On Secure Degrees of Freedom of the MIMO Interference Channel with Local Output Feedback

To help the readers to establish the sense of our transmission scheme, we begin this part with an illustrative example, where we set $M=2$ and $N=3$. We will show that we can securely deliver a total of $6$ data symbols over $5$ TSs, i.e., $6/5$ sum-SDoF lower bound. The transmission scheme has three phases, and is elaborated as follows:

Phase-I has $3$ TSs and is used to send AN signals. In each TS of Phase-I, each transmitter sends $2$ fresh AN symbols simultaneously. After Phase-I transmission, each receiver has $9$ equations about $12$ AN symbols. Thus, none of AN symbols can be decoded. After Phase-I transmission, the receivers 1 and 2’s output signals of Phase-I are fed back to receivers 1 and 2, respectively.

Phase-II has $1$ TS and is used to deliver $2$ fresh data symbols for transmitter 1 and $1$ fresh data symbol for transmitter 2. In each TS of Phase-II, transmitter 1 sends $2$ fresh data symbols along with the receiver 1’s output signals of Phase-I, and transmitter 2 sends $1$ fresh data symbol along with the receiver 2’s output signals of Phase-I. Since each receiver has $3$ antennas, it can decode all the received signals. Thus, after Phase-II transmission, receiver 1 can obtain $2$ fresh data symbols and receiver 2 can obtain $1$ fresh data symbols. Meanwhile, receiver 1 acquires extra $1$ equation about $12$ AN symbols, and receiver 2 acquires extra $2$ equations about $12$ AN symbols.

Phase-III has $1$ TS and is used to deliver $1$ fresh data symbol for transmitter 1 and $2$ fresh data symbols for transmitter 2. In each TS of Phase-III, transmitter 1 sends $1$ fresh data symbol along with the receiver 1’s output signals of Phase-I, and transmitter 2 sends $2$ fresh data symbols along with the receiver 2’s output signals of Phase-I. Since each receiver has $3$ antennas, it can decode all the received signals. Thus, after Phase-III transmission, receiver 1 can obtain $1$ fresh data symbol and receiver 2 can obtain $2$ fresh data symbols. Meanwhile, receiver 1 acquires extra $2$ equations about $12$ AN symbols, and receiver 2 acquires extra $1$ equation about $12$ AN symbols.

Overall, each receiver obtains $12$ equations about $12$ AN symbols. This implies no extra decoding ability can be used for eavesdropping on the other receiver’s data symbols. On the other hand, each receiver can obtain $3$ equations about $3$ desired data symbols from Phase-II and Phase-III transmissions, by subtracting the Phase-I output signals. Therefore, a total of $6$ data symbols are securely delivered over $5$ TSs.

To generalize the above example, the major difficulty is how to design the phase duration. This is because, heuristic phase duration assignment cannot guarantee any optimality. To tackle this problem, we first assume the undermined phase duration for the general scheme and then derive the security and decoding constraints with the undermined phase duration. Finally, we determine the optimal phase duration by maximizing the sum-SDoF lower bound with the security and decoding constraints. The flow of the proposed solution technique is given in Fig. 5. In the following, the general transmission scheme with undetermined phase duration is elaborated below, whose flow-diagram is depicted in Fig. 6.

First of all, we define the effective CSI matrices of Phase-I, Phase-II, and Phase-III as follows:

where $i,j=1,2$. In addition, we define the following full rank and pre-assigned matrices for compressing the output signals: ${\bf{\Phi}}_{1}^{a}\in\mathbb{C}^{M\tau_{2}\times N\tau_{1}}$,${\bf{\Phi}}_{2}^{a}\in\mathbb{C}^{(N-M)\tau_{2}\times N\tau_{1}}$,${\bf{\Phi}}_{1}^{b}\in\mathbb{C}^{(N-M)\tau_{2}\times N\tau_{1}}$,${\bf{\Phi}}_{2}^{b}\in\mathbb{C}^{M\tau_{2}\times N\tau_{1}}$. Each matrix can be generated by the standard Gaussian distribution and pre-stored at all transmitters and receivers.

