Differential Space-Time Modulation Using DAPSK Over Rician Fading Channels

Abstract

This paper studies differential space-time modulation using diversity-encoded differential amplitude and phase shift keying (DAPSK) for the multiple-input multiple-output (MIMO) system over independent but not identically distributed (inid) time-correlated Rician fading channels. An asymptotic maximum likelihood (AML) receiver is developed for differentially detecting diversity-encoded DAPSK symbol signals by operating on two consecutive received symbol blocks sequentially. Based on Beaulieu’s convergent series, the bit error probability (BEP) upper bound is analyzed for the AML receiver over inid time-correlated Rician fading channels. Particularly, an approximate BEP upper bound of the AML receiver is also derived for inid time-invariant Rayleigh fading channels with large received signal-to-noise power ratios. By virtue of this approximate bound, a design criterion is developed to determine the appropriate diversity encoding coefficients for the proposed DAPSK MIMO system. Numerical and simulation results show that the AML receiver for diversity-encoded DAPSK is nearly optimum when the average received signal-to-noise power ratios are high and the channel is heavily correlated fading and can provide better error performance than conventional noncoherent MIMO systems when the effect of non-ideal transmit power amplification is taken into account.

Appendix

1.1 Derivation of (18)

Given $\mathcal {S}$ and $\mathcal {S}_{d},\, Q_{kl}$ in (17) is a conditionally GQS of $U_{kl}$ and $V_{kl}$ which have means $\overline{U} _{kl}\triangleq \mathbb {E}\{U_{kl}|\mathcal {S},\mathcal {S}_{d}\}=\sqrt{ \gamma _{kl}/\varGamma _{kl}}\tau _{kl}(d_{k}-\widehat{d}_{k})x_{k}^{(i-1)}$ and $\overline{V}_{kl}\triangleq \mathbb {E}\left\{ V_{kl}|\mathcal {S}, \mathcal {S}_{d}\right\} =0$ and second central moments

$$\begin{aligned} \phi _{kl}^{(1)}&\triangleq \mathbb {E}\left\{ \left. \left| U_{kl}- \overline{U}_{kl}\right| ^{2}\right| \mathcal {S},\mathcal {S} _{d}\right\} \nonumber \\&= \frac{\gamma _{kl}}{1+\varOmega _{kl}}|x_{k}^{(i-1)}|^{2}\left[ \left| d_{k}\right| ^{2}+|\widehat{d}_{k}|^{2}-2\text{ Re }\left\{ \rho _{kl}d_{k}\widehat{d}_{k}^{*}\right\} \right] +|\widehat{d} _{k}|^{2}+1 \end{aligned}$$

(25)

$$\begin{aligned} \phi _{kl}^{(2)}&\triangleq \mathbb {E}\left\{ \left. \left| V_{kl}- \overline{V}_{kl}\right| ^{2}\right| \mathcal {S},\mathcal {S} _{d}\right\} =\frac{2\left( 1-\text{ Re }\left\{ \rho _{kl}\right\} \right) \gamma _{kl}}{1+\varOmega _{kl}}\left| d_{k}\right| ^{2}|x_{k}^{(i-1)}|^{2}+\left| d_{k}\right| ^{2}+1 \quad \quad \end{aligned}$$

(26)

$$\begin{aligned} \phi _{kl}^{(3)}&\triangleq \mathbb {E}\left\{ (U_{kl}-\overline{U} _{kl})(V_{kl}-\overline{V}_{kl})^{*}|\mathcal {S},\mathcal {S}_{d}\right\} \nonumber \\&= \frac{\gamma _{kl}|x_{k}^{(i-1)}|^{2}}{1+\varOmega _{kl}}\left[ \left( 1-\rho _{kl}\right) d_{k}+\left( 1-\rho _{kl}^{*}\right) \widehat{d}_{k}\right] d_{k}^{*}+\widehat{d}_{k}d_{k}^{*}+1. \end{aligned}$$

(27)

With reference to [27, eq. (B-5)], the conditional MGF $\varPhi _{Q_{kl}}(s|\mathcal {S},\mathcal {S}_{d})$ is given by (18) with

$$\begin{aligned}&v_{kl}^{(1)}\triangleq \sqrt{v_{kl}^{2}+u_{kl}}-v_{kl},\quad v_{kl}^{(2)}\triangleq -\sqrt{v_{kl}^{2}+u_{kl}}-v_{kl} \end{aligned}$$

(28)

$$\begin{aligned}&v_{kl}\triangleq \frac{(1+\left| d_{k}\right| ^{2})\phi _{kl}^{(1)}-(1+|\widehat{d}_{k}|^{2})\phi _{kl}^{(2)}}{2\left( \phi _{kl}^{(1)}\phi _{kl}^{(2)}-|\phi _{kl}^{(3)}|^{2}\right) },\quad u_{kl}\triangleq \frac{(1+\left| d_{k}\right| ^{2})(1+|\widehat{d} _{k}|^{2})}{\phi _{kl}^{(1)}\phi _{kl}^{(2)}-|\phi _{kl}^{(3)}|^{2}} \end{aligned}$$

