Delay-Phase Precoding for Wideband THz Massive MIMO
11 Delay-Phase Precoding for Wideband THzMassive MIMO
Linglong Dai, Jingbo Tan, and H. Vincent Poor
Abstract
Benefiting from tens of GHz bandwidth, terahertz (THz) communication is considered to be apromising technology to provide ultra-high speed data rates for future 6G wireless systems. To com-pensate for the serious propagation attenuation of THz signals, massive multiple-input multiple-output(MIMO) with hybrid precoding can be utilized to generate directional beams with high array gains.However, the standard hybrid precoding architecture based on frequency-independent phase-shifterscannot cope with the beam split effect in THz massive MIMO systems, where the directional beamswill split into different physical directions at different subcarrier frequencies. The beam split effect willresult in a serious array gain loss across the entire bandwidth, which has not been well investigatedin THz massive MIMO systems. In this paper, we first reveal and quantify the seriousness of thebeam split effect in THz massive MIMO systems by analyzing the array gain loss it causes. Then,we propose a new precoding architecture called delay-phase precoding (DPP) to mitigate this effect.Specifically, the proposed DPP introduces a time delay network as a new precoding layer betweenradio-frequency chains and phase-shifters in the standard hybrid precoding architecture. In this way,conventional phase-controlled analog beamforming can be converted into delay-phase controlled analogbeamforming. Unlike frequency-independent phase shifts, the time delay network introduced in the DPPcan realize frequency-dependent phase shifts, which can be designed to generate frequency-dependentbeams towards the target physical direction across the entire THz bandwidth. Due to the joint control ofdelay and phase, the proposed DPP can significantly relieve the array gain loss caused by the beam spliteffect. Furthermore, we propose a hardware structure by using true-time-delayers to realize frequency-dependent phase shifts for realizing the concept of DPP. Theoretical analysis and simulation results show
A part of this paper was presented in the IEEE Global Communications Conference (GLOBECOM’19) [1].Linglong Dai and Jingbo Tan are with the Beijing National Research Center for Information Science and Technology(BNRist) as well as the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (E-mails:[email protected]; [email protected]).H. Vincent Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA (E-mail:[email protected]).This work was supported in part by the National Key Research and Development Program of China (Grant No.2020YFB1807201), in part by the National Natural Science Foundation of China (Grant No. 62031019), and in part by the U.S.National Science Foundation under Grants CCF-0939370 and CCF-1908308. a r X i v : . [ c s . I T ] F e b that the proposed DPP can significantly mitigate the beam split effect across the entire THz bandwidth,and it can achieve near-optimal achievable rate performance with higher energy efficiency than theconventional hybrid precoding architecture. Index Terms
THz communication, massive MIMO, hybrid precoding, beam split.
I. I
NTRODUCTION
Because of the tenfold increase in bandwidth it can provide, terahertz (THz) communicationis considered to be a promising technology to support the rapid growth of the data traffic infuture 6G wireless systems [2]–[7]. Compared with the typical bandwidth of several GHz in themillimeter-wave (mmWave) band ( - GHz) for 5G, the THz band ( . - THz) for 6G iscapable of providing at least GHz bandwidth or even much larger [6], [8]. However, THzsignals suffer from severe propagation attenuation due to the very high carrier frequencies, whichis a major obstacle to practical THz communications [2]. Massive multiple-input multiple-output(MIMO), which utilizes a large antenna array to generate directional beams with high arraygains, can be used to compensate for such severe signal attenuation in THz communications.Consequently, THz massive MIMO is very promising for future 6G wireless communications[9]–[11]. Similar to mmWave massive MIMO, hybrid precoding has been recently consideredfor THz massive MIMO to relieve the substantial power consumption of THz radio-frequency(RF) chains [12]. The key idea of hybrid precoding is to decompose the high-dimensional fully-digital precoder into a high-dimensional analog beamformer realized by phase-shifters (PSs) anda low-dimensional digital precoder [13]. Thus, a significantly reduced number of RF chains canbe used. Thanks to the sparsity of THz channels [14], a small number of RF chains are stillsufficient to fully achieve the multiplexing gain in THz massive MIMO systems [13]–[15].
A. Prior works
In the conventional hybrid precoding architecture, the analog beamformer will generate direc-tional beams aligned with the physical directions of the channel path components to realize thefull array gain [15]. Such an analog beamformer works well for narrowband systems. However,for the wideband 5G mmWave massive MIMO systems, the beams at different subcarrier fre-quencies will point to different physical directions due to the use of the frequency-independent
PSs [15], which results in an array gain loss. To deal with the array gain loss incurred by thiseffect, called beam squint [16], several methods has been proposed for mmWave massive MIMOsystems [17]–[20]. Specifically, the hybrid precoding problem in orthogonal frequency divisionmultiplexing (OFDM) based wideband massive MIMO systems was formulated in [17], wherea near-optimal closed-form solution was developed. Aiming at improving the performance ofhybrid precoding, an alternating optimization algorithm was proposed in [18], which iterativelyoptimized the analog beamformer and digital precoder to achieve near-optimal achievable rateperformance across the entire bandwidth. In addition, codebooks containing beams with widebeamwidths were designed to reduce the array gain loss caused by the beam squint effect in[19], [20]. Specifically, the wide beams were designed in [19] by maximizing the minimum arraygain achieved at all subcarriers, while, a semidefinite relaxation method was utilized in [20] tomaximize the total array gain across the entire bandwidth achieved by the wide beams. Thesemethods [17]–[20] are effective for improving the achievable rate performance, as the beamsonly slightly squint and the array gain loss is not serious in wideband mmWave massive MIMOsystems.However, the methods of [17]–[20] are not valid for wideband THz massive MIMO systems.Due to the much wider bandwidth of THz signals and much larger number of antennas to be used,beams at different subcarriers will split into totally separated physical directions. This effect,called beam split in this paper, which is a key difference between mmWave and THz massiveMIMO systems, brings a new fundamental challenge for THz communications. Specifically,unlike the situation under the beam squint effect where the beams can still cover the user acrossthe entire bandwidth, the beams generated by frequency-independent PSs can only be alignedwith the target user over a small portion of all subcarriers around the central frequency due tobeam split effect. This indicates that only the beams around the central frequency can achievehigh array gain, while the beams at most other subcarriers suffer from a serious array gain loss.Therefore, the beam split effect will result in a severe achievable rate degradation, and counteractthe achievable rate gain benefiting from the bandwidth increase in THz massive MIMO systems.To our best knowledge, the beam split effect has not been investigated in THz massive MIMOsystems, and there are no existing solutions to this fundamental challenge.
B. Our contributions
In this paper, we first analyze the performance loss caused by the beam split effect, and thenwe propose the delay-phase precoding (DPP) architecture to mitigate the beam split effect inTHz massive MIMO systems. Specifically, the contributions of this paper can be summarized asfollows. • We first reveal and quantify the beam split effect, i.e., the THz rainbow, in wideband THzmassive MIMO systems. Specifically, the relationship between the array gain loss and thesystem parameters, including the bandwidth, the central carrier frequency, path directions,and the number of antennas, is analyzed. Based on this analysis, we define a metric calledthe beam split ratio to evaluate the degree of the beam split effect, which clearly showshow serious the beam split effect is in wideband THz massive MIMO systems. • We propose a new precoding architecture, namely DPP, to mitigate the beam split effect.In the proposed DPP, a new precoding layer, which is a time delay (TD) network, isintroduced between the RF chains and frequency-independent PSs in the conventional hybridprecoding architecture. The PSs are still used to generate beams aligned with the targetphysical direction, while the time delays in the TD network are designed to make thebeams aligned with the target physical directions across the entire bandwidth. In this way,the DPP architecture can convert frequency-independent phase controlled beamforming into frequency-dependent delay-phase controlled beamforming, which can significantly alleviatethe array gain loss caused by the beam split effect. • We further prove that beams generated by the DPP architecture are aligned with the targetphysical direction at different subcarriers, and they can achieve near-optimal array gainacross the entire bandwidth. • Finally, a hardware structure called true-time-delayers based DPP (TTD-DPP) is proposedto realize the concept of DPP, where the TD network is realized by a small number ofTTDs between the RF chains and the PS network. The analysis illustrates that the proposedTTD-DPP structure is able to achieve a near-optimal achievable rate, which is supportedby extensive simulation results.
C. Organization and notationOrganization:
The remainder of the paper is organized as follows. In Section II, a systemmodel of wideband THz massive MIMO with the conventional hybrid precoding architecture
Baseband Processing RF ChainRF ChainMultiplestreams Analog BeamformerDigitalPrecoder
Fig. 1. The classical hybrid precoding architecture [15]. is provided. In Section III, we analyze the array gain loss caused by the beam split effect,and propose the DPP architecture, together with the array gain performance analysis. Then, ahardware structure based on true-time-delayers is proposed to realize the concept of DPP inSection IV. In Section V, the simulation results are provided. Finally, conclusions are drawn inSection VI.
