On the integration of SIC and MIMO DoF for interference cancellation in wireless networks

Abstract

Recent advances in MIMO degree-of-freedom (DoF) models allowed MIMO research to penetrate the networking community. Independent from MIMO, successive interference cancellation (SIC) is a powerful physical layer technique used in multi-user detection. Based on the understanding of the strengths and weaknesses of MIMO DoF and SIC, we propose to have DoF-based interference cancellation (IC) and SIC help each other so that (i) precious DoF resources can be conserved through the use of SIC and (ii) the stringent SINR threshold criteria can be met through the use of DoF-based IC. In this paper, we develop the necessary mathematical models to realize the two ideas in a multi-hop wireless network. Together with scheduling and routing constraints, we develop a cross-layer optimization framework with joint DoF IC and SIC. By applying the framework on a throughput maximization problem, we find that SIC and DoF IC can indeed work in harmony and achieve the two ideas that we propose.

Appendix: Reformulation

Reformulation of (20) and (21). First, we introduce binary variables $\kappa _{ji}[t]$ and $\beta _{ji}[t]$ to replace $\theta _{ji}[t] \cdot \gamma _{ij}[t]$ and $\theta _{ji}[t] \cdot \gamma _{ji}[t]$. That is, $\kappa _{ji}[t] = \theta _{ji}[t] \cdot \gamma _{ij}[t]$ and $\beta _{ji}[t] = \theta _{ji}[t] \cdot \gamma _{ji}[t]$. This change of variables will introduce the following new constraints for $\kappa _{ji}[t]$ and $\beta _{ji}[t]$:

$$\begin{aligned} \kappa _{ji}[t]\ge & {} \theta _{ji}[t] + \gamma _{ij}[t]-1, (1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T), \end{aligned}$$

(30)

$$\begin{aligned} \theta _{ji}[t]\ge & {} \kappa _{ji}[t], (1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T), \end{aligned}$$

(31)

$$\begin{aligned} \gamma _{ij}[t]\ge & {} \kappa _{ji}[t], (1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T), \end{aligned}$$

(32)

$$\begin{aligned} \beta _{ji}[t]\ge & {} \theta _{ji}[t] + \gamma _{ji}[t]-1, (1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T), \end{aligned}$$

(33)

$$\begin{aligned} \theta _{ji}[t]\ge & {} \beta _{ji}[t], (1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T), \end{aligned}$$

(34)

$$\begin{aligned} \gamma _{ji}[t]\ge & {} \beta _{ji}[t], (1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T). \end{aligned}$$

(35)

Now we can rewrite constraints (20) and (21) for $(1\le i\le N, 1\le t \le T)$ as

$$\begin{aligned} \sum _{l\in {{\mathcal {L}}}_i^{\mathrm {out}}}z_{(l)}[t]+\sum _{j \in {{\mathcal {I}}}_i} \kappa _{ji}[t] \sum _{k\in {{\mathcal {L}}}_j^{\mathrm {in}}}^{{\mathrm {Tx}}(k) \ne i}z_{(k)}[t]\le & {} M \cdot x_i[t]+(1-x_i[t])B_i\;. \end{aligned}$$

(36)

$$\begin{aligned} \sum _{k\in {{\mathcal {L}}}_i^{\mathrm {in}}}z_{(k)}[t]+\sum _{j \in {{\mathcal {I}}}_i} \beta _{ji}[t] \sum _{l\in {{\mathcal {L}}}_j^{\mathrm {out}}}^{{\mathrm {Rx}}(l) \ne i}z_{(l)}[t]\le & {} M \cdot y_i[t]+(1-y_i[t])B_i\;. \end{aligned}$$

(37)

