Joint timing synchronization and channel estimation based on ZCZ sequence set in SCMIMOFDE system
 Yu Wang^{1},
 Shiwen He^{1}Email author,
 Yu Sun^{1},
 Qinzhen Xu^{1},
 Haiming Wang^{1} and
 Luxi Yang^{1}
https://doi.org/10.1186/s1363801605423
© Wang et al. 2016
Received: 28 August 2015
Accepted: 28 January 2016
Published: 16 February 2016
Abstract
In this paper, we investigate the channel estimation and time synchronization problem based on the zero correlation zone (ZCZ) sequence set for single carrier multipleinput multipleoutput frequency domain equalization (SCMIMOFDE). A factorized construction of ZCZ sequence set considering the properties of both ZCZ and nonzero correlation zone (NCZ) is proposed with efficient generator and correlator, which can be in favor of timing synchronization and channel estimation both in performance and computation complexity. Using the ZCZ sequence set, a new algorithm called twice sectionmaximum algorithm is put forward to eliminate energy interference among channels from different transmitting antennas to the same receiving antenna. The performance achieved by the developed algorithm which performs joint timing synchronization and channel estimation with lower computational cost is better than that of the conventional method.
Keywords
1 Introduction
In wireless communication system, it is well known that timing synchronization (TS) and channel estimation (CE) are two main tasks achieved by training signal in the receiver. Since accurate TS and CE play important roles in improving the overall system performance, the design of the training sequence set which is known to both transmitter and receiver is the crucial point. Motivated by this, this paper takes TS and CE of single carrier multipleinput multipleoutput frequency domain equalization (SCMIMOFDE) [1, 2] transmission mode into consideration. In [3–6], the periodic complementary sequence set (PCSS) is chosen to be the training sequence set. In the PCSS, there are N _{ T } groups of sequences which are transmitted from N _{ T } antennas, separately, and each group contains N _{ T } different sequences. However, N _{ T } sequences need N _{ T } cyclic prefixes (CP) to counteract interference of multipath time delay channel, which exaggerates overload. This problem is solved by generating special PCSS [6]; but it could not get rid of the restriction of the relation between length of CP L _{CP}, the length of PCSS L, and number of PCSS N _{ S } transmitted in one antenna, which is L _{CP}≤L/N _{ S }. In [7], the ZadoffChu (ZC) sequence is selected as the training sequence; but ZC sequence is multiphase and introduces multiply operation in the receiver which costs much computational resource. This paper chooses zero correlation zone (ZCZ) sequence set as the training sequence set for the SCMIMOFDE system to overcome all drawbacks mentioned above.
ZCZ sequence set is first introduced to enhance TS robustness in code division multiple access (CDMA) system [8, 9]. It has also shown that ZCZ is the optimal CE training sequence for MIMO system [10–12]. However, few works use ZCZ sequence set to perform TS due to the existence of the uncertain side lobes of autocorrelation and crosscorrelation in nonzero correlation zone (NCZ). The author pointed out that in his construction, there are only m shifting points where the correlation values are not equal to zero in NCZ area, where m is the size of the ZCZ sequence set; but the magnitude of those values are not mentioned, which is also one of the properties in NCZ [13].
There are two kinds of constructions of ZCZ sequence set mainly, whose optimized objects are the length of ZCZ. One is based on the properties of ZCZ sequence set either in its direct domain or transform domain [14]. The other is based on the fundamental sequence set, such as the following: those constructed in [15, 16] are based on complementary sequence set (CSS) [17], and those generated in [18, 19] are based on perfect sequence (PS) [20] using interleaving technique. All of these ZCZ sequence sets \(\left (L,M,Z\right)\) generated above are bounded by a general bound Z<Z _{ g } and achieve expected bound Z=Z _{ e }, where L, M, and Z denote sequence length, sequence number, and ZCZ length, separately; Z _{ g }=L/M [21], \(Z_{e}=\left (k^{2}2\right)L/\left (k^{2}M\right)\) [18], k is the number of polyphase. In this paper, we propose the factorized construction of ZCZ sequence set considering both ZCZ and NCZ, which is an extension of above constructions based on fundamental sequence set.
In order to achieve high speed of signal processing, efficient correlator is put forward for Golay sequence set [22, 23]. In addition, several efficient generations and correlations are also developed for PCSS [24–28]. The authors of [29, 30] have proposed efficient calculation of fast Fourier transform (FFT) of ZC sequence. Also, an efficient correlation method for the corresponding ZCZ sequence set is developed in [13].
