Massive MIMO Systems

Abstract

The development of massive multiple-input multiple-output (MaMIMO) techniques is motivated by the requirements of large spectral efficiency and reduced power consumption. The implementation of a large number of antennas offers a large spectral efficiency and link reliability. Moreover, the use of MaMIMO allows scaling down the transmitted power proportionally to the number of antennas used, which may lead to a significant improvement in terms of energy efficiency. Orthogonal frequency division multiplexing (OFDM) in combination with MaMIMO has a considerable potential to obtain very high data rates and high quality of service (QoS). However, MaMIMO systems require a mobile equipped with multiple antennas at the transmitter. This is a challenging issue in mobile devices mostly due to their size, cost, and computing power limitations. In MaMIMO, the radiated power per antenna decreases linearly with the number of antennas. Moreover, the effects of small-scale fading, non-coherent interference, and receiver noise are minimized. However, a massive number of antennas require a separate transceiver chain and power amplifier (PA) for each antenna (unless analog or hybrid analog-digital structures are used for beamforming purposes, in which case the number of RF chains can be reduced). In this situation, the size and costs of the analog front-end become a critical issue. The cost and size optimization implies the use of low-cost components which increase the imperfections that degrade the system performance. In this chapter, MaMIMO system performance, considering front-end RF imperfections and ADC/DAC with limited resolution, is carefully studied. Spectral and energy efficiency for the uplink and downlink scenarios are evaluated to quantify the overall system performance. Finally, low-resolution precoding techniques, antenna coupling, channel non-reciprocity, and channel estimation errors are also addressed.

1 Introduction

The requirements of communication systems with large spectral and energy efficiency motivate the development of MaMIMO. It offers large spectral efficiency and link reliability, and allows each transmitter chain to scale down the transmitted power proportionally to the number of antennas. This scaling potentially leads to a significant improvement in terms of energy efficiency [1, 2].

Massive MIMO consists of hundreds of antenna elements. Due to technological constraints, the implementation of multiple antennas in mobile devices is a challenging issue, and the use of large number of antennas is only affordable in base stations. In mobile devices only a few number of antennas are available [3]. The use of large number of antennas at the base station is essential to enhance the capacity without additional spectral resources [4, 5]. The combination of OFDM and MaMIMO has a huge potential to obtain very high data rates and high quality of service, and are promising candidates for future wireless cellular systems.

MaMIMO average out the effect of receiver noise, small-scale fading, and also the distortion generated by RF impairments [6]. The natural robustness against RF impairments is an excellent characteristic of MaMIMO that allows the implementation of RF chains with low cost and low power consumption components. This feature is fundamental in the search of power efficient and affordable wireless communication systems.

2 Single-Cell Massive MIMO System

In this chapter, a single-cell MaMIMO system based on OFDM is considered. It is composed of a base station with M antennas with a total power constraint P _tr, that serves L single-antenna users (M ≫ L) operating in a time-division duplex (TDD) mode. It is assumed, for simplicity, that all users are equidistant in a radius r to the base station located at the cell center. Slow-fading Rayleigh channel model is assumed, and there is no coupling between antenna elements. The scenario is depicted in Fig. 8.1.

Fig. 8.1

MaMIMO scenario where a base station with M antennas communicates with L single antennas terminals

2.1 Downlink

The signals received by each user at subcarrier k, with 0 ≤ k ≤ N − 1 where N is the OFDM symbol length, are grouped in the vector y _k, defined by

$$\displaystyle \begin{aligned} \mathbf{y}(k)= \rho(k) \mathbf{H}(k) \mathbf{A}(k) \mathbf{s}(k) + \mathbf{w}(k), {} \end{aligned} $$

(8.1)

where y(k) = [y ¹(k), …, y ^L(k)]^T, the wireless channel response at subcarrier k is modeled by the matrix H(k) = [h ¹(k), …, h ^L(k)]^T of L × M, with elements h ^{l, m}(k) for 1 ≤ l ≤ L and 1 ≤ m ≤ M. A(k) = [a ¹(k), …, a ^L(k)] is the precoding matrix for subcarrier k of size M × L, with elements a ^{m, n}(k). The data symbols vector s(k) = [s ¹(k), …, s ^L(k)], and w(k) = [w ¹(k), …, w ^L(k)]^T, where w ^l(k) are zero-mean white complex Gaussian processes with variance $\sigma _{w}^{2}$. The transmitted power is subject to a constraint given by $\rho (k) = \sqrt {p(k)}$, and it is assumed that the power per user is normalized, i.e., E[∥s ^l(k)|²] = 1. Additionally, the channel is assumed constant during an OFDM block frame (block fading) and its impulse response shorter than the cyclic prefix length.

2.1.1 Precoding Techniques

The downlink transmission requires a precoding operation that combines the messages to be transmitted to the L users and maps them into the M transmission antennas.

