On The Compound MIMO Wiretap Channel with Mean Feedback
Amr Abdelaziz C. Emre Koksal Hesham El Gamal Ashraf D. Elbayoumy














































































On The Compound MIMO Wiretap Channel with Mean Feedback
    ^†Amr Abdelaziz, ^†C. Emre Koksal and ^†Hesham El Gamal     ^*Ashraf D. Elbayoumy ^†Department of Electrical and Computer Engineering     ^*Department of Electrical Engineering

The Ohio State University     Military Technical College

Columbus, Ohio 43201     Cairo, Egypt

    December 30, 2023
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Compound MIMO wiretap channel with double sided uncertainty is considered under channel mean information model. In mean information model, channel variations are centered around its mean value which is fed back to the transmitter. We show that the worst case main channel is anti-parallel to the channel mean information resulting in an overall unit rank channel. Further, the worst eavesdropper channel is shown to be isotropic around its mean information. Accordingly, we provide the capacity achieving beamforming direction. We show that the saddle point property holds under mean information model, and thus, compound secrecy capacity equals to the worst case capacity over the class of uncertainty. Moreover, capacity achieving beamforming direction is found to require matrix inversion, thus, we derive the null steering (NS) beamforming as an alternative suboptimal solution that does not require matrix inversion. NS beamformer is in the direction orthogonal to the eavesdropper mean channel that maintains the maximum possible gain in mean main channel direction. Extensive computer simulation reveals that NS performs very close to the optimal solution. It also verifies that, NS beamforming outperforms both maximum ratio transmission (MRT) and zero forcing (ZF) beamforming approaches over the entire SNR range. Finally, An equivalence relation with MIMO wiretap channel in Rician fading environment is established.  




MIMO Wiretap Channel, Compound Wiretap Channel, Mean Channel Information, Saddle point, Worst Case Capacity.





















§ INTRODUCTION

A key consideration in determining the secrecy capacity of the MIMO wiretap channel is the amount of information available at the transmitter, not only about the eavesdropper channel, but also about the main channel. In principle, assuming perfect knowledge about the main channel, either full eavesdropper's channel state information (CSI) or, at least, its distribution are required to determine the secrecy capacity. The secrecy capacity of the general MIMO wiretap channel has been studied in <cit.> assuming perfect knowledge of both channels.
In practical scenarios, having even partial knowledge on the eavesdropper channel is typically not possible, especially, when dealing with strictly passive eavesdroppers. Further, in fast fading channel, it may also be unreasonable to have perfect main CSI at the transmitter. Compound wiretap channel <cit.> is a model that tackle these limitations in which CSI is known only to belong to a certain class of uncertainty. This assumption can be used to model eavesdropper only CSI <cit.> (single sided uncertainty) or both main and eavesdropper channels (double sided uncertainty)<cit.>. Depending on the considered class of uncertainty, the secrecy capacity of the compound wiretap channel can be characterized. 
Classes of uncertainity in the compound wiretap channel can be characterized in two different categories as it pertains to the set that the main and/or eavesdropper channels belong to: 1) Finite state channels 2) Continous set. The discrete memoryless compound wiretap channel with countably finite uncertainty set was studied in <cit.>. Meanwhile, the corresponding compound Gaussian MIMO wiretap channel with countably finite uncertainty set is analyzed in <cit.>. In both cases, the secrecy capacity is established only for the degraded case (i.e. main channel is stronger in all spatial directions). Meanwhile, the secrecy capacity itself remains unknown for the general indefinite case (i.e. main channel is stronger in subset of the available spatial directions). A closed form solution was obtained either in case of an isotropic eavesdropper <cit.> or the degraded case in the high SNR regime <cit.>. Although the optimal signaling scheme for the non-isotropic non-degraded case still not known to date in general, necessary conditions for optimality were derived in <cit.> and <cit.> for the deterministic known channel case. Recently in <cit.>, the compound Gaussian MIMO wiretap channel was studied under spectral norm constraint (maximum channel gain) and rank constraint for both single and double sided classes of uncertainty without the degradedness assumption.
 In <cit.> (Theorem 3) it was shown that, the secrecy capacity of the compound MIMO wiretap channel is upper bounded by the worst case capacity over the considered class of uncertainty. The term worst case capacity is established by optimizing the input signal covariance for all possible main and eavesdropper channel, then, taking the minimum over all main and eavesdropper channels over the considered class of uncertainty. Moreover, it was also shown that, the compound secrecy capacity is lower bounded by the capacity of the worst possible main and eavesdropper channels. Here, the saddle point needs to be considered, i.e. maxmin = minmax, where the max is taken over non-negative definite input covariance matrices subject to an average power constraint and the min is taken over the classes of channel uncertainty. If the saddle point property holds, the compound capacity is fully characterized and is known to match the worst case one. 
 In this paper, we consider the class of channels with double sided uncertainty under channel mean information model. In mean information model, channel is centered around a mean value which is fed back to the transmitter. An example for the mean information model is the channel with a strong Line-of-Sight (LOS) component, the gain of which is known at the transmitter. While it is unlikely to expect the eavesdropper to share its CSI (even its mean channel) in some scenarios, a secure communication system may be designed in a way that puts physical restrictions on the locations of possible attacker. These physical restrictions can be informative to the transmitter and may enable to achieve better secrecy rates by designing its signaling scheme accordingly. We first establish the worst case secrecy capacity of the compound MIMO wiretap channel under mean information model, then, we show that the saddle point property holds. We show that, the worst case main channel is anti-parallel to the channel mean information resulting in an overall unit rank channel. Further, the worst eavesdropper dropper channel is shown to be isotropic around its mean information. Accordingly, generalized eigenvector beamforming is known to be the optimal signaling strategy <cit.><cit.>. We show that the saddle point property holds under mean information model, and thus, compound secrecy capacity equals to the worst case capacity over the class of uncertainty. Further, as the generalized eigenvector solution requires matrix inversion, we introduce null steering (NS) beamforming, that is, transmission in the direction orthogonal to the eavesdropper mean channel direction maintaining the maximum possible gain in mean main channel direction, as an alternative suboptimal solution. Extensive computer simulation reveals that NS performs extremely close to the optimal solution. It also verifies that, NS beamforming outperforms both maximum ratio transmission (MRT) and zero forcing (ZF) beamforming approaches over the entire SNR range. Finally, An equivalence relation with MIMO wiretap channel in Rician fading environment is established.













