On the Design of Artificial-Noise-Aided
Secure Multi-Antenna Transmission
in Slow Fading Channels
In this paper, we investigate the design of artificial-noise-aided secure multi-antenna transmission in slow fading channels. The primary design concerns include the transmit power allocation and the rate parameters of the wiretap code. We consider two scenarios with different complexity levels: i) the design parameters are chosen to be fixed for all transmissions, ii) they are adaptively adjusted based on the instantaneous channel feedback from the intended receiver. In both scenarios, we provide explicit design solutions for achieving the maximal throughput subject to a secrecy constraint, given by a maximum allowable secrecy outage probability. We then derive accurate approximations for the maximal throughput in both scenarios in the high signal-to-noise ratio region, and give new insights into the additional power cost for achieving a higher security level, whilst maintaining a specified target throughput. In the end, the throughput gain of adaptive transmission over non-adaptive transmission is also quantified and analyzed.
With the rapid development of wireless technology, an ever-increasing amount of sensitive data (e.g., private conversation, credit card information) is transmitted over wireless networks. However, the broadcasting nature of the wireless medium makes it especially vulnerable to malicious interception. Currently, cryptographic algorithms, typically designed without exploiting the physical properties of the wireless medium, are used to keep broadcasted messages confidential. For such techniques, although the expenditure of interception may be very high, providing robust encryption algorithms is becoming ever more challenging, due to the continuing development of computing devices. By taking the physical properties of the wireless channels into consideration, the recently developed physical-layer security techniques can guarantee secure transmission regardless of the eavesdropper’s computational capability. As such, these techniques have drawn a lot of recent attention from the research community.
I-a Background and Previous Work
The notion of perfect secrecy was first introduced by Shannon . Subsequently, pioneering works on physical-layer security [2, 3] proved that there exist coding schemes which can ensure transmission reliability and perfect secrecy simultaneously. Many recent papers have expanded upon these initial contributions, considering different system configurations and assumptions. Particularly, multi-antenna techniques have been extensively studied as a means for achieving security enhancements [4, 5, 6, 7, 8]. However, in much of the literature on physical-layer security, the eavesdropper’s channel state information (CSI) was assumed to be available at the transmitter, which is usually impractical. To relax this strong assumption, the authors in  proposed a multi-antenna transmission scheme that inserts artificial noise into the transmitted signal in a controlled manner, thus to confuse the malicious eavesdropper. This transmission scheme requires the instantaneous CSI feedback from the intended receiver, but not the eavesdropper, which is a major advancement toward practical secure communications.
Building on the ideas from , the design and analysis of artificial-noise-aided transmission has been further studied for both fast and slow fading channels [10, 11, 12, 13, 14, 15, 16]. For fast fading channels, the channel coherence time is much shorter than the codeword length and the ergodic secrecy rate is often used as the performance metric for designing beamforming and power allocation strategies [10, 11, 12]. For slow fading channels, the channel coherence time is usually longer than the codeword length, and in such scenarios outage-based formulations become more appropriate. To this end, various secrecy outage formulations were proposed first in [17, 18] and recently in [19, 20]. In particular, the first formulation developed in [17, 18] has been used for studying artificial-noise-aided multi-antenna transmission schemes in [13, 14, 15, 16]. This secrecy outage formulation characterizes the possibility of having a secure and reliable transmission, without distinguishing secrecy from reliability. In other words, a secrecy outage event defined therein may occur due to either an insecure link to the eavesdropper or an unreliable link to the intended receiver. To better assist the secure transmission design, revised secrecy outage formulations were independently developed in  and . In these revised outage formulations, a secrecy outage event arises solely due to an insecure link to the eavesdropper; thus, the secrecy and reliability performance can be measured separately. These revised secrecy outage formulations can be utilized to obtain a better understanding and more practically-oriented designs for the artificial-noise-aided secure multi-antenna transmission.