Note that we still have nonlinear terms in (36) and (37), i.e., $\kappa _{ji}[t]\sum _{k\in {{\mathcal {L}}}_j^{\mathrm {in}}}^{{\mathrm {Tx}}(k) \ne i}z_{(k)}[t]$ and $\beta _{ji}[t]\sum _{l\in {{\mathcal {L}}}_j^{\mathrm {out}}}^{{\mathrm {Rx}}(l) \ne i}z_{(l)}[t]$. To reformulate these nonlinear terms, we again introduce new variables and adding new constraints. Specifically, we define new integer variable $\psi _{ji}[t]=\kappa _{ji}[t] \cdot \sum _{k\in {{\mathcal {L}}}_j^{\mathrm {in}}}^{{\mathrm {Tx}}(k) \ne i}z_{(l)}[t]$. Then (36) can be rewritten as

$$\begin{aligned} \sum _{i\in {{\mathcal {L}}}_i^{\mathrm {out}}}z_{(l)}[t]+\sum _{j \in {{\mathcal {I}}}_i} \psi _{ji}[t] \le M \cdot x_i[t]+(1-x_i[t])B_i \ \ \ (1\le i\le N, 1\le t \le T)\;, \end{aligned}$$

(38)

along with new constraints for $\psi _{ji}[t]$ for $(1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T)$.

$$\begin{aligned} \psi _{ji}[t]\le & {} \sum _{k\in {{\mathcal {L}}}_j^{\mathrm {in}}}^{{\mathrm {Tx}}(k) \ne i}z_{(k)}[t]\;, \end{aligned}$$

(39)

$$\begin{aligned} \psi _{ji}[t]\le & {} M \cdot \kappa _{ji}[t]\;, \end{aligned}$$

(40)

$$\begin{aligned} \psi _{ji}[t]\ge & {} M \cdot \kappa _{ji}[t] + \sum _{k\in {{\mathcal {L}}}_j^{\mathrm {in}}}^{{\mathrm {Tx}}(k) \ne i}z_{(k)}[t]-M\;. \end{aligned}$$

(41)

Similarly, for (37), we define new variable $\epsilon _{ji}[t]= \beta _{ji}[t]\sum _{l\in {{\mathcal {L}}}_j^{\mathrm {out}}}^{{\mathrm {Rx}}(l) \ne i}z_{(l)}[t]$. Then (37) can be rewritten as:

$$\begin{aligned} \sum _{i\in {{\mathcal {L}}}_i^{\mathrm {in}}}z_{(l)}[t]+\sum _{j \in {{\mathcal {I}}}_i} \epsilon _{ji}[t] \le M \cdot y_i[t]+(1-y_i[t])B_i \ \ \ (1\le i\le N, 1\le t \le T)\;, \end{aligned}$$

(42)

along with new constraints for $\epsilon _{ji}[t]$ for $(1 \le i \le N, j \in {{\mathcal {I}}}_i, 1\le t \le T)\;,$

$$\begin{aligned} \epsilon _{ji}[t]\le & {} \sum _{l\in {\mathcal {L}}_j^{\mathrm {out}}}^{{\mathrm {Rx}}(l) \ne i}z_{(l)}[t]\;, \end{aligned}$$

(43)

$$\begin{aligned} \epsilon _{ji}[t]\le & {} M \cdot \beta _{ji}[t]\;, \end{aligned}$$

(44)

$$\begin{aligned} \epsilon _{ji}[t]\ge & {} M \cdot \beta _{ji}[t] + \sum _{l\in {\mathcal {L}}_j^{\mathrm {out}}}^{{\mathrm {Rx}}(l) \ne i}z_{(l)}[t]-M\;. \end{aligned}$$

(45)

Reformulation of (24) and (25). The two sets of constraints in (24) and (25) are stated in the form of sufficient conditions rather than mathematical programming. To reformulate both, we first move $\eta _{ji}[t]=1$ out of the range in (24) by treating it as part of the sufficient condition. That is, if ($\eta _{ji}[t]=1$ and $\lambda _{ni}[t]=1$) then $\text{ r-SINR }_{ji}[t]\ge \beta$ for $(1 \le i \le N, j \in {\mathcal {I}}_i, p_j L_{ji}^2 \cdot C_{ji}> p_n L_{ni}^2 , 1\le t\le T)$. To combine $\eta _{ji}[t]=1$ and $\lambda _{ni}[t]=1$ into one condition, we introduce a binary variable $\delta _{(ji),(ni)}[t]$, where $\delta _{(ji),(ni)}[t]=1$ if and only if $(\eta _{ji}[t]=1$ and $\lambda _{ni}[t]=1)$ for $( 1 \le i \le N,\, (n,j) \in \mathcal {I}_i, p_j L_{ji}^2 \cdot C_{ji}>p_n L_{ni}^2 , 1\le t\le T)$. This logical condition can be expressed in mathematical form as following:

$$\begin{aligned} \delta _{(ji),(ni)}[t]\ge & {} \eta _{ji}[t] + \lambda _{ni}[t]-1\;, \end{aligned}$$

