In this series:

- Massive MIMO – Part 1. Introduction: From theory to implementation
- Massive MIMO – Part 2: A few lessons learned from asymptotic information theoretic analysis
- Massive MIMO – Part 3: Capacity, coherence time and hardware capability
- Massive MIMO – Part 4: Massive MIMO and small cells, the next generation network
- Massive MIMO – Part 5: The need for practical prototyping and implementation
- Massive MIMO – Part 7: Research issues

Massive multiple-input/multiple-output (MIMO) is a new network architecture that has the potential to remarkably increase both spectral and energy efficiencies. Also known as large-scale MIMO, massive MIMO is based on having a very large number of antennas at each base station (BS) and exploiting channel reciprocity in time-division duplex (TDD) mode

The primary benefits are as follows:

- inter-user interference is easily mitigated by the high beamforming resolution;
- low-complexity signal processing algorithms are asymptotically optimal;
- propagation losses are mitigated by a large array gain due to coherent beamforming/combining, and
- interference-leakage due to channel estimation errors vanish asymptotically in the large-dimensional vector space.

However, the authors in [3] showed that, when rejecting the conventional assumption of ideal, theoretical hardware, real-world hardware impairments create finite limitations on the channel estimation accuracy and the downlink/uplink capacity of each user equipment (UE) device. Additionally, capacity is limited mainly by the UE hardware, while the impact of impairments in large-scale arrays vanishes asymptotically and inter-user interference becomes negligible [3]. In other words, a non-zero capacity can be retained while the hardware quality is decreased as the array grows (thus enabling the use of inexpensive antenna elements).

The impact of transceiver hardware impairments on massive MIMO systems has received little attention—despite the possibility that large arrays might only be attractive for network deployment if they use low-cost antenna elements. Inexpensive hardware components are particularly prone to the impairments that exist in any transceiver, like I/Q-imbalance and phase noise [4]. The impact of hardware impairments is usually mitigated by compensation algorithms [5]. But, because the time-varying hardware characteristics cannot be fully parameterized and estimated, and due to the randomness inherent in different types of noise, these compensation algorithms cannot completely remove the impairments.

In this technical note, we reply on the generalized signal/system model developed in [3]. The system model captures the main characteristics of non-ideal hardware, in the sense that it lets us identify some fundamental differences in the behavior of massive MIMO systems as compared to ideal hardware. The additive distortion noises can describe, for instance, inter-carrier interference due to phase noise, leakage from the mirror subcarrier due to IQ imbalance, and amplitude non-linearities in power amplifiers.

For the sake of simplicity, we present a closed-form expression for the linear minimum mean square error (LMMSE) estimator’s error covariance matrix under the special case of identity channel and transmitted symbols covariance matrixes when the user equipment’s power *p*^{UE} grows indefinitely large as [3]:

where √(ℵ_{t}^{UE} ) and √(ℵ_{r}^{BS} ) represent the UE transmitter and BS receiver error vector magnitudes respectively [6].

The equation (1) shows that the error variance does not converge to zero even asymptotically, unlike in ideal hardware. Unfortunately, there is a positive error floor due to the transceiver’s hardware impairments.

On the other hand, closed-form upper bounds on channel capacity provide insight into the achievable downlink (DL) and uplink (UL) performance under transceiver hardware impairments. In particular, the following two corollaries from [3] provide some ultimate capacity limits in the asymptotic regimes of many BS antennas or large UE transmit powers.

The downlink upper capacity bound has the following asymptotic properties [3]:

Equations 2 and 3 show that the DL capacity has finite ceilings when either the DL transmit power or the number of BS antennas *N* become large. The ceiling depends on the impairment parameters ℵ_{t}^{UE} and ℵ_{r}^{BS}, which for the remainder are related to error vector magnitude (EVM), but the UE’s receiver impairments are clearly *N* times more influential. Therefore, increasing the capacity observed in high signal-to-noise ratio (SNR) and large antenna array regimes is not easily achievable in practice.

As far as the UL is concerned, the following closed form DL upper capacity bounds can be discussed [3]:

One can draw the same conclusion that the UE’s transmitter impairments are more influential as well. Also, only the quality of the UE’s transceiver limits the DL and UL capacities as the antenna array increases in size.

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