Noncoherent Massive MIMO with Embedded One-Way Function Physical Layer Security

N-DSTC [18] was proposed as a method to circumvent the requirement for perfect CSI estimation of classic coherent systems and to increase the transmission rate of square-matrix based DSTC [16, 21, 24, 25], which become particularly challenging in high-speed mobile and massive MIMO scenarios.

In N-DSTC schemes, $B$ bits of information are mapped into a data matrix $\mathbf{X}(i)\in\mathbb{C}^{M\times M}$ selected from a codebook of $2^{B}$ matrices. Among others, two space-time codebook construction methods are representative: a) the DUC method of [21], and b) the algebraic differential spatial modulation (ADSM) scheme of [24].

In the DUC method [21], the $B$ input bits are mapped into the integers $b=0,1,\cdots,2^{B}-1$, and the corresponding unitary matrix is generated by

where $j$ denotes the elementary imaginary number.

In this scheme, diversity is maximized by ensuring that the $M$ factors $u_{m}$ are designed to maximize the diversity product

while satisfying the condition $0<u_{1}\leq\cdots u_{M}\leq 2^{B}/2\in\mathbb{Z}$.

Notice that the search space size of problem (4) can be calculated as ${2^{B-1}+M-1\choose M}$, such that for $M=32,64,128,$ and $256$ transmit antennas, we have ${3M-1\choose M}$ candidates to map $B=\mathrm{log}_{2}(M)+2$ bits, which corresponds to $2\times 10^{25},5\times 10^{51},4\times 10^{104},$ and $4\times 10^{210}$ candidates, respectively, making problem (4) highly complex.

In turn, in the ADSM method [24], the $B$ input bits are partitioned into two sequences of length $B_{1}=\log_{2}(M)$ and $B_{2}=\log_{2}(L)$ bits, respectively, with the first $B_{1}$ bits used to select a specific dispersion matrix (DM) out of a codebook of DM matrices $\mathbf{A}_{m}$ given by

which implies that $\mathcal{A}$ is a set of unitary matrices.

At each $i$-th transmission block, a given DM $\mathbf{A}_{m}(i)$ is selected to encode the first $B_{1}$ bits, while the remaining $B_{2}$ bits are mapped to a symbol $s(i)\in\mathcal{S}$, with $\mathcal{S}$ denoting the $L$-PSK constellation, such that the following data matrix is generated

To elaborate, in the N-DSTC scheme, at the beginning of a transmission stream, the set of projection matrices $\{\mathbf{E}_{1},\cdots,\mathbf{E}_{M/T}\}$ are transmitted. During this initial stage, the received symbol is expressed as

for $1\leq i\leq M/T$. Notice that this requires $M$ timeslots.

After the projection matrices are communicated to the receiver, and assuming a transmission frame of $W$ blocks, the subsequent data matrices $\mathbf{X}(i)$ are differentially encoded as

mapped onto $M\times T$ nonsquare matrices via multiplication with the projection matrix $\mathbf{E}_{1}$, and transmitted such that the corresponding received signal becomes

for $M/T<i\leq W$.

The ratio between the number of transmit antennas $M$ and the frame length $W$, $i.e.,\eta\triangleq M/W$, is referred to as the reference insertion ratio. For the sake of performance assessment, we will set $\eta$ to 5%, such that $W=20M$, but we remark that the performance of N-DSTC schemes remain robust in high-speed mobile scenarios [19] even with $\eta=1\%$ and $\eta=0.1\%$, which yield frames of length $W=100M$ and $W=1000M$, respectively.

The N-DSTC scheme relies on a basis of projection matrices $\mathbf{E}_{i}\in\mathbb{C}^{M\times T}$, constructed to satisfy the constraints [18]:

for all $1\leq i\neq i^{\prime}\leq M/T$.

