Capacity Scaling of Single-source Wireless Networks: Effect of Multiple Antennas

We consider a single-source wireless network. There exists a single source having $m$ antennas at the center of the network and $n$ destinations each having a single antenna are uniformly distributed over a unit square area. The relation between $m$ and $n$ is given by

where $\beta>0$. Let $\mathbf{r}_{0}$ denote the position of the source and $\mathbf{r}_{i}$ denote the position of the $i$-th destination, where $i\in\{1,\cdots,n\}$.

We consider a far-field channel model. The $1\times m$ channel vector from the source to the $i$-th destination at time $t$ is given by

where $\alpha>2$ is the path-loss exponent and $\theta_{0i,j}(t)$ is the phase at time $t$ from the $j$-th transmit antenna of the source to the $i$-th destination. Now consider the channel from the $i$-th to the $j$-th destinations at time $t$ that is given by

where $\theta_{ij}(t)$ is the phase at time $t$ from the $i$-th to the $j$-th destinations. We assume fast fading in which $\theta_{0i,j}(t)$ and $\theta_{kl}(t)$ are uniformly distributed within $[0,2\pi)$ independent for different $i$, $j$, $k$, $l$, and $t$. We further assume that CSI is available only at the receivers.

The received signal of the $j$-th destination at time $t$ is given by

where $\mathbf{x}_{0}(t)$ is the $m\times 1$ transmit signal vector of the source, $x_{i}(t)$ is the transmit signal of the $i$-th destination, and $z_{i}\sim\mathcal{N}_{\mathbb{C}}(0,1)$ is the noise of the $i$-th destination that is independent for different $i$ and $t$. Note that not all destinations transmit simultaneously at a given time and some $x_{i}(t)$’s in (4) may be zero. The source and each destination have power constraints $P_{0}$ and $P_{1}$, respectively. Thus $\mathbb{E}\left(\|\mathbf{x}_{0}(t)\|^{2}\right)\leq P_{0}$ and $\mathbb{E}\left(|x_{i}(t)|^{2}\right)\leq P_{1}$ for $i\in\{1,\cdots,n\}$. For notational convenience, we will omit time index $t$ in the rest of the paper.

Throughout the paper, we will analyze the capacity scaling of the single-source wireless network, which could be a function of $m$ and $n$. We define the individual rate $R_{\operatorname{ind}}$ such that for suffuciently large $m$ the source can transmit with at least $R_{\operatorname{ind}}$ bps/Hz to each of $n$ destinations with high probability (whp). Then the achievable sum-rate is simply given by $R_{\operatorname{sum}}=nR_{\operatorname{ind}}$. For notational simplicity, ‘whp’ used in the paper means that an event occurs with high probability as $m\to\infty$¹¹1Since $n=m^{\beta}$, $n$ also tends to infinity as $m\to\infty$..

Using the $\Delta$ fraction of time, Phase $1$ performs the following procedure.

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  $n_{1}$-TDMA is used among $n_{1}$ groups.

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  $n_{2}$-TDMA is used among $n_{2}$ destinations in the same group.

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  The source transmits the message of destination $(k,i)$ via Gaussian signaling with covariance matrix $\frac{P_{0}\|\mathbf{r}_{0}-\mathbf{r}^{k}_{n_{2}}\|^{\alpha}}{m}\mathbf{I}_{m}$ to the nodes in the $k$-th group.

Since only the source transmits in Phase $1$, from (4), the received signal of destination $(k,j)$ in Phase $1$ is given by

where the superscript denotes the group index and $\mathbf{s}^{k}_{0}$ follows a complex Gaussian distribution with $\mathcal{N}_{\mathbb{C}}(0,\mathbf{I}_{m})$. From (2), $\mathbf{h}^{k}_{0i}$ is given by $\|\mathbf{r}_{0}-\mathbf{r}^{k}_{i}\|^{-\alpha/2}\big{[}e^{j2\pi\theta^{k}_{0i,1}},\cdots,e^{j2\pi\theta^{k}_{0i,m}}\big{]}$, where $\theta^{k}_{0i,j}$ is the phase from the $j$-th antenna of the source to destination $(k,i)$. Let $\mathbf{H}^{k}_{0}=\left[\mathbf{h}^{kT}_{01},\cdots,\mathbf{h}^{kT}_{0n_{2}}\right]^{T}$, $\mathbf{y}^{k}_{P1}=\left[y_{P1,1}^{k},\cdots,y_{P1,n_{2}}^{k}\right]^{T}$, and $\mathbf{z}^{k}_{0}=\left[z^{k}_{01},\cdots,z^{k}_{0n_{2}}\right]^{T}$. Then

