# Asymptotic Bound of Secrecy Capacity for MIMOME-Based Transceivers: A Suboptimally Tractable Solution for Imperfect CSI

Authors: Makan Zamanipour
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The paper principally proposes a suboptimally closed-form solution in terms of a general asymptotic bound of the secrecy capacity in relation to MIMOME-based transceivers. Such pivotal solution is essentially tight as well, fundamentally originating from the principleconvexity. The resultant novelty, per se, is strictly necessary since the absolutely central criterionimperfect knowledge of the wiretap channel at the transmitter should also be highly regarded. Meanwhile, ellipsoidal channel uncertainty set-driven strategies are physically taken into consideration. Our proposed solution is capable of perfectly being applied for other general equilibria such as multiuser ones. In fact, this in principle addresses an entirely appropriate alternative for worst-case method-driven algorithms utilising some provable inequality-based mathematical expressions. Our framework is adequately guaranteed regarding a totally acceptable outage probability (as 1 − preciseness coefficient). The relative value is almost 10% for the estimation error values (EEVs) ⩽0.5 for 2×2-based transceivers, which is noticeably reinforced at nearly 5% for EEVs  ⩽0.9 for the case 4×4. Furthermore, our proposed scheme basically guarantees the secrecy outage probability (SOP) less than 0.05% for the case of having EEVs ⩽0.3, for the higher power regime.


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## Body

## 1. Introduction

In order to highlight a significantly more favourable system performance in terms of higher security in wireless networks, a programmable beamformer is set up at the transmitter [1–4]. Such technologically advanced equipment (precoder) is undoubtedly able to manage the users’ links in totally different paradigms, originated from the fact ofresource allocation. In the context of the forenamed widely supported principle, it is also highly motivated to provide the fashions convex, without loss of optimality.The convexity cited above has a vast range of beneficial knock-on effects such as markedly more adequateconvergence in design problem. Thus, the enlightening concept convexity is highly emphasised in order to guarantee an absolute robustness in terms of tractability in association with the optimisation problem.The optimisation problem (where the precoder is computed) optimally/suboptimally features a guarantee of the quality of service (QoS) in multiuser games, for example, multi-input multioutput multiantenna eavesdropper- (MIMOME-) based ones. QoS should be fairly taken into account among allplayers, since eavesdroppers (Eaves) may be licensed or unauthorised ones in the mentioned futuristic transceivers.The suboptimality expressed above advances a trade-off between two criteria such as convexity which suffers from, for example, channel uncertainties (originated from imperfect channel state information, which necessarily causes information deficiency) and a tractable response. In this paper, such trade-off is addressed as well.


### 1.1. Related Work

In this part, we generally provide a comprehensive overview over the existing work in relation to secrecy enhancement.


#### 1.1.1. General Perspective

Chiefly according to the principleConvexification [5, 6], security is an absolutely central criterion in designing transceivers, in the context of telemedicine games [7, 8]; cooperative relay networks [9, 10];wireless information power transfer andenergy harvesting scenarios [11, 12]; orlow-rank beamforming issue [13].


#### 1.1.2. Algorithms: An Overview

In [14], a secure game theory-driven transmission framework was conducted for MIMOMEs proposing three algorithms aimed at maximising sum secrecy rate. A trade-off was theoretically defined in [15] for the principles sum secrecy rate reinforcement andenergy efficiency which could provide a balance in physical-layer-security-based transceiver designs. In [16], three general approaches were proposed in order to choose a paradigm fortwo-objective design problems, namely,Nash bargaining (the most efficient one),Jain’s index, and theKalai-Smorodinsky bargaining. In [17], utilising a dynamic programming algorithm, a multilayer wiretap coding framework was fundamentally proposed.


