Robust adaptive multi-service transmission with hierarchical modulation for OFDM systems in high mobility scenarios

Abstract

Wireless communications play a key role in intelligent transportation systems for bearing multiple application services with different quality of service (QoS) requirements. Hierarchical modulation (HM) is an effective technique to increase the spectral efficiency (SE) of differentiated services multiplexing transmissions. In high mobility scenario, due to the impact of the feedback delay and channel estimation error, only the imperfect channel state information is available at the transmitter. Thus, the performance of the existing HM schemes degrades severely. In this paper, we propose a robust adaptive HM scheme to increase the overall SE in the mobile orthogonal frequency division multiplexing system. The bit error rate of the HM symbol is derived for the reliable QoS-aware multi-service transmission, and the strategies of bit assignment and modulation parameters selection for different services are designed. Finally, comparing with the traditional uniform modulation scheme, the proposed robust adaptive HM scheme increases the overall SE of multiplexing transmissions by $15\%$ at $300\,\hbox { km}/\hbox {h}$.

Appendices

Appendix

Proof of Eq. (34)

Proof

According to (8), (26), considering the channel estimation error and feedback delay, the ICI power of the kth subcarrier with the equal average transmit power constraint ${\bar{P}} = E\{ |s_k|^2\}$ can be expressed as

$$\begin{aligned} \begin{aligned} {\mathbb {E}} \{ |\hat{I}_k|^2\}&= \frac{1}{N^2} \sum _{q=0, q \ne k}^{N-1} {\mathbb {E}}\{|s_q|^2\} {\mathbb {E}}\{{\hat{H}}_q(n_1) {\hat{H}}_q(n_2)^*\} \\&\quad \times \exp \left( \frac{j 2\pi (n_1 - n_2) (q-k)}{N} \right) \\&= \frac{1}{N^2} \sum _{q=0, q \ne k}^{N-1} {\bar{P}} \sum _{n_1=0}^{N-1} \sum _{n_2=0}^{N-1} {\mathbb {E}} \big \{ \big (H_q(n_1 - \varDelta n) \\&\quad + \varepsilon _q(n_1 - \varDelta n) \big ) \big (H_q(n_2 - \varDelta n)^* + \varepsilon _q(n_2 - \varDelta n) \big ) \big \} \\&\quad \times \exp \left( \frac{j 2\pi (n_1 - n_2) (q-k)}{N} \right) \\&= \frac{1}{N^2} \sum _{q=0, q \ne k}^{N-1} {\bar{P}} \sum _{n_1=0}^{N-1} \sum _{n_2=0}^{N-1} E\{ H_q(n_1 - \varDelta n) \\&\quad \times H_q(n_2 - \varDelta n)^* \} \exp \left( \frac{j 2\pi (n_1 - n_2) (q-k)}{N} \right) \\&\quad + \frac{1}{N^2} \sum _{q=0, q \ne k}^{N-1} {\bar{P}} N \sigma _{\varepsilon }^2 \\&= {\bar{P}} \left( \sum _{q=0, q \ne k}^{N-1} \varphi _{k,q} + \frac{1}{N^2} \sum _{q=0, q \ne k}^{N-1} N \sigma _{\varepsilon }^2 \right) \\&= {\bar{P}} \left( P_N + \frac{N-1}{N} \sigma _{\varepsilon }^2 \right) . \end{aligned} \end{aligned}$$

We have the normalized ICI power: $\hat{P}_N = P_N + \frac{N-1}{N} \sigma _{\varepsilon }^2$. According to (18), the instantaneous received SINR can be formulated as

$$\begin{aligned} \begin{aligned} \hat{\eta }_k&= \frac{\gamma _k}{ (P_N + \frac{N-1}{N} \sigma _{\varepsilon }^2 ) {\bar{\gamma }} +1} \\&= \frac{\gamma _k}{\hat{P}_N {\bar{\gamma }} +1}, \end{aligned} \end{aligned}$$

