Skip to main content# Low Complexity 1-bit Analog-to-Digital Converters based Massive MIMO

# Abstract

# Introduction

## Our contribution

# Massive MIMO uplink with 1-bit ADC and no AGC

## System model

## Receive filters

## Performance metrics

### Mutual information

### SER

# Simulation results

# Conclusions

Receivers that rely on 1-bit Analog-to-Digital Converters (ADC) do not need energy-consuming interfaces such as automatic gain control (AGC), thus decreasing the ADC building and operational costs as well as saving energy.

Published onJul 31, 2024

Low Complexity 1-bit Analog-to-Digital Converters based Massive MIMO

We investigate massive multiple-input-multiple-output (MIMO) uplink systems with 1-bit analog-to-digital converters (ADCs) on each receiver antenna. Receivers that rely on 1-bit ADC do not need energy-consuming interfaces such as automatic gain control (AGC). This decreases both ADC building and operational costs. Our design is based on maximal ratio combining (MRC), zero-forcing (ZF), and least squares (LS) detection, taking into account the effects of the 1-bit ADC on channel estimation. Through numerical results, we show good performance of the system in terms of mutual information and symbol error rate (SER). Furthermore, we provide an analytical approach to calculate the mutual information and SER of the MRC receiver. The analytical approach reduces complexity in the sense that a symbol and channel noise vectors Monte Carlo simulation is avoided.

**Keywords**: Massive MIMO, large-scale antenna systems, analog-to-digital converter, 1-bit ADC.

MIMO systems have attracted significant research interest during the last decade, and are incorporated into emerging wireless broadband standards like Long-Term Evolution (LTE) [1]. In order to perform the receive processing, the received analog baseband signal is converted into digital form using a couple of analog-to-digital converters per antenna, i.e., one sampler each for the in-phase and quadrature components.

There are several types of ADC. One ADC type is the flash. It consists of

Irrespectively of ADC technology, more output bits requires more operational power. There are several MIMO studies that take into account the effects of the ADC on the performance evaluation of the system. The paper [3] examines the ADC effects with a ZF filter at the receiver, while [4] and [5] explore adaptation of the linear minimum mean square error (MMSE) receiver and the non-linear MMSE-decision feedback receiver to take into account the ADC presence. Maximum likelihood detection with ADC is investigated in [6], while [7] focuses on beamforming techniques to improve the performance of a system with low precision ADC.

The special case of 1-bit ADC is particularly interesting, since no AGC is needed. While 1-bit ADC is advantageous in terms of hardware complexity and energy consumption, it generally has a severe impact on performance. Ultra-wideband MIMO systems with 1-bit ADC are studied in [8]. Rayleigh fading MIMO channels with 1-bit ADC are studied in [9]. In [10], an analysis of binary space-time block codes with optimum decoding is provided for systems with 1-bit receive signal quantization. All the above MIMO ADC treatments express the performance in terms of bit-error-rate and/or mutual information.

Massive MIMO as systems, also known as very large MIMO and large-scale antenna systems, have base stations (BSs) equipped with several hundred antennas, which simultaneously serve many tens of terminals in the same time-frequency resource [11]. To be more specific, we define massive MIMO as a system with

This paper considers the uplink of a massive MIMO system employing 1-bit ADCs. The main difference between our work and the works cited above is that we consider the massive MIMO case, i.e.,

A discussion of maximum a posteriori probability (MAP) channel estimation with 1-bit ADC is provided. We quantify the computational complexity to be exponential in

$K$ . Since this computational complexity is high in a setting with many users, we propose a sub-optimal LS-channel estimation approach.We suggest MRC and ZF filters based on the LS channel estimate described under the previous point. We further derive an LS detection filter, which is calculated directly from the uplink training sequences without relying on an intermediate channel estimate.

We derive an analytical expression for the probability distribution of the MRC filter soft symbol estimates. Using this probability distribution, closed form expressions are developed for both the mutual information between the transmitted symbols and hard symbol estimates and the mutual information between the transmitted symbols and the soft symbol estimates with MRC. The closed-form expression reduces computational complexity in the sense that a Monte Carlo simulation is avoided when computing the mutual information.

The proposed systems are evaluated by numerical experiments. Both mutual information and SER are investigated. Numerical evaluations of the mutual information and SER are also compared to the their closed form expressions. We conclude that massive MIMO provides excellent SER and mutual information performance for wide ranges of system parameters.

The rest of the paper is organized as follows. In Section 2, we describe the system model, detection filters, channel estimation methods, and analyze the performance of the system. Our proposed techniques are explored by experiments in Section 3. The paper is concluded in Section 4.

*Notation:* We use boldface lowercase and uppercase letters to denote vectors and matrices respectively.

