Near-Field Rainbow: Wideband Beam Training for XL-MIMO

As the antenna number is not very large for current 5G massive MIMO communications [11], existing works mainly focus on a simplified far-field beam split effect. To be specific, the channels are modeled under the planar wavefronts, and the beams over different frequencies are split into different spatial angles, where the distance information is ignored. These works can be generally classified into two categories, i.e., techniques overcoming the far-field beam split effect [12, 13, 14, 15, 16] and those taking advantage of it [17, 18, 19]. The first category desires to mitigate the array gain loss caused by the far-field beam split. To realize this target, introducing time-delay circuits into beamforming structures, such as true-time-delay array [12, 13] or delay-phase precoding structure [15, 16], is considered to be promising. Thanks to the frequency-dependent phase shift provided by the time-delay circuit, the generated beams over the entire bandwidth can be aligned with a certain spatial angle, and thus the far-field beam split effect can be alleviated. For the second category of taking advantage of the far-field beam split, it has been proved that time-delay circuits can not only mitigate the far-field beam split, but also flexibly control its degree [17, 19, 18]. By carefully designing the delay parameters, the covered angular range of the beams over different frequencies can be controlled. Benefiting from this fact, very fast channel station information (CSI) acquisition in the far-field, such as fast beam training or beam tracking, can be realized. In [17, 18], based on the true-time-delay array architecture, fast beam training schemes with only one pilot overhead were studied by generating frequency-dependent beams to simultaneously search multiple angles. In [19], a fast beam tracking scheme based on the delay-phase precoding architecture was proposed by adaptively adjusting the degree of the far-field beam split according to the user mobility. From the discussion above, we can find that although far-field beam split results in a severe beamsteering gain loss that should be addressed, it can also benefit the fast far-field CSI acquisition in massive MIMO systems.

When it comes to the XL-MIMO systems, the more realistic near-field beam split effect should be considered, since the antenna number is huge. Recently, researchers in [5] have tried to compensate for the corresponding array gain loss in the near-field range. Specifically, the near-field beam split effect was defined and analyzed in our previous work [5], and the time-delay (TD) based beamformer was also utilized to overcome this effect. We have proposed to partition the entire array into multiple sub-arrays, and then the user can be assumed to be located within the near-field range of the entire array but in the far-field range of each sub-array. Based on this partition, time-delay circuits can also be utilized to compensate for the group delays across different sub-arrays induced by near-field spherical wavefronts. As a result, the beams over the entire bandwidth can be focused on the desired spatial angle and distance, and the near-field beam split effect is alleviated accordingly.

Efficient design of wideband XL-MIMO beamfocusing to alleviate the near-field beam split effect requires accurate near-field CSI. To meet this requirement, an intuitive way is to directly utilize the existing wideband far-field CSI acquisition schemes [17, 19, 18] to estimate the near-field wideband channel. However, since the far-field planar wavefront mismatches the near-field spherical wavefront, these methods [17, 19, 18] may be not valid in the near-field range. To cope with this problem, inspired by the classical far-field hierarchical beam training scheme [20], a near-field hierarchical beam training method is proposed in [21], which uniformly searches multiple angles and distances to obtain the near-field CSI. Moreover, from the perspective of near-field array gain, [6] proved that the distances should be non-uniformly searched for improving channel estimation accuracy. Nevertheless, existing near-field CSI acquisition methods assume that, the bandwidth is not very large, so the near-field beam split effect is not considered. Moreover, unlike the angle-dependent far-field CSI, to obtain the near-field CSI, the angle and distance information should be estimated simultaneously, which may result in unacceptable pilot overhead for near-field beam training [21]. Unfortunately, to the best of our knowledge, determining how to obtain accurate wideband XL-MIMO near-field CSI with acceptable pilot overhead has not been studied in the literature.

To fill in this gap, inspired by the two categories of research on the far-field beam split, i.e., overcoming and taking advantage of it, we unveil that although the near-field beam split effect degrades the array gain, it can also provide a new possibility to benefit the near-field CSI acquisition. Based on this new finding, we propose a fast wideband near-field beam training scheme by taking advantage of the near-field beam split effect. Specifically, our contributions are summarized as follows.

• •

  Firstly, we prove the effect of near-field controllable beam split, i.e., time-delay circuits can flexibly control the degree of near-field beam split. Specifically, it has been proved in far-field scenarios that, time-delay circuits can control the covered angular range of the beams over different frequencies when the distance is very large [17, 19, 18]. By contrast, we will reveal that not only the covered angular range, but also the covered distance range of the beams over different frequencies can be controlled by the elaborate design of time delays, and thus the control of the near-field beam split is achievable. A simple analogy of the near-field controllable beam split effect is the dispersion of white light with a large bandwidth caused by a prism, thus in this paper, this effect is also termed as “near-field rainbow”. This near-field rainbow effect allows us to generate multiple beams focusing on multiple locations simultaneously by only one radio-frequency (RF) chain, which is not achievable for existing schemes.