§ SYSTEM MODEL AND PROBLEM STATEMENT




 §.§ Notations

In the rest of this paper we use boldface uppercase letters for random matrices, uppercase letters for their realizations, bold face lowercase letters for random vectors and lowercase letters for its realizations. Meanwhile, (.)^† denotes conjugate transpose, 𝐈_N denotes identity matrix of size N, (.) denotes matrix determinant operator and 1_m × n denotes a m × n matrix of all 1's. 



 §.§ System Model

We consider the MIMO wiretap channel scenario in which a transmitter 𝒜 with N_a > 1 antennas amounts to transmit a confidential message massage to a receiver, ℬ, having N_b > 1 antennas over an unsecure channel in the presence of a passive adversary, ℰ, equipped with N_e > 1 antennas. The discrete baseband equivalent channels for the signal received by each of the legitimate destination, 𝐲, and the adversary, 𝐳, are as follows:



    𝐲 = 𝐇_b   𝐱 + 𝐧_b,           𝐳 = 𝐇_e   𝐱 + 𝐧_e,
 

where  𝐱∈ℂ^N_a× 1 is the transmitted signal vector constrained by an average power constraint 𝔼[𝐭𝐫(𝐱𝐱^†)] ≤ P. Also, 𝐇_b∈ℂ^N_b × N_a and 𝐇_e  ∈ℂ^N_e × N_a are the channel coefficients matrices between message source, destination and adversary respectively. Finally, 𝐧_b ∈ℂ^N_b× 1 and  𝐧_e ∈ℂ^N_e× 1 are independent zero mean normalized to unit variance circular symmetric complex random vectors for both destination and adversary channels respectively, where, 𝐧_b∼𝒞𝒩(0,𝐈_N_b) and 𝐧_e∼𝒞𝒩(0,𝐈_N_e). 





