Sensing-Aided Hybrid Precoding for Efficient Terahertz Wideband Communications in Multiuser High-Data-Rate IoT

Terahertz (THz) massive MIMO with wideband hybrid precoding has been considered one of the crucial techniques to compensate for the high-path loss in 6G high-data-rate Internet of Things (IoT). However, the beam split in wideband hybrid precoding makes the beam of different subcarriers aiming at different directions, which results in only partial channel state information (CSI) from the users to the BS. The efficiency of the CSI-based terahertz (THz) wideband beamforming scheme which is more efficient than the hardware-based scheme in narrow-band would degrade severely. To address the degradation, in this article, we first propose a sensing-aided THz wideband hybrid precoding which restores the full CSI. Through sensing and deducing the angle-frequency information, we construct a channel-selecting matrix and inverse the full CSI from our complete channel dictionary. Moreover, in order to satisfy the multiuser access requirements in IoT, we also propose dynamic radio frequency (RF) chains and dynamic power allocation schemes to further enhance the performance in multiuser scenarios based on a new precoding perspective in which each RF chain serves only one user. This benefits from the highly sparse THz channel characteristic. The spectral efficiency and energy efficiency are employed to validate that the proposed is efficient. The numerical results demonstrate that our proposed sensing-aided wideband hybrid precoding scheme achieves similar performance to the optimal precoding and much better performance to the true time delay scheme and the full CSI-based scheme.

Abstract-Terahertz (THz) massive MIMO with wideband hybrid precoding has been considered one of the crucial techniques to compensate for the high-path loss in 6G high-data-rate Internet of Things (IoT).However, the beam split in wideband hybrid precoding makes the beam of different subcarriers aiming at different directions, which results in only partial channel state information (CSI) from the users to the base station.The efficiency of the CSI-based THz wideband beamforming scheme which is more efficient than the hardware-based scheme in narrow-band would degrade severely.To address the degradation, in this article, we first propose a sensing-aided THz wideband hybrid precoding which restores the full CSI.Through sensing and deducing the angle-frequency information, we construct a channel-selecting matrix and inverse the full CSI from our complete channel dictionary.Moreover, in order to satisfy the multiuser access requirements in IoT, we also propose dynamic radio frequency (RF) chains and dynamic power allocation schemes to further enhance the performance in multiuser scenarios based on a new precoding perspective in which each RF chain serves only one user.This benefits from the highly sparse THz channel characteristic.The spectral efficiency and energy efficiency are employed to validate that the proposed is efficient.The numerical results demonstrate that our proposed sensing-aided wideband hybrid precoding scheme achieves similar performance to the optimal precoding and much better performance to the true time delay scheme and the full CSI-based scheme.

I. INTRODUCTION
W ITH the unprecedented popularity of smart devices and the rapid development of the Internet of Things (IoT), 6G will require a Tbps-level data rate, which is far beyond the capacity limit of millimeter-wave (mmWave) bands (30-100 GHz) within 5G [1], [2].Compared with the mmWave band, the terahertz (THz) band (0.1-10 THz) can provide abundant frequency spectrum resources to support Tbps-level data rates [3], [4].Therefore, THz communication has been considered as one of the crucial technologies to achieve Internet of Everything for future 6G networks [5], [6], [7], [8].Nevertheless, it is well known that the THz electromagnetic wave exhibits high-path loss [1], [9].Massive multiple inputmultiple output (MIMO) can focus transmitting power in a specific direction to compensate for the high loss in the THz band without increasing the transmitting power [10], [11], and it is employed in the THz band communications [12].Traditional MIMO adopts a fully digital structure, i.e., each radio frequency (RF) chain is connected with one antenna.Due to the abundant antennas in THz MIMO, the fully connected structure will result in unaffordable hardware costs and power consumption.To address this issue, a hybrid analog and digital structure is proposed, which reduces the number of RF chains significantly as well as satisfies the demand for multiuser communication in IoT [10], [13].