Notation:
Lower-case and upper-case boldface letters represent vectors and matrices, respec-tively; ( · ) T , ( · ) H , (cid:107) · (cid:107) F , and (cid:107) · (cid:107) k denote the transpose, conjugate transpose, Frobenius norm,and k -norm of a matrix, respectively; H [ i,j ] denotes the element of matrix H at the i -th rowand the j -th column; E ( · ) denotes the expectation; | · | denotes the absolute value; I N representsthe identity matrix of size N × N ; blkdiag( A ) denotes a block diagonal matrix where eachcolumn of A represents the diagonal blocks of the matrix blkdiag( A ) ; CN ( µ, Σ ) denotes theGaussian distribution with mean µ and covariance Σ ; and finally, U ( a, b ) represents the uniformdistribution between a and b .II. S YSTEM M ODEL OF TH Z M ASSIVE
MIMOWe first consider a THz massive MIMO system with the conventional hybrid precoding asshown in Fig. 1. The base station (BS) employs N RF RF chains and an N t -antenna uniform lineararray (ULA) . An N r -antenna user is served and N s data streams are transmitted simultaneouslyby the BS (usually we have N s = N r ≤ N RF (cid:28) N t ). To realize wideband transmission,the widely used orthogonal frequency division multiplexing (OFDM) with M subcarriers is In this paper, we consider the ULA for simplicity, but the analysis of the beam split effect and the correspondingly proposedDPP architecture can be easily extended to the uniform planar array (UPA) [21], which has similar channel form as ULA. considered. The downlink received signal y m ∈ C N r × at the m -th subcarrier ( m = 1 , , · · · , M )can be expressed as [15] y m = √ ρ H Hm AD m s m + n m , (1)where H m ∈ C N t × N r denotes the frequency-domain channel at the m -th subcarrier, A ∈ C N t × N RF ,with constraint | A [ i,j ] | = √ N t due to the use of frequency-independent PSs [22], is the frequency-independent analog beamformer identical over all M subcarriers, D m ∈ C N RF × N s is the frequency-dependent digital precoder at the m -th subcarrier satisfying the transmission power constraint (cid:107) AD m (cid:107) = N s , s m ∈ C N s × represents the transmitted signal at the m -th subcarrier withthe normalized power E ( s m s Hm ) = N s I N s , ρ is the average received power, and n m ∈ C N s × denotes the additive white Gaussian noise (AWGN) at the m -th subcarrier following the Gaussiandistribution CN (0 , σ I N s ) with σ being the noise power.In this paper, we consider the widely used wideband ray-based channel model [9] for THzcommunications. We denote f c as the central frequency and B as the bandwidth. Then, thetime-domain channel h n t ,n r between the n t -th antenna of the BS and the n r antenna of the userwith n t ∈ , , · · · , N t and n r ∈ , , · · · , N r can be denoted as h n t ,n r = L (cid:88) l =1 g l δ ( t − τ l − ( n t − dc sin ˜ θ l − ( n r − dc sin ˜ φ l ) , (2)where L denotes the number of resolvable paths, g l and τ l represent the path gain and pathdelay of the l -th path, ˜ θ l , ˜ φ l ∈ [ − π/ , π/ are the frequency-independent physical directions ofthe l -th path at the BS side and the user side respectively, d is the antenna spacing usually setaccording to the central frequency f c as d = λ = c f c with λ denoting the wavelength at thecentral frequency f c and cannot be changed after the antenna array has been fabricated, and c denotes the light speed in the free space. In (2), ( n t − dc sin ˜ θ l and ( n r − dc sin ˜ φ l denote thetime delays caused by the physical directions of the l -th path at the n t -th antenna of the BS andthat at the n r -th antenna of the user, respectively.For the m -th subcarrier with the frequency f m = f c + BM ( m − − M − ) , the frequency-domainchannel H m can be presented by the discrete Fourier transform (DFT) of the time-domain channelin (2) as H m = L (cid:88) l =1 g l e − j πτ l f m f t (¯ θ l,m ) f r ( ¯ φ l,m ) H , (3)where f t (¯ θ l,m ) and f r ( ¯ φ l,m ) are the array responses at the BS side and the user side, and they can be presented by setting d = c f c as f t (¯ θ l,m ) = 1 √ N t (cid:104) , e jπ ¯ θ l,m , e jπ θ l,m , · · · , e jπ ( N t − θ l,m (cid:105) H , f r ( ¯ φ l,m ) = 1 √ N r (cid:104) , e jπ ¯ φ l,m , e jπ φ l,m , · · · , e jπ ( N r −
1) ¯ φ l,m (cid:105) H , (4)where ¯ θ l,m and ¯ φ l,m ∈ [ − , denote the spatial directions at the BS side and the user side of the l -th path component at the m -th subcarrier, respectively. The spatial directions are the directionsof the channel path components in the spatial domain. Since (3) is obtained by the DFT of (2),we can obtain the relationship between the spatial directions ( θ l,m , φ l,m ) in (3) and the physicaldirections ( ˜ θ l , ˜ φ l ) in (2) as follows ¯ θ l,m = 2 d f m c sin ˜ θ l , ¯ φ l,m = 2 d f m c sin ˜ φ l . (5)For simplification, in this paper, we use θ l = sin ˜ θ l and φ l = sin ˜ φ l to denote the physicaldirections, where θ l , φ l ∈ [ − , .By considering the channel model (3)-(5), we will introduce the beamforming mechanism inthe next section, based on which the beam split effect in THz massive MIMO will be revealedand the DPP architecture will be proposed.III. D ELAY -P HASE P RECODING FOR TH Z M ASSIVE
MIMOIn this section, we will first introduce the beamforming mechanism in massive MIMO systems.Based on the beamforming mechanism, the beam split effect in THz massive MIMO will berevealed by analyzing the severe array gain loss caused by it. Then, to mitigate the severearray gain loss in THz massive MIMO systems, the DPP architecture will be proposed, and thecorresponding array gain performance analysis will also be provided.
A. Beamforming mechanism
Generally, in THz massive MIMO systems with the hybrid precoding architecture, the analogbeamformer is designed to generate beams towards the physical directions of the channel pathcomponents, to compensate for the severe path loss [15], while, the digital precoder is designedbased on the determined analog beamformer to realize the spatial multiplexing gain. Therefore,whether the beams generated by the beamformer can precisely point to the physical directions of c c =2 2 cd f λ= c f l sin l d θ l θ l θ Fig. 2. Beamforming mechanism. channel path components has a crucial impact on the achievable rate performance. Here, takingthe narrowband THz massive MIMO system as an example, we will introduce the beamformingmechanism.Without loss of generality, we consider the l -th path component with the physical direction θ l of the channel in (3). Usually, the l -th column of the analog beamformer A in (1), i.e., theanalog beamforming vector a l = A [: ,l ] , is used to generate a directional beam towards the l -thpath’s physical direction θ l , which has been proved to be near-optimal [15]. Specifically, thebeamforming mechanism is to make electromagnetic waves that transmitted by different antennaelements form an equiphase surface, which is perpendicular to the target physical direction θ l , as shown in Fig. 2. To achieve this goal, different phase shifts provided by PSs should becompensated at different antenna elements. For instance, the distance difference between adjacentantenna elements reaching the equiphase surface is d sin ˜ θ l = dθ l . Therefore, for narrowbandsystems where f m ≈ f c , the phase difference that should be compensated between adjacentantenna elements is − π dλ c θ l = − π dc f c θ l , where λ c denotes the wavelength at the centralfrequency f c . As a result, the analog beamforming vector a l should be a l = 1 N t (cid:104) , e − j π dc f c θ l , e − j π dc f c θ l , · · · , e − j π ( N t − dc f c θ l (cid:105) T = f t (cid:18) dc f c θ l (cid:19) = f t ( θ l ) . (6)Based on the analog beamforming vector designed in (6), the normalized array gain η ( a l , θ l , f c ) achieved by a l in the physical direction θ l at the central frequency f c is η ( a l , θ l , f c ) = (cid:12)(cid:12)(cid:12)(cid:12) f t (2 d f c c θ l ) H a l (cid:12)(cid:12)(cid:12)(cid:12) ( a ) = (cid:12)(cid:12) f t ( θ l ) H a l (cid:12)(cid:12) ≈ (cid:12)(cid:12) f t ( θ l ) H f t ( θ l ) (cid:12)(cid:12) = 1 , (7)where (a) comes from d = λ c = c f c . It is clear from (7) that by setting the analog beamformingvector a l = f t ( θ l ) , the optimal normalized array gain of can be achieved at the centralfrequency f c . Thus, considering f m ≈ f c , m = 1 , , · · · , M , the narrowband systems can enjoythe satisfying normalized array gain across the entire bandwidth B . B. Beam split effect