 §.§ Problem Statement


 In this paper, we consider the case where the transmitter does not know the exact realizations of both 𝐇_b and 𝐇_e. Rather, it only knows that they both belong to a known compact (closed and bounded) uncertainty sets. Under the considered channel mean feedback model, we define channel uncertainty sets as follows: 

    𝒮_b    = {𝐇_b:𝐇_b = 𝐇_μ b + Δ𝐇_b, Δ𝐇_b_2 ≤ϵ_b,
    
               𝐇_μ b= λ_μ b^1/2v_bu_b^†} ,


    𝒮_e    = {𝐇_e:𝐇_e = 𝐇_μ e + Δ𝐇_e, Δ𝐇_e_2 ≤ϵ_e,
    
               𝐇_μ e= λ_μ e^1/2v_eu_e^† , u_e ∈𝒰} ,

where, 𝐇_μ∘ is the channel mean information which is assumed to be of unit rank and v_∘∈ℂ^N_∘× 1 , u_∘∈ℂ^N_a × 1. We assume that the transmitter knows u_b, meanwhile, it knows only that u_e ∈𝒰 where 𝒰 is the set of uncertainty about the eavesdropper mean information. In the extreme case when u_e is know exactly at the transmitter, we simply write 𝒰={u_e}.
Further, Δ𝐇_∘ is the channel uncertain part which is assumed to satisfy the bounded spectral norm condition Δ𝐇_∘_2 ≤ϵ_∘. In the compound wiretap channel, channels realizations are assumed to be fixed over the entire transmission duration. Therefore, Δ𝐇_∘ is considered fixed once it has been realized. This model is the scenario in which the eavesdropper can approach the transmitter up to a certain distance and from limited range of directions, see Fig. (<ref>).

First, let us define

    C(𝐖_b,𝐖_e,𝐐) = log(𝐈_N_a + 𝐖_b𝐐)(𝐈_N_a + 𝐖_e𝐐),


where 𝐖_∘≜𝐇_∘^†𝐇_∘, ∘∈{e,b}  is the channel Gram matrix and 𝐐=𝔼[𝐱𝐱^†] is the input signal covariance matrix. The capacity of the worst case main and eavesdropper channels can be defined as follows:

    C_w =   min_𝐖_b : 𝐇_b∈𝒮_b 
    𝐖_e : 𝐇_e∈𝒮_emax_𝐐≽0
    𝐭𝐫(𝐐) ≤ P  C(𝐖_b,𝐖_e,𝐐).

The following lower bound on the compound secrecy capacity was established in <cit.>:

    C_l =    max_𝐐≽0
    𝐭𝐫(𝐐) ≤ Pmin_𝐖_b : 𝐇_b∈𝒮_b 
    𝐖_e : 𝐇_e∈𝒮_e C(𝐖_b,𝐖_e,𝐐).

thus, the following bounds on the compound capacity holds <cit.>:

    C_l ≤ C_c ≤ C_w

The problem under consideration is first to evaluate the lower bound on the compound capacity over the uncertainty sets by solving (<ref>). To solve (<ref>) we need to identify the worst case main and eavesdropper channels (i.e. main and eavesdropper channel realizations that minimize the lower bound), and then, determining the optimal signaling scheme accordingly. Further, we need to check whether the saddle point property, in the form maxmin = minmax, for the considered class of channels.







§ COMPOUND SECRECY CAPACITY WITH KNOWN EAVESDROPPER MEAN INFORMATION


In this section we characterize the secrecy capacity of the considered compound wiretap channel when the mean information of the eavesdropper, u_e, is known exactly at the transmitter, i.e., 𝒰={u_e}. To proceed, we first need to identify the worst main channel Gram matrix, 𝐖_bw which is evaluated in the following proposition:


For the considered compound wiretap channel, for any non negative definite matrix 𝐐 and any 𝐖_e such that 𝐇_e ∈𝒮_e we have

    C(𝐖_b,𝐖_e,𝐐) ≥ C(𝐖_bw,𝐖_e,𝐐)

where 𝐖_bw = (λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^† is the worst main channel Gram matrix where (x)_+ = max (0,x).


The proof is give in Appendix <ref>.

The result of Proposition <ref> can interpreted as follows, the worst main channel is the channel that has lost all, but one, of its degrees of freedom, meanwhile, the only left degree of freedom happens with its maximum possible strength in the direction that is anti-parallel to the mean channel. An obvious direct consequence of Proposition <ref> is that the optimal input covariance, 𝐐^*, has to be of unit rank. That is because 𝐖_bw is shown to be of unit rank. Therefore, we conclude that beamforming is the optimal transmit strategy for the considered compound wiretap channel. Thus, we can restrict our analysis to a unit rank 𝐐. Next, we need to identify the worst eavesdropper Gram matrix.


