The paper principally proposes a suboptimally closedform solution in terms of a general asymptotic bound of the secrecy capacity in relation to MIMOMEbased transceivers. Such pivotal solution is essentially tight as well, fundamentally originating from the principle convexity. The resultant novelty, per se, is strictly necessary since the absolutely central criterion imperfect knowledge of the wiretap channel at the transmitter should also be highly regarded. Meanwhile, ellipsoidal channel uncertainty setdriven 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 worstcase methoddriven algorithms utilising some provable inequalitybased 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 2×2based 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.
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 of resource 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 knockon effects such as markedly more adequate convergence 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, multiinput multioutput multiantenna eavesdropper (MIMOME) based ones. QoS should be fairly taken into account among all players, since eavesdroppers (Eaves) may be licensed or unauthorised ones in the mentioned futuristic transceivers.
The suboptimality expressed above advances a tradeoff 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 tradeoff 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 principle Convexification [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 and energy harvesting scenarios [11, 12]; or lowrank beamforming issue [13].
1.1.2. Algorithms: An Overview
In [14], a secure game theorydriven transmission framework was conducted for MIMOMEs proposing three algorithms aimed at maximising sum secrecy rate. A tradeoff was theoretically defined in [15] for the principles sum secrecy rate reinforcement and energy efficiency which could provide a balance in physicallayersecuritybased transceiver designs. In [16], three general approaches were proposed in order to choose a paradigm for twoobjective design problems, namely, Nash bargaining (the most efficient one), Jain’s index, and the KalaiSmorodinsky bargaining. In [17], utilising a dynamic programming algorithm, a multilayer wiretap coding framework was fundamentally proposed.
1.1.3. ClosedForm Solutions: An Overview
Regarding maintoeavesdropper ratio, SOP’s diversity order remaining fixed was proven in [18]. In [19], taking into account various receiver arrangements for MIMOME transceivers over Nakagamim fading channels, SOP was precisely computed. In [20], it was proven that the intercept probability does not rely on Eave’ numbers in a cooperativebased 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’s exponential 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 of treating interference as noise over M×2Xchannels in [23]. Integrating a chaos stream cipher with the secure 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 closedform solution in relation to the channel uncertaintybased 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 closedform solution for the secrecy capacity while simultaneously guaranteeing
convexity which highlights a significantly more adequate response,
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 discovered tradeoff irrespective of the kind of fading channels, the type of strategies, and so on. Such novel tradeoff 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 tradeoff is interpreted with the aim of nearoptimally finding the upperbound (worstcase) 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 Section 2. Additionally, the simulationbased 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 MIMOMEbased transceiver is literally given in Figure 1. The MIMO channels between the legitimate transmitter and the legitimate user as well as Eave are principally denoted by H and G (please note that our scheme undoubtedly satisfies H and G of sizes M×N and M≠N; however, we considered them as N×N (square) for more convenience), in a Downlink paradigm. For G, 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=logdetI+1δr2HQHH,Ce=logdetI+1δe2GQGH,with respect to the noise variance terms δr2 and δe2 at the receiver sides and the transmit covariance matrix Q≜E{xxH}.
MIMOMEbased transceiver.
2.1. Our Proposed SolutionProperty 1.
Since we technically aim at maximising the secrecy capacity CS≜CrCe which can be logically interpreted as max{0,CrCe}, 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 decomposition E{xxH}↦Q cited above fundamentally requires the constraint rank(Q)=1 [28–31]. Inversely, Q being rank1 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% rank1 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 rank1 constraint can be basically relaxed as the dual constraints (2a)A1QxQHA2xxHxH1⩾0,(2b)TrA1xxH⩽0,where A1 and A2 are slack variable matrices, correspondingly.
Fully take into account the optimisation problem below as Problem P1(3a)maximiseQ⩾0CS,(3b)TrQ⩽PTransmit,(3c)2a,2b,where PTransmit is a threshold for the transmit power.
Lemma 1 (see [<xref reftype="bibr" rid="B35">35</xref>]).
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 Lemma 1, rank(M)=rank(Q)=1, while M≜M1QM1H, ∀M1∈{H,G} from (1). See, for example, [36, 9.6.10] as the positive definite property.
Corollary 3.
After carefully writing the dual Lagrange problem, then principally taking the derivative with respect to the defined Lagrange variables, then unifying the resultant term by zero, and finally right multiplying both sides of the resultant mathematical expression by Q, 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 [<xref reftype="bibr" rid="B37">37</xref>]).
If and only if rank(M)⩽1,(4)detI+M⩾1+TrM.
Corollary 5.
Simultaneously using Corollary 2 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 signaltonoise 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 costfunction as a penalty: namely, IOld(Q,G1;Γ). According to the penalty method (see, e.g., [17]), Problem P3 is alternatively expressed as(6a)maximiseQ⩾0SNRD1,(6b)5c,5d,for which we have(7)D1≜minξ,maxΓ2⩽θTrGQGH︸IOldQ,G1;Γ.
P3 is nonconvex, per se, owing to the lack of control on the error in the estimation of G, originated from the channel uncertainty parameter Γ. Since P3 has no closedform solution with respect to its existing format, consequently, it needs an entirely appropriate alternative. See the following theorem.
Theorem 6.
P3 unhesitatingly has a solvable and relaxed alternative as Problem P4 as(8a)maximiseQ⩾0SNRD2,(8b)5c,5d,in which(9)D2≜minξ,INewQ,G1;θ,(10)INewQ,G1;θ≜maxΓ2⩽θTrGQGH,while the upperbound of (10) can be conveniently expressed as(11)B≜TrG1QG1H+TrQ⊙Q×maxΓ4︸θ2.
