3D Beam Training in Terahertz Communication:
A Quadruple-UPA Architecture

We consider a point-to-point THz communication system with four half-wave spaced UPAs, a quadruple-UPA architecture, equipped at the transmitter and receiver, respectively. Without loss of generality, we assume that both the transmitter and the receiver have the same architecture where four identical UPAs with $N_{a}$ elements are equipped around a cube. As shown in Fig. 1, we use $x$, $y$, and $z$-axes to refer to the axes of the standard Cartesian coordinate system for all the UPAs. In the case of the $1$st UPA, with $N_{y}$ and $N_{z}$ elements on the $y$ and $z$-axes respectively ($N_{a}=N_{y}N_{z}$), the array response vector can be expressed in a conventional form¹¹1We assume the signal phase at the center of the UPA is zero., i.e.,

where $\phi$ and $\theta$ are the azimuth angle to $x$-axis and the elevation angle to $z$-axis respectively, ${n_{y}}=-\frac{{({N_{y}}-1)}}{2}+1,-\frac{{({N_{y}}-1)}}{2}+2,...,\frac{{({N_{y}}-1)}}{2}-1$, ${n_{z}}=-\frac{{({N_{z}}-1)}}{2}+1,-\frac{{({N_{z}}-1)}}{2}+2,...,\frac{{({N_{z}}-1)}}{2}-1$. Note that the perpendicular direction of the $k$th array is $(\frac{{(k-1)\pi}}{2},\frac{\pi}{2})$, thereby the response vector of the $k$th array can be written as

To provide omni-directional communication with adequate array gains, four UPAs are tailored for beamforming in four different space ranges by using the directional antenna elements respectively. As shown in Fig. 2, each array is dedicated to transmitting and receiving signals only in the range within $\pm\frac{\pi}{4}$ to the perpendicular direction of the array, in both azimuth and elevation space. As such, the transmit/receive range of $k$th array is denoted by

Let $s$ denote a transmitted symbol with unit power to the $k$th transmit UPA, the processed received signal from the $m$th receive UPA can be expressed as

where $P$ represents the transmit power, ${{\bf{H}}_{k,m}}\in{\mathbb{C}^{N_{a}\times N_{a}}}$ is the channel matrix between the $k$th transmit UPA and $m$th receive UPA, $s$ is the data symbol, ${\bf{f}}_{k}\in{\mathbb{C}^{N_{a}\times 1}}$ (resp. ${\bf{w}}_{m}\in{\mathbb{C}^{N_{a}\times 1}}$) is the normalized beamforming precoder (resp. decoder) at $k$th (resp. $m$th) UPA, and ${\bf{n}}\sim\mathcal{CN}({\bf{0}},\sigma^{2}{\bf{I}})$ is the zero-mean additive Gaussian noise with power $\sigma^{2}$.

To enable a reliable THz communication, the precoder ${\bf{f}}_{k}$ at the $k$th transmit UPA and the decoder ${\bf{w}}_{m}$ at the $m$th receive UPA are needed to be optimized to maximize the decoding SNR, which is equivalent to solving the following problem

Provided that ${{\bf{H}}_{k,m}}$ is perfectly known at the transmitter and receiver, the optimal precoder ${\bf{f}}_{k}^{{\rm{opt}}}$ and the decoder ${\bf{w}}_{m}^{{\rm{opt}}}$ can be easily derived by applying the singular value decomposition (SVD) on ${{\bf{H}}_{k,m}}$. However, the conventional channel estimation methods tailored for micro-wave and millimeter-wave communications may not apply to THz system since their pilot signals suffer from severe path loss in THz channels and can not be effectively detected by the receiver before efficient beamforming. Considering this fact, we need to find ${\bf{f}}_{k}^{{\rm{opt}}}$ and ${\bf{w}}_{m}^{{\rm{opt}}}$ by testing the precoder-decoder pairs predefined in a codebook, without the channel state information. From this perspective, THz channel estimation relies on effective beamforming. Note that the severe path loss also significantly limits scattering in THz communication, where the gain of the NLoS paths is much lower (more than 20dB) than that of the LoS counterpart. Therefore, in this paper, we mainly consider the LoS component in the beam training, i.e.,

where ${\alpha}_{{\rm{L}}}$ is the complex gain of the LoS path, ${{\bf{a}}_{k}}({\phi_{t}},{\theta_{t}})$ and ${{\bf{a}}_{m}}({\phi_{r}},{\theta_{r}})$ are the normalized transmit and receive array response vectors which both follow the definition given in (2). $({\phi_{t}},{\theta_{t}})$ and $({\phi_{r}},{\theta_{r}})$ are the LoS path’s azimuth and elevation angles of departure and arrival (AoDs/AoAs), respectively. $\mathcal{F}_{k}$ and $\mathcal{W}_{m}$ are predefined codebooks for $k$th precoder and $m$th decoder, respectively. In fact, if the AoD and AoA of the LoS path are within the ranges of the $i$th UPA at the transmitter and the $j$th UPA at the receiver, other UPA pairs can not establish an effective communication link. This allows us to find a valid UPA pairs first, and then only considering the beam training problem on this pair.

