Abstract
High power consumption and hardware cost are the two challenges for practical massive MIMO systems. One promising solution is to employ low resolution analog to digital converter (ADC). In this paper, massive MIMO relaying systems are investigated with double resolution ADCs at base station receiver, where some antennas are connected to low resolution ADCs, while the rest of the antennas are connected to high resolution ADCs. Closed form expression for uplink spectral efficiency (SE) is derived using the maximum ratio combining receiver assuming perfect channel state information. Some asymptotic analysis is discussed to explore the effects of system parameters on SE such as; the number of receive antennas, ADC quantization bits, user transmit power and the rate of high precision ADC in double resolution ADC structure. Moreover, energy efficiency with some system parameters is studied. It is shown that the performance wastage owing to low precision ADCs can be recompensed by growing the number of receive antennas \( \varvec{N} \), following a logarithmic scaling law instead of increasing the transmit power at both of source and relay. The double resolution ADCs can achieve a better tradeoff between power consumption and SE when using 4 quantization bits. By using only around 30 antennas, the double resolution ADCs achieves its optimal energy efficiency for selecting system parameters.
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Appendix A
Appendix A
1.1 Proof of Theorem 1
Proof
We approximate \( \varvec{R}_{\varvec{k}} \) as follows:
where \( \psi_{1} \) is defined in (12). The Expectation terms are derived term by term.
First: calculate the nominator in (28)
By using the distribution of \( g_{kH}^{\dag } g_{kH} = {\mathbb{a}}_{k}^{\dag } B_{H}^{\dag } B_{H} {\mathbb{a}}_{k} \) conditionally on \( {\mathbb{a}}_{k} \) and \( B_{H}^{\dag } B_{H} \sim {\mathcal{W}}_{M} \left( {\beta_{R} I_{M} ,N_{H} } \right) \) is Wishart matrix according to [14].
We have \( \left| {g_{kH}^{\dag } g_{kH} } \right|{\mathbb{a}}_{k} = \frac{1}{2}\beta_{R} \left\| {{\mathbb{a}}_{k} } \right\|^{2} \chi_{{2N_{H} }}^{2} \) by utilizing conditional expectation \( E\left\{ {\left| {g_{kH}^{\dag } g_{kH} } \right|^{2} {\mathbb{a}}_{k} } \right\} = \frac{1}{4}\beta_{R}^{2} \left\| {{\mathbb{a}}_{k} } \right\|^{4} E\left\{ {\left( {\chi_{{2N_{H} }}^{2} } \right)^{2} } \right\} = \beta_{R}^{2} \left\| {{\mathbb{a}}_{k} } \right\|^{4} N_{H} \left( {N_{H} + 1} \right) \). Then by using expectation over \( {\mathbb{a}}_{k} \), it produces
where \( \left\| {{\mathbb{a}}_{k} } \right\|^{2} \) follows gamma distribution \( \Gamma \left( {M,\beta_{k} } \right) \) and \( E \{ \left\| {{\mathbb{a}}_{k} } \right\|^{4} \} = \beta_{k}^{2} M\left( {M + 1} \right) \).
By similarity, \( E \left\{ {\left| {g_{kL}^{\dag } g_{kL} } \right|^{2} } \right\} = \beta_{R}^{2} \beta_{k}^{2} M\left( {M + 1} \right)N_{L} \left( {N_{L} + 1} \right) \) and also
By combining all above results, it produces
Second calculate denominator\( \varvec{E}\left( {\varvec{\psi}_{1} } \right) \)in (28)
\( \psi_{1} \) consists of three terms multi user interference \( I_{1} \), additive Gaussian noise \( I_{2} \) and quantization noise \( I_{3} \).
Calculate \( \varvec{E}\left\{ {\varvec{I}_{1} } \right\} \)
By analogues to (30), we have
By similarity, \( E\left\{ {\varvec{g}_{{\varvec{kl}}}^{\dag } \varvec{g}_{{\varvec{kL}}} \varvec{g}_{{\varvec{jL}}}^{\dag } \varvec{g}_{{\varvec{kL}}} } \right\} = \beta_{R}^{2} \beta_{k} \beta_{j} N_{L} M\left( {N_{L} + M} \right) \) and \( E\left\{ {\varvec{g}_{{\varvec{kH}}}^{\dag } \varvec{g}_{{\varvec{jH}}} \varvec{g}_{{\varvec{jL}}}^{\dag } \varvec{g}_{{\varvec{kL}}} } \right\} = E\left\{ {\varvec{g}_{{\varvec{kL}}}^{\dag } \varvec{g}_{{\varvec{jL}}} \varvec{g}_{{\varvec{jH}}}^{\dag } \varvec{g}_{{\varvec{kH}}} } \right\} = \beta_{R}^{2} \beta_{k} \beta_{j} N_{H} N_{L} M \).
Finally, \( E\left\{ {I_{1} } \right\} \) is given by
By utilizing the same procedure of deriving \( E\left\{ {I_{1} } \right\} \), we produce
Calculate \( \varvec{E}\left\{ {\varvec{I}_{3} } \right\} \)
We refer to \( mth \) column of \( B_{L} \) with \( b_{m}^{\dag } \) and \( jth \) column of \( A \) with \( {\mathbb{a}}_{j} \) and \( \varvec{I}_{3} = \frac{1}{{\varvec{G}_{\varvec{R}}^{2} }}\varvec{g}_{{\varvec{kL}}}^{\dag } {\mathbf{\mathcal{R}}}_{{\varvec{n}_{\varvec{q}} }} \varvec{g}_{{\varvec{kL}}} \)
\( {\mathbf{E}}\left( {\varvec{I}_{3} } \right) = \frac{1}{{\varvec{G}_{\varvec{R}}^{2} }}\mathop \sum \limits_{{\varvec{m} = 1}}^{{\varvec{N}_{\varvec{L}} }} \varvec{E}\left\{ {\varvec{d}_{\varvec{m}} \left| {b_{m}^{\dag } {\mathbb{a}}_{k} } \right|^{2} } \right\} \) and \( \varvec{d}_{\varvec{m}} \) is the \( \varvec{mth} \) entry of \( {\mathbf{\mathcal{R}}}_{{\varvec{n}_{\varvec{q}} }} \) and is given by
\( d_{m} = \alpha \left( {1 - \alpha } \right)\left( {\varvec{G}_{\varvec{R}}^{2} \varvec{P}_{\varvec{U}} \mathop \sum \limits_{{\varvec{j} = 1}}^{\varvec{K}} b_{m}^{\dag } {\mathbb{a}}_{\varvec{j}} {\mathbb{a}}_{\varvec{j}}^{\dag } \varvec{b}_{\varvec{m}} + \varvec{G}_{\varvec{R}}^{2} b_{m}^{\dag } b_{m} + 1} \right) \) by utilizing \( d_{m} \) in \( {\mathbf{E}}\left( {\varvec{I}_{3} } \right) \), we produce
where
Finally,
By combining (31), (33), (34), (35) into (28) with some parameters \( N = {\Re }M \),\( N_{H} = \kappa N \) and \( N_{L} = \left( {1 - \kappa } \right)N \),we complete the proof.□
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Said, S., Saad, W., Shokair, M. et al. On the performance of double resolution ADC receivers for massive MIMO relaying systems. Telecommun Syst 73, 143–154 (2020). https://doi.org/10.1007/s11235-019-00591-7
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DOI: https://doi.org/10.1007/s11235-019-00591-7