On the Spatial-Wideband Effects in Millimeter-Wave Cell-Free Massive MIMO

As mobile communication systems evolved, and the demands for communication technologies utilizing extremely high-frequency bands such as millimeter-wave (mmWave) and terahertz (THz) bands have been aroused. For the utilization of the higher-frequency band, we should consider that the signal can be easily attenuated by the obstacle or distance between transceivers. Attenuating signals by distance may decrease cell coverage, and communication systems require densified networks, namely, ultra-dense networks (UDN), to maintain communication performance. However, in the UDN, some potential drawbacks of the multi-cell networks may be deepened, such as inter-cell interference (ICI) or pilot contamination.

Several cooperative communication methods, such as the coordinated multi-point (CoMP) and network MIMO system, have been studied to mitigate ICI and improve communication performance in the cell-edge region. However, cooperative communications suffer from synchronizing all serving BSs because the serving BSs share the channel state information and independently process the received signals by their own RF chain. Cell-free massive MIMO (CF-mMIMO) has been proposed as the complete form of cooperative communication to address the synchronization problem. Specifically, in CF-mMIMO, a single central processing unit (CPU) processes the transceived signals from the users by the distributed BSs called the access points (APs) connected by the fronthaul links. The distributed structure of CF-mMIMO yields another advantage for the higher-frequency band communications to reduce the non-line-of-sight (NLOS) probability. As the frequency band increases, the diffraction becomes more severe due to the shorter wavelength, and even signal blockage may occur. The CF-mMIMO system places APs in closer proximity to the user compared to the conventional massive MIMO BS in cellular networks. It can decrease the distance between AP and the user and consequently provides the same effect as removing the obstacles.

Most existing studies on CF-mMIMO systems tend to assume perfect synchronization, but in higher frequency bands, the synchronization issues may still remain. In massive MIMO systems of cellular networks, the signal processing only considers the phase difference because distances between UE and antennas are almost the same. Adversely, CF-mMIMO is a form in which numerous antennas in the antenna array of cellular massive MIMO are distributed as the APs throughout the system. To consider this distributed nature of CF-mMIMO, the signal processing for CF-mMIMO systems should consider phase differences caused not only by the antenna in each AP but also by the different distances between UE and APs.

In this letter, we first investigate the synchronization problem due to the time delays between different APs. The time delays are not negligible and may cause severe inter-symbol interference (ISI) in the OFDM system. We provide the minimum length of the cyclic prefix for reducing the ISI. Furthermore, we investigate the spatial correlations for the channel responses in the frequency-phase domain. We decompose the spatial correlation matrix into two types of correlations, such as the micro-correlation in the antenna of each AP and the macro-correlation among APs. Consequently, we investigate the effect of correlations by introducing the numerical results.

Notation: Boldface lowercase and uppercase letters, such as x and X, denote column vectors and matrices, respectively. The superscripts T, $*$, and H denote the transpose, conjugate, and conjugate transpose, respectively. The matrix $\textrm{diag}(\textrm{x})$ denotes the diagonal matrix with the elements in the vector x on the diagonal. The $M$-dimensional identity matrix and the vector with all-one elements denote $\mathbf{I}_{M}$ and $\mathbf{1}_{M}$, respectively. We use $\triangleq$ for definitions. The complex set with the $M\times N$ dimensional elements denotes $\mathbb{C}^{M\times N}$. The function $\textrm{Re}\{\cdot\}$ stands for the real part of a complex signal. The operators $\otimes$ and $\circ$ stand for the Kronecker and Hadamard products, respectively. The expectation of the matrix X denotes $\mathbb{E}\{\textrm{X}\}$.

Considering a CF-mMIMO system consisting of $L$ APs and $K$ UEs equipped with a single antenna as illustrated in the left-hand side of Figure 1. Each AP is equipped with a $M$-antenna uniform linear array (ULA) and connected to a CPU via a fronthaul link. Each antenna is separated by an antenna spacing $\triangle$. We assume that the CF-mMIMO system works in a time division duplex (TDD) protocol. In a TDD protocol, we only consider the uplink channel training because the uplink channel state information (CSI) can be utilized for both the uplink and downlink transmission by channel reciprocity. Orthogonal frequency-division multiplexing (OFDM) with $P$ subcarriers is adopted. In the OFDM system, the carrier frequency and wavelength denote $f_{c}$ and $\lambda_{c}$, respectively. The subcarrier spacing will be $\eta=W/P$ when the transmission bandwidth is $W$. We consider that UE $k$ sends the planar wave to the AP $l$ with the $N$ physical paths. The direction of arrival (DoA) of the signal from UE $k$ to the AP $l$ is denoted by $\theta_{k,l,n}$ for the $n$-th physical path. The schematic descriptions of the signal model in AP $l$ are illustrated by the right-hand side of Figure 1.

First of all, we assume that the first antenna of the nearest AP to UE $k$ is perfectly synchronized, and other antennas and APs are received the delayed version of the signal. we define the time delay $\tau_{k,l,m,n}$ of the $m$th antenna of the AP $l$. Specifically, $\tau_{k,l,m,n}$ consists of two types of delay caused by the distance $d_{k,l}$ between user $k$ and AP $l$ and the adjacent antennas at AP $l$ that can be explicitly computed as $\frac{d_{k,l}}{c}$ and $m\frac{\triangle\sin{\theta_{k,l,n}}}{c}$, respectively. Note that $c$ denotes the speed of light. Therefore, we can represent the time delay as follows:

The baseband signal generated by the user $k$ can be represented as

where $\delta(\cdot)$ is the pulse shaping function. The user transmits this signal by modulating to the corresponding passband signal as $\textrm{Re}\{x_{k}(t)e^{j2\pi f_{c}t}\}$. The antenna $m$ at the AP $l$ receives the delayed passband signal in the $n$th path as

where $\bar{\alpha}_{k,l,n}$ is the corresponding complex channel gain. Then, we can formulate the corresponding received baseband signal at the $t$ time instant as

In this section, we investigate the spatial-wideband effect of cell-free massive MIMO systems utilizing mmWave bands. Specifically, we formulate the channel model with the spatial-wideband effects. Based on the channel model, we analyze the spatial-wideband effects in three different aspects: delay spread in the angular-delay domain, frequency spread in the frequency-phase domain, and spatial correlations. In the analysis, we confirm the synchronization problem and errors in the frequency channel responses.

As discussed previously, if the transmission bandwidth and antenna spacing are narrow or the number of antennas is small, the ISI can be negligible due to the sufficiently short time delay compared to the symbol period. However, the cell-free massive MIMO system has a similar structure where the existing massive MIMO antennas are spread throughout the network in the form of APs, resulting in a large number of antennas and very wide spacing between them. Furthermore, the mmWave band can be worked as an obstacle by amplifying the effect of spatial-wideband effects although the cell-free massive MIMO is one of the key technologies to enable the high-frequency bands.

We employ the UMi-Street Canyon NLOS channel model introduced in 3GPP TR 38.901 [4].