Based on (6), the beam training problem is equivalent to

One can notice that without the codebook constraint, an optimal solution to (7) is given by $\{{\bf{w}}_{m}^{{\rm{opt}}}={{\bf{a}}_{m}}({\phi_{r}},{\theta_{r}}),{\bf{f}}_{k}^{{\rm{opt}}}={{\bf{a}}_{k}}({\phi_{t}},{\theta_{t}})\}$. Since the optimal precoder and decoder follow the form of array response vector, a straightforward method to reach a desired solution is to traverse all precoder-decoder pairs from codebooks composed of array response vectors with different angles. This method is also referred to as exhaustive 3D beam training[15]. As for our considered system with four UPAs, the transmitter and receiver have $4N^{2}$ narrow beams on each. Thus, there are $16N^{4}$ beam pairs to be tested in the exhaustive beam training, which is quite time consuming when $N$ is large.

Next, we propose a novel 3D hierarchical codebook to realize the beam training in a fast way, where the codewords in different stages represent beams with different beam width. Due to the common structure, we define $\mathcal{C}_{k}$ as the codebook for $k$th UPA of both the transmitter and the receiver. Firstly, we design ${N^{2}}={2^{S}}$ narrow beams which can cover $\Omega_{k}$ in union, where $S$ is the number of stages of our proposed hierarchical codebook. These narrow beams lie in the bottom stage, i.e., stage $S$, of the proposed hierarchical codebook and one narrow beam among will be selected as the optimal precoder/decoder after the hierarchical beam training. Thus, based on (7), all the narrow beams ought to follow the form of array response vector. As such, the design of the ${N^{2}}$ narrow beams is reduced to determining their directions, i.e., $\phi$ and $\theta$ in ${{\bf{a}}_{k}}(\phi,\theta)$. Assume that an optimal decoder is used in the beam training, based on (7), the received normalized decoding power is given by

where $\mathcal{C}_{k}^{S}$ represents the $N^{2}$ narrow beams to be designed in the stage $S$ of $\mathcal{C}_{k}$. Due to the randomness of the wireless channel (random ${\phi_{t}}$ and ${\theta_{t}}$), the quality of the codewords $\mathcal{C}_{k}^{S}$ can be judged by its one-side worst-case performance, i.e.,

Thus, we endeavour to design the directions for narrow beams of $\mathcal{C}_{k}^{S}$, to guarantee the highest worst-case performance. As a result, in our proposed hierarchical codebook, the $N^{2}$ narrow beams of $\mathcal{C}_{k}^{S}$ are given by

It is worth mentioning that the advantage of the proposed hierarchical codebook is to reduce the time complexity while its performance is the same as that of exhaustively testing the $N^{2}$ narrow beams of $\mathcal{C}_{k}^{S}$. As for our considered system, we have ${N^{2}}={2^{S}}$ which indicates that $N$ is even. Thus, Proposition 1 provides the guarantee of the normalized one-side worst-case performance of the proposed hierarchical codebook, which is given by (11) with $\beta=\sin\Big{(}{\arccos\frac{{\sqrt{2}}}{{2N}}}\Big{)}$.

Next, we propose an approach to design the wide beams of upper stages, i.e., stage $s=0,1,...,S-1$, of the hierarchical codebook $\mathcal{C}_{k}$. Each wide beam in the stage $s$ covers two beams in the stage $s+1$ while the beam in stage $0$ covers the whole range of $\Omega_{k}$. As such, we have $2^{s}$ beams in the stage $s$. The codewords for wide beams are no longer array response vectors and we use $\bm{\omega}_{i}^{k,s}$ to represent the $i$th codeword in the stage $s$ of the hierarchical codebook $\mathcal{C}_{k}$.

