# Deterministic Pilot Design and Channel Estimation for Downlink Massive MIMO-OTFS Systems in Presence of the Fractional Doppler

Although the combination of the orthogonal time frequency space (OTFS) modulation and the massive multiple-input multiple-output (MIMO) technology can make communication systems perform better in high-mobility scenarios, there are still many challenges in downlink channel estimation owing to inaccurate modeling and high pilot overhead in practical systems. In this paper, we propose a channel state information (CSI) acquisition scheme for downlink massive MIMO-OTFS in presence of the fractional Doppler, including deterministic pilot design and channel estimation algorithm. First, we analyze the input-output relationship of the single-input single-output (SISO) OTFS based on the orthogonal frequency division multiplexing (OFDM) modem and extend it to massive MIMO-OTFS. Moreover, we formulate an accurate model for the practical system in which the fractional Doppler is considered and the influence of subpaths is revealed. A deterministic pilot design is then proposed based on the model and the structure of the pilot matrix to reduce pilot overhead and save memory consumption. Since channel geometry changes very slowly relative to the communication timescale, we put forward a modified sensing matrix based channel estimation (MSMCE) algorithm to acquire the downlink CSI. Simulation results demonstrate that the proposed downlink CSI acquisition scheme has significant advantages over traditional algorithms.

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03/22/2019

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## I Introduction

To meet the needs of future wireless communications, especially various new services and scenarios that are continually emerging, the fifth-generation (5G) mobile systems come into being. One objective of 5G is to achieve reliable communication for the scenarios with high Doppler spread, such as high-mobility scenarios [39, 8]. Orthogonal frequency division multiplexing (OFDM) technology has been widely used to combat the inter-symbol interference (ISI) in time-invariant channels [6, 33, 21, 7]. However, when there is a high Doppler spread in time-variant channels, which would lead to severe inter-carrier interference (ICI), the performance of OFDM degrades significantly[36].

To cope with this problem, orthogonal time frequency space (OTFS) modulation technology was proposed [12, 25, 11]

and attracted much attention due to significant advantages in time-variant channels. OTFS converts the signal in the time-frequency domain into a delay-Doppler domain and performs modulation accordingly. In particular, each information symbol in the delay-Doppler domain is expressed by a pair of orthogonal basis functions in the time-frequency domain, i.e., full diversity in the time-frequency domain. Furthermore, transmitted symbols experience roughly constant channels even with high Doppler spread. Another advantage is that the implementation of OTFS can be achieved simply as an overlay of the OFDM systems. Farhang

et al. investigated the OTFS system based on OFDM [9] and designed a modem scheme with low complexity, where the cyclic prefix (CP) is added in front of each OFDM symbol in an OTFS symbol.

Channel estimation is crucial in OTFS systems. In [26] and [28], training-based channel estimation methods were investigated. In [26], Murali et al. used the pseudo-random (PN) sequence as pilots to acquire the channel state information (CSI). In [28], Raviteja et al. arranged the pilots and data in the same OTFS symbol and estimated the delay-Doppler channel by a threshold method. After the acquisition of the CSI, the symbol detection can be performed. In [26]

, the Markov chain Monte Carlo (MCMC) sampling was utilized for the low-complexity symbol detection. In

[29], an explicit derivation of the input-output relationship of OTFS systems was given, and the effect of fractional Doppler caused by insufficient sampling of Doppler-dimension was analyzed. They also developed a message passing (MP) method to detect OTFS symbols. However, when they used the rectangular pulse-shaping waveforms, the CP was not considered, which would cause ISI and might make the receiver more complicated. Different from the symbol spaced sampling framework in [29], Ge et al. designed a fractionally sampling scheme for symbol detection [10], where only one CP is inserted for the whole OTFS symbol. Compared with [9], less CP overhead in [28, 29, 10] can improve spectral efficiency. In [37], Wei et al. reduced the channel spread caused by the fractional Doppler and improved the channel estimation performance through the design of transmitter and receiver windows. In [18], the embedded pilot-aided CSI acquisition scheme proposed in [28] and the MP algorithm proposed in [29] were extended to multiple-input multiple-output (MIMO) OTFS systems.