I-B Our Approach and Contributions
In this paper, we provide new design guidelines for artificial-noise-aided secure multi-antenna transmission in slow fading channels, based on the recently developed secrecy outage formulation in . This formulation allows us to measure the secrecy and reliability performance for any given rate parameters of the wiretap code. In turn, we are able to set the rate parameters to achieve a target secrecy level, given by a maximum allowable secrecy outage probability. To the best of the authors’ knowledge, no prior work on artificial-noise-aided multi-antenna transmission has considered the rate parameters of the wiretap code as design parameters.
Our main contributions include explicit design solutions and new performance analysis results for the throughput-maximizing transmission schemes with either fixed-rate or adaptive-rate encoder, under a constraint on the level of secrecy. The design concerns include the rate parameters of the wiretap code, as well as the transmit power allocation between the information-bearing signal and the artificial noise. We consider two scenarios with different system complexities:
In the first scenario, the design parameters are optimized off-line and remain fixed for all transmissions. We divide our design into two steps: The first step minimizes the transmission delay for a given data rate, while the second step maximizes the average throughput. In the first step, closed-form solutions are derived for the optimal system parameters and the secrecy-delay tradeoff for fixed-rate transmission is captured. In the second step, the average throughput is maximized by numerically optimizing the data rate with the optimal designs of all other system parameters already given in closed form. To obtain further analytical insights, we derive a high signal-to-noise ratio (SNR) approximation of the optimal data rate to maximize the average throughput. Focusing on the high SNR region, we further investigate the additional power cost incurred by imposing or strengthening the secrecy constraint, while guaranteeing a specified target throughput.
In the second scenario, the design parameters are dynamically adjusted based on the instantaneous channel feedback from the intended receiver. We provide an analytical solution to the optimal system parameters that maximize the achievable data rate for each realization of the intended channel, under the secrecy constraint, such that the average throughput is also maximized. In the high SNR region, we present accurate approximations for the maximal throughput, which enable us to study the additional power cost incurred by imposing or strengthening the secrecy constraint, while achieving a specified target throughput. Finally, we analyze the throughput gain of adaptive-rate transmission over fixed-rate transmission. Whilst doing adaptation is always beneficial in terms of increasing the throughput, our analysis shows that in the high SNR region, the throughput gain is most significant when the number of transmit antennas is small. Moreover, we show that the throughput gain brought by doing adaption is not very sensitive to the required secrecy level.
We point out that, similar to our work, a very recent contribution  also used a secrecy outage formulation along the lines of [19, 20] to study artificial-noise-aided multi-antenna transmission. The main focus therein was to minimize the power consumption for given levels of secrecy and quality-of-service performance, while the transmission rates were not part of the design consideration. In contrast, we consider a transmission system with a fixed transmit power, and optimize the rate parameters of the wiretap code as well as the transmit power allocation to maximize the average throughput, subject to (s.t.) a secrecy outage constraint.
Ii System Model and Performance Metric
We consider the transmission from Alice to Bob in the presence of an eavesdropper Eve. Alice is equipped with multiple transmit antennas () while Bob and Eve each has one receive antenna. Thus, the channel from Alice to Bob is multiple-input and single-output (MISO). We assume a non-line-of-sight rich scattering environment, and as such, model all channels as uncorrelated Rayleigh fading. It is also assumed that Bob can estimate his channel accurately and use a perfect feedback link to inform Alice about his instantaneous CSI. This feedback link is not secure and can be intercepted by Eve. We further assume that the coherence time is long enough to support the wiretap code  and the time used for learning the channel and feeding back the CSI is negligible. Assuming Eve is a passive eavesdropper, the instantaneous CSI of Eve is thereby unavailable to Alice.
The dimensional symbol vector to be transmitted is defined as and the received signal at Bob is given by
where the vector is the channel fading gain from Alice to Bob and is the receiver noise at Bob. The entries of and are assumed to be independent and identically distributed (i.i.d.) zero-mean complex Gaussian variables with unit variance.
Similarly, the received signal at Eve is given by
where the vector captures the channel fading gain from Alice to Eve and is the receiver noise at Eve. The entries of are assumed to be i.i.d. zero-mean complex Gaussian variables each with variance .
