-
遍历容量、误码率、中断概率等参数常用于表征多输入多输出(multi-input multi-output, MIMO)无线通信系统的性能,但它们不能用于解决MIMO无线系统在给定可靠性(差错概率)下的复杂度折衷问题。文献[1]提出的随机编码错误指数(RCEE)为解决以上问题提供了一种有效的途径,并在MIMO无线通信系统[2-6]和无线中继系统[7-9]中获得了应用。文献[2]首次将离散无记忆信道上的RCEE推广到MIMO衰落信道。文献[3-6]分别推导了瑞利乘性衰落信道、瑞利多匙孔衰落信道、Nakagami-m衰落信道、
$\eta - \mu $ 或$\kappa - \mu $ 衰落信道上MIMO无线系统或正交空时分组编码(OSTBC)无线系统的RCEE、遍历容量、截止速率、删改指数的闭合表达式。文献[7-9]分别研究了两跳或多跳放大前传无线中继系统在瑞利衰落信道、Nakagami-m衰落信道、$\eta - \mu $ 或$\kappa - \mu $ 衰落信道上的RCEE。文献[2-6]的研究结果只适用于接收端具有理想信道状态信息(channel state information, CSI)、平均发射功率受限且各发射天线等功率分配的块衰落信道应用场景。然而,在实际的无线通信系统中,由于反馈时延等因素的影响,往往只能使用过期的CSI进行译码与解调,从而导致系统性能恶化[10]。最小均方误差(MMSE)维纳信道预测器能够改善MIMO块衰落信道上反馈时延引起的无线系统误码性能恶化问题[11-14]。文献[14]采用MMSE信道预测器改善发射天线选择(TAS)/OSTBC编码无线系统的误码性能,但没有研究其中的收发天线数、发送块长、MMSE维纳滤波信道预测器长度等参数在给定传输可靠性下的实现复杂度折衷问题。本文将从RCEE的视角通过对RCEE及与之有关系的遍历容量、截止速率、删改指数等参数的推导与仿真对其进行研究。 -
考虑如图1所示的时间选择性MIMO瑞利块衰落信道上采用信道预测和发射天线选择(TASP)、接收端采用最大比合并(maximal ratio combining, MRC)和最大似然译码的(
${L_t},N;{L_r}$ )TAS/OSTBC编码无线系统。表1给出文中用到的一些系统与信道参数定义。表 1 系统与信道参数
参数 定义/解释 ${L_t}$ 发射天线总数 ${L_r}$ 接收天线数 ${N_c}$ 块长 $N$ 选择使用的发射天线数 ${T_s}$ 符号周期 ${E_p}$ 导频符号发射功率 ${f_D}$ 多普勒频移 $D$ 时延块数 P 平均发送功率 假定输入信号先经符号周期为Ts的数字调制,之后经码率为
${R_s} = M/T$ 的OSTBC码进行编码,即将M个调制信号OSTBC编码成T个OSTBC符号,最后由Lt根发射天线中使接收端MRC合并器输出信噪比为最大的$N$ 根天线按块发送出去,其中$1 \leqslant N \leqslant {L_t}$ ,传输帧结构如图2所示,其中帧(块)长${N_c} = {L_t} + {L_M}T$ ,${L_M}$ 为正整数。每帧的前Lt个符号为按正交设计[12]的用于信道估计与预测的导频信号,导频符号的发射功率为${E_p}$ ,每帧的后(${N_c} - {L_t}$ )个符号为OSTBC编码后的信号,且OSTBC编码矩阵的元素为调制符号及其共轭的线性组合。TASP/OSTBC编码无线系统的输入与输出之间关系为:
$${{Y}} = \sqrt {\frac{P}{N}} {{HX}} + {{V}}$$ (1) 式中,X、Y、H、
${{V}}$ 分别为$N \times {N_c}$ 维的发送信号矩阵、${L_r} \times {N_c}$ 维的接收信号矩阵、经天线选择后的${L_r} \times N$ 维实际信道矩阵以及${L_r} \times {N_c}$ 维的加性白高斯噪声矩阵。假定矩阵${{V}}$ 的各个元素相互独立且均服从${\rm C}{\rm N}(0,{N_0}$ )分布。考虑
${N_b}$ 个独立的相干间隔,假定系统的发射功率在N根发射天线上平均分配且在块编码长度为${N_b}{N_c}$ 的链路上能够实现可靠通信,则有${\rm E}\{ {\rm{tr}}({{X}}{{{X}}^{\rm H}})\} \leqslant{N_c}P $ 。表2中给出了文中用到的一些数学符号或函数定义。表 2 一些数学符号/函数的定义
符号 定义/解释 ${\rm{tr} }( \cdot )$ 矩阵求迹 ${( \cdot )^{\rm{*}}}$ 复共轭 ${\rm{E} }( \cdot )$ 求数学期望 ${( \cdot )^{\rm{H}}}$ 共轭转置 ${J_0}(\cdot )$ 第一类零阶Bessel函数 $\left\| \cdot \right\|_{\rm F}^2$ Frobenius范数 $\left\lceil x \right\rceil $ 大于或等于x的最小整数 ${\tilde{ H}}(u) \in {{\mathbb{C}}^{{L_r} \times {L_t}}}$ 实际信道矩阵 ${\hat{ H}}(u + D) \in {{\mathbb{C}}^{{L_r} \times {L_t}}}$ 信道预测矩阵 采用文献[11-12]的Jakes信道模型,假定
$ {\tilde{ H}}(u)$ 为第u个数据块的实际信道矩阵,其元素${\tilde h_{wi}}\left( u \right)$ 为第i根发射天线到第w根接收天线之间的实际链路增益,且相互独立,均服从$ {\rm C}{\rm N}$ (0,1)分布,则${\rm{E}}\left[ {{{\tilde h}_{wi}}\left( u \right)\tilde h_{wi}^ * \left( {u - \tau } \right)} \right] = {J_0}\left( {2{{\text π}}{f_d}\tau } \right)$ ,τ=DNcTs为反馈时延;且假定$ {\tilde{ H}}(u)$ 按块进行变化。假定采用文献[12]中预测长度为Lp的MMSE信道预测器进行