A. Prior Work
Over the years, various hybrid precoding schemes have been proposed to generate high-gain beam in the mmWave and the THz band [14], [15], [16], [17], [18], [19], [20], [21], [22].Most of the schemes are only applicable to narrow-band communication systems.However, with the ultra bandwidth characteristic of THz massive MIMO systems, the narrow band assumption is no long suitable for THz hybrid precoding.Because the analog beamformer would make beams at different subcarrier frequencies point to different directions, which is called THz beam split [23].Considering the wideband influence, several dynamic-subarray structures were proposed to generate a near digital precoding performance, while it brought an additional array loss and the hardware complexity is greatly increased [16], [24], [25].Meanwhile, the adaptive subarray also brought additional array loss and hardware costs.To solve the beam split fundamentally, a hybrid true-time 2327-4662 c 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
delayers (TTD)-based delay-phased precoding architecture was proposed in [23], which can reduce antenna gain loss over the whole bandwidth.However, the hybrid structure of TTD and phase shift networks makes the precoding algorithm more complicated and the power consumption is extremely increased.Considering this, Yan et al. [26] utilized a group of fixed TTD to replace TTD, which is much more low-cost than the structure based on TTD.However, fixed TTD can only process a fixed delay, which means it needs switch networks to choose the adaptable delay.And the structure of fixed TTD also has limited accuracy which depends on the number of fixed TTD.The works mentioned above mostly give solutions and corresponding algorithms from the hardware structure.In practice, these tasks are still challenging and hard to achieve because of the limitations of existing technological processes.And the additional hardware will cause extra power consumption, which is unacceptable, especially in the case of high loss in the THz band.So it is attractive to make full use of channel state information (CSI) for hybrid precoding without extra hardware cost and power consumption.Earlier works on hybrid precoding are based on the premise that the perfect CSI is known and the bandwidth is mostly narrow [14], [18], [19], [20], [21], [27], [28].Chen et al. [29] proposed a hybrid precoding scheme for IoT which is based on spatial lobes division method and maximum ratio combining-based diversity combining scheme.But the bandwidth is limited to 100MHz.Several works are oriented toward wideband MIMO systems.The scheme in [30] considered a wideband MIMO system but still utilized the full CSI of the whole bandwidth and constructs the common analog matrix for all subcarriers.However, with beam split and high loss of THz wideband MIMO system, it is challenging acquiring the full CSI perfectly.Chen et al. [31] also considered the influence of beam split and proposed a wideband hybrid precoding scheme which exploits the long-term channel's covariance matrix and the Angle of Departure (AoD) information.Similarly, due to the effects of beam split, the long-term channel's covariance matrix is also hard to obtain.Although several channel estimation techniques for hybrid precoding have been proposed in the last few years [32], [33], [34], wideband THz channel estimation remains an arduous task due to the limitation of hybrid structure.Some prior works also consider hybrid precoding based on partial CSI.In [35], a novel practical subspace construction (SC) algorithm based on partial CSI is proposed to estimate the required spatial covariance matrix instead of full CSI to complete hybrid precoding.Pradhan et al. [36] utilized a min-max nonconvex optimization to obtain a fully digital precoder first and then design a hybrid precoding algorithm based on the gradient-projection method.A limited feedback strategy via exploiting the dominant channel subspace is proposed in [37] to overcome the influence of imperfect CSI.Sun et al. [38] made use of the principal component analysis (PCA) method to acquire simplified CSI.Most aforementioned methods are based on narrow-band systems, which are not applicable in the THz band.Besides, the optimization line of all prior works is to fully excavate the information in partial CSI.But with limited feedback, it is extremely difficult to design a reliable hybrid precoding scheme.So in this article, our purpose is to dig out the internal connection between the subchannel spaces caused by beam squint and supplement the received partial CSI to improve the spectral efficiency and energy efficiency.
Motivated by the above discussion, in this article, we focus our attention on hybrid precoding with partial CSI in THz wideband communication.Different from the researches before, our technical route is to supplement partial CSI instead of using partial CSI for precoding directly.We first sense the frequency and angle of received subcarriers which exploits the angular and frequency domain properties of wideband THz channels.Utilizing the sensed value, a CSI reconstruction scheme is also proposed to supplement the partial CSI and complete the hybrid precoding.Furthermore, we expand our sensing-aided hybrid precoding based on partial CSI to a multiuser scenario which is necessary for IoT, and propose an efficient beam split solution for multiuser wideband THz massive MIMO systems.The proposed solution for the multiuser is high-efficient to support IoT.In this article, our main contributions to this article can be summarized as follows.

1)
We propose a sensing-aided THz wideband hybrid precoding scheme based on partial CSI.Specifically, we present a 2-D sensing algorithm to obtain the frequencyangle information of the received partial split beams.Then, according to the angular frequency relation of the subcarriers, the angle and frequency inversion of the subcarrier is carried out to obtain all the angular frequency information, and the channel selection matrix is constructed.Finally, we establish a THz complete channel dictionary containing frequency and angle information.With the constructed channel selection matrix, we retrieve the nearly complete CSI in the complete channel dictionary, and use the nearly complete CSI to complete the THz wideband hybrid precoding which obtains a near-optimal precoding performance.2) Based on the strong sparse channel characteristics of THz massive MIMO, we propose a dynamic RF access scheme in the multiuser beam split scenarios of IoT and make improvements to traditional hybrid precoding model.Considering the different channel characteristics between each user and the base station (BS), we obtain the main orthographic subchannels of each user by the method of PCA (these orthographic subchannels contain more than 90% channel information).With the main subchannels of different users, we consider two different application scenarios: a) actual number of RF chains > users' RF chain demand and b) actual number of RF chains < users' RF chain demand.When actual number of RF chains > users' RF chain demand, the RF chain is allocated using a demand-based allocation scheme for different users and the excess RF chains are closed, which greatly reduce the power consumption.In another case, a greedy allocation method is proposed to maximum the mutual information under the premise of satisfying the minimum demands of each user.Our scheme effectively improves spectral efficiency and energy efficiency in multiuser beam split scenarios.3) We propose a dynamic power allocation scheme for further improving all users reachable rate sum.As the different subcarriers possess distinct subspace in THz wideband massive MIMO systems, the subcarrier channel qualities also exist differences.As far as we know, no work has been done aimed at subcarrier power allocation in multiuser hybrid precoding.Therefore, we optimize the power allocation of all users' subcarriers to maximize the total reachable rate.Considering the asymptotic orthogonality of THz massive MIMO channels, the coupled power allocation problem is transformed into a convex optimization problem by decoupling the power allocation factors between different users, and the closed-form solution of the optimal solution is derived by the Lagrange multiplier method.Finally, a low-complexity half algorithm is designed to solve the numerical optimal solution.4) We evaluate the performances of our proposed hybrid precoding algorithm through numerical simulation.
The simulation results demonstrate that the proposed sensing-aided wideband THz hybrid precoding based on partial CSI can achieve similar performance to the scheme relying on full CSI.Our proposed scheme achieves almost 50% improvement comparing with the TTD scheme which do not take partial CSI into account.
With the increase in the number of antennas and RF chains, our scheme has always had great advantages.When extended to multiuser scenarios, our dynamic RF chain allocation and dynamic power allocation schemes still achieve significant advantages in terms of energy efficiency performance, which proves the effectiveness of the proposed schemes.The remainder of this article is organized as follows.Section II presents our system model, channel model, power consumption, and some of our assumptions.The sensingaided hybrid precoding scheme is introduced in Section III.In Section IV, two dynamic allocation schemes are proposed to expand the sensing-aided hybrid precoding scheme to multiuser scenarios.Section V discusses our simulation results.Finally, we conclude the work of this article in Section VI.
Notations: Uppercase boldface and lowercase boldface stand for matrices and vectors, respectively.Superscripts (•) T , (•) H , and (•) −1 denote the transpose, conjugate transpose, and matrix inversion.• F and • 2 represent F norm operation and two norm operations, respectively.| • | denotes the determinant of a matrix or the module of a vector.diag{a} denotes the diagonal matrix with the elements of a. [A] i,j is the (i, j) t h element of A. CN (a, A) denotes a complex Gaussian vector with mean a and covariance A. A(i, :) and A(:, j) represent the i th row of A and the j th column of A, respectively.• is the ceiling integer function.I N denotes the identity matrix with size of N × N.
denotes the Hadamard product.