This section describes the channel model, suggested solutions, and performance analysis. The system model is detailed in Section 2.1, and Section 2.2 describes the employed detection filters and channel estimation methods. Numerical and analytical procedures for estimating the mutual information are discussed in Section 2.3.

We consider a single-cell uplink, where there are

$\mathbf{r}=\sqrt{P_t}\mathbf{H}\mathbf{x}+\mathbf{n}, \tag{1}$

where

Let

$\mathbf{y}=\operatorname{Q}(\mathbf{r}) \tag{2}$

A soft estimate of the transmitted symbols is obtained by processing the quantized received vector through the receive filter

$\mathbf{\widetilde{x}}=\mathbf{A}^H\mathbf{y}. \tag{3}$

Finally, it is possible to perform a QPSK demodulation that gives as output a hard estimate

In order to derive the soft symbol estimates, the receive filter

To compute the receive filter

$\begin{aligned}
&\mathbf{A}^H=
\underset{\mathbf{\widetilde{A}}}{\operatorname{argmin}} \left( \frac{1}{N}\sum_{n=1}^N \bigg|\bigg|
\mathbf{\widetilde{A}}^H \mathbf{y}^{(n)} - \mathbf{x}^{(n)} \bigg|\bigg|^2 \right) \\
&=\left(\sum_{n=1}^N \mathbf{x}^{(n)} \left({\mathbf{y}^{(n)}}\right)^H \right)\left(\sum_{n=1}^N \mathbf{y}^ {(n)} \left({\mathbf{y}^{(n)}}\right)^H \right)^{-1}.
\end{aligned} \tag{4}$

In (4),

For MAP-optimal channel estimation, we calculate

$\begin{aligned}
\mathbf{\widehat{H}}&=
\underset{\mathbf{H}}{\operatorname{argmax}} \; p(\mathbf{H}|\mathbf{Y})=
\underset{\mathbf{H}}{\operatorname{argmax}} \; p(\mathbf{Y}|\mathbf{H})p(\mathbf{H}),
\end{aligned} \tag{5}$

where

$\mathbf{\widehat{h}}_i=\underset{\mathbf{h}_i}{\operatorname{argmax}} \; p\left(\mathbf{Y}_i |\mathbf{h}_i \right)p(\mathbf{h}_i), \: i=1,...,M,
\tag{6}$

where

To reduce the computational complexity, we use the LS estimator. The LS estimate of

$\begin{aligned}
&\mathbf{\widehat{H}}=
\underset{\mathbf{\widetilde{H}}}{\operatorname{argmin}} \sum_{n=1}^N \bigg|\bigg|
\mathbf{y}^{(n)} - \sqrt{P_t} \mathbf{\widetilde{H}} \mathbf{x}^{(n)} \bigg|\bigg|^2 \\
&=\left(\sum_{n=1}^N \sqrt{P_t} \mathbf{y}^{(n)} {\mathbf{x}^{(n)}}^H \right)\left(\sum_{n=1}^N P_t \mathbf{x}^{(n)} {\mathbf{x}^{(n)}}^H \right)^{-1}.
\end{aligned} \tag{7}$

Note that, in this case, we need at least

The next step is to derive the receive filter

$\mathbf{a}^i=\frac{{\mathbf{\widehat{h}}^i}}{\parallel{\mathbf{\widehat{h}}^i}\parallel^2}, \tag{8}$

where

$\mathbf{A}^H=\mathbf{\widehat{H}}^\dagger, \tag{9}$

where

In this section, we consider two performance metrics: the mutual information per user and the SER per user. We show how they can be calculated numerically and analytically.

The average mutual information of the discrete channel between the transmitted QPSK symbol

$\begin{aligned}
&\operatorname{I}(x_k;\widehat{x}_k)= \operatorname{E}_\mathbf{H} \left[ \sum_{x_k,\widehat{x}_k} p(\widehat{x}_k \rvert x_k ,\mathbf{H}) p(x_k) \log_2 \frac{p(\widehat{x}_k\rvert x_k, \mathbf{H})}{p(\widehat{x}_k \rvert \mathbf{H})}\right], \\
&
\end{aligned} \tag{10}$

where *In a case where the mutual information is evaluated numerically, ** is estimated through Monte Carlo simulations with many realizations of the transmit vector ** and the channel noise vector **. In what follows, we give an alternative to the Monte Carlo evaluation of **.*

To calculate the transition probabilities

$\begin{aligned}
\widetilde{x}_k &=\sum_{i=1}^M \frac{h_{ik}^*}{\lVert \mathbf{h}_k \rVert^2 } y_i \\
&= \sum_{i=1}^M \left(\frac{h_{ik}^*}{\lVert \mathbf{h}_k \rVert^2 } y_i -\mu_i + \mu_i\right) \\
&=\sum_{i=1}^M \left( \frac{h_{ik}^*}{\lVert \mathbf{h}_k \rVert^2 } y_i -\mu_i \right)+ \sum_{i=1}^M \mu_i,
\end{aligned} \tag{11}$

where

In order to calculate

every component

$u_i$ has a zero mean value,every component

$u_i$ has a finite variance$a_i^2=\operatorname{E}[\lvert u_i\rvert^2]$ ,$\frac{a_i}{s_L}\xrightarrow{L\rightarrow \infty} 0$ and$s_L \xrightarrow{L\rightarrow \infty} \infty$ , where$s_L=\sum_{i=1}^L a_i^2$ .