In wideband systems, since the PSs are frequency-independent and the antenna spacing d is setaccording to the central frequency f c , the analog beamforming vector a l is usually set the sameas (6), which is also frequency-independent . However, considering the frequency-independentphase shifts will cause frequency-dependent time delays, i.e., phase shift ∆ θ corresponds to timedelay − πf m ∆ θ at subcarrier frequency f m , the equiphase surfaces generated by the frequency-independent analog beamforming vector a l will be separated at different subcarriers. Therefore,the beams generated by the frequency-independent analog beamforming vector a l will point todifferent physical directions surrounding the target physical direction θ l at different subcarriers.This effect is called beam squint in mmWave massive MIMO systems [16]. Fortunately, sincethe beams at different subcarriers can still cover the user by their mainlobes, the array gaindegradation caused by the beam squint is small in mmWave massive MIMO systems and canbe solved by several existing methods [17]–[20].However, the beam squint effect will be severely aggregated in THz massive MIMO systemsfor two reasons. Firstly, due to the much larger bandwidth of THz massive MIMO systems,the physical direction deviations between the physical directions that the beams at differentsubcarriers are aligned with and the target physical direction will significantly increase. Secondly,the much larger number of antennas in THz massive MIMO systems leads to an extremely narrowbeamwidth. Due to the above two reasons, the beams at different subcarrier frequencies may betotally split into separated physical directions in THz massive MIMO systems as shown in Fig.3 (a). Thus, unlike the beam squint effect, an unacceptable array gain loss occurs since mostof the beams at different subcarriers cannot cover the user in their mainlobes. Unfortunately,this aggregated effect cannot be solved by the existing methods based on the classical hybridprecoding architecture [17]–[20], and it has not been well addressed in the literature. To this c f l m f l , | | l m l θ θ− Different equiphasesurface at different subcarriers Physical direction deviation Wideband signals Wideband White Light Frequency-independent PSs Beam at 𝑓𝑓 Beam at 𝑓𝑓 𝑐𝑐 Beam at 𝑓𝑓 𝑀𝑀 Beam at 𝑓𝑓 𝑐𝑐 Beam at 𝑓𝑓 𝑚𝑚 Frequency-independent PSs Tiny water dropletsin the air (a) (b)
Rainbow l θ , l m θ c c =2 2 cd f λ= l θ , l m θ Fig. 3. The mechanism of the beam split effect: (a) The beam split at subcarrier frequency f m . ˜ θ l,m denotes the physicaldirection that the beam at subcarrier frequency f m is aligned with; (b) The analogy between the beam split effect and therainbow over the large bandwidth. f and f M denote the lowest subcarrier frequency and the highest subcarrier frequency,respectively. end, we define the effect that the beams at different subcarriers are totally separated as the beamsplit effect . A simple analogy between the beam split effect and the rainbow can illustrate themechanism of the beam split effect. As shown in Fig. 3 (b), because the tiny water dropletsin the air have different refractive indices for the wideband white light composed of differentfrequencies, the pure light of different frequencies will totally separate and eventually producethe rainbow. Just like the rainbow, frequency-independent PSs cause different “refractive indices”for THz signals at different frequencies, and thus leads to totally separated beams at differentfrequencies. Therefore, we can also call the beam split effect as “ THz rainbow ”.The beam split effect will result in an unacceptable array gain loss, which is mainly deter-mined by the bandwidth and the number of antennas. Specifically, the following
Lemma 1 willtheoretically quantify the relationship between the array gain loss caused by the beam split effectand the system parameters such as bandwidth and the number of antennas. Notice that we utilize θ l,m = sin ˜ θ l,m ∈ [ − , denotes the physical direction that the beam at subcarrier frequency f m is aligned with for simplifying the expression. Lemma 1.
The beam generated by the analog beamforming vector a l = f t ( θ l ) is aligned withthe physical direction θ l,m satisfying θ l,m = θ l ξ m , where ξ m = f m f c is the normalized frequency.Furthermore, when | ( ξ m − θ l | ≥ N t , the normalized array gain η ( a l , θ l , f m ) achieved by the analog beamforming vector a l in the physical direction θ l at subcarrier frequency f m satisfies η ( a l , θ l , f m ) ≤ N t sin π N t . (8) Proof:
The normalized array gain achieved by the analog beamforming vector a l in an arbitraryphysical direction θ ∈ [ − , at subcarrier frequency f m can be denoted as η ( a l , θ, f m ) = (cid:12)(cid:12) f t (2 d f m c θ ) H a l (cid:12)(cid:12) . Then, we have η ( a l , θ, f m ) ( a ) = (cid:12)(cid:12)(cid:12)(cid:12) f t (2 d f m c θ ) H f t ( θ l ) (cid:12)(cid:12)(cid:12)(cid:12) ( b ) = 1 N t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N t − (cid:88) n =0 e jnπ ( ξ m θ − θ l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( c ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin N t π ( ξ m θ − θ l ) N t sin π ( ξ m θ − θ l ) e − j ( Nt − π ( ξ m θ − θ l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (9)where (a), (b), and (c) come from a l = f t ( θ l ) , (4), and the equation Σ N − n =0 e jnπα = sin Ntπ αN t sin π α e − j ( N − π α ,respectively. Therefore, the array gain achieved by the analog beamforming vector a l in anarbitrary physical direction θ at subcarrier frequency f m can be denoted as η ( a l , θ, f m ) = 1 N t | Ξ N t (( ξ m θ − θ l )) | , (10)where Ξ N t ( x ) = (cid:0) sin N t π x (cid:1) / (cid:0) sin π x (cid:1) is the Dirichlet sinc function [23]. It is known that theDirichlet sinc function has the power-focusing property, where the maximum is | Ξ N t (0) | = N t and the value of | Ξ N t ( x ) | decreases sharply as | x | increases [23].Denote θ l,m as the physical direction that the analog beamforming vector a l is aligned withat subcarrier frequency f m . The analog beamforming vector a l should achieve the largest arraygain in the physical direction θ l,m as θ l,m = arg max θ η ( a l , θ, f m ) . Therefore, considering theproperty of Dirichlet sinc function that the maximum is | Ξ N t (0) | = N t , the physical direction θ l,m should satisfy ξ m θ l,m − θ l = 0 according to (10). Hence, we can obtain the physical direction θ l,m that the analog beamforming vector a l is aligned with at subcarrier frequency f m shouldsatisfy θ l,m = θ l ξ m . (11)Moreover, by substituting θ = θ l into (10), the normalized array gain η ( a l , θ l , f m ) achieved bythe analog beamforming vector a l in the physical direction θ l at subcarrier frequency f m can bedenoted as η ( a l , θ l , f m ) = N t | Ξ N t (( ξ m − θ l ) | . According to the power-focusing property of theDirichlet sinc function, when | ( ξ m − θ l | ≥ N t , | ( ξ m − θ l | locates out of the mainlobe of the Dirichlet sinc function | Ξ N t ( x ) | , which means the beam generated by the analog beamformingvector a l cannot cover the user with its mainlobe at the subcarrier frequency f m , and a severearray gain loss will occur. Specifically, considering that the maximum value of | Ξ N t ( x ) | when | x | ≥ N t is (1 /N t ) sin π N t , we have | η ( a l , θ l , f m ) | = 1 N t | Ξ N t (( ξ m − θ l ) | ≤ N t sin π N t , (12)when | ( ξ m − θ l | ≥ N t , which means the beam generated by the analog beamforming vector a l cannot cover the user with its mainlobe at the subcarrier frequency f m . (cid:4) Lemma 1 has revealed two major parameters that affect the array gain loss caused by thebeam split effect, i.e., the bandwidth B and the number of antennas N t . Specifically, when thebandwidth B and the number of antennas N t are large, the condition | ( ξ m − θ l | ≥ N t canbe easily satisfied at most of subcarriers, which indicates that beams at most of subcarrierscannot cover the user with their mainlobes. As a result, the achieved array gains at most of thesubcarriers are upper bounded by N t sin π N t as shown in (12), which are extremely small with alarge number of antennas N t . This implies that the severe array gain loss will occur at most ofsubcarriers due to the beam split effect. The array gain loss caused by the beam split effect canbe explained from the physical perspective. Firstly, according to (11), the physical direction θ l,m that the beam generated by the analog beamforming vector a l is aligned with, is determined bythe subcarrier frequency f m due to ξ m = f m /f c . Hence, when the bandwidth B becomes larger,the physical direction deviation, i.e., the interval | θ l,m − θ l | between the target physical direction θ l and the actual physical direction θ l,m , increases with the subcarrier index m . Secondly, a largernumber of antennas N t will result in a narrower beamwidth as N t . Consequently, considering thelarge bandwidth B and the large number of antennas N t in THz massive MIMO systems, therewill be a large physical direction deviation and an extremely narrow beamwidth at most of thesubcarrier frequencies. Under this circumstances, the beams at different subcarriers will becometotally separated, which means the beam split effect as shown in Fig. 3 occurs. Due to the beamsplit effect, the beams at different subcarriers cannot cover the user with their mainlobes, whichwill induce a severe array gain loss.From Lemma 1 , we know that the array gain loss caused by the beam split effect is notonly determined by the bandwidth B , but also related to the number of antennas N t . Hence, itis important to define a single metric to evaluate the degree of the beam split effect. Actually, from the description above, we can conclude that the degree of the beam split effect or the arraygain loss caused by the beam split effect, is determined by the “relatively offset” between thephysical direction deviation and the beamwidth. When the relatively offset is large, beams tendto split at different subcarriers, and cannot cover the user with their mainlobes, which finallyresults in a severe array gain loss. Following this idea, we can define a metric called beam splitratio (BSR) to evaluate the degree of the beam split effect. Specifically, the BSR is defined asthe expectation of the ratio between the physical direction deviation and half of the beamwidthsfor all subcarrier frequencies and physical directions as BSR = 12 M (cid:90) − M (cid:88) m =1 | θ l,m − θ l | /N t d θ l = (cid:90) − M M (cid:88) m =1 | N t ( ξ m − θ l | d θ l , (13)where | θ l,m − θ l | is the physical direction deviation, and /N t denotes half of the beamwidth.Note that the condition | ( ξ m − θ l | ≥ N t or | N t ( ξ m − θ l | ≥ means the beam at thesubcarrier frequency f m cannot cover the user with its mainlobe, and the BSR is defined as theaverage of | N t ( ξ m − θ l | in (13). Therefore, we can suppose that if BSR > , the beams atdifferent subcarriers cannot cover the user with their mainlobes on average and the beam spliteffect occurs. On the contrary, when BSR < , the beams will slightly squint and only the beamsquint effect occurs. By using the proposed metric BSR, we can simply evaluate the degreeof the beam split effect. A larger BSR means a stronger degree of the beam split effect and amore serious array gain loss. For instance, for a THz massive MIMO system with parameters f c = 300 GHz, N t = 256 , M = 128 and B = 30 GHz,