II. SYSTEM MODEL, CHANNEL MODEL, AND POWER CONSUMPTION
In this section, we first describe the system model and channel model of the THz wideband massive MIMO system in a fixed multiuser communication environment.Then the power consumption of the system is analyzed in the following part.

A. System Model
As shown in Fig. 1, we consider a wideband MIMO system with orthogonal frequency-division multiplexing (OFDM) modulation aided in THz IoT scenario, which is set to a f c center frequency and K subcarriers.The BS with a uniform linear array (ULA) of N t antennas serves U users (USs) simultaneously.Each US is equipped with N r antennas.Respectively, the number of RF chains is N t RF and N r RF .We construct a new hybrid precoding structure with a switching network between the baseband and the RF chains, which means N t RF can be adjusted dynamically according to actual demand.The number of switches N SW is equal to N t RF .The fully connected structure is adopted at both the BS and the USs in our work, where each RF chain is shared with all antennas.The signal is transmitted from BS to USs through N s data streams.Considering a hybrid structure at the base, the number of antennas is always much larger than the number of RF chains, which satisfies N t RF N t and N r RF N r .Moreover, for multiple streams transmission, the number of data streams has to be smaller than RF chains and be larger than the users, which satisfies It should be noticed that our hybrid precoding model is different from any other before because we consider the highly sparse characteristic of the THz channel.With the highly sparse channel, the zero-forcing precoding is difficult to eliminate multiuser interference because of matrix singularity.Therefore, we adopt the method that different RF chains serve different users to avoid the interference of multiuser precoding.This ensures that data streams are transmitted to the designated users.Next, we will introduce our hybrid precoding model applicable to our dynamic RF chain allocation.
In hybrid precoding structure, the N s data streams signals RF , where f RF,u represents the analog precoding matrix for the u th user and u ∈ {1, . . ., U}.It should be noted that the analog precoding is frequency-flat, which means F RF provides the same phase shift for all subcarriers.Another limitation imposed by the phase shift array is the constant modulus (CM) constraint of F RF which satisfies |[F RF ] i,j | 2 = 1.Therefore, the transmitted signal x u [k] for the u th user can be expressed as where s u [k] is the (N s,u × 1)-element original signal conveyed by the k th subcarrier.The original signal is imposed according To the comparison of different hybrid precoding schemes, the precoding power is normalized according to Transmitted over a broadband channel, the transmitted signal is decoded to be received by the US, which can be formulated as where H u [k] is the subchannel matrix of the k th subcarrier between the transmitter and the receiver whose size is N r ×N t , n[k] is the additive white Gaussian noise vector satisfying CN (0, σ 2 n ) and ρ is the received power of each subcarrier which considers the path loss.Moreover, At this point, we have completed modeling the transmitted and received signals for each user.In this article, what we are concerned about is the beam split that happens at the BS.According to (1) and the hybrid structure, the complete transmitter signal vector x[k] from the BS at the k th subcarrier can be described as ( It is obvious the multiuser interference is avoided in the precoding process.

B. Channel Model
Due to the short wavelength of THz, the transmitted signal in the THz band is subject to high loss, which leads to limited scattering.As a result, the THz wave is spread in the line of sight (LOS) path.In this work, we assume that the channel consists of 1 cluster and each cluster contains N ray propagation paths.Therefore, the u th user's channel matrix of wideband THz MIMO systems at the kth subcarrier can be expressed as where α m,u is the gain at the mth path, β m,k,u = e −j2πτ m f k denotes the delay component (τ m represents the delay of the mth path).Because of the sparsity of the THz channel, the number of paths is usually no more than 20 and the first path is always the LOS path [23].The LOS path typically exhibits a gain that is more than 10 dB higher than NLOS paths.
Reference [39] a r (φ r m,k,u ) and a t (φ t m,k,u ) are the normalized array vectors which can be formulated as [30] 1, e jπφ r m,k,u , . . ., e j(N r −1)πφ r m,k,u T (5) 1, e jπφ r m,k,u , . . ., e j(N t −1)πφ r m,k,u T (6) where φ r m,k,u and φ t m,k,u represents the phase of the mth path at the kth subcarrier, which can be presented as where f k is the frequency at the kth subcarrier, θ r m,u ∈ [0, π) and θ t m,u ∈ [0, π) are the angle of arrival (AoA) and the AoD at the BS and USs.Moreover, c is the speed of light and d = (c/2f c ) represents the distance between two array elements.