Phase-I (Joint AN Transmission): This phase spans $\tau_{1}$ TSs with the aim of joint AN symbol transmission. In each TS, transmitter 1 sends $M\tau_{1}$ fresh AN symbols by $M$ transmit antennas, and transmitter 2 sends $M\tau_{1}$ fresh AN symbols by $M$ transmit antennas. As a holistic view, we denote all the AN symbols sent from transmitter 1 by $\textbf{u}_{1}\in\mathbb{C}^{M\tau_{1}}$ and all the AN symbols sent from transmitter 2 by $\textbf{u}_{2}\in\mathbb{C}^{M\tau_{1}}$. Consequently, the transmitted signals of Phase-I are expressed as

where we denote the transmitter 1’s transmit signals of Phase-I by $\textbf{x}_{1}^{\text{I}}\in\mathbb{C}^{M\tau_{1}}$ and the transmitter 2’s transmit signals of Phase-I by $\textbf{x}_{2}^{\text{I}}\in\mathbb{C}^{M\tau_{1}}$. At the receivers, the output signals are written by

where we denote the receiver 1’s output signals of Phase-I by $\textbf{y}_{1}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$, the receiver 2’s output signals of Phase-I by $\textbf{y}_{2}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$, and the AWGN at receivers 1 and 2 by $\textbf{z}_{1}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$ and $\textbf{z}_{2}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$, respectively. Besides, the dimension of $\textbf{H}_{i,j}^{\text{I}},i,j=1,2$ is $N\tau_{1}\times M\tau_{1}$. After Phase-I transmission, the output signals $\textbf{y}_{1}^{\text{I}}$ and $\textbf{y}_{2}^{\text{I}}$ are fed back to receivers 1 and 2, respectively.

Phase-II (Simultaneous Data Transmission with Emphasis on Receiver 1): This phase spans $\tau_{2}$ TSs with the aim of fresh data transmission for receivers 1 and 2. In each TS, $M$ fresh data symbols are sent from transmitter 1, and $N-M$ fresh data symbols are sent from transmitter 2. As a holistic view, we denote all the fresh data symbols for receiver 1 by $\textbf{s}_{1}^{a}\in\mathbb{C}^{M\tau_{2}}$ and all the fresh data symbols for receiver 2 by $\textbf{s}_{2}^{a}\in\mathbb{C}^{(N-M)\tau_{2}}$. To match the dimension of data symbols, we also compress the output signals into ${\bf{\Phi}}_{1}^{a}\textbf{y}_{1}^{\text{I}}\in\mathbb{C}^{M\tau_{2}}$ and ${\bf{\Phi}}_{2}^{a}\textbf{y}_{2}^{\text{I}}\in\mathbb{C}^{(N-M)\tau_{2}}$. For security, the transmitted signals of Phase-II are designed by

where we denote the transmitter 1’s transmit signals of Phase-II by $\textbf{x}_{1}^{\text{II}}\in\mathbb{C}^{M\tau_{2}}$ and the transmitter 2’s transmit signals of Phase-II by $\textbf{x}_{2}^{\text{II}}\in\mathbb{C}^{(N-M)\tau_{2}}$. At the receivers, the output signals are written by

where we denote the receiver 1’s output signals of Phase-II by $\textbf{y}_{1}^{\text{II}}\in\mathbb{C}^{N\tau_{2}}$, the receiver 2’s output signals of Phase-II by $\textbf{y}_{2}^{\text{II}}\in\mathbb{C}^{N\tau_{2}}$, and the AWGN at receivers 1 and 2 by $\textbf{z}_{1}^{\text{II}}$ and $\textbf{z}_{2}^{\text{II}}$, respectively. Besides, the dimension of $\textbf{H}_{1,j}^{\text{II}},j=1,2$ is $N\tau_{2}\times M\tau_{2}$ and the dimension of $\textbf{H}_{2,j}^{\text{II}},j=1,2$ is $N\tau_{2}\times(N-M)\tau_{2}$.