In this work, we consider the fractional Doppler caused by insufficient sampling of Doppler-dimension in practical systems. Moreover, for massive MIMO-OTFS systems, we propose a downlink CSI acquisition scheme, including deterministic pilot design and channel estimation algorithm. The contributions are summarized as follows:

• In the single-input single-output (SISO) case, we analyze the input-output relationship of the OTFS system, where the OFDM modem is chosen as the time-frequency modem. Specifically, we consider that the Doppler frequency of each path is mapped to an integer and a fractional Dopper tap and reveal the influence of the subpaths contained in each dominant path. Moreover, the characteristics of subpaths are used to simplify such a system model. Then we extend it to the massive MIMO-OTFS system and establish the downlink CSI acquisition model.

• To reconstruct the channel accurately, we design a Zadoff-Chu (ZC) sequence based deterministic pilot, which can achieve better sensing performance and memory consumption saving than the random pilot used in [32]. We first analyze the relationship between the pilot matrix and the position of pilots at each beam. Then we give two conditions that the deterministic pilots need to satisfy to ensure the low coherence between the pilot matrix columns. Next, ZC sequences are chosen as pilots, whose performance is also discussed.

• Based on the modified model, we propose a modified sensing matrix based channel estimation (MSMCE) algorithm to acquire CSI. Due to the slow change of delay-Doppler channels, path delays and Doppler frequencies of all dominant paths are extracted from the previous channel estimation result and used to modify the sensing matrix for more accurate CSI acquisition. Then we can utilize the estimated CSI to update the path delays and Doppler frequencies, which can be used in the next channel estimation.

The rest of the paper is organized as follows. In Section II, we investigate models for SISO OTFS systems and massive MIMO-OTFS systems. Section III establish a downlink CSI acquisition model. Based on the model, we design a deterministic pilot matrix and propose a MSMCE algorithm. Section IV presents the performance of the proposed CSI acquisition scheme by simulation results, and the paper is concluded in Section V.

Notations: The superscripts and

denote the conjugate and conjugated-transpose operations, respectively. The uppercase (lowercase) boldface letters denote matrices (column vectors).

denotes the imaginary unit. and denote the -norm of and , and denotes the -norm of . denotes the smallest integer that is not less than , while denotes the largest integer that is not greater than . is the Hadamard product operator, and is the Kronecker product operator. denotes mod , and denotes . The notation is used for definitions, and is used to indicate equivalence or approximate equivalence. denotes the Dirac delta function. denotes the -th row of , wihle denotes the -th colomn of . denotes the -th element of . denotes the -th element of .

## Ii System Model

In this section, some basic definitions and concepts of OTFS are reviewed first. Then we analyze the input-output relationship of the SISO OTFS system, where the OFDM modem is selected as time-frequency modem, and the fractional Doppler is considered. Then we extend it to the case of massive MIMO systems.

### Ii-a SISO OTFS System

For the sake of the subsequent derivation, we first define a lattice in the time-frequency domain as , where (seconds) and (Hz) are sampling intervals of the time-dimension and the frequency-dimension, respectively, , , . Similarly, the lattice in the delay-Doppler domain is defined as where and are sampling intervals of the delay-dimension and Doppler-dimension, respectively, and , .

We then arrange a set of quadrature amplitude modulated (QAM) symbols in the delay-Doppler domain on the lattice

. The inverse symplectic finite Fourier transform (ISFFT) is utilized at OTFS transmitter to convert

to the symbols in the time-frequency domain as [2]

 (1)

Next, an OFDM modulator can be used as a time-frequency modulator to convert to a transmitted signal with a transmitted waveform as

 s(t) =M−1∑m=0N−1∑n=0XTF[n,m]e¯ȷ2πmΔf(t−MCPTM−nTsym) ×gtx(t−nTsym), (2)

where is the length of CP, is time duration of an OFDM symbol without CP, and is defined as

 gtx(t)Δ={1√T,0≤t≤Tsym0,otherwise. (3)

Note that the CP is added in this step.