(46)

$$\begin{aligned} \eta _{ji}[t]\ge & {} \delta _{(ji),(ni)}[t] \;, \end{aligned}$$

(47)

$$\begin{aligned} \lambda _{ni}[t]\ge & {} \delta _{(ji),(ni)}[t]\;. \end{aligned}$$

(48)

Now, we can re-write MIMO SIC sequential SINR constraints derived in (24) and (25) based on the above newly defined variables and substituting $\text{ r-SINR }$ definitions for intended and unintended transmissions in (23) and (22), respectively. For $( 1 \le i \le N,\, (n,j) \in \mathcal {I}_i, p_j L_{ji}^2 \cdot C_{ji}>p_n L_{ni}^2 , 1\le t\le T)$,

$$\begin{aligned} \text{ if } \delta _{(ji),(ni)}[t]=1 \text{ then } \frac{p_j \cdot L_{ji}^2 \cdot C_{ji}}{ \sum _{k \in \mathcal {I}_i, k \ne j, \eta _{ki}[t]=1 \scriptstyle \mathrm {\ or\ } \lambda _{ki}[t]=1}^{p_k L_{ki}^2 \cdot C_{ki} \le p_j L_{ji}^2 \cdot C_{ji}} p_k \cdot L_{ki}^2 \cdot D_{jki}+N_0} \ge \beta \;, \end{aligned}$$

(49)

and for $(1 \le i \le N, n \in \mathcal {I}_i, 1\le t\le T)$,

$$\begin{aligned} \text{ if } \lambda _{ni}[t]=1 \text{ then } \frac{p_n L_{ni}^2 }{\sum _{k \in \mathcal {I}_i, k \ne n, \eta _{ki}=1}^{p_k L_{ki}^2 \cdot C_{ki} \le p_n L_{ni}^2 } p_k \cdot L_{ki}^2 \cdot D_{nki}+N_0} \ge \beta \;. \end{aligned}$$

(50)

The logical constraints (49) and (50) can now be reformulated into mathematical form. For $( 1 \le i \le N,\, (n,j) \in \mathcal {I}_i, p_j L_{ji}^2 \cdot C_{ji}>p_n L_{ni}^2 , 1\le t\le T)$,

$$\begin{aligned} \frac{ \delta _{(ji),(ni)}[t] \cdot p_j \cdot L_{ji}^2 \cdot C_{ji}+(1- \delta _{(ji),(ni)}[t]) \cdot G^\prime }{\sum _{k \in {\mathcal {I}}_i, k \ne j}^{p_k L_{ki}^2 \cdot C_{ki} \le p_j L_{ji}^2 \cdot C_{ji}} p_k \cdot L_{ki}^2 \cdot \eta _{ki}[t] \cdot D_{jki}+ \lambda _{ni}[t] \cdot p_n \cdot L_{ni}^2 \cdot D_{jni}+N_0} \ge \beta \;, \end{aligned}$$

(51)

and for $( 1 \le i \le N, n \in \mathcal {I}_i, 1\le t\le T)$,

$$\begin{aligned} \frac{\lambda _{ni}[t] \cdot p_j \cdot L_{ji}^2 + (1-\lambda _{ni}[t]) \cdot G}{\sum _{k \in {\mathcal {I}}_i k \ne n}^{p_k L_{ki}^2 \cdot C_{ki} \le p_n L_{ni}^2} p_k \cdot L_{ki}^2 \cdot \eta _{ki}[t] \cdot D_{nki}+N_0} \ge \beta \;. \end{aligned}$$