A strategy to obtain the projection matrix basis is to partition a unitary matrix $\mathbf{U}_{M}\in\mathbb{C}^{M\times M}$, such that $\mathbf{E}_{i}$ is given by the $(i-1)T$-th to the $iT$-th columns of $\mathbf{U}_{M}$, that is

In turn, in [18] it was proposed to construct the unitary matrix $\mathbf{U}_{M}$ for a system with $M$ transmit antennas, and given a number $N_{b}$ of nonzero components in each column, as follows

with $\mathbf{W}$ denoting the discrete Fourier transform matrix

where $\omega\triangleq\mathrm{exp}(-2\pi j/N_{b})$.

Notice that $\mathbf{E}_{1}$ becomes more sparse as $N_{b}\to 1$, and denser as $N_{b}\to M$. Also, since $M/N_{b}$ has to be an integer, we shall set $N_{b}=1,2^{1},2^{2},\cdots,2^{\mathrm{log_{2}}(M)}=M$ in this paper.

In light of the design described above, a simple (if suboptimal) noncoherent N-DSTC receiver is [18]

where the key component $\hat{\mathbf{Y}}(i)$ is defined by

while the constant matrices $\mathbf{E}^{(\alpha)}$ and $\mathbf{E}^{(1-\alpha)}$ are defined as

In equations (19a) and (19b), $\alpha(i)$ is a forgetting factor that determines the ratio of $\mathbf{Y}(i)$ and $\hat{\mathbf{Y}}(i-1)$ added together, which can be updated via the closed-form expression [19]

where

with the adaptive forgetting factor $\alpha_{v}(i)$ changed depending on the SNR and the difference between $\mathbf{H}(i)$ and $\mathbf{H}(i-1)$ [19], and $\mathbf{D}(i)\triangleq\mathbf{Y}(i)-\hat{\mathbf{Y}}(i-1)\hat{\mathbf{X}}(i)\mathbf{E}_{1}$.

For starters, let us analyze the performance of N-DSTC when $\mathbf{E}_{1}$ satisfies only the transmit power constraint of equation (13a), and identify the minimum condition under which the maximum performance is achieved. To that end, first consider the simplest when $(N_{b},T)=(M,1)$. In this case, $\mathbf{E}_{1}\in\mathbb{C}^{M\times 1}$ is a vector, which for the sake of clarity is denoted $\mathbf{e}_{1}\in\mathbb{C}^{M\times 1}$. When $\mathbf{e}_{1}$ satisfies the power constraint and the power is equally distributed among transmit antennas, $\mathbf{e}_{1}$ can be expressed as

where each $e_{m}=e^{j\theta_{m}}$, with $1\leq m\leq M$ and $0\leq\theta_{m}<2\pi$, lies on the unit-radius circle on the complex plane.

A good performance metric to evaluate the projection vector $\mathbf{e}_{1}$ is the coding gain defined as the minimum, among all pairs, Euclidean distance between projected data matrices $\mathbf{X}_{p}\mathbf{e}_{1}$ and $\mathbf{X}_{q}\mathbf{e}_{1}$, that is [26]

where $\mathcal{X}\triangleq\{\mathbf{X}_{1},\cdots,\mathbf{X}_{2^{B}}\}\in\mathbb{C}^{M\times M}$ is the codebook of all ${2^{B}}$ possible data matrices $\mathbf{X}$.

The above implies that the optimal projection vector $\mathbf{\mathbf{e}_{1}}$ is the solution of the problem

which is highly complex, as it requires the evaluation of the Euclidean distances of all pairs of $M\times M$ matrices in the set $\mathcal{X}$, of cardinality $2^{B}$, at the cost of $\mathcal{O}(M^{3})$ per pair.

Thanks to the fact that DUC codewords are diagonal and already optimized via discrete combinatorial optimization [21], the coding gain defined in equation (23) does not depend on $\mathbf{e}_{1}$ in the particular case of DUC mapping. More generally, and in the case of ADSM, however, the coding gain of equation (23) varies with $\mathbf{e}_{1}$, requiring a solution of the highly complex problem (24). In order to avoid the associated complexity, we first show in Appendix A, that equation (23) can be simplified to

with the functions $g_{1}(\mathbf{e}_{1})$ and $g_{2}(\mathbf{e}_{1})$ respectively given by

Thanks to equation (25), problem (24) can be significantly simplified as follows.