The use of nonlinear precoding techniques in systems that verify M ≫ L achieves the maximum system capacity. However, the practical implementation of nonlinear precoding techniques cannot be afforded for large M. Furthermore, the gain obtained by using nonlinear precoding methods comes at the cost of a considerable additional implementation complexity, which makes it impractical. This motivates the use of linear precoding techniques that provide a good performance with reduced implementation complexity.

To describe the different precoding techniques, initially it is assumed a perfect channel state information (CSI) at the MaMIMO base station (BS).

Considering a linear precoder, the transmit vector x _(k) is given as:

$$\displaystyle \begin{aligned} \mathbf{x}(k)= \sum_{l=1}^{L} {\mathbf{a}}^{l}(k) s^{l}(k)=\mathbf{A}(k)\mathbf{s}(k). {} \end{aligned} $$

(8.2)

Then, the received signal at terminal l is given by

$$\displaystyle \begin{aligned} y^{l}(k) = \underbrace{\rho^{l}(k) {\mathbf{h}}^{lT}(k) {\mathbf{a}}^{l}(k) s^{l}(k)}_{{}_{\mathrm{desired}}^{\mathrm{signal}}} + \underbrace{\rho^{l}(k) \sum^{L}_{{}_{i \neq l}^{i = 1}} {\mathbf{h}}^{l T}(k) {\mathbf{a}}^{i}(k) s^{i}(k) }_{\mathrm{Interference}} + \underbrace{w^{n}(k)}_{\mathrm{noise}}. {} \end{aligned} $$

(8.3)

A good selection of the weight vectors a ^l(k) allows to minimize the multiuser interference , improving the signal-to-interference-plus-noise ratio (SINR). Considering a given channel realization, the SINR is given by

$$\displaystyle \begin{aligned} \mathit{SINR}^{l}(k) = \frac{ p^{l}(k) |{\mathbf{h}}^{lT}(k) {\mathbf{a}}^{l}(k) |{}^2}{ p^{l}(k) \sum^{L}_{{}_{i \neq l}^{i = 1}} |{\mathbf{h}}^{lT}(k) {\mathbf{a}}^{i}(k) |{}^2 + \sigma_{w}^{2}(k)}, {} \end{aligned} $$

(8.4)

where p(k) = ρ ²(k). The maximum achievable rate for user l at subcarrier k can be approximated by

$$\displaystyle \begin{aligned} R^{l}(k) \cong \mathrm{E} [\log_2(1+\mathit{SINR}^{l}(k))]. {} \end{aligned} $$

(8.5)

The total achievable rate of the system is the sum of the capacity of each user, $R_{\mathrm{total}}(k)=\sum _{l=1}^{L} R^{l}(k)$ and the achievable average rate is $R_{a}(k)= \frac {R_{\mathrm{total}}(k)}{L}$.

In the following, the existing linear precoding techniques are described.

Zero forcing (ZF) precoding is designed to achieve the zero multiuser interference (MUI). The precoding matrix for each subcarrier k is given by the pseudo-inverse of the channel gain matrix H(k),

$$\displaystyle \begin{aligned} {\mathbf{A}}_{ZF}(k) = \beta_{ZF}^{-1} {\mathbf{H}}^{H}(k) \left( \mathbf{H}(k) {\mathbf{H}}^{H}(k) \right)^{-1} \end{aligned} $$

(8.6)

$$\displaystyle \begin{aligned} \beta_{ZF} = \sqrt{\|{\mathbf{H}}^{H}(k) \left( \mathbf{H}(k) {\mathbf{H}}^{H}(k) \right)^{-1}\|{}^{2}}. \end{aligned} $$

(8.7)

Zero forcing precoding avoids the MUI and thus the system can be treated as a set of MIMO single user systems at each subcarrier k. However, the noise is amplified when the channel has large fades in its frequency response. In this case, particularly for frequency-selective channels, the noise severely degrades the system performance.

Maximum ratio transmission (MRT) precoding maximizes the signal-to-interference ratio (SIR) at the receiver. For each subcarrier k, MRT weights are derived by maximizing the ratio between the power of the desired signal and the power of the received signal subject to power restriction,

$$\displaystyle \begin{aligned} {\mathbf{A}}_{\mathrm{MRT}}(k) = \beta_{\mathrm{MRT}}^{-1} {\mathbf{H}}^{H}(k) \end{aligned} $$

(8.8)

$$\displaystyle \begin{aligned} \beta_{\mathrm{MRT}} = \sqrt{\|{\mathbf{H}}^H(k) \|{}^{2}}. \end{aligned} $$

(8.9)

Although the MRT precoder creates MUI, the residual MUI is minimized because channels of different users tend to be quasi-orthogonal for large number of antennas [1]. This scheme avoids matrix inversion leading to the simplest precoding solution.