(52)

where $G^\prime$ is an upper bound of $\beta \cdot (\sum\nolimits _{k \in {\mathcal {I}}_i, k \ne j}^{p_k L_{ki}^2 \cdot C_{ki} \le p_j L_{ji}^2 \cdot C_{ji}} \eta _{ki}[t] \cdot p_k \cdot L_{ki}^2 \cdot D_{jki}+ \lambda _{ni}[t] \cdot p_n \cdot L_{ni}^2 \cdot D_{jni} +N_0)$ to ensure that the constraint holds whenever $\delta _{(ji),(ni)}[t]=0$. Define $G^\prime =\beta \cdot (\sum _{k \in {\mathcal {I}}_i, k \ne j} p_k \cdot L_{ki}^2 \cdot D_{jki} +N_0)$. Then $G^\prime \ge \beta \cdot (\sum _{k \in {\mathcal {I}}_i, k \ne j}^{p_k L_{ki}^2 \cdot C_{ki} \le p_j L_{ji}^2 \cdot C_{ji}} \eta _{ki}[t] \cdot p_k \cdot L_{ki}^2 \cdot D_{jki}+ \lambda _{ni}[t] \cdot p_n \cdot L_{ni}^2 \cdot D_{jni} +N_0).$ Similarly, G is an upper bound of $\beta \cdot (\sum\sum\nolimits _{k \in {\mathcal {I}}_i, k \ne n}^{p_k L_{ki}^2 \cdot C_{ki} \le p_n L_{ni}^2}\eta _{ki}[t] \cdot p_k \cdot L_{ki}^2 \cdot D_{nki} +N_0)$ to ensure that the constraint holds whenever $\lambda _{ni}[t]=0$. Define $G=\beta \cdot (\sum\nolimits _{k \in {\mathcal {I}}_i, k \ne n} p_k \cdot L_{ki}^2 \cdot D_{nki} +N_0)$. Then $G \ge \beta \cdot (\sum\nolimits _{k \in {\mathcal {I}}_i, k \ne n}^{p_k L_{ki}^2 \cdot C_{ki} \le p_n L_{ni}^2}\eta _{ki}[t] \cdot p_k \cdot L_{ki}^2 \cdot D_{nki} +N_0).$

In summary we replace (20), (21), (24), and (25) with (30)–(35), (38)–(41), (42)–(45), (51), and (52) in the original formulation TMP. The resulting optimization problem which we denote R-TMP, can be written as

$$\begin{aligned} \begin{array}{lll} \mathbf {R-TMP} &{}\max &{} r_{\min } \\ &{}\hbox { s.t}. &{}r_{\min } \le r(f) \qquad \qquad (f \in \mathcal {F});\\ &{} &{}\hbox {Half duplex constraint}: (1); \\ &{} &{}\hbox {Node activity constraints}: (2), (3);\\ &{} &{}\hbox {Node ordering constraints}: (4), (5);\\ &{} &{}\hbox {DoF consumption with SIC}: (14)-18), (30)-(35), (38)-(45);\\ &{} &{}\hbox {Sequential SIC with IC}: (19), (46)-(48), (51),(52);\\ &{} &{}\hbox {Flow balance constraints}: (26), (27); \\ &{} &{}\hbox {Link capacity constraints}: (29); \\ &{} &{}\hbox {Variables}: x_i[t],y_i[t],z_{(l)}[t],\pi _i[t],\theta _{ji}[t],\eta _{ji}[t],\gamma _{ji}[t],\lambda _{ji}[t],\\ &{} &{} \psi _{ji}[t],\kappa _{ji}[t],\epsilon _{ji}[t],\beta _{ji}[t],\delta _{(ji),(ni)}[t],r_l(f),r(f);\\ &{} &{}\hbox {Constants}: M,N,T,B_i,p_j,L_{ji}^2,\beta ,N_0,G,G^\prime ,C_{ji}, D_{jki}. \end{array} \end{aligned}$$

R-TMP is a mixed-integer linear problem (MILP). Therefore, we can apply a solver such as CPLEX [40] to obtain a solution efficiently.