First, notice that $g_{1}(\mathbf{e}_{1})$ is not actually a function of $\mathbf{e}_{1}$ and therefore is irrelevant to the maximization of coding gain via selection of the projection vector $\mathbf{e}_{1}$ as per equation (24). In turn, maximizing $g_{2}(\mathbf{e}_{1})$ directly is challenging, because the term $\Re\{s\,\mathbf{e}_{1}^{H}\mathbf{M}^{n}\mathbf{e}_{1}\}$ rotates both with $n$ and $s$, such that a given $L$-PSK symbol $s$ that minimizes $\Re\{s\,\mathbf{e}_{1}^{H}\mathbf{M}^{n}\mathbf{e}_{1}\}$ for a given value of $n$, maximizes the same term for another value of $n$ corresponding to a rotation of $\pi/2$ radians, and vice-versa.

In light of the above, and noticing that the term $\{s\,\mathbf{e}_{1}^{H}\mathbf{M}^{n}\mathbf{e}_{1}\}$ is a complex number of radius $|\mathbf{e}_{1}^{H}\mathbf{M}^{n}\mathbf{e}_{1}|$, it follows that a robust strategy to maximize $g_{2}(\mathbf{e}_{1})$ for all values of $n$ and $s$ is to minimize the average amplitude of the term $\mathbf{e}_{1}^{H}\mathbf{M}^{n}\mathbf{e}_{1}$ over $n$, such that the following alternative problem may be proposed as a relaxation of problem (24)

where the objective function $f(\mathbf{e}_{1})$ is defined as

Unlike the original problem (24), which is highly complex and dependent on a combinatorially large number of evaluation of a cost function of cubic complexity order $\mathcal{O}(M^{3})$, the latter problem (28) is, although not convex, a continuous manifold optimization problem, based on an objective of complexity order $\mathcal{O}(M^{2})$, which can be solved efficiently. In fact, in some cases, solutions of problem (28) can be obtained in closed form, as is the case for $(M,L)=(2,4)$. Specifically, in this case we can set $\mathbf{e}_{1}=\left[e^{j\theta_{1}}~{}e^{j\theta_{2}}\right]^{T}/\sqrt{2}$, without loss of generality, and search for the pair $(\theta_{1},\theta_{2})$ that minimize $f(\mathbf{e}_{1})$, which leads to the solution $(\theta_{1},\theta_{2})=(0,\pi/4)$. This solution yields the optimum projection vector $\mathbf{e}_{1}=[1~{}e^{j\frac{\pi}{4}}]^{T}/\sqrt{2}\in\mathbb{C}^{2\times 1}$, which in fact results in $f(\mathbf{e}_{1})=0$.

In the more general case, the optimization problem (28) can be solved via any number of publicly available standard optimization methods such as COBYLA, SLSQP [27], BFGS, L-BFGS, and Powell [28]. The performance of these methods will be compared in Subsection IV-C, and examples of angle vectors resulting in optimal $\mathbf{e}_{1}$ designs for various values of $M$ are given in Table I.

Notice that the projection vector design of Subsection IV-A generally yields fully dense solutions ($i.e.,|\mathbf{e}_{1}|_{0}=M$). Allowing for sparsity in the projection may be desired, especially in large scale MIMO systems, as it enables a reduction of the number of RF chains required to implement the scheme. In addition, it is also know that the coding gain of ADSM-based schemes increases with sparsity [24] in the projection. These facts motivate us to seek a sparse alternative to the design described in Subsection IV-A.

