**3.2. Large-system results**

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 of users.

An expression for the secrecy sum-rate *<sup>R</sup>*◦ *<sup>s</sup>* in the large-system regime is given by [19]

$$R\_s^\diamond = \max\left\{ K \log\_2 \frac{1 + \mathcal{g}\left(\mathcal{J}, \xi\right) \frac{\rho + \frac{\mathcal{G}}{\beta} \left[1 + \mathcal{g}\left(\beta, \xi\right)\right]^2}{\rho + \left[1 + \mathcal{g}\left(\beta, \xi\right)\right]^2}}{1 + \frac{\rho}{\left(1 + \mathcal{g}\left(\beta, \xi\right)\right)^2}}, 0 \right\} \tag{8}$$

with

4 Recent Trends in Multiuser MIMO Communications

an achievable secrecy sum-rate *Rs* is given by

*K* ∑ *k*=1

max log2 

the intended receiver *<sup>k</sup>* and the eavesdropper *<sup>k</sup>*, respectively, given by

**<sup>W</sup>** <sup>=</sup> <sup>1</sup> √*γ*

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

*γ* = tr 

is optimized to maximize the secrecy sum-rate.

**H***<sup>H</sup>*

SINR*<sup>k</sup>* <sup>=</sup> *<sup>ρ</sup>*

*Rs* =

can form an alliance ˜

where SINR*<sup>k</sup>* and SINR˜

RCI precoding matrix is given by

and

row.

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

noting that each user *k*, along with its own eavesdropper ˜

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

equivalent multi-input, single-output, multi-eavesdropper (MISOME) wiretap channel [10],

 <sup>−</sup> log2 

 **h***H <sup>k</sup>* **w***<sup>k</sup>* 2

> **h***<sup>H</sup> <sup>k</sup>* **w***<sup>j</sup>*

<sup>1</sup> + *<sup>ρ</sup>* <sup>∑</sup>*j*�=*<sup>k</sup>*

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

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

**HH***<sup>H</sup>* + *Mξ***I***<sup>K</sup>*

**<sup>H</sup>***H***H**(**H***H***<sup>H</sup>** <sup>+</sup> *<sup>M</sup>ξ***I***M*)−<sup>2</sup>

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

<sup>−</sup><sup>1</sup>

1 + SINR*<sup>k</sup>*

*k*, and cooperate to jointly eavesdrop on the message *uk*. By

*<sup>k</sup>* are the signal-to-interference-plus-noise ratios for the message *uk* at

<sup>1</sup> <sup>+</sup> SINR*<sup>k</sup>*

SINR*<sup>k</sup>* <sup>=</sup> *<sup>ρ</sup>* �**H***k***w***k*�<sup>2</sup> , (5)

 , 0 

*k* and the transmitter, forms an

<sup>2</sup> (4)

, (6)

. (7)

, (3)

$$\log\left(\beta\_{\prime}\xi\right) = \frac{1}{2}\left[\text{sgn}(\xi)\cdot\sqrt{\frac{\left(1-\beta\right)^{2}}{\xi^{2}} + \frac{2\left(1+\beta\right)}{\xi} + 1} + 1 + \frac{1-\beta}{\xi} - 1\right].\tag{9}$$

In [19], a closed form expression was also derived for the optimal regularization parameter *<sup>ξ</sup>*⋆◦, given by

$$\mathfrak{F}^{\star \flat} = \frac{-2\rho^2 \left(1 - \beta\right)^2 + 6\rho\beta + 2\beta^2 - 2\left[\beta\left(\rho + 1\right) - \rho\right] \cdot \sqrt{\beta^2 \left[\rho^2 + \rho + 1\right] - \beta\left[2\rho\left(\rho - 1\right)\right] + \rho^2}}{6\rho^2 \left(\beta + 2\right) + 6\rho\beta}. \tag{10}$$

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]

$$
\zeta^{\star \circ} = \frac{1}{3\rho + 1 + \sqrt{3\rho + 1}}, \quad \text{for } \beta = 1. \tag{11}
$$

We note that the value of the regularization parameter *<sup>ξ</sup>*⋆◦ that maximizes the secrecy sum-rate differs from the value *<sup>ξ</sup>*⋆◦ ns = *<sup>β</sup>*/*<sup>ρ</sup>* that maximizes the sum-rate without secrecy requirements [28].

