Which Channel in 6G, Low-rank or Full-rank, more needs RIS from a Perspective of DoF?

With the development of sixth-generation (6G) communication, there are higher requirements for system capacity, security, and reliability. A secure and precise transmission scheme by combining random subcarrier selection and directional modulation (DM) was proposed in [1], which enhances system security by concentrating signal power in the small neighbouring region around the legitimate user. Low-cost RISs can be deployed in wireless environments to provide cascaded links, which is a key technology for improving system performance. It is particularly noted that the conventional DM is very suitable for line-of-sight (LoS) channel with rank-one, which will result in a fact that DM only transmits single bit stream from base station (BS) to user. In [2], a RIS-assisted DM network was proposed to transmit two confidential data streams from Alice to Bob, achieving twice the secrecy rate of single-stream transmission in the high signal-to-noise ratio (SNR) region. In [3], a distributed RISs-assisted transmission rate enhancement method in LoS channel was studied to increase channel rank from one to $K$. By dividing a large RIS into multiple small RISs for multi-stream transmission, the transmission rate was improved compared to a large RIS-assisted transmission. In [4], near-field communication (NFC) was discussed as an essential technology for 6G networks, which leveraged near-field spherical-wave propagation to enhance beam focusing and improve performance metrics such as SNR and channel capacity. Similarly, the authors in [5] studied a joint active and passive beamforming design in RIS-aided systems, which showed that RIS can achieve better performance to massive MIMO systems but with reduced hardware complexity and energy consumption. Furthermore, RIS deployment at cell boundaries was analyzed to mitigate inter-cell interference in [6], which highlighted its role in enhancing spectral efficiency through spatial multiplexing in multicell MIMO systems. In [7], RIS could be combined with relay stations to improve the coverage and transmission rate of the wireless networks. Here, a dual-slot system model with simultaneous deployment of relay stations and RIS was studied, and compared with the single-antenna relay network, a 85% performance gain was achieved in the high SNR region.

As studied above, RIS can improve the performance of wireless network, which is the result of the degrees of freedom (DoF) increased by introducing RIS. A joint small-scale spatiotemporal correlation model was proposed and analyzed for DoF under isotropic scattering in [8]. Active RIS can improve the sum of DoFs (sum-DoF) of dual-user MIMO systems with an equal number of antennas at both the transmitting and receiving ends. In [9, 10], the sum-DoF was further studied for any number of receiving and transmitting antennas, and it was proved that the existence of DoF gain was related to the rate and the number of transmitting and receiving antennas. Based on information theory, the DoF of the joint and independent transmission of information for RIS and transmitter was investigated in[11]. The channel rank of the non-zero direct path can eliminate the phase ambiguity between the radio frequency signal and RIS phase shift caused by multiplicative fading. For the rank-deficient channel, when the rank of the channel from the transmitter to the receiver is large and/or the rank of the channel from the transmitter to the RIS is small, the DoF can be significantly improved [12]. Reference [13] studied the reconfigurable distributed antenna and reflecting surface assisted MIMO system and obtained additional DoF through mode selection of each reflection element. To fully exploit the DoF of RIS, a multi-layer refractive RIS-assisted receiver was proposed [14], and a robust optimization framework of a low-complexity synchronous wireless information and power transmission scheme was designed to enhance receiver performance. In what follows, we will focus on the invesitgation of DoF and rate performance in two typical low-rank channels such as no link and LoS. With the help of RIS, their channel ranks may be augmented from one/zero to full-rank.. This will harvest an extremely large rate gains.

Fig. 1 shows a typical three-node wireless network aided by RIS. In this figure, the number of antennas at BS, the number of RIS elements, and the number of antennas at user are $M$, $N$ and $K$, respectively. The corresponding channel matrices from BS to RIS, RIS to user, and BS to user are denoted as $\mathbf{G}_{BR}$, $\mathbf{G}_{RU}$ and $\mathbf{H}$, respectively. Meanwhile, the channel phase shifting matrix is defined as $\mathbf{\Theta}$.

