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Performance analysis of a MIMO-RFID system in Nakagami-m fading channels

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Abstract

We analyze the bit error rate (BER) performance of a multiple-input multiple-output (MIMO) radio-frequency-identification (RFID) system employing orthogonal space-time block codes when the forward and backward channels exhibit independent but not necessarily identically distributed Nakagami-m fading. A closed-form upper bound on the BER performance is derived, and the corresponding diversity order is quantified. Numerical results provide some insight into the impact of several different parameters on the system performance.

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Notes

  1. Recent measurement campaigns, e.g., [2–4], have revealed that multiple antennas can be accommodated on small RFID tags.

  2. The integer assumption on m h is necessary for the theoretical analysis hereafter (e.g., Eq. 14), whereas that on m g is just for analytical convenience.

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Acknowledgments

This work has been supported by DFG project “MIMO Backscatter-Übertragung auf Basis von Mehrantennen-Transpondern in RFID-basierten Funksystemen” with grant no. KA 1154/30-1.

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Authors and Affiliations

  1. College of Information and Communication Technology, Rangsit University, Pathumthani, Thailand

    Kiattisak Maichalernnukul

  2. Institute of Digital Signal Processing, University of Duisburg-Essen, Duisburg, Germany

    Kiattisak Maichalernnukul, Feng Zheng & Thomas Kaiser

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  1. Kiattisak Maichalernnukul
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  2. Feng Zheng
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  3. Thomas Kaiser
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Correspondence to Kiattisak Maichalernnukul.

Appendix

Appendix

Let us define \(\eta _{j}={\sum }_{i=1}^{L}\big {|}g_{i,j}\big {|}^{2}\) and \(\varphi _{j}=\big {|}{\sum }_{l=1}^{L}h_{j,l}\nu _{l}\big {|}^{2}\). Hence, the variable β j defined in Section 4 can be rewritten as β j = η j φ j . For notational simplicity, the subscript j in these three variables is omitted hereafter. One can show that η follows a Gamma distribution whose probability density function (PDF) is

$$ f_{\eta}(z)=\frac{m_{\text{g}}^{Lm_{\text{g}}}z^{Lm_{\text{g}}-1}}{{\Omega}_{\text{g}}^{Lm_{\text{g}}}{\Gamma}(Lm_{\text{g}})}\,\exp\!\left( \!-\frac{m_{\text{g}}z}{{\Omega}_{\text{g}}}\right);\hspace{3mm}z\geq{0}. $$
(13)

Using [16, Eq. (8)] and with the help of [17, p. 62], the PDF of φ can be obtained as

$$\begin{array}{@{}rcl@{}} f_{\varphi}(z)&=&\exp\!\left( \!-\frac{m_{\text{h}}z}{L{\Omega}_{\text{h}}}\!\right)\!\sum\limits_{p_{1}=0}^{m_{\text{h}}-1}\!\cdots\!\sum\limits_{p_{L}=0}^{m_{\text{h}}-1}\sum\limits_{q=0}^{\mathcal{S}}\prod\limits_{k=1}^{L}\left[\frac{(1\,-\,m_{\text{h}})_{p_{k}}}{(p_{k}!)^{2}}\!\left( \!\frac{{\Omega}_{\text{h}}}{4m_{\text{h}}}\!\right)^{\!p_{k}}\!\right] \\&&\quad \times\left( \!\frac{4m_{\text{h}}}{L{\Omega}_{\text{h}}}\!\right)^{\!\mathcal{S}+q+1}\!\frac{(-\mathcal{S})_{q}\,\mathcal{S}!\,z^{q}}{2^{2q+2}(q!)^{2}};\hspace{3mm}z\geq{0}. \end{array} $$
(14)

