Low-Complexity Butterfly Integration Structure for MMSE-SIC SISO Detector

Abstract

In this paper, we propose a low-complexity butterfly integration structure (LCBIS) for the minimum mean square error-soft interference cancellation (MMSE-SIC) soft-in soft-out (SISO) detector, which is widely used for MIMO systems. Unlike the conventional MMSE-SIC SISO detector which performs matrix inverse operations, the LCBIS performs butterfly integration operations with low complexity. To develop the LCBIS SISO detector, we derive an integral expression for the extrinsic information of the conventional MMSE-SIC SISO detector, and then propose a butterfly integration structure to compute the integral expression efficiently. Without matrix inverse operations, LCBIS significantly reduces the complexity of the MMSE-SIC SISO detector. In addition, simulation results show that LCBIS can offer much better BER than the other SISO detectors which do not perform matrix inverse operations.

Appendix

In this appendix, we develop the recursive relations that efficiently compute $E\left[ {x_k \left( q \right) } \right]$ and $\text{ var}\left[ {x_k \left( q \right) } \right]$ in (13). From (4), we obtain

$$\begin{aligned} E\left[ {x_k \left( q \right) } \right]&= \sum \limits _{\mathbf{v}_1, \ldots ,\mathbf{v}_{2Q} \in \mathrm{B}^{2Q}} \left( \left( {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=0}^j {\left( {2v_{i+1} -1} \right) } } } \right) \right. \nonumber \\&\left. +\,\sqrt{-1}\left( {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=0}^j {\left( {2v_{Q+i+1} -1} \right) } } } \right) \right) \cdot \prod _{i=1}^{2Q} {p(d_{2Q(k-1)+i} ( q )=v_i )} \nonumber \\ \end{aligned}$$

(22)

and

$$\begin{aligned} \text{ var}\left[ {x_k \left( q \right) } \right]&= E\left[ {\left| {x_k \left( q \right) } \right| ^{2}} \right] -\left| {E\left[ {x_k \left( q \right) } \right] } \right| ^{2} \nonumber \\&= \sum \limits _{\mathbf{v}_1, \ldots ,\mathbf{v}_{2Q} \in \mathrm{B}^{2Q}} \left| \left( {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=0}^j {\left( {2v_{i+1} -1} \right) } } } \right) \right. \nonumber \\&\left. +\sqrt{-1}\left( {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=0}^j {\left( {2v_{Q+i+1} -1} \right) } } } \right) \right| ^{2} \prod _{i=1}^{2Q} {p(d_{2Q(k-1)+i} \left( q \right) =v_i )} \nonumber \\&-\left| \sum \limits _{\mathbf{v}_1,\ldots ,\mathbf{v}_{2Q} \in \mathrm{B}^{2Q}} \left( \left( {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=0}^j {\left( {2v_{i+1} -1} \right) } } } \right) \right. \right. \nonumber \\&\left. \left. +\sqrt{-1}\left( {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=0}^j {\left( {2v_{Q+i+1} -1} \right) } } } \right) \right) \prod _{i=1}^{2Q} {p(d_{2Q(k-1)+i} \left( q \right) =v_i )} \right| ^{2} \nonumber \\ \end{aligned}$$

(23)

According to (22) and (23), computing $E\left[ {x_k \left( q \right) } \right]$ and computing $\text{ var}\left[ {x_k \left( q \right) } \right]$ are as follows.

1.1 Computing $E\left[ {x_k \left( q \right) } \right]$

Define

$$\begin{aligned}&\!\!\!\!\mu \left( s \right) \nonumber \\&\!=\!\sum \limits _{\mathbf{v}_{s+1} ,\ldots ,\mathbf{v}_Q \in \mathrm{B}^{Q-s}} {\left\{ {\left\{ {\sum \limits _{j=s}^{Q-1} {2^{Q-j-1}\prod _{i=s}^j {(2v_{i+1}\! -\!1)} } } \right\} \left\{ {\prod _{n=s}^{Q-1} {p(d_{2Q(k-1)+n+1} \left( q \right) \!=\!v_{n+1} )} } \right\} } \right\} } \nonumber \\ \end{aligned}$$

(24)

and

$$\begin{aligned}&\lambda \left( s \right) =\sum \limits _{\mathbf{v}_{s+1} ,\ldots ,\mathbf{v}_Q \in \mathrm{B}^{Q-s}} \left\{ \left\{ {\sum \limits _{j=s}^{Q-1} {2^{Q-j-1}\prod _{i=s}^j {\left( {2v_{i+1} -1} \right) } } } \right\} \right. \nonumber \\&\quad \qquad \quad \left. \times \left\{ {\prod _{n=s}^{Q-1} {p(d_{2Q(k-1)+Q+n+1} \left( q \right) =v_{n+1} )} } \right\} \right\} , \end{aligned}$$

