_r.Math.}^2{}_r+{.Math.G_{j_r}.Math.}^2{}_r+N_0\right)}{{.Math.G_{j_q}.Math.}^2}\right].& \left(32a\right)\end{array}$

(108) The numerator and the denominator inside the first expectation term in (32a) correspond to a central Wishart matrix. The numerator inside the second expectation term corresponds to the quadratic form of a Wishart matrix, which preserves Wishartness property [23a]. Hence, both of the expectations correspond to the ratio of two Wishart matrices. Since we assumed a MISO system, all of the Wishart matrices are in fact scalars. Hence, the expectations in (32) can be computed using the result in [24a, section 1]. Computing the expectation w.r.t. d.sub.q and d.sub.r can be tackled the same as what we will do for computation of C.sub.eq (i.e., equation (29a)) in simulations. Since (27a) and (32a) are computable, then the objective function and third constraint of (26a) are defined without the knowledge of the eavesdropping channel. Hence, we can establish Table I to solve (26a). The following describes an algorithm that achieves a robust solution for (26a).

(109) Algorithm Design

(110) The pseudo code of our algorithm that achieves a robust solution for the jamming control is shown below. In order to approximate the expectation of C.sub.eq w.r.t. the distances, the location of Eve will be assigned as a uniform distribution within a circle with radius {circumflex over (r)}.sub.e and the center coordinates of ({circumflex over (x)}.sub.e, .sub.e). Computation of line 5 and line 6 can be done using the method mentioned in (32a) (only for when .sub.r=.sub.r which requires both users to measure the interference at their receivers and exchange the values of .sub.q and .sub.q for q=1, 2. Note that the result of proposition 2 suggests that the computation of line 5 only sets two power levels for .sub.r (i.e., the loop in line 3 will be run once when .sub.r=.sub.r and once when .sub.r=P.sub.r.sup.jam) For the case of exhaustive search, instead of two power levels, we should search in all of the power levels in the interval [.sub.r, P.sub.r.sup.jam].

(111) Line 7 ensures that the selected power in line 4 results in a non-zero utility for the qth user. If the condition on line 7 is not satisfied the probability assigned to that power level (i.e., .sub.i.sup.q) is zero. Hence, one term will be removed from the objective and constraints of (26a). The operation in line 8 (computed in (27a)) firstly requires both users to compute their own secrecy rate using their local channel and the method mentioned for (27a). Then, the rth user should send the value of its own secrecy rate to the qth user in order to compute R.sub.q. After doing the operations of lines 2-12, two probability sets (i.e., two different vectors corresponding to .sup.q) will be found for the qth user (one for when .sub.r=.sub.r and one for when .sub.r=P.sub.r.sup.jam). As R.sub.q is already stored in line 8, line 14 chooses the probability set corresponding to the largest expected utility. Assuming that Alice is capable of generating random samples of a continuous-time uniform distribution, creating a probabilistic jamming power assignment is done by converting the uniform distribution to a probability mass function corresponding to .sup.q for q=1; 2 [25a]. Comparing to the pricing with complete knowledge of the eavesdropping channel, the robust jamming control only needs exchanging secrecy rates and jamming powers.

(112) The lines 16 to 25 constitute the outer loop of the algorithm that corresponds to satisfying rate constraints of the users. For some choice of and , since the information rate is a nonnegative monotonically increasing function of .sub.q and the fact that p.sub.q is a finite value, as long as the rate requirements are feasible, the linear (margin-adaptive) adjustment used in algorithm 1 converges without the need for central control (similar procedure can be found in [26a, Algorithm 1]). Hence, this linear adjustment ensures each user achieves its minimum target rate. If the target rate is not achievable, then the line 18 limits the users to their maximum information signal's power.

(113) Algorithm 1. Robust Jamming Control

(114) TABLE-US-00002 $\mathrm{Input}\text{:}\{\begin{array}{c}{N}_{T_q},P_q,{}_q=\frac{p_q}{{}_q\left(N_{T_q}-1\right)},c_q,Mq\left\{1,2\right\}\\ {\hat{r}}_e\%\mathrm{The}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{circle}\mathrm{within}\mathrm{which}\\ \begin{array}{c}\%\mathrm{Eve}\mathrm{is}\mathrm{uniformly}\mathrm{distributed}\\ \left({\hat{x}}_e,{\hat{y}}_e\right)\%\mathrm{The}\mathrm{center}\mathrm{of}\mathrm{the}\mathrm{circle}\mathrm{within}\mathrm{which}\end{array}\\ \%\mathrm{Eve}\mathrm{is}\mathrm{uniformly}\mathrm{distributed}.\end{array}\mathrm{Initialize}\text{:}0<{}_q<p_q,{}_q=\frac{p_q^{\mathrm{jam}}}{M}q$ 1: repeat 2: for q = 1 to 2 do 3: for i = 1 to M do 4: Set .sub.q = i.sub.q. 5: Compute .sub.r = .sub.r, r q. 6: Compute .sub.q. 7: if .sub.q < .sub.q then Set .sub.q.sup.i = 0. 8: else Compute and store R.sub.q. 9: end if 10: end for % do the same loop again by change 11: % line 5 to Set .sub.r = P.sub.r.sup.jam. 12: Find .sub.q by solving (26), % once for .sub.r = .sub.r and 13: % once for .sub.r = P.sub.r.sup.jam. 14: Choose the probability set (i.e., .sub.q) that corresponds to the largest expected utility. 15: end for 16: for q = 1 to 2 do 17: if C.sub.q < c.sub.q then Set .sub.q = .sub.q + . 18: if .sub.q > p.sub.q then Set .sub.q = p.sub.q. 19: end if 20: else 21: if .sub.q > c.sub.q + then Set .sub.q = .sub.q . 22: end if 23: end if 24: end for 25: until C.sub.q > c.sub.q .sub.q.

(115) Numerical Results

(116) In this part, we simulate the methods mentioned so far. In all of the simulations, the variance of additive noise at both receivers and at Eve (i.e., noise floor) is set to be N.sub.0=50 dBm. Also, the information rate constraints are chosen such that the users use the maximum information signal's power. The horizontal axis in all of the figures is the horizontal coordinate for the center of the circle within which Eve is uniformly distributed. Each point on the curves is the result of averaging over 10 random locations of Eve (in order to approximate (27a) w.r.t. distances). At each random location of Eve, 500 channel realizations are simulated and then averaged.

(117) FIG. 3 shows the variation of secrecy sum-rate in terms of Eve's location for a given total power constraint. It can be seen that greedy friendly jamming is outperforming no-jamming for all of Eve's locations. This due to the fact that in no-jamming scenario, Eve can decode both of the signals using SIC technique, which reduces the secrecy rates. Furthermore, when x.sub.e [8, 2], the performance of price-based friendly jamming is equal to the performance of greedy friendly jamming and that of exhaustive search, which indicates the optimality of greedy friendly jamming in these areas. It can also be seen that in all of Eve's locations, the pricing scheme has a performance close to the exhaustive search approach. As far as what we simulated, the optimality of greedy friendly jamming can be guaranteed only for very low power constraints that are not practical.