On the other hand, recall that the utilization of a projection vector $\mathbf{e}_{1}$, as opposed to a projection matrix $\mathbf{E}_{1}$, represents already a significant compression of transmission instances to $T=1$. With the additional insertion of sparsity into $\mathbf{e}_{1}$, the total number of degrees of freedom of the system is further reduced, which therefore motivates us also to extend the design of Subsection IV-A to the case when $T>1$.

These two modifications are the objectives of this subsection. For the sake of clarity, we shall hereafter denote the original projection vector $\mathbf{e}_{1}$ of Subsection IV-A by $\mathbf{e}_{1}(M,M)$, so as to explicitly indicate that it is designed for $M$ transmit antennas, with $M$ nonzero entries. Accordingly, a sparse projection vector with $|\mathbf{e}_{1}|_{0}=N_{b}\leq M$ shall be denoted by $\mathbf{e}_{1}(M,N_{b})\in\mathbb{C}^{M\times 1}$, which can be constructed as follows

For example, if we consider the case with $M=4$ and $N_{b}=2$, the corresponding extended basis is given by

The sparse projection vector described by equation (30) is shown in Appendix B to retain the same coding gain as that maximized under the solution of problem (28), and therefore can be said to be the optimal sparse solutions of the latter.

Finally, a projection vector $\mathbf{e}_{1}(M,N_{b})$ can be extended into a projection matrix denoted by $\mathbf{E}_{1}(M,N_{b},T)\in\mathbb{C}^{M\times T}$, which can be constructed as

where $\mathbf{P}$ is a permutation matrix defined as

Notice that left-multiplication of the matrix $\mathbf{P}^{n}$ onto a vector $\mathbf{v}$ yields the $n$-rotation of the elements of $\mathbf{v}$ in the downward direction.

For example, if $\mathbf{v}=[1~{}2~{}3~{}4]^{T}$, then $\mathbf{P}\mathbf{v}=[4~{}1~{}2~{}3]^{T}$, $\mathbf{P}^{2}\mathbf{v}=[3~{}4~{}1~{}2]^{T}$, and so on. It follows that the projection matrix designed via equation (32) consists of column-wise rotations of the design of equation (30). For example, with $M=4,N_{b}=2$, and $T=2$ we obtain

As mentioned above, the design of dense projection vectors based on the solution of problem (28) can be accomplished by standard optimization methods such as COBYLA, SLSQP, BFGS, L-BFGS, and Powell [27, 28]. In this subsection we offer a numerical comparison of the performance of these methods.

We start with Fig. 1, which shows histograms of the coding gains $g(\mathbf{e}_{1})$ obtained from equation (23) with dense projection vectors $\mathbf{e}_{1}$ obtained by solving problem (28) via various optimization methods, for systems with $M=64$ transmit and $N=2$ receive antennas. It is found that all methods yield gains $g(\mathbf{e}_{1})>1.26$, with the BFGS method proving most effective in achieving the highest gains, $g(\mathbf{e}_{1})\approx 1.41$, with the highest probabilities, followed closely by the SLSQP method.

Next, we compare the convergence time of each of the latter optimization methods as a function of the system size. The results are shown in Fig. 2 and indicate that all methods are relatively fast, with the solutions for systems with $M=64$ antennas obtained in around 10 seconds with the slowest approach (Powell method), and in less than 1 second with the fastest (COBYLA method). Considering both sets of results, we hereafter adopt the SLSQP method as it is found to be both fast and effective in achieving a high coding gain with a high probability.

Finally, Fig. 3 shows the initial and converged points of the optimization problem (28) solved by SLSQP method, with $M=2$. It can be seen that the objective function has many local minima, such that the final convergence point is highly distinct. Some initial points converge to the nearest minima, some do not, which makes it difficult for eavesdroppers to estimate the initial value from the converged value, even if $M=2$.

Straightforwardly, the proposed N-DSTC scheme can be summarized as follows:

1. 1.