$ {\tilde{ H}}(u)$ 的估计和预测,于是得到信道预测矩阵$ {\hat{ H}}(u+D)$ 。基于信道预测矩阵,可以完成发送端的发射天线选择,具体过程如下:计算
$ {\hat{ H}}(u+D)$ 各列的Frobenius范数,并按从大到小进行重新排序,删除重新排序后矩阵的后(Lt−N)列,将与重新排序后矩阵前N列对应的原N根发射天线选定为OSTBC码的发射天线并进行OSTBC编码。由文献[12]可得经发射天线选择后的归一化预测信道矩阵
$ {\hat{ H}}'(u+D)$ 和实际信道矩阵$ {{ H}}(u+D)$ 之间的关系为:$${ H}(u + D){\rm{ = }}\sqrt {{\rho _p}} {{\hat{ H}}^{'}}(u + D){\rm{ + }}\sqrt {1 - {\rho _p}} {{N}}(u + D)$$ (2) 式中,
$\, {\rho _p} = {{{r}}^{\rm H}}{{{R}}^{ - 1}}{{r}}$ 为功率相关系数[12];Lp维矢量r、Lp×Lp维矩阵R的各元素取值可由文献[12]或文献[14]给出;$ { N}(u + D) \in {{\mathbb{C}}^{{L_r} \times N}}$ 为元素相互独立且均服从$ {\rm C}{\rm N}$ (0,1)分布的加性白高斯噪声矩阵。若直接使用
$ {\tilde{ H}}(u)$ 选择发射天线,则图1所示的系统即变为文献[10]的反馈时延发射天线选择(TASD)/OSTBC编码无线系统,此时有:$${\tilde{ H}}(u + D){\rm{ = }}\sqrt {{\rho _d}} {\tilde{ H}}(u){\rm{ + }}\sqrt {1 - {\rho _d}} { N}(u + D)$$ (3) 比较式(2)、式(3)不难发现:两式的唯一区别仅在于功率相关系数不同,因此以下关于TASP/OSTBC编码无线通信系统的有关推导结果均适用于TASD/OSTBC编码无线系统,只需将其中的ρp用功率相关系数
$ {\rho _d} = J_0^2\left( {2{\text π} {f_d}\tau } \right)$ 替代即可。图3给出了ρp和ρd随归一化时延fdτ变化的性能曲线。由图3可知:1) 当fdτ一定时,ρp>ρd;2) 若不考虑反馈时延,即理想CSI情况,此时令fdτ=0,则
${\rho _p}{\rm{ = }}1$ ;3) 当fdτ越小,$\,{\rho _p}$ 越接近于1,即越接近于理想CSI时的情况。基于以上分析,可以预期采用信道预测的TAS/OSTBC编码无线系统的RCEE性能优于TASD/OSTBC编码无线系统。假定接收端采用最大似然译码,令平均接收信噪比
$ \bar \gamma = P/{N_0}$ ,$ c=1 / R_{s} N$ ,则MRC合并器的输出符号信噪比为$ {\gamma _s} = c\bar \gamma \left\| {{H}} \right\|_{\rm F}^2$ 。令$ \gamma={\gamma _s} /(c\bar \gamma)= \left\| {{H}} \right\|_{\rm F}^2$ ,利用文献[10],经推导可得γ的矩生成函数表达式:$$\begin{split} {\varPhi _{\gamma} }(s) =& N\left( {\begin{array}{*{20}{c}} {{L_t}} \\ N \end{array}} \right)\frac{1}{{\varGamma {{({L_r})}^N}}}{\rm{ }}\sum\limits_{{t_1},{t_2}, \cdots ,{t_N}} {n({L_r},{t_1}, \cdots ,{t_N})} \prod\limits_{k = 1}^{N - 1} {\dfrac{{{t_k}!}}{{{k^{{t_k}}}}}} \times \\ & \sum\limits_{j = 0}^{{L_t} - N} {\left( {\begin{array}{*{20}{c}} {{L_t} - N} \\ j \end{array}} \right){{( - 1)}^j}} \sum\limits_{\varLambda \in B} {\left( {\begin{array}{*{20}{c}} j \\ {{\varLambda _0},{\varLambda _1}, \cdots ,{\varLambda _{{L_r} - 1}}} \end{array}} \right)} \times \\ & \frac{{({c_{\varLambda j}} + {t_N})!}}{{{{(N + j)}^{{c_N}}}{A_{\varLambda j}}}}\left[ {\frac{{{{\left( {1 + (1 - {\rho _p})s} \right)}^{{c_{\varLambda j}}}}}}{{{{(1 + s)}^{{r_N}}}}}\frac{1}{{{{\left( {1 + {\eta _j}s} \right)}^{{c_N}}}}}} \right] \end{split} $$ (4) 式中,