(118) FIG. 4 and FIG. 5 show the secrecy rates of the users for another configuration. In this configuration, we simulated the pricing method for when we take the constraint (14a) into account (indicated as Pricing (Full CSI)) and for when we do not (indicated as Pricing (No Positive Secrecy)). It can be seen that considering (14a) in our jamming control significantly affects the secrecy sum-rate such that if it is overlooked, the performance of pricing method would be even lower than greedy approach in some points. Furthermore, satisfying the condition in (14a) guarantees non-zero secrecy rate for each user. However, if it is ignored, zero secrecy-rate happens to one or both users at some locations.

(119) In FIG. 6, we compared the performance of Algorithm 1 (indicated as Robust) to other approaches. The geometry distribution for Eve is the same as previous simulations, but P.sub.q=10 dBm. For the case of robust jamming control, we assumed that we use 8 bits for quantization of power levels. After finding the probability set that maximizes the expected utility in (26a), the probabilistic assignment of jamming power in robust jamming control is as follows. The qth user generates a sample of a continuous-time uniformly distributed random variable with zero mean and unit variance. These samples are then converted to some probability mass .sub.i.sup.q. Then, the probability mass is translated to i.sub.q and the user starts data transmission with that jamming power. Lastly, the achievable rate is computed using the method in (27a). This procedure is repeated 50 times per each channel realization and the expected utility in (26a) is approximated by averaging over these 50 samples of data transmission. It can be seen that the robust approach is 25 percent better than the greedy approach. When the eavesdropping channel is known, the advantage of price-based jamming becomes larger.

(120) The expected value in (26a) must be computed after averaging over several samples of data transmissions within one channel realization. However, in practical scenarios, the coherence time is not long enough to accommodate more than a very small number of data transmissions. In order to test this vulnerability, we compared the performance of robust optimization between 50 data transmissions and 1 data transmission per each channel realization so as to compute the expected utility in (26a). Furthermore, in order to fix the other parameters that might affect this comparison, we simulated 50 channel realizations at each location of Eve. It can be seen in FIG. 7 that averaging over 1 data transmission (indicated as Robust(1)) does not affect the secrecy sum-rate very much comparing to averaging over 50 data transmissions (indicated as Robust(50)). Therefore, the robust jamming control can also be implemented in channels with low coherence times.

(121) In this embodiment, the present invention showed that the greedy friendly jamming is not an optimal approach to realize a secure network. This embodiment of the present invention features a price-based jamming control that guarantees a local optimum for secrecy sum-rate. Simulations demonstrated a noticeable improvement in secrecy sum-rate by using pricing in jamming control. Uncertainty was then introduced in the eavesdropping channel and a robust method was designed. The robust jamming control can be used when the eavesdropper cannot be monitored in the network.

Example 2

(122) The following is a non-limiting example of a scenario where a transmitter sends two independent confidential data streams, intended to two legitimate users, in the presence of an eavesdropper of an unknown location.

(123) System Model

(124) As shown in FIG. 8, we consider a two-user broadcast channel in which Alice transmits two independent confidential data streams to Bob and Charlie in the presence of Eve. Let the number of antennas at Alice and Eve be N.sub.A and N.sub.E, respectively. The intended receivers, Bob and Charlie, have FD radios, each with a single antenna [16b]. Let x.sub.A .sup.N.sup.A.sup.1 be Alice's transmit signal, which includes two information messages plus TxFJ. Let x.sub.B and x.sub.C denote the transmit RxFJ signals from Bob and Charlie, respectively. The signals received by Bob, Charlie, and Eve are, respectively, given by:
y.sub.B=h.sub.ABx.sub.A+h.sub.BBx.sub.B+h.sub.CBx.sub.C+n.sub.B(1b)
y.sub.C=h.sub.ACx.sub.A+h.sub.BCx.sub.B+h.sub.CCx.sub.C+n.sub.C(2b)
y.sub.E=H.sub.AEx.sub.A+h.sub.BEx.sub.B+h.sub.CEx.sub.C+n.sub.E(3b)
where h.sub.AB.sup.1N.sup.A, h.sub.AC.sup.1N.sup.A, h.sub.BE.sup.N.sup.E.sup.1, and h.sub.CE.sup.N.sup.E.sup.1 are the channel vectors between Alice and Bob, Alice and Charlie, Bob and Eve, and Charlie and Eve, respectively. h.sub.BB and h.sub.CC are the self-interference channel gains, whereas h.sub.CB and h.sub.BC are the channel gains between Charlie and Bob, and between Bob and Charlie, respectively. H.sub.AE .sup.N.sup.E.sup.N.sup.A is the channel matrix between Alice and Eve. n.sub.B(0, .sub.B.sup.2), n.sub.C(0, .sub.C.sup.2), and n.sub.E(0, I.sub.N.sub.E.sub.E.sup.2) represent AWGN at Bob, Charlie and Eve, respectively. We assume block fading (the indices representing fading blocks and time instants are suppressed to improve readability). Furthermore, we assume that Eve's CSI is not known to Alice, Bob, or Charlie. However, Eve may know her own channels and other channels by overhearing exchanged control packets between Alice and Bob/Charlie. We impose the following instantaneous power constraints:
[x.sub.A.sup.x.sub.A]P.sub.A(4b)
[|x.sub.i|.sup.2]P.sub.i, i{B,C}(5b)
where P.sub.A, P.sub.B, and P.sub.C are given constant's.

(125) Rx-Based FJ with Zero-Forcing

(126) A. Communication Scheme

(127) The transmit signal at Alice can be expressed as:
x.sub.A=v.sub.Bs.sub.B+v.sub.Cs.sub.C+z.sub.Aw.sub.A(6b)
where s.sub.B(0,.sub.S.sub.B.sup.2) and s.sub.C(0,.sub.S.sub.C.sup.2) are the information signals, v.sub.B.sup.N.sup.A.sup.1 and v.sub.C .sup.N.sup.A.sup.1 are normalized precoding vectors for Bob and Charlie, respectively, such that v.sub.B.sup.v.sub.B=1 and v.sub.C.sup.v.sub.C=1, w.sub.A(0,.sub.J.sub.A.sup.2) is the TxFJ signal, and z.sub.A .sup.N.sup.A.sup.1 is its precoding vector. We let z.sub.A.sup.z.sub.A=1. (Note that we assume N.sub.A=3. However, this scheme can be easily extended to a scenario where N.sub.A>3. In that case, N.sub.A2 independent TxFJ signals, w.sub.A.sup.(1), . . . , w.sub.A.sup.(N.sup.A.sup.2), are generated at Alice, and z.sub.Aw.sub.A is replaced by .sub.(k=1).sup.(N.sup.A.sup.2)z.sub.A.sup.(k)w.sub.A.sup.(k).) Moreover, the RxFJ signals transmitted by Bob and Charlie are given by x.sub.i=w.sub.i, i{B, C}, where w.sub.i(0,.sub.J.sub.i.sup.2). Given the above, the received signals at Eve, Bob, and Charlie reduce to:
y.sub.E=H.sub.AEv.sub.Bs.sub.B+H.sub.AEv.sub.Cs.sub.C+H.sub.AEz.sub.Aw.sub.A+h.sub.BEw.sub.B+h.sub.CEw.sub.C+n.sub.E(7b)
y.sub.i=h.sub.Aiv.sub.is.sub.i+h.sub.Aiv.sub.js.sub.j+h.sub.Aiz.sub.Aw.sub.A+h.sub.iiw.sub.i+h.sub.jiw.sub.j+n.sub.i(8b)
where in (8b) {i, j}{B, C} and ij.