#### 1.1.3. Closed-Form Solutions: An Overview

Regardingmain-to-eavesdropper ratio, SOP’s diversity order remaining fixed was proven in [18]. In [19], taking into account various receiver arrangements for MIMOME transceivers over Nakagami-m fading channels, SOP was precisely computed. In [20], it was proven that the intercept probability does not rely on Eave’ numbers in a cooperative-based network; instead, it relies specifically upon the number of relays. Asymptotic bounds for the outage probability defined for the secrecy rate and the achievable diversity were explored in [21]. Investigating the information’sexponential decay rate intercepted by Eave when using an average wiretap channel code, a tight and achievable secrecy rate was mathematically derived in [22]. Taking into account the generalised degrees of freedom, an investigation was basically provided in relation to the optimality oftreating interference as noise over M×2X-channels in [23]. Integrating achaos stream cipher with thesecure polar coding strategy, a general secure framework was technically fulfilled in [24].


### 1.2. Motivation

Regarding the literature, still suboptimal solutions are needed as favourable as possible, which basically guarantee a closed-form solution in relation to the channel uncertainty-based scenarios.


### 1.3. Contribution

We generally calculate the asymptotic bound of the secrecy capacity which is totally acceptable providing an adequately solvable and convex equivalent problem, correspondingly.Main Contributions. Overall, our main contribution over the existing work is to achieve a generally closed-form solution for the secrecy capacity while simultaneously guaranteeing(i)
convexity which highlights a significantly more adequate response,(ii)
near optimality in terms of a logically deniable outage probability of error (a principally acceptable preciseness coefficient, which also guarantees an acceptable SOP), which are absolutely defined in parallel with each other.In other words, we mathematically provide a general solution as a newly discoveredtrade-off irrespective of the kind of fading channels, the type of strategies, and so on. Such novel trade-off is able to be deployed to other schemes and has not been examined so far, which simultaneously is satisfied with a tight (correct) response and a deniable SOP (computation error). The mentioned trade-off is interpreted with the aim of near-optimally finding the upper-bound (worst-case) of Eave’s received power.


### 1.4. Notation and Organisation

(
·
)
↘ means that a term falls, whereas (·)↗ means that a term rises. Additionally, A, a, and a stand for, respectively, matrices, vectors, and scalars. ⊗ stands for the Kronecker product. The other symbols and notations utilised throughout the paper are listed in Notations. Finally, all matrices and vectors are, respectively, N×N and N×1, without loss of generality.The rest of the paper is organised as follows. Firstly, the main problem and the proposed solution are addressed in Section2. Additionally, the simulation-based analytical results are discussed in more detail in Section 3. Finally, conclusion and proofs are given in Section 4 and the appendix, respectively.


## 1.1. Related Work

In this part, we generally provide a comprehensive overview over the existing work in relation to secrecy enhancement.


### 1.1.1. General Perspective

Chiefly according to the principleConvexification [5, 6], security is an absolutely central criterion in designing transceivers, in the context of telemedicine games [7, 8]; cooperative relay networks [9, 10];wireless information power transfer andenergy harvesting scenarios [11, 12]; orlow-rank beamforming issue [13].


### 1.1.2. Algorithms: An Overview

In [14], a secure game theory-driven transmission framework was conducted for MIMOMEs proposing three algorithms aimed at maximising sum secrecy rate. A trade-off was theoretically defined in [15] for the principles sum secrecy rate reinforcement andenergy efficiency which could provide a balance in physical-layer-security-based transceiver designs. In [16], three general approaches were proposed in order to choose a paradigm fortwo-objective design problems, namely,Nash bargaining (the most efficient one),Jain’s index, and theKalai-Smorodinsky bargaining. In [17], utilising a dynamic programming algorithm, a multilayer wiretap coding framework was fundamentally proposed.


### 1.1.3. Closed-Form Solutions: An Overview

Regardingmain-to-eavesdropper ratio, SOP’s diversity order remaining fixed was proven in [18]. In [19], taking into account various receiver arrangements for MIMOME transceivers over Nakagami-m fading channels, SOP was precisely computed. In [20], it was proven that the intercept probability does not rely on Eave’ numbers in a cooperative-based network; instead, it relies specifically upon the number of relays. Asymptotic bounds for the outage probability defined for the secrecy rate and the achievable diversity were explored in [21]. Investigating the information’sexponential decay rate intercepted by Eave when using an average wiretap channel code, a tight and achievable secrecy rate was mathematically derived in [22]. Taking into account the generalised degrees of freedom, an investigation was basically provided in relation to the optimality oftreating interference as noise over M×2X-channels in [23]. Integrating achaos stream cipher with thesecure polar coding strategy, a general secure framework was technically fulfilled in [24].