In this paper, we focus on the investigation of the joint TS and CE based on ZCZ sequence by taking the advantage of the ZCZ in CE and overcoming the shortcoming in the TS in multipath communication system. At the same time, in order to reduce the computational complexity and speed up the implementation, an efficient generator and correlator construction is obtained by extending the existing generation methods.
This paper is organized as follows. In Section 2, we present system model and state problems we need to address. In Section 3, a factorized construction of ZCZ sequence set is proposed to respond stating problem in Section 2 with efficient generator and correlator. Then, in Section 4, the constructed ZCZ sequence set is applied to joint MIMO TS and CE. At the same time, a new algorithm called twice sectionmaximum algorithm is put forward to eliminate channel energy interference. In Section 5, performance and computational complexity is analyzed in the simulation based on the selected ZCZ sequence set and the given communication scenario. And the last, conclusions are provided in Section 6.
2 System model and problem statement
where \(E_{\mathbf {s}_{i},\mathbf {s}_{j}}={\mathbf {s}_{i}}^{\mathrm {H}}\mathbf {s}_{j}\). The sequence set satisfying (5) is called an optimal training sequence set.
then it is clear that \(\mathcal {Z}\left (L,N_{T},Z_{D}\right)\) meets (6) based on the definition of ZCZ sequence set [18], where \(\mathcal {Z}\left (L,M,Z\right)\) is a ZCZ sequence set with the period of sequences L, the number of sequences M, and the length of ZCZ Z; and it means that \(\mathcal {Z}\left (L,N_{T},Z_{D}\right)\) can serve as optimal training sequence set for CE in MIMO system.
There are already many different ways to construct ZCZ sequence set used for CE as discussed in Section 1, but few of them consider the properties of NCZ which need to be taken into account for TS. In the following section, we give a construction of the ZCZ sequence set which is suitable for joint TS and CE considering the properties of NCZ.
3 Factorized construction of ZCZ sequence set
The method of the factorized construction is introduced to apply to the generation of ZCZ sequence set based on base ZCZ sequence set, which has the shortest length according to the size of the set, with interleaving technique.
At the same time, the corresponding efficient correlator is proposed to speed up the implementation.
3.1 Efficient generator of ZCZ sequence set
 ■Step 1: Let the base ZCZ sequence set be \(\mathcal {A}\,=\,\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1} =\mathcal {Z}\left (L_{a},M,Z_{a}\right)\), where\(\left a_{m,l}\right =1\). We enlarge the length of above sequences from L _{ a } to L=M ^{ N } L _{ a } by inserting M ^{ N }−1 zeros after each element, \(N\in \mathbb {N}\), defined by$$\mathbf{a}_{m}=\left[a_{m,0},\cdots,a_{m,l},\cdots,a_{m,L_{a}1}\right]^{\mathrm{T}}, $$Then, let the initial matrix be \(\mathbf {A}^{0}\,=\,\left [\mathbf {A}^{0}_{0},\cdots,\mathbf {A}^{0}_{m},\cdots,\mathbf {A}^{0}_{M1}\right ]\), where \(\mathbf {A}^{0}_{m}=\text {Circ}\left \{\mathbf {a}^{0}_{m},M^{N1},mM^{N1}\right \}\).$$\mathbf{a}^{0}_{m}=\left[a_{m,0},\mathbf{0}_{1\times \left(M^{N}1\right)},\cdots,a_{m,L_{a}1},\mathbf{0}_{1\times \left(M^{N}1\right)}\right]^{\mathrm{T}}. $$