To apply the Cramer’s central limit theorem to (11), we define *Further in this section, we will show that, with Assumption (A), the approximated pdf of ** derived below fits well with the real probability distribution, and in Section **3**, we will show that the performance evaluation based on the approximated pdf closely match the Monte Carlo symbol and noise vectors simulation results*.

Applying the Cramer’s central limit theorem with (11) and Assumption (A) above, we have that

$https://researcherstore.com/ \tag{12}$

where

$y_i=\operatorname{sign}\left(\overbrace{ \sqrt{P_t} h_{ik} x_k+ \underbrace{\sqrt{P_t}\sum_{j=1, j\neq k}^K h_{ij} x_j}_I + n_i }^{r_i}\right). \tag{13}$

Rewriting

$\begin{aligned}
I&=\sqrt{P_t} \sum_{j=1, j\neq k}^K h_{ij} x_j\\
&=\sqrt{P_t} \sum_{j=1, j\neq k}^K ( h_{ij} x_j -\mu^{\rm{I}}_{j}) +\sqrt{P_t} \sum_{j=1, j\neq k}^K \mu^{\rm{I}}_{j},
\end{aligned} \tag{14}$

with

$\begin{aligned}
&r_i \rvert x_k, \mathbf{H} \xrightarrow{K\rightarrow \infty} \mathcal{CN}\left(\sqrt{P_t}h_{ik} x_k ,\underbrace{ P_t \sum_{j=1, j\neq k}^K |h_{ij}|^2}_{\left({\sigma^{\rm{I}}}\right)^2} + \sigma_N^2 \right).\\
&
\end{aligned} \tag{15}$

The superscript

Noticing that the real and imaginary parts of

$\begin{aligned}
p_{i1}&=\operatorname{Prob} \left(y_i=1+j \Big\rvert x_k=\frac{1}{\sqrt{2}}+j\frac{1}{\sqrt{2}},\mathbf{H}\right) \\
&= \frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Re} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2} }\right) \\
& \;\;\cdot \frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Im} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2} }\right),
\end{aligned} \tag{16}$

$\begin{aligned}
p_{i2}&=\operatorname{Prob} \left(y_i=1-j \Big\rvert x_k=\frac{1}{\sqrt{2}}+j\frac{1}{\sqrt{2}},\mathbf{H}\right)\\
&= \frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Re} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2}} \right) \\
&\;\cdot \left(1-\frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Im} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2}} \right) \right),
\end{aligned} \tag{17}$

$\begin{aligned}
p_{i3}&=\operatorname{Prob} \left(y_i=-1-j \Big\rvert x_k=\frac{1}{\sqrt{2}}+j\frac{1}{\sqrt{2}},\mathbf{H}\right) \\
&= \left(1-\frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Re} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2} }\right) \right) \\
& \;\; \cdot \left(1-\frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Im} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2}} \right) \right),
\end{aligned} \tag{18}$

$\begin{aligned}
p_{i4}&=\operatorname{Prob} \left(y_i=1-j \Big\rvert x_k=\frac{1}{\sqrt{2}}+j\frac{1}{\sqrt{2}},\mathbf{H}\right) \\
&= \left(1-\frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Re} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2}} \right) \right) \\
&\quad \cdot \frac{1}{2} \operatorname{erfc} \left(-\frac{\sqrt{P_t }\operatorname{Im} (h_{ik}x_k)}{\sqrt{\sigma_N^2 + \left({\sigma^{\rm{I}}}\right)^2}} \right),
\end{aligned} \tag{19}$

and similarly for other possible values of

In order to validate the model (12) derived using Assumption (A) for the soft symbol estimate, we provide the results in Figure [1].

