The Generalized Degrees-of-Freedom of the Asymmetric Interference Channel with Delayed CSIT

The degrees-of-freedom (DoF) characterization with delayed channel state information at the transmitter (CSIT) has attracted a plenty of research interests in the past decade. For example, the DoF region of multiple-input multiple-output (MIMO) interference channel with delayed CSIT was derived in [1]. The study of DoF of MIMO broadcast channel with delayed CSIT can be found in [2, 3, 4, 5], whose exact value was still not completely obtained. The linear DoF region of MIMO X channel with delayed CSIT was characterized in [6].

One limitation of DoF is that the strength of each link is assumed to be equal, whereas the link strength can vary tremendously in wireless. Nevertheless, the generalized degrees-of-freedom (GDoF) overcomes this drawback by considering the different strength of each link, which was first proposed in [7]. The GDoF characterization with delayed CSIT can be found in [8, 9, 10, 11]. In [8], the GDoF was studied in the two-user multiple-input single-output (MISO) broadcast channel under alternating delayed, perfect, and no CSIT, where sum-GDoF upper and lower bounds were shown to be partially coincided. In [9], the secure GDoF was investigated in the two-user MISO broadcast channel with an external eaversdropper and alternating delayed, perfect, and no CSIT. The GDoF region of the MIMO Z channel with delayed CSIT was characterized in [10]. For MIMO interference channel with delayed CSIT and symmetric interfering link strengths, symmetric GDoF upper and lower bounds were derived in [11], where the symmetric GDoF was characterized partially, and the upper and lower bounds do not change with antenna ratio, i.e., number of antenna at each transmitter over number of antennas at each receiver, if the antenna ratio is equal to or larger than two¹¹1A representative antenna setting for this antenna configuration case is that the transmitter has 2 antennas and the receiver has 1 antenna..

In this paper, we study the GDoF of the asymmetric interference channel with delayed CSIT, where the transmitter has two antennas, the receiver has one antenna, and the strength for each interfering link can vary. We characterize the sum-GDoF by presenting matched converse and achievability proof. Then, via this result, we reveal that in our antenna setting, the symmetric GDoF lower bound in [11] can be elevated, and the symmetric GDoF upper bound in [11] is tight in fact. The idea of elevating the GDoF lower bound comes from [9], where the authors in [9] fully exploit the receiver signal space by sending new fresh symbols in each time slot. Our transmission scheme generalizes the scheme in [9, Appendix C], which was designed for achieving the corner points of GDoF region of MISO broadcast channel with delayed CSIT.

The considered asymmetric interference channel has two transmitters, denoted by $\text{Tx}_{1}$ and $\text{Tx}_{2}$, each with two antennas, and two receivers, denoted by $\text{Rx}_{1}$ and $\text{Rx}_{2}$, each with one antenna, as depicted in Fig. 1. The transmitter $\text{Tx}_{i},\,i=1,2,$ sends private message $W_{i}$ to the receiver $\text{Rx}_{i}$. The received signals at two receivers and time slot $t$ can be written as