• •

  Then, based on the mechanism of the near-field rainbow, a wideband near-field beam training scheme is proposed to realize fast near-field CSI acquisition. In the proposed scheme, multiple beams focusing on multiple angles in a given distance range are generated by time-delay circuits in each time slot. Next, for different time slots, different distance ranges are measured by fine-tuning the time-delay parameters. By this means, the optimal spatial angle can be searched in a frequency-division manner, while the optimal distance is obtained in a time-division manner. Unlike the exhaustive near-field beam training scheme that searches only one location in each time slot, the proposed scheme is able to search multiple locations in each time slot, thus the beam training overhead can be significantly reduced.

• •

  Finally, we provide simulation results to demonstrate the mechanism of the near-field rainbow and verify the advantages of the proposed beam training scheme. We demonstrate that our scheme is able to achieve a satisfactory average rate performance at a significantly reduced training overhead. We also show that our beam training scheme outperforms the existing far-field schemes in the near-field range. Additionally, our scheme also works well in the far-field range, since the proposed near-field beam training scheme is able to automatically decay to the far-field beam training in the far-field scenarios.

Organization: The rest of this paper is organized as below. In Section II, the system model is introduced. We first introduce the near-field beam split effect in wideband XL-MIMO systems and then discuss how to mitigate it with a time-delay beamformer. Then in Section III, we prove the mechanism of the near-field controllable beam split. In Section IV, the exhaustive near-field beam training is defined and the near-field rainbow based beam training is proposed. Section V provides the simulation results. Finally, conclusions are drawn in Section VI.

Notation: Lower-case and upper-case boldface letters represent vectors and matrices, respectively; $[{\mathbf{x}}]_{n}$ denotes the $n$-th element of the vector ${\mathbf{x}}$; $(\cdot)^{*}$, $(\cdot)^{T}$, and $(\cdot)^{H}$ denote the conjugate, transpose, and conjugate transpose, respectively; $|\cdot|$ and $\text{Tr}(\cdot)$ denote the absolute and trace operator; $\lceil x\rceil$ denotes the nearest integer greater than or equal to $x$; $\mathcal{CN}(\mu,\Sigma)$ and $\mathcal{U}(a,b)$ denote the Gaussian distribution with mean $\mu$ and covariance $\Sigma$, and the uniform distribution between $a$ and $b$, respectively.

In this paper, a wideband XL-MIMO system is considered. We assume the base station (BS) is equipped with a uniform linear array (ULA) to serve a single omnidirectional antenna user using orthogonal frequency division multiplexing (OFDM) with $M$ subcarriers. For expression simplicity, we assume the number of BS antennas is $N_{t}=2N+1$. We denote $B$, $c$, $f_{c}$, $\lambda_{c}=\frac{c}{f_{c}}$ as the bandwidth, the light speed, the central carrier frequency, and the central wavelength, respectively. The antenna spacing is presented as $d=\frac{\lambda_{c}}{2}$.

Due to the severe path loss incurred by the scatters, mmWave and THz communications heavily rely on the line-of-sight (LoS) path [4]. Therefore, we mainly focus on the near-field LoS channel, while the discussions in this paper can be straightforwardly extended to the non-LoS (NLoS) scenarios. As shown in Fig. 1, the user is located at $(r_{0}\cos\vartheta_{0},r_{0}\sin\vartheta_{0})$ with $\theta_{0}=\sin\vartheta_{0}$. Taking into account the near-field spherical wave characteristic [22], the channel between the $n$-th BS antenna and the user at the $m$-th subcarrier with $n=-N,\cdots,0,\cdots,N$ and $m=1,2,\cdots,M$ can be represented as

where $k_{m}=\frac{2\pi f_{m}}{c}$ denotes the wavenumber at subcarrier frequency $f_{m}=f_{c}+\frac{B}{M}\left(m-1-\frac{M-1}{2}\right)$ and $r^{(n)}_{0}$ represents the distance between the user and the $n$-th BS antenna. As the coordinate of the $n$-th BS antenna is $(0,nd)$, $r_{0}^{(n)}$ can be derived from the geometry as $r_{0}^{(n)}=\sqrt{(r_{0}\cos\vartheta_{0})^{2}+(r_{0}\sin\vartheta_{0}-nd)^{2}}=\sqrt{r_{0}^{2}+n^{2}d^{2}-2r_{l}\theta_{0}nd}$. The path gain $\beta^{(n)}_{m}$ can be modeled as [23]

where wavelength $\lambda_{m}=\frac{f_{m}}{c}$ and $\vartheta^{(n)}_{0}$ denotes the angle between the user and the $n$-th BS antenna. Moreover, $G_{t}$ is the antenna gain and $F\left(\vartheta\right)$ denotes normalized power radiation pattern, satisfying $\int_{\vartheta=0}^{\pi/2}G_{t}F(\vartheta)\sin\vartheta d\vartheta=1$. An example of normalized power radiation pattern is $F(\vartheta)=\cos^{3}\vartheta,\vartheta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ [23]. Generally, the distance $r_{0}$ between the user and BS is larger than the array aperture $D=N_{t}d$. For example, for a 256-element ULA working at 30 GHz, $r_{0}$ is very likely to be larger than $D=256\times 0.5\times 10^{-3}=1.28$ meters. With $r_{0}>D$, we can assume $\beta^{(-N)}_{m}\approx\cdots\approx\beta^{(N)}_{m}\approx\beta_{m}=\sqrt{G_{t}F\left(\vartheta_{0}\right)}\frac{\lambda_{m}}{4\pi r_{0}}$ based on the Fresnel approximation [24]. As a result, the near-field LoS channel ${\mathbf{h}}_{m}\in\mathbb{C}^{N_{t}\times 1}$ can be represented as

where ${\mathbf{a}}_{m}(\theta_{0},r_{0})$ is the near-field array response vector.