(128) To increase the communication rates, we consider zero-forcing precoding for the information signal intended to Bob such that it is cancelled out at Charlie, and vice versa. (This technique also provides confidentiality for Bob's message at Charlie and vice versa.) Accordingly, we consider the following zero-forcing constraints.
h.sub.ABv.sub.C=0(9b)
h.sub.ACv.sub.B=0.(10b)
We note that h.sub.AB and h.sub.AC should be linearly independent (otherwise, the cancellation will occur at the intended receivers as well), and the independence occurs with probability 1 due to fading. Constraint in (9b) (also, (10b)) reduces the degrees of freedom for the selection of the precoder v.sub.C (V.sub.B) by one, leaving N.sub.A1 degrees of freedom. As will be described herein, we discuss how to uniquely determine the optimal v.sub.C (V.sub.B) that maximizes the information rate at Charlie (Bob).

(129) The TxFJ signal coming from Alice to Bob is designed to be orthogonal to the channel between them in order to improve SINR level at Bob. A similar constraint is also imposed on the TxFJ signal observed by Charlie. In other words, we require
h.sub.ABz.sub.A=0(11b)
h.sub.ACz.sub.A=0.(12b)
It follows that z.sub.A[span(h.sub.AB, h.sub.AC)].sup..

(130) We consider full-duplex radio design as introduced in [16b] to eliminate the self-interference arising from the transmission of RxFJ signal W.sub.B at Bob (w.sub.C at Charlie, respectively). In particular, we incorporate into the model a residual self-interference term using SIS ratio, defined as the portion of self-interference left after suppression. This residual term is denoted with the scale factor [0,1]. Accordingly, (8b) becomes:
y.sub.i=h.sub.Aiv.sub.is.sub.i+h.sub.iiw.sub.i+h.sub.jiw.sub.j+n.sub.i.(13b)
With this communication scheme, by controlling the RxFJ powers at Bob and Charlie, we can manage the interference they impose on each other.

(131) B. Achievable Secrecy Sum-Rate

(132) Let (X;Y) refer to the mutual information between any two signals X and Y. We let s.sub.B S.sub.B, S.sub.C S.sub.C, y.sub.B Y.sub.B, y.sub.C Y.sub.C, and y.sub.E Y.sub.E. Given the communication scheme described in the previous section, the Alice.fwdarw.Bob and Alice.fwdarw.Charlie links can support the following instantaneous mutual information expressions:

(133) $\begin{array}{cc}{R}_B\stackrel{\mathrm{def}}{=}\left(S_B;Y_B\right)=\log \left(1+{\mathrm{SINR}}_B\right)& \left(14b\right)\\ R_C\stackrel{\mathrm{def}}{=}\left(S_C;Y_C\right)=\log \left(1+{\mathrm{SINR}}_C\right)& \left(15b\right)\end{array}$
where i{B, C},

(134) ${\mathrm{SINR}}_i=\frac{{}_{S_i}^2{.Math.h_{\mathrm{Ai}}{v}_i.Math.}^2}{{.Math.h_{\mathrm{ii}}.Math.}^2{}_{J_i}^2+{.Math.h_{\mathrm{ji}}.Math.}^2{}_{J_i}^2+{}_i^2}.$

(135) Remark 1:

(136) Later on, we incorporate the constraint SINR.sub.iT, where T is a required minimum SINR level at Bob/Charlie. In that case, we assume R.sub.i=log(1+T), if SINR.sub.iT, and zero otherwise, for i{B, C}.

(137) The received signal at Eve becomes:
y.sub.E=H.sub.AEv.sub.Bs.sub.B+H.sub.AEv.sub.Cs.sub.C+n(16b)
where

(138) $n\stackrel{\mathrm{def}}{=}{H}_{\mathrm{AE}}{z}_A{w}_A+h_{\mathrm{BE}}{w}_B+h_{\mathrm{CE}}{w}_C+n_E.$
Regarding this signal, we utilize the following mutual information expressions:

(139) $\begin{array}{cc}\begin{array}{c}{R}_{E,B}\stackrel{\mathrm{def}}{=}\left(S_B;Y_E\right)\\ =\log (1+{}_{S_B}^2{h_{{\mathrm{AE}}_B}^{}\left({}_{S_C}^2{h}_{{\mathrm{AE}}_c}{h}_{{\mathrm{AE}}_C}^{}+K\right)}^{}{h}_{{\mathrm{AE}}_C}+\\ {K)}^{-1}{h}_{{\mathrm{AE}}_B})\end{array}& \left(17b\right)\\ \begin{array}{c}{R}_{E,C}\stackrel{\mathrm{def}}{=}\left(S_C;Y_E|S_B\right)\\ =\log \left(1+{}_{S_C}^2{h}_{{\mathrm{AE}}_C}^{}{K}^{-1}{h}_{{\mathrm{AE}}_C}\right)\end{array}& \left(18b\right)\end{array}$
Where

(140) $h_{{\mathrm{AE}}_B}\stackrel{\mathrm{def}}{=}{H}_{\mathrm{AE}}{v}_B,h_{{\mathrm{AE}}_C}\stackrel{\mathrm{def}}{=}{H}_{\mathrm{AE}}{v}_C\mathrm{and}K\stackrel{\mathrm{def}}{=}{}_{J_A}^2{H}_{\mathrm{AE}}{z}_A{z}_A^{}{H}_{\mathrm{AE}}^{}+{}_{J_B}^2{h}_{\mathrm{BE}}{h}_{\mathrm{BE}}^{}+{}_{J_C}^2{h}_{\mathrm{CE}}{h}_{\mathrm{CE}}^{}+{}_E^2{I}_{N_E},\left\{i,j\right\}\left\{B,C\right\}\mathrm{and}ij.$
These expressions correspond to employing an MMSE-SIC decoder at Eve (a sum-rate optimal receiver strategy), and are utilized in the proof of secrecy. In particular, secrecy precoding for the signals intended to Bob and Charlie are designed according to the leakage seen by the eavesdropper over the fading channels, i.e., the required amount of randomization. The following theorem provides the resulting sum-rate.

(141) Theorem 2:

(142) An achievable secrecy sum-rate is given by
R.sub.sum=[[R.sub.BR.sub.E,B].sup.++[R.sub.CR.sub.E,C].sup.+](19b)
where the expectation is defined over different fading blocks.