 ■Step 2: Let U ^{ n } (n=0,1,⋯,N−1) be the nth matrix given as \(\mathbf {U}^{n}=\left [\mathbf {u}^{n}_{0},\cdots,\mathbf {u}^{n}_{m},\cdots,\mathbf {u}^{n}_{M1}\right ]\), which satisfies \(\left (\mathbf {U}^{n}\right)^{\mathrm {H}}\mathbf {U}^{n}=M\mathbf {I}_{M}\), where \(\mathbf {u}^{n}_{m}=\left [u_{0,m}^{n},\cdots,u_{i,m}^{n},\cdots,u_{M1,m}^{n}\right ]^{\mathrm {T}}\), \(\left u_{i,m}^{n}\right =1\); and \(\mathbf {u}^{n}_{m}\) is enlarged by filling in M ^{ N−1−n }−1 zeros asThen, let unitarylike matrix be \(\mathbf {V}^{n}\,=\,\left [\mathbf {V}^{n}_{0},\cdots \!,\mathbf {V}^{n}_{m},\cdots \!,\mathbf {V}^{n}_{M1}\right ]\), where \(\mathbf {V}^{n}_{m}=\)$${\fontsize{8.8pt}{9.6pt}{\begin{aligned} \mathbf{v}^{n}_{m}=\left[u_{0,m}^{n},\mathbf{0}_{1\times \left(M^{N1n}1\right)},\cdots,u_{M1,m}^{n},\mathbf{0}_{1\times \left(M^{N1n}1\right)}\right]^{\mathrm{T}} \end{aligned}}} $$$${\fontsize{8.8pt}{9.6pt}{\left\{ \begin{aligned} &\text{Circ}\left\{\mathbf{v}^{n}_{m},M^{N2n},mM^{N2n}\right\}&,&n=0,1,\cdots,N2\\ &\mathbf{v}^{n}_{m}&,&n=N1 \end{aligned} \right..}} $$
 ■Step 3: Let W ^{ n } be the nth coefficient matrix, given as W ^{ n }=where \(\left {w_{i}^{n}}\right =1\).$${\fontsize{8.8pt}{9.6pt}{\left\{ \begin{aligned} &\mathbf{I}_{M^{N2n}}\otimes \text{diag}\left\{{w_{0}^{n}},{w_{1}^{n}},\cdots,w_{M1}^{n}\right\}&,&n=0,\cdots,N2\\ &\text{diag}\left\{{w_{0}^{n}},{w_{1}^{n}},\cdots,w_{M1}^{n}\right\}&,&n=N1 \end{aligned}\right.,}} $$
 ■Step 4: A recursive sequence generation method is proposed by A ^{ n+1}=A ^{ n } V ^{ n } W ^{ n }. Then,$$ \begin{aligned} \mathbf{A}^{N}&=\left[\mathbf{a}^{N}_{0},\cdots,\mathbf{a}^{N}_{m},\cdots,\mathbf{a}^{N}_{M1}\right]\\ &=\mathbf{A}^{0}\mathbf{V}^{0}\mathbf{W}^{0}\mathbf{V}^{1}\mathbf{W}^{1}\cdots \mathbf{V}^{N1}\mathbf{W}^{N1}. \end{aligned} $$(7)
Note that the matrix V ^{ n } is the interleaving matrix, which interleave the columns of A ^{ n }; and the matrix W ^{ n } gives an coefficient to each column of A ^{ n } V ^{ n }.
Lemma 1.
In the above generation, \(\left \{\mathbf {a}^{N}_{m}\right \}_{m=0}^{M1}=\mathcal {Z}\left (L,M,Z\right)\), and the length of ZCZ satisfies Z≥M ^{ N } Z _{ a }. Specifically, if the ZCZ length between a _{ M−1} and a _{0} is equal to Z _{ a }, which is the ZCZ length of \(\mathcal {A}=\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1}=\mathcal {Z}\left (L_{a},M,Z_{a}\right)\), then Z=M ^{ N } Z _{ a }.
Proof 1.
See Appendix 1. □
There have been many similar properties of the length of ZCZ being obtained so far. For example, the bound of ZCZ length is deduced in [21], and the expected bound of ZCZ length of Nphase sequence set is shown in [18]. This lemma shows the worst case of the length of ZCZ of the interleaving constructed sequence set. In fact, the ZCZ length is not only determined by the ZCZ length of base sequence set but also the order of each ZCZ length between every two sequence among the set. Considering the complex cases of the order, we only give the best case using the shifting PS as base ZCZ sequence set in the following theorem. At the same time, the property in NCZ area is given and proofed.
Theorem 1.
Let PS \(\mathbf {a}=\left [a_{0},a_{1},\cdots, a_{L_{a}1}\right ]^{\mathrm {T}}\), where \(\left a_{i}\right =1\); and let \(\left \{\mathbf {a}_{m}=\text {Circ}\left \{\mathbf {a},1,{mM}_{1}\right \}\right \}_{m=0}^{M1}\), where L _{ a }=M M _{1}. Then, for \(\left \{\mathbf {a}^{N}_{m}\right \}_{m=0}^{M1}=\mathcal {Z}\left (L,M,Z\right)\), the length of ZCZ satisfies \(Z\geq \left (M_{1}1\right)M^{N}+\left (M2\right)M^{N1}\); and in NCZ, the area with nonzero value is bounded by \({mM}_{1}M^{N}+\left (M_{1}1\right)M^{N}\leq \left l\right \leq \left (m+1\right)M_{1}M^{N}+M^{N}1\), where m=0,1,⋯,M−2, and l is the shifting point in correlation.
Proof 2.