2.2 Uplink

In the uplink scenario , it is considered that user terminals only weight their symbols by a power normalization factor. At the base station, the signal received at each antenna is linearly processed and decoded.

Considering an OFDM system, the signal received at the base station m-th antenna at the k-th subcarrier is the combination of the signal transmitted by all the user terminals.

$$\displaystyle \begin{aligned} {y}^m(k)= \rho_{UL} \sum_{l=1}^L h^{l,m}(k) x^l(k) +w^m(k){}, \end{aligned} $$

(8.10)

where $w^m_k$ is the receiver noise, x ^l(k) is the weighted symbol transmitted by user l, and h ^{l, m}(k) is the channel between user l and antenna m.

Assuming that the base station has knowledge of the channel response, it is possible to recover the symbols transmitted by each user. In the uplink scenario, the decoding process is implemented using the precoding matrices employed for data transmission in the downlink operation. The decoding matrix calculated using (8.8) is referred as maximum ratio combiner (MRC).

3 Precoding/Decoding Techniques with Imperfect Channel State Information

The precoder designs (8.6)–(8.8) are based on the assumption of perfect channel knowledge. However, in real systems the information at the transmitter is limited by multiple factors: (a) errors in the estimation due to pilot contamination, (b) delayed channel estimation, (c) channel reciprocity mismatch, and (d) limited feedback [7]. These factors influence the performance of precoders and must be contemplated.

The effect of imperfect CSI is considered using a simple model of hardware impairments [6]. Amplitude and phase deviations from the actual channel are modeled by

$$\displaystyle \begin{aligned} \tilde{\mathbf{H}}(k) = \boldsymbol{\Delta}(k)\odot \mathbf{H}(k), \end{aligned} $$

(8.11)

where H(k) is the true channel, $\tilde {\mathbf {H}}(k)$ is the perturbed channel, ⊙ is the Hadamard product, and Δ(k) is the uncertainty matrix of L × M, with elements defined as:

$$\displaystyle \begin{aligned} \varDelta^{l,m} (k)= (1+g^{l,m}(k))e^{j \phi^{n,m}(k)}. \end{aligned} $$

(8.12)

The amplitude error g ^{l, m}(k) and phase error ϕ ^{l, m}(k) are independent zero-mean Gaussian random variables with variances $\sigma _g^{2}$ and $\sigma _{\phi }^{2}$, respectively. The gain $\sigma ^2_g$ and phase $\sigma ^2_{\phi }$ power errors are measured in dB and degrees, respectively. This type of multiplicative model of imperfect CSI includes most of the errors affecting the MIMO channel model. Matrix A(k) is directly affected by $\tilde {\mathbf {H}}(k)$, according to the designed precoders (8.6)–(8.8) and normalization factors (8.7)–(8.9).

3.1 Channel Non-reciprocity and Antenna Coupling

There are unavoidable mismatches between transmitter and receiver chains that degrade the system performance. Downlink precoding in MaMIMO TDD systems assumes channel reciprocity , i.e., that uplink and downlink channels are identical. However, the end-to-end channels for the transmitter and the receiver are affected by non-symmetrical hardware impairment s since radio chains of different characteristic are used for transmission and reception. E.g., high quality components are used in the BS, where implementation cost is not a restriction, but off-the-shelf components prone to errors are used in user terminals . On the other hand, mutual coupling effects between antenna elements cannot be prevented [8, 9]. The equivalent downlink and uplink channels between the BS and user l are a cascade of the transceiver front-end, the antenna mutual coupling at transmitter side, the physical channel, the antenna mutual coupling at the receiver, and the receiver front-end frequency response. In that model, it is assumed that L single antenna users are served by an M-antenna BS.

The equivalent channels of the downlink and the uplink are illustrated in Figs. 8.2 and 8.3, respectively.

Fig. 8.2

Equivalent downlink channel

Fig. 8.3

Equivalent uplink channel

The channel considering a BS with M antennas and L single antenna users can be written as [10]:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \bar{\mathbf{H}}_{DL}(k)={\mathbf{A}}_{U,R}(k) {\mathbf{H}}_{DL}(k) {\mathbf{C}}_{B,T}(k) {\mathbf{A}}_{B,T}(k) \end{array} \end{aligned} $$

(8.13)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \bar{\mathbf{H}}_{UL}(k)={\mathbf{A}}_{B,R}(k) {\mathbf{C}}_{B,R}(k) {\mathbf{H}}_{UL}(k) {\mathbf{A}}_{U,T}(k), \end{array} \end{aligned} $$

(8.14)

where the antenna coupling at the base station is expressed by the M × M matrices, C _B,T and C _B,R. H _DL and H _UL are the wireless channels, and assuming TDD verifies ${\mathbf {H}}_{DL}={\mathbf {H}}^T_{UL}$.