(143) We note the followings:

(144) In the proposed coding scheme, the achievable secrecy rates at Bob and Charlie are given by

(145) $R_B^{\left(s\right)}\stackrel{\mathrm{def}}{=}\left[{\left[R_B-R_{E,B}\right]}^{+}\right]\mathrm{and}{R}_C^{\left(s\right)}\stackrel{\mathrm{def}}{=}\left[{\left[R_C-R_{E,C}\right]}^{+}\right]$
respectively. As the sum-rate is considered, the asymmetric nature of secrecy rate penalties ([R.sub.B]R.sub.B.sup.(s) vs. [R.sub.C]R.sub.C.sup.(s)) is immaterial. If equal rates are desired, for instance due to fairness constraints, a time-division scheme can be implemented. That is, for half of the fading blocks, the scheme above can be used; for the remaining blocks, one can use the same scheme but with R.sub.E,B=I(S.sub.B;Y.sub.E|S.sub.C) and R.sub.E,C=I(S.sub.C;Y.sub.E). This way, each user would achieve a secrecy rate given by R.sub.sum/2.

(146) Information leakage is bounded by the designed rate of the signal. For instance, for fading blocks where Eve's information rate on Bob's signal (R.sub.E,B) is higher than the signal rate R.sub.B, there is no extra leakage (additional rate penalty) on the rate of the intended message. On the other hand, for the remaining fading blocks, where R.sub.B>R.sub.E,B, the coding scheme capitalizes upon the inferior signal quality at the eavesdropper. This mechanism occurs on the fly, i.e., without knowledge of Eve's CSI at Alice.

(147) If the network includes multiple eavesdroppers (say, with different channel fading distributions), the achieved sum rate can be modified as:

(148) $R_{\mathrm{sum}}=\underset{E\mathrm{.Math.}}{\min }\left\{\left[{\left[R_B-R_{E,B}\right]}^{+}+{\left[R_C-R_{E,C}\right]}^{+}\right]\right\}$
where denotes the set of eavesdroppers. Later on, we consider different channel models (protocol vs. path loss models), and we report the worst-case achieved secrecy rates (best possible eavesdropping).

(149) C. Optimization Formulation

(150) Given the achievable secrecy rate defined in Theorem 2, our objective in this section is to maximize this rate by optimizing the power allocation to data and jamming signals, and designing the best possible beamforming vectors. The corresponding optimization formulation is given by:

(151) $\begin{array}{cc}\underset{\left\{{}_{S_B}^2+{}_{S_C}^2+{}_{J_A}^2{P}_A,{}_{J_B}^2{P}_B,{}_{J_C}^2{P}_C\right\}}{\mathrm{maximize}}{R}_{\mathrm{sum}}& \left(20b\right)\end{array}$
and subject to constraints (9b), (10b), (11b), (12b), and v.sub.B.sup.v.sub.B=v.sub.C.sup.v.sub.C=z.sub.A.sup.z.sub.A=1.

(152) At this point, we consider a practical assumption, which enables us to solve the above optimization problem. We assume that the SINR at Bob/Charlie must be greater than or equal to T, otherwise R.sub.B and R.sub.C will be equal to zero. As a result, we should allocate just enough power for the information signals to satisfy this SINR threshold. The rest of the power budget at Alice can be used for TxFJ in order to decrease the SINR level at Eve as much as possible. Therefore, we set SINR.sub.i=T for i{B, C}.

(153) We note that there is more than one possible linear precoding vectors that satisfy the constraint h.sub.Aiv.sub.j=0, for {i, j}{B,C} and ij. In this case, v.sub.j should be chosen such that |h.sub.Ajv.sub.j| takes its maximum value. With this maximization (and the norm constraints on the beamforming vectors), we can write .sub.J.sub.B.sup.2 and .sub.J.sub.C.sup.2 as functions of .sub.S.sub.B.sup.2 and .sub.S.sub.C.sup.2 for a given T and . Then, .sub.J.sub.A.sup.2 is given by:
.sub.J.sub.A.sup.2=P.sub.A.sub.S.sub.B.sup.2.sub.S.sub.C.sup.2.(21b)

(154) Now, the question is which beamforming vector for TxFJ should be chosen. If Alice has 3 antennas (N.sub.A=3), then there will be only one possible dimension such that z.sub.A [span(h.sub.AB, h.sub.AC)].sup.. If N.sub.A>3, we end up with a multidimensional solution space for z.sub.A. In this case, .sub.J.sub.A.sup.2 can be evenly distributed among randomly chosen normalized vectors that are orthogonal to each other and that fully represent this space. This way, we increase the effective region of TxFJ. Note that in this case, N.sub.A2 independent TxFJ signals are generated at Alice, i.e., one TxFJ signal for each orthogonal normalized vector.

(155) Given the above setup, we can derive the optimal R.sub.sum in terms of .sub.S.sub.B.sup.2 and .sub.S.sub.C.sup.2 for a given T and .

(156) Simulation Results

(157) To demonstrate the efficacy of our design, we provide simulation results using N.sub.A=3 and N.sub.E=4. The carrier frequency is set 2.4 GHz. Alice, Bob, and Charlie are located at points (3,5), (7,7), and (7,3), respectively, in a 10 meter10 meter area. The transmit power budgets at Alice, Bob, and Charlie, normalized to the noise power, are taken as P.sub.A=100 dB, P.sub.B=10 dB, and P.sub.C=10 dB, respectively. Unless stated otherwise, we set SINR.sub.B=SIN.sub.C=5 dB and =0.1 (partial SIS). Two interference models are considered: protocol model and SINR model.

(158) A. Protocol Model

(159) In the protocol model, Eve's location is not known; however, if she is located inside a vulnerability zone of the legitimate receivers, she is immune to TxFJ. Since Alice employs zero forcing to cancel the TxFJ signal at both Bob and Charlie, this signal will be weak in that area. The authors in [17b] showed that a guard (vulnerability) zone of 19 wavelengths around a legitimate receiver is required to prevent eavesdropping. In our simulations, the guard zone is set to 10 wavelengths. We further assume that RxFJ has no effect outside this guard zone since the power of the RxFJ signal has to be small by design (especially, when SIS is imperfect). Rayleigh fading is assumed, so all channel entries are i.i.d. Circularly Symmetric Gaussian random variables (0, 1). When we only use TxFJ, Alice's normalized power budget is taken as 120 dB to make a fair comparison.

(160) With the above setting, the simulation run is repeated 10000 times, each time with a different channel entries. FIG. 9 shows the secrecy sum-rate (R.sub.sum) versus for three different scenarios. The highest secrecy rate is achieved when both RxFJ and TxFJ is used. R.sub.sum decreases with since RxFJ affects the legitimate receivers' SINR. On the other hand, the value of does not have much impact on the secrecy rate when only TxFJ or RxFJ is used. Interestingly, a slightly higher secrecy rate is achieved with the RxFJ-only scheme compared to the TxFJ-only scheme, because there is only one possibility for power allocation in the latter scheme. Moreover, with the protocol model, only eavesdropping locations inside the guard zoned of Bob and Charlie impact R.sub.sum, (since TxFJ is assumed to be nulled inside these zones). Thus, increasing the power of TxFJ will have no impact on R.sub.sum. On the other hand, when only RxFJ is used, the allocation of powers for the data and RXFJ signals can be optimized to achieve a higher R.sub.sum. We will later see a different behavior under the SINR model.

(161) To investigate the effect of the minimum required SINR T, we set to 0.1. As seen from FIG. 10, increasing T results in higher R.sub.sum for the three FJ schemes. Again, the combined Tx/Rx FJ scheme achieves the highest rate. The RxFJ-only scheme achieves a slightly better rate than that of the TxFJ only scheme, due to the same reasons mentioned before.