   Assuming that the transmitter Alice and the receiver Bob share a secret key⁴⁴4For a full physical-layer implementation, $z$ can be assumed to be extracted from the channel between Alice and Bob [8]., $z\in\mathbb{Z}$, the pair locally generates an identical pseudo-random initial vector $\mathbf{e}_{1}^{(0)}\in\mathbb{C}^{M\times 1}$ based on the seed $z$.

2. 2.

   Starting from $\mathbf{e}_{1}^{(0)}$, Alice and Bob locally solve problem (28) using the same optimization method and parameters ($e.g.$ number of iterations, step sizes, etc.) which may also be pseudo-randomly determined by the shared seed $z$, thus obtaining solution vector $\mathbf{e}_{1}$ common to the pair.

3. 3.

   Using the common vector $\mathbf{e}_{1}$, Alice and Bob locally produce $\mathbf{E}_{1}$ via equations (30) and (32).

4. 4.

   During the first $T$ instances Alice transmits a random unitary matrix $\mathbf{U}_{M}=[\mathbf{U}_{1},\cdots,\mathbf{U}_{M/T}]\in\mathbb{C}^{M\times M}$, subsequently transmitting its payload data matrices $\mathbf{X}(i)$, differently encoded and encrypted by $\mathbf{E}_{1}$, yielding at Bob the received symbol blocks

   $$\mathbf{Y}(i)\!=\!\begin{cases}\mathbf{H}(i)\mathbf{U}_{i}+\mathbf{V}(i)&\!\!\text{for }1\!\leq\!i\!\leq\!M/T,\\ \mathbf{H}(i)\underbrace{\mathbf{X}(i\!-\!1)\mathbf{X}(i)}_{\triangleq\mathbf{S}(i)}\mathbf{E}_{1}\!+\!\!\mathbf{V}(i)&\!\!\text{for }i>M/T,\\[-12.91663pt] \end{cases}$$

   (46)

   where $\mathbf{S}(i)$ are the differentially-encoded symbol blocks equivalent to those constructed via equation (11) in conventional schemes, and $\mathbf{X}(M/T)\triangleq\mathbf{I}_{M}$.

5. 5.

   Bob differentially decodes and decrypts the messages from Alice by computing

   $$\hat{\mathbf{X}}(i)=\mathop{\rm arg~{}min}\limits_{\mathbf{X}}||\mathbf{Y}(i)-\hat{\mathbf{Y}}(i-1)\,\mathbf{X}\,\mathbf{E}_{1}||^{2}_{F},$$

   (47)

   with $\hat{\mathbf{Y}}(i)$ defined as

   $$\hat{\mathbf{Y}}(i)\!\triangleq\!\begin{cases}\sum\limits_{k=1}^{M/T}\!\mathbf{Y}(k)\mathbf{U}^{H}_{k}&\!\!\!\text{for }i\!=\!M/T,\\ \mathbf{Y}(i)\mathbf{E}^{(1-\alpha)}\!+\!\!\hat{\mathbf{Y}}(i\!-\!1)\hat{\mathbf{X}}(i)\mathbf{E}^{(\alpha)}&\!\!\!\text{for }i\!>\!M/T,\end{cases}$$

   (48)

   the constant matrices $\mathbf{E}^{(\alpha)}$ and $\mathbf{E}^{(1-\alpha)}$ defined as

   	$\displaystyle\mathbf{E}^{(\alpha)}\triangleq\mathbf{I}_{M}-\mathbf{E}_{1}\mathbf{E}^{(1-\alpha)},$		(49a)
   	$\displaystyle\mathbf{E}^{(1-\alpha)}\triangleq(1-\alpha(i))\mathbf{E}_{1}^{H},$		(49b)

   and $\alpha(i)$ as given in equation (20).

6. 6.

   Restart the procedure from step 1) periodically.