See Appendix 2. □

(p1) \(Z\geq Z_{a}{N_{T}}^{N}+\left (N_{T}2\right){N_{T}}^{N1}\);

(p2) \(Z_{\text {MN}}\leq \left (\left (m+1\right)M_{1}M^{N}+M^{N}1 \right)  \left ({mM}_{1}M^{N}+\left (M_{1}1\right)M^{N}\right) +1 \leq 2{M}^{N}\);

(p3) \(Z_{\text {MZ}} = L_{a}M^{N1}  Z_{\text {MN}} \geq \left (L_{a}/M2\right){M}^{N} = \left (M_{1}2\right)M^{N}\);

(p4) R _{ A }≤δ and R _{ C }≤δ.
Since this factorized construction of ZCZ sequence set is an extension of existing construction based on interleaving method, here, we state the distinctive points which are different from the conventional one using the same technique. First, the interleaving construction is factorized by the matrix multiplication. This factorized formation is used widely in the construction and application of CSS [4–6, 25, 27, 28]. Note that it is the basis to deduce the efficient correlator in the next subsection. Second, since the ZCZ part is well qualified to perform CE thanks to the researches which have been done before, our works focus on the properties of NCZ part which contributes to TS: The coefficient matrix, which can be changed easily in application to generate different ZCZ sequence sets with different R _{ A } and R _{ C }, is added to the iteration construction; and the area with nonzero value in NCZ is determined by Theorem 1, given particular shifting PS as base ZCZ sequence set.
3.2 Efficient correlator of proposed ZCZ sequence set
Based on the proposed generation of ZCZ sequence set, an efficient correlator corresponding is derived and expressed in the following theorem.
Theorem 2.
where the complexity of complex multiplication is j L L _{ a }+N L M ^{2} and addition is \(jL\left (L_{a}1\right)+NLM\left (M1\right)\) in the condition that \(\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1}\) are results of the circulation of j (j=1,2,⋯,M) sequences in \(\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1}\).
Proof 3.
See Appendix 3. □
Computation complexity comparison of polyphase correlators
Correlator  Complex multiplication  Complex addition 

Direct  M L×L  \(ML \times \left (L1\right) \) 
Radix2 FFT  \(\frac {1}{2}\left (M+1\right)L \log _{2}L+ML\)  \(\left (M+1\right)L \times \log _{2}{L}\) 
Efficient  \(ML \left (L_{a}/M+NM\right) \)  \(ML \left (\frac {L_{a}1}{M}+N\left (M1\right)\right) \) 
[13]  \(ML \times \left (L/M\right)\)  \(ML \times \left (L1\right)/M \) 
Computation complexity comparison of binary/ quadriphase correlators
Correlator  Complex multiplication  Complex addition 

Direct  0  \(ML \times \left (L1\right) \) 
Radix2 FFT  \(\frac {1}{2}\left (M+1\right)L \log _{2}L+ML\)  \(\left (M+1\right)L \times \log _{2}{L}\) 
Efficient  0  \(ML \left (\frac {L_{a}1}{M}+N\left (M1\right)\right) \) 
[13]  \(ML \times \left (L/M\right)\)  \(ML \times \left (L1\right)/M \) 
4 Joint MIMO TS and CE using ZCZ sequence set
In this section, we introduce how to use the ZCZ sequence as a training sequence for joint TS and CE. Using the base ZCZ sequence set \(\mathcal {Z}\left (L_{a},N_{T},Z_{a}=L_{a}/N_{T}1\right)\) which is obtained from shifting PS, the proper ZCZ sequence set \(\mathcal {Z}\left (L={N_{T}}^{N}L_{a},N_{T},Z\right)=\left \{\mathbf {z}_{p}\right \}_{p=0}^{N_{T}1}\) is generated as training sequence set, namely, s _{ p }=z _{ p }, which has the properties of (p1), (p2), (p3), and (p4). In order to achieve effectively the TS and CE, here, we further give two assumptions: (a1) \(Z_{a}{N_{T}}^{N}+\left (N_{T}2\right){N_{T}}^{N1}\geq Z_{D}\); (a2) \(\frac {L_{a}/N_{T}2}{L_{a}/N_{T}}\approx 1\).
4.1 Mechanism of joint MIMO TS and CE
where \(E_{\mathbf {z}}=E_{\mathbf {z}_{i},\mathbf {z}_{i}}=L\left (i=0,\cdots,N_{T}1\right)\), and \(\mathbf {D}_{\mathbf {z}_{p},\mathbf {z}_{i}}=\text {Circ}\left \{\mathbf {d}_{\mathbf {z}_{p},\mathbf {z}_{i}},Z_{D},0\right \}\) is a \(\left (LZ_{D}\right)\times Z_{D}\) matrix called disturbance impulse responses (DIRs), where \(\mathbf {d}_{\mathbf {z}_{p},\mathbf {z}_{i}}=\left [R_{\mathbf {z}_{p},\mathbf {z}_{i}}\left (Z_{D}\right),R_{\mathbf {z}_{p},\mathbf {z}_{i}}\left (Z_{D}+1\right),\cdots,R_{\mathbf {z}_{p},\mathbf {z}_{i}}\left (L1\right)\right ]^{\mathrm {T}}\).