BSR = 1 . > , which indicates thebeam split effect really happens in THz massive MIMO systems. While, for a mmWave massiveMIMO, the BSR is usually smaller than , e.g., BSR = 0 . for a mmWave massive MIMOsystem with parameters f c = 28 GHz, N t = 64 , M = 128 and B = 2 GHz. This indicates thatthe beam squint rather than the beam split happens in mmWave massive MIMO. Consideringthat the performance loss caused by the beam squint is limited compared with the performanceloss caused by the beam split, we can conclude that the beam split effect is one of the keydifferences between mmWave and THz massive MIMO systems.To better illustrate the beam split effect, we provide the normalized array gain comparisonachieved by the analog beamforming vector a l between sub-6G MIMO system, mmWave massiveMIMO system, and THz massive MIMO system in Fig. 4. The BSRs of them are . , . ,and . , respectively. We can observe from Fig. 4 that in sub-6G and mmWave systems with (a) (b) (c) Fig. 4. Normalized array gain achieved by the classical hybrid precoding architecture with analog beamforming vector a l = f t ( θ l ) with respect to the physical direction θ , where f and f M denote the lowest subcarrier frequency and the highest subcarrierfrequency: (a) Sub-6GHz MIMO system with θ l = 0 . , f c = 3 . GHz, B = 0 . GHz, N t = 16 and M = 128 ; (b) MmWavemassive MIMO system with θ l = 0 . , f c = 28 GHz, B = 2 GHz, N t = 64 and M = 128 ; (c) THz massive MIMO systemwith θ l = 0 . , f c = 300 GHz, B = 30 GHz, N t = 256 and M = 128 . BSR < , the beams at subcarrier frequencies f and f M slightly squint from the beam at thecentral frequency f c . The array gain loss across the entire bandwidth is limited. Nevertheless, forTHz massive MIMO systems, the beams at subcarrier frequencies f and f M are totally separatedfrom the beam at the central frequency f c , and point to the physical direction far away from θ l .This beam split effect will cause serious array gain loss across the entire bandwidth. To betterreveal the array gain loss caused by the beam split effect, Fig. 5 illustrates the normailized arraygain achieved by the analog beamforming vector a l at different subcarriers m for sub-6G MIMOsystem, mmWave MIMO system, and THz massive MIMO system. The system parameters arethe same as these in Fig. 4. We can see from Fig. 5 that the analog beamforming vector a l suffers from a severe array gain loss in wideband THz massive MIMO system, while, the arraygain losses in sub-6G and mmWave systems are not serious. These results are consistent withthe BSRs of these systems as . < . < < . , which indicates the beam split effectoccurs in THz system but not in sub-6G or mmWave system. Particularly, for more than of subcarriers, e.g., the subcarriers m ≤ or m ≥ in THz massive MIMO system, the userwill suffer from more than array gain loss.Such a serious array gain loss incurred by the beam split effect is not acceptable for THz
20 40 60 80 100 120
Subcarrier (m) N o r m a li z ed a rr a y ga i n Sub-6G MIMO M mWave massive MIMO THz massive MIMO
Fig. 5. Normalized array gain achieved by the classical hybrid precoding architecture with analog beamforming vector a l = f t ( θ l ) in different systems. The system parameters are the same as these in Fig. 4. communications. However, the existing hybrid precoding methods with frequency-independentPSs cannot solve this problem. In ultrawideband radar systems, the array gain loss incurredby the beam split effect can be solved by utilizing a TTD for each antenna element [24], butthis solution is unpractical for hybrid precoding, since a large number of TTDs may bring theunacceptable power consumption and high hardware cost [16]. To our best knowledge, there areno practical solutions to solve the beam split effect in THz massive MIMO systems. To fill inthis gap, in the next section we will propose a new precoding architecture called DPP for THzmassive MIMO systems. C. Delay-phase precoding (DPP)
As discussed in Subsection III-B above, due to the beam split effect, the frequency-independent beamformer generated by the frequency-independent PSs in the classical hybrid precoding ar-chitecture, will result in the severe array gain loss. In this subsection, we will propose a newprecoding architecture called DPP to solve this problem. As shown in Fig. 6, compared withthe classical hybrid precoding architecture, a TD network is introduced as a new precodinglayer between the RF chains and the frequency-independent PS network in the proposed DPP. RF ChainRF Chain Analog BeamformerTD NetworkBaseband Processing RF ChainRF ChainMultiplestreams Analog BeamformerDigitalPrecoder PSsBaseband ProcessingMultiplestreams DigitalPrecoder (a)(b)
Fig. 6. Precoding architecture comparison: (a) The classical hybrid precoding architecture for mmWave massive MIMO; (b)The proposed DPP architecture for THz massive MIMO.
Specifically, each RF chain is connected to a TD network with K TD elements , and theneach TD element is connected to P = N t K PSs in a sub-connected manner, similar to the sub-connected hybrid precoding architecture [25]. Therefore, each RF chain still connects to everyantenna element through the PSs. The TD network can realize frequency-dependent phase shiftsthrough time delays, e.g., the phase shift − πf m t can be achieved by the time delay t at thesubcarrier frequency f m . Thus, by utilizing the TD network, the proposed DPP converts thetraditional phase-controlled beamformer into delay-phase jointly controlled beamformer, whichcan realize the frequency-dependent beamforming.Without loss of generality, we consider the l -th channel path component. Since the TD networkcan provide the frequency-dependent phase shifts, we now utilize the frequency-dependent a l,m instead of the frequency-independent a l to represent the analog beamforming vector generated bythe DPP for the l -th path component at the m -th subcarrier. Specifically, the frequency-dependent K is a variable parameter in the proposed DPP architecture, which should satisfy that P = N t /K is an integer. The designprinciple of K will be discussed later. analog beamforming vector a l,m can be denoted as a l,m = blkdiag (cid:0) [¯ a l, , ¯ a l, , · · · , ¯ a l,K ] (cid:1) p l,m , (14)where ¯ a l,k ∈ C P × with k = 1 , , · · · , K denotes the analog beamforming vector realized by PSsconnected to the k -th TD element, so we have | ¯ a l,k, [ j ] | = √ N t as usual due to the constraint ofconstant modulus, and p l,m ∈ C K × composes of the frequency-dependent phase shift realizedby K TD elements. Specifically, the k -th element p l,m, [ k ] in p l,m with k = 1 , , · · · , K satisfiesthe form p l,m, [ k ] = e − j πf m t l,k , where the time delay provided by the k -th TD element is t l,k .Aiming to compensate for the severe array gain loss caused by the beam split effect, the analogbeamforming vector a l,m should generate beams aligned with the target physical direction θ l atall M subcarriers. In this way, the user can be covered by the beams across the entire bandwidth,and the near-optimal array gain can be achieved. To realize this design goal, we will firstly usethe frequency-independent PSs to generate a beam aligned with the target physical direction θ l as [¯ a Tl, , ¯ a Tl, , · · · , ¯ a Tl,K ] T = f t ( θ l ) . (15)Then, we utilize the frequency-dependent p l,m in (14) to rotate the physical direction that thebeam [¯ a Tl, , ¯ a Tl, , · · · , ¯ a Tl,K ] T = f t ( θ l ) is aligned with from θ l,m to θ l . To maintain the directivityof the frequency-dependent beam generated by the analog beamforming vector a l,m , we set p l,m share the same form as the array response in (4). Specifically, by making the frequency-dependentphase shift − f m t l,k = − ( k − β l,m with k = 1 , , · · · , K , p l,m satisfies p l,m = [1 , e − jπβ l,m , e − j πβ l,m , · · · , e − jπ ( K − β l,m ] T , (16)where we define β l,m as the direction rotation factor at the m -th subcarrier for the l -th pathcomponent. Without loss of generality, we set the value range of the direction rotation factor as β l,m ∈ [ − , , due to the periodicity of the p l,m in (16) with respect to the direction rotationfactor β l,m . By adjusting the direction rotation factor β l,m , the beam generated by the analogbeamforming vector a l,m can be made to be aligned with the target physical direction θ l at all M subcarriers. The following Lemma 2 provides the specific design principle of the directionrotation factor β l,m , where β l,m is determined based on the physical direction θ l and the subcarrierfrequency f m . Lemma 2.