As mentioned above, to design the wide beams in stage $S-1$, we need to know the coverage of the beams in stage $S$, i.e., the narrow beams. Note that the directions of the narrow beams are non-uniformly distributed in the 3D space, i.e., these beams have smaller coverage in the center of $\Omega_{k}$ but larger coverage around the edge of $\Omega_{k}$. For ease of illustrating their range, we define two functions as $\Theta(\theta)=-\cos\theta$ and $\Phi(\phi)=\sin(\phi-\frac{{\pi(k-1)}}{2})+\sqrt{2}(k-1)$. By this means, as shown in Fig. 3, the range of the narrow beams can be represented as squares on $\Theta(\theta)$ and $\Phi(\phi)$, where the beam direction is in the center of the square. Based on this representation, Fig. 4 shows our proposed beams’ distribution as well as their coverage in different stages. According to the beams’ distribution shown, the codewords of narrow beams are given by

where $\bmod_{N}(i)$ returns the remainder after division of $i$ by $N$, and ${\rm{ceil}}(\cdot)$ returns the nearest integer greater than or equal to its argument. To develop the codewords of wide beams for $\mathcal{C}_{k}$, we have to construct a dense grid that represents all directions in front of the $k$th UPA, i.e., $\hat{\Omega}_{k}=\left\{{(\phi,\theta)|\phi\in[-\frac{\pi}{2}+\frac{{\pi(k-1)}}{2},\frac{\pi}{2}+\frac{{\pi(k-1)}}{2}],\theta\in[0,\pi]}\right\}$, which is larger than $\Omega_{k}$. As there are $N\times N$ narrow beams within range $\Omega_{k}$, as shown in Fig. 5, we construct $2N\times 2N$ grid blocks within this range, and total $4N\times 4N$ grid blocks within $\hat{\Omega}_{k}$. We should mention that if the rest $4N\!\times\!4N\!-\!2N\!\times\!2N$ blocks each has the same size of that within $\Omega_{k}$, the total coverage is beyond $\hat{\Omega}_{k}$. Thus, we set them smaller and uniformly distributed on $\Theta(\theta)$ and $\Phi(\phi)$ to exactly cover $\hat{\Omega}_{k}$. As such, the center directions of the grid blocks for $k$th UPA can be represented as

with piecewise $j=1,...,N$, $j=N\!+\!1,...,3N$, and $j=3N\!+\!1,...,4N$, respectively.

with piecewise $l=1,...,N$, $l=N\!+\!1,...,3N,$, and $l=3N\!+\!1,...,4N$, respectively. According to the proposed beams’ distribution as well as their coverage shown in Fig. 4, the set of grid directions/blocks covered by $\bm{\omega}_{i}^{k,s}$ can be expressed as

where $\nu$ represents the number of grid blocks at the same elevation covered by $\bm{\omega}_{i}^{k,s}$, $\delta$ represents the number of grid blocks at the same azimuth covered by $\bm{\omega}_{i}^{k,s}$, and $\mu$ represents the number of beams in stage $s$ at the same elevation. Regarding the wide beams in stage $s$, prior works [12, 13, 14] expect that $\bm{\omega}_{i}^{k,s}$ can only achieve beam gain within its coverage $\Upsilon_{i}^{k,s}$ and can not achieve gain in other range, i.e.,

holds true for all $j=1,2,...,4N$ and $l=1,2,...,4N$. However, it is worth mention that the feasible wide beam realized by the beamformer can not exactly fit (15), and only result in an approximate beam, which occurs notable trenches between adjacent ones. This is because the requirement of drastic jump/drop between $0$ and $1$ in (15) may squeeze the resulting beam patterns for minimizing the approximation error. These trenches bring dead zone and impair the overall performance of beam training. To eliminate them, we propose to modify the criterion given in (15) by adding a buffer zone, which is effective and will be validated in Section IV-B. The buffer zone ${\rm B}_{i}^{k,s}$ is the periphery of $\Upsilon_{i}^{k,s}$ with width of $w$. To write it in mathematical form, we first set an enlarged zone of $\Upsilon_{i}^{k,s}$, denoted by $\hat{\Upsilon}_{i}^{k,s}$ as

Then, we have ${\rm B}_{i}^{k,s}=\hat{\Upsilon}_{i}^{k,s}-\Upsilon_{i}^{k,s}$. Thus, in our proposed criterion, we expect that

where $\chi\in(0,1)$ is the expected beam gain in the buffer zone. Define a matrix as

Then, we can rewrite (17) in a more compact form as

where ${\bm{\Xi}_{s}}$ is a $16N^{2}\times 2^{s}$ matrix, which has an element of 1 in the coverage zone, an element of $\chi$ in the dead zone, and an element of $0$ in other zone. As a result, the $i$th codeword in the stage $s=0,1,...,S-1$ of $\mathcal{C}_{k}$ can be obtained as

Now, we have obtained all the codewords in $\mathcal{C}_{k}$. The narrow beams in stage $S$ are given by (13) and the wide beams in stage $s=0,1,...,S-1$ are given by (20).