C. Power Consumption
In this part, we will introduce the power consumption of our proposed dynamic RF chains (DRF) hybrid precoding architecture and the normal hybrid precoding structure separately.
The power consumption of individual devices at 0.3 THz and the number of devices we actually used in the normal architecture and the DRF architecture are given in Table I.For Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I SIMULATION PARAMETERS
normal THz hybrid precoding, the overall power consumption is formulated as Here, P BB denotes the power consumption of the baseband, P PA the power consumption of each antenna, P PC the power consumption of each power combiner, P PS the power consumption of each phase shifter, and P RF the power consumption of the RF chains.
Considering a switching network for dynamic adjustment of RF chains.the overall power consumption of our proposed architecture can be calculated as P SW all = P BB + N t P PA + N t P PC + N RF N t P PS + N RF P SW + N RF P RF (10) where P SW represents the power consumption of each switch.N RF is the number of all accessed RF chains, which satisfies N RF = U u=1 N RF,u .Note that the number of phase shifters is related to the number of accessed RF chains.Therefore, the energy consumption of the system can be greatly reduced by reducing the number of accessed RF chains.
The purpose of introducing a switch network is to reduce the power consumption of the whole THz MIMO systems, i.e., P all ≥ P SW all .When the condition is met, the relation between 1 and 2 can be expressed as The number of N t is extremely large in THz massive MIMO systems, typically 256, 512, or even larger, while the number of N t is small, generally only a dozen.Therefore, based on the power consumption of the devices in Table I and the number of devices in the actual system, the value of [(N t P PS + P RF )/(N t P PS + P SW + P RF )] approximately equals to 1.So we can re-express the relationship between N RF and N RF as N RF ≤ N RF − 1 ( N RF and N RF are all integers).As discussed above, our proposed architecture has lower power consumption in most RF chain demand scenarios.

III. SENSING-AIDED WIDEBAND HYBRID PRECODING BASED ON PARTIAL CSI
In this section, we propose a new wideband hybrid precoding scheme based on partial CSI.We first analyze the severe effect of beam split on complete CSI acquisition in wideband hybrid precoding, which will provide strong evidence for the value of our search.Then based on the fact that it is impossible to obtain complete CSI, we introduce a sensing-aided wideband hybrid precoding scheme in the second part which enhances hybrid precoding performance by sensing angle-frequency information and inverting channel information.

A. Effect of Beam Split
In multiuser wideband THz massive MIMO systems, beam split is generated between each US and BS.To better demonstrate the effect of beam split, we focus on the beam split of a single user in a multiuser scenario, shown in Fig. 2. Considering a FDD system, the full reciprocity property is not satisfied [40].Therefore, it is impossible to estimate the full CSI of the downlink channel through the uplink channel in FDD systems.Another way to obtain the full CSI is to estimate the full CSI of the downlink channel through feedback from the user side.However, with beam split, some subcarriers are lost in the downlink transmission which means it is also impractical to estimate the full CSI of the downlink channel by the US's feedback, shown as Fig. 2.Under the influence of beam split, different beams will point to different directions when subjected to the same array vector.In order to better describe the degree of beam dispersion, we define a sub-beam as the beam at a certain carrier frequency.The beam loss level is related to the transmission distance, beam width, and the aperture of the array antenna.We assume that the US-side linear array antenna and the BS-side array antenna are perpendicular to each other.Then the received subcarriers under beam split can be approximated as [41] where B is the set of received beams and B m is the m th subbeam produced by the beam split.D is the aperture of the US array antenna.L is the distance between the US and the BS.θ m is the AoD of the m th sub-beam and θ L is the AoD of the sub-beam corresponding to the central subcarrier, as well as the direction of the US.Fig. 3 shows an example of angular domain gain plot of beam split with L = 20 m, N r = 16, We assume that the THz wideband system works at a 300-GHz center frequency with 30-GHz bandwidth.We observe that Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.with a large bandwidth, most of the subcarriers are propagated to nontarget directions and only partial subcarriers near the center frequency are received by the US, shown as Fig. 3.
The achievable rate and energy efficiency of the single user can be formulated as Due to the single-user scenario, interference between USs is not considered.According to ( 13) and ( 14), the subcarrier loss reduces the system's effective bandwidth and the channel capacity heavily which results in severe performance degradation.Meanwhile, the loss of subcarriers makes limited feedback (only partial CSI near the center frequency) from the USs to BS, which leads to the hybrid precoding based on perfect CSI being impossible to achieve.