The matrices A _B,T and A _B,R refer to the frequency response of the TX and RX chains, and are defined as:

(8.15)

(8.16)

where α _B,T,m(k), α _B,R,m(k) denote the response at subcarrier k of the transmitter and receiver chains in the m transceiver chain of the BS, respectively.

A _U,T and A _U,R denote the equivalent RX-TX mismatch matrices, and are given by

(8.17)

(8.18)

where α _l,R(k) and α _l,T(k) denote the response at subcarrier k of the receiver and transmitter chains of the l-th user.

In the following, the performance of ZF and MRT precodings without the assumption of channel reciprocity and antenna coupling is evaluated. It is assumed that the mutual antenna coupling exists only between neighboring antennas, i.e., |C _B,R|_(i,j) = 0 if |i − j| > 1. The mutual coupling value, the frequency responses of the RX and TX chains, and the channel estimation error are modeled as zero-mean Gaussian process with respective variances ρ ², $\delta _a^2$, and $\sigma _e^2$. The effect of imperfect CSI is also included. Simulation parameters are summarized in Table 8.1.

Table 8.1 Simulation parameters MaMIMO with antenna coupling

Figure 8.4 illustrates the system capacity of MaMIMO system with M = 200 and 10 active users for ZF and MRT precodings. In these curves, the channel estimation error $\sigma ^2_e$ is fixed to − 30 dB, the TX-RX frequency response mismatch δ ² varies from − 30 to − 10 dB, and the antenna coupling at the BS ρ ² ranges from − 30 to − 10 dB.

Fig. 8.4

Rate per subcarrier using ZF and MRT precoding. The rate was evaluated for a MaMIMO system with M = 10 to 200 and 10 active users

It can be observed that channel estimation error affects the system performance. Moreover, high antenna coupling (− 10 dB) and high levels of TX-RX mismatch also degrade the MaMIMO performance. In this case, in order to reduce the loss in terms of capacity, channel calibration techniques can be applied [11,12,13].

Capacity as a function of antenna coupling and the number of antennas in the base station is shown in Figs. 8.5 and 8.6. In these curves, it can be observed that increasing the number of antennas at the BS allows to reduce the SNR, i.e., decreasing TX power but maintaining identical data rate. This effect is even more pronounced for large antenna coupling. For large levels of mismatch, i.e., δ ² = −10 dB, illustrated in Fig. 8.6, the system capacity decreases and the harmful effect is even more pronounced when MRT precoders are employed.

Fig. 8.5

Rate per subcarrier using ZF and MRT precoding varying the BS antenna coupling. The capacity was evaluated for a MaMIMO system with M = 10 to 200 and 10 active users. The channel estimation error is fixed to $\sigma ^2_e=-30$ dB and the TX-RX frequency response mismatch is δ ² = −30 dB

Fig. 8.6

Rate per subcarrier using ZF and MRT precoding varying the BS antenna coupling. The capacity was evaluated for MaMIMO system with M = 10 to 200 and 10 active users. The channel estimation error is fixed to $\sigma ^2_e=-30$ dB and the TX-RX frequency response mismatch is δ ² = −10 dB

3.2 Channel Estimation and Pilot Contamination

Channel state information is obtained at the BS using training sequences sent by each user. During the training, the L active users transmit orthogonal sequences and the BS estimates each channel. Then, assuming channel reciprocity, the uplink channel estimates are employed to create the precoding matrix.

Two channel characteristics are important to design the massive MIMO system and the channel estimation algorithm: (a) channel coherence bandwidth and (b) channel coherence time . The normalized channel coherence bandwidth is defined as the number of subcarriers for which the channel frequency response is constant, and is denoted as $N_{coh}=\frac {B_{c}}{\varDelta _f}$, where $B_{c}=\frac {1}{T_d}$ is the channel coherence bandwidth defined in Sect. 2.3.1, T _d is the channel delay spread, and Δ _f is the OFDM subcarrier spacing. The normalized channel coherence time defines the amount of OFDM symbols for which the channel is considered time-invariant, and is expressed by $\tau _c=\frac {T_c}{T_{symb}}$, where T _c is the channel coherence time, also defined in Sect. 2.3.1, and T _symb is the duration of an OFDM symbol. The base station is able to serve a limited number of users determined by the coherence time and coherence bandwidth of the channel, considering that a fraction τ _p of the coherence time is employed to transmit pilots, and the rest, denoted τ _d, is used to transmit information.