Ii-a Transmit Beamforming with Artificial Noise Generation
The authors in  introduced the concept of generating artificial noise to guarantee secure transmission. The key idea is outlined as follows. Alice generates an orthonormal basis of as , where . Then she mixes some artificial noise with the message symbol as
where is complex Gaussian distributed and is the artificial noise vector.
while the received signal at Eve becomes
where and .
The total transmit power available at Alice is denoted as . The power allocation ratio is defined as the fraction of the information-bearing signal power to the total transmit power. Thus, the variance of is set to . By (3), it is clear that Alice is performing maximal ratio transmission on the channel to Bob and the instantaneous effective channel gain is . Therefore, the instantaneous channel capacity to Bob is , where . The complementary cumulative distribution function (c.c.d.f.) of is , where is the regularized upper incomplete gamma function. All transmit power not allocated to is equally assigned to the entries of , due to the absence of Eve’s CSI. Therefore, the entries of are set to be i.i.d. zero-mean complex Gaussian variables each with variance . The power allocation ratio may be dynamically adjusted based on .
Since the noise power at Eve is typically unknown to Alice, a robust approach, as done in , is to design for the worst-case scenario and assume that there is no receiver noise at Eve (i.e., ). Therefore, from (4), for a given realization of the channel fading gains (i.e., and ), the instantaneous SNR at Eve is given by
Since has i.i.d. complex Gaussian entries each with variance and is unitary, also has i.i.d. complex Gaussian entries each with variance . Using [22, Eq. 19], the c.c.d.f. of can be characterized as
As , this converges to the c.c.d.f. of an exponential distribution with mean . Note that if is adaptively adjusted based on , the distribution of the received SNR at Eve would be changed dynamically.
Ii-B Secure Transmission and Throughput
Using the well-known wiretap code , data is encoded before transmission. The rates of the transmitted codeword and the secret message are denoted as and , respectively. In this paper, we refer to the rates and as the rate parameters of the wiretap code. The rate redundancy is intentionally added in order to provide secrecy against eavesdropping. If the instantaneous channel capacity to Eve is larger than , perfect secrecy cannot be achieved and a secrecy outage is deemed to occur.
We consider an on-off transmission scheme  where Alice decides to transmit or not, based on her knowledge of Bob’s channel . To be specific, Alice transmits only if the effective channel gain exceeds a threshold . As will be seen, this on-off scheme is adopted to prevent undesirable transmissions which incur capacity outages (i.e., ) or unacceptably high risk of secrecy outage (which occurs when ). With the on-off transmission strategy, the transmit probability is
Since the channel fading gain changes from time to time, Alice would transmit once the effective channel gain exceeds the preselected threshold (i.e., ). In this way, the value of the transmit probability is directly related to the average delay, i.e., the larger , the shorter the expected delay.
Since the rates of the wiretap code may be adaptively adjusted based on Bob’s channel feedback, the rates and are potentially functions of . The average throughput is then defined as
where for , as a consequence of the on-off transmission protocol. Note that this throughput definition is meaningful only when: i) the risk of secrecy outage is under control, ii) the intended receiver can decode the messages correctly. As will be seen later, our designs can satisfy these two conditions simultaneously; thus, it is meaningful to apply this throughput definition.
Ii-C Secrecy Performance Characterization
From (5), it is clear that changing the power allocation would effectively alter the distribution of the received SNR at Eve. With on-off transmission, for a given , if , Alice would specify three parameters: , and for transmission; otherwise, she stops transmission. Therefore, the secrecy outage probability is given by
evaluated from (5).
Since we assumed that , the above-mentioned is an upper bound on the actual secrecy outage probability. It depends on the power allocation ratio , but not the transmit power .
The overall secrecy outage probability of this transmission system is given by
In sequel, we consider the optimization of the throughput performance, given a maximum allowable secrecy outage probability .
Iii Secure Transmission Design
with Non-Adaptive Encoder
In this section, we consider the scenario where a single codebook is used by Alice and Bob, thus and are carefully chosen to have fixed values which do not change with . In addition, the value of is also optimized off-line and remains fixed. We refer to this as the non-adaptive encoding (NAE) scheme.