Notice that unlike conventional schemes reviewed in Subsection III-C, the proposed scheme does not rely on an entire set of compression matrices $\mathbf{E}_{1},\mathbf{E}_{2},\cdots,\mathbf{E}_{M/T}$, but rather on a single projection/encryption matrix $\mathbf{E}_{1}$. This feature not only simplifies the ML detection process, as can be inferred by direct comparison of equations (19) and (49), but also is crucial for security, since $\mathbf{E}_{1}$ is uniquely determined by $\mathbf{E}_{2},\cdots,\mathbf{E}_{M/T}$.

Furthermore, also unlike conventional schemes, the secret projection/encrypting matrix $\mathbf{E}_{1}$ is never exposed to Eve, nor is it related to the unitary matrix transmitted during the initialization phase of the differentially encoding procedure, in contrast to conventional schemes, $e.g.$ as in equation (14).

In light of the features highlighted above, in order to decrypt the secret messages exchanged between Alice and Bob, an eventual eavesdropper Eve must extract $\mathbf{E}_{1}$ directly from its own received signals, subject to a distinct channel. In the next subsections, we formulate the best attack Eve can inflict onto the proposed scheme and subsequently analyze the secrecy rate of the method.

In order to assess the secrecy performance of the N-DSTC scheme with embedded one-way function physical layer security under the most strenuous conditions, an idealized scenario highly beneficial to Eve is considered, namely:

• •

  Perfect CSI: the channel matrix $\mathbf{H}_{\mathrm{Eve}}$ from Alice to Eve is assumed to be perfectly known, despite the fact that the N-DSTC scheme does not incorporate any procedure for channel estimation.

• •

  Noise-free Prior Information: it is assumed that Eve is in knowledge of the first transmitted message $\tilde{\mathbf{X}}\triangleq\mathbf{X}(M/T+1)$ and in possession of a noise-free copy of the corresponding received signal, namely

  $$\tilde{\mathbf{Y}}_{\mathrm{Eve}}=\mathbf{H}_{\mathrm{Eve}}\tilde{\mathbf{X}}\mathbf{E}_{1}.$$

  (50)

• •

  Coherent Detection: combining the perfect CSI assumption with the prior information summarized in equation (50), and in view of the N-DSTC transmission model of equation (46), it is assumed that Eve can attempt to detect Alice’s messages coherently from the signal model

  $$\mathbf{Y}_{\mathrm{Eve}}(i)=\mathbf{H}_{\mathrm{Eve}}\,\mathbf{X}(i)\,\mathbf{E}_{1}+\mathbf{V}_{\mathrm{Eve}}(i).$$

  (51)

• •

  System Information: it is assumed that Eve is in full knowledge of the fact that Alice and Bob generate $\mathbf{E}_{1}$ via the solution of problem (28) with the objective function given in equation (29).

Under the highly favorable conditions described above, Eve can mount the following idealized attack⁵⁵5Referring to equation (22), notice that the search space of $\mathbf{E}_{1}$ is of dimension $2^{\beta M}$, where $\beta$ denotes the resolution of $\theta_{m}$, and that under practical conditions Eve must also estimate $\mathbf{H}_{\mathrm{Eve}}$ and infer $\mathbf{X}(M/T+1)$ while searching for $\mathbf{E}_{1}$, making brute-force approaches virtually impossible with sufficiently large $\beta$ and/or $M$. aiming at extracting the encrypting projection matrix $\mathbf{E}_{1}$

and subsequently attempt to decipher $\mathbf{X}(i)$ by solving

The resilience of the proposed PLS-protected N-DSTC scheme described in Subsection V-A to the eavesdropping attack described above can be assessed via its secrecy rate under a finite-alphabet signaling which is given by [30, 31, 32, 33]

where $I_{\mathrm{Bob}}$ and $I_{\mathrm{Eve}}$ denote the average mutual information (AMI) between the information source Alice, and Bob or Eve, respectively, which are given by [34]

with $\mathbf{Y}_{\mathrm{Bob}}$ simplified, for analytical purposes to

and

where we remind that $L\triangleq 2^{B}$, and the auxiliary indices $p$ and $q$, with $1\leq p,q\leq L$ are used to indicate distinct symbol matrices in the codebook $\mathcal{X}\triangleq\{\mathbf{X}_{1},\cdots,\mathbf{X}_{2^{B}}\}\in\mathbb{C}^{M\times M}$.