For the considered compound wiretap channel, for any unit rank matrix 𝐐 with λ(𝐐) ≤ P, we have

    C(𝐖_bw,𝐖_e,𝐐) ≥ C(𝐖_bw,𝐖_ew,𝐐)

where 𝐖_ew = (λ_μ e+ 2 λ_μ e^1/2ϵ_e) u_e u_e^† + ϵ_e^2 𝐈 is the worst eavesdropper channel Gram matrix.


The proof is given in Appendix <ref>.



















The statement of proposition <ref> states that the worst eavesdropper channel is isotropic around its mean channel. This means that, the worst eavesdropper dropper channel happens with its maximum strength in the direction parallel to its mean channel. The main result of this paper is given in the following theorem. We give the compound secrecy capacity of the considered class of channel and the capacity achieving input signal covariance 𝐐^*.


The secrecy capacity for the compound wiretap channel defined in (<ref>) and (<ref>) is equal to the worst case capacity, the saddle point property holds

    C_c^*    = max_𝐐≽0
    𝐭𝐫(𝐐) ≤ Pmin_𝐖_b : 𝐇_b∈𝒮_b 
    𝐖_e : 𝐇_e∈𝒮_e C(𝐖_b,𝐖_e,𝐐) 
       = min_𝐖_b : 𝐇_b∈𝒮_b 
    𝐖_e : 𝐇_e∈𝒮_emax_𝐐≽0
    𝐭𝐫(𝐐) ≤ P C(𝐖_b,𝐖_e,𝐐)
       = C(𝐖_bw,𝐖_ew,𝐐^*)= C_w

where 𝐖_bw and 𝐖_ew are as given in Propositions <ref> and <ref> respectively. Moreover, beamforming is the optimal signaling strategy:

    𝐐^* = P q_* q_*^†,

where q_* is the eigenvector associated with the maximum eigenvalue of (𝐈_Na + P 𝐖_ew)^-1(𝐈_Na + P 𝐖_bw).












The proof is given in Appendix <ref>. 

Theorem <ref> proves that the saddle point property holds for the class of channels described in (<ref>) and (<ref>), and thus, the secrecy capacity of the compound wiretap channel is equal to the worst case capacity. Further, since the worst case main channel is of unit rank, accordingly, generalized eigenvector beamforming is known to be the optimal signaling strategy <cit.><cit.>.




 §.§ Null Steering Beamforming as an Alternative Solution

 As can be seen from Theorem <ref>, the generalized eigenvector solution requires matrix inversion which may require a considerably high computational complexity especially when the number of transmitting antennas gets large. Therefore, we introduce the null steering (NS) beamforming <cit.> as an alternative suboptimal solution. In our case, the NS beamforming matrix is given as follows:

    𝐐_ns = P q_ns q_ns^†,          q_ns= [𝐈-u_eu_e^†]u_b[𝐈-u_eu_e^†] u_b.

NS beamformer can recognized as the projection of u_b onto the null space of u_w. In particular, q_ns maximizes the gain in the direction u_b while creating a null notch in the direction u_e. Thus, it can understood as the transmission in the direction orthogonal to the eavesdropper mean channel direction maintaining the maximum possible gain in mean main channel direction. We give justifications for the choice of NS beamforming as a candidate suboptimal solution for our problem in Appendix <ref>. Extensive computer simulation provided in section <ref> reveals that NS performs extremely close to the optimal solution, yet, with no need for matrix inversion.  



    






§ COMPOUND SECRECY CAPACITY WITH EAVESDROPPER MEAN UNCERTAINTY


Unlike the previous section where u_e is assumed to be known at the transmitter, in this section we characterize the secrecy capacity of the considered compound wiretap channel when the eavesdropper mean direction, u_e, is known only to belong to the set 𝒰.

 A key step toward the characterization of the secrecy capacity is to find the worst eavesdropper channel Gram matrix 𝐖_ew with that assumption. We give 𝐖_ew in the following proposition.


For the considered compound wiretap channel with u_e ∈𝒰, for all 𝐖_b ∈𝒮_1 and any non negative definite matrix 𝐐 we have

    C(𝐖_b,𝐖_e,𝐐) ≥ C(𝐖_b,𝐖_ew,𝐐)

where 𝐖_ew = (λ_μ e+ 2 λ_μ e^1/2ϵ_e) u_* u_*^† + ϵ_e^2 𝐈 is the worst eavesdropper channel Gram matrix where:

    u_* = _u ∈𝒰 u_b^†u,



The proof is given in appendix <ref>.