B. Sensing-Aided Wideband Hybrid Precoding Scheme With Partial CSI
As analyzed in Sections I-A and III-A, only partial CSI can be fed back from the US with the effect of beam split and the weakness of the conventional CSI estimation, which results that existing full CSI-based schemes are no longer applicable.So in this section, we propose a new hybrid precoding scheme that senses and restores the full CSI utilizing the limited feedback.It should be noticed that our work focus on reconstructing full CSI based on partial CSI, rather than estimating the partial CSI (the partial CSI can be easily obtained from the conventional CSI estimation).Therefore, in this article, we assume that the partial CSI of the receivable subcarriers is known and has been estimated through unbiased estimation.With the restored full CSI, we achieve the wideband hybrid precoding.In the following part, we present the details of our scheme.
In order to recover the full CSI from the partial CSI, the first thing we need to do is to perform angular and frequency sensing of the beam split.Motivated by this, we first propose a wideband angle-frequency sensing algorithm.To the best of our knowledge, there is no advanced estimation algorithm that can simultaneously sense the angle and frequency of the receiving subcarriers.For a better expression, we represent the transmitting channels of all subcarriers as follows: sin θ T (16) where H t represents the transmitting channels.α m is the gain at the mth path, β m,k = e −j2πτ m f k denotes the delay component (τ m represents the delay of the mth path).s n,k is channel response of the n th element at the k th subcarrier.a k (k ∈ [1, . . ., K]) represents the array steering vector at the k th subcarrier which adopts a symmetrical expression.It should be noticed that H t is different from the expression of channel in (4).H t can be considered as a multiple-antenna transmit but single-antenna receive channel.As well known, the beam squint is generated by the transmitter, shown as Fig. 2. So the angle-frequency estimation for beam squint does not take into account the number of receiver antennas.Therefore, it is acceptable to use only the transmission channel H t .This expression can effectively reduce the complexity of the anglefrequency estimation.Moreover, it is obvious that a k can be further decomposed into two parts, one containing only angle information and the other containing both angle and frequency information.Then a k can be written as where the size of S(θ ) is N t × 1 and the size of h(θ, With the transmit channel matrix, we next estimate the angle and frequency corresponding to the partial CSI.The BS first sends sensing signal s to the US.The received sensing signal can be expressed as where Y is the received sensing signals of partial subcarriers and the s represents the sensing signal which consists of different frequency sine signals with random phases satisfying uniform distribution.E n is Gaussian noise in different frequency.The covariance matrix of the received sensing signal can be expressed as where U c is the matrix composed of eigenvectors corresponding to K larger eigenvalues and U n is the matrix composed of eigenvectors corresponding to the remaining smaller eigenvalues. According to principle of subspace and ( 17), (20), the eigenvalue matrix U n and array steering vector a k have the constraint which can be formulated as where θ c is the DoA of the center frequency.According to ( 17), ( 22) can be rewritten as We can see that C(θ c ) only includes the DoA of the center frequency while h(θ c , f k ) includes both the DoA of the center frequency and the frequency of the k th subcarrier.According to (20), h H (θ, f k ) = 0 and C(θ ) is nonnegative definite conjugate symmetric matrix.So the sufficient conditions for (21) to hold is that C(θ ) is a singular matrix.When K ≤ N t , rank{U n } is no less than N t and usually C(θ ) is a full rank matrix.Only when θ = θ c , will rank{U n } reduce which means det(C(θ )) = 0. Based on this, we can estimate the DoA of the center frequency, which can be expressed as With θc , the frequencies of the received subcarriers can be calculated as This formula represents selecting the subcarriers corresponding to the maximum value of ãH k ( θc , f k )U n U H n ãk ( θc , f k ) within a 3dB attenuation range.The estimated frequencies correspond to the frequencies of the received at the US.The whole angle-frequency information can be deduced backward from the estimated values.Obviously, according to (7) and (8) there is an approximate linearity between f k and the normalized angle φ k of the k th subcarrier in the beam split model.Based on this relationship, the whole angle-frequency information can be formulated as where φc is the normalized angle which can be written as φc = sin θc .B is the bandwidth of the downlink channel and f c is the known system center frequency.
Then we supplement the CSI depending on the obtained angle and frequency.In particular, we construct a frequencyindependent channel matrix dictionary used to deduce the channel shown in Fig. 4, which can be formulated as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where L is the angular resolution.The size of the dictionary is solely determined by the angular resolution, without considering the frequency.This keeps the dictionary dimensionality confined to one dimension, effectively reducing the complexity of the dictionary.The complete H can be chosen from D f c (θ) via a selection matrix Z, whose ith element in kth subcarrier is where I N r ∈ C N r ×N r is identity matrix.With the effect of beam split, the selection matrix Z can be expressed as It follows that the compensated CSI about all frequencies can be approximated as H is entirely retrieved from the complete channel dictionary.However, as analyzed in Section III-A, under beam split, there are still some subcarriers that can be received, indicating that the CSI of partial subcarriers can indeed be obtained in practice.We assume that the CSI of the received subcarriers is perfectly estimated, as already assumed at the beginning of this section.We substitute the subcarrier CSI that has been actually perfectly estimated for the subcarrier CSI retrieved through channel dictionary indexing.The final compensated CSI can be formulated as Invert θk and fk based on Equation ( 26), ( 27) where kmin is the minimum of { k} and kmax is the minimum of { k}.{ k} is calculated by (25).
Then we can utilize the compensated full CSI H to complete the wideband hybrid precoding.The analog precoding matrix and digital precoding matrix are designed respectively.The analog precoding is frequency-flat while the wideband channel is frequency-selective.So we construct an average channel using the compensated full CSI to apply to the wideband channel.The average channel and corresponding SVD can be formulated as [30] where U[k] ∈ C N r ×N r and V[k] ∈ C N t ×N t represent the left and the right singular component, respectively.Utilizing V[k], the F RF can be expressed as After the analog precoding, we continue the digital precoding utilizing the analog precoding matrix.The channel capacity reaches the maximum only while N s streams do not interfere with each other, in which F BB can be formulated as The algorithmic steps of our proposed approach are summarized in Algorithm 1.