When ¯ a l,k satisfies [¯ a Tl, , ¯ a Tl, , · · · , ¯ a Tl,K ] T = f t ( θ l ) as shown in (15) and p l,m =[1 , e jπβ l,m , · · · , e jπ ( K − β l,m ] T as shown in (16), the beam generated by the frequency-dependentanalog beamforming vector a l,m can be aligned with the physical direction θ opt at the subcarrierfrequency f m as θ opt = arg max θ | η ( a l,m , θ, f m ) | = θ l ξ m + β l,m ξ m P , (17) where P = N t /K , and the normalized array gain achieved by the analog beamforming vector a l,m in the physical direction θ opt is η ( a l,m , θ opt , f m ) = KN t Ξ P ( β l,m P ) .Proof : See Appendix.We can know from Lemma 2 that the direction rotation factor β l,m can change the physicaldirection of the beam from θ l,m = θξ m achieved by the classical hybrid precoding architectureto θ opt = θ l ξ m + β l,m ξ m P achieved by the proposed DPP architecture. Therefore, to compensate forthe array gain loss caused by the beam split effect across the entire bandwidth, we should make θ opt = θ l , i.e., θ l ξ m + β l,m ξ m P = θ l . (18)Then, we can easily obtain the direction rotation factor β l,m as β l,m = ( ξ m − P θ l . (19)By setting the direction rotation factor β l,m as (19), the array gain loss incurred by the beam spliteffect can be efficiently eliminated, since the beam generated by the analog beamforming vector a l,m is aligned with the physical direction θ l across the entire bandwidth for any subcarrierfrequency f m . It should be noted that as we can see from the Appendix, the beamwidths ofbeams generated by a l,m is approximately decided by | Ξ K ( P ( θ l − ξ m θ ) + β l,m ) | , which is equalto the original beam generated by a l = f t ( θ l ) as N t .An important problem in the proposed DPP is that how many TD elements are sufficient tomitigate the beam split at all subcarriers in all possible physical directions. Note that the valueof the direction rotation factor is restricted by β l,m ∈ [ − , . Thus, the direction rotation factor β l,m calculated as (19) should lie in [ − , in all possible physical directions at all subcarrierfrequencies f m . This means − ≤ ( ξ m − P θ l ≤ . (20) Recalling that θ l ∈ [ − , and f f c ≤ ξ m ≤ f M f c , we have P ≤ f M f c − . (21)Substituting K = N t /P into (21), we can obtain the constraint on the number of the TDelements K to compensate for the array gain caused by the beam split effect at all subcarriersin all possible physical directions, which should be K ≥ ( f M f c − N t . (22)From (22), we can observe that the number of TD elements K increases linearly with the ratiobetween the maximum subcarrier frequency f M and the central frequency f c . Since f M /f c isproportional to the bandwidth B , we can conclude that the number of TD elements K increaseslinearly with the bandwidth B . This means that when the bandwidth is narrow, a small numberof TD elements K is enough to compensate for the array gain loss. While, the number of TDelements should be accordingly increased with the serious beam split effect. Specifically, innarrowband system with the assumption f m ≈ f c , the number of TD elements K becomes according to (22). This means that the proposed DPP architecture degenerates into the classicalhybrid precoding architecture in the narrowband case, which indicates that the classical hybridprecoding architecture is a special case of the proposed DPP architecture. D. Array gain performance of DPP
In this subsection, we will provide the theoretical analysis of the array gain achieved by theproposed DPP architecture. By adapting the constraint of K in (22), the array gain performanceachieved by the proposed DPP can be derived. From (57), for the m -th subcarrier, we have η ( a l,m , θ l , f m ) = KN t (cid:12)(cid:12)(cid:12)(cid:12) Ξ P ( β l,m P ) (cid:12)(cid:12)(cid:12)(cid:12) , (23)where the θ opt is replaced by θ l , because we have θ opt = θ l by the proposed DPP. By substituting β l,m = ( ξ m − P θ l in (19) into (23), the expectation of the array gain achieved by thebeamforming vector a l,m at all subcarriers m = 1 , , · · · , M in all possible physical direction θ l,c ∈ [ − , can be denoted as E ( η ( a l,m , θ l , f m ) = K M N t M (cid:88) m =1 (cid:90) − | Ξ P (( ξ m − θ l ) | d θ l . (24) Since it is difficult to calculate the integration of the Dirichlet sinc function, we utilize apolynomial to fit it by three points ( − , | Ξ P (1 − ξ m ) | ) , (0 , P ) and (1 , | Ξ P ( ξ m − | ) . Then,we have (cid:90) − | Ξ P (( ξ m − θ l ) | d θ l ≈ (cid:90) − (cid:2) (Ξ P ( ξ m − − P ) θ l + P (cid:3) d θ l = 23 | Ξ P ( ξ m − | + 43 P. (25)By substituting (25) into (24), we have E ( η ( a l,m , θ l , f m )) ≈ KM N t M (cid:88) m =1 (cid:18) | Ξ P ( ξ m − | + 23 P (cid:19) . (26)It is clear from (26) that the expectation of the array gain achieved by the proposed DPP overall subcarriers in all possible physical directions is mainly decided by the normalized frequency ξ m . Considering the constraint of (22), ξ m − always locates in the mainlobe of the Dirichlet sincfunction Ξ P . This guarantees the array gain achieved by the DPP is larger than KP N t = 0 . ,which is much higher than the array gain achieved by the classical hybrid precoding architectureas shown in Fig. 5. For instance, when f c = 300 GHz, B = 15 GHz, M = 128 , K = 8 and N t = 256 , we have E ( | η ( a l,m , θ l , f m ) | ) ≈ . , which means the proposed DPP is able toapproach the near-optimal array gain at all subcarriers in all possible physical directions.IV. H ARDWARE I MPLEMENTATION OF THE
DPPIn the previous section, we have proposed the DPP architecture, where a new TD networkis introduced between the RF chains and the PS network in the classical hybrid precodingarchitecture to provide the near-optimal array gain over the whole wide bandwidth. Moreover,through theoretical analysis, we have shown that the proposed DPP can eliminate the array gainloss caused by the beam split effect and achieve the near-optimal array gain performance. Thehardware implementation of the DPP architecture is important to make the DPP concept practicalin real THz massive MIMO systems. In this section, we will propose a practical hardwarestructure and the corresponding precoding algorithm to realize the concept of the DPP based ontrue-time-delayers (TTDs). Furthermore, the theoretical analysis of achievable rate is provided,which illustrates the proposed TTD based DPP (TTD-DPP) structure can realize the near-optimalachievable rate performance. RF ChainRF Chain Analog BeamformerTTDsBaseband ProcessingMultiplestreams DigitalPrecoder PSs
Fig. 7. The proposed TTD-DPP structure.
A. True-time-delayers based DPP
Based on the intuitive idea that utilizing TTDs can be directly used to realize the TD networkin the proposed DPP architecture, we propose a hardware structure called TTD-DPP, as shownin Fig. 7. In the TTD-DPP structure, each RF chain is connected to K TTDs, and each TTD isconnected to P = N t K PSs. The TTDs can realize the phase shift − πf m t by the time delay t atfrequency f m . Therefore, the received signal y m at the m -th subcarrier in (1) can be denoted as y m = √ ρ H Hm A u A TTD m D m s m + n m , (27)where A u ∈ C N t × KN RF denotes the analog beamformer provided by the frequency-independentPSs with the form as A u = [ A u , , A u , , · · · , A u ,N RF ] , (28)where A u ,l = blkdiag([¯ a l, , ¯ a l, , · · · , ¯ a l,K ]) denotes the analog beamformer realized by the PSsconnected to the l -th RF chain through TTDs, and A TTD m ∈ C KN RF × N RF denotes the frequency-dependent phase shifts realized by TTDs, which satisfies A TTD m = blkdiag (cid:18) [ e − j πf m t , e − j πf m t , · · · , e − j πf m t N RF ] (cid:19) , (29)where t l ∈ C K × = [ t l, , t l, , · · · , t l,K ] T denotes the time delays realized by K TTDs for the l -thpath component.Based on the notation above, the beamforming vector for the l -th path component a l,m = (cid:2) A u A TTD m (cid:3) [ l, :] = A u ,l e − j πf m t l = diag([¯ a l, , ¯ a l, , · · · , ¯ a l,K ]) e − j πf m t l . Recalling (14), (15), (16),and Lemma 2 in subsection III-C, to compensate for the beam split effect for the l -th path component, the phase shifts provided by PSs and the time delays realized by K TTDs shouldsatisfy [¯ a Tl, , ¯ a Tl, , · · · , ¯ a Tl,K ] T = f t ( θ l ) , (30) e − j πf m t l = (cid:2) , e − jπβ l,m , e − jπ β l,m , · · · , e − jπ ( K − β l,m (cid:3) T , (31)where the direction rotation factor is β l,m = ( ξ m − P θ l according to (19). Therefore, notingthat the value difference between phase shifts of adjacent TTDs is equal, the time delay vector t l should be set as t l = [0 , s l T c , s l T c , · · · , ( K − s l T c ] T , (32)where T c is the period of the carrier frequency f c , and s l denotes the number of periods thatshould be delayed for the l -th path component. Thus, s l should satisfy − πf m s l T c = − πβ l,m . (33)Then, substituting T c = f c , ξ m = f m f c and β l,m = ( ξ m − P θ l in (19) into (33), we have s l = ( ξ m − P θ l ξ m . (34)Note that in (34), the number of periods s l is not only decided by the fixed P and the targetphysical direction θ l , but also decided by the variable relative frequency ξ m . This makes (34)hard to realize for all M subcarriers, since s l must be fixed due to the hardware constraint ofTTDs. To solve this problem, we divide the phase shift − πβ l,m = − π ( ξ m − P θ l into twoparts − πξ m P θ l and πP θ l . The first part is frequency-dependent and can be realized by TTDsby setting s l as s l = P θ l . (35)Then, the second part πP θ l is frequency-independent, which can be realized by PSs by addingan extra phase shift. Specifically, the phase shifts provided by PSs ¯ a l,k , k = 1 , , · · · , K shouldbe changed from (30) to (cid:2) ¯ a Tl, , ¯ a Tl, , · · · , ¯ a Tl,K (cid:3) T = (cid:2) f t ( θ l ) [1: P ] , e jπP θ l f t ( θ l ) [ P +1:2 P ] , · · · , e jπ ( K − P θ l f t ( θ l ) [( K − P +1: N t ] (cid:3) . (36)Considering the time delays t l,i should be larger than , a small modification is required to be operated on (35). Finally, the time delay of the i -th delayer t l,i should be t l,i = ( K − (cid:12)(cid:12)(cid:12)(cid:12) P θ l (cid:12)(cid:12)(cid:12)(cid:12) T c + i P θ l T c , θ l < ,i P θ l T c , θ l ≥ , (37)We can observe from (37) that the value range of t l,i is t l,i ∈ [0 , N t T c ] with θ l ∈ [ − , and ≤ i ≤ K . For example, when the central frequency f c is GHz, the number of antennas N t is , and the number of TTDs K is K = 16 , the range of time delays provided by the TTDsis between and ps. It should be noted that this range of time delays can be realized bymany existing efficient TTDs [26]–[29], e.g., the TTD designed based on inductance-capacitanceartificial transmission lines in [29] can realize a maximum time delay of ps with a pstime delay step, and it can support GHz bandwidth, which are able to support the practicalhardware implementation of the TTD-DPP.