4.2 Solutions of the energy interference among channels
When the energy interference among channels is severe, just in the case like Fig. 3, (16) works well. However, when that interference is mild, meaning that energies of all channels are similar, (16) loses nearly \(\left (N_{T}1\right)/N_{T}\) information which is useful for TS. This deficiency of (16) makes the performance of TS decline.
Taking advantage of the idea of solution 2, we propose an algorithm called twice sectionmaximum algorithm to perform TS and CE jointly. In the following algorithm, we use \(\mathcal {C}\left \{\mathbf {r}_{q},\mathcal {Z}\left (L,N_{T},Z\right)\right \}\) to denote the correlation operation between column vector r _{ q } and N _{ T } ZCZ sequences in the set \(\mathcal {Z}\left (L,N_{T},Z\right)\) utilizing the efficient correlator obtained from Theorem 2.
5 Simulations
5.1 Communication scenario setting
Simulation parameters
Parameter  mmWave  Channel model 

channel  D of 802.11ac  
Number of transmitting antenna  4  4 
Number of receiving antenna  4  4 
Number of channel path  25  18 
Carrier frequency  45 GHz  5.25 GHz 
Symbol rate  440 MHz  12.5 MHz 
Maximum multipath time delay  100 ns  400 ns 
Number of transmit frames  10,000  10,000 
5.2 ZCZ sequence set selection

(1) Base ZCZ sequence set is \(\mathcal {A}=\left \{\mathbf {a}_{m}\right \}_{m=0}^{3}\), where \(\mathbf {a}_{m}=\text {Circ}\left \{\mathbf {a},1,4m\right \}\), and \(\mathbf {a}=\left [1,1,1,1,1,j,1,j,1,1,1,1,1,j,1,j \right ]^{\mathrm {T}}\) is a PS.

(2) Number of iterations is N=2.

(3) U ^{0} and U ^{1} are$$\mathbf{U}^{0}= \left[ \begin{array}{cccc} 1& 1& 1& 1 \\ 1& 1 &1 &1 \\ 1& 1& 1 &1 \\ 1 &1 &1 &1 \end{array} \right]; \mathbf{U}^{0}= \left[ \begin{array}{cccc} 1& 1& 1& 1 \\ 1& j &1 &j \\ 1& 1& 1 &1 \\ 1 &j &1 &j \end{array} \right]. $$

(4) Coefficients \({w_{m}^{n}}\) (m=0,1,2,3; n=0,1) are$$\left[ \begin{array}{cc} {w_{0}^{0}}& {w_{0}^{1}} \\ {w_{1}^{0}}& {w_{1}^{1}}\\ {w_{2}^{0}} & {w_{2}^{1}} \\ {w_{3}^{0}} &{w_{3}^{1}} \end{array} \right] = \left[ \begin{array}{cc} 1& 1 \\ 1& 1\\ j & j \\ j &j \end{array} \right]. $$
ZCZ sequence set
\(\mathcal {Z}\left (256,4,56\right)\)  

z _{0}  00310233330213220031130011200211 
00312011330231000031312211202033  
00310233330213221102201122311322  
22130233112013223320201100131322  
00310233330213222213312233022033  
00312011330231002213130033020211  
00310233330213223320023300133100  
22130233112013221102023322313100  
z _{1}  01100312302110010110102312030330 
01102130302132230110320112032112  
01100312302110011221213023101001  
23320312120310013003213001321001  
01100312302110012332320130212112  
01102130302132232332102330210330  
01100312302110013003031201323223  
23320312120310011221031223103223  
z _{2}  02330031310011200233110213220013 
02332213310033020233332013222231  
02330031310011201300221320331120  
20110031132211203122221302111120  
02330031310011202011332031002231  
02332213310033022011110231000013  
02330031310011203122003102113302  
20110031132211201300003120333302  
z _{3}  03120110322312030312122110010132 
03122332322330210312300310012310  
03120110322312031023233221121203  
21300110100112033201233203301203  
03120110322312032130300332232310  
03122332322330212130122132230132  
03120110322312033201011003303021  
21300110100112031023011021123021 
5.3 Performance and computational complexity analysis
where \({\sigma _{D}^{2}}\) is the energy of transmitting signal, and it is normalized by \(N_{T}{\sigma _{D}^{2}}=1\), so the last equation in (21) is demonstrated; and the NMSE of CE is \(\mathrm {{NMSE}_{\textit {CE}}}=\mathrm {E}\left \{\sum _{q=0}^{N_{R}1}\hat {\mathbf {h}}_{q}^{\mathrm {H}}\hat {\mathbf {h}}_{q}\right \}/\mathrm {E}\left \{\sum _{q=0}^{N_{R}1}\mathbf {h}_{q}^{\mathrm {H}}\mathbf {h}_{q}\right \}\). It indicates that both of the CE methods attain CRLB.
