**3. Physical layer security with multi-user MIMO linear precoding**

Although suboptimal, linear precoding schemes are of particular interest because of their low-complexity implementations and because they can control the amount of crosstalk between the users to maintain a high sum-rate in the broadcast channel [20–27]. In the MISO BCC, linear precoding can be employed to control the amount of interference and information leakage to the unintended receivers introduced by the transmission of each confidential message [17–19].

Let the transmitted signal be denoted by **x**, then the received signal is given by

$$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n} \tag{1}$$

where **<sup>H</sup>** <sup>=</sup> [**h**1,..., **<sup>h</sup>***K*] is the *<sup>K</sup>* <sup>×</sup> *<sup>M</sup>* channel matrix, **<sup>h</sup>***<sup>k</sup>* is the *<sup>k</sup>*-th column of **<sup>H</sup>** and it represents the channel between the BS and the *k*-th user, and **n** is complex Gaussian noise. In linear precoding, the transmitted vector **x** is derived from the vector containing the confidential messages **u** = [*u*1,..., *uK*] *<sup>T</sup>* through a deterministic linear transformation (precoding) [22–25]. Let **<sup>W</sup>** <sup>=</sup> [**w**1,..., **<sup>w</sup>***K*] be the *<sup>M</sup>* <sup>×</sup> *<sup>K</sup>* precoding matrix, where **<sup>w</sup>***<sup>k</sup>* is the *k*-th column of **W**. Then the transmitted signal is

$$\mathbf{x} = \mathbf{W}\mathbf{u} = \sum\_{k=1}^{K} \mathbf{w}\_k u\_k. \tag{2}$$

#### **3.1. Achievable secrecy sum-rates with linear precoding**

The secrecy sum-rates achievable by linear precoding were obtained in [18] by considering the worst-case scenario, where for each intended receiver *k* the remaining *K* − 1 users can form an alliance ˜ *k*, and cooperate to jointly eavesdrop on the message *uk*. By noting that each user *k*, along with its own eavesdropper ˜ *k* and the transmitter, forms an equivalent multi-input, single-output, multi-eavesdropper (MISOME) wiretap channel [10], an achievable secrecy sum-rate *Rs* is given by

$$R\_s = \sum\_{k=1}^{K} \max\left\{ \log\_2 \left( 1 + \text{SINR}\_k \right) - \log\_2 \left( 1 + \text{SINR}\_{\tilde{k}} \right), 0 \right\}, \tag{3}$$

where SINR*<sup>k</sup>* and SINR˜ *<sup>k</sup>* are the signal-to-interference-plus-noise ratios for the message *uk* at the intended receiver *<sup>k</sup>* and the eavesdropper *<sup>k</sup>*, respectively, given by

$$\text{SINR}\_{k} = \frac{\rho \left| \mathbf{h}\_{k}^{H} \mathbf{w}\_{k} \right|^{2}}{1 + \rho \sum\_{j \neq k} \left| \mathbf{h}\_{k}^{H} \mathbf{w}\_{j} \right|^{2}} \tag{4}$$

10.5772/57130

145

http://dx.doi.org/10.5772/57130

(8)

(10)

**3.2. Large-system results**

An expression for the secrecy sum-rate *<sup>R</sup>*◦

*R*◦ *<sup>s</sup>* = max

*<sup>g</sup>* (*β*, *<sup>ξ</sup>*) <sup>=</sup> <sup>1</sup>

sum-rate differs from the value *<sup>ξ</sup>*⋆◦

regime. The secrecy sum-rate *<sup>R</sup>*⋆◦

to obtain the optimal secrecy sum-rate *<sup>R</sup>*⋆◦

2

sgn(*ξ*) ·

 

*K* log2

�

<sup>2</sup> + 6*ρβ* + 2*β*<sup>2</sup> − 2 [*β* (*ρ* + 1) − *ρ*] ·

For the specific case *<sup>β</sup>* = 1, i.e. *<sup>M</sup>* = *<sup>K</sup>*, the value of *<sup>ξ</sup>*⋆◦ reduces to [18]