Algorithm 1
Hybrid precoding for TTD-DPP.
Inputs:
Channel H m ; Physical direction θ l Output:
Hybrid precoder A u , A TTD m , and D m for l ∈ { , , · · · , N RF } do (cid:2) ¯ a Tl, , · · · , ¯ a Tl,K (cid:3) T = (cid:2) f t ( θ l ) [1: P ] , e jπP θ l f t ( θ l ) [ P +1:2 P ] , · · · , e jπ ( K − P θ l f t ( θ l ) [( K − P +1: N t ] (cid:3) A u ,l = blkdiag([¯ a l, , ¯ a l, , · · · , ¯ a l,K ]) A u = [ A u , , A u , , · · · , A u ,N RF ] s l = P θ l / t l,i = ( K − (cid:12)(cid:12)(cid:12)(cid:12) P θ l (cid:12)(cid:12)(cid:12)(cid:12) T c + i P θ l T c , θ l < ,i P θ l T c , θ l ≥ , t l = [ t l, , t l, , · · · , t l,K ] end for A u = [ A u, , A u, , · · · , A u,N RF ] for m ∈ { , , · · · , M } do A TTD m = blkdiag (cid:18) [ e − j πf m t , · · · , e − j πf m t N RF ] (cid:19) H m, eq = H Hm A u A TTD m D m = µ V m, eq , [: , N RF ] , H m, eq = U m, eq Σ m, eq V Hm, eq end for return A u , A TTD m and D m By setting ¯ a l,i and t l,i for i = 1 , , · · · , K as (36) and (37), the TTD-DPP structure cancompensate for the array gain loss caused by the beam split effect for the l -th path component.Based on the derivation above, we propose a hybrid precoding algorithm for the TTD-DPPstructure, where the key idea is to generate beams towards different physical directions of pathcomponents at first, and then the time delays are calculated accordingly to make the beamaligned with the physical direction at each subcarrier. Specifically, the pseudo-code is shown in Algorithm 1 . At first, for each path component, the analog beamformer A u ,l is calculated insteps - . Then, the time delays that should be delayed by K TTDs are generated in steps - .After that, for each subcarrier frequency, the analog beamformer A TTD m is generated in step .Finally, the digital precoder D m is also calculated based on singular value decomposition (SVD)precoding in steps and , where µ is the power normalization coefficient.In the TTD-DPP structure, K × N RF TTDs are used based on the classical hybrid precodingstructure. Although the power consumption of a TTD is much higher than a PS, the increasedpower consumption of TTD-DPP compared with the classical hybrid precoding structure islimited since K × N RF (cid:28) N t . This will also be verified in simulation results that a smallnumber of TTDs is able to compensate for the beam split effect across the entire bandwidth.In addition, it should be emphasized that we only provide one practical hardware implementa-tion of DPP in this paper. Actually, any hardware component that can realize frequency-dependentphase shifts is able to realize the concept of DPP as shown in Subsection III-C. For instance,multiple RF chains can be used to realize frequency-dependent phase shifts in the baseband.Specifically, N RF RF chain group with each group containing K RF chains can be utilized,where each RF chain group connects to all antenna elements through PSs in a sub-connectedmanner. In this way, when the PSs generate frequency-independent beams according to (15) andthe baseband signal processing realize the frequency-dependent phase shifts β m according to(19), the mechanism of the DPP proved in Lemma 2 can also be realized, and thus the beamsplit effect can be eliminated. In the next subsection, the achievable rate performance of theTTD-DPP will be provided though theoretical analysis.
B. Achievable rate performance
In this subsection, we will derive the achievable rate of the proposed TTD-DPP structure. Forthe wideband THz massive MIMO systems with M subcarriers, the achievable rate R can be presented as [21] R = M (cid:88) m =1 R m = M (cid:88) m =1 log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12) I N r + ρN s σ H m A m D m D Hm A Hm H Hm (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) , (38)where R m denotes the achievable rate at the m -th subcarrier, and A m = A u A TTD m for the TTD-DPP. By utilizing the ordered SVD of H m as H m = U m Σ f m V Hm , the achievable rate R m can beconverted into [15] R m = log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12) I d m + ρN s σ Σ f m V f mH A m D m D Hm A Hm V f m (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) , (39)where the diagonal matrix Σ f m = diag([ λ , λ , · · · , λ d m ]) ∈ C d m × d m ( λ i , i = 1 , , · · · , d m )representing the singular value of H m , and the matrix V f m ∈ C N t × d m with V f mH V f m = I d m are obtained from the ordered SVD of the channel H m , where d m denotes the rank of H m .Without loss of generality, we assume that the parameters ( N t , N RF , N s ) for DPP are delicatelydesigned so that the multiplexing gain from the multi-path channel can be fully exploited. Thisassumption can be easily satisfied in practical THz massive MIMO systems [15]. Under thisassumption, according to [15], (39) can be rewritten as R m = log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12) I N s + ρN s σ Σ m V mH A m D m D Hm A Hm V m (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) , (40)where Σ m = Σ f m, [1: N s , N s ] and V m = V f m, [: , N s ] . Note that V m is the optimal precoding matrixwithout any hardware constraint. Considering the fact that the steering vectors f t (¯ θ l,m ) in thechannel model (3) are approximately orthogonal due to the large number of antennas [30], thesevectors f t (¯ θ l,m ) can form a set of orthogonal basis of HH H . Since the columns of V f m are alsoa set of orthogonal basis for HH H , the columns of V m can be approximately seen as the linearcombination of f t (¯ θ l,m ) as V m ≈ A t D m, opt , (41)where A t = [ f t (¯ θ ,m ) , f t (¯ θ ,m ) , · · · , f t (¯ θ N RF ,m )] with ¯ θ l,m being sorted by path gains | g | > | g | > · · · > | g N RF | , and D m, opt ∈ C N RF × N s . Note that in (41), V m and A t satisfy V mH V m = I N s and A H t A t = I N RF . Therefore, the optimal digital precoder D m, op also satisfies D Hm, opt D m, opt = I N s .Obviously, when A m = A t and D m = D m, opt , the optimal achievable rate R m, opt can be derived as R m, opt = log (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) I N s + ρN s σ Σ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (42)However, for the proposed TTD-DPP, the near-optimal precoder A m = A t and D m = D m, opt cannot be achieved at all subcarriers, since the analog beamforming vectors a l,m as shown in (14)in DPP cannot be equal to f t (¯ θ l,m ) at all subcarriers. Specifically, we can obtain the achievablerate R m, TTD of the proposed TTD-DPP by substituting (41) into (40) as R m, TTD = log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12) I N s + ρN s σ Σ m V Hm, eq V m, eq (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) , (43)where R m, TTD is the achievable rate achieved by the TTD-DPP, and V m, eq = D Hm A Hm A t D m, opt .Considering that the columns of A m are the analog beamforming vectors generated by the DPP,which can generate beams aligned with the physical direction θ l , and the steering vectors towardsdifferent physical directions are approximately orthogonal [30], we have A Hm A t = blkdiag (cid:0)(cid:2) a H ,m f t (¯ θ ,m ) , a H ,m f t (¯ θ ,m ) , · · · , a HN RF ,m f t (¯ θ N RF ,m ) (cid:3)(cid:1) , (44)where A Hm A t is a diagonal matrix. Recalling the definition of the array gain η ( a l,m , θ l , f m ) = | f t (2 d f m c θ l ) H a l,m | and (5), we know that Φ m = A Hm A t satisfies Φ Hm Φ m = blkdiag (cid:0)(cid:2) η ( a ,m , θ , f m ) , η ( a ,m , θ , f m ) , · · · , η ( a N RF ,m , θ N RF , f m ) (cid:3)(cid:1) . (45)Therefore, based on (45), the achievable rate R m, TTD becomes R m, TTD = log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N s + ρ Σ m D Hm, opt Φ Hm D m D Hm Φ m D m, opt N s σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:33) ( a ) = log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N s + ρ Σ m D Hm, opt Φ Hm Φ m D m, opt N s σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:33) ( b ) ≈ log (cid:32) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N s + ρ E ( η ( a l,m , θ l , f m ) ) Σ m D Hm, opt D m, opt N s σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:33) ( c ) = log (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) I N s + ρN s σ E ( η ( a l,m , θ l , f m )) Σ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (46)where ( a ) is