(29)

$$\begin{aligned}&\alpha _{kl}^{(1)}\triangleq \frac{\left| \overline{U}_{kl}\right| ^{2}\phi _{kl}^{(2)}}{(1+\left| d_{k}\right| ^{2})(1+|\widehat{d} _{k}|^{2})},\quad \alpha _{kl}^{(2)}\triangleq \frac{\left| \overline{U} _{kl}\right| ^{2}}{1+|\widehat{d}_{k}|^{2}}. \end{aligned}$$

(30)

Note that $v_{kl}^{(1)}v_{kl}^{(2)}=-u_{kl}$. Using (25)–(27 ), $\phi _{kl}^{(1)}\phi _{kl}^{(2)}-|\phi _{kl}^{(3)}|^{2}$ can be further expressed as

$$\begin{aligned} \phi _{kl}^{(1)}\phi _{kl}^{(2)}\!-\!|\phi _{kl}^{(3)}|^{2}\!=\!|d_{k}\!-\!\widehat{d} _{k}|^{2}\left[ \frac{\gamma _{kl}^{2}\left| d_{k}\right| ^{2}|x_{k}^{(i-1)}|^{4}}{\left( 1\!+\!\varOmega _{kl}\right) ^{2}}\left( 1\!-\!\left| \rho _{kl}\right| ^{2}\right) \!+\!\frac{\left( 1\!+\!\left| d_{k}\right| ^{2}\right) \gamma _{kl}}{1\!+\!\varOmega _{kl}} |x_{k}^{(i-1)}|^{2}\!+\!1\right] . \end{aligned}$$

For $d_{k}\ne \widehat{d}_{k},\, \phi _{kl}^{(1)}\phi _{kl}^{(2)}-|\phi _{kl}^{(3)}|^{2}$ is always positive, and then $v_{kl}^{(1)}>0$ and $v_{kl}^{(2)}<0$ from (28).

1.2 Derivations of (23) and (24)

When $\varOmega _{kl}=0$ and $\rho _{kl}=1$ for all $k\in \mathcal {Z} _{L_{t}}^{+}$ and $l\in \mathcal {Z}_{L_{r}}^{+}$, the conditional MGF $\varPhi _{Q}(s|\mathcal {S},\mathcal {S}_{d})$ in (18) specializes to

$$\begin{aligned} \varPhi _{Q}(s|\mathcal {S},\mathcal {S}_{d})=\prod \limits _{k=1}^{L_{t}}\prod \limits _{l=1}^{L_{r}}\frac{-u_{kl}}{\left( s-v_{kl}^{(1)}\right) \left( s-v_{kl}^{(2)}\right) } \end{aligned}$$

(31)

where $v_{kl}$ and $u_{kl}$ in (29) simplify, respectively to

$$\begin{aligned} v_{kl}&= \frac{\left( 1+\left| d_{k}\right| ^{2}\right) |x_{k}^{(i-1)}|^{2}g_{k}\varGamma _{kl}\gamma _{t}}{2\left[ 1+\left( 1+\left| d_{k}\right| ^{2}\right) |x_{k}^{(i-1)}|^{2}g_{k}\varGamma _{kl}\gamma _{t}\right] },\\ u_{kl}&= \frac{(1+\left| d_{k}\right| ^{2})(1+|\widehat{d}_{k}|^{2})}{|d_{k}-\widehat{d}_{k}|^{2} \left[ 1+\left( 1+\left| d_{k}\right| ^{2}\right) |x_{k}^{(i-1)}|^{2}g_{k}\varGamma _{kl}\gamma _{t}\right] }. \end{aligned}$$

When $\gamma _{t}\gg 1,\, v_{kl}$ and $u_{kl}$ can be further approximated as $v_{kl}\approx 1/2$ and $u_{kl}\approx (1+|\widehat{d}_{k}|^{2})/(|d_{k}- \widehat{d}_{k}|^{2}|x_{k}^{(i-1)}|^{2}g_{k}\varGamma _{kl}\gamma _{t})$, and thus $v_{kl}^{(1)}$ and $v_{kl}^{(2)}$ in (28) are approximated as $v_{kl}^{(1)}\approx 0$ and $v_{kl}^{(2)}\approx -1$ for $k\in \mathcal {Z} _{L_{t}}^{+}$ and $l\in \mathcal {Z}_{L_{r}}^{+}$ because $u_{kl}$ is much smaller than $v_{kl}$ for large $\gamma _{t}$ values. Using this approximation, $\varPhi _{Q}\left( s|\mathcal {S},\mathcal {S}_{d}\right)$ in ( 31) can be approximated for $\gamma _{t}\gg 1$ as

$$\begin{aligned} \varPhi _{Q}\left( s|\mathcal {S},\mathcal {S}_{d}\right) \approx \left( \prod \limits _{k=1}^{L_{t}}\prod \limits _{l=1}^{L_{r}}\frac{1+|\widehat{d} _{k}|^{2}}{|d_{k}-\widehat{d}_{k}|^{2}|x_{k}^{(i-1)}|^{2}g_{k}\varGamma _{kl}\gamma _{t}}\right) \left[ \frac{-1}{s(s+1)}\right] ^{L_{t}L_{r}}. \end{aligned}$$