In this subsection, we propose a low-complexity 3D training procedure for our considered system. We name the two terminals as Alice and Bob respectively. To find the optimal narrow-beam pair between Alice and Bob, two phases are developed to achieve different groups of measurements. Due to the page limit, we omit the details in the description, and the overall procedure is sketched in Fig. 6 and Fig. 7.

We first consider the beam patterns of our proposed narrow beams, which are in the bottom stage of the hierarchical book. For comparison, we present two benchmarks as follows.

• •

  Uniform Real Angles: For the $k$th UPA, we extend the method in [15] by setting $N$ azimuth angles uniformly distributed in $[-\frac{\pi}{4}+(k-1)\frac{\pi}{2},\frac{\pi}{4}+(k-1)\frac{\pi}{2}]$ and $N$ elevation angles uniformly distributed in $[\frac{\pi}{4},\frac{3\pi}{4}]$.

• •

  Uniform Virtual Angles: For the $1$th UPA, we consider the simplified array response vector with virtual angles (also named spatial angles), i.e.,

  $$\begin{split}&{{\bf{a}}_{1}}(\widetilde{\phi},\widetilde{\theta})=\frac{1}{{\sqrt{{N_{a}}}}}[1,...,{e^{j\pi({n_{y}}\widetilde{\phi}+{n_{z}}\widetilde{\theta})}},\\ &\qquad\qquad\qquad\qquad...,{e^{j\pi[({N_{y}}-1)\widetilde{\phi}+({N_{y}}-1)\widetilde{\theta}]}}]^{T},\end{split}$$

  (21)

  where $\widetilde{\phi}$ and $\widetilde{\theta}$ are the virtual azimuth and elevation angles within $[-1,1]$. We set $N$ virtual azimuth angles and $N$ virtual elevation angles uniformly distributed $[-\frac{{\sqrt{2}}}{2},\frac{{\sqrt{2}}}{2}]$. For the $k$th UPA ($k=2,3,4$), we rotate the beam patterns of the $1$th UPA $(k-1)\frac{\pi}{2}$ in azimuth.

Fig. 8 plots the narrow-beam patterns of our proposed distribution and the benchmarks, where each UPA has $8\times 8$ antenna elements and uses $8\times 8$ narrow beams to cover its range. It is observed that the trenches of the narrow beams with uniform real angles are deeper than those with our proposed distribution. Note that the azimuth coverage of the narrow beams of benchmarks expand at high elevation angle, which cause a certain overlap between adjacent UPAs. This indicates that our proposed narrow beams have better worst-case performance, and yield more uniform beam patterns, especially for the beams at the edge of the coverage of each UPA.

Next, we consider the beam patterns of our proposed wide beams $\{\bm{\omega}_{i}^{k,s}\}_{s=0}^{S-1}$, which are in the stage $0$ to $S-1$ of the hierarchical book. For comparison, we extend the inverse approach adopted in [12] and [14] to the 3D scenario as a benchmark, the wide beam $\bm{\omega}_{i}^{k,s}$ are designed for covering the range $\Upsilon_{i}^{k,s}$.

Fig. 9 plots the wide-beam patterns in stage $0$ to $3$ of our proposed codebook and the benchmark, where each UPA has $16\times 16$ antenna elements and the codebook has $16\times 16$ narrow beams in the bottom stage. It is observed that in stage $0$, the wide beams realized by both approaches all have notable trenches between adjacent UPAs. In contrast, the trenches of our proposed wide beams are relatively smaller. In stage $1$ to $3$, the wide beams realized by the benchmark have remarkable trenches even within the coverage range of each UPA. By comparison, there are no trenches within it in the patterns of our proposed wide beams. The beam patterns imply that using our proposed wide beams will have less dead zone during the beam training, and thus are expected to achieve better performance.

To evaluate the performance of the hierarchical beam training, we consider looking into the average/worst-case performance and the successful alignment rate. With a successful alignment, the final performance is determined by the narrow beams in the bottom stage. Fig. 10 shows the normalized performance, i.e., normalized beam gain, of using different narrow beams, in which the worst-case performance is presented as a baseline. As can be seen, the performance of all schemes increases with the number of narrow beams, and our proposed narrow beams outperform the benchmarks.

Fig. 11 shows three sets of results (denoted by different colors, respectively) of the successful alignment rate by using our proposed hierarchical codebook as well as the benchmark shown in Fig. 9. It can be observed that both schemes can achieve 100% successful alignment rate with sufficiently high SNR, and our proposed codebooks can outperform the benchmark codebook in different setups. Thus, our proposed scheme can find a high-quality beamforming solution, i.e., the optimal narrow-beam pair, even in the severe-path-loss THz channel, and the performance is comparable compared to optimal beamforming.