Notice that since the near-field spherical wave characteristic is considered, the array response vector ${\mathbf{a}}_{m}(\theta_{0},r_{0})$ is significantly different from the classical far-field array response vector [25], where the latter is based on the planar wave assumption and ignores the influence of distance $r_{0}$. The radius of the near-field region is determined by the Rayleigh distance $\text{RD}=\frac{2D^{2}}{\lambda_{c}}$ [8]. For a 1-meter diameter array operating at 30 GHz, its Rayleigh distance reaches up to 200 meters. Therefore, in such a mmWave XL-MIMO system, the near-field spherical wave characteristic becomes essential and the distance information cannot be ignored.

However, since the distance $r_{0}^{(n)}$ is a complicated radical function with respect to the antenna index $n$, it is difficult to analyze the property of the near-field spherical wavefront directly from (3). To deal with this problem, the Fresnel approximation [24] can be adopted to approximate $r_{0}^{(n)}$ as

where $(a)$ is based on the second-order Taylor expansion $\sqrt{1+x}\approx 1+\frac{1}{2}x-\frac{1}{8}x^{2}$. For expression simplicity, we denote $\alpha_{0}=\frac{1-\theta_{0}^{2}}{2r_{0}}$ and $\alpha=\frac{1-\theta^{2}}{2r}$. As shown in Fig. 2, the curve $\alpha=\frac{1-\theta^{2}}{2r}$ corresponds to a ring in the physical space, and thus the curve $\alpha=\frac{1-\theta^{2}}{2r}$ is termed as distance ring $\alpha$. Then, according to (4), the $n$-th element of ${\mathbf{a}}_{m}(\theta_{0},r_{0})$ can be approximated as

where $[{\mathbf{b}}_{m}(\theta_{0},\alpha_{0})]_{n}=\frac{1}{\sqrt{N}}e^{jk_{m}(nd\theta_{0}-n^{2}d^{2}\alpha_{0})}$ denotes the $n$-th element of vector ${\mathbf{b}}_{m}(\theta_{0},\alpha_{0})$. Since the residue phase $e^{-jk_{m}r_{0}}$ in (5) has no relationship with the antenna index $n$, we only need to focus on the vector ${\mathbf{b}}_{m}(\theta_{0},\alpha_{0})$.

At mmWave or THz band, frequency-independent PS-based beamformer is widely considered to serve the user by generating a focused beam aligned with location $(r_{0},\theta_{0})$ in the polar coordinate. Generally, the beamfocusing vector ${\mathbf{w}}_{\text{PS}}$ generated by frequency-independent PSs is usually set as ${\mathbf{w}}_{\text{PS}}={\mathbf{b}}_{c}^{*}(\theta_{0},\alpha_{0})$ to focus the beam energy on the location $(r_{0},\theta_{0})$ [26], where ${\mathbf{b}}_{c}(\theta_{0},\alpha_{0})$ represents the array response vector (5) at frequency $f_{c}$.

However, the array response vector (5) of the wideband channel is frequency-dependent, which mismatches the frequency-independent beamfocusing vector ${\mathbf{w}}_{\text{PS}}={\mathbf{b}}_{c}^{*}(\theta_{0},\alpha_{0})$. This mismatch leads to the fact that, the beams generated by ${\mathbf{w}}_{\text{PS}}$ at different frequencies will be focused on different locations, which is detailed in the following Lemma 1.

Proof: At frequency $f_{m}$, the array gain achieved by ${\mathbf{w}}_{\text{PS}}$ on an arbitrary user location $(r,\theta)$ with $\alpha=\frac{1-\theta^{2}}{2r}$ is

where we define $G(x,y)=\frac{1}{N_{t}}\left|\sum_{n}e^{jndx-jn^{2}d^{2}y}\right|$. Apparently, the beam at frequency $f_{m}$ is focused on the location $(r_{m},\theta_{m})$ corresponding to the maximum array gain $G(k_{m}\theta_{m}-k_{c}\theta_{0},k_{m}\alpha_{m}-k_{c}\alpha_{0})$, i.e.,

where $\alpha_{m}=\frac{1-\theta_{m}^{2}}{2r_{m}}$. Since $G(0,0)=1$ and $|G(x,y)|\leq 1$, it is obvious that $(x,y)=(0,0)$ is an optimal solution to maximize the function $G(x,y)$. Therefore, we have $k_{m}\theta_{m}-k_{c}\theta_{0}=0$ and $k_{m}\alpha_{m}-k_{c}\alpha_{0}=0$, which leads to the results of (6) and (7). $\hfill\blacksquare$