With the aid of [17, p. 140], the PDF of β can be derived as follows:

$$\begin{array}{@{}rcl@{}} f_{\beta}(z)&=&{\int}_{0}^{\infty}\frac{1}{x}\,f_{\eta}\!\left( \frac{z}{x}\right)\!f_{\varphi}(x)\,\text{d}x \\& =&\frac{m_{\text{g}}^{Lm_{\text{g}}}z^{Lm_{\text{g}}-1}}{{\Omega}_{\text{g}}^{Lm_{\text{g}}}{\Gamma}(Lm_{\text{g}})}\sum\limits_{p_{1}=0}^{m_{\text{h}}-1}\!\cdots\!\sum\limits_{p_{L}=0}^{m_{\text{h}}-1}\sum\limits_{q=0}^{\mathcal{S}}\prod\limits_{k=1}^{L}\left[\frac{(1\,-\,m_{\text{h}})_{p_{k}}}{(p_{k}!)^{2}}\left( \!\frac{{\Omega}_{\text{h}}}{4m_{\text{h}}}\!\right)^{\!p_{k}}\right]\!\left( \!\frac{4m_{\text{h}}}{L{\Omega}_{\text{h}}}\!\right)^{\!\mathcal{S}+q+1} \\&&\quad \times\frac{(-\mathcal{S})_{q}\,\mathcal{S}!}{2^{2q+2}(q!)^{2}}{\int}_{0}^{\infty}x^{q-Lm_{\text{g}}}\exp\!\left( \!-\frac{m_{\text{h}}x}{L{\Omega}_{\text{h}}}-\frac{m_{\text{g}}z}{{\Omega}_{\text{g}}x}\!\right)\text{d}x \\& =&\frac{2}{\Gamma(Lm_{\text{g}})}\sum\limits_{p_{1}=0}^{m_{\text{h}}-1}\!\cdots\!\sum\limits_{p_{L}=0}^{m_{\text{h}}-1}\sum\limits_{q=0}^{\mathcal{S}}\prod\limits_{k=1}^{L}\left[\frac{(1\,-\,m_{\text{h}})_{p_{k}}}{(p_{k}!)^{2}}\left( \!\frac{{\Omega}_{\text{h}}}{4m_{\text{h}}}\!\right)^{\!p_{k}}\right]\!\left( \!\frac{4m_{\text{h}}}{L{\Omega}_{\text{h}}}\!\right)^{\!\mathcal{S}} \\&&\quad \times\left( \!\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}}\!\right)^{\!(Lm_{\text{g}}+q+1)/2}\frac{(-\mathcal{S})_{q}\,\mathcal{S}!\,z^{(Lm_{\text{g}}+q-1)/2}}{(q!)^{2}}\;\mathcal{K}_{q-Lm_{\text{g}}+1}\!\left( \!2\sqrt{\frac{m_{\text{h}}m_{\text{g}}z}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}}}\right) \end{array} $$
(15)

where the last equality is obtained by using [10, Eq. (3.471.9)]. Since the MGF of β, denoted by \(\mathcal {M}_{\beta }(s)={\int }_{0}^{\infty }\exp (sz)\,f_{\beta }(z)\,\text {d}z\), is the Laplace transform of f β (z) with the exponent reversed in sign [13, p. 100], we have