C. Complexity Analysis of Angle-Frequency Estimation
In the angle-frequency estimation, the complexity is primarily manifested in three aspects: 1) the computation and SVD of the signal covariance matrix; 2) the calculation of intermediate variable C(θ ); and 3) the angle-frequency search.The complexity of computation and SVD of the signal covariance matrix can be expressed as O(N 2 t M +N 3 t ).C(θ ) is obtained by the multiplication of several matrices, and its complexity can Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

be described as O(N t (M−1)+N
What we want to get is the actual received signal bandwidth and AoD.Therefore, in the end, we determine the degree of beam split through angle and frequency searches, the complexity of which is The whole complexity can be given by O

IV. WIDEBAND HYBRID PRECODING FOR MULTIUSER
In Section III, we show a sensing-aided hybrid precoding for a single US which achieves an excellent effect in simulation.However, it is impossible for a BS to serve only one US due to the demand for massive access in 6G IoT.Considering this, in this section, we propose a dynamic allocation wideband hybrid precoding scheme which is suitable for multiuser scenarios based on the sensing-aided hybrid precoding scheme in Section III.The dynamic connected hybrid precoding structure has been proven to effectively enhance the system's performance [39], [42].In [39], a Dynamic Array-of-Subarrays (DAoSA) hybrid precoding architecture is proposed, which effectively improves the system's spectral efficiency.Another interesting work is to dynamically adjust the RF chain-to-antenna connection configuration to improve the system's energy efficiency [42].Our work takes a different approach.Upon a fully connected architecture, we control the access of RF chains and the allocation of power based on the USs' channel strengths.This approach directly enhances both energy and spectral efficiency while simultaneously reducing the number of switches required.More importantly, it can eliminate interuser interference and ensure data streams to the intended users.Our dynamic allocation can be divided into two parts-dynamic RF chain allocation and dynamic power allocation.The details of the dynamic allocation scheme will be detailed in the following part.

A. Dynamic RF Chain Allocation
Because of the high-power consumption, the number of RF chains in hybrid precoding is small, usually 4, 8, or 16.However, the number of user accesses is not a definite value in the actual application scenario of the IoT, which means that the number of RF chains required by users is greater than the actual number of RF chains at certain times (N RF,need > N RF ) while it is opposite at some moments (N RF,need < N RF ).Motivated by this, we propose a dynamic RF chain allocation scheme which considers the actual needs of USs.
Well known the sparse channel characteristic in THz, the channel can be decomposed into several orthogonal subchannels.This process is implemented through SVD according to (33).V consists of the orthogonal subchannel vector.is the eigenvalue diagonal matrix in which the magnitude of the eigenvalue corresponds to the strength of the subchannel.We observe that though the size of is great, only several values are very large.Therefore, we only distribute the RF chains to the main subchannels.Specifically, we use the idea of PCA to select subchannels that occupy more than 90% of the total channels.So that each user channel can be fully expressed through precoding.We describe our approach in two scenarios.
1) N RF,need < N RF : In this case, there are usually few USs.We can express the number of RF chains required of the u th user as The total demand for the RF chains can be expressed as As the number of RF chains is sufficient, we assign the RF chain as needed.The excess RF chains are closed controlled by the switch network.
2) N RF,need > N RF : As the number of USs becomes larger, we must consider maximizing the benefits of distribution within limited RF chains.Our purpose is to maximize the strength of selected subchannels, which can be formulated as where S H,u is the whole strength of the u th US's subchannel.The first constraint makes sure each US's connection without interruption.We define the subchannel strength of the u th US as Here, N t RF,u can be considered integer variables.As the columns of V avg,m (1 : N t RF,u ) are orthogonal unit vectors, the above formula can be further simplified as According to (41), it is easy to prove that the optimal solution of S all can be obtained by a greedy algorithm.So we just need to sort the subchannels of all the USs and take the first N t RF − U subchannels.We count the USs corresponding to the subchannels and get the optimal solution of S all .Note that the number of users cannot exceed the number of RF chains.
We summarize the dynamic RF chain allocation algorithm in this section as Algorithm 2. Calculate the optimal solution of S all by greedy algorithm.The optimal solution is the RF chain allocation scheme.8: else 9: The RF chain allocation is the demand calculated as (37) 10: end if

B. Dynamic Power Allocation
As the channel is frequency-selective, the different frequency subcarriers of the OFDM system are subject to different declines.Moreover, the fading of the different subcarriers is independent.Therefore, it is almost impossible for all subcarrier channels to have large simultaneous fading.In this article, according to the instantaneous the characteristic of subcarrier channel, we adopt an adaptive power allocation scheme to maximize the transmission rate of the system.
In a multiuser scenario, the achievable rate of the whole system can be formulated as where represents the precoding for the u th user at the k th subcarrier.The purpose of dynamic power allocation is to maximize the achievable rate of the whole system within limited transmitting power, which can be expressed as The second condition ensures that some information is carried by each subcarrier.In other words, it can ensure each user is allocated a minimum power even if all of its channels are bad.The Lagrangian optimal function of ( 43) is given by μ u,k P u,k − P min (44) where λ ≥ 0 is the Lagrangian multipliers.Before we get the KKT conditions, we first simplify the optimization problem.
As the interuser interference is not ignored, we use diagonal precoding to reduce the multiuser interference which has been shown in (3).Moreover, THz massive MIMO channel is sparse and asymptotic orthogonal as the array is large.So the interuser interference γ 2 is approximately 0 according to the properties of zero-forcing beamforming and asymptotic orthogonality [43].Based on the function (44) and the above analysis, the KKT conditions of the optimization problem ( 43) are derived to be μ u,k P u,k − P min = 0 (47) The function ( 45) is the matrix derivative.We assume that each US transmits data through half of the data streams, i.e., N s = [(N RF,u )/2].Then the derivative can be given by We get the P u,k by solving the KKT condition (51) which can be expressed as where As the KKT condition (47) requires that μ u,k and P u,k − P min cannot be 0 at the same time.When μ u,k = 0, (52) can be rewritten as When P u,k − P min = 0, P u,k can only equal P min .Combining the above two situations, P u,k can be formulated as The optional solution is similar to water filling in form.The value of λ is calculated through the condition (46).In order to speed up the convergence, we design a binary iterative algorithm.The specific solving process is shown in Algorithm 3.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.end while 13: end if 14: Calculate the P u,k via (53).15: Allocation the residual power all P u,k equally.