The primary design objective is to meet the secrecy requirement given on the secrecy outage probability. For the NAE scheme, the design parameters , and have fixed values for all transmissions (i.e., they do not change with ); thus, (8) becomes independent of . Consequently, the conditional expectation in (9) can be ignored, and the overall secrecy outage probability in this case is described by (8). In other words, for the NAE scheme, the overall secrecy outage probability is a function of , and , and the secrecy requirement can be written as .
Meanwhile, as already mentioned, on-off transmission with threshold is adopted to prevent undesirable transmissions which incur capacity outages at Bob. When Alice decides to transmit (i.e., ), the channel capacity to Bob is
Thus, with satisfying , the transmitted messages can be decoded at Bob with an arbitrarily low error probability.
Now we consider throughput optimization of this NAE scheme under the secrecy constraint. Combining the above discussions, the throughput optimization problem is posed as follows:
We solve the problem in two steps. In the first step, we fix and maximize with respect to (w.r.t.) , and . Since is fixed, by maximizing , this step can be viewed as a delay-minimizing design, while satisfying the secrecy constraint. With a minor abuse of notation, the maximal for a given is denoted as , where is optimized already. Then, in the second step, we consider the throughput maximization, i.e., maximizing w.r.t. .
Iii-a Delay Minimization
From (11), the delay minimization problem is posed as
To simplify the expressions throughout the paper, we define the quantity:
With a non-adaptive encoder, for a prescribed message rate and under the secrecy constraint , the maximal transmit probability in (12) leading to the minimal average delay is given by:
This is achieved with the following choice of system parameters:
and are independent of . This is explained by noting that with the zero-noise assumption of Eve, the secrecy outage probability depends only on and . Therefore, for a given , whether the secrecy constraint can be satisfied depends only on . Since the latter optimization w.r.t. also does not involve , the independence is then observed.
increases with (since decreases with , as shown in Appendix -B), approaching the constant:
This result follows the intuition that when more transmit antennas are installed, a smaller fraction of power can be used for generating artificial noise while still ensuring the required security level.
The points above focus on the optimal system parameters. Some useful insights regarding the system performance and design implications may also be concluded:
For a given transmission system (i.e., given and ) and a prescribed message rate , the optimal tradeoff curve between the secrecy and delay performance is completely specified by (14). This is observed by noting that is achieved when , while varying effectively traces the optimal tradeoff. In , we considered the minimization of the overall secrecy outage probability under an average delay constraint (i.e., ), which is the symmetric problem to (12). Since the tradeoff curve obtained in  coincides with (14), we conclude that each point on the optimal tradeoff curve is Pareto optimal . As shown in Fig. 1, the area under each curve represents the feasible region of , points of which are achievable.
If the system is designed to operate under a joint secrecy and delay constraint (i.e., and ), the minimum required transmit power can be computed by replacing in (14) with and solving the obtained equality w.r.t. . This comes from the Pareto optimality indicated above and the fact that increasing would effectively expand the feasible region. If extra antennas may be added, the decreasing rate of is most significant for small . For the special case , it can be proved that decreases inverse linearly as goes large, using [25, Eq. 6.2]. As shown in Fig. 2, this inverse linear behavior holds for a wide range of (here, the dotted lines are properly scaled inverse linear curves, shown for reference).
Iii-B Throughput Optimization and Analysis
and the corresponding maximal throughput is given by
From (14) and (17), it can be shown that if is too small, even though may be high (i.e., close to one), the value of is still very small; whereas, if is too large, the value of and therefore will also become very small. Moreover, by differentiating w.r.t. , one can verify that the derivative is first positive and then negative with increasing ; thus is unique. As such, even if the objective function is non-concave, as shown in Fig. 3, we still can solve the derivative to find numerically.
where is the principle branch of the Lambert-W function and is defined in (13).