where $\mathbf{\Gamma}^{k}_{0}=\operatornamewithlimits{diag}\left(\frac{\|\mathbf{r}-\mathbf{r}^{k}_{n_{2}}\|^{\frac{\alpha}{2}}}{\|\mathbf{r}-\mathbf{r}^{k}_{1}\|^{\frac{\alpha}{2}}},\cdots,\frac{\|\mathbf{r}-\mathbf{r}^{k}_{n_{2}}\|^{\frac{\alpha}{2}}}{\|\mathbf{r}-\mathbf{r}^{k}_{n_{2}}\|^{\frac{\alpha}{2}}}\right)$ and $\left[\mathbf{\Theta}^{k}_{0}\right]_{ij}=e^{j2\pi\theta^{k}_{0i,j}}$. Although the Gaussian signaling may not be optimal because the elements of $\mathbf{H}^{k}_{0}$ are not $i.i.d.$ [5], we will show that it achieves the optimal capacity scaling in the next section.

Because each destination should collect the quantized received signals from the other $n_{2}-1$ nodes in the same group, $n_{2}(n_{2}-1)$ transmission pairs should be served in each group during Phase $2$. Notice that by choosing $n_{2}$ transmission pairs such that each of $n_{2}$ nodes becomes a transmitter and a receiver of two different pairs, we can construct $n_{2}-1$ scheduling sets as shown in Fig. 1. Now consider the transmission of $n_{2}$ pairs in each scheduling set. For the communcation, we can use transmission schemes proposed for the ad hoc network model, for example the hierarchical cooperation in [2], as well as the simple TDMA to serve the pairs in each scheduling set. Using the $1-\Delta$ fraction of time, Phase $2$ performs the following procedure.

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  $4$-TDMA is used among adjacent groups, that is, one out of $4$ groups are activated simultaneously.

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  $(n_{2}-1)$-TDMA is used among $(n_{2}-1)$ scheduling sets in the same group.

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  One of the following two is performed.

  1. TDMA-based cooperation: $n_{2}$-TDMA is used among $n_{2}$ transmission pairs in the same scheduling set.

  2. Hierarchical cooperation: Hierarchical cooperation is used among $n_{2}$ transmission pairs in the same scheduling set.

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  For each transmission pair, the transmitter quantizes its received signal and transmits it to the receiver with power $P_{1}$.

Let $h^{k}_{ij}$ denote the channel from destination $(k,i)$ to destination $(k,j)$ and $h^{lk}_{ij}$ denote the channel from destination $(l,i)$ to destination $(k,j)$. Suppose that $\mathcal{A}(t)$ is the set of active groups at time $t$, which is determined by $4$-TDMA. Then the received signal of destination $(k,j)$ when destination $(k,i)$ transmits is given by

where we assume that the $i$-th destination also transmits in the other active groups.

Fig. 2 illustrates the message transmission from the source to destination $(k,j)$. Let $T$ denote the block length and $\underline{y}^{k}_{P1,i}$ denote $[y^{k}_{P1,i}(1),\cdots,y^{k}_{P1,i}(T)]$. In Phase 1, the source sends a message $W(k,j)$ to the destinations in the $k$-th group through the channel $p_{Y^{k}_{P1,1},\cdots,Y^{k}_{P1,n_{2}}|\mathbf{X}^{k}_{0}}(\cdot)$. In Phase 2, destination $(k,i)$ quantizes its received signal $\underline{y}^{k}_{P1,i}$ and transmits the quantized signal $\underline{\hat{y}}^{k}_{P1,ij}$ to destination $(k,j)$ through the link having a finite capacity of $C(k,i,j)$ for all $i\in\{1,\cdots,n_{2}\}$. Destination $(k,j)$ finally decodes $W(k,j)$ based on the quantized outputs $\underline{\hat{y}}^{k}_{P1,1j},\cdots,\underline{\hat{y}}^{k}_{P1,n_{2}j}$. Therefore, the source and destination $(k,j)$ have $(2^{TR(k,j)};T)$ channel encoder and decoder, respectively, where $R(k,j)$ denotes the data rate. Destination $(k,i)$ and destination $(k,j)$ have $(2^{TR_{Q}(k,i,j)};T)$ quantizer and dequantizer, respectively, where $R_{Q}(k,i,j)$ denotes the quantization rate. The achievable rates can be derived by modifying the result in [2].