V. SIMULATION RESULTS AND DISCUSSION
In this section, we will present extensive numerical simulation results to prove the effectiveness of our proposed wideband hybrid precoding scheme in THz wideband multiuser massive MIMO systems.We assume a multiuser communication scenario with THz massive MIMO systems which work at 300-GHz center frequency (f c = 300 GHz).The users with simultaneous access to the BS is no more than 10 (N u ≤ 10).ULAs are deployed at the users and BS with λ/2 elementspacing where λ is the wavelength of the center subcarrier.The bandwidth varies from 0.5 to 30 GHz.The number of subcarriers is set to 128 (K = 128).Our proposed scheme is applicable to the whole path angle (−(π/2), (π/2)].For a better performance presentation, both the angle of arrival (AoA) and AoD are randomly generated between [ ± 30 • , ±60 • ] in the following simulation.Well known for the high directivity of THz wave, the angle spread in our simulation is within 3 • and the path delay is uniformly from 0-1 ns [44].In other words, the multipath is inconspicuous so the maximum of N t RF is set to 16 in our work and the N r RF is 4. The number of data streams of all users is set to 1 (N s = 1).The distance between BS and the user is 20 m, which satisfies the condition of the far field.Moreover, it should be noted that our proposed scheme and the benchmarks are all based on the fully connected structure.More details of simulation parameters can be found in Table I.
Benchmark: We highlight the advantages of our schemes from two aspects.The first aspect we want to display is the effect of beam split compensation of our scheme and the other aspect is the ability to compensate for beam split in multiuser scenarios.Notice that we mark the scheme we proposed in Section III as "SA-PC." 1) To highlight the first aspect, we set several benchmarks to analyze the performances in detail.First, the scheme based on full CSI is considered as an upper bound of our scheme (marked as "FC") [30].This scheme assumes that all CSI can be received successfully so this scheme has the best performance without adding other hardware structures.The second comparison scheme is hybrid precoding based on TTD with partial CSI (marked as "TTD-PC").The partial CSI in the TTD-based scheme is assumed to be estimated by the conventional CSI estimation without any error but only about the received subcarriers.The number of TTD is set to N t T = 16.More details about the TTD-based scheme can be found in [23].Finally, hybrid precoding based on only partial CSI is set as another comparison (marked as "PC") [30].The hybrid precoding algorithm of the third comparison scheme is the same as the first group.The only difference is that the CSI of the third comparison scheme is also assumed to be estimated by the conventional CSI estimation without any error while the first is ideal full CSI.
2) To demonstrate the advantages of our dynamic allocation scheme in multiuser scenarios, we set several benchmarks for comparison.Notice that the following benchmarks are all based on the hybrid precoding scheme proposed in Section III while the power and RF chain allocation are different.The first scheme we compare is hybrid precoding for multiuser in which the RF chains are distributed equally to each US but our proposed dynamic power allocation algorithm is used (marked as "With DP").The second comparison scheme has an equal allocation of power and RF chains, and the number of access RF chains is the same as in our scheme (marked as "No DRF-DP(Partial RF chains)").The allocation scheme of the third benchmark is the same as the second comparison but RF chains all access (marked as "No DRF-DP(Full RF chains)").

A. Performance of Sensing-Aided Hybrid Precoding in Single User Scenario
In Figs. 5 and 6, we evaluate the spectral efficiency and energy efficiency of our proposed sensing-aided wideband hybrid precoding scheme and the three benchmarks for different signal-to-noise ratio (SNR), where we have N t = 256, N r = 64, N t RF = 16, N r RF = 4, and B = 30 GHz.We can observe that the performance of our proposed sensing-aided scheme always performs better than the scheme with only partial CSI, which confirms the efficiency of our solution.We can also see that our scheme obtains a close efficiency to the scheme based on full CSI.That is because there will be only a little estimation bias in our scheme so our scheme can be close to the full CSI scheme but cannot achieve a better performance.Furthermore, it can also be observed that our scheme performs much better than the TTD-based scheme with partial CSI though the TTD-based scheme is always considered a useful line to relieve the beam squint.With SNR increasing, all scheme spectral efficiency and energy efficiency increases in the same trend.We also observe that the scheme based on full CSI and our proposed sensing-aided scheme are much better than the benchmarks based on partial CSI with no sensing, which indicates the necessity of supplementing the partial CSI by sensing.In Fig. 7, we show the comparison of different scheme performances under different bandwidth.The parameters are set the same as the last paragraph.With the increase of transmission bandwidth, we can observe that the performances of all schemes decline.However, our proposed scheme and the scheme based on full CSI decline slowly while the other schemes are fast.It is worth to be noted that the cause of performance degradation is different.Our proposed scheme and the scheme based on full CSI decline because the hybrid precoding algorithm only based on CSI cannot resolve the beam squint fundamentally.While the TTD-based scheme declines because of the CSI loss caused by beam squint.So the schemes do not fall at the same rate.
Figs. 8 and 9 illustrate the performances of different schemes in spectral efficiency and energy efficiency versus  the number of antennas at the BS, where we have N r = 64, N t RF = 16, N r RF = 4, N s = 1, and B = 30 GHz.We observe that our proposed scheme achieves better spectral efficiency and energy efficiency than the TTD-based scheme and the  scheme only based on partial CSI against different numbers of N t .Moreover, an upward trend in spectral efficiency can be found in FC and SA-PC as the number of N t increases.The schemes based on partial CSI (TTD-PC and PC) go down and then go up.That is because that the results of the two schemes are first influenced by the beam split and then benefit from the narrow beam.As the additional antennas result that the power consumption rises, all schemes show an energy efficiency degradation.But our scheme still has a higher efficiency under different numbers of antennas.In Figs. 10 and 11, we present the spectral efficiency and energy efficiency for different numbers of RF chains.We can observe that our proposed scheme always performs much better than the TTD-based scheme and the scheme only based on partial CSI.Moreover, another interesting phenomenon is found in the simulation of different numbers of RF chains.With the number of RF chains increasing, the spectral efficiency of both our proposed scheme and the full CSI-based scheme increases slowly and then becomes smooth.The performance of the TTD-based scheme and the scheme based on partial CSI are always smooth.In general, all schemes' performances are barely growing.That is because the wave  in the THz band has a low-angle spread and depends on the LOS path to transmit, which means the spacial paths is sparse.In (33) and (34), we get the orthogonal basis of CSI to precode F t RF .But with sparse spatial paths, the dimension of the orthogonal basis will be smaller than the dimension of N t RF when the number of RF chains increases.That means that it is unnecessary to set up so many RF chains.So the spectral and energy efficiency increase slowly or even keep smooth.This analysis can provide a design criterion for THz wideband hybrid precoding to reduce the system power consumption.