(48)

where $\hat{P}_N = P_N + \frac{N-1}{N} \sigma _{\varepsilon }^2$ is the normalized ICI power with the imperfect CSI. $\square$

Proof of Eq. (36)

Proof

According to (36), we have the following expression

$$\begin{aligned} \begin{aligned}&{{\widetilde{BER}}}_4^{I, 1}(\hat{\gamma }_k, \alpha _1) \\&\quad = \int _0^\infty \kappa s_4^{I, 1} \exp \left( {t_4^{I, 1}(\alpha _1) \gamma _k} \right) f(\gamma _k | \hat{\gamma }_k) ~d\gamma _k \\&\quad = \int _0^\infty \kappa s_4^{I, 1} \frac{1}{{\bar{\gamma }} \sigma ^2} \exp \left( \frac{\rho ^2 \hat{\gamma }_k}{- {\bar{\gamma }} \sigma ^2 (\sigma _{{\hat{H}}_k}^2)^2} \right) \\&\qquad \times \exp \left( {t_4^{I, 1}(\alpha _1) \gamma _k} + \frac{\gamma _k (\sigma _{{\hat{H}}_k}^2)^2}{- {\bar{\gamma }} \sigma ^2 (\sigma _{{\hat{H}}_k}^2)^2}\right) \mathrm {I}_0 \left( \frac{2 \rho \sqrt{\hat{\gamma }_k \gamma _k}}{{\bar{\gamma }} \sigma ^2 \sigma _{{\hat{H}}_k}^2} \right) d\gamma _k , \end{aligned} \end{aligned}$$

(49)

By means of the integral formula in [35], we have the following result

$$\begin{aligned} \begin{aligned}&\int _{0}^{+\infty } x^{\mu - \frac{1}{2}} \exp (- \alpha x) \mathrm {I}_{2 v} (2 \beta \sqrt{x}) ~dx\\&\quad = \frac{\varGamma \left( \mu + v + \frac{1}{2}\right) }{\varGamma \left( 2 v + \frac{1}{2}\right) } \beta ^{-1} \exp \left( \frac{\beta ^2}{2 \alpha }\right) \alpha ^{- \mu } \mathrm {M}_{(-\mu , v)} \left( \frac{\beta ^2}{\alpha }\right) , \end{aligned} \end{aligned}$$

(50)

where $\mathrm {M}_{(a, b)}(y) = y^{(b+ \frac{1}{2})} \exp (\frac{-y}{2}) \varPhi (b- a+ \frac{1}{2}, 2 b+ 1, y)$, and $\varPhi (d, d, z) = \exp (z)$.

Considering the expression of the average BER (49), we have $\mu = \frac{1}{2}$, $v = 0$, $\alpha = -t_4^{I, 1}(\alpha _1) + {(\sigma _{{\hat{H}}_k}^2)^2} / {{\bar{\gamma }} \sigma ^2 (\sigma _{{\hat{H}}_k}^2)^2}$, and $\beta = {\rho \sqrt{\hat{\gamma }_k}} / {{\bar{\gamma }} \sigma ^2 (\sigma _{{\hat{H}}_k}^2)^2}$. Substituting (50) into (49), the approximation of the average BER can be derived as

$$\begin{aligned} \begin{aligned}&{{\widetilde{BER}}}_4^{I, 1}(\hat{\gamma }_k, \alpha _1) \\&\quad = \frac{\kappa s_4^{I, 1}}{1- {\bar{\gamma }} \sigma ^2 t_4^{I, 1}(\alpha _1) } \\&\qquad \exp \bigg (\frac{\rho ^2 \hat{\gamma }_k }{(\sigma _{{\hat{H}}_k}^2)^2} \frac{t_4^{I, 1}(\alpha _1)}{1- {\bar{\gamma }} \sigma ^2 t_4^{I, 1}(\alpha _1)} \bigg ), \end{aligned} \end{aligned}$$