According to Lemma 1, the beam at $f_{m}$ is focused on the location $(\theta_{m},r_{m})=(\theta_{m},\frac{1-\theta_{m}^{2}}{2\alpha_{m}})=(\frac{\theta_{0}}{\eta_{m}},r_{0}\frac{\eta_{m}-\frac{1}{\eta_{m}}\theta_{0}^{2}}{1-\theta_{0}^{2}})$. As shown in Fig. 3, since the desired user is located at $(r_{0},\theta_{0})\neq(r_{m},\theta_{m})$, the beam at frequency $f_{m}$ cannot be aligned with the desired user location. This misalignment is termed as the “near-field beam split” effect [5], which will result in a severe array gain loss when $f_{m}$ is far away from $f_{c}$ in wideband XL-MIMO systems. For example, if the carrier frequency and bandwidth are $30$ GHz and $1$ GHz, and the BS is equipped with a 256-element ULA, then around 50% subcarriers will suffer from more than 50% array gain loss [5]. Therefore, the near-field beam split effect should be elaborately addressed, especially when the bandwidth is very large.

To mitigate the near-field beam split effect, a method is to utilize TD based beamformer rather than PS based beamformer to generate frequency-dependent beams to match the frequency-dependent channels. Notice that the TD based beamsteering has been studied in the far-field wideband systems [12, 13, 14, 15, 16, 17, 18, 19], while in this study, we consider utilizing it to overcome and control the near-field beam split effect.

As shown in Fig. 4, we assume the BS is equipped with an $N_{t}$-element TD beamformer¹¹1The main purpose of this paper is to derive that time-delay circuits are able to control the near-field beam split effect. Therefore, we directly utilize the time-delay beamformer as an example, while the discussion on some other low-power consumption architectures, such as delay-phase precoding, is left for future works.. Then the frequency response at frequency $f_{m}$ of the $n$-th antenna is $[{\mathbf{w}}_{m}]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-j2\pi f_{m}\tau^{(n)^{\prime}}}$, where ${\mathbf{w}}_{m}$ denotes the frequency-dependent beamforming vector, and $\tau^{(n)^{\prime}}$ denotes the adjustable time-delay parameter of the $n$-th time-delay circuit. For expression simplicity, we denote $r^{(n)^{\prime}}=c\tau^{(n)^{\prime}}$ as the adjustable distance, and then $[{\mathbf{w}}_{m}]_{n}$ becomes $\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}r^{(n)^{\prime}}}$. Notice that $[{\mathbf{w}}_{m}]_{n}$ has a similar form with the array response vector $[{\mathbf{b}}_{m}(\theta_{0},r_{0})]_{n}=\frac{1}{\sqrt{N_{t}}}e^{jk_{m}(nd\theta_{0}-n^{2}d^{2}\alpha_{0})}$. Thus, the $n$-th adjustable distance $r^{(n)^{\prime}}$ can be set as $r^{(n)^{\prime}}=nd\theta^{\prime}-n^{2}d^{2}\alpha^{\prime}$, while $\theta^{\prime}$ and $\alpha^{\prime}$ are adjustable delay parameters of a TD beamformer. Then, the corresponding beamfocusing vector at $f_{m}$ can be presented as

In this case, at frequency $f_{m}$, the array gain on an arbitrary location $(r,\theta)$ with $\alpha=\frac{1-\theta^{2}}{2r}$ is given by

Obviously, the beam at $f_{m}$ is focused on $(r_{m},\theta_{m})=\arg\max_{r,\theta}G(k_{m}(\theta-\theta^{\prime}),k_{m}(\alpha-\alpha^{\prime}))$. Similar to the derivation of the optimal solution of (9), $G(k_{m}(\theta-\theta^{\prime}),k_{m}(\alpha-\alpha^{\prime}))$ approaches its maximum value when $k_{m}(\theta-\theta^{\prime})=0$ and $k_{m}(\alpha-\alpha^{\prime})=0$. Therefore, the beam at frequency $f_{m}$ is focused on $\theta_{m}=\theta^{\prime}$ and $r_{m}=r^{\prime}=\frac{1-\theta^{\prime 2}}{2\alpha^{\prime}}$, which is not relevant to $f_{m}$. Then, once the LoS path information $(r_{0},\theta_{0})$ is available at the BS, and the delay parameters are set as $\theta^{\prime}=\theta_{0}$ and $\alpha^{\prime}=\frac{1-\theta_{0}^{2}}{2r_{0}}$, then the beams across the entire bandwidth are able to be focused on the location $\theta_{m}=\theta^{\prime}=\theta_{0}$ and $r_{m}=\frac{1-\theta^{\prime 2}}{2\alpha^{\prime}}=r_{0}$. As a consequence, the near-field beam split effect can be mitigated by the TD beamformer.

On the other hand, efficient wideband beamfocusing requires that the LoS path information $(r_{0},\theta_{0})$ is available at the BS side. To realize this requirement, in the following discussions, we prove that a TD beamformer can not only mitigate the near-field beam split effect, but also flexibly control its degree. Then, we will further utilize this property to achieve efficient near-field beam training to obtain $(r_{0},\theta_{0})$.