After the transmitted signal passing through the multipath time-variant channel, the received signal can be obtained as

 r(t)=∬hc(τ,ν)e¯ȷ2πν(t−τ)s(t−τ)dτdν+n(t), (4)

where is the delay, is the Doppler frequency, is impulse response in the delay-Doppler domain[17], and is the additive Gaussian noise. Usually, there are only a few scatterers in the transmission environment. Thus the channel can be represented in a sparse way [29]. We assume that the number of the dominant paths between the transmitter and the receiver is , and each consists of subpaths. Hence, is given by

 hc(τ,ν)=P−1∑i=0S−1∑si=0hsiδ(τ−τi)δ(ν−νsi), (5)

where and are the complex path gain and the Doppler frequency of the -th subpath of the -th dominant path, respectively. All of subpaths in the -th dominant path have the same delay [15]. We define the delay and Doppler taps for subpath as

 τi=liMΔf,νsi=ksi+~ksiNTsym, (6)

where and are integers and represent the indexes of delay and Doppler taps. The real number , whose value range is , is defined as the fractional Doppler. The fractional delay is not considered since the resolution of the delay axis is sufficient so that each path delay can be mapped to an integer delay tap in typical wide-band systems [35].

At the receiver, the cross-ambiguity function between a received waveform and the received signal is computed by the matched filter as

 Agrx,r(τ,ν)=∫e−¯ȷ2πν(t−τ)g∗rx(t−τ)r(t)dt, (7)

where is defined as

 grx(t)Δ=⎧⎨⎩1√T,MCPTM≤t≤Tsym0,otherwise. (8)

Note that the CP is removed in this step. By sampling the cross-ambiguity function, the received data in the time-frequency domain is given by

 YTF[n,m]=Agrx,r(τ,ν)∣∣τ=nTsym,ν=mΔf. (9)

It is worth noting that when the time duration of CP, i.e., is beyond the maximum path delay of all dominant paths, there is no ISI between OFDM symbols within an OTFS symbol at the receiver, which is different from [28, 29, 10].

Next, the symplectic finite Fourier transform (SFFT) can be utilized to map to the symbols in the delay-Doppler domain as

 YDD[k,l]=1√NMN−1∑n=0M−1∑m=0YTF[n,m]e¯ȷ2π(mlM−nkN). (10)

We define the complex gain of the time-variant channel on the delay tap at time as

 hρ,l=P−1∑i=0S−1∑si=0hsie¯ȷ2π(ρ−l)Tsνsiδ(lTs−τi), (11)

where is the system sampling interval, and we define as the complex gain of the -th subpath of . Through the above discussion, we give the input-output relationship of the SISO OTFS system, as shown in Proposition 1.

Proposition 1: For a SISO OTFS system, when the OFDM modem is used as the time-frequency modem, the input-output relationship is expressed as

 YDD[k,l] ×XDD[⟨k−k′⟩N,(l−li)M]+ZDD[k,l], (12)

where

 HDD[k′,li,l] Δ=1√NN−1∑j=0⎛⎜ ⎜⎝S−1∑si=0hsiMCP+j(M+MCP),lie¯ȷ2πl(ksi+~ksi)(M+MCP)N⎞⎟ ⎟⎠e−¯ȷ2πk′jN, (13)

and is the additive noise in the delay-Doppler domain.

Proof: See Appendix.

From (II-A), we have the following two findings. First, is related to , which is the received data position along the delay-dimension. Second, for the -th dominant path, is affected by all subpaths due to the fractional Doppler. These two points make the system model very complicated, which is not conducive to subsequent analysis. Hence, we utilize the characteristics of subpaths to simplify it. Since all subpaths of one dominant path usually originate from the same scattering cluster, the directions of arrival deviation of these subpaths at the user terminal are usually slight [16]. Therefore, we approximate the Doppler frequency of all subpaths of the -th dominant path to the same value , i.e., , . Such an approximation simplifies the model in (II-A) to the follows

 YDD[k,l] ×XDD[⟨k−k′⟩N,(l−li)M]+ZDD[k,l], (14)

where is the delay-Doppler domain channel, which is defined as

 HDD[k,l]=1√NN−1∑j=0hMCP+j(M+MCP),le−¯ȷ2πkjN. (15)

Equality in (II-A) holds precisely if there is only one subpath in a dominant path (i.e., ).