achieved by setting D m = D m, opt and D Hm, opt D m, opt = I N s , ( b ) comes from(45) and the assumption η ( a ,m , θ , f m ) ≈ η ( a ,m , θ , f m ) ≈ · · · ≈ η ( a N RF ,m , θ N RF , f m ) ≈ E ( η ( a l,m , θ l , f m ) ) which is reasonable since the beams generated by the DPP can achieve near- optimal array gain across the entire bandwidth as will be verified by simulation results in SectionV, and ( c ) comes from D Hm, opt D m, opt = I N s .We can observe from (46) that the achievable rate R m, TTD is mainly decided by the arraygain obtained in different physical directions at different subcarriers. This indicates that the arraygain loss caused by the beam split effect is vital to the achievable rate in wideband THz massiveMIMO systems. Based on (46) and (50), the ratio between the achievable rate achieved by theTTD-DPP and the optimal achievable rate can be denoted as R m, TTD R m, opt = log (cid:16)(cid:12)(cid:12)(cid:12) I N s + ρN s σ E ( η ( a l,m , θ l , f m )) Σ m (cid:12)(cid:12)(cid:12)(cid:17) log (cid:16)(cid:12)(cid:12)(cid:12) I N s + ρN s σ Σ m (cid:12)(cid:12)(cid:12)(cid:17) = Σ N s l =1 log (1 + ρN s σ E ( η ( a l,m , θ l , f m ) ) λ l )Σ N s l =1 log (1 + ρN s σ λ l ) ( a ) > Σ N s l =1 ρN s σ E ( η ( a l,m , θ l , f m ) ) λ l Σ N s l =1 ρN s σ λ l , (47)where the derivation ( a ) is based on log (1 + x ) < x and η ( a l,m , θ l , f m ) ≤ . According to(47), the ratio R m, TTD /R m, opt can be denoted as R m, TTD R m, opt > Σ N RF l =1 ρN s σ E ( η ( a l,m , θ l , f m ) ) λ l Σ N RF l =1 ρN s σ λ l = E (cid:16) η ( a l,m , θ l , f m ) (cid:17) . (48)In Subsection III-D, the expectation of the normalized array gain E (cid:16) η ( a l,m , θ l , f m ) (cid:17) has beenprovided in (26). Similar to the process to compute E (cid:16) η ( a l,m , θ l , f m ) (cid:17) in (25), the polynomialfitting with three points ( − , Ξ P (1 − ξ m ) ) , (0 , P ) and (1 , Ξ P ( ξ m − ) can be also utilized tocalculate E ( η ( a l,m , θ l , f m ) ) in (48) as E (cid:16) η ( a l,m , θ l , f m ) (cid:17) = K M N M (cid:88) m =1 (cid:90) − [Ξ P (( ξ m − θ l )] d θ l ≈ K M N M (cid:88) m =1 (cid:90) − (cid:2) (Ξ P ( ξ m − − P ) θ l + P (cid:3) d θ l = K M N M (cid:88) m =1
13 Ξ P ( ξ m − + 23 P . (49)It is clear from (49) that the achievable rate R m, TTD of the TTD-DPP structure will increase as K increases, since the mainlobe of Ξ P ( x ) become wider, which results in the fact that | Ξ P ( ξ m − | is closer to . For example, when f c = 300 GHz, B = 15 GHz, M = 128 , K = 8 and N t = 256 ,we have E ( η ( a l,m , θ l , f m ) ) = 0 . . This means that by adopting the proposed TTD-DPP, which N o r m a li z ed a rr a y ga i n N o r m a li z ed a rr a y ga i n Fig. 8. Normalized array gain of beams generated by the proposed DPP structure. is able to efficiently mitigate the beam split effect, so the near-optimal achievable rate can beachieved. V. S
IMULATION R ESULTS
In this section, we provide simulation results to verify the performance of the proposed TTD-DPP to realize the concept of DPP for wideband THz massive MIMO. The main simulation
TABLE IS
YSTEM P ARAMETERS FOR S IMULATIONS
The number of the BS antennas N t The number of the user antennas N r , , The number of channel paths L The central frequency f c GHzThe bandwidth B GHzThe number of the subcarriers M The number of RF chains N RF The number of TD elements K Physical directions of the paths ˜ θ l , ˜ φ l U [ − π , π ] The transmission SNR ρ/σ − ∼ dB
20 40 60 80 100 120
Subcarrier (m) N o r m a li z ed a rr a y ga i n Hybrid Precoding, B=0.1 GHzHybrid Precoding, B=1 GHzHybrid Precoding, B=10 GHzProposed DPP, B=10 GHz
Fig. 9. Normalized array gain comparison between the classical hybrid precoding architecture and the proposed DPP architectureacross the entire bandwidth (128 subcarriers). parameters are shown in Tabel I. The ULAs are considered at the BS and the user, with N t = 256 and N r = 1 , , . Without loss of generality, we assume that the number of streams is equal tothe number of receive antennas, i.e., N s = N r . The wideband spatial channel model (3) isadopted in the simulation with L = 4 , f c = 300 GHz, B = 30 GHz and M = 128 . Foreach path, the physical directions of the transmitter and the receiver are generated randomly as ˜ θ l , ˜ φ l ∼ U [ − π/ , π/ . The performance of classical hybrid precoding methods using PSs is alsoprovided as the benchmark for comparison. In the proposed TTD-DPP, the number of TTDs forone RF chain is set as K = 16 . We utilize (38) to calculate the achievable rate, in which thetransmission signal-to-noise ratio (SNR) is defined as ρ/σ .Fig. 8 shows the normalized array gain against the physical direction of the beamformingvector a l, at the minimum subcarrier frequency f , a l,M at the maximum subcarrier frequency f M , and the beamforming vector at the central frequency f c , which are generated by the proposedDPP with the target physical direction θ l = 0 . . We can observe from Fig. 8 that at the subcarrierswith the minimum frequency f and the maximum frequency f M , the beamforming vector a l, and a l,M generated by the proposed DPP can be aligned with the target physical direction θ l .Thus, we can conclude that by using the proposed DPP, the user can be covered by beams atdifferent subcarrier frequencies, which efficiently mitigates the array gain loss caused by thebeam split effect. For example, more than of the optimal array gain can be achieved by the s N s N s N Fig. 10. Achievable rate performance versus the transmission SNR for the proposed TTD-DPP. proposed DPP at f and f M as shown in Fig. 8.To better illustrate the array gain achieved by the proposed DPP across the entire bandwidth,we provides the normalized array gain performance of the proposed DPP at different subcarriersin Fig. 9. The target physical direction is still set as θ l = 0 . , and the bandwidth is B = 30 GHz.The array gains achieved by the classical hybrid precoding structure with different bandwidths B = 0 . , , GHz are also provided for comparison. We can observe from Fig. 9 that when B = 30 GHz, the classical hybrid precoding structure which solely controls the phase throughfrequency-independent PSs [15] suffers from the severe array gain loss due to the beam spliteffect, e.g., array gain loss at most of subcarriers. On the contrary, the proposed DPP,which jointly controls the delay and phase, can realize almost flat array gain across the entirebandwidth of B = 30 GHz, which is much better than the classical hybrid precoding structure,and even better than the performance achieved by the classical hybrid precoding structure witha much smaller bandwidth of B = 3 GHz. From Fig. 8 and Fig. 9, we can conclude that for acertain target physical direction for beamforming, the proposed DPP can efficiently mitigate thearray gain loss caused by the beam split effect through the joint control of delay and phase.Fig. 10 illustrates the average achievable rate of the proposed TTD-DPP, where differentnumbers of data streams N s = 1 , , are considered. We also provide the achievable rateperformance of the optimal unconstrained fully-digital precoding [8] as the upper bound and -20 -15 -10 -5 0 5 10 15 SNR (dB) A c he i v abe l r a t e pe r s ub c a rr i e r ( b i t/ s / H z ) Optimal precodingProposed TTD-DPPOptimization based hybrid precoding [1 ] Wide beam hybrid precoding [ ]Spartially sparse precoding [1 ] Fig. 11. Achievable rate performance comparison between the proposed scheme and existing schemes. the theoretical result in (49) as the lower bound for comparison. We can find that the proposedTTD-DPP can achieve more than % of the optimal achievable rate, and the actual achievablerate is always larger than the lower bound in (49), which is consistent with the analysis in thesubsection IV-B.Fig. 11 compares the average achievable rate performance between the proposed TTD-DPPand other existing hybrid precoding schemes when N s = 4 . The existing solutions includethe wideband hybrid precoding with the spatially sparse precoding [15], the achievable rateoptimization [17], and the wide beam based hybrid precoding [20]. Specifically, we can observefrom Fig. 11 that the spatially sparse precoding [15] suffer a nearly achievable rate losscaused by the beam split effect. Although the achievable rate optimization [17] and wide beambased hybrid precoding [20] designed for mmWave massive MIMO systems can partially relievethe achievable rate loss incurred by the beam split effect, the performance is still unacceptabledue to the very large bandwidth and antenna number in wideband THz massive MIMO systems.On the contrary, the proposed TTD-DPP scheme can significantly outperforms these existingschemes and can achieve the near-optimal achievable rate, e.g., more than % of the optimalachievable rate.To better show the effect of the number of TTDs K on the proposed TTD-DPP, Fig. 12gives the average achievable rate performance against K , where N s = 4 and SNR = 10 dB are