The BS can serve a maximum of [14]

$$\displaystyle \begin{aligned} L_{max}=\tau_p N_{coh}=\frac{T_p}{T_{symb}\varDelta_f T_d} \end{aligned} $$

(8.19)

users, where T _p = τ _pT _symb is the time employed to send pilot symbols. From this equation can be observed the dependence of the number of users to be served with the channel delay spread. To increase the amount of served users in case of channel with large delay spread, the time assigned to transmit pilots, T _p, can be increased. However, the energy spent on pilots is also increased reducing the energy efficiency of the system.

The most direct way to obtain channel estimations of different users is by employing orthogonal sets of pilot tones. However, in the case of multicell systems, where the same frequency band is reused in several cells, pilot contamination occurs [15]. If each cell is serving the maximum number of users, the BS will receive pilot tones contaminated by pilots transmitted by users of neighbor cells [14]. Consider a scenario of two cells as depicted in Fig. 8.7. Due to channel contamination , channel estimates calculated by the base station 1 are affected by the pilots transmitted in cell 2. In this case, the precoding matrix obtained from the contaminated channel estimates will generate an additional beam in the direction of the interfering user [16]. Pilot contamination limits the quality of channel estimate and therefore the ability to minimize the interuser interference. Pilot contamination decreases the expected gain of MaMIMO systems, and is considered one of the major impediment to its development [17, 18].

Fig. 8.7

Pilot contamination effect. During the training period, the BS of cell 1 estimates the channel using pilot transmitted by users k of cell 1 and cell 2

An alternative to mitigate the effect of channel contamination is the use of superposed pilots [19]. This technique uses an iterative data-aided channel estimation that outperforms the conventional time multiplexing of information and pilots, at the cost of an increase in the computational complexity. Other approaches have also been proposed to solve this problem in multicell scenarios [20,21,22]. The trade-off between additional complexity and obtained gain needs to be considered to choose the most adequate pilot decontamination technique for each case.

4 RF Front-End Minimum Requirements

A critical component of MaMIMO front-end , in terms of cost and power consumption, is the ADC due to the large number required when a large number of antenna elements are used.

The power consumption of the ADCs grows exponentially with the number of quantization bits and also grows with the sampling rates. Power consumption on the order of several watts are reported for ADCs with large resolution, e.g. 12 bits, and high sampling rate (wideband applications) [23], what is prohibitive for MaMIMO applications where hundreds of RF+ADCs chains are required.

From several works [24,25,26] it can be concluded that for large number of antennas at the base station (BS), the system capacity is limited by the impairments at the user equipment and RF impairments at BS can be supported with low performance degradation. These results allow the use of low-resolution ADCs at the receiver front-end (BS).

Low-resolution ADCs are proposed to minimize the cost and power consumption of each RF chain [27]. The trade-off between quantization noise and power consumption needs to be evaluated in order to find an optimal resolution that minimizes the SNR degradation. In the following section the use of low-resolution ADCs and its effect on the system performance, capacity, and effective SNR are addressed. Spectral and energy efficiency , both affected by the ADCs resolution, are studied in Sect. 8.5. The following sections are focused on the uplink context where the harmful effects of low-resolution ADCs are important.

4.1 Low-Resolution ADCs

Several aspects must be considered in the selection of the ADC. It is clear that there is a trade-off between the allowed quantization distortion and the consumed power. Moreover, the use of large signal bandwidth requires high sampling rate that inexorably increases the ADC power consumption. To evaluate the behavior of an ADC there are two fundamental parameters to be taken into account: (a) quantization noise and (b) power consumption.

Quantization Noise

Based on the Bussgang theorem [28], the output of an ADC can be expressed by a scaling of the input signal and a distortion noise as x _adc = Q(x) = κ _adx + w _ad, where Q(x) denotes the quantization function, κ _ad is a scaling factor, and w _ad is the quantization noise.

Power Consumption

The ADC power consumption can be modeled as:

$$\displaystyle \begin{aligned} P_{adc}=FoM 2^{2b}f_s, {}\end{aligned} $$

(8.20)

where FoM is a constant that depends on the ADC architecture and the technology, b is the resolution, and f _s is the sampling frequency.

The use of low-resolution ADCs affects in two ways the performance of MaMIMO systems:

• Receiver quantization noise: the quantization of the received information generates distortion, increasing the receiver noise floor that degrades the link throughput.

• Noisy channel estimation: The terminals transmit pilot symbols and, due to the ADC low-resolution, a noisy channel estimate is obtained.

The channel estimate can be expressed as:

$$\displaystyle \begin{aligned} \hat{\mathbf{H}}(k)=\mathbf{H}(k)+\boldsymbol{\Delta}{\mathbf{H}}_{q}(k), {} \end{aligned} $$

(8.21)

where ΔH _q(k) is the channel estimation error due to quantization noise. A noisy channel estimate will affect the response of the downlink precoding matrix (transmission) and the transmitted signal will cause leakage over other users generating interuser interference.