For the far-field scenarios, the distances $r$ and $r^{\prime}$ are assumed to be larger than the Rayleigh distance RD, so that the spherical wavefront can be approximated as a planar wavefront. In this case, all of the distance-related parameters $\alpha=\frac{1-\theta^{2}}{2r}$ and $\alpha^{\prime}=\frac{1-\theta^{\prime 2}}{2r^{\prime}}$ reliably approach 0. Then, the array response vector becomes $[{\mathbf{b}}_{m}(\theta,\alpha)]_{n}=[{\mathbf{b}}_{m}(\theta,0)]_{n}=\frac{1}{\sqrt{N_{t}}}e^{jk_{m}nd\theta}$, and the beamfocusing vector realized by a TD beamformer becomes $[{\mathbf{w}}_{m}(\theta^{\prime},0)]_{n}=\frac{1}{\sqrt{N_{t}}}e^{-jk_{m}nd\theta^{\prime}}$. Therefore, the array gain in (II-C) can be simplified as

where $G(x,0)=\left|\frac{\sin\left(\frac{N_{t}}{2}dx\right)}{N_{t}\sin\left(\frac{1}{2}dx\right)}\right|$. It has been previously proved that the beams over the entire bandwidth generated by ${\mathbf{w}}_{m}(\theta^{\prime},0)$ are focused on the spatial angle $\theta_{m}=\theta^{\prime}$, with $m=1,2,\cdots,M$. However, notice that the spatial angle $\theta_{m}$ is corresponding to an actual physical angle $\vartheta_{m}=\arcsin\theta_{m}$ as shown in Fig. 1, which implies the value range of $\theta_{m}$ is restrained by $\theta_{m}\in[-1,1]$. By contrast, $\theta^{\prime}$ is an adjustable parameter of a TD beamformer. A question naturally arises that, if the adjustable parameter is set as $\theta^{\prime}\notin[-1,1]$, then what it has to do with the actual spatial angle $\theta_{m}$? Actually, the answer to this question is exactly the mechanism of the far-field controllable beam split.

For expression simplicity, parameter $\theta^{\prime}\notin[-1,1]$ is termed as the abnormal value, while parameter $\theta^{\prime}\in[-1,1]$ is termed as the actual value. Restrained by the value range of $\theta_{m}$, if $\theta^{\prime}$ is an abnormal value, it is obvious that $\theta_{m}\neq\theta^{\prime}$. To acquire the relationship between $\theta_{m}$ and $\theta^{\prime}$, we observe that the far-field array gain $G(x,0)$ is a periodic function against $x$ with a period $\frac{2\pi}{d}$. For any integer $p\in\mathbb{Z}$, we have

In Section II, we have indicated that the optimal solution $x^{\text{opt}}$ to maximize $G(x,0)$ is $(x^{\text{opt}},0)=(0,0)$. However, according to (13), we find that $(0,0)$ is just one of the optimal solutions for maximizing $G(x,0)$. The periodicity of $G(x,0)$ implies that the optimal solutions should satisfy $(x^{\text{opt}},0)=(\frac{2p\pi}{d},0)$, $p\in\mathbb{Z}$. As a consequence, by solving $\theta_{m}=\arg\max_{\theta}G(k_{m}(\theta-\theta^{\prime}),0)$, the focused spatial angle $\theta_{m}$ at frequency $f_{m}$ is derived as

where the function of $p\in\mathbb{Z}$ is to ensure $\theta_{m}\in[-1,1]$ becomes an actual spatial angle. From (14), the mechanism of the far-field controllable beam split is acquired, which has the following features.

Feature 1: Since $p$ is an integer, if $p\neq 0$, the spatial angle $\theta_{m}$ is related to the frequency $f_{m}$, which means beams at different frequencies will split towards different spatial angles. Therefore, despite the TD based beamsteering architecture being utilized, the far-field beam split effect can also be induced.

Feature 2: The specific value of $p\in\mathbb{Z}$ is determined by $\theta^{\prime}$. If $\theta^{\prime}\in[-1,1]$ belongs to the actual value range of $\theta_{m}$, then $p$ can be zero and $\theta_{m}$ exactly equals to $\theta^{\prime}$. That is to say, with $\theta^{\prime}\in[-1,1]$, the far-field beam split effect is eliminated. While if $\theta^{\prime}\notin[-1,1]$ becomes an abnormal value, then to guarantee that $\theta_{m}$ is an actual spatial angle, it is obvious that $p$ cannot be zero. For instance, if $\theta^{\prime}$ is set as $1.5$, at the central frequency $f_{c}$ with $\eta_{c}=\frac{f_{c}}{f_{c}}=1$, the corresponding spatial angle is $\theta_{c}=1.5+2p/\eta_{c}=1.5+2p$, $p\in\mathbb{Z}$. Only if $p=-1$ can $\theta_{c}=-0.5\in[-1,1]$ be an actual spatial angle. Therefore, with $\theta^{\prime}\notin[-1,1]$, $p$ is not zero and thus $\theta_{m}$ is a function of the frequency $f_{m}$, where the beam split effect is induced again.