(25)

for $s=Q-1,Q-2,\ldots 0$. By using (24) and (25), we rewrite $E\left[ {x_k \left( q \right) } \right]$ in (22) as

$$\begin{aligned} E\left[ {x_k \left( q \right) } \right] =\mu \left( 0 \right) +\sqrt{-1}\lambda \left( 0 \right) . \end{aligned}$$

(26)

To compute $\mu \left( 0 \right)$ in (26), we rewrite (24) as

$$\begin{aligned}&\mu \left( s \right) = \tanh \left( {{L_a (d_{2Q(k-1)+s+1} \left( q \right) )}/2} \right) \nonumber \\&\qquad \qquad \cdot \left\{ 2^{Q-(s+1)}+\sum \limits _{v_{s+2},\ldots , v_Q \in \mathrm{B}^{Q-(s+1)}} \left\{ {\sum \limits _{j=s+1}^{Q-1} {2^{Q-j-1}\prod _{i=s+1}^j {(2v_{i+1} -1)} } } \right\} \right. \nonumber \\&\qquad \qquad \times \left. \left\{ {\prod _{n=s+1}^{Q-1} {p(d_{2Q(k-1)+n+1} \left( q \right) =v_{n+1} )} } \right\} \right\} . \end{aligned}$$

(27)

Since (24) also implies

$$\begin{aligned}&\mu \left( {s+1} \right) =\sum \limits _{v_{s+2} ,\ldots ,v_Q \in \mathrm{B}^{Q-(s+1)}} \left\{ \left\{ {\sum \limits _{j=s+1}^{Q-1} {2^{Q-j-1}\prod _{i=s+1}^j {(2v_{i+1} -1)} } } \right\} \right. \nonumber \\&\qquad \qquad \qquad \left. \times \left\{ {\prod _{n=s+1}^{Q-1} {p(d_{2Q(k-1)+n+1} \left( q \right) =v_{n+1} )} } \right\} \right\} , \end{aligned}$$

(28)

we substitute (28) into (27) and obtain the recursive relation

$$\begin{aligned} \mu \left( s \right) =\tanh \left( {{L_a (d_{2Q(i-1)+s+1} \left( q \right) )}/2} \right) \left( {2^{Q-(s+1)}+\mu \left( {s+1} \right) } \right) \end{aligned}$$

(29)

for $s=Q-2,Q-3,\ldots ,0$. The initial value for (29) is by (24) equal to $\mu \left( {Q-1} \right) =\tanh \left( {{L_a (d_{2Q(k-1)+Q} \left( q \right) )}/2} \right)$. By using (29) with the initial value $\mu \left( {Q-1} \right) =\tanh \left( {{L_a (d_{2Q(k-1)+Q} \left( q \right) )}/2} \right)$, we can find $\mu \left( s \right)$ recursively from $s=Q-2$ to $s=0$.

To compute $\lambda \left( 0 \right)$ in (26), we follow the same procedure of finding (29) to obtain the recursive relation

$$\begin{aligned} \lambda \left( s \right) =\tanh \left( {{L_a (d_{2Q(k-1)+Q+s+1} \left( q \right) )}/2} \right) \left( {2^{Q-(s+1)}+\lambda \left( {s+1} \right) } \right) \end{aligned}$$

(30)

for $s=Q-2,Q-3,\ldots ,0$. The initial value for (30) is by (25) equal to $\lambda \left( {Q-1} \right) =\tanh \left( {{L_a (d_{2Q(k-1)+2Q} \left( q \right) )}/2} \right)$. By using (30) with the initial value $\lambda \left( {Q-1} \right) =\tanh \left( {{L_a (d_{2Q(k-1)+2Q} \left( q \right) )}/2} \right)$, we can find $\lambda \left( s \right)$ recursively from $s=Q-2$ to $s=0$.