## 1.1.1. General Perspective

Chiefly according to the principleConvexification [5, 6], security is an absolutely central criterion in designing transceivers, in the context of telemedicine games [7, 8]; cooperative relay networks [9, 10];wireless information power transfer andenergy harvesting scenarios [11, 12]; orlow-rank beamforming issue [13].


## 1.1.2. Algorithms: An Overview

In [14], a secure game theory-driven transmission framework was conducted for MIMOMEs proposing three algorithms aimed at maximising sum secrecy rate. A trade-off was theoretically defined in [15] for the principles sum secrecy rate reinforcement andenergy efficiency which could provide a balance in physical-layer-security-based transceiver designs. In [16], three general approaches were proposed in order to choose a paradigm fortwo-objective design problems, namely,Nash bargaining (the most efficient one),Jain’s index, and theKalai-Smorodinsky bargaining. In [17], utilising a dynamic programming algorithm, a multilayer wiretap coding framework was fundamentally proposed.


## 1.1.3. Closed-Form Solutions: An Overview

Regardingmain-to-eavesdropper ratio, SOP’s diversity order remaining fixed was proven in [18]. In [19], taking into account various receiver arrangements for MIMOME transceivers over Nakagami-m fading channels, SOP was precisely computed. In [20], it was proven that the intercept probability does not rely on Eave’ numbers in a cooperative-based network; instead, it relies specifically upon the number of relays. Asymptotic bounds for the outage probability defined for the secrecy rate and the achievable diversity were explored in [21]. Investigating the information’sexponential decay rate intercepted by Eave when using an average wiretap channel code, a tight and achievable secrecy rate was mathematically derived in [22]. Taking into account the generalised degrees of freedom, an investigation was basically provided in relation to the optimality oftreating interference as noise over M×2X-channels in [23]. Integrating achaos stream cipher with thesecure polar coding strategy, a general secure framework was technically fulfilled in [24].


## 1.2. Motivation

Regarding the literature, still suboptimal solutions are needed as favourable as possible, which basically guarantee a closed-form solution in relation to the channel uncertainty-based scenarios.


## 1.3. Contribution

We generally calculate the asymptotic bound of the secrecy capacity which is totally acceptable providing an adequately solvable and convex equivalent problem, correspondingly.Main Contributions. Overall, our main contribution over the existing work is to achieve a generally closed-form solution for the secrecy capacity while simultaneously guaranteeing(i)
convexity which highlights a significantly more adequate response,(ii)
near optimality in terms of a logically deniable outage probability of error (a principally acceptable preciseness coefficient, which also guarantees an acceptable SOP), which are absolutely defined in parallel with each other.In other words, we mathematically provide a general solution as a newly discoveredtrade-off irrespective of the kind of fading channels, the type of strategies, and so on. Such novel trade-off is able to be deployed to other schemes and has not been examined so far, which simultaneously is satisfied with a tight (correct) response and a deniable SOP (computation error). The mentioned trade-off is interpreted with the aim of near-optimally finding the upper-bound (worst-case) of Eave’s received power.


## 1.4. Notation and Organisation

(
·
)
↘ means that a term falls, whereas (·)↗ means that a term rises. Additionally, A, a, and a stand for, respectively, matrices, vectors, and scalars. ⊗ stands for the Kronecker product. The other symbols and notations utilised throughout the paper are listed in Notations. Finally, all matrices and vectors are, respectively, N×N and N×1, without loss of generality.The rest of the paper is organised as follows. Firstly, the main problem and the proposed solution are addressed in Section2. Additionally, the simulation-based analytical results are discussed in more detail in Section 3. Finally, conclusion and proofs are given in Section 4 and the appendix, respectively.