(51)

where $s_4^{I, 1} = 1/2$, and $t_4^{I, 1}(\alpha _1) = {-\alpha _1^2} / {(K_4 (\hat{P}_N {\bar{\gamma }} +1)})$. $\square$

Proof of Eq. (43)

Proof

According to (37), assuming the average BER denoted by $p(s, t, \hat{\gamma }_k)$ should to meet the target BER $\varepsilon$, we have the following equation

$$\begin{aligned} \frac{\kappa s}{1- {\bar{\gamma }} \sigma ^2 t} \exp \left( \frac{\rho ^2 \hat{\gamma }_k }{\left( \sigma _{{\hat{H}}_k}^2\right) ^2} \frac{t}{1- {\bar{\gamma }} \sigma ^2 t} \right) = \varepsilon . \end{aligned}$$

(52)

Let $x = {t}/{(1- {\bar{\gamma }} \sigma ^2 t)}$, (52) can be reformed as

$$\begin{aligned} {\kappa s} (1+ {\bar{\gamma }} \sigma ^2 x) \exp \left( \frac{\rho ^2 \hat{\gamma }_k}{\left( \sigma _{{\hat{H}}_k}^2\right) ^2} x \right)&= \varepsilon \nonumber \\ \frac{\kappa s}{\epsilon } + \frac{\kappa s}{\epsilon } {\bar{\gamma }} \sigma ^2 x - \exp \left( \frac{- \rho ^2 \hat{\gamma }_k}{\left( \sigma _{{\hat{H}}_k}^2\right) ^2} x\right)&= 0. \end{aligned}$$

(53)

Many equations involving exponentials can be solved using the Lambert function $z = W(z) e^{W(z)}$, i.e.

$$\begin{aligned} c z +d -\exp \left( a z +b \right) =0, \end{aligned}$$

(54)

which yields the final solution

$$\begin{aligned} z = \frac{- W\left( \frac{-a}{c} \exp (b- \frac{a d}{c}) \right) }{a} - \frac{d}{c}. \end{aligned}$$

(55)

According to (53), we have $a = {- \rho ^2 \hat{\gamma }_k} / {(\sigma _{{\hat{H}}_k}^2)^2}$, $b = 0$, $c = {\bar{\gamma }} \sigma ^2 {\kappa s} / {\epsilon }$, and $d = {\kappa s} / {\epsilon }$. Hence, the solution of (53) can be obtained as

$$\begin{aligned} x = \frac{W\left( \frac{\rho ^2 \hat{\gamma }_k }{(\sigma _{{\hat{H}}_k}^2)^2} \frac{\epsilon }{{\bar{\gamma }} \sigma ^2 {\kappa s} } \exp \Big ( \frac{\rho ^2 \hat{\gamma }_k }{(\sigma _{{\hat{H}}_k}^2)^2} \frac{1}{{\bar{\gamma }} \sigma ^2}\Big ) \right) }{{\rho ^2 \hat{\gamma }_k } / {(\sigma _{{\hat{H}}_k}^2)^2} } - \frac{1}{{\bar{\gamma }} \sigma ^2}. \end{aligned}$$

(56)

Substituting $x = {t}/{(1- {\bar{\gamma }} \sigma ^2 t)}$ into the equation (56), the solution of modulation parameter t can be derived as

$$\begin{aligned} t = \frac{1}{{\bar{\gamma }} \sigma ^2} - \frac{\hat{\gamma }_k {\rho ^2} / {({\bar{\gamma }} \sigma ^2 \sigma _{{\hat{H}}_k}^2)^2}}{W \left( \frac{\rho ^2 \hat{\gamma }_k }{(\sigma _{{\hat{H}}_k}^2)^2}\frac{\epsilon }{\kappa s {\bar{\gamma }} \sigma ^2} \exp \Big (\frac{\hat{\gamma }_k \rho ^2}{(\sigma _{{\hat{H}}_k}^2)^2} \frac{1}{{\bar{\gamma }} \sigma ^2} \Big ) \right) }. \end{aligned}$$

(57)

$\square$