where $\textbf{x}_{i}[t]\in\mathbb{C}^{2\times 1}$ denotes the transmitted signal from transmitter $\text{Tx}_{i}$ at time slot $t$, $y_{j}[t]\in\mathbb{C}^{1},\,j=1,2,$ denotes the received signal at receiver $\text{Rx}_{j}$ and time slot $t$, $\textbf{h}_{ji}\in\mathbb{C}^{1\times 2}$ denotes the channel matrix from transmitter $\text{Tx}_{i}$ to receiver $\text{Rx}_{j}$, $n_{j}\sim{\cal{CN}}(0,\sigma^{2})$ denotes the additive White Gaussian noise (AWGN) at receiver $\text{Rx}_{j}$, the channel gains of desired links and interfering links are denoted by $\sqrt{\rho}$ and $\sqrt{\rho^{\alpha_{i}}}$ with $0<\rho$ and $0\leq\alpha_{i}$, respectively. Without loss of generality, we assume that $\alpha_{2}\leq\alpha_{1}$. The transmit signals follow an average power constraint, given by $\frac{1}{n}\sum_{t=1}^{n}\text{tr}\left(\mathbb{E}\{\textbf{x}_{i}[t]\textbf{x}_{i}[t]^{H}\}\right)\leq 1$. All entries of channel matrices are independent and identical distributed (i.i.d.) across space and time slot. The signal-to-noise ratio (SNR) at each receiver is $\rho$, and interference-to-noise ratio (INR) at receivers $\text{Rx}_{1}$ and $\text{Rx}_{2}$ are $\rho^{\alpha_{2}}$ and $\rho^{\alpha_{1}}$, respectively. We define the following assembly of channel matrices: $\mathcal{H}[t]\triangleq\{\textbf{h}_{ji}[t]\}_{i,j=1,2}$, and $\mathcal{H}^{\tau}\triangleq\{\mathcal{H}[t]\}_{t=1}^{\tau}$.

Due to feedback delay, the transmitter has delayed channel state information. Specifically, at time slot $t$, the transmitters know $\mathcal{H}^{\tau}$, namely all the channel matrices up to time slot $t-1$. Two receivers have instantaneous knowledge of channel matrices. The encoding function at transmitter $\text{Tx}_{i}$ and time slot $t$ is denoted by $e_{i,t}(W_{i},\mathcal{H}^{t-1})$. The decoding function at receiver $\text{Rx}_{j}$ after $n$ time slots is denoted by $c_{j,n}(W_{i},\mathcal{H}^{n})$.

The rate tuple is written as $(R_{1}(\rho,\alpha_{1},\alpha_{2}),R_{2}(\rho,\alpha_{1},\alpha_{2}))$, where rate $R_{i}=\frac{\log|\mathcal{W}_{i}|}{n}$ is the cardinality of message set $\mathcal{W}_{i}$. The rate is achievable, if there are a sequence of codebook pairs $\{\mathcal{C}_{1,t},\mathcal{C}_{1,t}\}_{t=1}^{n}$ and decoding functions $\{c_{1,n},c_{2,n}\}$ such that the error probability $P_{e}^{(n)}$ goes to zero when $n$ goes to infinity. The sum-capacity is defined as the supremum of sum of achievable rates, i.e., $C_{\text{sum}}=\sup\sum_{i=1}^{2}R_{i}(\rho,\alpha_{1},\alpha_{2})$. Then, the sum-GDoF is defined as the pre-log factor of sum-capacity, i.e., $\sum_{i=1}^{2}d_{i}=\lim_{\rho\rightarrow{\cal{1}}}\frac{C_{\text{sum}}}{\log\rho}.$

Theorem 1: For the considered asymmetric interference channel with delayed CSIT, defined in Section-II, the sum-GDoF is given as follows:

Remark 1: This sum-GDoF in (2) degenerates to optimal sum-DoF in [1], by setting $\alpha_{1}=\alpha_{2}=1$, whose value is the celebrated $4/3$. Furthermore, in our antenna setting, it can be verified that the symmetric GDoF upper bound in [11] is tight.

The key steps of this proof follows the that in [11]. To begin with, we define the following virtual received signals, which are obtained from removing the impact of $\textbf{x}_{1}[t]$ in the received signals at each receiver:

Before the next step, we define the following assembly of channel matrices: $\overline{\mathcal{Y}}_{i}^{\tau}\triangleq\{\overline{y}_{i}[t]\}_{t=1}^{\tau}$, $\mathcal{Y}_{i}^{\tau}\triangleq\{y_{i}[t]\}_{t=1}^{\tau}$, and $\mathcal{X}_{i}^{\tau}\triangleq\{\textbf{x}_{i}[t]\}_{t=1}^{\tau}$, $i=1,2$. Since the error probability $P_{e}^{(n)}$ goes to zero as $n$ goes to infinity, we denote $n\epsilon_{n}\triangleq 1+nR_{i}P_{e}^{(n)}$ so that $\lim_{n\rightarrow{\cal{1}}}\epsilon_{n}=0$. According to [11], the rate of receiver 1 can be bounded as

where $\mathcal{U}[t]\triangleq\{\overline{\mathcal{Y}}_{1}^{t-1},\overline{\mathcal{Y}}_{2}^{t-1},\mathcal{H}^{t-1}\}$. Next, according to [11], the rate of receiver 2 can be bounded as

Henceforth, we define $\textbf{S}[t]\triangleq[\sqrt{\rho^{\alpha_{2}}}\textbf{h}_{12}[t],\sqrt{\rho}\textbf{h}_{22}[t]]^{T}$, $\textbf{K}[t]\triangleq\mathbb{E}\{\textbf{x}_{2}[t]\textbf{x}_{2}[t]^{H}|\mathcal{U}[t]\}$, $\textbf{L}[t]\triangleq\mathbb{E}\{\textbf{x}_{1}[t]\textbf{x}_{1}[t]^{H}|\mathcal{H}^{t-1}\}$, $\mathcal{V}[t]\triangleq\{\mathcal{U}[t],\mathcal{H}[t]\}$, where the transmit covariance matrix $\textbf{K}[t]$ is independent of $\textbf{h}_{12}[t],$ $\textbf{h}_{22}[t]$, and $\textbf{S}[t]$. Applying the extremal inequality in [12] for the physically degraded channel $\mathcal{X}_{2}^{\tau}\rightarrow(\overline{\mathcal{Y}}_{1}^{n},\overline{\mathcal{Y}}_{2}^{n})\rightarrow\overline{\mathcal{Y}}_{1}^{n}$, we have the following inequality:

To proceed, we can approximate the first term of (6) as

where (a) is from SVD of $\textbf{K}[t]$, i.e., $\textbf{K}[t]=\textbf{U}[t]{\bf{\Sigma}}[t]\textbf{U}[t]^{H}$ with unitary matrix $\textbf{U}[t]\in\mathbb{C}^{2\times(2-k_{t})}$ and diagonal matrix ${\bf{\Sigma}}[t]\in\mathbb{C}^{(2-k_{t})\times(2-k_{t})}$, and $\widetilde{\textbf{h}}_{j2}[t]\triangleq\textbf{h}_{j2}[t]\textbf{U}[t]{\bf{\Sigma}}[t]^{1/2}$, where $k_{t}\in\{0,1,2\}$ denotes the number of zero singular values; (b) is from $|\textbf{I}+\textbf{AB}|=|\textbf{I}+\textbf{BA}|$; (c) is from [11, Lemma 1]. The second term of (6) can be approximated as

where (a) is from SVD of $\textbf{K}[t]$ and [11, Lemma 1]. Next, we can approximate the first term of (4) as

where (a) is from Gaussian input maximizing the entropy with covariance constraints; (b) is from $\textbf{L}[t]\preceq\textbf{I}_{2}$ for $\text{tr}\{\textbf{L}[t]\}\leq 1$ and the SVD of $\textbf{K}[t]$; and (c) is from [11, Lemma 1].

As such, the upper bound of weighted sum of achievable rates from (4) and (5) is given in (10), shown on the top of next page,

where (a) is from (4), (5) and (9); (b) is from (7) and (8); (c) is from maximizer is $k_{t}=0$ by exhausting $k_{t}\in\{0,1,2\}$. Then, we rewrite (10) into GDoF expression,

Due to the symmetry, we have another GDoF inequality, i.e.,

Moreover, considering single-user GDoF bound for MIMO point-to-point channel, we have

Combing (11) with (12) and (13), we derive the sum-GDoF upper bound in Theorem 1 (see (2)). This ends the proof.