$n({L_r},{t_1}, \cdots ,{t_N}){\rm{ }}$ 为表达式${({y_1} + {y_2} + \cdots + {y_N})^{{L_r} - 1}}$ ${({y_2} + \cdots + {y_N})^{{L_r} - 1}} \cdots {y_N}^{{L_r} - 1}$ 中${y_1}^{{t_1}}{y_2}^{{t_2}} \cdots {y_N}^{{t_N}}$ 的系数;集合$B = \{ ({\varLambda _0},{\varLambda _1},\cdots ,{\varLambda _{{L_r} - 1}}):{\varLambda _n} \in \{ 0,1, \cdots ,j\} ,\displaystyle\sum\limits_{n = 1}^{{L_r} - 1} {{\varLambda _n} = j} \}$ ;$\left(\!\!\! {\begin{array}{*{20}{c}} j \\ {{\varLambda _0},\cdots ,{\varLambda _{{L_r} - 1}}} \end{array}}\!\!\! \right) = $ $\frac{{j!}}{{{\varLambda _0}!\cdots {\varLambda _{{L_r} - 1}}!}}$ ;${\eta _j} = \left[ {N + j(1 - {\rho _p})} \right]/(N + j)$ ;${c_{\varLambda j}} = \displaystyle\sum\limits_{{k_1} = 1}^{{L_r} - 1} {{k_1}{\varLambda _{{k_1}}}} $ ;${A_{\varLambda {\rm{ }}j}} = \prod\limits_{{k_2} = 1}^{{L_r} - 1} {{{({k_2}!)}^{{\varLambda _{{k_2}}}}}} $ ;${r_N} = r + N - 1$ ;$r = \displaystyle\sum\limits_{{k_3} = 1}^{N - 1} {{t_{{k_3}}}} $ ;${c_N} = {c_{\varLambda j}} + {t_N} + 1$ 。令
${\rho _p}{\rm{ = }}1$ ,式(4)可退变为文献[15]中的式(7),即理想CSI下的TAS/OSTBC编码无线通信系统为本文的特殊情况。利用部分分式展开定理和拉普拉斯反变换后,可得
$\gamma $ 的概率密度函数(probability density function, PDF)为:$$\begin{split} {p_{\gamma} }(z) =& \varSigma_1 {\varphi ({L_r}N,1)} +\varSigma_2 {\left( {\sum\limits_{m = 1}^{r + N - 1} {{P_m}\varphi (m,1)} } \right.} + \\ & \quad \quad \left. {\sum\limits_{l = 1}^{{c_{\varLambda j}} + {t_N} + 1} {{Q_l}\varphi (l,{\eta _j})} } \right) \end{split} $$ (5) 式中,
$${P_m} = \frac{1}{{({r_N} - m)!}}\frac{{{\partial ^{{r_N} - m}}}}{{\partial {s^{{r_N} - m}}}}{\left[ {\frac{{{{(1{\rm{ + }}s(1 - {\rho _p}))}^{{c_{\varLambda j}}}}}}{{{{(1 + {\eta _j}s)}^{{c_N}}}}}} \right]_{s = - 1}}{\rm{ }}$$ (6) $${Q_l} = \frac{{{{({\eta _j})}^{ - {c_N}}}}}{{({c_N} - l)!}}\frac{{{\partial ^{{c_N} - l}}}}{{\partial {{{}_s}^{{c_N} - l}}}}{\left[ {\frac{{{{\left( {1{\rm{ + }}s(1 - {\rho _p})} \right)}^{{c_{\varLambda j}}}}}}{{{{(1 + s)}^{{r_N}}}}}} \right]_{s = - \frac{1}{{{\eta _j}}}}}{\rm{ }}$$ (7) $$ \begin{split} \varSigma_1 =& N\left( {\begin{array}{*{20}{c}} {{L_t}} \\ N \end{array}} \right)\frac{1}{{\varGamma {{({L_r})}^N}}}{\rm{ }}\sum\limits_{{t_1},{t_2}, \cdots ,{t_N}} {n({L_r},{t_1}, \cdots ,{t_N})\times}\\ & \quad \quad \quad \prod\limits_{k = 1}^{N - 1} {\frac{{{t_k}!}}{{{k^{{t_k}}}}}} \frac{{({t_N})!}}{{{N^{{t_N} + 1}}}} \end{split} $$ (8) $$\begin{split} & \varSigma_2 = N\left( {\begin{array}{*{20}{c}} {{L_t}} \\ N \end{array}} \right)\frac{1}{{\varGamma {{({L_r})}^N}}}{\rm{ }}\sum\limits_{{t_1},{t_2}, \cdots ,{t_N}} {n({L_r},{t_1},\cdots ,{t_N})} \times \\ & \qquad\quad \prod\limits_{k = 1}^{N - 1} {\frac{{{t_k}!}}{{{k^{{t_k}}}}}} \sum\limits_{j = 1}^{{L_t} - N} {\left( {\begin{array}{*{20}{c}} {{L_t} - N} \\ j \end{array}} \right){{( - 1)}^j}} \times \\ &\qquad \sum\limits_{\varLambda \in B} {\left( {\begin{array}{*{20}{c}} j \\ {{\varLambda _0},{\varLambda _1}, \cdots ,{\varLambda _{{L_r} - 1}}} \end{array}} \right)} \frac{{({c_{\varLambda j}} + {i_N})!}}{{{{(N + j)}^{{c_N}}}{A_{\varLambda j}}}} \end{split} $$ (9) $$\varphi (L,\mu ) = \frac{{{z^{L - 1}}}}{{(L - 1)!}}{{\rm e}^{ - \frac{z}{\mu }}}$$ (10) -