Now consider the Gaussian channel in (9) and (10). Let $N^{k}_{ij}$ be the variance of the quantization noise given by $N^{k}_{ij}=\mathbb{E}(|Y^{k}_{P1,i}-\hat{Y}^{k}_{P1,ij}|^{2})$. If we set the conditional probability distributions as $p_{\hat{Y}^{k}_{P1,ij}|Y^{k}_{P1,i}}(\cdot)\sim\mathcal{N}_{\mathbb{C}}(y^{k}_{P1,i},N^{k}_{ij})$ for all $i\in\{1,\cdots,n_{2}\}$, then

where we use the fact that the Gaussian distribution maximizes entropy for a given received power [10]. From (12) and (13), if the following condition is satisfied

then we can find $R(k,i,j)$ satisfying (12) and (13) simultaneously. Thus we set $N^{k}_{ij}$ as the minimum value that satisfies (16) with equality such that

Let $\hat{\mathbf{y}}^{k}_{P1,j}=[\hat{y}_{P1,1j},\cdots,\hat{y}_{P1,n_{2}j}]^{T}$. Then $\hat{\mathbf{y}}^{k}_{P1,j}=\mathbf{H}^{k}_{0}\mathbf{x}^{k}_{0}+\mathbf{z}^{k}_{0}+\hat{\mathbf{z}}^{k}_{0j}$, where $\mathbb{E}(\hat{\mathbf{z}}^{k}_{0j}\hat{\mathbf{z}}^{k\dagger}_{0j})=\operatornamewithlimits{diag}(N^{k}_{1j},\cdots,N^{k}_{n_{2}j})$. Therefore, from (14), we conclude

is achievable, where $\mathbf{Q}^{k}_{j}=\operatornamewithlimits{diag}(1+N^{k}_{1j},\cdots,1+N^{k}_{n_{2}j})$.

There exists a trade-off between the size of MIMO and the variance of the quantization noise $N^{k}_{ij}$. If we set $n_{2}$ as a small value from (17), we can make $N^{k}_{ij}$ small. The reason is that as $n_{2}$ decreases, that is $n_{1}$ increases, we have more spatially reusable groups in Phase $2$ so that the aggregate rate of Phase $2$ increases and, as a result, we can decrease $N^{k}_{ij}$. Although small $n_{2}$ has an advantage in terms of quantization noises, which increases $R(k,j)$, it also decreases the size of $\mathbf{H}^{k}_{0}$, which decreases $R(k,j)$. In the next subsection, we will show that the optimal rate scaling is achievable by choosing $n_{1}$ and $n_{2}$ properly.

In this subsection, we derive $R_{\operatorname{sum}}$ of the proposed scheme in the limit of large $m$. To specify $n_{1}$ and $n_{2}$, we divide the network into small square cells of area $n^{-q}$, where $q\in(0,1)$. Then $n_{1}$ is given by $n^{q}$ and $n_{2}$ is approximately given by $n^{1-q}$ whp, which will be proved in Lemma 1. Before deriving the main results, we consider the following lemmas, which will be used to prove the main results.

The following theorem shows the scaling behavior of $N\times M$ MIMO in which the channel matrix has $i.i.d.$ elements. For our far-field channel, on the other hand, the channel elements are not $i.i.d.$ due to the different path-loss terms. Thus we further lower the achievable rate in (18) to make the channel elements $i.i.d$. and apply the result of the following theorem.

Consider the effect of multiple antennas at the destinations. Let $l=m^{\gamma}$ be the number of antennas at each destination, where $\gamma\in[0,1]$. Then the upper bound is given by $R_{\operatorname{sum}}=\operatorname{O}(m^{\operatornamewithlimits{min}\{1,\beta+\gamma\}}\operatorname{log}m)$, which is directly obtained from Theorem 1 by regarding each receive antenna as a node. If we apply the TDMA-based cooperation, the rate of Phase $2$ scales as $\Theta(\frac{l}{ln_{2}})=\Theta(\frac{1}{n_{2}})$ since the rate increases proportionally to $l$ and there exist $ln_{2}$ receive antennas in a group, meaning $ln_{2}$-TDMA is needed in a scheduling set. Thus $R_{\operatorname{sum}}$ scales whp as $\Theta(m\operatorname{log}m)$ for $\beta>2(1-\gamma)$ and $\Theta(m^{\beta/2+\gamma})$ for $\beta\leq 2(1-\gamma)$. If we apply the hierarchical cooperation, the rate of Phase $2$ is the same as Lemma 4. Thus $R_{\operatorname{sum}}$ scales whp as $\Theta(m\operatorname{log}m)$ for $\beta>(1-\gamma)(1-\epsilon)^{-1}$ and $\Theta(m^{\beta(1-\epsilon)+\gamma})$ for $\beta\leq(1-\gamma)(1-\epsilon)^{-1}$. We omit the detailed procedure, which is the same as Theorems 4 and 5.