From (19) and [26, Eq. 65], we know that increases with . Therefore, from (15) and (16), we can observe that for the throughput-maximizing scheme, as , the optimal power allocation ratio and the added rate redundancy converge to
Here is the high SNR throughput approximation for the NAE scheme when the system is operating without any secrecy constraint (i.e., ), while reflects the throughput loss for realizing secure transmission. Interestingly, the throughput loss in (24) is independent of and this is a direct consequence of the zero-noise assumption of Eve.
The benefits of adding extra transmit antennas are two-fold: i) the system is more capable of achieving a larger throughput, ii) the throughput loss for securing the transmission will be reduced. However, the benefits in terms of reducing by adding extra antennas are limited, i.e.,
which depends only on , as we may expect.
Even in the high SNR region, the second term in (23) is usually insignificant due to the double logarithm. Therefore, to achieve a specified target throughput, the additional power cost for imposing or strengthening the secrecy constraint can be approximated as
which is independent of the targeted throughput. It is clear that the power cost in both cases in (25) can be effectively reduced by adding extra transmit antennas.
In Fig. 4, we show that the throughput approximation in (22) is quite accurate by comparing with the exact maximal throughput, which is obtained by numerically optimizing (17). When the secrecy constraint is strengthened, in order to maintain a specified target throughput, for , the first arrow shows that with and , the additional power cost is dB, while the second arrow suggests that with and , the additional power cost is dB. These two values are very close to the approximation provided in (25). Note that when is very small (e.g., ), the approximation in (22) becomes inaccurate due to the approximation procedure used in Appendix -D and a better approximation can be obtained by plugging (19) into (18).
Iv Secure Transmission Design
with Adaptive Encoder
In this section, we consider the scenario where Alice dynamically adjusts the system parameters: , and for each realization of the channel to Bob . In other words, Alice performs adaptive encoding (AE). As before, the target is to maximize the throughput under the secrecy constraint given by a maximum allowable secrecy outage probability . In particular, the values of , and are dynamically chosen to meet the requirement on the secrecy outage probability for each transmission, i.e., . This design will in turn satisfy the requirement on the overall secrecy outage probability. With the secrecy constraint satisfied, the problem of maximizing the throughput is equivalent to maximizing for each . Intuitively, the AE scheme can achieve a larger throughput compared with the NAE scheme, but demands a higher complexity.
Iv-a Message Rate Maximization
From (8), the problem of maximizing for each is given by
where the second constraint eliminates the capacity outages at the intended receiver.
Intuitively, we may expect that a positive message rate can always be achieved by properly adjusting the transmission system; thus, it may not be necessary to introduce the on-off transmission protocol. However, the derivation in Appendix -E shows that after guaranteeing the secrecy performance, a positive can be achieved only when . In other words, when Bob’s channel is not sufficiently good, a positive and the required secrecy performance cannot be achieved simultaneously, no matter how Alice adjusts the power allocation or encoding. Therefore, to maximize the throughput whilst guaranteeing the required secrecy level, on-off transmission with is introduced, as stated in (28).
The optimal power allocation ratio in (29) decreases with increasing , approaching the constant:
which is independent of and is identical to that for the NAE scheme in (20). This convergence suggests that in the high SNR region, near optimal performance can be obtained by employing a non-adaptive power allocation strategy.
With optimized power allocation, is set to the instantaneous capacity of Bob’s channel to guarantee successful decoding, as shown in (30). From (27) and (30), we can observe that as , the added rate redundancy converges to
which is independent of and is identical to that for the NAE scheme in (21).
Fig. 5 illustrates our analysis, showing for the AE scheme for different . The value of for the NAE scheme is also shown for comparison and it is found by numerically optimizing in (17), followed by plugging it into (15). As we may expect, the optimal power allocation ratios all converge to the same limit with increasing . This observation is similar to the one in  where the channel gain was found to be irrelevant in finding the optimal power allocation in the high SNR region for maximizing the ergodic secrecy rate in fast fading channels. However, besides the similarity, the difference is also significant. In the high SNR region, the optimal power allocation ratio that maximizes the ergodic secrecy rate for fast fading channels  is around , while the limiting value of in (31) that maximizes the throughput for slow fading channels depends strongly on the required security level and the number of transmit antennas .