By substituting the optimal value of the regularization parameter (10) in (8), it is possible to obtain the optimal secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* achievable by RCI precoding in the large-system regime. The secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* is a function of *<sup>K</sup>*, *<sup>β</sup>* and *<sup>ρ</sup>*. When *<sup>β</sup>* = 1, the optimal secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* has a simple expression, given by

$$R\_s^{\star \diamond} = K \log\_2 \frac{9\rho + 2 + (6\rho + 2)\sqrt{3\rho + 1}}{4\left(4\rho + 1\right)}, \quad \text{for } \beta = 1. \tag{12}$$

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

0 5 10 15 20

<sup>s</sup> /M <sup>β</sup> = 1

β = 1.2

*<sup>s</sup>* becomes simpler in the limit of large SNR. In

*<sup>s</sup>* /*M*. It is possible to see from Fig. 3 that *<sup>β</sup>*opt is an

, as *ρ* → ∞. (13)

for *β* > 1

β = 0.8

Physical Layer Security for Multiuser MIMO Communications

ρ [dB]

**Figure 2.** Comparison between the secrecy sum-rate with RCI precoding in the large-system regime (8) and the simulated ergodic secrecy sum-rate for finite *M*. Three sets of curves are shown, each one corresponds to a different value of *β*.

*<sup>β</sup>* <sup>+</sup> *<sup>K</sup>* log2 *<sup>ρ</sup>* for *<sup>β</sup>* <sup>&</sup>lt; <sup>1</sup>

*<sup>β</sup>*−<sup>1</sup> <sup>−</sup> *<sup>K</sup>* log2 *<sup>ρ</sup>*, 0�

We note from (13) that for high SNR, the behavior of the secrecy sum-rate significanly depends on the ratio *β* between the number of users *K* and the number of transmit antennas *M*. When *K* < *M*, the secrecy sum-rate scales linearly with the factor *K*. If *K* = *M*, the scaling factor reduces to *K*/2. When the number of users *K* exceeds the number of antennas *M*, then the secrecy sum-rate decreases with the SNR *ρ*, and there is a value of *ρ* beyond which the

Fig. 3 depicts the asymptotic secrecy sum-rate per transmit antenna as a function of *β*, for several values of the SNR. We denote by *β*opt the value of the ratio *β* that maximizes the

increasing function of the SNR. The value of *β*opt falls between 0 and 1, and it tends to 1 in

We denote by *β*max the maximum value of *β* allowed for non-zero secrecy sum-rates. The value of *β*max represents the maximum number of users per transmit antenna that can be

<sup>2</sup> log2 *<sup>ρ</sup>* for *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

Rs/M (M = 10) Rs/M (M = 20) Rs/M (M = 40)

R<sup>⋆</sup>◦

0

The expression of the secrecy sum-rate *<sup>R</sup>*⋆◦

 

*K* log2

max �

*K* <sup>2</sup> log2 27 <sup>64</sup> <sup>+</sup> *<sup>K</sup>*

1−*β*

3*K* log2

*β*

achievable secrecy sum-rate becomes zero.

secrecy sum-rate per transmit antenna *<sup>R</sup>*⋆◦

**3.3. Effect of the network load**

the limit of large SNR.

fact, it can be approximated by

*<sup>R</sup>*⋆◦ *<sup>s</sup>* <sup>≈</sup> 1

Per-antenna

2

 secrecy sum-rate

3

4

Although the optimal value of the regularization parameter *<sup>ξ</sup>*⋆◦ in (10) was derived in [19] in the large-system regime, using *<sup>ξ</sup>*⋆◦ in a finite-size system does not cause a significant loss in the secrecy sum-rate compared to using a regularization parameter *<sup>ξ</sup>*fs(**H**) optimized for every channel realization.