(162) B. SINR Model

(163) In this section, we consider the so-called SINR interference model, where the channel gain from each transmit antenna to each receive antenna is given by:

(164) 0$\begin{array}{cc}H=\left(\sqrt{{10}^{\frac{{\mathrm{SL}}_{dB}+{\mathrm{PL}}_{dB}}{10}}}\right)G& \left(22b\right)\end{array}$
where the quantity between the parenthesis represents the large-scale fading effects, with SL.sub.dB and PL.sub.dB representing the loss in dB due to shadowing and the path loss, respectively. The second term, G(0,1), represents small-scale fading effects. Shadowing is assumed to be log-normal with 8 dB standard deviation, SL.sub.dB8N(0,1); on the other hand, the path loss is modeled as PL.sub.dB=20 log.sub.10(d) where d is the distance between the transmit and receive antennas. We set P.sub.A=100 dB, P.sub.B=10 dB, and P.sub.C=10 dB, are normalized to the noise power. We consider full self-interference suppression and T=5 dB. In TxFJ-only case, the receivers have no power, and P.sub.A is set to 120 dB to maintain the total power budget in the network.

(165) FIGS. 11, 12 and 13 show the contours of the achievable secrecy sum-rate according to Eve's location under three different FJ schemes. In each figure, the locations of Alice, Bob and Charlie are specified. We make a few observations. First, in FIG. 12, the contour lines have a circular shape around Alice, as expected, since only TxFJ is used. On the other hand, in FIG. 11, the contour lines are not symmetric around Alice. Rather, they are ellipsoid-like with right centres around Alice, since RxFJ pushes the contour lines towards Alice. As a result, the performance of RxFJ+TxFJ is always better than the others even if the total power budget is kept constant. In FIG. 13, the contours have a circular shape around Alice except for when they are near to Bob and Charlie. When Eve is between Alice and Bob/Charlie or farther from Bob/Charlie, the TxFJ-only scheme achieves a higher secrecy sum-rate than the RxFJ-only scheme. However, if Eve is around the receivers, the performance of both schemes is similar since RxFJ degrades the SINR of Eve.

(166) Proof of Theorem 2

(167) Encoder:

(168) Here, we design the codewords (signals) s.sub.B.sup.N and s.sub.C.sup.N carrying the secure messages to Bob and Charlie (M.sub.B and M.sub.C), respectively. The encoding is designed such that P.sub.e,i=Prob{{circumflex over (M)}.sub.iM.sub.i}.fwdarw.0 as N.fwdarw. where {circumflex over (M)}.sub.i is the estimate of M.sub.i at receiver i, and

(169) $\begin{array}{cc}\frac{1}{N}\left(M_B,M_C;Z^N|H\right).fwdarw.0,& \left(23b\right)\end{array}$
as N.fwdarw. (here, H refers to channel states). This is the secrecy constraint implying that the mutual information leakage rate to the eavesdropper is made vanishing as the number of channel uses, N, gets large.

(170) The channels given in (13b) and (16b) can be summarized as
y.sub.B.sup.(j,t)=h.sub.B.sup.(j,t)s.sub.B.sup.(j,t)+n.sub.B.sup.(j,t)(24b)
y.sub.C.sup.(j,t)=h.sub.C.sup.(j,t)s.sub.C.sup.(j,t)+n.sub.C.sup.(j,t)(25b)
y.sub.E.sup.(j,t)=h.sub.1.sup.(j,t)s.sub.B.sup.(j,t)+h.sub.2.sup.(j,t)s.sub.C.sup.(j,t)+n.sub.E.sup.(j,t),(26b)
where (j,t) indicates channel coherence interval (fading block) j{1, . . . , J} and time instant t{1, . . . , T}. That is, channel use index n=(j1)T+t for (j, t)-th channel use, and total number of channel uses is N=JT. This is a block fading interference channel where interference links are removed, and we'll use techniques in [15b] and [14b] to build encoder. In particular, the signals s.sub.B and s.sub.C share the randomness needed to confuse the eavesdropper as in [14b] and designed as in [15b] to overcome fading limitations and absence of eavesdropper CSI.

(171) Let R.sub.B=(S.sub.B; Y.sub.B|H)=[log(1+SINR.sub.B)]. We generate all binary sequences {v.sub.B} each of length N. R.sub.B, and then, independently assign each of them uniformly randomly to one of 2.sup.nR.sup.B.sup.s groups. Given the secret message M.sub.B {1, . . . , 2.sup.nR.sup.B.sup.s}, a sequence v.sub.B is randomly selected from the group M.sub.B. This v.sub.B is divided into J blocks v.sub.B=[v.sub.B(1), . . . , v.sub.B(j), . . . , v.sub.B(J)], where each block v.sub.B(j) has T log(1+SINR.sub.B.sup.(j)) bits, and transmitted in j-th fading block. To transmit these bits in block j, Alice uses random Gaussian codebook consisting of 2.sup.T log(1+SINR.sup.B.sup.(j)) codewords s.sub.B.sup.(j) each of length T. Then, the transmitted signal is given by s.sub.B.sup.N=(s.sub.B.sup.(1), . . . , s.sub.B.sup.(J)). A similar scheme is used to construct the signal S.sub.C.sup.N.

(172) Decoder:

(173) Bob can decode each message for j-th fading block (as the rate supports channel capacity), and jointly typical decoding will succeed with high probability as T.fwdarw.. Then, union bound will imply that all messages can be recovered, from which v.sub.B can be reconstructed and the bin index M.sub.B can be declared as the decoded message. Charlie will use the same scheme to reliably decode M.sub.C.

(174) Security:

(175) (Proof Sketch) Consider the following:

(176) $\begin{array}{cc}\begin{array}{c}\left(M_B,M_C;Y_E^N|H\right)=H\left(M_B,M_{C}H\right)-H\left(M_B,M_C|Y_E^N,H\right)\\ \stackrel{\left(a\right)}{}H\left(M_B,M_C|H\right)-\left(M_B,M_C;X_S^N|Y_E^N,H\right)\\ =H\left(M_B,M_C\right)-H\left(X_S^N|Y_E^N,H\right)+\\ H\left(X_S^N|Y_E^N,M_B,M_C,H\right)\\ \stackrel{\left(b\right)}{=}H\left(M_B,M_C\right)-\underset{j=1}{\overset{J}{.Math.}}H\left(X_S^{\left(j,1:T\right)}|Y_E^{\left(j,1:T\right)},H\right)+\\ H\left(X_S^N|Y_E^N,M_B,M_C,H\right)\\ =H\left(M_B,M_C\right)-\underset{j=1}{\overset{J}{.Math.}}[H\left(X_S^j|H\right)-\\ \left(X_S^j;Y_E^j|H\right)]+H\left(X_S^N|Y_E^N,M_B,M_C,H\right)\\ H\left(M_B,M_C\right)-\underset{j=1}{\overset{J}{.Math.}}(T\left({\left[R_B^j-R_{E,B}^j\right]}^{+}\right)+\\ T\left({\left[R_C^j-R_{E,C}^j\right]}^{+}\right))+H\left(X_S^N|Y_E^N,M_B,M_C,H\right)\\ \stackrel{\left(c\right)}{=}H\left(X_S^N|Y_E^N,M_B,M_C,H\right)\\ \stackrel{\left(d\right)}{}{N}_{},\end{array}& \left(27b\right)\end{array}$