Feature 3: Notice that the beam split pattern (14) of a TD beamformer is different from (6) of a PS beamformer. For a PS beamformer, the degree of beam split is fixed and uncontrollable. However, for a TD beamformer, the degree of beam split is controllable by adjusting $\theta^{\prime}$. For instance, considering the central frequency $f_{c}$, if $\theta^{\prime}$ is set as $1.5$, then $p$ should be $-1$ and $\theta_{c}=1.5-2=-0.5$. On the other hand, if $\theta^{\prime}$ is set as $3.5$, then $p$ should be $-2$ and $\theta_{c}=3.5-4=-0.5$. For the above two examples, although the spatial angles $\theta_{c}$ are the same, the integers $p$ are different, which indicates the degree of the beam split $\theta_{m}=\theta^{\prime}+2p/\eta_{m}$ is different. Therefore, by adjusting the delay parameter $\theta^{\prime}$, the integer $p$ is indirectly controlled, then the degree of beam split is adjusted. This property is termed as the far-field controllable beam split. Benefiting from this property, the adjustment on the angular coverage range of the beams is achieved, which can be utilized to realize fast far-field CSI acquisition [17, 19, 18].

In the next sub-section, we will extend this conclusion to a more general near-field scenario.

For the near-field scenario, the distances $r$ and $r^{\prime}$ are lower than the Rayleigh distance. Then $\alpha$ and $\alpha^{\prime}$ can not be zero, and the near-field spherical wave characteristics should be considered. To prove the mechanism of the near-field controllable beam split, analog to the derivation in the far-field scenarios, we consider the abnormal values for the adjustable parameters $\theta^{\prime}$ and $\alpha^{\prime}$ in (10) simultaneously.

Specifically, it has been discussed before $\theta^{\prime}\notin[-1,1]$ is an abnormal value for the angle. As for the distance-related parameter $\alpha=\frac{1-\theta^{2}}{2r}$, since $1>\theta^{2}$ and $r>0$, a realistic $\alpha$ should be larger than zero. On this condition, $\alpha^{\prime}<0$ is obvious the abnormal value for the distance ring. If $\theta^{\prime}\in[-1,1]$ and $\alpha^{\prime}>0$ are actual values, it was proved in Section II-C that the near-field beam split effect is eliminated, i.e., beams are focused on $\theta_{m}=\theta^{\prime}$ and $\alpha_{m}=\alpha^{\prime}$ for all frequencies $f_{m}$. On the other hand, for the abnormal value $\theta^{\prime}\notin[-1,1]$ or $\alpha^{\prime}<0$, the following Lemma 2 explains the behaviors of the beams generated by ${\mathbf{w}}_{m}(\theta^{\prime},\alpha^{\prime})$.

Proof: Similar to the derivation of (14), the periodicity of the near-field array gain function $G(x,y)$ should be considered. It is proved in Appendix A that $G(x,y)$ is a periodic function against the vector variable $(x,y)$ with a period $(\frac{2\pi}{d},\frac{2\pi}{d^{2}})$. Therefore, for any integers $p\in\mathbb{Z}$ and $q\in\mathbb{Z}$, $G(x,y)$ can be rewritten as

It has been indicated in Section II that the optimal solution $(x^{\text{opt}},y^{\text{opt}})$ to maximize the array gain $G(x,y)$ is $(x^{\text{opt}},y^{\text{opt}})=(0,0)$. The periodicity in (17) implies $(0,0)$ is just one of the optimal solutions for maximizing $G(x,y)$. Besides, the optimal solutions should satisfy $(x^{\text{opt}},y^{\text{opt}})=(\frac{2p\pi}{d},\frac{2q\pi}{d^{2}})$ with $p\in\mathbb{Z}$ and $q\in\mathbb{Z}$. Accordingly, to obtain the focused location of the beam at frequency $f_{m}$ by solving $(\theta_{m},r_{m})=\arg\max_{\theta,r}G(k_{m}(\theta-\theta^{\prime}),k_{m}(\alpha-\alpha^{\prime}))$, we have $k_{m}(\theta_{m}-\theta^{\prime})=\frac{2p\pi}{d}$ and $k_{m}(\alpha_{m}-\alpha^{\prime})=\frac{2q\pi}{d^{2}}$. As a consequence, $\theta_{m}$ and $\alpha_{m}$ become $\theta^{\prime}+\frac{2p}{\eta_{m}}$ and $\alpha^{\prime}+\frac{2q}{d\eta_{m}}$, respectively, where the function of integers $p$ and $q$ is to ensure that $\theta_{m}\in[-1,1]$ and $\alpha_{m}>0$ are an actual spatial angle and an actual distance ring. $\hfill\blacksquare$

According to Lemma 2, the mechanism of the near-field controllable beam split is acquired, which has the following features.

Feature 1: If $p\neq 0$ or $q\neq 0$, then $\theta_{m}$ or $\alpha_{m}$ is related to the frequency $f_{m}$, which means beams at different frequencies will be focused on different locations. Therefore, despite the TD beamformer being utilized, the near-field beam split effect can also be induced.

