.. _sphx_glr_auto_examples_mimo:

MIMO
----

Detection, space-time codes, capacity, message passing for large MIMO.


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    <div class="sphx-glr-thumbcontainer" tooltip="Four QPSK streams sent from four antennas and received on four, over i.i.d. Rayleigh fading with a new channel for every vector: y = H x + w. The detectors trade complexity for diversity:">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_01_mimo_detection_thumb.png
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  :doc:`/auto_examples/mimo/plot_01_mimo_detection`

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      <div class="sphx-glr-thumbnail-title">MIMO detection: linear, successive cancellation and maximum likelihood</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Diversity is the slope of the error rate at high SNR: with L independent fading branches, P_b \propto \bar\gamma^{-L}. Receive diversity is obtained by maximal-ratio combining (MRC); Alamouti&#x27;s code obtains the same order at the transmitter, without channel knowledge there, at the price of splitting the power over the two antennas (3 dB). BPSK, i.i.d. Rayleigh fading constant over a code block; the lines are the closed form of MRC with L branches of mean SNR \gamma (Proakis &amp; Salehi, eq. 13.4-15):">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_02_space_time_diversity_thumb.png
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  :doc:`/auto_examples/mimo/plot_02_space_time_diversity`

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      <div class="sphx-glr-thumbnail-title">Transmit diversity: Alamouti and space-time block codes</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 4 of S. M. Alamouti, &quot;A simple transmit diversity technique for wireless communications&quot;, IEEE J. Sel. Areas Commun. 16(8), 1998, compares the BER of uncoded coherent BPSK over Rayleigh fading for five systems: no diversity, maximal ratio receive combining (MRRC) with two and four receive antennas, and the two-branch transmit diversity scheme of the paper with one and two receive antennas. The assumptions of the paper&#x27;s Sec. IV are reproduced: the total transmit power of the two antennas equals that of the single antenna of MRRC, the fading amplitudes of every transmit-receive pair are uncorrelated Rayleigh with the same average power, and the receiver knows the channel.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_03_alamouti1998_fig4_thumb.png
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  :doc:`/auto_examples/mimo/plot_03_alamouti1998_fig4`

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      <div class="sphx-glr-thumbnail-title">Alamouti 1998, Fig. 4: transmit diversity against MRRC</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The capacity of a channel realisation with equal power on the n_t antennas is \log_2\det(I + \frac{\rho}{n_t} H H^H); its mean over Rayleigh fading grows like \min(n_t, n_r) \log_2 \rho at high SNR — one bit per 3 dB and per spatial stream (Foschini &amp; Gans 1998, Telatar 1999). Water-filling on the eigenmodes, which needs the channel at the transmitter, helps at low SNR, where only the strongest modes deserve power. Antenna correlation (exponential, \rho = 0.9) removes part of the multiplexing gain.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_04_mimo_capacity_thumb.png
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  :doc:`/auto_examples/mimo/plot_04_mimo_capacity`

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      <div class="sphx-glr-thumbnail-title">Ergodic capacity of MIMO channels</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="With tens of streams, the maximum-likelihood search is out of reach and the linear detectors lose several dB. Approximate message passing (AMP) alternates a linear step on the whole vector and a scalar MMSE denoiser on every stream: in the large-system limit, it decouples the MIMO channel into parallel AWGN channels whose noise variance a scalar recursion, the state evolution, predicts exactly.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_05_large_mimo_message_passing_thumb.png
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  :doc:`/auto_examples/mimo/plot_05_large_mimo_message_passing`

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      <div class="sphx-glr-thumbnail-title">Large MIMO: message passing (LAMA and OAMP)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Fig. 4 of M. Biguesh and A. B. Gershman, &quot;Training-based MIMO channel estimation: a study of estimator tradeoffs and optimal training signals&quot;, IEEE Trans. Signal Process. 54(3), 2006, compares four estimators of a flat r \times t channel from S = H P + V, each needing more prior knowledge than the last: least squares (none), scaled LS (the trace of R_H = E\{H^H H\}), the relaxed MMSE (the same trace, R_H replaced by a scaled identity) and the LMMSE (all of R_H). The training is the DFT matrix of orthogonal rows (their eq. (10)), the transmit correlation exponential with \varepsilon = 0.8 (eq. (60)), the abscissa the training power over the noise in dB.">

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  .. image:: /auto_examples/mimo/images/thumb/sphx_glr_plot_06_biguesh2006_fig4_thumb.png
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  :doc:`/auto_examples/mimo/plot_06_biguesh2006_fig4`

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      <div class="sphx-glr-thumbnail-title">Biguesh & Gershman 2006, Fig. 4: LS, scaled LS, relaxed MMSE and LMMSE</div>
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.. toctree::
   :hidden:

   /auto_examples/mimo/plot_01_mimo_detection
   /auto_examples/mimo/plot_02_space_time_diversity
   /auto_examples/mimo/plot_03_alamouti1998_fig4
   /auto_examples/mimo/plot_04_mimo_capacity
   /auto_examples/mimo/plot_05_large_mimo_message_passing
   /auto_examples/mimo/plot_06_biguesh2006_fig4