(177) where (a) follows as H(M.sub.B, M.sub.C|X.sub.S.sub.N, Z.sup.N, H)0, where X.sub.S.sup.Nv.sub.B.sup.Ns.sub.B.sup.N+v.sub.C.sup.Ns.sub.C.sup.N. (b) follows due to memoryless channel and independence signals. (c) follows by choosing secrecy rates H(M.sub.B)=JT[R.sub.B.sup.jR.sub.E,B.sup.j] and H(M.sub.C)=JT[R.sub.C.sup.jR.sub.E,C.sup.j].sup.+ and by taking J.fwdarw. and T.fwdarw., as time average converges to expected value due to ergodicity of the channel. (d) follows by Fano's inequality as the eavesdropper can decode signals s.sub.e and s.sub.C by employing a list decoding (similar to [14b]). In particular, consider H(X.sub.S.sup.N|Z.sup.N, M.sub.B, M.sub.C, H)=H(S.sub.B.sup.N|Z.sup.N, M.sub.B, M.sub.C, H)+H (S.sub.C.sup.N|Z.sup.N, S.sub.B.sup.N, M.sub.B, M.sub.C, H). Here, both terms on the right can be shown to be vanishing if the secrecy rates R.sub.B.sup.S and R.sub.C.sup.S satisfy R.sub.B.sup.s=[R.sub.B.sup.jR.sub.E,B.sup.j].sup.+ and R.sub.C.sup.s=[R.sub.C.sup.jR.sub.E,C.sup.j].sup.+. Then, Fano's inequality will imply that

(178) $\begin{array}{cc}\frac{1}{N}\left(M_B,M_C;Z^N|H\right)& \left(28b\right)\end{array}$
for arbitrarily small as N.fwdarw., which proves the secrecy.

(179) In this embodiment, we considered the scenario where a transmitter sends two independent confidential data streams, intended to two legitimate users, in presence of an eavesdropper of an unknown location. With the knowledge of that the security applications require guard zones around receivers up to 19 wavelengths, we proposed using RxFJ along with TxFJ. That way, even if an eavesdropper has a highly correlated channel with that of any legitimate receiver and is able to cancel out TxFJ, RxFJ keeps providing confidentiality for the information messages. TxFJ is still needed since a transmitter generally has much more power than mobile receivers. Therefore, it can provide PHY security over a larger area. To be able to send RxFJ from the receivers, we considered full-duplex receivers, but with one antenna per receiver. These receivers are capable of partial/complete self-interference suppression.

(180) We used zero-forcing technique not only to remove the TxFJ interference at intended receivers but also to hide the information messages from the unintended receivers. An optimization problem was formulated for the power allocations of the two information signals, the TxFJ signal, and the two RxFJ signals with the goal of maximizing the secrecy sum-rate. Assuming that the legitimate links demand a certain SINR such that their achieved data rates remain constant, and they achieve no data rate below this SINR threshold, we provided the optimal solution for this problem.

Example 3

(181) The following is a non-limiting example of friendly jamming for PHY-layer security in small-scale multi-link wireless networks in the presence of eavesdroppers.

(182) Distributed MIMO for FJ Nullification

(183) If the FJ signals are to be generated by the same MIMO node (e.g., Alice), then the phases of these signals can be easily controlled to provide the desired nullification. A set of FJ signals can add destructively and nullify each other at an intended receiver if these signals, each of which traverses a different channel, are received out-of-phase and sum up to zero. To achieve this, techniques such as zero-forcing beamforming are employed to determine the phase and amplitude of each FJ signal at the transmit antennas. However, in general, FJ signals may be produced by different devices that do not share a reference clock and so are not synchronous. Hence, the signals transmitted from distributed FJ nodes may experience unknown random delays. In this section, we explain how we synchronize FJ devices, each equipped with a single antenna, and establish the sufficient conditions on the jamming signals to ensure nullification of FJ signals at all legitimate receivers.

(184) A. Synchronization of FJ Devices

(185) To enable synchronized FJ devices, we exploit SourceSync's synchronization protocol proposed and empirically demonstrated in [9c] for OFDM systems. According to this method, a set of distributed cooperative transmitters chooses a leader, who initiates the synchronization process by transmitting an OFDM-based sync header. Using the phase offsets measured across different subcarriers, each cooperating transmitter can estimate the exact arrival time of the sync header. Based on the estimated RTT between each transmitter and the leader, and the switching time from Rx mode to Tx, each transmitter synchronizes in time with the leader. Finally, considering the propagation delay of the cooperative transmitters to the receiver, each transmitter selects a transmission time so as to synchronize the arrival of all the transmissions at the receiver.

(186) B. Nullification of FJ Signals

(187) Assuming that the distributed FJ nodes have been synchronized, the amplitudes/phases of their signals must be adjusted to cancel out at the legitimate receivers. Consider M legitimate receivers and N FJ nodes. The channel is characterized by an MN channel matrix H=[h.sub.ij]; where h.sub.ii is the channel coefficient between receiver i and transmitter j. By setting N=M+1, we can guarantee a nonempty null space for the channel matrix H [3c]. Let y be an M-by-1 vector that represents only the received FJ signals at the M receivers, let t be an N-by-1 vector that represents the transmitted signals from the N FJ antennas, and let F represent the N-by-1 precoding vector (precoder) of the FJ signal.

(188) At any time instant (time index is dropped for simplicity) and ignoring the effect of noise, we have:
y=Ht=HFm(1c)
where m denotes a random complex scalar at the current time and m.sup.2=1. The Singular Value Decomposition (SVD) of H can be obtained as
H=U.sub.MM.sub.MNV.sub.NN.sup..(2c)
Thus, y can be expressed as:
y=UV.sup.t.(3c)
If the jamming precoder F lies in the null space of H, then y=Ht=0. In our design, we select F as the rightmost column of the matrix V, i.e., the kernel of H. For a given total budget on the jamming power and a given m, the precoder F determines the phase of each of the FJ signals and implies some dependencies between their jamming powers so that they add up destructively at the legitimate receivers. Let P.sub.j=t.sub.j.sup.2=F.sub.j.sup.2 be the jamming power of the jth FJ device. We explicitly derive these dependencies by solving .sub.j=1.sup.Nh.sub.ijt.sub.j=0, i=1, . . . , M. It turns out that each jamming power must be a linear function of P.sub.1 as follows:
P.sub.j=.sub.jP.sub.1, j=2, . . . , N(4c)
where .sub.j is the scalar ratio between P.sub.j and P.sub.1.

(189) Network Model and Problem Formulation

(190) We consider a static multi-link network, consisting of an arbitrary number of legitimate nodes, each equipped with an omni-directional antenna. These nodes form a set of links . Each link l consists of a source (Alice) and a destination (Bob). This general multi-link network model accommodates both P2P and multihop scenarios. For the P2P scenario, consists of several independent single-hop links, connecting different Alice-Bob pairs. In the multihop case, contains specific links that form several paths between various pairs of nodes. Along with the set , there is a finite set of eavesdropping locations and a set of FJ nodes. We adopt a 2-D discrete model for the points in [6c], [7c]. The probability that an eavesdropper is in location e is denoted by p.sub.e. Even though this model assumes some (probabilistic) knowledge of the eavesdroppers' locations, it can represent numerous scenarios by adjusting the number of locations and the probability assigned to each location.