In light of equations (51) and (56), the conditional probabilities in equations (55) and (57) are respectively given by

Substituting equations (58) and (59) into (55) and (57), respectively, yields

where

In this subsection we compare the proposed nonsquare coding (ADSM with the proposed projection vector/matrix designed in Subsections IV-A and IV-B) and the conventional nonsquare coding (ADSM and DUC with the conventional projection vector/matrix designed in Subsection III-C) in terms of the coding gain and the bit error rate (BER).

Fig. 4 compares the coding gain achieved with projection vectors obtained from equation (30), systems with parameters $M=64,N=2,T=1$, and $B=8$, as a function of the sparsity of the vector, ranging from $N_{b}=1$ to $64$. As references, curves corresponding to the conventional nonsquare ADSM and DUC scheme are also included⁶⁶6In the DUC case, the factors $u_{1},u_{2},\cdots,u_{M}$ are optimized as described in [21] and indicated by equation (4), but only $N_{b}$ elements are utilized..

It can be seen that the coding gain of conventional nonsquare ADSM schemes decreases as the projection vector becomes more dense, with the opposite occurring in DUC approach. In turn, the coding gain achieved using projection vectors designed via the method proposed in Subsections IV-A and IV-B is found to be independent of sparsity, and to upper-bound those achieved by the latter schemes.

Fig. 5 compares the coding gain achieved with projection matrices obtained from equation (32), systems with parameters $M=64,N=2,N_{b}=64$, and $B=8$, as the number of timeslots, ranging from $T=1$ to $16$. Notice that with such parameterization, the matrices are actually dense. It can be seen that in this case the proposed dense matrix design via (32) achieves max coding gain, still outperforming the DUC approach. It is also seen, furthermore, that the gain achieved under maximum compression ($i.e.$, with $T=1$) is not much lower than that achieved with $T=16$, which, taken together with the implications of the value of $T$ onto the rate of the system, indicates the desirability of designs with the $T\to 1$. The latter feature will also be exploited in Section V, where the proposed scheme is further extended to incorporate physical layer security.

Next, Fig. 6 shows the BER comparison of the proposed and conventional codings, system with parameters $M=64,N=2,N_{b}=4$ or $64,T=1$, and $B=8$. The results corroborate those of Figs. 4 and 5, by showing that the BER performance of the proposed schemes is, unlike that of conventional methods, independent of sparsity.

Finally, Fig. 7 shows the BER comparison of the proposed and conventional schemes, system with parameters $M=64,N=2,N_{b}=64,T=2$, and $B=8$. It is seen that the proposed method achieves the same performance as the DUC, despite the significantly lower complexity. Notice that the BER of the proposed method, with $T=2$ is much better than that with $T=1$, however, the transmit rate $R\triangleq B/T~{}\text{[bit/symbol]}$ decreases from $R=8$ to $R=4$. These results reflect the results of Fig. 5 for the $T=2$ case. Further BER performance improvement can be obtained, albeit at the sacrifice of transmission rate, by setting $T>2$.

For the benefit of the eavesdropper, we evaluate in this subsection the secrecy performance of the proposed N-DSTC scheme with embedded security with parameters $N_{b}=M$ and $T=1$, since the sparsity resulting from setting $N_{b}<M$ and $T>1$ on the one hand was shown in Subsection VI-A not to negatively impact the BER performance of the proposed scheme, and on the other hand only increases the complexity of solving equation (52). In addition, in what follows, we denote the number of receive antennas at Bod and Eve respectively by $N_{\mathrm{Bob}}$ and $N_{\mathrm{Eve}}$.