## 2. Problem Description

A MIMOME-based transceiver is literally given in Figure1. The MIMO channels between the legitimate transmitter and the legitimate user as well as Eave are principally denoted byH andG (please note that our scheme undoubtedly satisfiesH andG of sizes M×N and M≠N; however, we considered them as N×N (square) for more convenience), in a Down-link paradigm. ForG, we experience imperfect knowledge (which degrades QoS): that is, G≜G1±Γ  (our scheme is not applicable for some other frameworks [25, 26] in which, for example, the channel matrix is an arbitrary known matrix physically multiplied by a clear matrix [27, p. 4]). Now, regarding the threshold θ, for G we merely define the ellipsoidal uncertainty set Γ≜{Γ∈CN×N:Γ2⩽θ}. Instantaneous Ergodic capacity and also the undesired capacity (in connection with Eave) are, respectively, determined as(1)Cr=log⁡detI+1δr2HQHH,Ce=log⁡detI+1δe2GQGH,with respect to the noise variance terms δr2 and δe2 at the receiver sides and the transmit covariance matrix Q≜E{xxH}.Figure 1
MIMOME-based transceiver.


### 2.1. Our Proposed Solution

Property 1.
Since we technically aim at maximising the secrecy capacityCS≜Cr-Ce which can be logically interpreted as max⁡{0,Cr-Ce}, we should theoretically provide the physical implication CS(Cr,Ce;θ)∈[0,R) instead of CS(Cr,Ce;Γ)∈[0,R). R is the target secrecy capacity.The mapping decompositionE{xxH}↦Q cited above fundamentally requires the constraint rank(Q)=1 [28–31]. Inversely, Q being rank-1 unfavourably causes nonconvexity [29–31]. Of course under some circumstances, the aforementioned constraint holds and it can be alternatively dropped (e.g., in [32] in which nearly 99.79% rank-1 results in the simulations were witnessed). However, in order to guarantee the considerably more acceptable system with the higher reliability, we should ensure the design problem further.As newly recently proposed in [33, 34], the rank-1 constraint can be basically relaxed as the dual constraints (2a)A1QxQHA2xxHxH1⩾0,(2b)TrA1-xxH⩽0,where A1 and A2 areslack variable matrices, correspondingly.Fully take into account the optimisation problem below as ProblemP1(3a)maximiseQ⩾0CS,(3b)TrQ⩽PTransmit,(3c)2a,2b,where PTransmit is a threshold for the transmit power.Lemma 1 (see [35]).
As described in [35], multiplying (pre or post one) by any matrix (of any rank) does not cause any change on positive semidefinite matrices’ rank (see, e.g., [36, 9.6.10] for the proof).Corollary 2.
With respect to Lemma1, rank(M)=rank(Q)=1, while M≜M1QM1H, ∀M1∈{H,G} from (1). See, for example, [36, 9.6.10] as thepositive definite property.Corollary 3.
After carefully writing thedual Lagrange problem, then principally taking the derivative with respect to the definedLagrange variables, then unifying the resultant term by zero, and finally right multiplying both sides of the resultant mathematical expression byQ, a positive semidefinite matrix would theoretically hold. Regarding Lemma 1, the exact value in relation to the rank can be conveniently calculated.Lemma 4 (see [37]).
If and only ifrank(M)⩽1,(4)detI+M⩾1+TrM.Corollary 5.
Simultaneously using Corollary2 and Lemma 4, and since the log⁡(·) function has no technical influence on minimisation or maximisation in principle, we can accordingly maximise Tr{HQHH}, as well as maximising -Tr{GQGH}, which is in correspondence with the signal-to-noise ratio (SNR) and Eave’s received power, respectively. Now, with the aid of SNR instead of Tr{HQHH}, and defining ξ as a threshold for the maximum of Eave’s received power, Problem P2 is(5a)maximiseQ⩾0SNR,(5b)maxΓ2⩽θTrGQGH⩽ξ,(5c)TrQ⩽PTransmit,(5d)2a,2b.Methodology. We remove the constraint (5b) and accurately add it to the cost-function as a penalty: namely, IOld(Q,G1;Γ). According to thepenalty method (see, e.g., [17]), Problem P3 is alternatively expressed as(6a)maximiseQ⩾0SNR-D1-,(6b)5c,5d,for which we have(7)D1-≜min⁡ξ,maxΓ2⩽θTrGQGH︸IOldQ,G1;Γ.P
3 is nonconvex, per se, owing to the lack of control on the error in the estimation ofG, originated from the channel uncertainty parameter Γ. Since P3 has no closed-form solution with respect to its existing format, consequently, it needs an entirely appropriate alternative. See the following theorem.Theorem 6.
P
3 unhesitatingly has a solvable and relaxed alternative as Problem P4 as(8a)maximiseQ⩾0SNR-D2-,(8b)5c,5d,in which(9)D2-≜minξ,INewQ,G1;θ,(10)INewQ,G1;θ≜maxΓ2⩽θTrGQGH,while the upper-bound of (10) can be conveniently expressed as(11)B≜TrG1QG1H+TrQ⊙Q×maxΓ4︸θ2.See the Appendix for the proof.Therefore, the upper-bound of (10) is recasted to B as well. See the following lemma.Lemma 7.
Defining the deterministic valueINew(Q,G1;θ) instead of IOld(Q,G1;Γ) as its worst-case (upper-bound) and simultaneously maximising -INew(Q,G1;θ) provide the asymptotic bound of the optimisation problem in terms of θ, which efficiently guarantees convexity and then tractability. Indeed, we merely provided a technical segregation between the terms qii and eii (in the Appendix), which chiefly provided an opportunity where the inaccessible and unrecognised (in relation to the ellipsoidal channel uncertainty sets) term eii is generally formulated from another viewpoint.Lemma 8.
According toSchur product theorem [38, Theorem 7.5.3], Q⊙Q would be inevitably a positive semidefinite matrix; therefore, the key features (intrinsic physical and statistical properties) sufficiently remain stable.