(32)

Putting the high SNR approximation of $\Pr \{Q<\vartheta |\mathcal {S}, \mathcal {S}_{d}\}$ in (32) into (21) yields

$$\begin{aligned}&\Pr \{Q<\vartheta |\mathcal {S},\mathcal {S}_{d}\}\approx \left\{ \begin{array}{l@{\quad }l} Res\left( \varPhi _{Q}\left( s|\mathcal {S},\mathcal {S}_{d}\right) \frac{\exp \left\{ -s\vartheta \right\} }{s},s=0,L_{t}L_{r}+1\right) , &{} \vartheta >0 \\ -Res\left( \varPhi _{Q}\left( s|\mathcal {S},\mathcal {S}_{d}\right) \frac{\exp \left\{ -s\vartheta \right\} }{s},s=-1,L_{t}L_{r}\right) , &{} \vartheta \le 0 \end{array} \right. \end{aligned}$$

(33)

$$\begin{aligned}&\quad =\left\{ \begin{array}{l@{\quad }l} \left( \prod \limits _{k=1}^{L_{t}}\prod \limits _{l=1}^{L_{r}}\frac{1+|\widehat{ d}_{k}|^{2}}{|d_{k}-\widehat{d}_{k}|^{2}|x_{k}^{(i-1)}|^{2}g_{k}\varGamma _{kl}\gamma _{t}}\right) \frac{(-1)^{L_{t}L_{r}}}{(L_{t}L_{r})!}\frac{ d^{L_{t}L_{r}}}{ds^{L_{t}L_{r}}}\left. \left( \frac{\exp \left\{ -s\vartheta \right\} }{(s+1)^{L_{t}L_{r}}}\right) \right| _{s=0}, &{} \vartheta >0 \\ \left( \prod \limits _{k=1}^{L_{t}}\prod \limits _{l=1}^{L_{r}}\frac{1+|\widehat{ d}_{k}|^{2}}{|d_{k}-\widehat{d}_{k}|^{2}|x_{k}^{(i-1)}|^{2}g_{k}\varGamma _{kl}\gamma _{t}}\right) \frac{(-1)^{L_{t}L_{r}+1}}{(L_{t}L_{r}-1)!}\frac{ d^{L_{t}L_{r}-1}}{ds^{L_{t}L_{r}-1}}\left. \left( \frac{\exp \left\{ -s\vartheta \right\} }{s^{L_{t}L_{r}+1}}\right) \right| _{s=-1}, &{} \vartheta \le 0 \end{array} \right. .\nonumber \\ \end{aligned}$$

(34)

Using Leibniz’s rule of differentiation in [28, eq. (0.42)], we have

$$\begin{aligned}&\frac{(-1)^{L_{t}L_{r}}}{(L_{t}L_{r})!}\frac{d^{L_{t}L_{r}}}{ds^{L_{t}L_{r}}} \left. \left( \frac{\exp \left\{ -s\vartheta \right\} }{(s+1)^{L_{t}L_{r}}} \right) \right| _{s=0}=\sum _{m=0}^{L_{t}L_{r}}\left( \begin{array}{l}{ 2L_{t}L_{r}-1-m}\\ {L_{t}L_{r}-1}\end{array}\right) \frac{\vartheta ^{m}}{m!},\quad \vartheta >0 \end{aligned}$$

(35)

$$\begin{aligned} \frac{(-1)^{L_{t}L_{r}+1}}{(L_{t}L_{r}-1)!}\frac{d^{L_{t}L_{r}-1}}{ ds^{L_{t}L_{r}-1}}\left. \left( \frac{\exp \left\{ -s\vartheta \right\} }{ s^{L_{t}L_{r}+1}}\right) \right| _{s=-1}&= \sum _{m=0}^{L_{t}L_{r}-1} \left( \begin{array}{l}{2L_{t}L_{r}-1-m}\\ {L_{t}L_{r}}\end{array}\right) \frac{\left( -\vartheta \right) ^{m}\exp \left\{ \vartheta \right\} }{m!},\nonumber \\&\quad \vartheta \le 0. \end{aligned}$$

(36)

Putting (35) and (36) into (34) results in (23) and (24).