A robust MMSE-DFE framework with joint time-frequency domain processing for UAV-to-ground SC-FDE communication systems

Abstract

In this paper, a robust block MMSE-DFE (joint minimum mean square error and decision feedback equalization) framework with joint time-frequency domain processing is developed for UAV-to-Ground single carrier modulation with frequency domain equalization (SC-FDE) communication systems. Two important issues related to iterative block decision feedback equalization (IBDFE) are discussed here, i.e., the complexity of iterative operations and the feedback symbols reliability. Firstly, in order to simplify the operation process, we directly use the results of MMSE to avoid the complexity of iterative operation. Then in the feedback loop of the MMSE-DFE, we proposed a criterion for feedback symbol metric, which could help us control the error propagation process. This results in the improvement of the error propagation, one of the common problems in a DFE or IBDFE. Finally, simulation results demonstrate the robustness of our designed block MMSE-DFE with joint time-frequency domain processing. Firstly, the feedback correlation measurement criteria proposed in this paper can greatly control the error propagation of DFE, and further improve the balancing performance of DFE equalizer. Second, the equalization framework proposed in this paper can adapt to different modulation modes, different equalization algorithms, and can be well combined with coding algorithms.

Appendix A: Section title of first appendix

For the IBDFE design, the Lagrange method can be used to solve the minimum solution of the Eq. (10). Under the constraint of Eq. (A1), the Lagrange function is defined as Eq. (A2):

$$\begin{aligned}&\sum \limits _{k = 0}^{N - 1} {B_k^{\left( l \right) }} = 0, \end{aligned}$$

(A1)

$$\begin{aligned}&\begin{array}{c} f\left( {{{\textbf{C}}^{\left( l \right) }},{{\textbf{B}}^{\left( l \right) }},{\lambda ^{\left( l \right) }}} \right) = \frac{1}{{{N^2}}}\sum \limits _{k = 0}^{K - 1} {{{\left| {C_k^{\left( l \right) }} \right| }^2}{M_w}} + {\left| {C_k^{\left( l \right) }{H_k} - 1} \right| ^2}{M_{{S_k}}} + {\left| {B_k^{\left( l \right) }} \right| ^2}{M_{\hat{S}_k^{\left( {l - 1} \right) }}}\\ + 2\Re \left[ {B_{k}^{{(l)^{*}}}\left( {C_k^{\left( l \right) }{H_k} - 1} \right) {r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}}} \right] + {\lambda ^{\left( l \right) }}\sum \limits _{k = 0}^{K - 1} {B_k^{\left( l \right) }} \end{array}, \end{aligned}$$

(A2)

where ${\lambda ^{\left( l \right) }}$ is the Lagrange multiplier. It is assumed that ${r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}}$, ${M_{{S_k}}}$, and ${M_{\hat{S}_k^{\left( {l - 1} \right) }}}$ are independent of each other with respect to different frequencies k.

Taking the derivative of $f\left( {{{\textbf{C}}^{\left( l \right) }},{{\textbf{B}}^{\left( l \right) }},{\lambda ^{\left( l \right) }}} \right)$ with respect to $B_k^{\left( l \right) }$, $C_k^{\left( l \right) }$, and ${\lambda ^{\left( l \right) }}$, and setting it equal to 0, yields:

$$\begin{aligned}&\frac{{\partial f\left( {{{\textbf{C}}^{\left( l \right) }},{{\textbf{B}}^{\left( l \right) }},{\lambda ^{\left( l \right) }}} \right) }}{{\partial C_k^{\left( l \right) }}} = 2C_k^{\left( l \right) }{M_w} \nonumber \\&\quad + 2\left[ {C_k^{\left( l \right) }{H_k} - 1} \right] H_k^{*}{M_{{S_k}}} + 2B_k^{\left( l \right) }H_k^{*}r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}^{*} = 0, \end{aligned}$$

(A3)

$$\begin{aligned}&\frac{{\partial f\left( {{{\textbf{C}}^{\left( l \right) }},{{\textbf{B}}^{\left( l \right) }},{\lambda ^{\left( l \right) }}} \right) }}{{\partial B_k^{\left( l \right) }}} = 2B_k^{\left( l \right) }{M_{\hat{S}_k^{\left( {l - 1} \right) }}} + {\lambda ^{\left( l \right) }} \nonumber \\&\quad + 2\left[ {C_k^{\left( l \right) }{H_k} - 1} \right] {r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}} = 0, \end{aligned}$$

(A4)

$$\begin{aligned}&\frac{{\partial f\left( {{{\textbf{C}}^{\left( l \right) }},{{\textbf{B}}^{\left( l \right) }},{\lambda ^{\left( l \right) }}} \right) }}{{\partial {\lambda ^{\left( l \right) }}}} = \sum \limits _{k = 0}^{N - 1} {B_k^{\left( l \right) }} = 0, \end{aligned}$$

(A5)

where $k = 0,\;1,\; \cdots ,\;N - 1$.

According to the Eq. (A4), it can be obtained:

$$\begin{aligned} B_k^{\left( l \right) } = - \frac{1}{{{M_{\hat{S}_k^{\left( {l - 1} \right) }}}}}\left[ {{r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}}\left[ {C_k^{\left( l \right) }{H_k} - 1} \right] - \frac{{{\lambda ^{\left( l \right) }}}}{2}} \right] . \end{aligned}$$

(A6)

For any k, the above expression can be rewritten as follows:

$$\begin{aligned} B_k^{\left( l \right) } = - \frac{{{r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}}}}{{{M_{\hat{S}_k^{\left( {l - 1} \right) }}}}}\left[ {C_k^{\left( l \right) }{H_k} - {\gamma ^{\left( l \right) }}} \right] , \end{aligned}$$

(A7)

where ${\gamma ^{\left( l \right) }}$ is to be able to hold the Eq. (A5), ${\gamma ^{\left( l \right) }}$ can be expressed as follows:

$$\begin{aligned} {\gamma ^{\left( l \right) }} = \sum \limits _{k = 0}^{N - 1} {C_k^{\left( l \right) }{H_k}}. \end{aligned}$$

(A8)

For the derivation of the forward filter coefficient, by bringing Eq. (A7) into Eq. (A3), we get:

$$\begin{aligned}&2C_k^{\left( l \right) }{M_w} + 2\left[ {C_k^{\left( l \right) }{H_k} - 1} \right] H_k^{*}{M_{{S_k}}} \nonumber \\&\quad - 2\frac{{{r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}}}}{{{M_{\hat{S}_k^{\left( {l - 1} \right) }}}}}\left[ {C_k^{\left( l \right) }{H_k} - {\gamma ^{\left( l \right) }}} \right] H_k^{*}r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}^{*} = 0. \end{aligned}$$

(A9)

By solving the above equation, we can get:

$$\begin{aligned} C_k^{\left( l \right) } = \frac{{H_k^{*}{M_{{S_k}}}\left( {1 - \frac{{{{\left| {{r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}}} \right| }^2}}}{{{M_{\hat{S}_k^{\left( {l - 1} \right) }}}{M_{{S_k}}}}}{\gamma ^{\left( l \right) }}} \right) }}{{{M_w} + {M_{{S_k}}}\left( {1 - \frac{{{{\left| {{r_{{S_k},\hat{S}_k^{\left( {l - 1} \right) }}}} \right| }^2}}}{{{M_{\hat{S}_k^{\left( {l - 1} \right) }}}{M_{{S_k}}}}}} \right) {{\left| {{H_k}} \right| }^2}}}. \end{aligned}$$

(A10)