Cell-Free Symbiotic Radio: Channel Estimation Method and Achievable Rate Analysis

Along with the rapid deployment of the fifth-generation (5G) mobile communication networks, researchers have started the investigation of 6G targeting for network 2030 [1, 2]. In order to support orders-of-magnitude performance improvement in terms of coverage, connectivity density, data rate, reliability, latency, etc., many promising technologies have been extensively studied, such as extremely large-scale MIMO/surface [3, 4], TeraHertz communication [5], non-terrestrial networks (NTN) [6, 7], and AI-aided wireless communications [8]. In particular, cell-free massive MIMO [9] and symbiotic radio [10] were recently proposed as promising networking architecture and transmission technology for beyond 5G (B5G), respectively.

Cell-free massive MIMO is different from the classical cellular networking architecture in the sense that it blurs the conventional concepts of cells or cell boundary [9]. Instead, distributed access points (APs), which are connected to the central processing unit (CPU), exploit their local channel state information (CSI) to simultaneously serve the users. As such, cell-free massive MIMO system is expected to mitigate the inter-cell interference issues in small cell systems and provides users with appealing uniform good service everywhere [11]. Meanwhile, no exchange of CSI is required among different APs, which enables low complexity and light backhaul load between APs and CPU. Therefore, significant research efforts have been recently devoted to the theoretical analysis and practical design of cell-free massive MIMO, e.g., precoding design [12], power optimization [9], and energy efficiency analysis [13, 14].

On the other hand, in terms of transmission technology for B5G, symbiotic radio, which combines the benefits of the conventional cognitive radio (CR) and ambient backscattering communications (AmBC), has been proposed for spectral- and energy-efficient communications [10]. In typical symbiotic radio systems, the secondary device not only utilizes the spectral but also the power of the primary system via the passive backscattering technology [15]. Based on the relationship of symbol durations of the primary and the secondary signals, symbiotic radio system can be classified as commensal symbiotic radio (CSR) and parasite symbiotic radio (PSR) [16]. In CSR, the secondary signals have much longer symbol duration than the primary signals, making the secondary backscattering transmission contribute additional multipath components to enhance the primary communication. As a result, the primary and secondary communications form a mutualism relationship [16]. By contrast, in PSR, the primary and secondary signals have equal symbol duration, and the secondary signals are often treated as interference to the primary signals. Significant research efforts have been recently devoted to the study of symbiotic radio systems, e.g., in terms of performance analysis [17] and resource allocations [18, 19].

However, all the aforementioned existing works studied cell-free massive MIMO and symbiotic radio separately, i.e., cell-free system with the conventional active transmission or symbiotic radio transmission in conventional cellular network or the basic point-to-point communications. As the promising B5G networking architecture and transmission technology, respectively, it is natural that cell-free networking and symbiotic radio communication would merge each other to reap the benefits of both. This thus motivates our current work to study cell-free symbiotic radio systems, which, to our best knowledge, have not been studied in the existing literature. In cell-free symbiotic radio systems, a number of distributed APs cooperatively send primary information to a receiver, and concurrently support the passive backscattering communication of the secondary backscatter device (BD). As such, the distributed cooperation gain by APs can be exploited to enhance both the primary and secondary communication rate. An efficient two-phase uplink-training based channel estimation method is proposed to estimate the direct-link channel and cascaded backscatter channel, respectively. Furthermore, the interrelationship between the primary and secondary transmission is revealed by deriving their achievable rates taking into account the channel estimation errors. Besides, to achieve a flexible trade-off between the primary and secondary communication rate, a low-complexity weighted-maximal-ratio transmission (weighted-MRT) beamforming scheme is proposed, which only requires local processing at each AP without having to exchange the estimated CSI among APs. Numerical results are provided to show the performance of the cell-free symbiotic radio system with different training lengths.

As shown in Fig. 1, we consider a cell-free symbiotic radio system, which consists of $M$ distributed APs, one information receiver, and one BD. The $M$ APs cooperatively send primary information to the receiver, and simultaneously support the BD for secondary communication via backscattering to the same receiver. The considered setup may model a wide range of applications, e.g., with the receiver corresponding to smartphones and the BD being the smart home sensor nodes.