It is assumed that $M$ and $N$ are far larger than $K$ with $N\geq M$ due to a small size of mobile terminal at user. In general, the rank of channel matrix $\mathbf{H}$ varies from 0 to $K$. Similarly, the ranks of channel matrices $\mathbf{G}_{BR}$ and $\mathbf{G}_{RU}$ range from 0 to $M$ and 0 to $K$ in accordance with their wireless environments. In a rich-scattered urban environment, if any two of all three channel matrices reach up to full-rank, then the total channel matrix $(\mathbf{G}_{BR}\mathbf{\Theta}\mathbf{G}_{RU}+\mathbf{H})$ may achieve a full-rank: $K$. For example, $\mathbf{G}_{BR}$ and $\mathbf{G}_{RU}$ are an independent and identically distributed (iid) complex Gaussian distributions with their elements being iid and zero mean, then rank$(\mathbf{G}_{BR}\mathbf{\Theta}\mathbf{G}_{RU}+\mathbf{H})=K$.

However, in this paper, we will focus on a very typical kind of channels, called rank-deficient channels like deep-space, marine, UAV and satellite communications with rank-deficient or even extremely low-rank near one. Such a channel has the following property: rank$(\mathbf{H})<K$. It is very evident that such a channel needs RIS to improve its channel rank from less than $K$ to $K$, i.e., full-rank. In particular, when rank$(\mathbf{H})=0$ or 1, i.e., an extremely low-rank scenario, the enhanced-DoF role of RIS will become a dominant factor to boost the overall system rank in order to achieve a high-rate transmission. The former rank$(\mathbf{H})=0$, i.e. $\mathbf{H}=\mathbf{0}$, means that the channel from BS to user is blocked fully with no direct signal energy received by user. The latter rank$(\mathbf{H})=1$ implies that the channel is LoS channel or additive white Gaussian noise (AWGN) channel. Even when BS and user are employed with multiple antennas, there is only a single message stream transmitted from BS to user in LoS channel. Clearly, the above two channels demand RIS to improve their channel ranks to implement a multiple-stream transmission from BS to user. In the next two sections, we will discuss how to make such a significant enhancement in rank or DoF with the help of RIS.

In this paper, we divide wireless channels with MIMO into two typical channels: rank-deficient and full-rank. In a rich-scatted environment, the channel from BS to user as shown in Fig. 1 is viewed as a full-rank channel. The conventional LoS channel without reflecting and scattered paths is a rank-deficient channel with rank-one as shown in Fig. 2a, which is an extremely low-rank channel. However, the lowest-rank channel as shown in Fig. 2d is a rank-zero channel, which is completely blocked such that user cannot receive any power from the signal transmitted by BS, called no-link channel in what follows. In Fig. 2, a far-field RIS-aided wireless network in LoS channel is plotted. Clearly, the channel rank increases from one (no RIS) in Fig. 2a to two (RIS-aided) in Fig. 2b, where both channels from BS to RIS and RIS to user are assumed to be LoS channels and RIS is not on the linear line from BS to user. Furthermore, if both reflected channel matrices $\mathbf{G}_{BR}$ and $\mathbf{G}_{RU}$ as shown in Fig. 2c are full-rank with their elements being iid Gaussian distributions, the channel rank is boosted up to $\min(M,N,K)=K$ considering $M$ and $N$ are far larger than $K$. Similarly, as shown in Figs. 2e and 2f, in a no-link channel, due to the introduction of RIS, the channel rank may be enhanced from 0 in Fig. 2d to 1 in Fig. 2e to $K$ in Fig. 2f. In summary, for a rank-deficient channel with rank $k$ more than LoS, its rank is also improved from $k$ to the range from $k+1$ to full-rank ($K$) with the aid of RIS. Additionally, it is particularly pointed out if the channel from BS to user is a typical rich-scattered urban channel, i.e. full-rank and rank$(\mathbf{H})=K$, then a RIS or even multiple RISs are introduced into such a system, there are no DoF gain achieved in a scenario considering rank$(\mathbf{G}_{BI}\mathbf{\Theta}\mathbf{G}_{RU}+\mathbf{H})=K\leq K$. However, reflecting array gain and diversity gain created by RIS always exist.