Fig. 1 shows the complementary cumulative distribution function (CCDF) of the normalized secrecy sum-rate difference between using *<sup>ξ</sup>*⋆◦ and *<sup>ξ</sup>*fs(**H**) as the regularization parameter of the RCI precoder for *K* = 4, 8, 16, 32 users, for *β* = 1 and at an SNR of 10dB. The difference is normalized by dividing by the secrecy sum-rate of the precoder that uses *<sup>ξ</sup>*fs(**H**). We observe that the average normalized secrecy sum-rate difference is less than 2.4 percent for all values of *<sup>K</sup>*. As a result, the large-system regularization parameter *<sup>ξ</sup>*⋆◦ may be used instead of the finite-system regularization parameter with only a small loss of performance. Moreover, the value of *<sup>ξ</sup>*⋆◦ does not need to be calculated for each channel realization.

**Figure 1.** Complementary cumulative distribution function (CCDF) of the normalized secrecy sum-rate difference between using *<sup>ξ</sup>*fs (**H**) and *<sup>ξ</sup>*⋆◦, with *<sup>β</sup>* <sup>=</sup> <sup>1</sup> and *<sup>ρ</sup>* <sup>=</sup> <sup>10</sup>dB.

Fig. 2 compares the secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* of the RCI precoder from the large-system analysis to the simulated ergodic secrecy sum-rate *Rs* with a finite number of users, for different values of *β*. We observe that as *M* increases, the simulated rates approach the curves from large-system analysis. For *<sup>β</sup>* ≤ 1, *<sup>R</sup>*⋆◦ *<sup>s</sup>* is always positive and monotonically increasing with the SNR *ρ*. However when *β* > 1, the secrecy sum-rate does not monotonically increase with *ρ*. There is an optimal value of the SNR beyond which the achievable secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* starts decreasing, until it becomes zero for large SNR. When *<sup>β</sup>* <sup>≥</sup> 2 no positive secrecy sum-rate is achievable at all [19].

**Figure 2.** Comparison between the secrecy sum-rate with RCI precoding in the large-system regime (8) and the simulated ergodic secrecy sum-rate for finite *M*. Three sets of curves are shown, each one corresponds to a different value of *β*.

The expression of the secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* becomes simpler in the limit of large SNR. In fact, it can be approximated by

$$R\_s^{\star \diamond} \approx \begin{cases} K \log\_2 \frac{1-\beta}{\beta} + K \log\_2 \rho & \text{for } \beta < 1\\ \frac{K}{2} \log\_2 \frac{2\gamma}{64} + \frac{K}{2} \log\_2 \rho & \text{for } \beta = 1\\ \max\left\{3K \log\_2 \frac{\beta}{\beta - 1} - K \log\_2 \rho, 0\right\} & \text{for } \beta > 1 \end{cases} \tag{13}$$

We note from (13) that for high SNR, the behavior of the secrecy sum-rate significanly depends on the ratio *β* between the number of users *K* and the number of transmit antennas *M*. When *K* < *M*, the secrecy sum-rate scales linearly with the factor *K*. If *K* = *M*, the scaling factor reduces to *K*/2. When the number of users *K* exceeds the number of antennas *M*, then the secrecy sum-rate decreases with the SNR *ρ*, and there is a value of *ρ* beyond which the achievable secrecy sum-rate becomes zero.

#### **3.3. Effect of the network load**

6 Recent Trends in Multiuser MIMO Communications

*<sup>R</sup>*⋆◦

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 2 compares the secrecy sum-rate *<sup>R</sup>*⋆◦

large-system analysis. For *<sup>β</sup>* ≤ 1, *<sup>R</sup>*⋆◦

sum-rate is achievable at all [19].

CCDF

*<sup>ξ</sup>*fs (**H**) and *<sup>ξ</sup>*⋆◦, with *<sup>β</sup>* <sup>=</sup> <sup>1</sup> and *<sup>ρ</sup>* <sup>=</sup> <sup>10</sup>dB.

*<sup>R</sup>*⋆◦

every channel realization.