## 2.1. Our Proposed Solution

Property 1.
Since we technically aim at maximising the secrecy capacityCS≜Cr-Ce which can be logically interpreted as max⁡{0,Cr-Ce}, we should theoretically provide the physical implication CS(Cr,Ce;θ)∈[0,R) instead of CS(Cr,Ce;Γ)∈[0,R). R is the target secrecy capacity.The mapping decompositionE{xxH}↦Q cited above fundamentally requires the constraint rank(Q)=1 [28–31]. Inversely, Q being rank-1 unfavourably causes nonconvexity [29–31]. Of course under some circumstances, the aforementioned constraint holds and it can be alternatively dropped (e.g., in [32] in which nearly 99.79% rank-1 results in the simulations were witnessed). However, in order to guarantee the considerably more acceptable system with the higher reliability, we should ensure the design problem further.As newly recently proposed in [33, 34], the rank-1 constraint can be basically relaxed as the dual constraints (2a)A1QxQHA2xxHxH1⩾0,(2b)TrA1-xxH⩽0,where A1 and A2 areslack variable matrices, correspondingly.Fully take into account the optimisation problem below as ProblemP1(3a)maximiseQ⩾0CS,(3b)TrQ⩽PTransmit,(3c)2a,2b,where PTransmit is a threshold for the transmit power.Lemma 1 (see [35]).
As described in [35], multiplying (pre or post one) by any matrix (of any rank) does not cause any change on positive semidefinite matrices’ rank (see, e.g., [36, 9.6.10] for the proof).Corollary 2.
With respect to Lemma1, rank(M)=rank(Q)=1, while M≜M1QM1H, ∀M1∈{H,G} from (1). See, for example, [36, 9.6.10] as thepositive definite property.Corollary 3.
After carefully writing thedual Lagrange problem, then principally taking the derivative with respect to the definedLagrange variables, then unifying the resultant term by zero, and finally right multiplying both sides of the resultant mathematical expression byQ, a positive semidefinite matrix would theoretically hold. Regarding Lemma 1, the exact value in relation to the rank can be conveniently calculated.Lemma 4 (see [37]).
If and only ifrank(M)⩽1,(4)detI+M⩾1+TrM.Corollary 5.
Simultaneously using Corollary2 and Lemma 4, and since the log⁡(·) function has no technical influence on minimisation or maximisation in principle, we can accordingly maximise Tr{HQHH}, as well as maximising -Tr{GQGH}, which is in correspondence with the signal-to-noise ratio (SNR) and Eave’s received power, respectively. Now, with the aid of SNR instead of Tr{HQHH}, and defining ξ as a threshold for the maximum of Eave’s received power, Problem P2 is(5a)maximiseQ⩾0SNR,(5b)maxΓ2⩽θTrGQGH⩽ξ,(5c)TrQ⩽PTransmit,(5d)2a,2b.Methodology. We remove the constraint (5b) and accurately add it to the cost-function as a penalty: namely, IOld(Q,G1;Γ). According to thepenalty method (see, e.g., [17]), Problem P3 is alternatively expressed as(6a)maximiseQ⩾0SNR-D1-,(6b)5c,5d,for which we have(7)D1-≜min⁡ξ,maxΓ2⩽θTrGQGH︸IOldQ,G1;Γ.P
3 is nonconvex, per se, owing to the lack of control on the error in the estimation ofG, originated from the channel uncertainty parameter Γ. Since P3 has no closed-form solution with respect to its existing format, consequently, it needs an entirely appropriate alternative. See the following theorem.Theorem 6.
P
3 unhesitatingly has a solvable and relaxed alternative as Problem P4 as(8a)maximiseQ⩾0SNR-D2-,(8b)5c,5d,in which(9)D2-≜minξ,INewQ,G1;θ,(10)INewQ,G1;θ≜maxΓ2⩽θTrGQGH,while the upper-bound of (10) can be conveniently expressed as(11)B≜TrG1QG1H+TrQ⊙Q×maxΓ4︸θ2.See the Appendix for the proof.Therefore, the upper-bound of (10) is recasted to B as well. See the following lemma.Lemma 7.
Defining the deterministic valueINew(Q,G1;θ) instead of IOld(Q,G1;Γ) as its worst-case (upper-bound) and simultaneously maximising -INew(Q,G1;θ) provide the asymptotic bound of the optimisation problem in terms of θ, which efficiently guarantees convexity and then tractability. Indeed, we merely provided a technical segregation between the terms qii and eii (in the Appendix), which chiefly provided an opportunity where the inaccessible and unrecognised (in relation to the ellipsoidal channel uncertainty sets) term eii is generally formulated from another viewpoint.Lemma 8.
According toSchur product theorem [38, Theorem 7.5.3], Q⊙Q would be inevitably a positive semidefinite matrix; therefore, the key features (intrinsic physical and statistical properties) sufficiently remain stable.