On the other hand, the noisy channel estimates will also affect the decoding matrix and the spatial resolution of the MaMIMO system in the uplink scenario. The effect of noisy channel estimates affects the linear precoding matrices A(k) for MRC and ZF implementations. In this section the uplink scenario with L active users and a BS with M antennas is analyzed. This model is depicted in Fig. 8.8.

Fig. 8.8

MaMIMO uplink scenario where L users communicate with a base station with M antennas

The split of channel estimates, Eq. (8.21), is reflected in the precoding matrices. The MRT matrix can be expressed as $\tilde {\mathbf {A}}_{MRC}(k)=\mathbf { A}_{MRC}(k)+{\mathbf {A}}_{MRC_q}(k)=(\mathbf {H}(k)+{\boldsymbol {\Delta } H}_{q}(k))^H$, and the ZF precoder is given by $\tilde {\mathbf {A}}_{ZF}(k)={\mathbf {A}}_{ZF}(k)+{\mathbf {A}}_{ZF_q}(k)=pinv(\mathbf {H}(k)+\boldsymbol {\Delta } H_{q}(k))$.

The signal recovered at the BS for the user l at subcarrier k considering the modified precoding matrices can be expressed as:

(8.22)

where the desired signal and several noise/interference contribution can be identified. The subscript q denotes a term that is generated by the low-resolution ADC. In case of an ideal ADC, these terms are zero. There are two sources of multiuser interference : (a) the MUI determined by the employed precoding method and (b) the MUI_q due to channel estimates affected by quantization noise. In Eq. (8.22), two scaled versions of channel Gaussian noise appear, denoted as w ₁ and w ₂. The effects of low-resolution ADC are given by the quantization noises, $w_{q_1}$ and $w_{q_2}$.

Considering the desired signal and interference terms expressed in (8.22), and based on the definition of SINR (8.4), the signal to interference and quantization noise ratio (SINRQ) for the user l can be obtained as:

$$\displaystyle \begin{aligned} \mathit{\mathrm{SINRQ}}^{l}(k) = \frac{|\tilde{x}^{l}(k)|{}^2}{|{\mathrm{MUI}}(k)|{}^2+ |{\mathrm{MUI}}_q(k)|{}^2+|w_1(k)|{}^2+|w_2(k)|{}^2+||{}^2+|w_{q_1}(k)|{}^2+|w_{q_2}(k)|{}^2}, {}\end{aligned} $$

(8.23)

where the effects of ADC low-resolution and channel estimation error are reflected. The uplink rate of user l can be calculated as:

$$\displaystyle \begin{aligned} R^{l}(k) \cong \mathrm{E}[\log_2(1+\mathit{\mathrm{SINRQ}}^{l}(k))]. {} \end{aligned} $$

(8.24)

On the other hand, DAC resolution generates distortion in the precoding matrix and also affects the system performance [29, 30]. The issue of low-resolution DACs is not addressed in this chapter since the main focus is in low-resolution ADCs placed on RF chains of MaMIMO base stations and its effects over the system performance considering an uplink scenario. It is also considered that the terminals employ ADCs with enough resolution to avoid a significant increment of the receiver noise figure.

4.2 Performance Evaluation of a MaMIMO Uplink with Low-Resolution ADC

To evaluate the performance of MaMIMO systems with low-resolution ADCs, a simulation scenario that consists of a BS equipped with M antennas and L single antenna users, employing MRC and ZF linear precoders and decoders is presented. The uplink scenario is studied when the MaMIMO BS obtains the channel estimates from orthogonal pilot symbols transmitted by each user terminal.

The recovered symbol constellation at the BS for ADCs with 1 to 4 bits of resolution is illustrated in Figs. 8.9 and 8.10, for ZF and MRC decoding techniques, respectively. In the simulation, L = 4 active users and M = 50 antennas at the BS are considered. The recovered constellations with identical system parameters but M = 250 are plotted in Figs. 8.11 and 8.12. In these figures, ideal constellations are included as reference.

Fig. 8.9

Recovered symbol constellation using ZF decoding with 1–4 bits of ADC resolution with M = 50 and L = 4 active users (red markers for low-resolution ADC and yellow markers without quantization error)

Fig. 8.10

Recovered symbol constellation using MRC decoding with 1–4 bits of ADC resolution with M = 50 and L = 4 active users (blue markers for low-resolution ADC and green markers without quantization error)

Fig. 8.11

Recovered symbol constellation using ZF decoding with 1–4 bits of ADC resolution with M = 250 and L = 4 active users (red markers for low-resolution ADC and yellow markers without quantization error)

Fig. 8.12

Recovered symbol constellation using MRC decoding with 1–4 bits of ADC resolution with M = 250 and L = 4 active users (blue markers for low-resolution ADC and green markers without quantization error)

Severe quantization noise can be observed when a 1 bit ADC is employed even for large number of BS antennas. Using the ZF decoder and 50 antennas, the residual MUI generated by quantization effects over channel estimates plus the receiver noise are severe and cannot be removed. The results are improved for the case of 250 antenna elements. When an ADC of 2 bits is adopted, the combination of quantization noise and interference is reduced.