1.2 Computing $\text{ var}\left[ {x_k \left( q \right) } \right]$

After $E\left[ {x_k \left( q \right) } \right]$ is found, the next step is to find $E\left[ {\left| {x_k \left( q \right) } \right| ^{2}} \right]$ since $\text{ var}\left[ {x_k \left( q \right) } \right]$ in (23) is equal to $\text{ var}\left[ {x_k \left( q \right) } \right] =E\left[ {\left| {x_k \left( q \right) } \right| ^{2}} \right] -\left| {E\left[ {x_k \left( q \right) } \right] } \right| ^{2}$. Define

$$\begin{aligned}&\alpha \left( s \right) =\sum \limits _{\mathbf{v}_{s+1} ,\ldots ,\mathbf{v}_Q \in \mathrm{B}^{Q-s}} \left\{ \left\{ {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=s}^j {(2v_{i+1} -1)} } } \right\} ^{2}\right. \nonumber \\&\qquad \qquad \left. \times \left\{ {\prod _{n=s}^{Q-1} {p(d_{2Q(k-1)+n+1} \left( q \right) =v_{n+1} )} } \right\} \right\} \end{aligned}$$

(31)

and

$$\begin{aligned}&\beta \left( s \right) =\sum \limits _{\mathbf{v}_{s+1} ,\ldots ,\mathbf{v}_Q \in \mathrm{B}^{Q-s}} \left\{ \left\{ {\sum \limits _{j=0}^{Q-1} {2^{Q-j-1}\prod _{i=s}^j {\left( {2v_{i+1} -1} \right) } } } \right\} ^{2}\right. \nonumber \\&\qquad \qquad \left. \times \left\{ {\prod _{n=s}^{Q-1} {p(d_{2Q(k-1)+Q+n+1} \left( q \right) =v_{n+1} )} } \right\} \right\} , \end{aligned}$$

(32)

for $s=Q-1,Q-2,\ldots 0$. By using (31) and (32), we rewrite $E\left[ {\left| {x_k \left( q \right) } \right| ^{2}} \right]$ in (23) as

$$\begin{aligned} E\left[ {\left| {x_k \left( q \right) } \right| ^{2}} \right] =\alpha \left( 0 \right) +\beta \left( 0 \right) . \end{aligned}$$

(33)

To compute $\alpha \left( 0 \right)$ in (33), we rewrite (31) as

$$\begin{aligned} \alpha \left( s \right)&= \sum \limits _{v_{s+2},\ldots , v_Q \in \mathrm{B}^{Q-(s+1)}} \left\{ 2^{2(Q-(s+1))}+2^{Q-s}\left( {\sum \limits _{j=s+1}^{Q-1} {2^{Q-j-1}\prod _{i=s+1}^j {(2v_{i+1} -1)} } } \right) \right. \nonumber \\&\left. +\left( {\sum \limits _{j=s+1}^{Q-1} {2^{Q-j-1}\prod _{i=s+1}^j {(2v_{i+1} -1)} } }\! \right) ^{2} \right\} \cdot \left\{ \! {\prod _{n=s+1}^{Q-1} {p(d_{2Q(k-1)+n+1} \left( q \right) \!=\!v_{n+1} )} } \right\} .\nonumber \\ \end{aligned}$$

(34)

Since (31) also implies

$$\begin{aligned}&\alpha \left( {s+1} \right) =\sum \limits _{v_{s+2}, \ldots ,v_Q \in \mathrm{B}^{Q-(s+1)}} \left( {\sum \limits _{j=s+1}^{Q-1} {2^{Q-j-1}\prod _{i=s+1}^j {(2v_{i+1} -1)} } } \right) ^{2}\nonumber \\&\qquad \qquad \qquad \times \left\{ {\prod _{n=s+1}^{Q-1} {p(d_{2Q(k-1)+n+1} \left( q \right) =v_{n+1} )} } \right\} , \end{aligned}$$

(35)

we substitute (35) and (28) into (34) and then obtain the recursive relation

$$\begin{aligned} \alpha \left( s \right) =2^{2(Q-(s+1))}+2^{Q-s}\mu \left( {s+1} \right) +\alpha \left( {s+1} \right) \end{aligned}$$

(36)

for $s=Q-2,Q-3,\ldots ,0$. The initial value for (36) is by (31) equal to $\alpha \left( {Q-1} \right) =1.$ By using (36) with the initial value $\alpha \left( {Q-1} \right) =1$, we can find $\alpha \left( s \right)$ recursively from $s=Q-2$ to $s=0$.

To compute $\beta (0)$ in (33), we follow the same procedure of finding (36) to obtain the recursive relation

$$\begin{aligned} \beta \left( s \right) =2^{2(Q-(s+1))}+2^{Q-s}\lambda \left( {s+1} \right) +\beta \left( {s+1} \right) \end{aligned}$$

(37)

for $s=Q-2,Q-3,\ldots ,0$. The initial value for (37) is by (32) equal to $\beta \left( {Q-1} \right) =1$. By using (37) with the initial value $\beta \left( {Q-1} \right) =1$, we can find $\beta \left( s \right)$ recursively from $s=Q-2$ to $s=0$.