Phase-III (Simultaneous Data Transmission with Emphasis on Receiver 2): This phase spans $\tau_{2}$ TSs with the aim of fresh data transmission for receivers 1 and 2. In each TS, $N-M$ fresh data symbols are sent from transmitter 1, and $M$ fresh data symbols are sent from transmitter 2. As a holistic view, we denote all the fresh data symbols for receiver 1 by $\textbf{s}_{1}^{b}\in\mathbb{C}^{(N-M)\tau_{2}}$ and all the fresh data symbols for receiver 2 by $\textbf{s}_{2}^{b}\in\mathbb{C}^{M\tau_{2}}$. To match the dimension of data symbols, we also compress the output signals into ${\bf{\Phi}}_{1}^{b}\textbf{y}_{1}^{\text{I}}\in\mathbb{C}^{(N-M)\tau_{2}}$ and ${\bf{\Phi}}_{2}^{b}\textbf{y}_{2}^{\text{I}}\in\mathbb{C}^{M\tau_{2}}$. For security, the transmitted signals of Phase-III are designed by

where we denote the transmitter 1’s transmit signals of Phase-III by $\textbf{x}_{1}^{\text{III}}\in\mathbb{C}^{(N-M)\tau_{2}}$ and the transmitter 2’s transmit signals of Phase-III by $\textbf{x}_{2}^{\text{III}}\in\mathbb{C}^{M\tau_{2}}$. At the receivers, the output signals are written by

where we denote the receiver 1’s output signals of Phase-III by $\textbf{y}_{1}^{\text{III}}\in\mathbb{C}^{N\tau_{2}}$, and the receiver 2’s output signals of Phase-III by $\textbf{y}_{2}^{\text{III}}\in\mathbb{C}^{N\tau_{2}}$, and the AWGN at receivers 1 and 2 by $\textbf{z}_{1}^{\text{III}}\in\mathbb{C}^{N\tau_{2}}$ and $\textbf{z}_{2}^{\text{III}}\in\mathbb{C}^{N\tau_{2}}$, respectively. Besides, the dimension of $\textbf{H}_{1,j}^{\text{III}},j=1,2$ is $N\tau_{2}\times(N-M)\tau_{2}$ and the dimension of $\textbf{H}_{2,j}^{\text{III}},j=1,2$ is $N\tau_{2}\times M\tau_{2}$.

The major concern of the transmission scheme is that we must ensure the security and decodability after transmission. To this end, we perform the security and decoding analysis in the following. Specifically, the key of security analysis is to verify the information leakage to the other receiver is zero, i.e., $I(\textbf{s}_{1}^{a},\textbf{s}_{1}^{b};\textbf{y}_{2}|\textbf{s}_{2}^{a},\textbf{s}_{2}^{b})=0$. However, the direct verification of zero information leakage is difficult. Hence, we resort to data processing inequality to transform it into a tractable one.

Security Analysis: To ensure security, zero information leakage to the eavesdropper should be guaranteed. For simplicity, we define $\textbf{y}_{2}=[\textbf{y}_{2}^{\text{I}};\textbf{y}_{2}^{\text{II}};\textbf{y}_{2}^{\text{III}}]$ and $\textbf{u}=[\textbf{u}_{1};\textbf{u}_{2}]$. The verification of information leakage $I(\textbf{s}_{1}^{a},\textbf{s}_{1}^{b};\textbf{y}_{2}|\textbf{s}_{2}^{a},\textbf{s}_{2}^{b})=0$ is presented in (16), shown at the top of next page, where the reason of each critical step is given as follows:

1. (a)

