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Successive optimization TomlinsonHarashima precoding strategies for physicallayer security in wireless networks
EURASIP Journal on Wireless Communications and Networking volume 2016, Article number: 259 (2016)
Abstract
In this paper, we propose novel nonlinear precoders for the downlink of a multiuser MIMO system in the existence of multiple eavesdroppers. The proposed nonlinear precoders are designed to improve the physicallayer secrecy rate. Specifically, we combine the nonlinear successive optimization TomlinsonHarashima precoding (SOTHP) with the generalized matrix inversion (GMI) technique to maximize the physicallayer secrecy rate. For the purpose of comparison, we examine different traditional precoders with the proposed algorithm in terms of secrecy rate as well as bit error rate (BER) performance. We also investigate simplified generalized matrix inversion (SGMI) and latticereduction (LR) techniques in order to efficiently compute the parameters of the precoders. We further conduct computational complexity and secrecyrate analysis of the proposed and existing algorithms. In addition, in the scenario without knowledge of the channel state information (CSI) to the eavesdroppers, a strategy of injecting artificial noise (AN) prior to the transmission is employed to enhance the physicallayer secrecy rate. Simulation results show that the proposed nonlinear precoders outperform existing precoders in terms of BER and secrecyrate performance.
1 Introduction
Data security in wireless systems has been traditionally dominated by encryption methods such as the Data Encryption Standard (DES) and the Advanced Encryption Standard (AES) [1]. However, these existing encryption algorithms suffer from high complexity and high latency. Furthermore, development in computing power also brings great challenges to existing encryption techniques. Therefore, the development of techniques that are capable of achieving secure transmission under high computingpower scenario with low complexity have become an important research topic.
From the viewpoint of information theory, Shannon established the theorem of cryptography in his seminal paper [2]. Wyner has subsequently posed the AliceBobEve problem and described the wiretap transmission system [3]. Furthermore, the system discussed in [3] suggests that physicallayer security can be achieved in wireless networks. Later on, another study reported in [4] proved that secrecy transmission is achievable even under the situation that the eavesdropper has a better channel than the desired user in a statistical sense. Furthermore, the secrecy capacity for different kinds of channels, such as the Gaussian wiretap channel and the multiinput multioutput (MIMO) wiretap channel, have been studied in [5, 6]. In some later works [7, 8], it has been found that the secrecy of the transmission can be further enhanced by adding artificial noise to the system.
1.1 Prior and related work
In recent years, precoding techniques, which rely on knowledge of channel state information (CSI), have been widely studied in the downlink of multiuser MIMO (MUMIMO) systems. Linear precoding techniques such as zeroforcing (ZF), minimum mean square error (MMSE), and block diagonalization (BD) have been introduced and studied in [9–11]. Furthermore, nonlinear precoding techniques like TomlinsonHarashima precoding (THP) [12] and vector perturbation (VP) precoding [13] have also been reported and investigated. In the previous mentioned works, the implementation of linear or nonlinear precoding techniques at the transmitter are considered with perfect knowledge of CSI to the users. In the scenario without knowledge of CSI to the eavesdroppers, one technique which is effective in improving the secrecy rate of the downlink of MUMIMO systems is the application of artificial noise (AN) at the transmitter [7]. Several criteria or strategies applying AN to wireless systems have been introduced in [14, 15]. In particular, the approaches reported in [8] have been applied to the downlink of MUMIMO systems. Apart from the studies in precoding techniques, there are also some works that introduce latticereduction (LR) strategies [16, 17]. The LR strategies are also implemented prior to the transmission, and it has been proved that the LRaided system can achieve full diversity in the downlink of MUMIMO systems.
1.2 Motivation and contributions
Prior work on precoding for physicallayer security systems has been heavily based on [7, 8], which can effectively improve the secrecy rate of wireless systems. However, it is well known in the wireless communications literature that nonlinear precoding techniques can outperform linear approaches. In particular, nonlinear precoding techniques require lower transmit power than linear schemes and can achieve higher sum rates. However, work on nonlinear precoding for physicallayer security in wireless systems is extremely limited even though there is potential to significantly improve the secrecy rate of wireless systems. The motivation for this work is to develop and study nonlinear precoding algorithms for MUMIMO systems that can achieve a secrecy rate higher than that obtained by linear precoders as well as a lower transmit power requirement and an improved bit error rate (BER) performance.
In our work, we develop and study successive optimization TomlinsonHarashima precoding (SOTHP) algorithms based on the generalized matrix inversion approach reported in [11]. Specifically, the proposed nonlinear precoders exploit both successive interference cancelation, lattice reduction, and block diagonalization, which can impose orthogonality between the channels of the desired users. This combined approach has not been considered previously in the literature and has the potential of achieving a higher secrecy rate than existing nonlinear and linear precoding algorithms as well as an improved BER performance as compared to prior art. The major contributions in our paper are summarized as follows:

A novel nonlinear precoding technique, namely SOTHP+GMI, is proposed for the downlink of MUMIMO networks in the presence of multiple eavesdroppers.

The proposed SOTHP+GMI algorithm combines the SOTHP with the generalized matrix inversion (GMI) technique to achieve a higher secrecy rate.

The proposed SOTHP+GMI precoding algorithm is extended to a simplified GMI (SGMI) version which aims to reduce computational complexity of the SOTHP+GMI algorithm.

An LR strategy is combined with the aforementioned SGMI version proposed algorithm, and this socalled LRaidedversion algorithm achieves full receive diversity.

An analysis of the secrecy rate achieved by the proposed nonlinear precoding algorithms is carried out along with an assessment of their computational complexity cost.

When different power is allocated to generate the artificial noise, an analysis of the power ratio which can achieve the optimal value in terms of secrecy rate is given
The rest of this paper is organized as follows. We begin in Section 2 by introducing the system model and the performance metrics. A brief review of the standard SOTHP algorithm is included in Section 3. In Section 4, we present the details of the proposed SOTHP+GMI, SOTHP+SGMI, and LRSOTHP+SGMI precoding algorithms. Next, in Section 5, the analysis of secrecy rate and the computational complexity of the precoding algorithms are carried out. In Section 6, numerical evaluation is conducted to show the advantage of the proposed precoding algorithms. Finally, some concluding remarks are given in Section 7.
1.3 Notation
Bold uppercase letters \({\boldsymbol {A}}\in {\mathbb {C}}^{M\times N}\) denote matrices with size M×N and bold lowercase letters \({\boldsymbol {a}}\in {\mathbb {C}}^{M\times 1}\) denote column vectors with length M. Conjugate, transpose, and conjugate transpose are represented by (·)^{∗}, (·)^{T} and (·)^{H}, respectively; I _{ M } is the identity matrix of size M×M; diag{a} denotes a diagonal matrix with the elements of the vector a along its diagonal; and \(\mathcal {CN}(0,{\sigma _{n}^{2}})\) represents complex Gaussian random variables with i.i.d entries with zero mean and \({\sigma _{n}^{2}}\) variance.
2 System model and performance metrics
In this section, we introduce the system model of the downlink of the MUMIMO network under consideration. The performance metrics used in the assessment of the proposed and existing techniques are also described.
2.1 System model
Consider a MUMIMO downlink wireless network consisting of one transmitter or Alice at the access point, T users or Bob, and K eavesdroppers or Eve at the receiver side as shown in Fig. 1. The transmitter is equipped with N _{ t } antennas. Each user and each eavesdropper node are equipped with N _{ r } and N _{ k } receive antennas, respectively. In this system, we assume that the eavesdroppers do not jam the transmission and the channel from the transmitter to each user or eavesdropper follows a flatfading channel model. The quantities \({\boldsymbol {H}}_{r}\in {\mathbb {C}}^{N_{r}\times N_{t}}\) and \({\boldsymbol {H}}_{k}\in {\mathbb {C}}^{N_{k}\times N_{t}}\) denote the channel matrix of the ith user and kth eavesdropper, respectively. Following [18], the number of antennas should satisfy \(N_{t}^{\text {total}} \geqslant T \times N_{r}\). During the transmission, N _{ t }=M×N _{ r } antennas at the transmitter are activated to perform the precoding procedure. In other words, the precoding matrix is assumed here for convenience to be always a square matrix.
We use the vector \({\boldsymbol {s}}_{r}\in {\mathbb {C}}^{N_{r}\times 1}\) to represent the data symbols to be transmitted to user r. An artificial noise (AN) can be injected before the data transmission to enhance the physicallayer secrecy. We use the vector \({\boldsymbol {s}}_{r}'\in {\mathbb {C}}^{m\times 1}\) with \(m \leqslant (N_{t}^{\text {total}}T \times N_{r})\) to denote the independently generated jamming signal. Assume the transmit power of user r is E _{ r }, and 0<ρ<1 is the power fraction devoted to the user. Then, the power of the user and the jamming signal can be respectively expressed as \(E[{\boldsymbol {s}_{r}^{H}}\boldsymbol {s}_{r}]=\rho E_{r}\) and \(E[{\boldsymbol {s}_{r}^{\prime H}}\boldsymbol {s}_{r}^{\prime }]=(1\rho) E_{r}\). Finally, the signal after precoding can be expressed as
where the quantities \({\boldsymbol {P}}_{r}\in {\mathbb {C}}^{N_{t}\times N_{r}}\) and \({\boldsymbol {P}}_{r}'\in {\mathbb {C}}^{N_{t}\times m}\) are the corresponding precoding matrices. Here, we take zeroforcing precoding as an instance. Given the total channel matrix \(\boldsymbol {H}=[\boldsymbol {H}_{1}^{T}\quad {\boldsymbol {H}_{2}^{T}}\quad \cdots \quad {\boldsymbol {H}_{r}^{T}}\quad \cdots \quad {\boldsymbol {H}_{M}^{T}}]^{T}\), the total precoding matrix can be obtained as P ^{ZF}=H ^{H}(H H ^{H})^{−1}. The precoding matrix P ^{ZF} can be expanded to P ^{ZF}=[P _{1} P _{2} ⋯ P _{ r } ⋯ P _{ M }]. Simultaneously, the precoding matrix P r′ can be generated from the null space of the rth user channel H _{ r } by singular value decomposition (SVD) [8]. As a result, we have H _{ r } P r′=0, which means the jamming signal does not interfere with the user’s signal. The received data for each user or eavesdropper can be described by
where \(\beta _{r}=\sqrt {\frac {E_{r}}{\boldsymbol {P}_{r}+\boldsymbol {P}_{r}'}}\) is used to ensure that the transmit power after precoding remains the same as the original transmit power E _{ r } for user r.
2.2 Secrecy rate and other relevant metrics
In this subsection, we describe the main performance metrics used in the literature to assess the performance of precoding algorithms.
2.2.1 Secrecy rate and secrecy capacity
According to [3], the level of secrecy is measured by the uncertainty of Eve about the message R _{ e } which is called the equivocation rate. With the total power equal to E _{ s }, the maximum secrecy capacity C _{ s } for the MIMO system without AN is expressed as [6]
where the quantity Q _{ s } is a positivedefined covariance matrix associated with the signal after precoding. E_{ s } is the total transmit power. \({\boldsymbol {H}}_{ba} \in \mathcal {CN}(0,1)\) and \({\boldsymbol {H}}_{ea} \in \mathcal {CN}(0,1/m)\) represent the channel to the users and eavesdroppers, respectively. Here, \(m=\frac {\sigma _{ea}^{2}}{\sigma _{ba}^{2}}\) represents the gain ratio between the main and wiretap channels. The secrecy capacity is defined as the maximization of the difference between two mutual informations. However, the channels are usually not perfectly known in reality. This situation is known as the imperfect channel state information (CSI) case in [7], which we will address in our studies.
2.2.2 Computational complexity
According to [18], nonlinear precoding techniques can approach the maximum channel capacity with high computational complexity. High complexity of the algorithm directly leads to a high cost of power consumption. In our research, however, novel nonlinear precoding algorithms with reduced complexity are developed.
2.2.3 BER performance
Ideally, we would like the users to experience reliable communication and the eavesdroppers to have a very high BER (virtually no reliability when communicating). The algorithm is supposed to achieve high diversity for the MIMO system.
3 Review of the SOTHP algorithm
In this section, a brief review of the conventional successive optimization THP (SOTHP) in [9] is given. The general structure of the SOTHP algorithm is illustrated in Fig. 2, and its main implementation steps are introduced in the following.
In Fig. 2, a modulo operation M(·) which is defined in [19] is employed to fulfill the SOTHP algorithm. Based on [18], THP can be equivalently implemented in a successive block diagonalization manner. In particular, the precoding matrix is given by
where \(\tilde {\boldsymbol {V}_{r}}^{(0)}\in {\mathbb {C}}^{N_{t}\times N_{r}}\) is the nullifying matrix of the rth user’s channel and V _{eff} is a unitary matrix of the corresponding effective channel, and the demodulation matrix of the rth user is chosen as \({\boldsymbol {D}_{r}}={\boldsymbol {U}_{\text {eff}}^{H}}\), where \({\boldsymbol {U}_{\text {eff}}^{H}}\) is also obtained from the effective channel. Given a channel matrix \(\tilde {\boldsymbol {H}_{r}}=[\tilde {\boldsymbol {H}_{1}}^{T}\quad \tilde {\boldsymbol {H}_{2}}^{T}\quad \cdots \quad \tilde {\boldsymbol {H}_{r1}}^{T}\quad \tilde {\boldsymbol {H}_{r+1}}^{T} \cdots \quad \tilde {\boldsymbol {H}}_{T}^{T}]^{T}\), \(\tilde {\boldsymbol {V}_{r}}^{(0)}\) can be obtained by the SVD operation \(\tilde {\boldsymbol {H}_{r}}={\tilde {\boldsymbol {U}}_{r}}{\tilde {\boldsymbol {\Sigma }}_{r}}[{\tilde {\boldsymbol {V}}_{r}}^{(1)} {\tilde {\boldsymbol {V}}_{r}}^{(0)}]^{H}\). Based on \(\tilde {\boldsymbol {V}_{r}}^{(0)}\), an effective channel can be calculated, and with a second SVD operation \({\boldsymbol {H}}_{\text {eff}}={\boldsymbol {H}}_{r}\tilde {\boldsymbol {V}_{r}}^{(0)}={{\boldsymbol {U}}_{\text {eff}}} {{\boldsymbol {\Sigma }}_{\text {eff}}}{{\boldsymbol {V}}_{\text {eff}}}^{H}\), we are capable of getting V _{eff} and \({\boldsymbol {U}_{\text {eff}}^{H}}\). For each iteration, the SOTHP algorithm selects the user with maximum capacity from the remaining users and processes it first. The selection criterion is described as
where C _{max,r } denotes the maximum capacity of the rth user and C _{ r } is the capacity considering the interference from the other users. If we assume there is no interference from other users and the capacity can be achieved by the SVD procedure, we have
In the scenario considering the interference from the other users, the BD decomposition is implemented on the channels of the remaining users in each iteration:
Therefore, the filters for the SOTHP algorithm can be obtained as
It is worth noting that F in (9) and D in (10) are calculated in the reordered way according to Eq. (5) and the scaling matrix \(\boldsymbol {G}=\text {diag}\left ([\boldsymbol {D}\boldsymbol {H}\boldsymbol {F}]_{ii}^{1}\right)\).
4 Proposed precoding algorithms
In this section, we present three nonlinear precoding algorithms SOTHP+GMI, SOTHP+SGMI, and LRSOTHP+SGMI for the downlink of MUMIMO systems and a selection criterion based on capacity is devised for these algorithms. We then derive filters for the three proposed precoding techniques, which are computationally simpler than SOTHP.
According to [18], the conventional SOTHP algorithm has the advantage of improving the BER and the sumrate performances; however, the complexity of this algorithm is high due to the successive optimization procedure and the multiple SVD operations. In [20], an approach called generalized MMSE channel inversion (GMI) is developed to overcome the noise enhancement drawback of BD caused by its focus on the suppression of multiuser interference. Later in [21], it has been shown that the complete suppression of multiuser interference is not necessary and residual interference is so small that it cannot affect the sumrate performance. This approach is called simplified GMI (SGMI). The proposed algorithms are inspired by dirty paper coding (DPC) [22] and other nonlinear precoding techniques [18, 23, 24] which have been investigated for the downlink of MUMIMO systems.
4.1 SOTHP+GMI algorithm
The proposed SOTHP+GMI algorithm mainly focuses on achieving high secrecyrate performance with lower complexity than the conventional SOTHP algorithm. In the conventional SOTHP algorithm, the precoding matrix as well as the receive filters are obtained with (4) using the BD algorithm without considering noise enhancement. In [20], the GMI scheme uses the QR decomposition to decompose the MMSE channel inversion \(\boldsymbol {\bar {H}} \in {\mathbb {C}}^{N_{t}\times {TN}_{r}}\) as expressed by
where \(\bar {\boldsymbol {H}_{r}}\in {\mathbb {C}}^{N_{t}\times N_{r}}\), \(\bar {\boldsymbol {Q}_{r}}^{(0)} \in {\mathbb {C}}^{N_{t}\times N_{t}}\) is an orthogonal matrix and \(\bar {\boldsymbol {R}_{r}}\in {\mathbb {C}}^{N_{t}\times N_{r}}\) is an upper triangular matrix. In (12), the noise is taken into account. As a result, the generation of the precoding matrix will mitigate the noise enhancement. When the GMIgenerated precoding matrix is used to calculate the channel capacity with (8), the reduced noise contributes to the increase of secrecy rate. Also with (13), the QR decomposition reduces the computational complexity as compared with the conventional SOTHP algorithm implementing the SVD decomposition. To completely mitigate the interference, a transmitcombining matrix T _{ r } given in [20] is applied to \(\bar {\boldsymbol {Q}_{r}}^{(0)}\). Once we have \(\bar {\boldsymbol {Q}_{r}}^{(0)}\) and T _{ r }, we can write the relation
Then, the precoding matrix and the receive filter for the GMI scheme are given by
where \({\boldsymbol {P}_{\text {GMI}}}\in {\mathbb {C}}^{N_{t}\times N_{t}}\) and \({\boldsymbol {M}_{\text {GMI}}}\in {\mathbb {C}}^{N_{t}\times N_{t}}\). The details of the proposed SOTHP+GMI algorithm to obtain the precoding and receive filter matrices are given in the table of Algorithm 1.
4.2 SOTHP+SGMI algorithm
Further development on SOTHP+GMI with complexity reduction leads to a novel SOTHP+SGMI algorithm. A simplified GMI (SGMI) has been developed in [21] as an improvement of the original RBD precoding in [9]. This is known as SGMI. In (14), a transmitcombining matrix is applied to achieve complete interference cancelation between different users. In this case, the interference will not be completely mitigated, resulting in a slight decrease of the sum rate even though the complexity will have a significant reduction [21]. Here, we incorporate the SGMI technique into an SOTHP scheme and devise the SOTHP+SGMI algorithm. The transmit and receive filters of the proposed SOTHP+SGMI algorithm are described by
where \({\boldsymbol {P}_{\mathrm {SGMI}}}\in {\mathbb {C}}^{N_{t}\times N_{t}}\) and \({\boldsymbol {M}_{\textrm {SGMI}}}\in {\mathbb {C}}^{N_{t}\times N_{t}}\).
With reduced computational complexity, the SOTHP+SGMI algorithm is capable of achieving better secrecyrate performance especially at lower signaltonoise ratio (SNR). The detailed SGMI procedure implemented in the proposed SOTHP+SGMI algorithm is shown in Algorithm 2. Cooperated with Algorithm 1, the precoding and receive filter matrices can be obtained.
4.3 LRSOTHP+SGMI algorithm
The development in linear algebra contribute to the latticereduction technique application in wireless networks. According to the study in [16], a basis change may lead to improved performance as corroborated by latticereduction techniques. The more correlated the columns of channel H, the more significant the improvements will be. To achieve full diversity of the system, with complex latticereduction (CLR) algorithm [25], the LRtransformed channel for the rth user is obtained as
where \({\boldsymbol {H}_{\text {red}_{r}}}\in {\mathbb {C}}^{N_{r}\times N_{t}}\) is the transposed reduced channel matrix. The quantity \({\boldsymbol {L}_{r}} \in {\mathbb {C}}^{N_{r}\times N_{r}}\) is the transform matrix generated by the CLR algorithm. Note that the transmit power constraint is satisfied since M _{ r } is a unimodular matrix.
Compared to the conventional SOTHP algorithm, the latticereduced channel matrix \(\phantom {\dot {i}\!}\boldsymbol {H}_{\text {red}_{n}}\) is employed in the conventional SGMI algorithm. The details of the LRaided SGMI procedure are given in Algorithm 3. Cooperated with Algorithm 1, we can complete the calculation of the precoding and receive filter matrices.
5 Analysis of the algorithms
In this section, we develop an analysis of the secrecy rate of the proposed precoding algorithms along with a comparison of the computational complexity between the proposed and existing techniques.
5.1 Computational complexity analysis
According to [25], it can be calculated that the cost of the QR in floatingpoint operations per second (FLOPS) is 22.4 % lower than BD. The results shown in Table 1 indicate that the complexity is reduced by about 22.4 % by the proposed SOTHP+GMI compared with the conventional SOTHP calculated in the same way. Based on the proposed SOTHP+GMI algorithm, further complexity reduction can be achieved by SOTHP+SGMI and the complexity is about 34.4 % less than that of the conventional SOTHP algorithm.
Figure 3 shows the required FLOPS of the proposed and existing precoding algorithms. Linear precoding gives lower computational complexity, but the BER performance is worse than nonlinear ones. The three proposed algorithms show an advantage over the conventional SOTHP algorithm in terms of complexity. Among all the three proposed algorithms, SOTHP+SGMI has the lowest complexity followed by LRSOTHP+SGMI. SOTHP+GMI requires the highest complexity. In Fig. 3, the SOTHP+SGMI algorithm has similar performance as the LRSOTHP+SGMI algorithm. Although the latticereduction procedure is implemented in the LRSOTHP+SGMI, the matrices produced in the lattice reduction can be also used in the SGMI algorithm so that the complexity of the LRSOTHP+SGMI algorithm is just slightly higher than SOTHP+SGMI.
5.2 Secrecyrate analysis
Theorem 1
In fullrank MUMIMO systems with perfect knowledge of CSI, the proposed algorithms are capable of achieving a high secrecy rate, and in the highSNR regime (i.e., E _{ s }→∞), the secrecy rate will converge to \(C_{\text {sec}}^{E_{s}\rightarrow \infty }\) which is given as (21),
Proof
Under the conditions
and based on (3), we can have the secrecy capacity expressed as (24). If \(\Gamma (\boldsymbol {P})={(\boldsymbol {H}_{ea}\boldsymbol {P} \boldsymbol {P}^{H}\boldsymbol {H}_{ea}^{H})^{1}}{(\boldsymbol {H}_{ba}\boldsymbol {P} \boldsymbol {P}^{H}} {\boldsymbol {H}_{ba}^{H})}\), (24) can be converted to (25).
In (25), (26), and (27), P is the precoding matrix derived from the legitimate users’ channel. With \(\boldsymbol {Q}_{s}=E[\boldsymbol {x}_{s} {\boldsymbol {x}_{s}^{H}}]=E[\boldsymbol {P} \boldsymbol {s} \boldsymbol {s}^{H} \boldsymbol {P}^{H}]\), E[s s ^{H}]=E _{ s }, and P P ^{H}=I, we can have
Then
In (25), the expectation value is given as (31). Substituting (29) into (31), the formula can be expressed as (33).
According to (33), in the highSNR regime and when SNR→∞, E _{ s }→∞, S A→I. Then, the secrecy rate expressed in (25) will result in (34).
To satisfy the power constraint, we always have E[P P ^{H}]=I, then the secrecy rate C _{sec} will converge to a constant, that is,
This completes the proof. □
In the following, the percentage of the injected artificial noise power is set to 40 % of the total transmit power. The percentage of the artificial noise power is determined to achieve an optimal result in terms of the secrecy rate. The details of the derivation are expressed in the Appendix. When AN is added during the transmission, Eq. (3) can be transformed to
To assess the influence of different channel gain ratios between legitimate users and the eavesdroppers, we fix the legitimate users’ channel gain and change the eavesdroppers. The above Eq. (36) can be further transformed to
In the highSNR regime, E _{ s }→∞, according to (28), Q _{ s },Q s′→∞, the term \(({\boldsymbol {H}_{ea}\boldsymbol {H}_{ea}^{H}})^{1}\) then can be omitted and the result is the following expression
Considering artificial noise, (Q s′)^{−1} Q _{ s }=ρ/(1−ρ)I. When ρ is fixed, log(det(I+(Q s′)^{−1} Q _{ s })) would be a constant. From (38), the secrecy rate will increase even when the eavesdroppers have better statistical channel knowledge than the legitimate users. Although, the secrecy rate can be positive in the scenario that the eavesdroppers have better statistical channel knowledge. With more power allocated to the artificial noise 1−ρ→1, less power will be available for the users ρ→0 which will lead to a fast decrease of the capacity to the intended users which is expressed as \(\log (\det (\boldsymbol {I}+\boldsymbol {H}_{ba} \boldsymbol {Q}_{s} \boldsymbol {H}_{ba}^{H}))\). As a result, the secrecy rate will finally fall down to zero. By changing the variable ρ from 0.1 to 0.9, the secrecy rate will rise to an optimal value, then converge to zero.
6 Simulation results
A system with N _{ t }=4 transmit antennas and T=2 users as well as K=1,2 eavesdroppers is considered. Each user or eavesdropper is equipped with N _{ r }=2 and N _{ k }=2 receive antennas.
6.1 Perfect channel state information
In Fig. 4, the proposed LRSOTHP+SGMI algorithm has the best BER performance. The SOTHP+SGMI algorithm has the exact same secrecyrate performance as the SOTHP+GMI algorithm. In Fig. 5, all of these nonlinear proposed algorithms can achieve a much higher secrecyrate performance especially in lowSNR scenarios. In the scenario where T>K, the secrecy rate of the proposed algorithms has around a 5 bits/Hz higher rate than the other precoding techniques. When T=K, Fig. 6 a shows that the proposed algorithms achieve a higher secrecy rate than the other techniques at low SNRs. And the secrecy rate will converge to a constant which will depend on the gain ratio between the main and the wiretap channels m.
6.2 Imperfect channel state information
In the simulations, the channel errors are modeled as a complex random Gaussian noise matrix E following the distribution \(\mathcal {CN}(0,{\sigma _{e}^{2}})\). Then, the imperfect channel matrix H ^{e} is defined as
We assume the channels of the legitimate users are perfect and the eavesdropper will have imperfect CSI.
In Fig. 6b, the secrecyrate performance is evaluated in the imperfect CSI scenario. Compared with the secrecyrate performance in Fig. 6 a, the secrecy rate will suffer a huge decrease in the imperfect CSI scenario. When T=K, Fig. 6 b shows that the secrecy rate at low SNR is degraded and the secrecy rate requires very high SNR to converge to a constant. It is worth noting that the proposed SOTHP+SGMI has the best secrecyrate performance among the studied precoding techniques.
6.3 Imperfect channel state information with artificial noise
In Fig. 6 c AN is added and the total transmit power E _{ s } is the same as before. Comparing the results in Fig. 6 b, c, it is clear that by injecting the artificial noise, the secrecy rate can achieve a much better performance in a highSNR scenario. In Fig. 6 c, the channel gain ratio is m=2. According to the secrecy performance of Fig. 6 d, 40 % of the transmit power E _{ s } is used to generate AN. In Fig. 6 d, it shows the secrecy rate with the change of the transmit signal power ratio to the artificial noise. The channel gain ratio is m=1. Comparing the theoretical result and the simulation result, the optimal value can be achieved when the transmit signal power ratio to the artificial noise is 0.6.
7 Conclusions
Precoding techniques are widely used in the downlink of MUMIMO wireless networks to achieve good BER performance. They also contribute to the improvement of the secrecy rate in the physical layer. The three proposed algorithms can all achieve higher secrecyrate performance than conventional techniques. Firstly, if we consider the complexity as the most important metric, among all the studied nonlinear precoding techniques, the proposed SOTHP+SGMI algorithm requires the lowest computational complexity which results in a significant improvement on the efficiency. Secondly, if the transmission accuracy comes first in the design, the LRSOTHP+SGMI algorithm is also superior to the existing linear and nonlinear algorithms considered.
8 Appendix
Based on Eq. (38), if we consider \(\rho \boldsymbol {A}=\boldsymbol {H}_{ba} \boldsymbol {Q}_{s} \boldsymbol {H}_{ba}^{H}\) and (Q s′)^{−1} Q _{ s }=ρ/(1−ρ)I, it can be rewritten as
Here, we first take the derivative of log(det(I+ρ A)). Firstly, we assume A is an m×m matrix. The eigenvalue of matrix I+ρ A is 1+ρ a _{1},1+ρ a _{2},⋯1+ρ a _{ m }. The eigenvalue of matrix A(I+ρ A)^{−1} is \(\frac {a_{1}}{1+\rho a_{1}},\quad,\frac {a_{2}}{1+\rho a_{2}},\quad \cdots \quad \frac {a_{m}}{1+\rho a_{m}}\). With these assumptions, we can have
and
The differentiate of log(det(I+ρ A)) can be expressed as
Similarly, the differentiate of log(det(I+ρ/(1−ρ)I) can be obtained as Tr(I) ln(2). The optimal value can be obtained by calculating
which is equivalent to
Substituting A with \(\frac {1}{\rho }\boldsymbol {H}_{ba} \boldsymbol {Q}_{s} \boldsymbol {H}_{ba}^{H}\), (45) can be rewritten as
Given the constraints Tr(Q _{ s })=ρ E _{ s } and \(\boldsymbol {H}_{ba} \in \mathcal {CN}(0,1)\), in the simulation scenario, we can solve (46) as
The optimal value is achieved at ρ=0.5. However, due to the existence of Gaussian noise, the optimal value is shifted. Based on (44), we focus on the first term Tr(A(I+ρ A)^{−1}) ln(2). It can be rewritten as Tr((A ^{−1})^{−1}(I+ρ A)^{−1}) ln(2). Finally, we can have
when considering Gaussian noise, \(\boldsymbol {A}=\frac {1}{{\sigma _{n}^{2}}\rho }\boldsymbol {H}_{ba} \boldsymbol {Q}_{s} \boldsymbol {H}_{ba}^{H}\). Here, \({\sigma _{n}^{2}}\) represents the variance of the noise. When the signal variance is set to 1, we can have \({\sigma _{n}^{2}}<1\). If we use the matrix N _{ a } to represent the effect of the noise, we can transfer \(\boldsymbol {A}=\frac {1}{\rho }\boldsymbol {H}_{ba} \boldsymbol {Q}_{s} \boldsymbol {H}_{ba}^{H}+\boldsymbol {N}_{a}\) where the elements in N _{ a } are positive. Substituting A with \(\frac {1}{\rho }\boldsymbol {H}_{ba} \boldsymbol {Q}_{s} \boldsymbol {H}_{ba}^{H}+\boldsymbol {N}_{a}\), (45) can be obtained as
Comparing (46) with (49), the optimal value of (49) is achieved at a higher value of ρ which is 0.6 in our simulation.
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Lu, X., de Lamare, R.C. & Zu, K. Successive optimization TomlinsonHarashima precoding strategies for physicallayer security in wireless networks. J Wireless Com Network 2016, 259 (2016). https://doi.org/10.1186/s1363801607555
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DOI: https://doi.org/10.1186/s1363801607555
Keywords
 Physicallayer security
 Precoding algorithms
 Successive optimization
 Secrecyrate analysis