It is noticed that when , (II-A) is transformed into

 YDD[k,l]N→∞=1√NM−1∑l′=0⌈N/N22⌉−1∑k′=⌈−N/−N22⌉HDD[k′,l′]e¯ȷ2πlk′(M+MCP)N (16)

which is the same as that in [32] and [22]. Therefore, the model in [32] and [22] can be seen as a special case of our proposed input-output relationship when . Moreover, comparing (II-A) with (II-A), we can find that the phase compensation in (II-A) uses the information of each dominant path, which makes the model more accurate in the practical system.

Note that we choose the OFDM-based OTFS system similar to [9] in this paper, which can be achieved simply as an overlay of the widely used OFDM system, instead of the spectral efficient OTFS model in [28, 29, 10]. If fewer CPs are used to improve spectral efficiency as in [28, 29, 10], the input-output relationship will have an additional case owing to the ISI between OFDM symbols within an OTFS symbol, and the only difference between the two cases is the exponential term, which implies that the basic idea of using the channel path information to modify the phase compensation matrix of the sensing matrix to improve the channel estimation performance, as shown later, is still valid in the spectral efficient OTFS model.

### Ii-B Massive MIMO-OTFS System

We consider the downlink transmission in a massive MIMO-OTFS system. The BS deploys antennas and serves single-antenna user terminals. We consider the centralized massive MIMO, where the uniform linear array (ULA) is equipped at the BS, and the antenna spacing is set to half wavelength. Without loss of generality, we focus on one user terminal and disregard the dependency of the channel on user index for simplicity.

Similar to (II-A), the symbols received at the user terminal in the delay-Doppler domain can be expressed as

 YDD[k,l] ≊1√NNt−1∑nt=0P−1∑i=0⌈N/N22⌉−1∑k′=⌈−N/N22⌉HDDS[k′,li,nt]e¯ȷ2πl(ki+~ki)(M+MCP)N ×XDDS[⟨k−k′⟩N,(l−li)M,nt]+ZDD[k,l], (17)

where is transmitted symbols in the delay-Doppler-space domain, is the delay-Doppler-space domain channel, which is defined as [13]

 HDDS[k,l,nt] =1√NN−1∑j=0P−1∑i=0S−1∑si=0hsie¯ȷ2π(MCP+j(M+MCP)−l)Tsνsi ×δ(ℓTs−τi)e¯ȷπntsinφsie−¯ȷ2πkjN, (18)

where is the angle of departure (AoD) of the -th subpath. To exploit the sparsity of the beam domain, the delay-Doppler-beam domain channel is obtained by applying normalized discrete Fourier transform (DFT) for the delay-Doppler-space domain channel along the space-dimension as [32, 34]

 HDDB[k,l,b] =1√NtNt−1∑nt=0HDDS[k,l,nt]e−¯ȷ2πbntNt =√NNtP−1∑i=0S−1∑si=0hsie¯ȷ2π(MCP−l)TsνsiΞN(k−NTsymνsi) (19)

where is the beam index, and . From (II-B), we can find that the dominant elements of are distributed in the positions where , and , which means that the channel has the 3D sparsity over the delay-Doppler-beam domain [32, 1]. By combining (II-B) and (II-B), the received symbols can be rewritten as

 YDD[k,l] ≊1√NNt/Nt22−1∑b=−Nt/Nt22P−1∑i=0⌈N/N22⌉−1∑k′=⌈−N/N22⌉HDDB[k′,li,b]e¯ȷ2πl(ki+~ki)(M+MCP)N ×XDDB[⟨k−k′⟩N,(l−li)M,b]+ZDD[k,l], (20)

where

 XDDB[k,l,b]Δ=1√NtNt∑nt=0e¯ȷ2πrntNtXDDS[k,l,nt]. (21)

Equality in (II-B) holds precisely if there is only one subpath in a dominant path. Similarly, when , the received symbols are given by

 YDD[k,l] ×XDDB[⟨k−k′⟩N,(l−l′)M,b]+ZDD[k,l]. (22)

According to downlink massive MIMO-OTFS system models and the sparsity of the delay-Doppler-beam domain channel , the downlink CSI acquisition is actually a sparse signal reconstruction problem [32]. A variety of compressive sensing (CS) algorithms can be used to solve this problem. Next, we will give the model of downlink CSI acquisition and propose the corresponding deterministic pilot design and MSMCE algorithm.