$$\begin{array}{@{}rcl@{}}&&\mathcal{M}_{\beta}(-s)={\int}_{0}^{\infty}\exp(-sz)\,f_{\beta}(z)\,\text{d}z \\&& =\frac{2}{\Gamma(Lm_{\text{g}})}\sum\limits_{p_{1}=0}^{m_{\text{h}}-1}\!\cdots\!\sum\limits_{p_{L}=0}^{m_{\text{h}}-1}\sum\limits_{q=0}^{\mathcal{S}}\prod\limits_{k=1}^{L}\left[\frac{(1\,-\,m_{\text{h}})_{p_{k}}}{(p_{k}!)^{2}}\left( \!\frac{{\Omega}_{\text{h}}}{4m_{\text{h}}}\!\right)^{\!p_{k}}\right]\!\left( \!\frac{4m_{\text{h}}}{L{\Omega}_{\text{h}}}\!\right)^{\!\mathcal{S}}\!\left( \!\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}}\!\right)^{\!(Lm_{\text{g}}+q+1)/2} \\&&\quad \times\frac{(-\mathcal{S})_{q}\,\mathcal{S}!}{(q!)^{2}}{\int}_{0}^{\infty}z^{(Lm_{\text{g}}+q-1)/2}\exp(-sz)\,\mathcal{K}_{q-Lm_{\text{g}}+1}\!\left( 2\sqrt{\frac{m_{\text{h}}m_{\text{g}}z}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}}}\right)\text{d}z \\&& =\frac{4}{\Gamma(Lm_{\text{g}})}\sum\limits_{p_{1}=0}^{m_{\text{h}}-1}\!\cdots\!\sum\limits_{p_{L}=0}^{m_{\text{h}}-1}\sum\limits_{q=0}^{\mathcal{S}}\prod\limits_{k=1}^{L}\left[\frac{(1\,-\,m_{\text{h}})_{p_{k}}}{(p_{k}!)^{2}}\left( \!\frac{{\Omega}_{\text{h}}}{4m_{\text{h}}}\!\right)^{\!p_{k}}\right]\!\left( \!\frac{4m_{\text{h}}}{L{\Omega}_{\text{h}}}\!\right)^{\!\mathcal{S}}\!\left( \!\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}}\!\right)^{\!(Lm_{\text{g}}+q+1)/2} \\&&\quad \times\frac{(-\mathcal{S})_{q}\,\mathcal{S}!}{(q!)^{2}}{\int}_{0}^{\infty}r^{Lm_{\text{g}}+q}\exp(-s{r^{2}})\,\mathcal{K}_{q-Lm_{\text{g}}+1}\!\left( 2r\sqrt{\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}}}\right)\text{d}r \\&& =\exp\!\left( \!\frac{m_{\text{h}}m_{\text{g}}}{2L{\Omega}_{\text{h}}{\Omega}_{\text{g}}s}\!\right)\sum\limits_{p_{1}=0}^{m_{\text{h}}-1}\!\cdots\!\sum\limits_{p_{L}=0}^{m_{\text{h}}-1}\sum\limits_{q=0}^{\mathcal{S}}\prod\limits_{k=1}^{L}\left[\frac{(1-m_{\text{h}})_{p_{k}}}{(p_{k}!)^{2}}\left( \!\frac{{\Omega}_{\text{h}}}{4m_{\text{h}}}\!\right)^{\!p_{k}}\right]\!\left( \!\frac{4m_{\text{h}}}{L{\Omega}_{\text{h}}}\!\right)^{\!\mathcal{S}} \\&&\quad \times{\left( \!\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}s}\!\right)^{\!(Lm_{\text{g}}+q)/2}\frac{(-\mathcal{S})_{q}\,\mathcal{S}!}{q!}\;W_{\frac{Lm_{\text{g}}+q}{2},\frac{q-Lm_{\text{g}}+1}{2}}\!\left( \!\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}s}\!\right)} \\&& =\left( \!\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}s}\!\right)^{\!Lm_{\text{g}}}\,\sum\limits_{p_{1}=0}^{m_{\text{h}}-1}\!\cdots\!\sum\limits_{p_{L}=0}^{m_{\text{h}}-1}\sum\limits_{q=0}^{\mathcal{S}}\prod\limits_{k=1}^{L}\left[\frac{(1-m_{\text{h}})_{p_{k}}}{(p_{k}!)^{2}}\left( \!\frac{{\Omega}_{\text{h}}}{4m_{\text{h}}}\!\right)^{\!p_{k}}\right]\!\frac{(-\mathcal{S})_{q}\,\mathcal{S}!}{q!}\left( \!\frac{4m_{\text{h}}}{L{\Omega}_{\text{h}}}\!\right)^{\!\mathcal{S}} \\&&\quad \times{U\!\left( \!Lm_{\text{g}},Lm_{\text{g}}-q;\frac{m_{\text{h}}m_{\text{g}}}{L{\Omega}_{\text{h}}{\Omega}_{\text{g}}s}\!\right)} \end{array} $$
(16)

where the third, fourth, and last equalities are obtained by using the change of variable \(r=\sqrt {z}\), the identity in [10, Eq. (9.232.1)], and the transformation in [11, Eq. (13.1.33)], respectively. Changing the sign of s in Eq. 16 yields (8).

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Maichalernnukul, K., Zheng, F. & Kaiser, T. Performance analysis of a MIMO-RFID system in Nakagami-m fading channels. Ann. Telecommun. 71, 151–156 (2016). https://doi.org/10.1007/s12243-015-0481-8

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