We assume that each AP is equipped with $N$ antennas, and the receiver and BD each has one antenna. Denote by $\mathbf{g}_{m}\in\mathbb{C}^{N\times 1}$ and $\mathbf{f}_{m}\in\mathbb{C}^{N\times 1}$ the multiple-input single-output (MISO) channels from the $m$th AP to the receiver and BD, respectively, where $m=1,...,M$. Further denote by $q\in\mathbb{C}$ the channel coefficient from the BD to the receiver. Then, the cascaded backscatter channel from the $m$th AP to the receiver via the BD is $q\mathbf{f}_{m}$.

In this paper, we focus on the PSR setup[16], where the symbol duration of the primary and secondary signals are equal. Let $s(n)\sim\mathcal{CN}(0,1)$ and $c(n)\sim\mathcal{CN}(0,1)$ denote the circularly symmetric complex Gaussian (CSCG) information-bearing symbols of the primary and secondary signals respectively. Further denote by $p$ the transmit power of each AP, and $\mathbf{w}_{m}\in\mathbb{C}^{N\times 1}$ with $\|\mathbf{w}_{m}\|^{2}=1$ denotes the transmit beamforming vector of the $m$th AP. Then the received signal at the receiver is

where $\alpha$ denotes the power reflection coefficient, $z(n)\sim\mathcal{CN}(0,\sigma^{2})$ is the additive white Gaussian noise. Based on the received signal $r(n)$ in (1), the receiver wishes to decode both the primary and secondary signals. To that end, since the backscatter link is typically much weaker than the direct link, the receiver may first decode the primary symbols $s(n)$, by treating the backscattered signals as noise, whose power is $\mathbb{E}[p\alpha|\sum_{m=1}^{M}q\mathbf{f}_{m}^{H}\mathbf{w}_{m}s(n)c(n)|^{2}]=p\alpha|q|^{2}|\sum_{m=1}^{M}\mathbf{f}_{m}^{H}\mathbf{w}_{m}|^{2}$. Therefore the signal-to-interference-plus-noise ratio (SINR) for decoding the primary information is

Note that due to the product of $c(n)$ and $s(n)$ in the second term of (1), the resulting noise for decoding $s(n)$ is no longer Gaussian. However, by using the fact that for any given noise power, Gaussian noise results in the maximum entropy and hence constitutes the worst-case noise [21, 20], the achievable rate of the primary signal in (1) is

After decoding the primary information, the first term in (1) can be subtracted from the received signal before decoding the secondary symbols $c(n)$. The resulting signal is

Note that since $s(n)$ varies across different secondary symbols $c(n)$, (4) can be interpreted as a fast-fading channel, whose instantaneous channel gain depends on $|s(n)|^{2}$ [22]. With $s(n)\sim\mathcal{CN}(0,1)$, its squared envelope follows an exponential distribution. Therefore, the ergodic rate of the backscatter communication (4) can be expressed as [16, 23]

where $\mathrm{Ei}(x)\triangleq\int_{-\infty}^{x}\frac{1}{u}e^{u}du$ is the exponential integral, and $\beta=\frac{p\alpha|q|^{2}|\sum_{m=1}^{M}\mathbf{f}_{m}^{H}\mathbf{w}_{m}|^{2}}{\sigma^{2}}$ is the average received SNR of the backscatter link.

Note that the above analysis is based on the assumption of perfect CSI on $\mathbf{f}_{m}$, $\mathbf{g}_{m}$, and $q$. In practical wireless communication systems, these channels need to be acquired via e.g., pilot-based channel estimation. In the following, we propose the channel estimation methods for cell-free symbiotic radio systems and analyze the achievable rates taking into account the channel estimation errors.

Similar to the extensively studied massive MIMO systems, efficient channel estimation for cell-free massive MIMO can be achieved by exploiting the uplink-downlink channel reciprocity [24, 25], i.e., the downlink channels can be efficiently estimated via uplink training. However, different from the existing cell-free massive MIMO systems [9], the channel estimation for cell-free symbiotic radio requires estimating not only the direct-link channels $\mathbf{g}_{m}$, but also the backscatter channels $q$ and $\mathbf{f}_{m},m=1,...,M$. To this end, in the following, we propose a two-phase based channel estimation method for cell-free symbiotic radio systems. In the first phase, pilot symbols are sent by the receiver while muting the BD, so as to estimate the direct-link channels $\mathbf{g}_{m},m=1,...,M$. In the second phase, pilots are sent both by the receiver and the BD so that, together with the estimation of the direct-link channels, the cascaded backscatter channels $q\mathbf{f}_{m}$, are estimated.