当输入信号服从高斯分布,在信息传输速率为R的连续输入输出MIMO信道上采用信道预测的TAS/OSTBC编码无线系统的RCEE可表示为[5]:
$${E_r}(R,{N_c}) = \mathop {\max }\limits_{0 \leqslant \rho \leqslant 1} \left( {\mathop {\max }\limits_{0 \leqslant \beta \leqslant N} {{\tilde E}_0}(\rho ,\beta ,{N_c}) - \rho R} \right)$$ (11) 式中,
$\lambda {\rm{ = }}\bar \gamma /{R_s}$ ;${p_{\gamma} }(z)$ 为$\left\| { H} \right\|_{\rm F}^2$ 的PDF;$$\begin{split} & {{\tilde E}_0}(\rho ,\beta ,{N_c})= \underbrace {(1 + \rho )(N - \beta ) + N(1 + \rho )\ln (\beta /N)}_{A(\rho ,\beta )} - \\ & \quad \quad \frac{1}{{{N_c}}}\ln \left( {{\rm{E}}\left\{ {\det {{\left( {{{ I}_{{L_r}}} + \frac{{\lambda { H}{{ H}^{\rm H}}}}{{\beta (1 + \rho )}}} \right)}^{ - {N_c}\rho }}} \right\}} \right) = \\ & \quad\;\; A(\rho ,\beta ) - \frac{1}{{{N_c}}}\ln \left\{\!\! {\int_0^\infty \!\!{{{\left(\!\! {1 + \frac{{\lambda z}}{{\beta (1 + \rho )}}} \right)}^{ - {N_c}\rho }}\!\!{p_{\gamma} }(z){\rm d}z} }\! \right\} \end{split} $$ (12) -
将式(4)代入式(10),可得:
$$\begin{split} & {E_r}(R,{N_c}\!) \!=\! \mathop {\max }\limits_{0 \leqslant \rho \leqslant 1} \!\!\left\{\! \mathop {\max }\limits_{0 \leqslant \beta \leqslant N} \left( {A(\rho ,\beta ) - \!\!\!}\right. \right. \frac{1}{{{N_c}}} \ln \Bigg(\! {\varSigma_1\!{\phi (b,\alpha ,{L_r}N,1)} } + \\ & \varSigma_2 {\left( {\sum\limits_{m = 1}^{r + N - 1} {{P_m}\phi (b,\alpha ,m,1)} } \right.} \!+\!\!\left.\left. \left. \sum\limits_{l = 1}^{{c_{\varLambda j}} + {t_N} + 1} {{Q_l}\phi (b,\alpha ,l,{\eta _j})}\! \right)\! \right)\! - \rho R\! \right\} \end{split} $$ (13) 式中,
$$\phi (b,\alpha ,L,\mu ) = \int_0^\infty {{{(1 + bz)}^{ - \alpha }}} \varphi (L,\mu ){\rm d}z$$ (14) 下面分别利用Tricomi超几何函数[16]和Meijer-G函数
$G_{p,q}^{u,n}\left[ {\omega x\left| {\begin{array}{*{20}{c}} {{a_1}, \cdots ,{a_p}} \\ {{b_1}, \cdots ,{b_q}} \end{array}} \right.} \right]$ 推导式(14)的解析表达式。利用文献[4],有:
$${(1 - x)^{ - \alpha }} = \frac{1}{{\varGamma (\alpha )}}G_{1,1}^{1,1}\left[ { - x\left| {\begin{array}{*{20}{c}} {1 - \alpha } \\ 0 \end{array}} \right.} \right]$$ (15) $$ \begin{split} & \int_0^\infty {{x^{ - \psi }}} {{\rm e}^{ - \xi x}}G_{p,q}^{u,n}\left[ {\omega x\left| {\begin{array}{*{20}{c}} {{a_1}, \cdots ,{a_p}} \\ {{b_1}, \cdots ,{b_q}} \end{array}} \right.} \right]{\rm{d}}x = \\ & \quad\;\; {\xi ^{\psi - 1}}G_{p + 1,q}^{u,n + 1}\left[ {\frac{\omega }{\xi }\left| {\begin{array}{*{20}{c}} {\psi ,{a_1}, \cdots ,{a_p}} \\ {{b_1}, \cdots ,{b_q}} \end{array}} \right.} \right] \end{split}$$ (16) 由式(14)可推得:
$$\phi (b,\alpha ,L,\mu ){\rm{ = }}\frac{{{\mu ^L}}}{{\varGamma (\alpha )(L - 1)!}}G_{2,1}^{1,2}{\rm{ }}\left[ {b\left| {\begin{array}{*{20}{c}} {1 - L,1 - \alpha } \\ 0 \end{array}} \right.} \right]$$ (17) 由文献[5]:
$$\int_0^\infty {\frac{{{{(1 + bx)}^{ - v}}}}{{{x^{1 - q}}{{\rm e}^{px}}}}} {\rm d}x = \frac{{{p^{v - q}}}}{{{b^v}}}\varGamma (q)U\left(v;v - q + 1;\frac{p}{b}\right)$$ (18) 式(14)可推得为:
$$\phi (b,\alpha ,L,\mu ) = \frac{{{\mu ^{l - \alpha }}}}{{{b^\alpha }}}{\rm{ }}U\left( {\alpha ;\alpha - L + 1;\frac{1}{{b\mu }}} \right)$$ (19) 式中,
$U({u_1};{u_2};y)$ 为Tricomi超几何函数[16]。将式(17)、式(19)分别代入式(13),可得时间选择性瑞利块衰落信道上采用MMSE信道预测的TAS/OSTBC编码无线系统的两种RCEE解析表达式。
令
$\,{\rho _p}{\rm{ = }}1$ ,将式(5)代入式(11)、式(12),利用式(19)可得瑞利块衰落信道上理想CSI下的TAS/OSTBC编码无线系统的RCEE表达式为:$$\begin{split} & {E_r}(R,{N_c}) = \mathop {\max }\limits_{0 \leqslant \rho \leqslant 1} \left\{ {\mathop {\max }\limits_{0 \leqslant \beta \leqslant N} \left( {A(\rho ,\beta ) - \frac{1}{{{N_c}}}} \right.} \right. \times \\ & \quad \left. {\left. {\ln \phi \left( {\frac{\lambda }{{\beta (1 + \rho )}},{N_c}\rho ,{L_t}{L_r},1} \right)} \right) - \rho R} \right\} \end{split} $$ (20) 不难发现,式(20)的结果与文献[5]中瑞利块衰落信道上理想CSI下的TAS/OSTBC编码无线系统的RCEE完全相同,即文献[5]的结果是本文的特殊情况。
-
利用RCEE与遍历容量的关系[2],可得瑞利块衰落信道上采用MMSE信道预测的TAS/OSTBC编码无线系统的遍历容量为:
$$ C = {R_s}{\left. {\left[ {\frac{{\partial {{\tilde E}_0}(\rho ,\beta ,{N_c})}}{{\partial \rho }}} \right]} \right|_{\rho = 0,\beta = N}} = {R_s}\int_0^\infty {\ln \left( {1 + \frac{\lambda }{N}z} \right){p_{\gamma} }(z)} {\rm d}z $$ (21) 令
$$J(L,\mu ) = \int_0^\infty {\ln \left( {1 + \frac{\lambda }{N}x} \right)} \varphi (L,\mu ){\rm d}x$$ (22) 利用文献[4]:
$$\ln \left( {1 + \frac{\lambda }{N}x} \right) = G_{2,2}^{1,2}\left[ {\frac{\lambda }{N}x\left| {\begin{array}{*{20}{c}} {1,1} \\ {1,0} \end{array}} \right.} \right]$$ (23) 可推得:
$$J(L,\mu ) = \int_0^\infty {G_{2,2}^{1,2}\left[ {\frac{\lambda }{N}x\left| {\begin{array}{*{20}{c}} {1,1} \\ {1,0} \end{array}} \right.} \right]} \varphi (L,\mu ){\rm d}x$$ (24) 再利用式(16),由式(24)可推得:
$$J(L,\mu ) = \frac{{{\mu ^L}}}{{(L - 1)!}}G_{3,2}^{1,3}\left[ {\frac{\lambda }{N}\mu \left| {\begin{array}{*{20}{c}} {1 - L,1,1} \\ {1,0} \end{array}} \right.} \right]$$ (25) 将式(5)代入式(21),并利用式(17)可得瑞利块衰落信道上采用MMSE信道预测的TAS/OSTBC编码无线系统的遍历容量为:
$$\begin{split} & \qquad\quad {{ C }} = {R_s}\varSigma_1 {J({L_r}N,1)} +\\ & {R_s}\varSigma_2 {\left( {\sum\limits_{m = 1}^{r + N - 1} {J(m,1)} } + {\sum\limits_{l = 1}^{{c_{\varLambda j}} + {t_N} + 1} {J(l,{\eta _j})} } \right) } \end{split} $$ (26) -