(191) The number of FJ devices can be less than or greater than ||. In this embodiment, however, we only consider the case when||>|J|, since the solution for the other cases is trivial: Assign an FJ node to each of the possible eavesdropping locations. In contrast to previous research [6c], [7c], we assume that there can be more than one eavesdropping location per active link, i.e., .sub.ep.sub.e can be greater than one.

(192) A. Problem Formulation

(193) We formulate the optimal placement and power allocation problem for the FJ devices such that the average SINR at each location e is less than a threshold , and FJ interference is nullified at legitimate receivers. We consider the case of cooperative FJ whereby FJ devices cooperatively nullify their jamming signals at all || receivers, even if only a subset of these receivers are active. We employ the SourceSync protocol [9c] to synchronize the FJ devices. SourceSync was initially designed to exploit sender diversity by transmitting the same packet from multiple cooperative senders. However, in our design, we leverage it to sync the FJ nodes. The leader will be an active data transmitter (Alice), who sends a sync header together with a random m before its main transmission. The power of the sync-header's transmission must be adjusted to reach all FJ nodes. Following the receipt of this header, FJ nodes calculate their transmission times to synchronize and create a null region around all || receivers.

(194) Henceforth, when we say Alice and Bob we mean the transmitter and respective receiver of a specific link l, respectively. Because the FJ signals are nullified at Bob, the transmission power of Alice is only a function of the SINR threshold at Bob and the distance between Alice and Bob of link l, denoted by d.sub.l (assuming a pathloss channel model). Therefore, to maintain the SINR at Bobsome threshold , the minimum transmission power at Alice of link l, denoted by P.sub.t,l, is given by:

(195) $\begin{array}{cc}{P}_{t,l}=\frac{N_o}{d_l^{-}}.& \left(5c\right)\end{array}$
where N.sub.o is AWGN power and is the pathloss exponent.

(196) For the case of cooperative jamming, the SINR at Bob (SINR.sub.b) is given by:

(197) $\begin{array}{cc}{\mathrm{SINR}}_b=\frac{P_{t,l}{d}_l^{-}}{N_o}.& \left(6c\right)\end{array}$
The optimization problem can now be stated as follows:

(198) $\begin{array}{cc}P1\text{:}\underset{\left\{x_j,y_j,P_{j}j\mathrm{��}\right\}}{\mathrm{minimize}}\underset{j\mathrm{��}}{.Math.}{P}_j& \left(7c\right)\\ \mathrm{subject}\mathrm{to}C1\text{:}{p}_e{\mathrm{SINR}}_e,e\mathrm{.Math.},l& \\ C2\text{:}\underset{j=1}{\overset{.Math.\mathrm{��}.Math.}{.Math.}}{h}_{\mathrm{ij}}{t}_j=0,i=1,\mathrm{.Math.},.Math..Math.& \end{array}$
where (x.sub.j, y.sub.j) are the Cartesian coordinates of FJ node j. Constraints C1 and C2 represent the secrecy and nullification constraints, respectively. When link l is active, the SINR at eavesdropper e (SINR.sub.e) is given by:

(199) $\begin{array}{cc}{\mathrm{SINR}}_e=\frac{P_{t,l}{d}_{ae,l}^{-}}{N_o+\underset{j\mathrm{��}}{.Math.}{P}_j{d}_{\mathrm{je}}^{-}}& \left(8c\right)\end{array}$
where is the pathloss exponent, d.sub.ae,l is the distance between Alice (of link l) and eavesdropping location e, and d.sub.je is the distance between the FJ node j and eavesdropper e. Note that j, e, and hence d.sub.je are not associated with a specific link l.

(200) We propose two schemes based on formulation P1: per-link and network-wide schemes. For the per-link scheme, the problem is solved independently for each link. In this case, we have || independent problems. For each of these problems, the secrecy and nullification constraints are considered only for a specific link l, i.e., ||=1. To ensure a nonempty nullspace for the channel matrix , |J| has to be greater than ||. This implies that in the per-link scheme, in each problem must contain a minimum of two FJ nodes. Hence, we need at least 2|| FJ nodes to secure all links. For the network-wide scheme, all links and FJ devices are simultaneously considered in the secrecy and nullification constraints, i.e., we jointly optimize over all links in the set . One advantage of this scheme is that we only need ||+1 FJ nodes to ensure that the jamming signals are nullified at all || legitimate receivers.

(201) Considering the network-wide scheme and assuming that the locations of legitimate nodes are known and P.sub.t,l is known l. Thus, the first constraint can be reformulated as:

(202) $\begin{array}{cc}C1\text{:}{p}_e\frac{\underset{l}{\max }{P}_{t,l}{d}_{\mathrm{ae},l}^{-}}{N_o+\underset{j\mathrm{��}}{.Math.}{P}_j{d}_{\mathrm{je}}^{-}},e\mathrm{.Math.}.& \left(9c\right)\end{array}$

(203) B. Signomial Programming

(204) Before we explain how the optimization problem P1 can be solved, we take a detour and discuss a class of optimization problems known as signomial programming. A function is said to be monomial in the variables x.sub.1, x.sub.2, . . . , n.sub.n if:

(205) $\begin{array}{cc}f\left(x_1,x_2,\mathrm{.Math.},x_n\right)=x_1^{a_1}{x}_2^{a_2}\mathrm{.Math.}{x}_n^{a_n}=\underset{i=1}{\overset{n}{.Math.}}{x}_i^{a_i}& \left(10c\right)\end{array}$
where a.sub.1, a.sub.2, . . . , a.sub.n an are some real-valued numbers. A function h is said to be posynomial in the variables x.sub.1, x.sub.2, . . . , x.sub.n if it can be written as a linear combination of monomials, i.e.,

(206) 0$\begin{array}{cc}h\left(x_1,x_2,\mathrm{.Math.},x_n\right)=\underset{k=1}{\overset{n}{.Math.}}{c}_k{f}_k\left(x_1,x_2,\mathrm{.Math.},x_n\right)& \left(11c\right)\end{array}$

(207) where c.sub.k .sup.+ and f.sub.k is monomial in the variables x.sub.1, x.sub.2, . . . , x.sub.n.

(208) Geometric program is a class of non-convex, non-linear optimization problems that can be written in the following standard form [10c, [11c]:

(209) $\begin{array}{cc}\underset{x}{\mathrm{minimize}}& h_o\left(x\right)\\ \mathrm{subject}\mathrm{to}& C1\text{:}{h}_i\left(x\right)f_i\left(x\right),i=1,2,\mathrm{.Math.},I\\ & C2\text{:}{g}_k\left(x\right)q_k\left(x\right),k=1,2,\mathrm{.Math.},K\end{array}$
where x(x.sub.1, x.sub.2, . . . , x.sub.n); h.sub.o and h.sub.i are posynomials; I and K are two arbitrary integer numbers; and f.sub.i, g.sub.k and q.sub.k are monomials. Note that by change of variables, the standard form geometric program can be transformed into a convex problem, thus a global optimum can be computed efficiently.