Computational cost comparison between joint TS and CE methods
Complex addition  Complex multiplication  

Twice sectionmaximum algorithm  \(N_{R} \left (\left (N_{T}1\right)N_{X} + N_{T}L\left (\frac {L_{a}1}{N_{T}}+N\left (N_{T}1\right)\right)\right)\)  0 
Wang’s method [33] (radix2 FFT)  \(N_{R}L \left (\left (N_{T}+1\right)Z_{D} 1+2\log _{2}L+\frac {N_{T}1}{L}\right)\)  \(N_{R}L \left (\left (N_{T}+1\right)Z_{D} +\log _{2}L+1\right) \) 
6 Conclusions
A factorized construction of ZCZ sequence set with the efficient generator and correlator has been investigated and applied to TS and CE jointly for the SCMIMOFDE system. Considering the properties of ZCZ and NCZ, the efficient generator can generate ZCZ sequence set suitable for CE and TS simultaneously. Further, the twice sectionmaximum algorithm is proposed and behaves well to eliminate the energy interference among channels; and the efficient correlator reduces complex multiplication, which can be avoided for efficient quadriphase or binary correlator, and complex addition from exponential order to linear order. Both efficient correlator and mechanism of joint TS and CE can reduce the processing time for the receiver.
7 Appendix 1
Proof of Lemma 1
 ■(a) \(\left l\right  \leq M^{N}Z_{a}\)where \(\hat {\mathbf {R}}_{l}=\text {diag}_{l}\left \{E_{\mathbf {a}_{0}}\mathbf {R}_{l},E_{\mathbf {a}_{1}}\mathbf {R}_{l},\cdots,E_{\mathbf {a}_{M1}}\mathbf {R}_{l}\right \}\) with the size of M ^{ N }×M ^{ N }, and \(\mathbf {R}_{l}=\text {diag}_{l}\left \{1,1,\cdots,1\right \}\) with the size of M ^{ N−1} × M ^{ N−1}. When \(\left l\right \!\leq \! M^{N1}\,\,1\), \(\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}=\)$$\left(\mathbf{T}_{l}\mathbf{A}^{0} \right)^{\mathrm{H}}\mathbf{A}^{0} =\left\{ \begin{aligned} &\hat{\mathbf{R}}_{l}&,&\leftl\right\leq M^{N1}1 \\ &\mathbf{0}_{M^{N}}&,& M^{N1}\leq \leftl\right \leq M^{N}Z_{a} \end{aligned} \right., $$When \(\left l\right  \geq M^{N1}\), \(\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}=\mathbf {0}_{M}\).$$\begin{aligned} &\left(\mathbf{V}^{N1}\mathbf{W}^{N1}\right)^{\mathrm{H}}\cdots\left(\mathbf{V}^{1}\mathbf{W}^{1}\right)^{\mathrm{H}} \hat{\mathbf{R}}_{l} \mathbf{V}^{0}\mathbf{W}^{0}\cdots \mathbf{V}^{N1}\mathbf{W}^{N1}\\ &=\left\{ \begin{aligned} &\mathbf{I}_{M}&,&l=0\\&\mathbf{0}_{M}&,&1\leq\leftl\right\leq M^{N1}1 \end{aligned} \right.; \end{aligned} $$
 ■(b) \(\left l\right = M^{N}Z_{a}+1\)$$ \left(\mathbf{T}_{l}\mathbf{A}^{0} \right)^{\mathrm{H}}\mathbf{A}^{0}\,=\,\left\{\!\! \begin{aligned} &\text{diag}_{1M^{N}}\left\{R_{\mathbf{a}_{M1},\mathbf{a}_{0}}\left(Z_{a}+1\right)\right\},l>0\\ &\text{diag}_{M^{N}1}\left\{R_{\mathbf{a}_{0},\mathbf{a}_{M1}}\left(Z_{a}1\right)\right\},l<0 \end{aligned} \right.. $$
If the length of ZCZ between a _{ M−1} and a _{0} is Z _{ a }, then \(R_{\mathbf {a}_{M1},\mathbf {a}_{0}}\left (Z_{a}+1\right)=R_{\mathbf {a}_{0},\mathbf {a}_{M1}}\left (Z_{a}1\right)\neq 0\), so \(\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}\neq \mathbf {0}_{M}\). Otherwise, the length of ZCZ between a _{ M−1} and a _{0} is larger than Z _{ a }, then \(R_{\mathbf {a}_{M1},\mathbf {a}_{0}}\left (Z_{a}+1\right)=R_{\mathbf {a}_{0},\mathbf {a}_{M1}}\left (Z_{a}1\right)= 0\), so \(\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}= \mathbf {0}_{M}\).