利用文献[2]可得瑞利块衰落信道上采用MMSE信道预测的TAS/OSTBC编码无线系统的截止速率为:
$$\begin{split} &\qquad\quad {R_0} = - \frac{1}{{{N_c}}}{\tilde E_0}(1,N,{N_c}) = \\ &\;\;\; - \frac{1}{{{N_c}}}\ln \int_0^\infty {{{\left( {1 + \frac{\lambda }{{2N}}z} \right)}^{ - {N_c}}}} \Bigg[ {\varSigma_1 {\varphi ({L_r}N,1)} + } \\ & \left. {\varSigma_2 {\left( {\sum\limits_{m = 1}^{r + N - 1} {{P_m}\varphi (m,1)} } \right.} + \left. {\sum\limits_{l = 1}^{{c_{\varLambda j}} + {t_N} + 1} {{Q_l}\varphi (l,{\eta _j})} } \right)} \right]{\rm d}z \end{split} $$ (27) 利用式(14)和式(17),可推得:
$$\begin{split} {R_0} =& - \frac{1}{{{N_c}}}\ln \left\{ {\varSigma_1 {\phi \left( {\frac{\lambda }{{2N}},{N_c},{L_r}N,1} \right)} + } \right.\varSigma_2 {\left( {\sum\limits_{m = 1}^{r + N - 1} {{P_m}} } \right.} \times \\ & \left. {\left. {\phi \left( {\frac{\lambda }{{2N}},{N_c},m,1} \right) + \sum\limits_{l = 1}^{{c_{\varLambda j}} + {t_N} + 1} {{Q_l}\phi \left( {\frac{\lambda }{{2N}},{N_c},l,{\eta _j}} \right)} } \right)} \right\} \end{split} $$ (28) -
RCEE认定所有的好码和坏码对平均错误概率的影响相同,因此,在低速率区域内可以通过删除所有的坏码来改善RCEE的性能限[5],由此得到删改指数。删改指数
${E_{\rm {ex}}}(R,{N_c})$ 定义为[5]:$${E_{\rm {ex}}}(R,{N_c}) = \mathop {\max }\limits_{\rho \geqslant 1} (\mathop {\max }\limits_{0 \leqslant \beta \leqslant N} {\tilde E_x}(\rho ,\beta ,{N_c}) - \rho R)$$ (29) 式中,
$$\begin{split} & {{\tilde E}_x}(\rho ,\beta ,{N_c}) = \underbrace {2\rho (N - \beta ) + 2\rho N\ln (\beta /N)}_{A'(\rho ,\beta )} - \\ & \qquad \frac{1}{{{N_c}}}\ln \left( {\int_0^\infty {{{\left( {1 + \frac{\lambda }{{2\rho \beta }}z} \right)}^{ - {N_c}\rho }}{p_{\gamma} }(z){\rm d}z} } \right) \end{split} $$ (30) 将式(5)代入上式,可得:
$$\begin{split} & {{\tilde E}_x}(\rho ,\beta ,{N_c}) = A'(\rho ,\beta ) - \frac{1}{{{N_c}}}\ln \left\{\!\! {\int_0^\infty {{{\left(\!\! {1 + \frac{\lambda }{{2\rho \beta }}z} \right)}^{ - {N_c}\rho }} \times } } \right. \\ & \qquad{\rm{ }}\left[ {\varSigma_1 {\varphi ({L_r}N,1)} + \varSigma_2 {\left( {\sum\limits_{m = 1}^{r + N - 1} {{P_m}} \varphi (m,1) + } \right.} } \right. \\ & \qquad\qquad \left. {\left. {\left. {\sum\limits_{l = 1}^{{c_{\varLambda j}} + {t_N} + 1} {{Q_l}\varphi (l,{\eta _j})} } \right)} \right]{\rm d}z} \right\} \end{split} $$ (31) 利用式(14)和式(17),可得瑞利块衰落信道上采用MMSE信道预测的TAS/OSTBC编码无线系统删改指数的表达式为:
$$\begin{split} & \qquad\qquad{\tilde E_x}(\rho ,\beta ,{N_c}) = \\ & A'(\rho ,\beta ) - \frac{1}{{{N_c}}}\ln \left\{ {\varSigma_1 {\phi \left( {\frac{\lambda }{{2\rho \beta }},{N_c}\rho ,{L_r}N,1} \right)} } \right. + \\ & \quad \varSigma_2 {\left[ {\sum\limits_{m = 1}^{r + N - 1} {{P_m}\phi \left( {\frac{\lambda }{{2\rho \beta }},{N_c}\rho ,m,1} \right)} } \right. + } \\ & \qquad \left. {\left. {\sum\limits_{l = 1}^{{c_{\varLambda j}} + {t_N} + 1} {{Q_l}\phi \left( {\frac{\lambda }{{2\rho \beta }},{N_c}\rho ,l,{\eta _j}} \right)} } \right]} \right\} \end{split} $$ (32) -