Iv-B Throughput Performance Analysis
Though it seems difficult to find a closed-form expression, similar to the NAE case, progress can be made by appealing to the high SNR region. Specifically, as , the following approximation is derived in Appendix -F:
Here is the high SNR throughput approximation for the AE scheme without any secrecy constraint, which coincides with the high SNR approximation of the ergodic capacity of the MISO Rayleigh fading channel , while reflects the throughput loss incurred by enforcing the secrecy constraint.
In the high SNR region, the throughput loss for imposing the secrecy constraint in (37) is independent of and it is identical to the throughput loss for the NAE scheme in (24). The throughput loss in (37) is actually twice the limiting value of the rate redundancy in (32). This can be explained by noting that in the high SNR region, the transmit threshold in (28) is close to zero and, in addition to the rate redundancy in (32), the transmission system also suffers a power loss since a certain fraction of power is used to generate artificial noise, as shown in (31). It is interesting that this power loss leads to a data rate loss which is identical with the rate redundancy in (32).
In the high SNR region, to achieve a specified target throughput, the additional power cost for imposing or strengthening the secrecy constraint can be approximated from (35) and the results are identical to those for the NAE scheme in (25). However, although the NAE and AE schemes suffer the same power cost when imposing the same secrecy constraint, the AE scheme still outperforms the NAE scheme in terms of the throughput performance. This is due to the fact that when the secrecy constraint is removed, the AE scheme is designed to achieve a better throughput performance, compared with the NAE scheme. Specifically, when , for both schemes, all the transmit power is allocated to the information-bearing signal and the data will be transmitted without wiretap coding. In this case, the NAE scheme still chooses a constant rate to transmit and the transmit threshold is set to the minimum required effective channel gain to support the selected rate; while the AE scheme always transmits at the instantaneous capacity of Bob’s channel and the transmit threshold is therefore set to zero.
In Fig. 6, we show that the throughput approximation in (35) is quite accurate by comparing with the exact maximal throughput, which is obtained by numerically evaluating (33). The additional power cost analysis is confirmed by the examples therein. Note that when is very small (e.g., ), (35) becomes less accurate due to the approximation procedure used in Appendix -F and (34) provides a better approximation.
Iv-C Throughput Gain of Adaptation: AE over NAE
In the following, we analyze the throughput gain brought by doing adaptation. With the throughput approximations for the NAE and AE schemes in (22) and (35), in the high SNR region, the throughput gain can be approximated by
With this approximation, several observations can be made:
In the high SNR region, the throughput gain increases with , albeit very slowly, as can be seen from the double logarithm in the first term of (38).
It turns out that as increases, the second term in (38) (i.e., the -independent term) increases slowly. The first term, on the other hand, decreases very fast with (i.e., linearly), and moreover, this term becomes dominant when is large. These observations imply that for high SNR, the throughput gap between the NAE and AE schemes shrinks when the number of antennas grows. Consequently, for systems operating at high SNRs, if the number of antennas is not small, it may be more preferable to employ the NAE scheme rather than the AE scheme, due to the complexity saving.
Fig. 7 compares the approximation in (38) with the exact throughput gain. It is clear that the exact throughput gain decreases with increasing . Note that the variation in the throughput gain is not very significant over a wide range of , though the gain does decrease with strengthening the secrecy constraint (i.e., reducing ). We note that the approximation is not very accurate, since the throughput gain is relatively small and thus the approximation error in (38) is relatively pronounced. What is important is that, as stated above, our analysis accurately predicts the varying trends of the exact throughput gain.
This paper investigated the design of artificial-noise-aided secure multi-antenna transmission in slow fading channels. We provided explicit design solutions to the secrecy-constrained throughput-maximization problem with either non-adaptive transmission (i.e., the NAE scheme) or adaptive transmission (i.e., the AE scheme). To facilitate practical system design, we examined the additional power cost for achieving a higher secrecy level for both schemes and analyzed the throughput gain of the AE scheme over the NAE scheme in the high SNR region. Our analysis obtained new insights on the design of artificial-noise-aided secure multi-antenna transmission in slow fading channels.