See the Appendix for the proof.
Therefore, the upperbound of (10) is recasted to B as well. See the following lemma.
Lemma 7.
Defining the deterministic value INew(Q,G1;θ) instead of IOld(Q,G1;Γ) as its worstcase (upperbound) 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 to Schur 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.
Figure 2 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.
Outage probability.
2 dimensions: against θ
3 dimensions: against θ and N
Analytically, the outage probability error is written as (12)OProb≔EΓ,θD2D1D1,according to which preciseness coefficient is calculated as 1OProb.
It should be noted that, for N=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 (SingleInput SingleOutput schemes) such closedfrom solution has a remarkably less reliable performance. The case 1×1 is discovered in detail in the second subsection.
Figure 3 repeats the previous one for 1OProb in the context of preciseness 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.
1 − outage probability.
2 dimensions: against θ
3 dimensions: against θ and N
Figure 4 shows the observably more efficient secrecy capacity for our approach compared to the worstcase 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. Worstcase method in our simulation is applied according to SLemma (see, e.g., [39, eq. 2021], [40, eq. 11–15], [41, eq. 58–61]) which indicates that (5b) can be relaxed as(13)vecG1HQH⊗INvecG1ξθA1vecG1HQH⊗INQH⊗INvecG1QH⊗IN+A1IN⩾0.In the linear matrix inequalitybased expression (13), A1 is the slack variable.
Secrecy capacity versus SNR, while ξ=50 mWatt.
N=4
N=4
N=8
N=8
In Figure 5, the secrecy capacity for our approach compared to the worstcase 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=a1xb1x, subjected to b1x⩽ξ. In general, M1↗ when ξ↗. It is sufficient to write Lagrange function for the problem, after taking the derivative with respect to one Lagrange variable in connection with the unique constraint and also x and then unifying the resultant terms by zero; M1=ξ(a1/a21). Indeed, M1∝ξ. In other words, the higher ξ physically causes the bigger search 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].
Secrecy capacity versus SNR, while ξ=100mWatt.
N=4
N=4
N=8
N=8
Figure 6 illustrates SOP against SNR (dB) as well for the different values of ξ, θ, and N. Analytically, SOP is expressed as (14)SOP≔EΓ,θCWCWOCWO,in which CW and CWO conventionally denote the optimised secrecy capacity achieved, respectively, from the case of applying our scheme (SNRD2) (see (8a)) and of not applying our scheme (SNRD1) (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,∞).
SOP versus SNR (dB).
θ=0.3, ξ=50 mWatt
θ=0.6, ξ=50 mWatt
θ=0.3, ξ=100 mWatt
θ=0.6, ξ=100 mWatt
Figure 6 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,∞).
SNRs needed for guaranteeing SOP⩽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
10
In Figure 7, 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.
Penalty versus SNR, while changing ξ.
Secrecy capacity in Figure 8 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 worstcase one [39–41].
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 flipflops, 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(2K1)). 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(·) is BigO function.
3.2. <inlineformula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M214"><mml:mn mathvariant="normal">1</mml:mn><mml:mo>×</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math></inlineformula> 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, 1OProb↗ 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 worstcase one).
4. Conclusion
An asymptotic bound for the secrecy capacity in the context of resource allocation was suboptimally defined for MIMOMEbased 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 tradeoff 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 almost 5% 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 5×104 for the case where ξ=50mWatt and θ=0.3, ∀SNR∈[12dB,∞).
AppendixProof of Theorem <xref reftype="statement" rid="thm1">6</xref>.
(10) is written as(A.1)TrG1QG1H+maxΓ2⩽θTrΓQΓH=D3+∑i=1Nqiieii+Leij,eji,qji,qiji≠j⪅aD3+∑i=1Nqiieii⩽bD3+∑i=1Nqiieii2⩽cD3+∑i=1Nqii2∑i=1Neii2⩽B,in which B is provided in (11) and(A.2)D3=TrG1QG1H.eii∈R (unclear) and qii are, respectively, the iith main diagonals elements of ΓHΓ and Q; also L(·) is a linear function of the offmain diagonal elements (∈R). (A.1) is analytically written by knowing(A.3)TrAB=TrBA,TrA+B=TrA+TrB.The proof can be conveniently completed regarding the detailed Remarks A.1, A.2, and A.3.
Remark A.1. The fourth line in (A.1) is literally approximated while generally ignoring L(·). Succinctly speaking, the general perspective and theoretical background behind the forenamed negligence lie mostly in the influence of the norms of the relative terms. The aforementioned logic can be theoretically observed in Figure 2.
Remark A.2. The logic behind the fifth line in (A.1) comes technically from the physical and casual behaviour of both precoder and error matrices. Indeed, since the forenamed matrices are positive semidefinite, hence, they should principally exist.
Remark A.3. The penultimate line in (A.1) straightforwardly originates from CauchySchwarzBunyakovsky inequality [43, eq. 11–112].
The proof of Theorem 6 is completed.
Notations(·)H:
Hermitian
det{·}:
Determinant
log{·}:
Logarithm
Tr(·):
Trace of matrix
max{·}:
Maximum value
min{·}:
Minimum value
·F:
Frobenius norm
0:
Allzeros matrix
IN:
NbyN identity matrix
A⩾0:
Hermitian positive semidefinite matrix
CN×N:
NbyN dimension complex space
⊙:
Hadamard product
·:
Absolute value
Ez(·):
Expected value over z.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
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