由文献[2]可得瑞利块衰落信道上采用MMSE信道预测的TAS/OSTBC编码无线系统的错误概率与其
${E_r}(R,{N_c})$ 的关系为:$$\begin{split} {P_e} =& \left( {{{8{\text π} } / N}} \right){\left( {N - {\beta ^*}\left( \rho \right)} \right)^2}{N_b}{N_c} \times \\ & \exp \left( {2 - {N_b}{N_c}{E_r}\left( {R,{N_c}} \right)} \right) \end{split} $$ (33) 式中,
$\,{\beta ^*}(\rho )$ 为式(11)的${\tilde E_0}\left( {\rho ,\beta ,{N_c}} \right)$ 取最大值时的$\,\beta $ 值,且$0 \leqslant \beta \leqslant N$ ;$L = {N_c} \times \left\lceil {{N_b}} \right\rceil $ 是系统为获得上述错误概率时所需的块编码长度。分析式(33)可知:当${P_e}$ 一定时,增大RCEE,则L会减小,即编解码的实现复杂度可以降低。
RCEE Analysis of Wireless Communication Systems with TAS/OSTBC on Rayleigh Block Fading Channel
-
摘要: 设计瑞利块衰落信道上采用信道预测和发射天线选择(TAS)的正交空时分组编码(OSTBC)无线通信系统,需要考虑收发天线数、发送块长、最小均方误差信道预测器长度等参数之间的折衷问题。随机编码错误指数(RCEE)作为一种理论分析工具,可以有效地解决上述问题。利用Meijer-G函数和Tricomi超几何函数推导瑞利块衰落信道上采用信道预测的TAS/OSTBC编码无线通信系统的RCEE解析表达式;利用遍历容量、截止速率、删改指数等参数与RCEE的关系推导其解析表达式。上述性能参数的数值计算与仿真证明了理论分析的正确性。结果还表明,采用信道预测的TAS/OSTBC编码无线系统的RCEE大于反馈时延下TAS/OSTBC编码无线通信系统的RCEE;在低信息速率下可采用删改指数计算使用信道预测的TAS/OSTBC编码无线系统所需的编码长度。Abstract: In order to design wireless communication systems with predictive transmit antenna selection (TAS) and orthogonal space-time block code (OSTBC) on Rayleigh block fading channel, the trade-off problem among the number of transmit/receive antennas, the block length, the length of minimum mean square error (MMSE) channel predictor needs to be considered. As a theoretical analysis tool, random coding error exponent (RCEE) can effectively solve the above problems. The closed-form expressions of RCEE of wireless communication systems with TAS/OSTBC on Rayleigh block fading channel are derived by using Meijer-G function and Tricomi hypergeometric function. The closed-form expressions of the relationship among RCEE and ergodic capacity, cut-off rate, expurgated exponent of the above systems are also developed. The numerical calculation and simulation results of the above performance parameters prove the correctness of the above theoretical analysis. The results also show that the RCEE of the TAS/OSTBC wireless communication system with channel prediction is larger than that of TAS/OSTBC wireless communication system under feedback delay. The required code length for the TAS/OSTBC wireless communication system can be calculated by using the expurgated exponent at the low information rate.
