Zadoff-Chu sequence design for random access initial uplink synchronization

Transcription

1 Zadoff-Chu sequence design for random access initia upink synchronization Md Mashud Hyder and Kaushik Mahata arxiv: v [cs.it] 6 Apr 206 Abstract The autocorreation of a Zadoff-Chu (ZC) sequence with a non-zero cycicay shifted version of itsef is zero. Due to the interesting property, ZC sequences are widey used in the LTE air interface in the primary synchronization signa (PSS), random access preambe (PRACH), upink contro channe (PUCCH) etc. However, this interesting property of ZC sequence is not usefu in the random access initia upink synchronization probem due to some specific structures of the underying probem. In particuar, the state of the art upink synchronization agorithms do not perform equay for a ZC sequences. In this work, we show a systematic procedure to choose the ZC sequences that yied the optimum performance of the upink synchronization agorithms. At first, we show that the upink synchronization is a sparse signa recovery probem on an overcompete basis. Next, we use the theory of sparse recovery agorithms and identify a factor that contros performance of the agorithms. We then suggest a ZC sequence design procedure to optimay choose this factor. The simuation resuts show that the performance of most of the state of the art upink synchronization agorithms improve significanty when the ZC sequences are chosen by using the proposed technique. Index Terms Random Access, initia synchronization, Zadoff- Chu sequence, sparse representation, OFDMA. A. Background I. INTRODUCTION Many wireess communication systems adopt orthogona frequency-division mutipe access (OFDMA) technoogy for data transmission such as LTE [], WiMAX [2] etc. To maintain orthogonaity among the subcarriers in the upink of OFDMA system, the upink signas arriving at the enodeb from different user equipments (UEs) shoud be aigned with the oca time and frequency references. For this purpose, a UEs who want to set up connection to the enodeb must go through an initia upink synchronization (IUS) procedure. The IUS aows the enodeb to detect these users, and estimate their channe parameters. These estimates are then used to timesynchronize UEs transmissions and adjust their transmission power eves so that a upink signas arrive at the enodeb synchronousy and at approximatey the same power eve [3]. The initia upink synchronization is a contention based random access (RA) process. The process starts with the aocation of a pre-defined set of subcarriers caed Physica Random Access Channe (PRACH) in some pre-specified time sots known as a random access opportunity []. The downink synchronized UEs wiing to commence communication, referred to as the synchronizing random access terminas (RTs), use this opportunity by moduating randomy Department of Eectrica Engineering, The University of Newcaste, Austraia. seected codes (aso caed RA preambes) from a pre-specified RA code matrix onto the PRACH. As the RTs are ocated in different positions within the radio coverage area, their signas arrive at the enodeb with different time deays. At the receiving side, the enodeb needs to detect the transmitted RA codes, and extract the timing and channe power information for each detected code [3] [6]. The success of LTE in 4G ceuar networks has made it popuar. It is aso envisaged that many of its key ingredients wi aso dominate the 5G systems. In this work, we consider the upink data transmission protoco of an LTEike system [6]. Zadoff-Chu sequences [7], [8] are used as random mutipe access codes in the LTE system due to their perfect autocorreation properties i.e., the autocorreation of a ZC sequence with a non-zero cycicay shifted version of itsef is zero. However, the enodeb cannot fuy expoit the perfect autocorreation property of ZC sequences in different types of synchronization probems due to some factors. Hence, designing appropriate ZC sequences for different types of synchronization probems have received ots of research interests. A training-aided ZC sequence design procedure has been proposed in [9] for the frequency synchronization probem. The frequency synchronization deas with the estimation and compensation of carrier frequency offset (CFO) between the transmitter and a receiver. The CFO arises mainy due to the Dopper shift in downink synchronization [3], [9]. The effect of carrier frequency offset on the autocorreation property of the ZC sequences has been addressed in [0], []. A cosedform expression of the autocorreation between two cycicay shifted ZC sequences in presence of frequency offset has been deveoped in []. Consequenty, an empirica method has been proposed to design two different sets of ZC sequences: one for high frequency offset scenario and other for ow frequency offset scenario, which can be used for the frequency offset estimation of UEs. The ZC sequence design for the timing synchronization probem has been addressed in [9], [2], [3]. The timing synchronization deas with the estimation of timing offset which arises due to the random propagation deay between enodeb and user [9]. The method deveoped in [2] transmits partia Zadoff-Chu sequences on disjoint sets of equay spaced subcarriers for timing offset estimation. The signature sequences design procedure proposed in [3] considers the time and frequency synchronization in presence of carrier frequency offset between the enodeb and singe UE. To the best of our knowedge, the ZC sequence design for the IUS probem has not been addressed before. The IUS probem is different from the frequency and timing synchronization probems. In the IUS, the enodeb has to detect mutipe