Feature 2: As the function of the abnormal value of $\theta^{\prime}$ has been discussed in the previous section, we mainly explain the impact of the abnormal value of $\alpha^{\prime}$ on the integer $q$ here. If $\alpha^{\prime}>0$ is an actual value, then $q$ could be zero, and for all beams, we have $\alpha_{m}=\alpha^{\prime}$. However, if $\alpha^{\prime}<0$ is an abnormal value, to guarantee $\alpha_{m}>0$, the value of $q$ must be larger than 0, and thus the beams at different subcarriers will be focused on different distances $r_{m}=\frac{1-\theta_{m}^{2}}{2\alpha_{m}}$. For instance, let the delay parameter $\alpha^{\prime}$ be $-\frac{2}{d}$, then we have $\alpha_{m}=\frac{2}{d}(\frac{q}{\eta_{m}}-1)$. To guarantee $\alpha_{m}\geq 0$ is an actual distance ring, the integer $q$ should be larger than $\eta_{m}$. Accordingly, when $f_{m}\leq f_{c}$ and $\eta_{m}=\frac{f_{m}}{f_{c}}\leq 1$, then the integer $q=1$ is enough to make $\alpha_{m}>0$ and the beam at $f_{m}$ is focused on the distance ring $\alpha_{m}=\frac{2}{d}(\frac{1}{\eta_{m}}-1)$. On the other hand, when $f_{m}>f_{c}$ and $\eta_{m}=\frac{f_{m}}{f_{c}}>1$, then the integer $q=2$ is enough to make $\alpha_{m}>0$ and the beam at $f_{m}$ is focused on the distance ring $\alpha_{m}=\frac{2}{d}(\frac{2}{\eta_{m}}-1)$. As a consequence, the abnormal value of $\alpha^{\prime}$ is able to introduce and control the beam split effect on the distance dimension.

Feature 3: The far-field scenario is a special case of the near-field controllable beam split by setting $\alpha_{m}=\alpha^{\prime}=0$. The far-field scenario assumes the distance is very large, and it can only control the beam split effect on the angle $\theta_{m}$. By contrast, the near-field controllable beam split effect can adjust the degree of the beam split effect on the angle $\theta_{m}$ and distance $r_{m}$ jointly. For instance, one can set $\theta^{\prime}$ as an abnormal value and $\alpha^{\prime}$ as an actual value, then we have $\theta_{m}=\theta^{\prime}+2p/\eta_{m}$, $\alpha_{m}=\alpha^{\prime}$ and $r_{m}=\frac{1-\theta_{m}^{2}}{2\alpha_{m}}=\frac{1-(\theta^{\prime}+2p/\eta_{m})^{2}}{2\alpha^{\prime}}$. Therefore, the beam at frequency $f_{m}$ is focused on a location satisfying $\left(\frac{1-\theta_{m}^{2}}{2r_{m}},\theta_{m}\right)=\left(\alpha^{\prime},\theta^{\prime}+\frac{2p}{\eta_{m}}\right)$, which indicates that the beams are focused on multiple angles in the distance ring $\alpha^{\prime}$ in the near-field, as shown in Fig. 5 (a). Moreover, we can set $\theta^{\prime}$ as an actual value and $\alpha^{\prime}$ as an abnormal value, then the beams are focused on multiple distances at the same angle as shown in Fig. 5 (b), satisfying $\left(\frac{1-\theta_{m}^{2}}{2r_{m}},\theta_{m}\right)=\left(\alpha^{\prime}+\frac{2q}{d\eta_{m}},\theta^{\prime}\right)$. As a result, by carefully designing the parameters $\alpha^{\prime}$ and $\theta^{\prime}$, the beams across the entire band could occupy multiple angles and distances simultaneously, and this property is more flexible than that in the far-field scenario.

In conclusion, the mechanism of near-field controllable beam split indicates that a TD beamformer can not only mitigate the near-field beam split effect, but also control its degree, i.e., one can flexibly adjust the covered angular range and distance range of the beams generated by TD beamformer over the entire bandwidth. A simple analogy of this effect is the dispersion of white light caused by a prism. Since a prism has different refractive indices for the wideband white light, the different frequency components of the pure light will disperse and eventually produce the rainbow. In our discussion, the function of time-delay circuits is similar to that of the prism. Therefore, the near-field controllable beam split effect is also termed as “near-field rainbow” in this paper.

By elaborately designing the pattern of the near-field rainbow as shown in Fig. 5, efficient near-field beam management can be realized, which will be discussed in the next section.

Beam training is a well-adopted scheme to obtain CSI for current 5G systems [27]. The concept of near-field beam training can be derived from the classical far-field beam training. For the classical far-field beam training, the spatial angle information $\theta_{0}$ is desired. The optimal beamsteering vector for a user is selected from a predefined far-field beam codebook through a training procedure between the BS and the user. Generally, each codeword in the far-field codebook determines a unique spatial angle, where the distance is assumed to be very large so that the near-field property is ignored. Therefore, the entire far-field codebook occupies all of the potential angles in the far-field.

Similarly, the near-field beam training desires to obtain the location information $(r_{0},\theta_{0})$ of the dominant path between the BS and the user. The optimal beamfocusing vector is selected from a predefined near-field beam codebook through a training procedure between the BS and the user. Each near-field codeword determines a unique location, and the entire near-field codebook occupies all of the desired angles and distances.