10 20 30 40 50 602426283032343638 A v e r age a c h i e v ab l e r a t e ( b i t/ s / H z ) Optimal precodingProposed TTD-DPP
Fig. 12. Achievable rate performance versus K . The number of RF chains E ne r g y E ff i c i en cy ( bp s / H z / W ) Hybrid precoding with TTDsProposed TTD-DPPClassical hybrid precoding
Fig. 13. Energy efficiency comparison versus the number of RF chains. considered. From (22), we know that to compensate for the array gain loss caused by the beamsplit effect across the entire bandwidth, K should satisfy K ≥ . for the parameters in TABLEI. We can observe from Fig. 12 that for the proposed TTD-DPP, the achievable rate performanceincreases as K becomes large, and it achieves the near-optimal achievable rate when K ≥ .This trend is consistent with the theoretical result in (49).Finally, Fig. 13 provides the energy efficiency comparison when N RF = N s varies from to . The energy efficiency is defined as the ratio between the achievable rate and the powerconsumption. Specifically, we compare the energy efficiency of the conventional PSs based hybridprecoding architecture [15], the TTDs based hybrid precoding where the PSs are replaced byTTDs, and the proposed TTD-DPP. The power consumption of these three schemes are denotedas P HP , P TTD , P
DPP , respectively, and we have P HP = P t + P BB + N RF P RF + N RF N P PS , (50) P TTD = P t + P BB + N RF P RF + N RF N P
TTD , (51) P DPP = P t + P BB + N RF P RF + N RF KP TTD + N RF N P PS , (52)where P t is the transmission power, and P BB , P RF , P PS , and P TTD denote the power consumptionof baseband processing, RF chain, PS, and TTD, respectively. Here, we adopt the practical valuesas ρ = 30 mW [25], P BB = 300 mW [31], P RF = 200 mW [25], P PS = 20 mW [31] and P TTD = 100 mW [29]. We can observe from Fig. 13 that the proposed TTD-DPP enjoys muchhigher energy efficiency than the conventional hybrid precoding architecture using PSs and TTDs.Compared with the conventional hybrid precoding architecture with PSs, the proposed TTD-DPPcan achieve much higher achievable rate by eliminating the achievable rate loss caused by thebeam split effect, with an acceptable power consumption increase caused by the additional useof limited TTDs. Moreover, the proposed TTD-DPP also has higher energy efficiency than theconventional hybrid precoding architecture with TTDs, since much less TTDs ( N RF K instead of N RF N , where K (cid:28) N ) are utilized in the TTD-DPP. This indicates that the TDD-DPP is ableto provide a better tradeoff between the achievable rate performance and power consumption,which is promising for future THz massive MIMO systems.VI. C ONCLUSIONS
In this paper, we have investigated the wideband hybrid precoding for future THz massiveMIMO systems. A vital problem called beam split, i.e., the THz rainbow, where the generatedbeams will split into separated physical directions at different subcarrier frequencies, has beenanalyzed. We revealed that the beam split effect may cause serious array gain loss and achievablerate degradation in wideband THz massive MIMO systems. To solve this problem, we haveproposed a DPP architecture by introducing a TD network as a new precoding layer into theconventional hybrid precoding architecture. By leveraging the TD network, the DPP architecture can realize a delay-phase jointly controlled beamformer, which can compensate for the array gainloss caused by the beam split effect. To realize the concept of DPP, we have further proposeda hardware structure called TTD-DDP, where the frequency-dependent phase shifts provided bythe TD network are realized by multiple TTDs. Theoretical analysis and simulation results haveshown that the proposed DPP can eliminate the array gain loss caused by the beam split effect,so it can achieve more than % of the optimal array gain and achievable rate performanceacross the entire bandwidth in wideband THz massive MIMO systems. Potential future worksmay include other feasible implementations of DPP, improved algorithm to realize DPP, channelestimation [32] and beam tracking [33] for DPP, and low-cost hardware solutions such as low-resolution PSs and low-resolution ADCs/DACs [34].A PPENDIX . P
ROOF OF L EMMA Proof:
The normalized array gain achieved by the analog beamforming vector a l,m on anarbitrary physical direction θ at the subcarrier frequency f m can be denoted as η ( a l,m , θ, f m ) = | f t (2 d f m c θ ) H a l,m | . With [¯ a Tl, , ¯ a Tl, , · · · , ¯ a Tl,K ] T = f ( θ l ) and p l,m = [1 , e − jπβ l,m , · · · , e − jπ ( K − β l,m ] T ,we have η ( a l,m , θ, f m ) = 1 N t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K (cid:88) k =1 P (cid:88) p =1 e − jπ [( k − P +( p − θ l e − jπ ( k − β l,m e jπ [( k − P +( p − ξ m θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (53)By seperating the summation on K and P , we have η ( a l,m , θ, f m ) = 1 N t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K (cid:88) k =1 e − jπ ( k − [ P ( θ l − ξ m θ )+ β l,m ] P (cid:88) p =1 e − jπ ( p − θ l − ξ m θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (54)Then, we can transform the summation in (54) as η ( a l,m , θ, f m ) = 1 N t | Ξ K ( P ( θ l − ξ m θ ) + β l,m )Ξ P ( θ l − ξ m θ ) | . (55)We can see that the array gain achieved by the analog beamforming vector a l,m is the productof two Dirichlet sinc functions, whose figures are shown in Fig. 14. Because the power ofthe Dirichlet sinc function is focused in the mainlobe, we can analyze the property of thearray gain η ( a l,m , θ, f m ) through the mainlobes of these two Dirichlet sinc functions. For thefunction | Ξ K ( P ( θ l − ξ m θ ) + β l,m ) | with respect to θ , the maximum value can be achievedby setting P ( θ l − ξ m θ ) + β l,m = 0 , i.e., θ = θ K, max = θ l ξ m + β l,m ξ m P , and the mainlobe width of | Ξ K ( P ( θ l,c − θ ) + β l,m ) | is N t . Similarly for | Ξ P ( θ l,c − ξ m θ ) | , the maximum value can be achieved Mainlobe ofMainlobe of Maximum point of K Σ P Σ K Σ Fig. 14. The two of Dirichlet sinc functions, where Ξ K denotes | K Ξ K ( P ( θ l − ξ m θ )+ β l,m ) | and | Ξ P | denotes | P Ξ P ( θ l − ξ m θ ) | . when θ = θ P, max = θ l ξ m and the mainlobe width is P . Considering that the available value rangeof the direction rotation factor is β l,m ∈ [ − , , we have θ K, max ∈ [ θ l ξ m − ξ m P , θ l ξ m + ξ m P ] , whichmeans that θ K, max locates in the mainlobe of | Ξ P ( θ l − ξ m θ ) | whose range is [ θ l ξ m − ξ m P , θ l ξ m + ξ m P ] .In addition, considering P = N t /K , the mainlobe width of | Ξ P ( θ l − ξ m θ ) | is P which is K times wider than that of | Ξ K ( P ( θ l − ξ m θ ) + β l,m ) | , whose mainlobe width is N t . Therefore, wecan conclude that the variation of | Ξ P ( θ l − ξ m θ ) | in the mainlobe of | Ξ K ( P ( θ l − ξ m θ ) + β l,m ) | is much smaller than the variation of | Ξ K ( P ( θ l − ξ m θ ) + β l,m ) | , which is shown in Fig. 14.Therefore, the maximum value of the array gain η ( a l,m , θ, f m ) can be approximately consideredto be decided by | Ξ K ( P ( θ l − ξ m θ ) + β l,m ) | . Thus, we have θ opt = arg max θ η ( a l,m , θ, f m ) = θ K, max = θ l ξ m + β l,m ξ m P . (56)Then, the array gain achieved by a l,m at this physical direction θ opt can be denoted by substituting(56) into (55) as | η ( a l,m , θ opt , f m ) | = 1 N t | Ξ K (0)Ξ P ( θ l − θ opt ) | = (cid:12)(cid:12)(cid:12)(cid:12) KN t Ξ P ( β l,m P ) (cid:12)(cid:12)(cid:12)(cid:12) , (57)which completes the proof. (cid:4) R EFERENCES [1] J. Tan and L. Dai, “Delay-phase precoding for THz massive MIMO with beam split,” in
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