To evaluate the rate performance enhancement achieved by DoF increasement due to the introduction of RIS, we will present numerical simulations as follows. System parameters are set as follows: the distances from BS to RIS, RIS to user, and BS to user are 82 m, 28 m, and 100 m, respectively. It is not specifically noted that thermal noise powers at RIS and user are -90 dBm and -70 dBm, respectively. The maximum ratio transmission (MRT), phase alignment, and maximizing the SNR at transmitter, RIS, and receiver are adopted in our simulation. Fig. 3 plots the achievable rate in the scenarios with $\text{rank}(\mathbf{H})=1$ including RIS-aided and no RIS. Here, the number of RIS elements is 1024 and $K=4$. When $M=64$, the rates of RIS-aided systems in all-LoS and LoS+Raleigh fading channels may about 1.6 times and 2.3 over that of LoS without RIS mainly due to the increase from one to two in system DoF. When the number of antennas at BS is increased from 64 to 128, we attain the same tendency in rate improvement.

Fig. 4 demonstrates the curves of achievable rate versus the number of RIS elements of no link with the aid of RIS, where no link means that the direct channel from BS to user is completely blocked. The rate without the RIS is zero due to no link between BS and user. By using only single RIS, the maximum DoF of the all-LoS system as shown in Fig. 2e is one, this means that it may transmit a single bit stream from BS to user, which will make a no-link become a single transmission. If both reflected channels $\mathbf{G}_{BR}$ and $\mathbf{G}_{RU}$ are a full-rank MIMO Rayleigh channels as shown in Fig. 2f, the maximum DoF of the system is up to $K$, i.e., full-rank. This implies that and a maximum of $K$ spatial parallel bit streams of data may be transmitted from BS to user, which will harvest a dramatic rate gain.

In this section, we will discuss a special scenario with both channels from BS to RIS and RIS to user as shown in Fig. 5 being LoS channels. Is there any way to further boost the DoF value of system? In the following, a distributed multi-RIS concept is proposed to make a dramatic DoF enhancement [3] in all-LoS scenario. However, it is particularly noted that the idea is also extended to the cases of all-reflected channels with channel rank being larger than or equal to one.

In general, a traditional low-rank network has one or two DoFs, which will seriously limit its rate performance. In accordance with basic principle of MIMO system, DoF is vital to boost rate performance. In this section, a novel distributed system model with $J$ RISs as shown in Fig. 5 are distributed deployed to create more DoFs. The corresponding channel matrices from BS to RIS $j$ and RIS $j$ to user are denoted as $\mathbf{G}_{BR}^{j}$ and $\mathbf{G}_{RU}^{j}$, respectively. If the alignment of spatial features between $\mathbf{H}$ and cascaded channels is low, $J\leq K-1$ RISs with low spatial feature alignment are required. If the spatial features of $\mathbf{H}$ and cascaded channels are aligned, $J\leq K$ RISs are divided. Specifically, the angle $\theta_{ij}$ between $\mathbf{G}_{BR}^{i}$ and $\mathbf{G}_{BR}^{j}$ and the angle $\phi_{ij}$ between $\mathbf{G}_{RU}^{i}$ and $\mathbf{G}_{RU}^{j}$ should be well designed to satisfy low spatial alignment, and then $\text{rank}(\sum_{j=0}^{J-1}\mathbf{G}_{BR}^{j}\mathbf{\Theta}^{j}\mathbf{G}_{RU}^{j}+\mathbf{H})=K$ is well-conditioned. The well-conditioned positions of the RISs can be obtained through geometric knowledge and MIMO theory. According to [15], it has been proved that $\mathbf{G}_{BR}^{i}\times\mathbf{G}_{BR}^{j}=\mathbf{0}$ $\mathbf{G}_{RU}^{i}\times\mathbf{G}_{RU}^{j}=\mathbf{0}$. Specifically, $\theta_{ij}=\theta_{j}-\theta_{i}\in[0,\pi/2]$ where $\theta_{j}=\arccos(\cos(\theta_{i})+{\lambda l}/{Md})$ , where $l$ is an integer, $\lambda$ denotes the wavelength, and $d$ is the minimum distance between antennas. In the extreme situation, the channel matrices $\mathbf{G}_{BR}^{j}$ and $\mathbf{G}_{RU}^{j}$ are rank-one, with each channel having only single eigen-vector. Then, $K=J+1$. Subsequently, a point-to-point $J+1$-stream transmission can be achieved in this rank-deficient channel scenario. Moreover, a significant rate enhancement can be achieved by this model.