To be more specific, we introduce the procedure of exhaustive near-field beam training realized by a TD beamformer. Notice that the procedure below is also valid for the PS beamformer when the bandwidth is not very large. We define $[\theta_{\min},\theta_{\max}]$ as the potential range of spatial angle, satisfying $-1\leq\theta_{\min}\leq\theta_{\max}\leq 1$. Moreover, we assume the minimum distance between the BS and the user is $\rho_{\min}$, so that the potential range of distance is $r\in[\rho_{\min},+\infty]$ or $\alpha=\frac{1-\theta^{2}}{2r}\in[0,\alpha_{\max}]$ with $\alpha_{\max}=\frac{1}{2\rho_{\min}}$. Then, multiple angles from $\theta\in[\theta_{\min},\theta_{\max}]$ and distances from $r\in[\rho_{\min},+\infty]$ are sampled simultaneously to construct the near-field codebook. As proved in [6], the sampled locations could satisfy

where $u=0,1,\cdots,U-1$ and $s=0,1,\cdots,S-1$. $U$ denotes the number of sampled angles and $S$ denotes the number of sampled distance rings. Notice that when $S=1$, we have $\alpha_{s}\equiv 0$, and then the near-field codebook is naturally decayed to the far-field codebook. The exhaustive near-field beam training scheme searches the entire codebook $\{\theta_{u}\}$ and $\{\alpha_{s}\}$ to obtain the optimal beamfocusing vector. Apparently, the overhead for exhaustive near-field beam training, i.e., the number of time slots used for beam training, is $T_{1}=US$.

In the $t$-th time slot, where $t=sU+u$ with $t=0,1,\cdots,T_{1}-1$, $u=0,1,\cdots,U-1$, and $s=0,1,\cdots,S-1$, we set the parameters of the TD beamformer as $\theta_{t}^{\prime}=\theta_{u}=\theta_{\min}+\frac{u}{U}(\theta_{\max}-\theta_{\min})$ and $\alpha_{t}^{\prime}=\alpha_{s}=\frac{s}{S}\alpha_{\max}$, and then generate the beamfocusing vector ${\mathbf{w}}_{m}(\theta_{t}^{\prime},\alpha_{t}^{\prime})$ according to (10). Since $\theta_{t}^{\prime}$ and $\alpha_{t}^{\prime}$ are actual values, the BS is able to transmit the pilot sequence to the user by the beam focused on the location $(r_{t}^{\prime},\theta_{t}^{\prime})=\left(\frac{1-\theta_{u}^{2}}{2\alpha_{s}},\theta_{u}\right)$. So the received signal $y_{m,t}$ in the $t$-th time slot at $f_{m}$ is

where $n_{m}\sim\mathcal{CN}(0,\sigma^{2})$ denotes the Gaussian noise, $P_{t}$ denotes the transmit power, and $x_{m}$ denotes the transmit pilot satisfying $\|x_{m}\|^{2}=1$. After $T_{1}$ time slots, the estimated physical location $(\hat{r},\hat{\theta})=(\frac{1-\hat{\theta}^{2}}{2\hat{\alpha}},\hat{\theta})$ corresponding to the largest user received power can be selected from the $T_{1}$ measured locations, where,

Finally, a near-field beam ${\mathbf{w}}_{m}(\hat{\theta},\hat{\alpha})$ aligned with the location $(\hat{r},\hat{\theta})$ can be generated to serve the user, and a near-optimal beamfocusing gain can be achieved.

Nevertheless, since the exhaustive near-field beam training scheme has to search the entire near-field codebook exhaustively, and the scale of the near-field codebook is much larger than that of the far-field, i.e. $US\gg U$, the training overhead is unacceptable in practice. Therefore, a near-field beam training scheme with low pilot overhead is essential for XL-MIMO systems.

The main reason for the high training overhead of exhaustive near-field beam training is that only one physical location can be measured in each time slot. By contrast, as we discussed in Section III-B, a TD beamformer is able to generate multiple beams focusing on multiple locations by only one RF chain. Taking advantage of the near-field rainbow, multiple physical locations can be measured simultaneously in each time slot. Inspired by this observation, we propose a near-field rainbow-based beam training method to significantly reduce the training overhead.

Generally, since the spatial resolution of an antenna array on the angle is much higher than that on the distance, the number of sampled angles $U$ is usually much larger than the number of sampled distances $S$ [6]. Thus, the training overhead is mainly determined by the search of angle $\theta_{0}$. Therefore, the near-field rainbow on the angle dimension, as shown in Fig. 5 (a), is utilized to avoid the exhaustive search of angle, i.e., the angle-related parameters $\theta^{\prime}$ are set as abnormal values, while the distance-related parameters $\alpha^{\prime}$ can be set as actual values. In other words, the proposed scheme searches the optimal angle in a frequency division manner, and searches the optimal distance ring in a time division manner. The specific procedure of the proposed beam training scheme is illustrated in Algorithm 1.

Firstly, in steps 1-2, we desire to design the abnormal parameter $\theta^{\prime}\notin[-1,1]$, so that the frequency-dependent angles $\theta_{m}=\theta^{\prime}+(2pf_{c})/f_{m}$ across the entire band is able to cover the entire potential angle range $[\theta_{\min},\theta_{\max}]$, as shown in Fig. 5 (a). To realize this target, we first assume that, at the central frequency $f_{c}$, the beam is aligned with the angle $\theta_{c}=\theta^{\prime}+2p$, where $\theta_{c}$ is a predefined spatial angle satisfying $-1\leq\theta_{\min}<\theta_{c}<\theta_{\max}\leq 1$. Then we have