The plot is divided in a

Once that

In what follows, we also introduce the mutual information between the transmitted QPSK symbol

$\begin{aligned}
&\operatorname{I}(x_k;\widetilde{x}_k)=\\ &=\int \operatorname{E}_\mathbf{H}\left[ \sum_{x_k} f(\widetilde{x}_k\rvert x_k,\mathbf{H}) p(x_k) \log_2 \frac{f(\widetilde{x}_k\rvert x_k,\mathbf{H})}{f(\widetilde{x}_k\rvert \mathbf{H})} \right] d\widetilde{x}_k.
\end{aligned} \tag{20}$

The integral over

$\begin{aligned}
&\operatorname{I}(x_k;\widetilde{x}_k^\Delta)=\\ &=\operatorname{E}_\mathbf{H}
\left[ \sum_{x_k,\widetilde{x}_k^\Delta} p(\widetilde{x}_k^\Delta \rvert x_k,\mathbf{H}) p(x_k) \log_2 \frac{p(\widetilde{x}_k^\Delta \rvert x_k,\mathbf{H})}{p(\widetilde{x}_k^\Delta\rvert \mathbf{H})} \right],
\end{aligned} \tag{21}$

where

The average SER of user

$\begin{aligned}
&\operatorname{SER}= p(\widehat{x}_k \neq x_k)\\%\operatorname{E}_\mathbf{H} [p(\widehat{x}_k \neq x_k \rvert) \mathbf{H}] =\\
%&= \operatorname{E}_\mathbf{H}\left[ \sum_{x_k} p(\widehat{x}_k \neq x_k \rvert x_k,\mathbf{H})p(x_k) \right]=\\
&=\operatorname{E}_\mathbf{H}\left[ \sum_{x_k} \sum_{\widehat{x}_k \neq x_k} p(\widehat{x}_k \rvert x_k,\mathbf{H})p(x_k) \right],
\end{aligned} \tag{22}$

where

In a case where the SER is evaluated numerically,

When the MRC filter is employed, we can alternatively use (23) to calculate the SER. The transition probabilities

We will compare the proposed schemes in terms of Monte Carlo channel simulations. If nothing else is mentioned, the results have been obtained numerically by Monte Carlo simulations, i.e., not by the analytical treatment employing (12). Moreover, if nothing else is mentioned, the mutual information based on (10) associated with the discrete channel between the transmitted and received QPSK symbols

Symbol and noise vectors Monte Carlo simulation mutual information is obtained by averaging the mutual information over

Figure 2 shows the mutual information per user versus the SNR for MRC, ZF, and LS receivers, with

[c] SNR (dB)

The MRC and the ZF filters are investigated for both the cases of full CSI and imperfect channel knowledge. In order to calculate the channel estimate in (7) and the LS filter matrix in (4), we choose pilot sequences of length

Figure 1 is organized in the same way, but considering a BS equipped with

Differently from the massive MIMO case, here the LS filter is performing better with respect to the ZF and MRC when

Focusing the attention on the MRC and the ZF receivers, in Figure 4 and Figure 5 the mutual information per user versus the SNR and the SER per user versus the SNR are depicted, respectively.

The aim of the graphs is to show how the length of the training sequences

In Figure 6 and Figure 7, we show the performance gap between the case in which the quantizer is considered in the system model and the case in which it is not considered.

We show the mutual information per user versus the SNR and the SER per user versus the SNR, respectively, for the ZF receiver. Both the cases of full CSI and imperfect CSI are depicted. For the channel estimation phase, a training sequence of length

All the above results have been obtained numerically by symbol and noise vectors Monte Carlo simulations. In order to verify the analytical analysis in Section 2, Figure 8 and Figure 9 compare the performance obtained using (12) with those obtained by Monte Carlo symbol and noise vectors simulations.

In particular, Figure 8 depicts the mutual information per user versus the SNR, while Figure 9 shows the SER per user versus the SNR. For the analytical curves, the transition probabilities

The graphs show that the results based on the analytical treatment closely match the symbol and noise vectors Monte Carlo simulation results. As expected in Figure 8, the mutual information soft symbol estimate (21) is larger than the mutual information in (10) associated with the discrete channel.

This paper has examined the performance of a massive MIMO uplink system that employs 1-bit ADCs. Numerical evaluation of the mutual information and the symbol error rate have been provided for MRC, ZF, and LS receive filters. While the LS filter has been directly calculated depending on the uplink training sequences, the MRC and ZF filters have been derived based on the CSI estimate. We provided a discussion of MAP channel estimation but, due to the the high computational complexity in a setting with many users, we suggested a sub-optimal LS-channel estimation approach. We have also shown how the training sequence length affects the performance, and the performance gap between the scenario with a quantized receive vector and the scenario with an unquantized receive vector. In general, the ZF filter shows better performance compared to the MRC and the LS filters. However, when the SNR is larger than a certain value, all the filters achieve the maximal possible QPSK capacity, whatever the training sequence length is, and for both the quantized and the unquantized cases. Further, an analytical analysis of the performance is provided for the MRC filters. We showed that the results based on the analytical analysis fit well to the Monte Carlo results. Thanks to the closed-form derivation of the symbol estimate pmf, the computational complexity is reduced in the sense that a symbol and channel noise vectors Monte Carlo simulation is avoided. Concluding, we have showed that massive MIMO systems exhibit good performance even when employing 1-bit receive signal quantization. Thus, the ADC implementation complexity and power consumption can be eliminated.