To evaluate the rate performance enhancement achieved by DoF increasement due to the introduction of distributed RISs, we will present numerical simulations as follows. System parameters are set as follows: the distances from BS to RIS, RIS to user, and BS to user are 82 m, 28 m, and 100 m, respectively; the number of antennas at BS, the number of RIS elements, and the number of antennas at user are $M=128$, $N=600$, and $K=4$, respectively. Fig. 6a illustrates the curves of achievable rate versus noise power $\sigma_{r}^{2}$ at RIS obtained by distributed RISs with $\sigma_{u}^{2}=-70$ dBm, where the transmitting beamforming vector, phase shifting matrix, and receive beamforming vector are designed by null-space-projection, phase alignment, and zero-forcing, respectively. Observing this figure, we find: as noise power $\sigma_{r}^{2}$ at RIS increases from -120 dBm to -60 dBm, the achieved rate gains over no RIS gradually decrease and converge to zero. When the number of distributed RISs is greater than or equal to 2, i.e., the system DoF increase by 2 or more, the performance is obviously improved compared with no RIS under high noise power, i.e., $\sigma_{r}^{2}=-60$ dBm. As the number of distributed RISs grows from one to four, the achieved rate gains are as follows: about 0.7, 1.9, 2.7, and 3.5 times over no RIS under the same power-sum constraint and $\sigma_{r}^{2}=-90$ dBm. According to this trend, it can be inferred that increasing the number of distributed RISs and reducing noise power $\sigma_{r}^{2}$ at RIS will further improve the rate performance.

Fig. 6b illustrates the curves of achievable rate versus noise power $\sigma_{u}^{2}$ at user obtained by distributed RISs with $\sigma_{r}^{2}=-90$ dBm. From this figure, it is seen: as noise power $\sigma_{u}^{2}$ at user varies from -120 dBm to -60 dBm, the achieved rate gains over no RIS gradually decrease.

In summary, we have the following conclusions: (1) Increasing the number of distributed RISs may boost the value of system rank or DoF such that a significant rate gain can be attained, (2) low-noise RIS and receiver are also crucial for an active RIS-aided wireless network to achieve a high rate gain.

Due to the above major advantages of RIS and its ability to make a striking DoF improvement in low-rank channels, RIS may extend its application to several new fields such as near-sea, near-space, and UAV communications. However, there still are several open challenging problems to address in RIS fields. Here, we list several important ones of them as follows:

1. 1.

   Similar to a movable antenna array, a movable RIS may be introduced to further optimize the space position of RIS over a given 3D area to harvest the performance gains. Additionally, how about its performance upper bounds?

2. 2.

   In near field, how to compute and evaluate the DoF or rank value of such a RIS-aided system when the distance from user to RIS is less than the Rayleigh distance? It is clear that this problem is hard, however, its DoF upper or lower bounds may be derived as a DoF performance benchmark. Moreover, the corresponding beamforming and phase adjusting methods is also designed to implement a multi-stream near-field transmission from BS to each user.

3. 3.

   In a distributed multi-RIS scenario, how to achieve an optimal matching between RISs and users in order to maximize the sum-rate by using some traditional methods of graph theory in a multi-user situation? Furthermore, similar to scheduling theory, how to make a fair use of RISs per user?

4. 4.

   For a RIS-assisted hybrid beamforming structure, how to jointly design the transmission digital precoding vector, transmission analog weight vector, RIS phase shifting matrix, and reception beamforming vector in order to achieve the maximum DoF of the system?

In this article, the great potential of RIS has been highlighted as a crucial DoF enhancement tool for several typical low-rank future application scenarios such as marine, low-altitude, UAV, satellite and deep-space communications. A distributed multi-RIS deployment was proposed to achieve a significant rank enhancement for low-rank wireless situations. There are still several challenging problems ahead to exploit its great potential of the technology. Also, we have raised several new open important research problems. Finally, in our view, a distributed multi-RIS will achieve wide diverse promising applications in the coming future.