Combining (a) and (b), we can obtain the conclusion: If the length of ZCZ between a _{ M−1} and a _{0} is Z _{ a }, then Z=M ^{ N } Z _{ a }; Otherwise, Z>M ^{ N } Z _{ a }.
8 Appendix 2
Proof of Theorem 1
 ■
(a) First, we prove \(Z\geq \left (M_{1}1\right)M^{N}+\left (M2\right) M^{N1}\).
 ■
(a. 1) \(N=1\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}=\left (\mathbf {V}^{0}\mathbf {W}^{0}\right)^{\mathrm {H}}\left (\mathbf {T}_{l}\mathbf {A}^{0} \right)^{\mathrm {H}} \mathbf {A}^{0}\mathbf {V}^{0}\mathbf {W}^{0}\) satisfies the two situations:
 ■
(a. 1. 1) \(\left l\right  \leq \left (M_{1}1\right)M\)
The proof is the same as case (a) in Appendix 1.
 ■(a. 1. 2) \(\left (M_{1}1\right)M+1\leq \left l\right  \leq \left (M_{1}1\right)M+M2 \left (\mathbf {T}_{l}\mathbf {A}^{0} \right)^{\mathrm {H}}\mathbf {A}^{0}=\)$${} \left\{\begin{aligned} \text{diag}_{lM_{1}M}&\left\{R_{\mathbf{a}_{M_{1}Ml},\mathbf{a}_{0}}\left(M_{1}\right),R_{\mathbf{a}_{M_{1}Ml+1},\mathbf{a}_{1}}\left(M_{1}\right),\right.&\\ & \left. \cdots,R_{\mathbf{a}_{Ml},\mathbf{a}_{l\left(M_{1}1\right)M1}}\left(M_{1}\right) \right\},l>0&\\ \text{diag}_{l+M_{1}M}&\left\{R_{\mathbf{a}_{0},\mathbf{a}_{M_{1}M+l}}\left(M_{1}\right),R_{\mathbf{a}_{1},\mathbf{a}_{M_{1}M+l+1}}\left(M_{1}\right), \right.&\\ & \left.\cdots,R_{\mathbf{a}_{l\left(M_{1}1\right)M1},\mathbf{a}_{Ml}}\left(M_{1}\right) \right\},l<0 & \end{aligned} \right.. $$
So, \(R_{\mathbf {a}_{M_{1}M\left l\right +k},\mathbf {a}_{k}}\left (M_{1}\right)=0\) and \(R_{\mathbf {a}_{k},\mathbf {a}_{M_{1}M\left l\right +k}}\left (M_{1}\right)=0\), \(k=0,1,\cdots,\left l\right \left (M_{1}1\right)M1\). Then, \(\left (\mathbf {T}_{l}\mathbf {A}^{0} \right)^{\mathrm {H}}\mathbf {A}^{0}=\mathbf {0}_{M}\), and \(\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}=\mathbf {0}_{M}\).
 ■
(a. 2) N>1
Using the ZCZ sequence set generated in (a. 1), we can easily get \(Z\geq \left (M_{1}1\right)M^{N}+\left (M2\right)M^{N1}\) according to Lemma 1.
Combining (a. 1) and (a. 2), \(Z\geq \left (M_{1}1\right)M^{N}+\left (M2\right)M^{N1}\) is proved.
 ■
(b) Now, we prove both the autocorrelation and crosscorrelation have \(Z_{N}\leq 4\left (M1\right)M^{N}\) nonzero values.
 ■
(b. 1) When \({mM}_{1}M^{N}+M^{N}\leq \left l\right \leq {mM}_{1}M^{N}+ \left (M_{1}1\right)M^{N}1\), \(\left (\mathbf {T}_{l}\mathbf {A}^{0} \right)^{\mathrm {H}}\mathbf {A}^{0}=\mathbf {0}_{M}\) is deduced based on (23), where m=0,1,⋯,M−1. Thus, \(\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}=\mathbf {0}_{M}\).