*<sup>ξ</sup>*⋆◦ <sup>=</sup> <sup>1</sup>

*<sup>s</sup>* has a simple expression, given by

3*ρ* + 1 + �3*ρ* + 1

We note that the value of the regularization parameter *<sup>ξ</sup>*⋆◦ that maximizes the secrecy

By substituting the optimal value of the regularization parameter (10) in (8), it is possible

(1 − *β*) 2 *<sup>ξ</sup>*<sup>2</sup> <sup>+</sup>

In [19], a closed form expression was also derived for the optimal regularization parameter

of users.

with

*<sup>ξ</sup>*⋆◦, given by

*<sup>ξ</sup>*⋆◦ <sup>=</sup> <sup>−</sup>2*ρ*<sup>2</sup> (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)

requirements [28].

secrecy sum-rate *<sup>R</sup>*⋆◦

The secrecy sum-rate achievable by the RCI precoder in the MISO BCC was obtained in [19] by large-system analysis, where both the number of receivers *K* and the number of transmit antennas *M* approach infinity, with their ratio *β* = *K*/*M* being held constant. Unless otherwise stated, the results presented in the following refer to the large-system regime. We note that these results are accurate even when applied to small systems with a finite number

1 + *g* (*β*, *ξ*)

*<sup>ρ</sup>*+ *ρξ*

2 (1 + *β*)

*<sup>ξ</sup>* <sup>+</sup> <sup>1</sup> <sup>+</sup>

<sup>1</sup> <sup>+</sup> *<sup>ρ</sup>* (1+*g*(*β*,*ξ*))<sup>2</sup>

*<sup>s</sup>* in the large-system regime is given by [19]

, 0

 

Physical Layer Security for Multiuser MIMO Communications

1 − *β <sup>ξ</sup>* <sup>−</sup> <sup>1</sup>

�*β*<sup>2</sup> [*ρ*<sup>2</sup> + *ρ* + 1] − *β* [2*ρ* (*ρ* − 1)] + *ρ*<sup>2</sup>

, for *β* = 1. (11)

<sup>6</sup>*ρ*<sup>2</sup> (*<sup>β</sup>* <sup>+</sup> <sup>2</sup>) <sup>+</sup> <sup>6</sup>*ρβ* .

ns = *<sup>β</sup>*/*<sup>ρ</sup>* that maximizes the sum-rate without secrecy

*<sup>s</sup>* is a function of *<sup>K</sup>*, *<sup>β</sup>* and *<sup>ρ</sup>*. When *<sup>β</sup>* = 1, the optimal

*<sup>s</sup>* achievable by RCI precoding in the large-system

. (9)

*<sup>β</sup>* [1+*g*(*β*,*ξ*)]<sup>2</sup> *ρ*+[1+*g*(*β*,*ξ*)]<sup>2</sup>

and

$$\text{SINR}\_{\vec{k}} = \rho \left\| \mathbf{H}\_k \mathbf{w}\_k \right\|^2,\tag{5}$$

and where *ρ* is the transmit SNR, and **H***<sup>k</sup>* is the matrix obtained from **H** by removing the *k*-th row.

Particular attention was given to the Regularized Channel Inversion (RCI) precoder, because it achieves better performance than the plain Channel Inversion precoder, especially at low SNR [24, 25]. A linear precoder based on RCI was proposed for the MISO BCC in [19]. The RCI precoding matrix is given by

$$\mathbf{W} = \frac{1}{\sqrt{\gamma}} \mathbf{H}^H \left( \mathbf{H} \mathbf{H}^H + M \tilde{\mathbf{y}} \mathbf{I}\_K \right)^{-1},\tag{6}$$

where *γ* is a long-term power normalization constant, given by

$$\gamma = \text{tr}\left\{ \mathbf{H}^H \mathbf{H} (\mathbf{H}^H \mathbf{H} + M \mathbf{\tilde{g}} \mathbf{I}\_M)^{-2} \right\}. \tag{7}$$

For each message, the function of the regularization parameter *ξ* is to achieve a tradeoff between maximizing the signal power at the intended user and minimizing the interference and information leakage at the other unintended users. In [19], the regularization parameter is optimized to maximize the secrecy sum-rate.