## Iii Downlink CSI acquisition for Massive MIMO-OTFS Systems

In this section, we first model the downlink CSI acquisition as a sparse signal reconstruction problem. To improve sensing performance, the deterministic pilots are designed, and the pilot overhead is also discussed. Then, we propose a MSMCE algorithm to reconstruct the sparse delay-Doppler-beam domain channel and analyze its performance.

### Iii-a Downlink CSI Acquisition Model

Fig. 1 shows the position of pilots in the delay-Doppler domain at one antenna. We consider that the position of pilots in the delay-Doppler domain at each transmit antenna is the same and given by

 k=kp,⋯,kp+Np−1,l=lp,⋯,lp+Mp−1, (23)

where and are the initial position of the pilots in the Doppler domain and delay domain, respectively, and and are the lengths of pilots. The guard intervals (i.e., zero symbols) are demanded to eliminate the interference between the data and pilots.

Delay-Doppler-beam domain channels have finite support and along the delay-dimension and the Doppler-dimension, respectively [12]. Hence for the delay-dimension, the guard intervals should be placed at the beginning and the end of the pilots with the length . However, for the Doppler-dimension, is typically non-integer in practical systems, which means that according to the characteristic of in (II-B), the magnitude at each is not zero and decreases as moves away from . Thus we only consider for and replace the values outside the range with zero, and the guard intervals should be placed at the beginning and the end of the pilots with the length . Therefore, the range of and of the delay-Doppler-beam domain channel are limited to and , respectively. Moreover, the data is placed in the position except for the pilot and the guard interval.

The position of received symbols for channel estimation is the same as (23). Therefore, (II-B) can be rewritten as

 yDD≊(Φ⊙XDDB)hDDB+zDD, (24)

where is the pilot matrix with element of index , where , , , and . In (24), and are the vector forms of received symbols and the additive noise with element and of index . is the vector form of the delay-Doppler-beam domain channel with element of index , is the phase compensation matrix with element of index , which is defined as

 ϕ(l,l′)=⎧⎪ ⎪⎨⎪ ⎪⎩e¯ȷ2πl(ki+~ki)(M+MCP)N,l′∈Del,l′=li1,l′∉Del, (25)

where Del is the tap index set of the delay of all dominant paths, i.e., . is selected from the set Dop which consists of the integer and fractional tap indexes of the Doppler frequency of all dominant paths, i.e., . When , according to (II-B), in (24) is converted to with element of index . And in this case, the model is the same as that in [32].

We can rewrite (24) as

 yDD≊ΘhDDB+zDD, (26)

where . Thus the downlink CSI acquisition of massive MIMO-OTFS systems is converted to a sparse signal reconstruction problem, where is the sparse vector to be recovered, and is the sensing matrix. When , is converted to . Hence we have two kinds of sensing matrix that can be used in CS, and the only difference between them is the phase compensation matrix. The phase compensation matrix is more accurate but requires both path delay and Doppler frequency of each dominant path, which can not be directly obtained. In contrast, the phase compensation matrix is irrelevant to the channel, although it is inaccurate. To combine the advantages of both, the MSMCE algorithm is proposed and will be discussed in detail later. Prior to that, we give the deterministic pilot design in the next subsection.

### Iii-B Deterministic Pilot Design

In order to recover the sparse vector reliably, the sensing matrix must be designed carefully and satisfies the restricted isometry property (RIP) proposed in [3]

. The RIP implies that the coherence (i.e., inner product) between columns in a sensing matrix should be as small as possible to obtain a good sensing performance. Although the random matrix is usually used as the sensing matrix due to its near-optimality

[4], it is hard to be implemented and costs a lot of memory consumption for its randomness. Therefore, we focus on the deterministic pilot design for practical communication systems.