B. Performance of Dynamic RF Chains and Power Allocation in Multiuser Scenario
Figs. 12 and 13 illustrate the performances of dynamic allocation schemes in spectral efficiency and energy efficiency versus different SNR, where we have N u = 3, N t = 256, N r = 64, N t RF = 16, N t RF = 4, N s,u = N RF /2 and B = 30 GHz.Compared with the scheme With DP and No DRF-DP (partial RF chains), we observe that our proposed dynamic power chains always perform better than the scheme with no dynamic power allocation versus the SNR.Moreover, the spectral efficiency and energy efficiency are further promoted after adding our proposed dynamic RF chains scheme.That is because the better subchannels are selected and precoded.We also compare our dynamic allocation schemes with the scheme with more RF chains [no DRF-DP (full RF chains)].We can see that though partial RF chains are used for precoding our dynamic allocation schemes all perform better in spectral efficiency.As the comparing scheme accesses more RF chains, its energy efficiency is lowest which confirms the effectiveness of dynamic access.

VI. CONCLUSION
A sensing-aided THz wideband hybrid precoding based on partial CSI for multiuser IoT systems is proposed in this work.The sensing-aided hybrid precoding scheme can restore the full CSI from the channel-selecting dictionary by sensing the angle-frequency information after beam split and constructing a channel selection matrix.Therefore, the approximate optimal precoding utilizing the restored full CSI would be performed.Moreover, the sensing-aided hybrid precoding scheme is expanded to a multiuser scenario of IoT large-scale access through a new precoding perspective (each RF chain serves only one user) and two dynamic allocation schemes, including dynamic RF chains allocation and power allocation.The two dynamic allocation schemes can avoid multiuser interference in precoding and improve the spectral and energy efficiency within limited power.The new precoding perspective and dynamic allocation schemes can avoid multiuser interference in precoding and improve the spectral and energy efficiency within limited power.Analytical and numerical results demonstrate the necessity of the proposed sensing-aided wideband hybrid precoding technique under beam split and the effectiveness of the dynamic RF chains and power allocation schemes within limited power.

Fig. 1 .
Fig. 1.Multiuser THz downlink system model, in which a BS with hybrid digital/analog precoding structure utilizes a large antenna array to serves U USs.Each US also employs hybrid digital/analog precoding structure and only has partial CSI to the BS.
K}, where f BB u represents the digital precoding matrix for the u th user and u ∈ {1, . . ., U}.The digital precoding is frequency selective, which means signals transmitted at different frequencies are precoded by different F BB [k].Then the signals are processed by K-dim IFFT modulation.Following adding CP, the signals are conveyed to the analog phase shift array and precoded by the analog matrix

Fig. 2 .Fig. 3 .
Fig. 2. Example of beam split between the BS and one of the USs.

T 2 (
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.2k−K+1) f +fc fc

Fig. 4 .
Fig. 4. Dictionary which is used to deduce the full CSI from the received partial CSI.

32 ) 1
Algorithm Sensing-Aided Hybrid Precoding Based on Partial CSI Initialize: Input the received sensing signal Y, θ = ∅, f = ∅ and D f c (θ ).Output: F RF , F BB 1: Sense the frequency f and angle θc of Y via (24) 2: Based on the previous sensing results, calculate the AoA θc of center frequency using LS method.3: for all k ∈ {1, ..., N f } do 4:

Algorithm 2
Dynamic RF Chains Allocation Initialize: Input the restored full CSI H avg,m and the number of RF chains N t RF .Output: {N t RF } 1: for all k ∈ {1, ..., N f } do

Algorithm 3
Dynamic Power AllocationInitialize: Input the full CSI {H u [k]}, the precoding matrix {F u,k } and the total power P all .Output: {P u,k }.1: Calculate Q u,k = H u [k]F u,k2 of each sub-carrier for each US and sort Q u,k in ascending order (Q sort = {Q u,k } sort ).2: Initialize the sum power P sum = P all , λ = Q s ort(1)flag down = U × K and count = 1.3: if all P u,k ≥ P all then