*<sup>s</sup>* = *K* log2

9*ρ* + 2 + (6*ρ* + 2)

Although the optimal value of the regularization parameter *<sup>ξ</sup>*⋆◦ in (10) was derived in [19] in the large-system regime, using *<sup>ξ</sup>*⋆◦ in a finite-size system does not cause a significant loss in the secrecy sum-rate compared to using a regularization parameter *<sup>ξ</sup>*fs(**H**) optimized for

Fig. 1 shows the complementary cumulative distribution function (CCDF) of the normalized secrecy sum-rate difference between using *<sup>ξ</sup>*⋆◦ and *<sup>ξ</sup>*fs(**H**) as the regularization parameter of the RCI precoder for *K* = 4, 8, 16, 32 users, for *β* = 1 and at an SNR of 10dB. The difference is normalized by dividing by the secrecy sum-rate of the precoder that uses *<sup>ξ</sup>*fs(**H**). We observe that the average normalized secrecy sum-rate difference is less than 2.4 percent for all values of *<sup>K</sup>*. As a result, the large-system regularization parameter *<sup>ξ</sup>*⋆◦ may be used instead of the finite-system regularization parameter with only a small loss of performance. Moreover, the

0 0.01 0.02 0.03 0.04 0.05

*<sup>s</sup>* of the RCI precoder from the large-system analysis

*<sup>s</sup>* is always positive and monotonically increasing with

Normalized throughput difference

**Figure 1.** Complementary cumulative distribution function (CCDF) of the normalized secrecy sum-rate difference between using

to the simulated ergodic secrecy sum-rate *Rs* with a finite number of users, for different values of *β*. We observe that as *M* increases, the simulated rates approach the curves from

the SNR *ρ*. However when *β* > 1, the secrecy sum-rate does not monotonically increase with *ρ*. There is an optimal value of the SNR beyond which the achievable secrecy sum-rate

*<sup>s</sup>* starts decreasing, until it becomes zero for large SNR. When *<sup>β</sup>* <sup>≥</sup> 2 no positive secrecy

value of *<sup>ξ</sup>*⋆◦ does not need to be calculated for each channel realization.

3*ρ* + 1

K = 4, mean = 0.024 K = 8, mean = 0.0084 K = 16, mean = 0.0023 K = 32, mean = 6.4 × 10<sup>−</sup><sup>4</sup>

<sup>4</sup> (4*<sup>ρ</sup>* <sup>+</sup> <sup>1</sup>) , for *<sup>β</sup>* <sup>=</sup> 1. (12)

Fig. 3 depicts the asymptotic secrecy sum-rate per transmit antenna as a function of *β*, for several values of the SNR. We denote by *β*opt the value of the ratio *β* that maximizes the secrecy sum-rate per transmit antenna *<sup>R</sup>*⋆◦ *<sup>s</sup>* /*M*. It is possible to see from Fig. 3 that *<sup>β</sup>*opt is an increasing function of the SNR. The value of *β*opt falls between 0 and 1, and it tends to 1 in the limit of large SNR.

We denote by *β*max the maximum value of *β* allowed for non-zero secrecy sum-rates. The value of *β*max represents the maximum number of users per transmit antenna that can be served with non-zero secrecy sum-rate. Fig. 3 shows that *β*max is a decreasing function of the SNR. The value of *β*max can be found by solving the following cubic equation [19]

$$\left(\left(\rho+1\right)\beta\_{\text{max}}^{\text{3}} - \left(3\rho+2\right)\beta\_{\text{max}}^{2} + 3\rho\beta\_{\text{max}} - \rho = 0.\tag{14}$$

The value of *β*max falls between 1 and 2. This means that if *K* ≥ 2*M*, i.e. if *β* ≥ 2, then the secrecy sum-rate is zero for all SNRs. In the limit of large SNR, equation (14) reduces to

$$(\beta\_{\text{max}} - 1)^3 = 0,\tag{15}$$

10.5772/57130

149

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*<sup>s</sup>* is zero

ρ = 5dB ρ = 10dB ρ = 15dB ρ = 20dB

Physical Layer Security for Multiuser MIMO Communications

1.48

1.66

*<sup>s</sup>* to the sum-rate *<sup>R</sup>*⋆◦ in the limit of large SNR.

