Massive-MIMO
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**High-dynamic-range and low-power M-MIMO Communications.** The massive multiple-input multiple-output architecture (M-MIMO) is a communication system that utilizes a large number of antennas and the base station, serving a large number of users. While M-MIMO offers substantial potential for spectral efficiency gains, it faces considerable practical challenges. Among the limitations are the power consumption and receiver saturation, which are particularly critical in M-MIMO since each antenna has a corresponding radio frequency (RF) chain.

:math:`\lambda`-MIMO is a new approach for holistically addressing the high power consumption and receiver saturation problems in M-MIMO architectures. Compared to de-coupled approaches, the overall performance can be optimized, as balancing trade-offs can be avoided. The uplink system architecture is constructed by replacing the ADC in each RG chain by the M-ADC.

 .. figure:: _static/img/LMIMO1.png
    :alt: M-MIMO uplink architecture.
    :align: center
    :width: 500

    M-MIMO uplink architecture.

Theoretical analysis and hardware experiments indicate that the proposed :math:`\lambda`-MIMO architecture achieves better energy efficiency compared to the conventional M-MIMO architecture. The reconstruction, detection and sum-rate performance are also improved.

**System Model.** At the transmitter side, the  Narrowband Single-Carrier Signal Model (NBSC) and  the wideband MIMO–OFDM were studied.
At the receiver side, the received signal at the :math:`n^{\mathsf{th}}` antenna, denoted by :math:`z_{n}(t)`
is amplified and de-modulated to the baseband. The quadrature and in-phase signal are then lowpass filtered, and sampled by the M-ADC.

.. :math:`z_{n}(t) = \sum_{m \in \lfloor M \rceil} \sqrt{p_u} \left( (h_{n,m} \ast_T x_m)(t) \right) + \epsilon_n` . where :math:`p_{u}` is the average transmit power at the user, and :math:`\epsilon_{n}(t) \sim \mathcal{CN} (0, 1), \quad \forall n \in \mathbb{Z}` is the additive Gaussian noise at the receiver.

Performance Analysis
++++++++++++++++++++++
The M-ADC achieves lower quantization noise (more information: :ref:`sampling_res`), and the impact on the performance of communication is shown by analyzing the analysis of the SQNR and achievable sum-rate for a maximum ratio combining (MRC), and zero forcing (ZF) combiners.

**SQNR**

The *baseband sampled single-carrier signal* is subject to uniform distribution :math:`r[k] \sim U(0,1)`.
The SQNR for a conventional ADC is: :math:`\text{SQNR}_{\text{unif}} = 10 \log_{10} \frac{\sigma^2_{r,\text{unif}}}{\sigma^2_{q,\text{unif}}} = 6.02 b \, (\text{dB})`.

The modulo folded samples follow a uniform distribution :math:`\mathscr{M}_{\lambda}(r(t))\vert_{t=kT} \sim U(0, \zeta^{2})`.
Consequentially, the quantization error is lowered by: :math:`\sigma^2_{q,\lambda} = \zeta^2 \sigma^2_{q,\text{unif}}`, and the SQNR becomes:

.. math::

   \text{SQNR}_{\text{unif}, \lambda} = 10 \log_{10} \frac{\sigma^2_{r,\text{unif}}}{\sigma^2_{q,\lambda}} = 6.02 b + 20 \log_{10}(1 / \zeta) \, (\text{dB}).

Notably, the M-ADC gives an :math:`20 \log_{10}(1/\zeta)` enhancement on the SQNR.

The OFDM signal is commonly assumed to be a centered, wide-sense stationary, ergodic Gaussian process :math:`r[k] \sim \mathcal{N}(0,1)`.
The SQNR for a conventional ADC is: :math:`\text{SQNR}_{\text{Gau}} = 10 \log_{10} \frac{\sigma^2_{r,\text{Gau}}}{\sigma^2_{q,\text{Gau},(m)}}\approx 6.02 b - 4.35`.
The modulo folded samples follow a wrapped distribution, which was empirically found to be uniform :math:`\mathscr{M}_{\lambda}(r(t))\vert_{t=kT} \sim U(0, \zeta^{2})` for :math:`\lambda` being a fraction of :math:`\sigma^{2}`.
The SQNR is,

 .. math::

   \text{SQNR}_{\text{Gau}, \lambda} = 10 \log_{10} \frac{\sigma^2_{r,\text{Gau}}}{\sigma^2_{q,\lambda}}
   \approx 6.02 b + 20 \log_{10}(1 / \zeta).


 .. figure:: _static/img/LMIMO2.png
    :alt: M-MIMO SQNR.
    :align: center
    :width: 500

    M-MIMO SQNR.

**Sum-Rate Analysis**

Analytical approximations for the achievable sum-rate are derived for two combiners.

*MRC Analytical Approximation of Achievable Sum-Rate*:

The uplink rate of the :math:`m`\th user with MRC is given by:

.. math::

   \widetilde{R}{m, (\text{MRC})} = \log_2 \left( 1 + \frac{p_u \gamma_{\lambda} \eta_{m} (N + 1)}{I_{\text{MRC}}} \right),

where :math:`I_{\text{MRC}}` is the expectation of the noise-plus-interference term, with:

.. math::

   I_{\text{MRC}} = p_u \gamma_{\lambda} \sum_{i=1, i \neq n}^{M} \eta_i + p_u (1 - \gamma_{\lambda}) \left( \sum_{i=1}^{M} \eta_i + \eta_m \right) + 1.

The sum-rate for the system is: :math:`R = \sum_{m=1}^{M} \widetilde{R}_{m}`.

*ZF Analytical Approximation of Achievable Sum-Rate*:

The uplink rate of the :math:`m`\th user with ZF is given by:

.. math::

   \widetilde{R}_{m,(\text{ZF})} = \log_2 \left( 1 + \frac{\gamma_{\lambda} p_u}{\frac{\gamma_{\lambda}}{(N - M) \eta_m} + (1 - \gamma_{\lambda}) I_{\text{ZF}}} \right),

where :math:`I_{\text{ZF}}` is the expectation of the noise-plus-interference term. Here, :math:`\delta < 1` is the attenuation coefficient.
The following figure present the comparison of the sum-rate for conventional and M-ADC on simulated data. As observed, by setting :math:`b` as low as :math:`1` or :math:`2` with M-ADC the up-link sum-rate approaches that of :math:`b=\infty` for conventional ADC.

 .. figure:: _static/img/LMIMO3.png
    :alt: M-MIMO Sum-Rate.
    :align: center
    :width: 500

    M-MIMO Sum-Rate.

The advantages of the M-ADC are also present via comparison of the energy efficiency, especially for low bit resolutions.