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(b. 2) When \(0<\left l\right <M^{N}\), \(\left (\mathbf {T}_{l}\mathbf {A}^{N} \right)^{\mathrm {H}}\mathbf {A}^{N}=\mathbf {0}_{M}\) according to case (a) in Appendix 1.
Except case (b. 1) and (b. 2) where ZCZ across whole correlation, the area with nonzero value is bounded by \({mM}_{1}M^{N}+\left (M_{1}1\right)M^{N}\leq \left l\right \leq \left (m+1\right)M_{1}M^{N} +M^{N}1\), m=0,1,⋯,M−2.
9 Appendix 3
Proof of Theorem 2
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(a) Computational complexity of A ^{0}:
Let \(\mathbf {Y}^{\mathrm {T}}\left (\mathbf {a}^{0}_{m}\right)^{*}=\mathbf {g}_{m}^{0}\), thenwhere \(\mathbf {G}_{m}^{0}=\text {Circ}\left \{\mathbf {g}_{m}^{0},M^{N1},mM^{N1}\right \}\). So, only \(\mathbf {g}_{m}^{0}\) is needed to construct \(\mathbf {g}_{m}^{0}\); and it takes L M L _{ a } complex multiplication and \(LM\left (L_{a}1\right)\) complex addition. In general, if \(\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1}\) are results of the circulation of j (j=1,2,⋯,M) sequences in \(\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1}\), \(\left \{\mathbf {a}_{m}^{0}\right \}_{m=0}^{M1}\) are results of the circulation of j sequences in \(\left \{\mathbf {a}_{m}^{0}\right \}_{m=0}^{M1}\), too. Thus, j sequences in \(\left \{\mathbf {g}_{m}^{0}\right \}_{m=0}^{M1}\) are needed to construct \(\left \{\mathbf {g}_{m}^{0}\right \}_{m=0}^{M1}\); and it takes j L L _{ a } complex multiplication and \(jL\left (L_{a}1\right)\) complex addition.$$ \mathbf{Y}^{\mathrm{T}}\left(\mathbf{A}^{0}\right)^{\ast}=\mathbf{G}^{0}=\left[\mathbf{G}_{0}^{0},\cdots,\mathbf{G}_{m}^{0},\cdots,\mathbf{G}_{M1}^{0}\right], $$  ■
(b) Computational complexity of V ^{ i } W ^{ i } (0≤i≤N−2):
Let \(\mathbf {G}^{i+1}=\mathbf {G}^{i}\left (\mathbf {V}^{i}\mathbf {W}^{i}\right)^{*}\). Ifwhere \(\mathbf {G}_{m}^{i}=\text {Circ}\left \{\mathbf {g}_{m}^{i},M^{N1i},mM^{N1i}\right \}\). Then,$$ \mathbf{G}^{i}=\left[\mathbf{G}_{0}^{i},\cdots,\mathbf{G}_{m}^{i},\cdots,\mathbf{G}_{M1}^{i}\right], $$where \(\mathbf {G}_{m}^{i+1}=\text {Circ}\left \{\mathbf {g}_{m}^{i+1},M^{N2i},mM^{N2i}\right \}\). So, only \(\mathbf {g}_{m}^{i+1}\) is needed to construct \(\mathbf {g}_{m}^{i+1}\); and it takes L M ^{2} complex multiplication and \(LM\left (M1\right)\) complex addition.$$ \mathbf{G}^{i+1}=\left[\mathbf{G}_{0}^{i+1},\cdots,\mathbf{G}_{m}^{i+1},\cdots,\mathbf{G}_{M1}^{i+1}\right], $$  ■
(c) Computational complexity of V ^{ N−1} W ^{ N−1}:
Since V ^{ N−1} W ^{ N−1} is a matrix with the size M×M and G ^{ N−1} is a matrix L×M, it is easy to see that \(\mathbf {G}^{N}=\mathbf {G}^{N1}\left (\mathbf {V}^{N1}\mathbf {W}^{N1}\right)^{*}\) takes L M ^{2} complex multiplication and \(LM\left (M1\right)\) complex addition.
Combining (a), (b), and (c), we can obtain the conclusion: The efficient correlator takes j L L _{ a }+N L M ^{2} complex multiplication and \(jL\left (L_{a}1\right)+NLM\left (M1\right)\) complex addition in the condition that \(\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1}\) are results of the circulation of j sequences in \(\left \{\mathbf {a}_{m}\right \}_{m=0}^{M1}\).
Declarations
Acknowledgments
This work was supported by 863 Program of China under Grant 2015AA01A703, National Natural Science Foundation of China under Grants 61471120, and 61372101.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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