ns = *<sup>β</sup>*/*<sup>ρ</sup>* we obtain the following large-SNR

, as *ρ* → ∞. (17)

1.34

0 0.5 1 1.5 2

1.23

0.70

0.75

0.65

0.57

approximation for the secrecy sum-rate without secrecy requirements [19]

1−*β*

*β*

β

**Figure 3.** Asymptotic secrecy sum-rate per transmit antenna with RCI as a function of *β*. Circles denote *β*opt, squares denote

for *β* ≥ *β*max, with *β*max given by (14). Therefore, introducing the secrecy requirements will limit to *β*max the number of users per transmit antenna that can be served with a non-zero

*<sup>β</sup>* <sup>+</sup> *<sup>K</sup>* log2 *<sup>ρ</sup>* for *<sup>β</sup>* <sup>&</sup>lt; <sup>1</sup>

*<sup>β</sup>*−<sup>1</sup> for *<sup>β</sup>* <sup>&</sup>gt; <sup>1</sup>

By comparing (17) to (13), we can draw the following conclusions regarding the large-SNR regime. If the number of transmit antennas *M* is larger than the number of users *K*, then

network. Therefore, by using *<sup>ξ</sup>*⋆◦ from (10) one can achieve secrecy while maintaining the same sum-rate, i.e. there is no secrecy loss. If *M* = *K*, then *β* = 1, the secrecy requirements only reduce the sum-rate by a constant value, and the scaling factor *K*/2 remains unchanged. Alternatively, one can achieve secrecy while maintaining the same sum-rate, by increasing the transmitted power by a factor 64/27 ≈ 3.75dB. If *M* < *K*, i.e. *β* > 1, then the secrecy

*<sup>s</sup>* <sup>=</sup> *<sup>R</sup>*⋆◦, and the secrecy requirements do not decrease the sum-rate of the

<sup>2</sup> log2 *<sup>ρ</sup>* for *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

ns <sup>=</sup> *<sup>β</sup>*/*ρ*. It is easy to show that *<sup>R</sup>*⋆◦ <sup>≥</sup> 0 for all values of *<sup>β</sup>* and *<sup>ρ</sup>*, with equality only for *<sup>ρ</sup>* = 0, and that *<sup>R</sup>*⋆◦ tends to zero as *<sup>β</sup>* → <sup>∞</sup>. Hence, there is no limit to the number of users per transmit antenna *β* that the system can accommodate with a non-zero sum-rate. However if we impose the secrecy requirements, the secrecy sum-rate *<sup>R</sup>*⋆◦

0

We now compare the secrecy sum-rate *<sup>R</sup>*⋆◦

*<sup>R</sup>*⋆◦ <sup>≈</sup>

Again by using the regularization parameter *<sup>ξ</sup>*⋆◦

  *K* log2

*K* log2

*K*

*β*max.

with *<sup>ξ</sup>*⋆◦

sum-rate.

*<sup>β</sup>* < 1, *<sup>R</sup>*⋆◦

1

2

R⋆◦

s /M 3

4

5

yielding to *β*max = 1. These results can be explained as follows. In the worst-case scenario, the alliance of cooperating eavesdroppers can cancel the interference, and its received SINR is the ratio between the signal leakage and the thermal noise. In the limit of large SNR, the thermal noise vanishes, and the only means for the transmitter to limit the eavesdropper's SINR is by reducing the signal leakage to zero by inverting the channel matrix. This can only be accomplished when the number of transmit antennas is larger than or equal to the number of users, hence only if *β* ≤ 1. When *β* > 1 this is not possible, and no positive secrecy sum-rate can be achieved. When *β* ≥ 2, the eavesdroppers are able to drive the secrecy sum-rate to zero irrespective of *ρ*. This is consistent with the results presented in [10] for a single-user system.