-a Proposition 1
In this proof, we derive the optimal values of the system parameters , and to maximize the transmit probability in (12) (i.e., minimize the average delay). We first optimize and for a given , and then optimize . The message rate is assumed to be at a prescribed value.
As we discussed at the beginning of Section III, for the NAE scheme, the overall secrecy outage probability in (9) reduces to (8). For a given , from (8), it is clear that the secrecy constraint in (12) can be satisfied by choosing a large enough . Substituting (8) into the secrecy constraint in (12) (i.e., ) and solving w.r.t. , we have
where is defined in (13) and its parameters are omitted for brevity.
To minimize the average delay, the transmit probability in (6) needs to be maximized. From (6), it is clear that is maximized when the transmit threshold is set to its smallest possible value. From the second constraint in (12), which eliminates capacity outages at Bob, we know that for a given , should be set to
in order to maximize the transmit probability, whilst satisfying the secrecy constraint.
Substituting into (6), the corresponding is given by
Since the regularized upper incomplete gamma function decreases with its second parameter, it is clear that in (39) achieves its maximum when takes its minimum. Note that is a convex function of ; thus, by solving the derivative of w.r.t. , the optimal power allocation ratio can be found as
and the corresponding maximal transmit probability is given by
Summarizing the obtained results, we have Proposition 1.
-B Monotonicity of Quantity in (13)
In this proof, we show that in (13) decreases with increasing . To this end, we temporarily assume that is a continuous variable. Taking the derivative of w.r.t. , we have
To show that the above derivative is negative, we need to show that the expression in the bracket is smaller than . Since the secrecy constraint is smaller than one, we know that . From the inequality for , letting , we see that the quantity in the bracket in (41) satisfies
Thus, we know that the derivative in (41) is negative. Since the discrete is just a sampled version of the continuous , the monotonic properties are the same. We then know that in (13) decreases with increasing .
-C Approximation of Optimal Message Rate in (19)
In this proof, we derive an approximation for the optimal message rate in (17). From the discussions after (18), we know that one can solve the derivative of the throughput w.r.t. to find . However, the equality obtained by setting the derivative to zero is quite complicated and seems not solvable. Hence, we consider first approximating the throughput and then optimizing the approximated throughput. The details can be found as follows.
Using the inequality , a lower bound to can be given by
where , and is defined in (13).
where in (43) we used the standard series expansion of the exponential function.
Ignoring the high order terms in the above series representation and taking the derivative w.r.t. , we have
We then set the above derivative to zero and solve the obtained equality w.r.t. . After summarizing the obtained results, we have the approximation in (19).
-D High SNR Throughput Approximation in (22)
In (19), we have obtained an approximation for the optimal message rate in (17). Plugging an approximation for (19) into a lower bound to the throughput in (44) and ignoring the high order terms, we can get an elegant approximation for the maximal throughput . The details can be found as follows.
In [26, Eq. 65], a series expansion is provided for the principle branch of the Lambert-W function , and it states that as ,
Thus, as , by ignoring the high order terms, (19) can be further approximated as
Taking the leading order term yields
Here , which can be ignored for high , thereby giving (22).
-E Proposition 2
In this proof, we derive the optimal values of the system parameters , and to maximize the message rate in (26) for a given realization of the intended channel , whilst satisfying the secrecy constraint. (Note that for the AE scheme, the design parameters , and are intrinsically functions of .) We first optimize and for a given , and then optimize .
For a given , to eliminate capacity outages at Bob (i.e., the second constraint in (26)), Alice sets to the capacity of Bob’s channel:
For a given , if Alice decides to transmit with above, and with , the secrecy outage probability for this transmission is given by (8) as
which simply increases with increasing . Thus, the maximal will be obtained when the secrecy constraint in (26) is met with equality, i.e., . Solving this equality gives
which is a function of , and is defined in (13).
From (45), we see that when (equivalently, ), it is impossible to adjust to achieve a positive . For the alternative case, when (i.e., ), the range of which can guarantee that is . Henceforth, we focus on the case where .