In case of MRC, the performance is severally degraded due to quantization effects and multiuser interference even for large number of antennas. ADC resolutions larger than 4 bits are required for MRC receivers.

The rate of the system using ZF and MRC decoding as a function of a varying number of antennas at the base station and different ADC resolutions is illustrated in Fig. 8.13. It can observed that the rate depends on the ADC resolution and the number of antennas. The use of low-resolution ADCs requires an increment in the number of antennas to reach similar rate. The rate is also illustrated in Fig. 8.14, using ZF and MRC decoding with M = 50,100, and 250 antennas.

Fig. 8.13

User rate using ZF and MRC decoding. The rate was evaluated considering an ADC with a resolution varying from 1 to 8 bits and with a BS with M = 10 to 250 and L = 4 active users

Fig. 8.14

Rate using ZF (blue line) and MRC (red line) decoding. The capacity was evaluated considering an ADC with a resolution varying from 1 to 8 bits and with a BS with M = 50, 100, 250 and 4 active users

A large number of antennas is useful to alleviate the quality requirements of the receiver chain components. It is verified that large levels of RF impairments are tolerated, which allows the implementation with low-cost off-the-shelf components. However, the use of a large amount of antennas requires more transceiver chains, increasing the implementation cost. The trade-off between the allowed level of imperfections and the required number of antennas needs to be evaluated.

5 Power Consumption Analysis

The power consumption of communication systems was studied in Chap. 3. In this section, the MaMIMO scenario is presented and the contribution in terms of power consumption of each transceiver component is described.

Two typical blocks that can be identified in a communication system are the RF and the DSP blocks. The power consumption of RF chain has two dominant elements: the power amplifier and the ADC. The DSP block includes several elements that need to be considered when the power consumption of the overall system is under study. The decoding processing, which depends of the chosen codification technique, demands high computational complexity (number of operations) that results in large power consumption. Channel estimation and FFT/IFFT are also significant tasks to be executed by the digital processor [31].

Considering the uplink, the ADCs are one of the main sources of distortion and also the main power consumers. The ADC resolution defines its own power consumption and also the computational complexity of the baseband operations that are proportional to the word length. Channel estimation and linear decoding are operations that depend on the ADC resolution b. The power consumption of the ADC and the DSP operations that depend on the ADC resolution are given by [24]

$$\displaystyle \begin{aligned} \tilde{P}_c(b)=2MP_{adc}(b)+P_{lp}(b)+P_{ce}(b), {} \end{aligned} $$

(8.25)

where P _lp and P _ce are the power consumed by linear processing and channel estimation, respectively. The number of operations required for channel estimation is given by

$$\displaystyle \begin{aligned} O_{ce}(b)=2 M L b \frac{B N_p}{T_c B_{c}} {} \end{aligned} $$

(8.26)

where N _p is the length of the pilot sequence, B is the system bandwidth, and T _c and B _c are the time and bandwidth channel coherence, respectively. The power consumed for channel estimation is obtained dividing the number of operations by the computational efficiency L _eff of the employed DSP, i.e., $P_{ce}=\frac {O_{ce}}{L_{eff}}$.

The linear decoding process involves the following operations:

$$\displaystyle \begin{aligned} O_{lp}(b)= 2 M L B b \left (1-\frac{N_p}{T_c B_{c}} \right ) {} \end{aligned} $$

(8.27)

and the power consumed is $P_{lp}=\frac {O_{lp}}{L_{eff}}$. Replacing (8.20), (8.26), and (8.27) in (8.25), the consumption that depends on the ADC resolution is given by

$$\displaystyle \begin{aligned} \tilde{P}_c(b)=2M FoM 2^{2b}f_s +\frac{2 M L B b}{L_{eff}}. {} \end{aligned} $$

(8.28)

The total power consumption of the uplink is obtained by adding the power required for signal transmission P _tu (single antenna users), the power consumption of RF chains of terminals P _U, and BS P _B [24]. They are given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} P_{tu}&\displaystyle =&\displaystyle \frac{L}{\rho_{pa}} p_{tx_{u}} \\ P_U &\displaystyle =&\displaystyle C_u L \\ P_B &\displaystyle =&\displaystyle C_B M, \end{array} \end{aligned} $$

(8.29)

where $p_{tx_{u}}$ is the power transmitted for each user, C _u and C _B are a set of constants that states for power consumption of RF chains of user terminals and base station, respectively.