In this section, simulation results are provided to evaluate the performance of cell-free symbiotic radio systems. We set up a Cartesian coordinate system, where the BD is located at the origin (0,0), and the receiver is located at (5m, 0). Furthermore, we assume that $M=16$ APs, each with $N=4$ antennas, are evenly spaced in a square area of size $750\mathrm{m}\times 750\mathrm{m}$, i.e., their locations correspond to the $4\times 4$ grid points, with the x- and y-coordinates chosen from the set {-375m, -125m, 125m, 375m}. The channels of all communication links are independent, where the small-scale fading coefficients follow the i.i.d. CSCG distribution with zero mean and unit variance. Furthermore, the large-scale channel gains of all links are modeled as $b=\beta_{0}d^{-\gamma}$, where $\beta_{0}=(\frac{\lambda}{4\pi})^{2}$ is the reference channel gain with $\lambda=0.0857$m denoting the wavelength, $d$ represents the corresponding link distance, and $\gamma$ denotes the path loss exponent. We set $\gamma=2.7$ for the AP-to-BD and AP-to-receiver channels, and $\gamma=2.1$ for BD-to-receiver channels. The power reflection coefficient is $\alpha=1$, and the transmitter-side SNR for both information and pilot transmission is set as $\frac{p}{\sigma^{2}}=\frac{p_{t}}{\sigma^{2}}=130$ dB, which may correspond to $p=p_{t}=20$ dBm and $\sigma^{2}=-110$ dBm. The simulation results are obtained by taking the average values over 1000 channel realizations.

Fig. 2 shows the achievable rate regions of the primary and secondary rates with different uplink training lengths $\tau_{1}$, and hence different training ENR ${e_{1}=\frac{p_{t}\tau_{1}}{\sigma^{2}}}$, while the pilot length in the second training phase is fixed to $\tau_{2}=100$. Note that each point of the curve corresponds to a primary-secondary rate pair with the weighted-MRT beamforming (33), by varying the weight $\rho$ from 0 to 1 with step size 0.1. It is observed from Fig. 2 that with the training SNR $\frac{p_{t}}{\sigma^{2}}$ and training length $\tau_{2}$ fixed, the achievable rate regions critically depend on the training length $\tau_{1}$. For $\tau_{1}=1$, which corresponds to low training ENR $e_{1}$ in the first phase, the secondary communication rate is almost zero, regardless of the beamforming weight $\rho$. This can be explained by the fact that when $e_{1}$ is low, there exists severe channel estimation error for the direct-link channel estimation, whose detrimental effect will be exacerbated for the estimation of the weaker cascaded backscatter channels in the second training phase. This thus severely limits the achievable rate of the secondary backscattering communication. As $\tau_{1}$ increases to 100 so that both the direct-link and backscatter channels are estimated more accurately, the rate region enlarges significantly.

By fixing $\tau_{1}=100$, Fig. 3 plots the achievable rate regions with different training lengths $\tau_{2}$ in the second phase. Similar to Fig. 2, it is observed from Fig. 3 that the rate region enlarges significantly as $\tau_{2}$ increases, as expected. It is also interesting to note that with larger $\tau_{2}$, the minimum primary communication rate (corresponding to $\rho\!=\!0$) actually reduces. This can be explained by the fact that as the cascaded backscatter channels are estimated more accurately with larger $\tau_{2}$, it results in stronger interference to the primary communication with MRT beamforming matched to the secondary link, which thus decreases the minimum primary rate. However, the maximum primary rate ($\rho=1$) is almost unaffected by $\tau_{2}$. By comparing Fig. 2 and Fig. 3, it is also observed that larger rate regions are achieved for $\tau_{2}\!=\!10$ and $\tau_{2}\!=\!1$ in Fig. 3 than its counterpart in Fig. 2. This implies that if the total training length $\tau_{1}\!+\!\tau_{2}$ is fixed, higher priority should be given to the first training phase. This is expected since the estimation of the direct-link channels in the first phase impacts not only the primary communication rate, but also the quality of the channel estimation of the backscatter channels.

In this paper, a novel cell-free symbiotic radio system was studied, in which a number of distributed APs cooperatively send primary information, while concurrently supporting the secondary backscattering communication. A two-phase uplink-training based channel estimation method was proposed to estimate the direct-link channel and cascaded backscatter channel. Furthermore, a low-complexity weighted-MRT beamforming scheme was proposed to achieve a flexible trade-off between the primary and secondary communication rate. Simulation results were provided to demonstrate the performance of the cell-free symbiotic radio systems.

This work was supported by the National Key R&D Program of China with Grant number 2019YFB1803400.