(210) The standard form geometric program requires that each and every posynomial be upper bounded by a monomial. Signomial programming is a generalized form of posynomial geometric programming in which the posynomials may be lower bounded by monomials. Signomial problems cannot be transformed to convex problems through a change of variables; they are non-linear, non-convex problems that are generally NP-hard. One approach of solving signomial programs is by using condensation techniques [12c], which approximate any posynomial function into a monomial function. This approximation reduces the problem to the standard geometric form.

(211) Proposition III.1.

(212) Problem P1 is a signomial programming problem with ||+|| signomial constraints.

(213) Proof.

(214) The objective function is a summation of linear variables (i.e., .sub.jJP.sub.j). As for the secrecy constraint, we have e:

(215) $\begin{array}{cc}{P}_e{\mathrm{SINR}}_e& \left(12c\right)\\ p_e\frac{P_{t,l}{d}_{\mathrm{ae},l}^{-}}{N_o+\underset{j\mathrm{��}}{.Math.}{P}_j{d}_{\mathrm{je}}^{-}}& \left(13c\right)\end{array}$

(216) $\begin{array}{cc}\frac{p_e{P}_{t,l}{d}_{\mathrm{ae},l}^{-}}{}{N}_o+\frac{P_1}{d_{1e}^{}}+\frac{P_2}{d_{2e}^{}}+\mathrm{.Math.}+\frac{P_{.Math.\mathrm{��}.Math.}}{d_{.Math.\mathrm{��}.Math.e}^{}}& \left(14c\right)\\ \left(\frac{p_e{P}_{t,l}{d}_{\mathrm{ae},l}^{-}}{}\right)d_{1e}^{}{d}_{2e}^{}\mathrm{.Math.}{d}_{.Math.\mathrm{��}.Math.e}^{}{N}_o\left(d_{1e}^{}{d}_{2e}^{}\mathrm{.Math.}{d}_{.Math.\mathrm{��}.Math.e}^{}\right)+P_1\left(d_{2e}^{}\mathrm{.Math.}{d}_{.Math.\mathrm{��}.Math.e}^{}\right)+\mathrm{.Math.}+P_{.Math.\mathrm{��}.Math.}\left(d_{1e}^{}{d}_{2e}^{}\mathrm{.Math.}{d}_{\left(.Math.\mathrm{��}.Math.-1\right)e}^{}\right)& \left(15c\right)\\ \left(\frac{p_e{P}_{t,l}{d}_{\mathrm{ae},l}^{-}}{}\right)\underset{j\mathrm{��}}{.Math.}{d}_{\mathrm{je}}^{}{N}_o\left(\underset{j\mathrm{��}}{.Math.}{d}_{\mathrm{je}}^{}\right)+\underset{j\mathrm{��}}{.Math.}{P}_j\underset{\underset{ij}{i\mathrm{��}}}{.Math.}{d}_{\mathrm{ie}}^{}& \left(16c\right)\\ \frac{\left(\frac{p_e{P}_{t,l}{d}_{\mathrm{ae},l}^{-}}{}\right)\underset{j\mathrm{��}}{.Math.}{d}_{\mathrm{je}}^{}}{N_o\left(\underset{j\mathrm{��}}{.Math.}{d}_{\mathrm{je}}^{}\right)+\underset{j\mathrm{��}}{.Math.}{P}_j\underset{\underset{ij}{i\mathrm{��}}}{.Math.}{d}_{\mathrm{ie}}^{}}1,& \left(17c\right)\end{array}$
which is in the form of:

(217) $\begin{array}{cc}\frac{Q\left(x\right)}{P\left(x\right)}1& \left(18c\right)\end{array}$
where Q(x) and P(x) are monomial and posynomial, respectively.

(218) It follows that our formulation belongs to the category of signomial programming, in which the posynomial is lower-bounded by a monomial. The same analysis can be applied to the nullification constraint to show that it also represents || reversed posynomial constraints.

(219) Solution Based on Condensation Techniques

(220) In this section, we attempt to solve P1 in a sub-optimal fashion by approximating the signomial problem as a standard geometric program using condensation techniques. The basic idea in condensation is to approximate a multi-term posynomial by a monomial or by a single-term function [12c]. The arithmetic/geometric mean inequality is the key element in the condensation process. It is given by the following expression:

(221) $\begin{array}{cc}{}_1{V}_1+{}_2{V}_2+\mathrm{.Math.}+{}_k{V}_k{V}_1^{{}_1}{V}_2^{{}_2}\mathrm{.Math.}{V}_k^{{}_k}=\underset{i=1}{\overset{k}{.Math.}}{V}_i^{{}_i}& \left(19c\right)\end{array}$
where k is the number of terms and

(222) $V_i,{}_i{+}^{}\mathrm{for}i=1,2,\mathrm{.Math.},k.\mathrm{Let}{U}_i\stackrel{\mathrm{def}}{=}{}_i{V}_i.$
Then,

(223) $\begin{array}{cc}\underset{i=1}{\overset{k}{.Math.}}{U}_i\underset{i=1}{\overset{k}{.Math.}}{\left(\frac{U_i}{{}_i}\right)}^{{}_i}& \left(20c\right)\\ U_i>0,{}_i>0& \left(21c\right)\\ \underset{i=1}{\overset{k}{.Math.}}{}_i=1.& \left(22c\right)\end{array}$

(224) Consider the following generalized reversed geometric program:

(225) $\begin{array}{cc}\begin{array}{cc}\mathrm{minimize}& x_o\\ \mathrm{subject}\mathrm{to}& \frac{Q\left(x\right)}{P\left(x\right)}1\end{array}& \left(23c\right)\end{array}$

(226) where x.sub.o is a linear objective function, and both Q(x) and P(x) are posynomials. The objective is to condense P(x) into a monomial. The resulting constraint is a posynomial that is upper-bounded by a monomial, i.e., it is in a standard convex form of a geometric program.

(227) Using condensation techniques, P(x) can be written as a weighted sum of monomials:

(228) $\begin{array}{cc}P\left(x\right)=\underset{i=1}{\overset{k}{.Math.}}{P}_i\left(x\right)=\underset{i=1}{\overset{k}{.Math.}}{c}_i\underset{n=0}{\overset{N}{.Math.}}{x}_n^{a_{in}}& \left(24c\right)\end{array}$
where c.sub.i is a positive number that represents the weight of the ith term, N denotes the number of variables, and a.sub.in is the exponent associated with the nth variable in the ith term of the posynomial.

(229) Let P.sub.i(x)=U.sub.i. It follows that P(x)=.sub.i=1.sup.kP.sub.i(x)=.sub.i=1.sup.kU.sub.i. Using the arithmetic/geometric inequality, P(x) can be written as:

(230) 0$\begin{array}{cc}P\left(x\right)\underset{i=1}{\overset{k}{.Math.}}{\left(\frac{U_i}{{}_i}\right)}^{{}_i}& \left(25c\right)\end{array}$
where .sub.i is defined as: