Joint SVDGSVD precoding technique and secrecy capacity lower bound for the MIMO relay wiretap channel
 Marouen Jilani^{1}Email author and
 Tomoaki Ohtsuki^{2}
https://doi.org/10.1186/168714992012361
© Jilani and Ohtsuki; licensee Springer. 2012
Received: 1 November 2011
Accepted: 15 November 2012
Published: 10 December 2012
Abstract
We consider a problem of secure communications for the communication system consisting of multiple inputs for a source and a relay and multiple outputs for the relay, a destination and an eavesdropper. For the abovementioned communication system, we establish a lower bound on the secrecy capacity at which secure communications between the source and the destination are attainable. We make use of the singular value decomposition (SVD) and its generalization to decompose the whole system into parallel independent channels. At the source, the generalized singular value decomposition (GSVD) is performed to simultaneously diagonalize the channel matrices of the relay and the destination and independently code across the resulting parallel channels. At the relay, the SVD is performed to beamform the signal towards the destination. The scalar case of what we are considering in this article has been investigated in previous literature, to prove that the introduction of a fourth party, the relay, in the wiretap channel facilitates secure wireless communications. Our simulation results are in line with the scalar case’s and prove to be successful in achieving secrecy capacity where the conventional model failed, i.e., when no relay is introduced and the eavesdropper’s channel incurs as little noise as the legitimate receiver.
Keywords
1 Introduction
Wireless communications are prone to eavesdropping by nature: it is inevitable for electromagnetic waves propagated over the public medium to be subject to wiretapping from an unwanted party, which makes the security one of the biggest challenges for the wireless community to ever encounter. However, owing to cryptography, wireless applications gained trust in the market. For instance, cryptosystems are deployed to prevent the computing powerlimited enemy from causing any threat. Nevertheless, today the statement about this limitation is being regarded as a somehow strong assumption amid technological advances in computing technologies. Hence, the blink future of this kind of security and the need for the focus on security methods that drop this unrealistic assumption.
When introducing the brilliant notion of informationtheoretic security [1], Shannon, the father of information theory, established the condition for a secure communication between legitimate parties to succeed: when an eavesdropper is no better informed about the transmit messages after intercepting them than he was before. By bringing the channel uncertainty into play, Wyner introduced the wiretap channel [2] where he gave a new form of the condition for perfect secrecy, when the eavesdropper’s equivocation about a message is equal to the entropy of the latter. For this to happen, the eavesdropper was assumed to incur a degraded version of the legitimate channel. From Wyner’s model spanned many studies that characterized the secrecy capacity of different channel models, namely the extension to the Gaussian channel [3], the broadcast channel [4] and the recent multipleinput multipleoutput (MIMO) channel [5].
Among studies to address the security issue in a relaynetwork scenario are [6–9]. In [6, 7], the authors address the problem of securing a communication, between a sender and a receiver assisted by a relay, from the relay itself. In [8, 9], the limits to the Gaussian wiretap model in ensuring secure communications were pushed further by the introduction of a relay in the communication system. The fourth party proved to be a key component in establishing a secure link between the source and the destination even when the latter’s channel is as noisy as the eavesdropper’s. Our work here is also motivated by the fact that the MIMO wiretap model is also insecure when the eavesdropper incurs as little noise as the destination. The behavior of the above defined model following the introduction of a multiantenna relay is to be analyzed in this article.
The generalized singular value decomposition (GSVD) will serve as a precoding technique in the model under investigation as did the singular value decomposition (SVD) for the Gaussian MIMO channel in [10]. While the SVD decomposes a system comprising a pair of sender/receiver into parallel independent subchannels, the GSVD decomposes a system comprising one sender and two receivers. Although in [10] it has been proved that the SVDbased precoding technique achieves capacity, proving the same for the GSVD in our model is beyond the scope of this article. GSVD precoding at the source in conjunction with SVD precoding at the relay allows for the transmitter (source and relay) to beamform the signals towards the legitimate receivers (relay and destination), thus providing the latter with an advantage over the eavesdropper in the reception. That being done, it becomes straightforward to transfer results from the scalar case [8, 9] and thus extend the proof, to the MIMO case, that a relayassisted communication achieves secrecy when the conventional scheme fails.
The rest of the article is organized as follows. In Section 2, we introduce the system model and give a brief statement about the GSVD and the secrecy capacity of the Gaussian relay wiretap channel. Our results are derived in Section 2 and analyzed in Section 2 by computer simulations. Finally, we conclude our study in Section 2.
Notations: For a given matrix A, trace(A), null(A), and rank(A) denote the trace, the null space and the rank, respectively. The superscript ^{⊥} denotes the orthogonal complement of a subspace. Finally, [x]^{+} is the maximum between x and 0.
2 System model and preliminaries
2.1 Channel model
where

${\mathbf{X}}_{i}\in {\mathbb{C}}^{{N}_{i}\times 1}$, ${\mathbf{X}}_{i}\backsim \mathcal{N}(0,{\mathbf{Q}}_{i})$, $\text{trace}\left({\mathbf{X}}_{i}{\mathbf{X}}_{i}^{\u2020}\right)\le {P}_{i}$, i = s, r, respectively, is the source transmit signal, relay transmit signal, respectively.

${\mathbf{Y}}_{r}\in {\mathbb{C}}^{{N}_{r}\times 1}$, ${\mathbf{Y}}_{d}\in {\mathbb{C}}^{{N}_{d}\times 1}$, and ${\mathbf{Y}}_{e}\in {\mathbb{C}}^{{N}_{e}\times 1}$ are the received signals at the relay, destination and eavesdropper nodes, respectively.

${\mathbf{H}}_{1}\in {\mathbb{C}}^{{N}_{r}\times {N}_{s}}$, ${\mathbf{H}}_{2}\in {\mathbb{C}}^{{N}_{d}\times {N}_{s}}$, ${\mathbf{H}}_{3}\in {\mathbb{C}}^{{N}_{e}\times {N}_{s}}$, ${\mathbf{H}}_{4}\in {\mathbb{C}}^{{N}_{d}\times {N}_{r}}$, and ${\mathbf{H}}_{5}\in {\mathbb{C}}^{{N}_{e}\times {N}_{r}}$ are the complexvalued channel gain matrices as depicted in Figure 1.

${\mathbf{Z}}_{r}\in {\mathbb{C}}^{{N}_{r}\times 1}$, ${\mathbf{Z}}_{d}\in {\mathbb{C}}^{{N}_{d}\times 1}$, and ${\mathbf{Z}}_{e}\in {\mathbb{C}}^{{N}_{e}\times 1}$ are independent complex Gaussian noise vectors with distribution $C\mathcal{N}(0,\mathbf{I}{\sigma}_{r}^{2})$, $C\mathcal{N}(0,\mathbf{I}{\sigma}_{d}^{2})$, and $C\mathcal{N}(0,\mathbf{I}{\sigma}_{e}^{2})$, respectively.
2.2 Problem statement
The source wishes to communicate with the destination. The relay takes part in the communication process by relaying data from the source to the destination. We assume the relay’s channel to be less noisier than the destination’s^{a}. Meanwhile, we do not exclude the case where a successful communication is feasible in the direct link (from source to destination). A question that arises here is: Why do we need a relay anyway?
To answer this question, we highlight the primary role of the relay in our model. The third legitimate party was not introduced for a primary goal to fill his traditional role [11] (to guarantee a successful communication when the direct link is too noisy to serve, alone), but to guarantee a secure communication when the direct link is compromised by eavesdropping. It has been proved that the relay assumes this new role in the scalar case [8]. Our goal here is to prove so for the MIMO case.
2.3 Generalized singular value decomposition
Definition 1
it follows that k = s_{1} + s_{2} + s_{12}.
s _{ 1 } , s _{ 2 } , s _{ 12 } , s _{ n } for different configurations of the fullrank pencil ( H _{ 1 } , H _{ 2 } )
Scenario  Configuration  s _{12}  s _{1}  s _{2}  s _{ n } 

1  N_{ r } + N_{ d } < N_{ s }  0  N _{ r }  N _{ d }  N_{ s } − (N_{ r } + N_{ d }) 
2  N_{ r } + N_{ d } = N_{ s }  0  N _{ r }  N _{ d }  0 
3  max(N_{ r }, N_{ d }) < N_{ s } < N_{ r } + N_{ d }  (N_{ r } + N_{ d }) − N_{ s }  N_{ s } − N_{ d }  N_{ s } − N_{ r }  0 
4  N_{ d } < N_{ s } ≤ N_{ r }  N _{ d }  N_{ r } − N_{ d }  0  0 
5  N_{ r } < N_{ s } ≤ N_{ d }  N _{ r }  0  N_{ d } − N_{ r }  0 
6  N_{ s } ≤ min(N_{ r }, N_{ d })  N _{ s }  0  0  0 
2.4 Lower bound on the secrecy capacity for the Gaussian relay wiretap channel
where X_{ s } and X_{ r } are the source and relay transmit signals. Y_{ r }, Y_{ d }, and Y_{ e } are the received signals at the relay, the destination and the eavesdropper, respectively.
3 Secrecy rate for the Gaussian MIMO relay wiretap channel
In the following, we derive a lower bound on the secrecy capacity for the Gaussian MIMO relay wiretap channel described by the system model in (1). The idea is to decompose the whole system into parallel independent channels, making it easy to transmit over interferencefree virtual channels. The duration of communicating a codeword spans two time slots, with the beginning of a next communication interleaving with the end of a previous one. For that, the destination needs to split his antennas (not physically) into two groups, for the reception from the source and the relay. Following this communication scheme, only Scenarios 3 and 4 arise as feasible ones. In Scenarios 1 and 2, the receiver exploits all its antennas for the reception from the source. Thus, no further antennas are spared for the second time slot (reception from the relay). Hence, these two scenarios are infeasible. In Scenarios 5 and 6, since s_{12} = 0 (i.e., no private channel exists between the source and the relay), relaying cooperation cannot be applied. Finally, in both Scenarios 3 and 4, we assume that the destination’s channel is knowledgeable to the source and the relay. We also assume that the source has perfect knowledge of the relay’s channel.
3.1 Scenario 3
Scenario 3 communication scheme
Step  Source  Relay 

1  performs a GSVD of (H_{1}, H_{2})  
2  performs a GSVDbased precoding to construct codewords that convey messages towards the relay and the destination over their respective private channels  
3  performs an SVDbased precoding to beamform the message towards the destination 
The destination processes his received signal in (24) ((25), respectively) by multiplying it by ${\mathbf{\Psi}}_{d}^{\u2020}$ (${\mathbf{U}}_{\text{rd}}^{\u2020}$, respectively).
3.2 Scenario 4
Scenario 4 communication scheme
Step  Source  Relay 

1  performs a GSVD of (H_{1}, H_{2})  
2  performs a GSVDbased precoding to construct a codeword that conveys the messages towards the relay over his respective private channels  
3  performs an SVDbased precoding to beamform the message towards the destination 
In (d) X_{ s } is nulled out by the relay’s channel since it lies on null(H_{2}).
The relay processes his received signal (45) by multiplying it by ${\mathbf{\Psi}}_{r}^{\u2020}$.
The destination processes his received signal in (46) by multiplying it by ${\mathbf{U}}_{\text{rd}}^{\u2020}$.
where ${\stackrel{~}{\mathbf{Z}}}_{r}={\mathbf{\Psi}}_{r}^{\u2020}{\mathbf{Z}}_{r}$ and ${\stackrel{~}{\mathbf{Z}}}_{d}={\mathbf{U}}_{\text{rd}}^{\u2020}{\mathbf{Z}}_{d}$.
4 Simulation results
In this section, we convey the secure communication performance of the proposed scheme by running two simulations. We compare our results to the MIMO wiretap channel’s, with no relay brought into play. We refer to it henceforth as the conventional model. The key to outperformance of one scheme over another is the secrecy rate achieved between the source and the destination.
Simulation settings
Figure5  Figure6  

Number of antennas  
N _{ s }  8,4  8 
N _{ r }  6,3  6 
N _{ d }  4,2  4 
N _{ e }  4,2  4 
Noise variances  
${\sigma}_{r}^{2}$  1  1 
${\sigma}_{d}^{2}$  10  10,5 
${\sigma}_{e}^{2}$  10  10,5 
Power allocation  
P _{sd}  $\frac{2}{3}{P}_{s}$  $\frac{2}{3}{P}_{s}$ 
P _{sr}  $\frac{1}{3}{P}_{s}$  $\frac{1}{3}{P}_{s}$ 
P _{ r }  $\frac{1}{2}{P}_{s}$  
Figure axes  
x  $10log\left(\frac{{P}_{\text{sd}}}{\underset{d}{\overset{2}{\sigma}}}\right)$ (dB)  $10log\left(\frac{{P}_{\text{sd}}}{\underset{d}{\overset{2}{\sigma}}}\right)$ (dB) 
y  (b/sec/Hz) 
5 Conclusion
In this article, the problem of securing a communication between a source and a destination with the help of a relay against a passive eavesdropper was considered. We referred to this model as the MIMO relay wiretap channel for which a closed form of the secrecy rate was derived. The key step to this result was a combination of SVD and GSVD to decompose the whole system into parallel independent channels, which allowed the source to beamform the communication simultaneously towards the relay and the destination and the relay to beamform the signal towards the destination, thus providing the legitimate parties with an advantages in the reception over the eavesdropper. The proposed model outperforms the MIMO wiretap channel with no relay assistance, when the eavesdropper’s channel incurs as little noise as the destination’s. This emphasizes the importance of cooperation in achieving secrecy. Future studies in this subject will be the enhancing of secrecy by deriving and adopting the optimal power allocation scheme.
Endnotes
^{a}It should be noted that the case when the relay’s channel is noisier than that of the destination’s is trivial and needless to mention. This is because, as demonstrated in [11], the introduction of the relay (with a noisier channel than the destination’s) has no benefit in enhancing the capacity between the source and the destination.^{b}We note that we particularly performed the GSVD on the pencil (H_{1}, H_{2}) and not on (H_{1}, H_{3}) or (H_{2}, H_{3}) (the pencils containing the eavesdropper’s channel). This is because in doing one of the latter GSVD decompositions, we need to assume that the source has perfect knowledge of the eavesdropper’s channel, which may not be available in many cases in realworld scenarios. However, for readers interested in a study that considered a pencil containing the eavesdropper’s channel, [13] is a good reference.
Declarations
Authors’ Affiliations
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