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Transmit Power Minimization for MIMO Systems of Exponential Average BER with Fixed Outage Probability

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Abstract

This paper is concerned with a wireless multiple-antenna system operating in multiple-input multiple-output (MIMO) fading channels with channel state information being known at both transmitter and receiver. By spatiotemporal subchannel selection and power control, it aims to minimize the average transmit power (ATP) of the MIMO system while achieving an exponential type of average bit error rate (BER) for each data stream. Under the constraints on each subchannel that individual outage probability and average BER are given, based on a traditional upper bound and a dynamic upper bound of Q function, two closed-form ATP expressions are derived, respectively, which can result in two different power allocation schemes. Numerical results are provided to validate the theoretical analysis, and show that the power allocation scheme with the dynamic upper bound can achieve more power savings than the one with the traditional upper bound.

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Acknowledgments

This work was supported in part by the Scheme of Research Exchanges with China and India, the Royal Academy of Engineering of UK, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20132125110006, and the open research fund of Zhejiang Provincial Key Lab of Data Storage and Transmission Technology, Hangzhou Dianzi University, under Grant 201401.

Author information

Authors and Affiliations

  1. College of Information Science and Technology, Dalian Maritime University, Dalian, 116026, Liaoning, China

    Dian-Wu Yue

  2. School of Communications Engineering, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Dian-Wu Yue

  3. School of Engineering and Technology, University of Hertfordshire, College Lane, Hatfield, Herts, AL10 9AB, UK

    Yichuang Sun

Authors
  1. Dian-Wu Yue
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  2. Yichuang Sun
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Corresponding author

Correspondence to Dian-Wu Yue.

Appendix

Appendix

The proof of Proposition 1

Since \(\overline{P}_b(i) \le \frac{\xi _i}{2} \frac{\overline{\lambda }_{\mathrm{out}}(i)}{\overline{\lambda }_{\mathrm{mea}}(i)}\), then it follows from Lemma 2 that \(\lambda _0(i)=\overline{\lambda }_{\mathrm{out}}(i)\). Furthermore, we can have

$$\begin{aligned} \hat{\rho }^{(i)} \left( \overline{P}_b(i),P_{\mathrm{out}}^{(i)}\right) =\int _{\overline{\lambda }_{\mathrm{out}}(i)}^{\infty }p_i f_i(\lambda _i)d\lambda _i. \end{aligned}$$
(60)

Substituting (29) into (60), we obtain

$$\begin{aligned} \hat{\rho }^{(i)} \left( \overline{P}_b(i),P_{\mathrm{out}}^{(i)}\right)= & {} \int _{\overline{\lambda }_{\mathrm{out}}(i)}^{\infty }\frac{\widehat{\hbox {SNR}}(i)}{\lambda _i} f_i(\lambda _i)d\lambda _i\nonumber \\&+\int _{\overline{\lambda }_{\mathrm{out}}(i)}^{\infty }\frac{2}{\beta _i\lambda _i} \ln \left( \frac{\lambda _i\Delta (i)}{\overline{\lambda }_{\mathrm{out}}(i)}\right) f_i(\lambda _i)d\lambda _i. \end{aligned}$$
(61)

With respect to (61), we define

$$\begin{aligned} \rho _s \left( \widehat{\hbox {SNR}}(i),\overline{\lambda }_{\mathrm{out}}(i)\right) =\int _{\overline{\lambda }_{\mathrm{out}}(i)}^{\infty }\frac{\widehat{\hbox {SNR}}(i)}{\lambda _i} f_i(\lambda _i)d\lambda _i \end{aligned}$$
(62)

and

$$\begin{aligned} \rho _{\Delta } \left( \overline{\lambda }_{\mathrm{out}}(i)\right) =\int _{\overline{\lambda }_{\mathrm{out}}(i)}^{\infty }\frac{2}{\beta _i\lambda _i} \ln \left( \frac{\lambda _i\Delta (i)}{\overline{\lambda }_{\mathrm{out}}(i)}\right) f_i(\lambda _i)d\lambda _i. \end{aligned}$$
(63)

In what follows, we consider to derive (62) and (63), respectively. \(\square\)

From Lemma 1 in [36], the marginal p.d.f. of the \(i\)-th largest eigenvalue \(\lambda _i\) can be written as

$$\begin{aligned} f_i(\lambda _i)=\sum _{k=i}^m(-1)^{k-i}{k-1 \atopwithdelims ()i-1}{m \atopwithdelims ()k}f_{\mathrm{min:}k}(\lambda _i) \end{aligned}$$
(64)

where \(f_{\mathrm{min:}k}(x)\) denotes the p.d.f. of the smallest random variable considered in a subset of k random variables over the set of all eigenvalues, and is given by

$$\begin{aligned} f_{\mathrm{min:}k}(x) &= \frac{kC}{m!}\sum _{{\mathcal {\alpha }}} \sum _{{\mathcal {\mu }}} \hbox {sgn}({\mathcal {\alpha }})\hbox {sgn} ({\mathcal {\mu }})A_{k}({\mathcal {\alpha }},{\mathcal {\mu }}) \nonumber \\&\times \,e^{-kx}{\mathbb {\sum }}_{{\mathcal {\tau }}} x^{\theta +\alpha _k+\mu _k-2+\sum _{\iota =1}^{k-1}\tau _{\iota }} \prod _{\iota =1}^{k-1}\frac{(\theta +\alpha _{\iota }+\mu _{\iota }-2)!}{\tau _{\iota }!}. \end{aligned}$$
(65)

With the help of the complementary incomplete gamma function \(\Gamma (q,x)\), thus we can obtain the desired result (43) after a simple derivation.

The derivation of (63) is similar, but involves a process employing the following special function \(\jmath _q(x)\) defined as [38]:

$$\begin{aligned} \jmath _q(x) &= \int _1^\infty t^{q-1}\ln t e^{-xt}dt\nonumber \\= & {} \frac{(q-1)!}{x^q}\sum _{k=0}^{q-1}\Gamma (k,x)/k!. \end{aligned}$$
(66)

Finally, again making use of (64), we can easily obtain the desired expression of \(\rho _{\Delta }(\overline{\lambda }_{\mathrm{out}}(i))\) (44).

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Yue, DW., Sun, Y. Transmit Power Minimization for MIMO Systems of Exponential Average BER with Fixed Outage Probability. Wireless Pers Commun 90, 1951–1970 (2016). https://doi.org/10.1007/s11277-016-3432-4

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