-
表 1 系统与信道参数
参数 定义/解释 ${L_t}$ 发射天线总数 ${L_r}$ 接收天线数 ${N_c}$ 块长 $N$ 选择使用的发射天线数 ${T_s}$ 符号周期 ${E_p}$ 导频符号发射功率 ${f_D}$ 多普勒频移 $D$ 时延块数 P 平均发送功率 表 2 一些数学符号/函数的定义
符号 定义/解释 ${\rm{tr} }( \cdot )$ 矩阵求迹 ${( \cdot )^{\rm{*}}}$ 复共轭 ${\rm{E} }( \cdot )$ 求数学期望 ${( \cdot )^{\rm{H}}}$ 共轭转置 ${J_0}(\cdot )$ 第一类零阶Bessel函数 $\left\| \cdot \right\|_{\rm F}^2$ Frobenius范数 $\left\lceil x \right\rceil $ 大于或等于x的最小整数 ${\tilde{ H}}(u) \in {{\mathbb{C}}^{{L_r} \times {L_t}}}$ 实际信道矩阵 ${\hat{ H}}(u + D) \in {{\mathbb{C}}^{{L_r} \times {L_t}}}$ 信道预测矩阵 -
[1] GALLAGER R G. Information theory and reliable communication[M]. New York, USA: Wiley, 1968. [2] SHIN H, WIN M Z. Gallager’s exponent for MIMO channel: A reliability-rate tradeoff[J]. IEEE Transactions on Communications, 2009, 57(4): 972-985. doi: 10.1109/TCOMM.2008.04.050673 [3] XUE J, SARKAR M Z I, RATNARAJAH T, et al. Error exponents for Rayleigh fading product MIMO channels[C]//IEEE International Symposium on Information Theory Proceedings. Cambridge, USA: IEEE, 2012: 2166-2170. [4] XUE J, SARKAR M Z I, RATNARAJAH T. Error exponents for Rayleigh fading multi-keyhole MIMO channels[C]//IEEE International Conference on Communications. Budapest, Hungary: IEEE, 2013: 3176-3180. [5] XUE J, SARKAR M Z I, RATNARAJAH T. Random coding error exponent for OSTBC nakagami-m fading MIMO channel[C]//IEEE Vehicular Technology Conference (VTC Spring). Yokohama, Japan: IEEE, 2011: 1-5. [6] ZHANG J, MATTHAIOU M, KARAGIANNIDIS G K, et al. Gallager's exponent analysis of STBC MIMO systems over η-μ and κ-μ fading channels[J]. IEEE Transactions on Communications, 2013, 61(3): 1028-1039. doi: 10.1109/TCOMM.2012.120512.120327 [7] NGO H Q, LARSSON E. Linear multihop amplify- and-forward relay channels: Error exponent and optimal number of hops[J]. IEEE Transactions on Wireless Communications, 2011, 10(11): 3834-3842. doi: 10.1109/TWC.2011.092011.102194 [8] NGO H Q, QUEK T , SHIN H. Random coding error exponent for dual-hop nakagami-m fading channels with amplify-and-forward relaying[J]. IEEE Communications Letters, 2009, 13(11): 823-825. doi: 10.1109/LCOMM.2009.091580 [9] ZHANG Y, XUE J, RATNARAJAH T, et al. Error exponents analysis of dual-hop η-μ and κ-μ fading channel with amplify-and-forward relaying[J]. IET Communications, 2015, 9(11): 1367-1379. doi: 10.1049/iet-com.2014.0918 [10] YU X, XIA X, LEUNG S, et al. Performance analysis of MIMO systems with arbitrary number transmit antenna selection and OSTBC in the presence of imperfect CSI[J]. Science China Information Sciences, 2016, 59(8): 1-16. [11] ZHOU S, GIANNAKIS G B. How accurate channel prediction needs to be for transmit-beamforming with adaptive modulation over Rayleigh MIMO channel?[J]. IEEE Transactions on Wireless Communications, 2004, 3(4): 1285-1294. doi: 10.1109/TWC.2004.830842 [12] PRAKASH S, MCLOUGHLIN I. Effects of channel prediction for transmit antenna selection with maximal-ratio combining in Rayleigh fading[J]. IEEE Transactions on Vehicular Technology, 2011, 60(6): 2555-2568. doi: 10.1109/TVT.2011.2157184 [13] 李光球, 王思婷. 信道预测联合收发分集的正交信号误码性能[J]. 电信科学, 2017, 33(5):29-38. LI Guang-qiu, WANG Si-ting. Performance analysis of orthogonal signals with TASP/MRC diversity[J]. Telecommunications Science, 2017, 33(5): 29-38. [14] 汪玲波, 李光球, 钱辉. 信道预测和天线选择的空时码误码性能分析[J]. 通信技术, 2018, 51(8):1791-1796. doi: 10.3969/j.issn.1002-0802.2018.08.007 WANG Ling-bo, LI Guang-qiu, QIAN Hui. Symbol error rate performance of DE-QPSK with predictive transmit antenna selection and orthogonal space-time block code[J]. Communications Technology, 2018, 51(8): 1791-1796. doi: 10.3969/j.issn.1002-0802.2018.08.007 [15] PHAN K T, TELLAMBURA C. Capacity analysis for transmit antenna selection using orthogonal space-time block codes[J]. IEEE Communications Letters, 2007, 11(5): 423-425. doi: 10.1109/LCOMM.2007.070062 [16] ABRAMOWITZ M, STEGUN I A. Handbook of mathematical functions with formulas, graphs, and mathematical tables[M]. New York, USA: Dover Publications, 1972.