## 3. Results

In this section, we analyse the system performance as well.Some parameters and assumptions in the simulations are as follows.PTransmit is equal to SNR. Correlations at the MIMO antenna array are defined as 0%. The number of randomly generated Rayleigh fading channel realisations is 1000. N, θ, and ξ are separately defined for each figure.Figure2 strongly proves the tremendously sensible reliability in terms of the noticeably acceptable outage probability error (which is mathematically defined in (12)), which is plotted versus the θ regime while changing N. The relative content is repeated in the subfigure (b) for 3 dimensions against θ and N. Meanwhile, we should inform that θ=0 technically means the perfect CSI case.Figure 2
Outage probability.
(a)
2 dimensions: againstθ
(b)
3 dimensions: againstθ and NAnalytically, the outage probability error is written as(12)OProb≔EΓ,θD2--D1-D1-,according to whichpreciseness coefficient is calculated as 1-OProb.It should be noted that, forN=4 or higher, our proposed solution is completely trustable or even for N=2 while having the lower error rates in the estimation, that is, ∀θ⩽0.5. Inversely, for N=1 (Single-Input Single-Output schemes) such closed-from solution has a remarkably less reliable performance. The case 1×1 is discovered in detail in the second subsection.Figure3 repeats the previous one for 1-OProb in the context ofpreciseness coefficient. As can be seen, for N=2 and ∀θ⩽0.5, preciseness coefficient is higher than 90%, which is enhanced at ⩾95% for higher Ns. Again, the relative content is repeated in the subfigure (b) for 3 dimensions against θ and N. Meanwhile, we should inform that θ=0 technically means the perfect CSI case.Figure 3
1 − outage probability.
(a)
2 dimensions: againstθ
(b)
3 dimensions: againstθ and NFigure4 shows the observably more efficient secrecy capacity for our approach compared to theworst-case method prevalently utilised in the literature [39–41], while changing N and θ and also having ξ=50mWatt. Such strong reinforcement is effectively justified by the markedly more adequate convergence for P4, analytically caused by convexity comprehensively originated from solvability. Worst-case method in our simulation is applied according toS-Lemma (see, e.g., [39, eq. 20-21], [40, eq. 11–15], [41, eq. 58–61]) which indicates that (5b) can be relaxed as(13)vecG1HQH⊗INvecG1-ξ-θA1vecG1HQH⊗INQH⊗INvecG1QH⊗IN+A1IN⩾0.In thelinear matrix inequality-based expression (13), A1 is the slack variable.Figure 4
Secrecy capacity versus SNR, whileξ=50 mWatt.
(a)
N
=
4
(b)
N
=
4
(c)
N
=
8
(d)
N
=