   Data processing inequality for Markov chain $(\textbf{s}_{1}^{a},\textbf{s}_{1}^{b},\textbf{u})\rightarrow([\textbf{H}^{\text{I}}_{1,2},\textbf{H}^{\text{I}}_{2,2}]\textbf{u},\textbf{H}_{1,2}^{\text{II}}\textbf{s}_{1}^{a}+\textbf{H}_{1,2}^{\text{II}}{\bf{\Phi}}_{1}^{a}[\textbf{H}_{1,1}^{\text{I}},\textbf{H}_{2,1}^{\text{I}}]\textbf{u}$, $\textbf{H}_{1,2}^{\text{III}}\textbf{s}_{1}^{b}+\textbf{H}_{1,2}^{\text{III}}{\bf{\Phi}}_{1}^{b}[\textbf{H}_{1,1}^{\text{I}},\textbf{H}_{2,1}^{\text{I}}]\textbf{u})\rightarrow\textbf{y}_{2}$.

2. (b)

   When input is circularly symmetric complex Gaussian, according to [30], rewriting into $\log\text{det}(\textbf{I}+\text{SNR}\textbf{A}\textbf{A}^{H})-\log\text{det}(\textbf{I}+\text{SNR}\textbf{B}\textbf{B}^{H})$, and using Lemma 2 in [8].

3. (c)

   The rank of matrix A is $N(\tau_{1}+\tau_{2})$. The rank of matrix B is $\min\{N(\tau_{1}+\tau_{2}),2M\tau_{1}\}$, whose reason is given in Appendix A.

Due to the symmetry of antenna configuration at receivers, the security analysis of the other receiver is similar. Therefore, to ensure security, according to (16), it should be followed that

which is the security constraint. Next, we perform the decoding analysis.

Decoding Analysis: The decoding equation for receiver 1 and 2 is given in (17a) and (17b), respectively, where $\overline{\textbf{z}}_{1}$ and $\overline{\textbf{z}}_{2}$ are the AWGN vectors, and $\textbf{H}_{1}$ and $\textbf{H}_{2}$ are effective decoding channel. Accordingly, the information rate for receiver 1 is evaluated as follows:

where the reason of critical step is given as follows:

1. (a)

   When input is circularly symmetric complex Gaussian, according to [30], rewriting into $\log\text{det}(\textbf{I}+\text{SNR}\textbf{H}_{1}\textbf{H}_{1}^{H})-\log\text{det}(\textbf{I}+\text{SNR}\text{bd}\{\textbf{H}_{2,1}^{\text{II}},\textbf{H}_{2,1}^{\text{III}}\}\text{bd}\{\textbf{H}_{2,1}^{\text{II}},\textbf{H}_{2,1}^{\text{III}}\}^{H})$, and using Lemma 2 in [8].

Therefore, $N$ data symbols desired by receiver 1 can be decoded. Due to the symmetry of antenna configuration at receivers, we have $I(\textbf{s}_{2}^{a},\textbf{s}_{2}^{b};\textbf{y}_{2})=N\log\text{SNR}$ as well. Consequently, the decodability of $N$ data symbols desired by receiver 2 can be guaranteed. With the security and decoding constraints, we are able to optimize the phase duration.

Optimal Phase Duration: We propose to optimize the phase duration by maximizing the sum-SDoF lower bound achieved by our scheme with the security constraint, as the decodability can be always guaranteed,. In the general scheme, we securely deliver $2N\tau_{1}$ data symbols over $\tau_{1}+2\tau_{2}$ TSs. Hence, we formulate a linear-fractional optimization problem, given by

To solve the problem, we propose to re-formulate it as follows:

Since the objective function is inversely proportional to $\tau_{1}/\tau_{2}$, the optimal solution is obtained when $\tau_{1}/\tau_{2}=N/(2M-N)$. Therefore, for the original problem, the optimal value is $2N(2M-N)/(4M-N)$ by setting $\tau_{1}=N$ and $\tau_{2}=2M-N$. This shows that the following sum-SDoF lower bound is attainable:

To help the readers to establish the sense of our transmission scheme, we begin this part with an illustrative example, where we set $M=3$ and $N=2$. We will show that we can securely deliver a total of $12$ data symbols over $7$ TSs, i.e., $12/7$ sum-SDoF lower bound. The transmission scheme has four phases, and is elaborated as follows:

Phase-I has $2$ TSs and is used to send AN signals with $2$ transmit antennas. In each TS of Phase-I, each transmitter sends $3$ fresh AN symbols simultaneously. After Phase-I transmission, each receiver has $4$ equations about $8$ AN symbols. Thus, none of AN symbols can be decoded. After Phase-I transmission, the receivers 1 and 2’s output signals of Phase-I are fed back to receivers 1 and 2, respectively.