Our first assessment, depicted in Fig. 8, is the information leakage probability to Eve, measured as $1-2\times\mathrm{BER}_{\text{Eve}}$ [35], with $\mathrm{BER}_{\text{Eve}}$ given by the average bits in error in the symbols $\hat{\mathbf{X}}(i)$ detected by Eve under equation (53), using projection matrices $\hat{\mathbf{E}}_{1}$ estimated by solving equation (52), and given received signals $\tilde{\mathbf{Y}}_{\mathrm{Eve}}$ obtained in the absence of noise, as described by equation (50). As can be seen from the definition of channel reliability [35], which in the context hereby can be understood as a leakage probability, when $\mathrm{BER}_{\text{Eve}}=0.5$ Eve’s entropy becomes zero.

The figure shows plots of such leakage as a function of the number of transmit antennas $M$, varied from $M=2$ to $128$, and for different configurations receive antennas at Eve, varying from $N_{\mathrm{Eve}}=2$ to $N_{\mathrm{Eve}}=16$.

It can be seen that even under the highly idealized conditions assumed in favor of Eve, zero leakage can be achieved by the proposed scheme, as long as the number of the transmit antennas $M$ of the system is significantly larger than the number of receive antennas employed by Eve since $N_{\mathrm{Eve}}$ degrees of freedom are not sufficient for estimating the $M$ variables in $\mathbf{e}_{1}$. We remark that the curves must be interpreted as fundamental bounds, not only because the assumption $\text{SNR}=\infty$ at Eve is impractical, but also because in realistic conditions it is very hard for Eve to estimate $\mathbf{H}_{\mathrm{Eve}}$ at all (let alone perfectly), since the N-DSTC scheme does not include a channel estimation procedure between Alice and Bob ($i.e.$, no pilot symbols are transmitted by the system).

Next, we compare in Fig. 9 average mutual information from Alice to Bob and Eve, respectively, as given by equations (60) and (61), with $M=16,~{}N_{\mathrm{Bob}}=2,~{}N_{\mathrm{Eve}}=8$ and as a function of the SNR, assumed to be the same both at Bob and Eve. For the sake of convenience, the secrecy rate obtained by equation (54) is also included. The results highlight the consistency of the proposed secure N-DSTC scheme, as it can be seen that under the evaluated conditions Bob enjoys monotonically increasing AMI and secrecy rate, as SNR increases, with the maximum rate of 6 bit/symbol (determined by $B$) achieved at around $\text{SNR}=20$ dB.

Finally, we study in Fig. 10 the evolution of the secrecy rate achieved when the number of receive antennas employed by Eve increased to $N_{\mathrm{Eve}}=4,8,16$, in a system with a fixed number of transmit antennas, set to $M=16$, and of receive antennas employed by Bob, set to $N_{\mathrm{Bob}}=2$. The results highlight the robustness of the scheme, as it can be seen that a secrecy rate equal to the achievable rate of the system, namely $B=6$ bit/symbol, can be reached at reasonable SNRs even when Eve has double the number of receive antennas as Bob, $i.e.$, $N_{\text{Eve}}=4$ and $N_{\text{Bob}}=2$, respectively.

In fact, it is found that even if Eve has four times the number of antennas employed by Bob ($N_{\text{Eve}}=8$ against $N_{\text{Bob}}=2$), the system still reaches a secrecy rate only a fraction of a bit away from the maximum achievable rate. It is only when Eve employs as many receive antennas as the transmitter itself, $i.e.$ $N_{\text{Eve}}=M=16$, that the secrecy rate of the system drops to a fraction of the total achievable rate.

We remark that the fact that secrecy performance of the proposed secure N-DSTC is higher with increasing $M$, combined with the fact that the BER performance of the system is unaffected by sparsity in the projection matrices $\mathbf{E}_{1}$, make the contribution here presented particularly attractive to massive MIMO systems, since systems to a large number of transmit antennas and smaller number of RF chains can take full advantage of the embedded physical security provided by the proposed scheme, without any compromise in rate-performance.