8In Figure5, the secrecy capacity for our approach compared to theworst-case method is depicted against SNR (dB), while changing N and θ and also having ξ=100mWatt. It is revealed that the higher ξ we adjust, the larger values for the secrecy capacity can be mathematically achieved. This point can be theoretically analysed as follows. For example, assume that the general convex optimisation problem maximisex⩾0M1=a1x-b1x,subjected tob1x⩽ξ. In general, M1↗ when ξ↗. It is sufficient to writeLagrange function for the problem, after taking the derivative with respect to oneLagrange variable in connection with the unique constraint and also x and then unifying the resultant terms by zero; M1=ξ(a1/a2-1). Indeed, M1∝ξ. In other words, the higher ξ physically causes the biggersearch region. Finally, how to take the derivative needed for our scheme is essentially obvious according to the fact (σ/σQ)Tr{M1QM1H}=M1HM1, while M1∈{H,G}, which is proven with regard to (σ/σX)Tr{AXB}=AHBH, which is given in [36, eq. 101].Figure 5
Secrecy capacity versus SNR, whileξ=100mWatt.
(a)
N
=
4
(b)
N
=
4
(c)
N
=
8
(d)
N
=
8Figure6 illustrates SOP against SNR (dB) as well for the different values of ξ, θ, and N. Analytically, SOP is expressed as (14)SOP≔EΓ,θCW-CWOCWO,in which CW and CWO conventionally denote the optimised secrecy capacity achieved, respectively, from the case of applying our scheme (SNR-D2-) (see (8a)) and of not applying our scheme (SNR-D1-) (see (6a)). SOP fairly seems acceptable even for the case 1×1, which is technically less than 0.0005 for the case where ξ=50mWatt and θ=0.3, ∀SNR∈[12dB,∞).Figure 6
SOP versus SNR (dB).
(a)
θ
=
0.3, ξ=50 mWatt
(b)
θ
=
0.6, ξ=50 mWatt
(c)
θ
=
0.3, ξ=100 mWatt
(d)
θ
=
0.6, ξ=100 mWattFigure6 is briefly analysed in Table 1. In this table, the values in connection with ξ are in terms of mWatt which are ignored (not written) for more convenience. Indeed, the asymptotic bound of the secrecy capacity is technically equal to SNR while mathematically subtracting Eave’s received power. The penalty in relation to Eave’s received power is basically equal to min⁡{ξ,INew(θ)}, which would be evidently fixed at ξ in our simulations in the high SNR regime; that is, -10dB for the case where ξ=100mWatt and ∀SNR∈[17dB,∞).Table 1
SNRs needed for guaranteeingSOP⩽0.1%.
-
ξ
=
50, θ=0.3
ξ
=
50, θ=0.6
ξ
=
100, θ=0.3
ξ
=
100, θ=0.6
N
=
1
7
11
13
∞
N
=
2
0
0
7
10In Figure7, the penalty function in connection with Eave’s received power (undesired power) against SNR (dB) is shown while changing ξ. It is in parallel with Figure 6.Figure 7
Penalty versus SNR, while changingξ.Secrecy capacity in Figure8 is depicted against SNR (dB) for the cases of perfect CSI (θ=0) and imperfect one (θ≠0), while ξ=100mWatt and N=8. In this figure, our method is compared with the worst-case one [39–41].Figure 8
Secrecy capacity versus SNR (dB) for the cases of perfect (θ=0) and imperfect CSI, while ξ=100 mWatt and N=8.