Phase-II has $2$ TSs and is used to deliver $6$ fresh data symbols for transmitter 1. In each TS of Phase-II, transmitter 1 sends $3$ fresh data symbols along with the receiver 1’s output signals of Phase-I, and transmitter 2 does not send any signals. Since receiver 1 has $2$ antennas, it cannot decode the fresh data symbols. Meanwhile, receiver 1 does not acquire new equations about $8$ AN symbols, and receiver 2 acquires extra $4$ equations about $8$ AN symbols. After Phase-II transmission, the receiver 2’s output signals of Phase-II are fed back to the receiver 2.

Phase-III has $2$ TSs and is used to deliver $6$ fresh data symbols for transmitter 2. In each TS of Phase-III, transmitter 2 sends $3$ fresh data symbols along with the receiver 2’s output signals of Phase-I, and transmitter 1 does not send any signals. Since receiver 2 has $2$ antennas, it cannot decode the fresh data symbols. Meanwhile, receiver 2 does not acquire new equations about $8$ AN symbols, and receiver 1 acquires extra $4$ equations about $8$ AN symbols. After Phase-III transmission, the receiver 1’s output signals of Phase-III are fed back to the receiver 1.

Phase-IV has $1$ TS and is used to simultaneously provide $1$ equation about data symbols for transmitters 1 and 2. In Phase-IV, transmitter 1 sends the receiver 1’s output signals of Phase-III, and transmitter 2 sends the receiver 2’s output signals of Phase-II. At the receiver 1, it can obtain $1$ equation about the data symbols, by subtracting the receiver 1’s output signals of Phase-III from the received signal. At the receiver 2, it can obtain $1$ equation about the data symbols, by subtracting the receiver 2’s output signals of Phase-II from the received signal. Meanwhile, receivers 1 and 2 do not acquire new equations about $8$ AN symbols.

Overall, each receiver obtains $8$ equations about $8$ AN symbols. This implies no extra decoding ability can be used for eavesdropping on the other receiver’s data symbols. On the other hand, each receiver can obtain $6$ equations about $6$ desired data symbols. Therefore, a total of $12$ data symbols are securely delivered over $7$ TSs.

To generalize the above example, we need to solve the problem of assigning the phase duration. To tackle this problem, we adopt the design technique presented in sub-section-A. In the following, the general transmission scheme with undetermined phase duration is elaborated below, whose flow-diagram is given in Fig. 7.

First of all, we define the effective CSI matrices of Phase-I, Phase-II, Phase-III, and Phase-IV as follows:

where $i,j=1,2$. In addition, we define the following full rank and pre-assigned matrices for shaping the received output at transmitters: ${\bf{\Phi}}\in\mathbb{C}^{M\tau_{2}\times N\tau_{1}}$, ${\bf{\Theta}}\in\mathbb{C}^{N\tau_{3}\times N\tau_{2}}$. Each matrix can be generated by the standard Gaussian distribution and pre-stored at all transmitters and receivers.