The total power consumption of the system under study can be expressed by

$$\displaystyle \begin{aligned} P_c=\tilde{P}_c(b)+P_{tu}+ P_U+P_B=2M FoM 2^{2b}f_s +\frac{2 M L B b}{L_{eff}}+ \frac{L}{\rho_{pa}} p_{tx_{u}} + C_u L + C_B M, {} \end{aligned} $$

(8.30)

where the dependence of the overall power consumption with the ADC resolution, number of active users L, and number of BS antennas M is observed.

A metric to make a decision of the optimal solution between the ADC resolution and number of antennas at the BS is the energy efficiency (EE) measured in [bits/joules]. The EE can be defined for the uplink case as:

$$\displaystyle \begin{aligned} \eta_{EE}^{UL}=\frac{E[R_{UL}]}{P_c}=\frac{E[\sum_{l=1}^L R^{l}]}{P_c}=\frac{E[\sum_{l=1}^L log_2(1+\mathit{SINRQ}^{l})]}{\tilde{P}_c(b)+P_{tu}+ P_U+P_B}, {} \end{aligned} $$

(8.31)

where R _UL, the total achievable rate of the system, is the sum of the rate of each terminal. From (8.31), it can be observed the dependence of EE with ADC resolution and the number of antennas. By increasing the ADC resolution, the effective SINRQ is improved, resulting in a larger user rate that increases the numerator of (8.31). However, a large resolution also increases the ADC power consumption and the consumption of the DSP block (linear processing and channel estimation) that results in a larger denominator of (8.31), reducing the EE. Similar results can be obtained by modifying the number of BS antennas M. A large M reduces the impact of ADC quantization noise and other impairments increasing the SINQR. On the other hand, the overall power consumption scales with M. There are a combination of ADC resolution and number of antennas that reaches the optimal results in terms of energy efficiency.

In the following, the EE of uplink MaMIMO system is studied using numerical evaluation. Table 8.2 resumes the simulation parameters employed in this section extracted from [31].

Table 8.2 Simulation parameters

EE results are illustrated in Fig. 8.15 considering MRC and ZF decoders. It can be observed that EE degrades for low (<3) and high (>8) ADC resolution obtaining the optimal results for ADC resolution of 5–6 bits for ZF decoders and 4–5 bits for MRC implementations. There is also a large improvement when the number of antennas is scaled from M = 20 to M = 40. The duplication in the number of antennas creates an improvement in data rate that is larger than the growth of power consumption. However, when the number of antennas are increased to large values, the increment in power consumption of the RF chains is not compensated with large rate gains, creating degradation of the system energy efficiency.

Fig. 8.15

Energy efficiency of MRC and ZF systems with an ADC resolution varying from 1 to 12 bits and with a BS with M = 10, 50, 200 and L = 4 active users

Surfaces plot are employed to illustrate the dependence of EE with ADC resolution and BS antennas. The results for ZF and MRC are shown in Figs. 8.16 and 8.17, respectively. In these figures, the maximum value of EE is indicated for both implementations. In case of ZF decoders, the maximum energy efficiency is reached using a BS with M = 40 antennas and ADC resolution of 6 bits. For MRC, 4 bits of resolution and 60 antennas are the optimal values. The low requirements in terms of number of BS antennas, 40–60, are justified by the fact that only L = 4 active users are considered. When the number of users is increased, it is demonstrated that a larger number of antennas are required [24].

Fig. 8.16

Energy efficiency of ZF systems with an ADC resolution varying from 1 to 12 bits and with a BS with M = 10–250 and L = 4 active users

Fig. 8.17

Energy efficiency of MRC systems with an ADC resolution varying from 1 to 12 bits and with a BS with M = 10–250 and L = 4 active users

It is also worth to mention that the use of large number of antennas is useful to mitigate the impairments of the RF front-end as is demonstrated for the case of ADC quantization noise . In this evaluation, only quantization noise is included. If other impairments are added, large number of antennas at the BS will be required to average out the harmful distortion.

The quantization noise employed in this section follows an additive model and its effects are similar to distortion noise generated by an LNA, inter-carrier interference (ICI) due to oscillator phase noise and carrier frequency offset all modeled by Gaussian noise (for large number of OFDM subcarriers). The mitigation of RF impairments provided by MaMIMO systems allows low-cost and low power consumption hardware that is mandatory when hundreds of receiver chains to be implemented.

6 Summary of the Key Points

Massive MIMO is a technology with high potential to reach energy-efficient wireless solutions. The use of large number of antennas minimizes the effect of the RF front-end imperfections and allows the implementation of low cost/low power consumption hardware. There are several issues that must be addressed to get a full expansion of MaMIMO systems. Pilot contamination in multicell scenarios, channel reciprocity, and antenna coupling are challenging topics to be investigated.