Observe that, the assumption that u_e ∈𝒰 does not affect 𝐖_bw, thus, the optimal covariance is again of unit rank as in the case 𝒰={u_e}. Therefore, beamforming is still the optimal transmit strategy under the assumption u_e ∈𝒰. Again, transmission in the direction of the eigenvector of (𝐈_N_a + P 𝐖_ew)^-1(𝐈_N_a + P 𝐖_bw) is the optimal solution where 𝐖_ew as given in proposition <ref>. We give the optimal 𝐐^* in the following corollary as a direct consequence of Theorem <ref>


The saddle point property (<ref>) holds for the considered compound wiretap channel with u_e ∈𝒰. 







 Moreover, the optimal signaling scheme is zero mean Gaussian with covariance matrix given by 

    𝐐^* = P q_* q_*^†,

where q_* is the eigenvector associated with the maximum eigenvalue of (𝐈_Na + P 𝐖_ew)^-1(𝐈_Na + P 𝐖_bw), where 𝐖_bw and 𝐖_ew are as given in propositions <ref> and  <ref>, respectively.












Follows immediately by Theorem <ref> while realizing that the worst eavesdropper mean channel is in the direction u_*. 
  
Corollary <ref> extends Theorem <ref> to the case of uncertainty about the eavesdropper mean channel direction. We can conclude that, since the transmitter does not know the eavesdropper mean channel, it design its signal assuming the worst eavesdropper mean channel. Again, we note that, NS beamforming still can be introduced as an alternative solution against an eavesdropper with mean direction uncertainty. For this particular scenario, q_ns takes the same form as in (<ref>), yet, in the direction u_* instead of u_e.


§ APPLICATION TO RICIAN FADING MIMO WIRETAP CHANNEL

In this section we consider a special class of MIMO wiretap channels which is well adopted to the class of compound wiretap channel considered in this paper. We study the Rician fading MIMO wiretap channel. In a Rician fading environment, the deterministic line of sight (LOS) component causes the channel variations to be centered around a mean matrix. This mean matrix is usually of unit rank whose gain depends mainly on the distance between transmitter and receiver, array configuration and respective array orientation. In the next section we give the Rician fading MIMO channel model, and then, in section <ref> we describe the relation between Rician MIMO wiretap channel and the compound wiretap channel described in (<ref>) and (<ref>).  


 §.§ Rician Fading MIMO Channel Model
 

Wireless MIMO channel with dominant LOS component is best described by the Rician fading model. In Rician fading model, the received signal can be decomposed into two components; one is the specular component originated from the LOS path and the other is the diffuse non-line of sight component (NLOS) component. Following, we give the mathematical model for the considered Rician MIMO wiretap channel with the subscript ∘∈{b,e} denotes the legitimate and eavesdropper channels respectively. 

    𝐇_∘ = 𝐇_∘^los + 𝐇_∘^nlos,

where 𝐇_∘^los and 𝐇_∘^nlos represents the LOS and NLOS components respectively and

    𝐇_∘^los   = √(γ_∘^2 k_∘1+k_∘)Ψ_∘,    𝐇_∘^nlos   = √(γ_∘^21+k_∘)𝐇̂_∘,

where γ_∘ quantifies the channel strength for both receiver and eavesdropper, k_∘ is the Rician factor that facilitates the contribution of the LOS component to the received signal, Ψ_∘ =𝐚(θ_∘)𝐚^†(ϕ_∘), 𝐚(θ_∘) and 𝐚(ϕ_∘) are the antenna array spatial signatures (steering vectors) at receiver (eavesdropper) and transmitter respectively, θ_∘ and ϕ_∘ are the angle of arrival (AoA) and angle of departure (AoD) of the transmitted signal respectively. Note that, AoD, ϕ, represents the azimuth angle of the receiver (eavesdropper) with respect to the transmitter antenna array. Meanwhile, 𝐇̂_∘ represents the channel coefficients matrix for the NLOS signal component.



 §.§ Relation to the Compound Wiretap Channel
 

In the previous section we gave the mathematical description of the Rician fading MIMO wiretap channel. In this section we highlight the equivalence relation between this class of MIMO wiretap channel and the compound wiretap channel studied in this paper. Recalling the definition of the compound wiretap channel given in (<ref>) and (<ref>), it is straight forward to see that the following analogies hold:

    𝐇_μ∘   ⇔𝐇_∘^los,    λ_μ∘   ⇔N_aN_∘γ_∘^2 k_∘1+k_∘, 
    
    v_∘   ⇔𝐚(θ_∘),     u_∘   ⇔𝐚(ϕ_∘), 
    Δ𝐇_∘   ⇔𝐇_∘^nlos,    ϵ_∘^2            ⇔N_∘γ_∘^21+k_∘.
   
Observe that in the settings of Rician fading MIMO wiretap channel, eavesdropper eigen direction, u_e, corresponds to the physical direction (in azimuth plane) of the eavesdropper. Therefore, the assumptions that u_e is known at the transmitter corresponds to the scenario in which the transmitter has a prior knowledge about the eavesdropper azimuth direction. Whereas, the assumption that u_e ∈𝒰 corresponds to the scenario in which the transmitter does not know exactly the azimuth direction of the eavesdropper, meanwhile, it knows that the eavesdropper, if any, has a restricted access to the communication area. That is, it can only approach the transmitter up to a certain distance and the receiver up to a certain azimuth direction.


 §.§ Numerical results


For the sake of numerical evaluation, we use the established equivalence relation between the considered compound wiretap channel and the MIMO channel with Rician fading. We compare the performance of the optimal solution to our proposed NS beamforming solution. Given the azimuth direction of both eavesdropper and legitimate receiver, another two possible transmission schemes may come to mind. First, beamforming toward the intended receiver which is well known as MRT. Second, creating a deep null notch in the direction of the eavesdropper which is well known as ZF. To evaluate the value of the mean information, we provide numerical simulation for an eavesdropper having the same parameters as of the main receiver, i.e. N_a=N_b=N_e=4 and γ_b=γ_e=1, for the same Rician k factor, and thus, we have ϵ_b=ϵ_e and λ_μ b=λ_μ e. We assume uniform linear array configuration at all nodes with antenna spacing of half wavelength. Whereas, we assume the receiver and eavesdropper not to share the same azimuth direction, ϕ_b = 25^∘ and ϕ_e = 60^∘. As can be seen in Fig. (<ref>), we compare the achievable secrecy rate for the NS, MRT and ZF beamforming approaches against the optimal solution for different values of Rician k factor. Simulation results shows that NS beamforming performs extremely close to optimal and outperforms both MRT and ZF over the entire SNR range. Although It may be seen that NS performance matches the optimal solution, we provide a zoom in picture at the upper left corner of Fig. (<ref>) to show that the acheivable rate by NS beamforming is slightly below the secrecy capacity of the channel. It is observed that it maintains a small gap to capacity in order of 10^-4 over the entire SNR range for all values of k.       



§ DISCUSSION AND FUTURE WORK

Compound MIMO wiretap channel with double sided uncertainty is considered under channel mean information model. The worst case main channel is shown to be anti-parallel to the channel mean information resulting in an overall unit rank channel. Further, the worst eavesdropper dropper channel is shown to be isotropic around its mean information. Accordingly, generalized eigenvector beamforming is shown to be the optimal signaling strategy. The saddle point property is shown to hold under mean information model, and thus, compound secrecy capacity equals to the worst case capacity over the class of uncertainty. Further, as the generalized eigenvector solution requires matrix inversion, we introduced NS beamforming, that is, transmission in the direction orthogonal to the eavesdropper mean channel direction maintaining the maximum possible gain in mean main channel direction, as an alternative suboptimal solution. Extensive computer simulation revealed that NS performs extremely close to the optimal solution. It also verified the superiority of NS beamforming to both MRT and ZF approaches over the entire SNR range.
It is worth noting that, the results for the compound wiretap channel are too conservative in general and, consequently, so is the result of this paper. That is due to the assumption that channel realizations remain constant over the entire transmission duration leading us to the worst case optimization. While this assumption simplifies the mathematical analysis, it does not usually hold in practice. More interesting scenario is to consider the compound wiretap channel with channel realizations allowed to change, possibly random, during transmission duration. 





 















IEEEtran
















§ PROOF OF PROPOSITION <REF>


We observe that

    C(𝐖_b,𝐖_e,𝐐)    = log(𝐈_N_a + 𝐖_b𝐐)(𝐈_N_a + 𝐖_e𝐐)
       (a)=∑_i=1^N_alog(1 + λ_i(𝐖_b𝐐)) 
       - log(𝐈_N_a + 𝐖_e𝐐) 
       (b)=∑_i=1^N_alog(1 + σ_i^2((𝐇_μ b+Δ𝐇_b)𝐐^1/2)) 
       - log(𝐈_N_a + 𝐖_e𝐐) 
       (c)≥log(1 + (λ_μ b^1/2 - ϵ_b)_+^2λ_1(𝐐)) 
       - log(𝐈_N_a + 𝐖_e𝐐) 
       (d)= C(𝐖_bw,𝐖_e,𝐐)

where (a) follows from determinant properties and (b) follows by recognizing that λ_i(𝐖_b𝐐) = σ_i^2((𝐇_μ b+Δ𝐇_b)𝐐^1/2) where σ_i(𝐀) is the i^th singular value of 𝐀. Meanwhile, (c) follows from the singular value inequality in Lemma 7 in <cit.>, that is, σ_i^2((𝐇_μ b+Δ𝐇_b)𝐐^1/2) ≥ (σ_i(𝐇_μ b) - σ_1(Δ𝐇_b))_+ λ_i(𝐐) and removed the summation due to the fact that σ_1(𝐇_μ b) = λ_μ b^1/2 and σ_i(𝐇_μ b) = 0   ∀ i>1.


§ PROOF OF PROPOSITION <REF>


It can be seen that

    C(𝐖_bw,𝐖_e,𝐐)     (a)=log(𝐈_N_a + (λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^†𝐐)(𝐈_N_a + 𝐖_e𝐐)
       (b)=log1 + λ((λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^†𝐐)1 + σ^2((𝐇_μ e + Δ𝐇_e)𝐐^1/2)
       (c)=log1 + λ((λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^†𝐐)1 + σ^2(𝐇_μ e𝐐^1/2 + Δ𝐇_e 𝐐^1/2)
       (d)≥log1 + λ((λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^†𝐐)1 + (σ(𝐇_μ e𝐐^1/2) + σ (Δ𝐇_e 𝐐^1/2))^2


where (a) follows by direct substitution in (<ref>) with 𝐖_bw given in proposition <ref>, (b) follows since 𝐐 is of unit rank and that λ(𝐖_b𝐐) = σ^2((𝐇_μ b+Δ𝐇_b)𝐐^1/2) and (c) is straightforward. Meanwhile, the upper bound in (d) follows since σ(A+B) ≤σ(A) + σ(B) for unit rank matrices A and B where the inequality holds with equality when A and B have the same singular vectors. Therefore, the inequality in (d) established with equality if 𝐇_μ e  and Δ𝐇_e have the same singular vectors. Let us write 𝐇_μ e = VΣ_μ eU^† where the first columns of V and U are v_e and u_e respectively, and Σ_μ e = diag{λ_μ e^1/2,0,..,0}. Hence to establish (d) with equality, Δ𝐇_e need to have V and U as its right and singular vectors respectively. Consequently, we can write 𝐇_e= VΣ_eU^†, and hence, 𝐖_e = UΣ_e^2U^†. But we have that Σ_μ e≼ϵ_e 𝐈, accordingly, Σ_e ≼Σ_ew where Σ_ew = diag{λ_μ e^1/2+ϵ_e, ϵ_e,...,ϵ_e}. Noting that the function log(𝐈+𝐖𝐐) is monotonically increasing in 𝐖, we conclude that 𝐖_ew = U Σ_ew^2U^†. However, we can write 𝐖_ew as (λ_μ e+ 2 λ_μ e^1/2ϵ_e) u_e u_e^† + ϵ_e^2 𝐈 as required. 



§ PROOF OF THEOREM <REF>



To establish the saddle point property, we give a proof similar to the one given in Theorem 6 in <cit.> while keeping in mind the difference between the compound wiretap channel defined there with the one defined in (<ref>) and (<ref>). Let 𝐐^* be the optimal solution for the left hand side max-min problem, we observe that, to show the saddle point property in (<ref>) is equivalent to show that <cit.>:

    C(𝐖_bw,𝐖_ew,𝐐)    (a)≤ C(𝐖_bw,𝐖_ew,𝐐^*) 
       (b)≤ C(𝐖_b,𝐖_e,𝐐^*),

where 𝐖_ew and 𝐖_bw are as given in propositions <ref> and <ref> respectively. Note that (a) follows since 𝐐^* is optimal for 𝐖_b = 𝐖_bw and 𝐖_e = 𝐖_ew. Now we write:

    C(𝐖_bw,𝐖_ew,𝐐^*)    (a)=log(1 + σ_i^2((λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^†𝐐^*1/2)) 
           - log(𝐈+𝐖_ew𝐐^*)  
       (b)≤∑_i=1^N_alog(1 + σ_i^2((𝐇_μ b+Δ𝐇_b)𝐐^*1/2)) 
           -log(𝐈+𝐖_ew𝐐^*)  
       (c)= C(𝐖_b,𝐖_ew,𝐐^*) 
       (d)≤ C(𝐖_b,𝐖_e,𝐐^*)

where (a) follows by direct substitution by 𝐖_bw, meanwhile, (b) and (c) follow from (<ref>) and we used (<ref>) to write (d). Since the difference channel is, at most, of unit rank, then, beamforming toward the eigenvector associated with the largest eigenvalue of (𝐈_Na + P 𝐖_ew)^-1(𝐈_Na + P 𝐖_bw) follows by corollary 1 in <cit.> and Theorem 6 in <cit.>.



§ PROOF OF PROPOSITION <REF>


Observe that 𝐇_μ e^†𝐇_μ e = λ_μ euu^† for some u ∈𝒰. Thus, the result of proposition <ref> can be established in a fashion similar to the proof of proposition <ref>, however, by realizing that

    min_𝐖_e : 𝐇_e ∈𝒮_e C(   𝐖_bw,𝐖_e(u),𝐐) = 
       min_u ∈𝒰 C(𝐖_bw,(λ_μ e+ 2 λ_μ e^1/2ϵ_e) u u^† + ϵ_e^2 𝐈,𝐐).

Thus, taking the minimum of (<ref>) over u and dropping the constraint u ∈𝒰, the minimum is attained when u=u_b. Meanwhile, with the constraint into action, the minimum is attained at u_* ∈𝒰 which has the minimum distance to u_b. Equivalently,

    u_*    = _u ∈𝒰u_b-u
       = _u ∈𝒰  u_b^† u
 
which agree with (<ref>).




§ JUSTIFICATION FOR NS BEAMFORMING


To understand the motivation behind introducing NS beamforming as an alternative solution, we write 𝐐 = P qq^†. Now, it can be seen that:

    C_c^*    = max_𝐐≽0
    𝐭𝐫(𝐐) ≤ P C(𝐖_bw,𝐖_ew,𝐐) 
       (a)=max_𝐐≽0
    𝐭𝐫(𝐐) ≤ Plog(𝐈+𝐖_bw𝐐)(𝐈+𝐖_ew𝐐)
       (b)=max_𝐐≽0
    𝐭𝐫(𝐐) ≤ Plog(𝐈+(λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^†𝐐)(𝐈+(λ_μ e+ 2 λ_μ e^1/2ϵ_e) u_e u_e^† + ϵ_e^2 𝐈 )𝐐)
       (c)=max_q 
    q=1log(𝐈+P(λ_μ b^1/2 - ϵ_b)_+^2 u_bu_b^†qq^†)(𝐈+P((λ_μ e+ 2 λ_μ e^1/2ϵ_e)u_eu_e^†qq^† + ϵ_e^2qq^† ))
       (d)=max_q 
    q=1log1 +P(λ_μ b^1/2 - ϵ_b)_+^2 u_b^†q1+P((λ_μ e+ 2 λ_μ e^1/2ϵ_e)u_e^†q + ϵ_e^2))

where (a) follows by direct substitution with 𝐖_bw and 𝐖_ew in (<ref>), (b) follows by substituting the values of 𝐖_bw and 𝐖_ew. Meanwhile, in (c) we used that 𝐐 is of unit rank and thus it has only one eigenvalue equals to P and its corresponding eigenvector q, also, we have removed the power constraint by introducing the constraint q=1. Since both of the numerator and the denominator are of unit rank, d follows from (c) by substituting the only eigenvalue of both of them.  Now observe that, the choice of q does not affect the eigenvalue of the matrix ϵ_e^2qq^†, rather, it do affect the eigenvalues of the other matrices. Clearly our objective is to find q that simultaneously maximizes the numerator and minimizes (optimally, nulling out) the denominator in (<ref>(d)).  The optimal q_* that maximizes C_c is given by Theorem <ref>. However, we note that q_ns in (<ref>) is the optimal solution to the following optimization problem

    max_q
    q=1 <q^†,u_b> 
    Subject to <q^†,u_e> = 0,
 
i.e., beamforming in the direction q_ns maximizes the gain in the direction u_b while creating a null notch in the direction u_e.