Phase-I (Joint AN Transmission): This phase spans $\tau_{1}$ TSs with the aim of joint AN symbol transmission. In each TS, transmitter 1 sends $N\tau_{1}$ fresh AN symbols by $N$ antennas, and transmitter 2 sends $N\tau_{1}$ fresh AN symbols by $N$ antennas. As a holistic view, we denote all the AN symbols sent from transmitter 1 by $\textbf{u}_{1}\in\mathbb{C}^{N\tau_{1}}$ and all the AN symbols sent from transmitter 2 by $\textbf{u}_{2}\in\mathbb{C}^{N\tau_{1}}$. Consequently, the transmitted signals of Phase-I are expressed as

where we denote the transmitter 1’s transmit signals of Phase-I by $\textbf{x}_{1}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$ and the transmitter 2’s transmit signals of Phase-I by $\textbf{x}_{2}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$. At the receivers, the output signals are written by

where we denote the receiver 1’s output signals of Phase-I by $\textbf{y}_{1}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$, the receiver 2’s output signals of Phase-I by $\textbf{y}_{2}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$, and the AWGN at receivers 1 and 2 by $\textbf{z}_{1}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$ and $\textbf{z}_{2}^{\text{I}}\in\mathbb{C}^{N\tau_{1}}$, respectively. Besides, the dimension of $\textbf{H}_{i,j}^{\text{I}},i,j=1,2$ is $N\tau_{1}\times M\tau_{1}$. After Phase-I transmission, the output signals $\textbf{y}_{1}^{\text{I}}$ and $\textbf{y}_{2}^{\text{I}}$ are fed back to receivers 1 and 2, respectively.

Phase-II (Data Transmission from Transmitter 1 to Receiver 1): This phase spans $\tau_{2}$ TSs with the aim of fresh data symbol transmission for receiver 1. In each TS, $M$ fresh data symbols are sent from transmitter 1, and transmitter 2 does not send any signals. As a holistic view, we denote all the fresh data symbols for receiver 1 by $\textbf{s}_{1}\in\mathbb{C}^{M\tau_{2}}$ and the empty signals by $\textbf{0}\in\mathbb{C}^{M\tau_{2}}$. To match the dimension of data symbols, we also compress the output signals into ${\bf{\Phi}}\textbf{y}_{1}^{\text{I}}\in\mathbb{C}^{M\tau_{2}}$. For security, the transmitted signals of Phase-II are designed by

where we denote the transmitter 1’s transmit signals of Phase-II by $\textbf{x}_{1}^{\text{II}}\in\mathbb{C}^{M\tau_{2}}$ and the transmitter 2’s transmit signals of Phase-II by $\textbf{x}_{2}^{\text{II}}\in\mathbb{C}^{M\tau_{2}}$. At the receivers, the output signals are written by

where we denote the receiver 1’s output signals of Phase-II by $\textbf{y}_{1}^{\text{II}}\in\mathbb{C}^{N\tau_{2}}$, the the receiver 2’s output signals of Phase-II by $\textbf{y}_{2}^{\text{II}}\in\mathbb{C}^{N\tau_{2}}$, and the AWGN at receivers 1 and 2 by $\textbf{z}_{1}^{\text{II}}\in\mathbb{C}^{N\tau_{2}}$ and $\textbf{z}_{2}^{\text{II}}\in\mathbb{C}^{N\tau_{2}}$, respectively. Besides, the dimension of $\textbf{H}_{i,j}^{\text{II}},i,j=1,2$ is $N\tau_{2}\times M\tau_{2}$. After Phase-II transmission, the receiver 2’s output signals of Phase-II are fed back to the receiver 2.

Phase-III (Data Transmission from Transmitter 2 to Receiver 2): This phase spans $\tau_{2}$ TSs with the aim of fresh data symbol transmission for receiver 2. In each TS, $M$ fresh data symbols are sent from transmitter 2, and transmitter 1 does not send any signals. As a holistic view, we denote all the fresh data symbols for receiver 2 by $\textbf{s}_{2}\in\mathbb{C}^{M\tau_{2}}$ and the empty signals by $\textbf{0}\in\mathbb{C}^{M\tau_{2}}$. To match the dimension of data symbols, we also compress the output signals into ${\bf{\Phi}}\textbf{y}_{2}^{\text{I}}\in\mathbb{C}^{M\tau_{2}}$. For security, the transmitted signals of Phase-III are designed by