### 3.1. Complexity

In relation to the complexity, we should compare (11) with (13). To this end, among the methods in the literature (e.g., [5, 32, 42]), we use the procedure of counting the flip-flops, as described in [5, Table 3]. In this method, for example, the complexity of the matrix product for two matrices of sizes, respectively, N×K and K×L is O(NL(2K-1)). Regarding an extreme range of operations in (13), our scheme has a more favourable complexity, which is ≈O(4N3) compared to ≈O(14N4) in which O(·) isBig-O function.


### 3.2.1×1 Case

Feature 1. According to OProb, the reliability for N=1 is crucially lower than that of the cases with larger values for N. However, because of the term SNR which is extremely larger than D2-(see (8a)),  SOP can be guaranteed instead, which perfectly highlights the importance of the outperformance, correctness, and effectiveness of our work.Feature 2. For the case 1×1, convexity theoretically is satisfied.


### 3.3. Quick Overview

Feature 3. In general, from (14) it can be seen that limSNR(dB)→∞SOP=0 since SNR≫D1-,D2-.Feature 4. In general, SOP↘ when N↗.Feature 5. In general, OProb↘ when N↗; accordingly, 1-OProb↗ when N↗.With respect to the main advantages thoroughly discussed in relation to the proposed scheme, its outperformance, correctness, and effectiveness are obviously highlighted over the existing methods (conventionally utilised in the literature, specifically the worst-case one).


## 3.1. Complexity

In relation to the complexity, we should compare (11) with (13). To this end, among the methods in the literature (e.g., [5, 32, 42]), we use the procedure of counting the flip-flops, as described in [5, Table 3]. In this method, for example, the complexity of the matrix product for two matrices of sizes, respectively, N×K and K×L is O(NL(2K-1)). Regarding an extreme range of operations in (13), our scheme has a more favourable complexity, which is ≈O(4N3) compared to ≈O(14N4) in which O(·) isBig-O function.


## 3.2.1×1 Case

Feature 1. According to OProb, the reliability for N=1 is crucially lower than that of the cases with larger values for N. However, because of the term SNR which is extremely larger than D2-(see (8a)),  SOP can be guaranteed instead, which perfectly highlights the importance of the outperformance, correctness, and effectiveness of our work.Feature 2. For the case 1×1, convexity theoretically is satisfied.


## 3.3. Quick Overview

Feature 3. In general, from (14) it can be seen that limSNR(dB)→∞SOP=0 since SNR≫D1-,D2-.Feature 4. In general, SOP↘ when N↗.Feature 5. In general, OProb↘ when N↗; accordingly, 1-OProb↗ when N↗.With respect to the main advantages thoroughly discussed in relation to the proposed scheme, its outperformance, correctness, and effectiveness are obviously highlighted over the existing methods (conventionally utilised in the literature, specifically the worst-case one).


## 4. Conclusion

An asymptotic bound for the secrecy capacity in the context of resource allocation was suboptimally defined for MIMOME-based systems for a general channel uncertainty fashion, supporting one theorem and also some lemmas and corollaries. The suboptimality cited above was principally highlighted in terms of a significantly acceptable trade-off between tractability and a less outage probability. Indeed, the deniable values obtained for the relative outage probability guaranteed our framework, which also fundamentally highlighted a tight and acceptable response for the system as well. The relative outage probability was almost5% for EEVs ⩽0.9 for 4×4 MIMO antenna arrays. Our proposed framework is completely implementable for other general equilibria, which basically provided SOPs less than 5×10-4 for the case where ξ=50mWatt and θ=0.3, ∀SNR∈[12dB,∞).



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*Source: 3656547-2017-12-14.xml*