Since the phase compensation matrix of the sensing matrix is related to the channel, which varies with time, the pilot matrix is what we analyzed exactly. The design of pilot matrix is equivalent to the design of transmitted pilots in the delay-Doppler-beam domain with their position in the delay-Doppler domain at each beam. Note that the pilots in the delay-Doppler-space domain (i.e., the pilots in the delay-Doppler domain at each BS transmit antenna) can be obtained using (21), and such a transformation will not affect the position of pilots in the delay-Doppler domain.

We first analyze the relationship between the pilot matrix and the pilot position at one beam. As shown in Fig. 2, each dashed box contains pilot symbols, which are used to construct the corresponding column of the pilot matrix in Fig. 2. The pilots at each beam can construct columns totally.

However, we can find that if we directly use the position of pilots shown in Fig. 1, there will be some zero symbols in the pilot matrix, which will affect the coherence between columns of the pilot matrix and make the pilot design difficult. Therefore, based on Fig. 1, we propose a new pilot position design shown in Fig. 3. Specifically, along the Doppler-dimension, the first pilots are added to the end (marked in yellow grids in Fig. 3), and the last pilots are added to the beginning (marked in pink grids in Fig. 3) while the guard intervals are placed at the beginning and the end of pilots with the length . Along the delay-dimension, the last pilots are added to the beginning (marked in thick-black grids in Fig. 3) while the guard intervals are placed at the end of the pilots with the length of . The data is placed in the position except for the pilot and the guard interval.

Next, we discuss the pilot design at a given beam and between different beams. For a given beam, we assume that the -th column of the pilot matrix (i.e., the black dashed box in Fig. 2) constructed by the pilots at this beam shown in Fig. 3 is defined as

 black: c(b+Nt2)MgNg+1=pMp0⊗pNp0, (27)

where denotes a sequence of length and is cyclically shifted by symbols. According to the designed pilot position and its relationship with the pilot matrix, different columns (i.e., dashed boxes with different colors in Fig. 2) can be expressed as

 red: c(b+Nt2)MgNg+2    =pMp0⊗pNp1green: c(b+Nt2)MgNg+Ng   =pMp0⊗pNpNg−1purple: c(b+Nt2)MgNg+Ng+1=pMp1⊗pNp0brown:c(b+Nt2)MgNg+MgNg=pMpMg−1⊗pNpNg−1. (28)

Therefore, according to the property of Kronecker product , the pilot sequence should be orthogonal to its cyclically shifted version to ensure the orthogonality between columns of the pilot matrix at the given beam, which is defined as the orthogonality condition.

The above pilot design can be used at up to beams, where and . The difference between these beams is that should be replaced with , or with , where and . Note that at least one of the two pilot sequences needs to be replaced to ensure the orthogonality between columns. However, for the beam other than these beams, a new pair of pilot sequences should be used. And the coherence of the columns constructed by this new pair of pilot sequences and the columns constructed by other pairs of pilot sequences should be as low as possible to ensure the sensing performance of the entire pilot matrix, which is defined as the low coherence condition.

To sum up, the designed deterministic pilots should satisfy both the orthogonality condition and the low coherence condition. ZC sequence is quite suitable for the proposed deterministic pilot design since it is a cyclic orthogonal sequence with good autocorrelation and low cross-correlation. We define the ZC sequence cyclically shifted by symbols with root and length as , and its -th element can be expressed as

 zL,rc[k]=1√Le¯ȷπr(k−c)L((k−c)L+(L)2)L, (29)

where . Note that in the current system should be or . The cyclic orthogonality of the ZC sequence makes it satisfy the orthogonality condition. Meanwhile, when the length of ZC sequence is prime, the magnitude of cross-correlation between two normalized ZC sequences with different roots is [5]. It means that when we need a new pair of pilot sequences, a pair of ZC sequences with a new pair of roots can be chosen, and they satisfy the low coherence condition. Therefore, based on the ZC sequence, the proposed deterministic pilot design is summarized in Algorithm 1, where the pilots at all beams are rearranged as a matrix of size .