2 2 RTs as we as extract their channe power and timing offset information. The effects of mutipe-access interference (MAI) and unknown mutipath channe impuse responses of RTs make the ZC sequence design probem chaenging. It this work, we consider the ZC sequence design probem for the IUS purpose. B. Some motivating exampes In this section, we observe that the performance of state of the art initia upink synchronization (IUS) agorithms [3] [6] can vary significanty depending on the RA code matrix. Moreover, the IUS agorithms may not yied the optimum performance with the RA code matrix generated by using the conventiona procedure. To expain the matter, we first describe the procedure of RA code generation [], [4]. To give exampe we consider the LTE system proposed in [6] where N = 644. Tota M = 839 number of adjacent subcarriers are aocated for the PRACH. The RA codes for the IUS are generated by cycicay shifting the Zadoff-Chu (ZC) sequence [7], [8]. The eements of the u-th root ZC sequence are given by Z u (k) = e iπuk(k+)/m, 0 k < M () where u is a positive integer with u < M. Different RA codes are obtained by cycicay shifting the u-th root ZC sequence. Let c (u) be the -th RA code. Then the (k+)-th eement of is given by c (u) + c (u) k+,+ = Zu {(k + n cs ) mod M}, 0 k < M. (2) Here ( mod M) := M /M, with r denoting the argest integer ess than or equa to r. In addition, n cs is an integer vaued system parameter which is reated to the wireess ce radius [0, eq. (7.0)]: n cs ( 20 3 γ τ d ) M T SEQ +n g, (3) where z is the smaest integer not ess than z, γ is the ce radius (km), τ d is the maximum deay spread (µs), T SEQ is the preambe sequence duration (µs), and n g is the number of additiona guard sampes. For a wireess ce radius of.3 km, typicay T SEQ = 800µs, and τ d < µs, which yieds n cs. The maximum number of RA codes that can be generated from a singe ZC root depends on the vaue of n cs. For exampe, setting n cs =, we can generate maximum M/n cs = 76 RA codes from the ZC root u. If we need more than 76 RA codes then we have to utiize mutipe roots. The procedure has been described expicity in Section-IV-B. Now, consider a RA upink synchronization scenario where K number of RTs are simutaneousy contending on the same PRACH channe. The channe impuse response of the RTs have a maximum order35 and totag = 50 codes are avaiabe in the RA code matrix : C = [c (u) c (u) 2 c (u) G ]. In practice, there are 64 RA codes avaiabe in each ce. However, some codes are reserved for contention free RACH [4]. We assume that there are 4 reserve codes. TABLE I IUS USER DETECTION PROBABILITIES BY SOME ALGORITHMS FOR DIFFERENT VALUES OF u AND n cs. SNR=0dB, TOTAL IUS USERS K = 3 AND P s DENOTES THE PROBABILITY OF SUCCESSFULLY DETECTING THE USERS. Agorithm P s with u = n cs = SMUD [4] SRMD [5] P s with u = 2 P s with u = 3 n cs = SMUD [4] SRMD [5] We can generate different RA code matrices by using different vaues of u and n cs. Tabe-I shows the user detection performances of some state of the art IUS agorithms for different code matrices 2. Note that the conventiona code design procedure (see Section-IV-B) suggests using n cs =. However, as can be seen in Tabe-I, with u = the probabiity P s of detecting a given number of codes successfuy by a agorithms are uniformy poor for n cs < 5. In contrast, P s for both SMUD [4] and SRMD [5] are above 0.9 for n cs 5. Tabe-I aso shows resuts for u = 2 and 3. However, comparing three different vaues of u, we see that the agorithms perform at their best with u = and n cs 5. This is interesting to note that just by taking some different combinations of n cs and u there is a significant variation of detection performance. Moreover, this happens uniformy for both the agorithms. The above exampe ceary demonstrates the importance of choosing the parametersuand n cs carefuy in order to ensure that the IUS agorithms can produce the best resuts. To the best of our knowedge, so far, there is no systematic study in the iterature addressing the issue. In the seque, we present a systematic procedure for finding good vaues of u and n cs that give the IUS agorithms an opportunity to perform at their best. C. Contributions In this work, we deveop a systematic procedure to study the dependency of code detection performance of the IUS agorithms on the code matrix and demonstrate an efficient code matrix design technique. The procedure can be outined as foows. We first deveop a data mode of the received signa at enodeb over the PRACH subcarriers. We represent the received signa as a inear combination of few coumns of a known matrix. This matrix is constructed by using the RA codes and some sub-fourier matrices. We further show that the data mode aows us to pose the IUS parameter estimation as a sparse signa representation probem on an overcompete basis [5], [6]. Thereby, sparse recovery agorithms can be used for the IUS parameter estimation probem. We then appy the compressive sensing theory 3 to recognize a factor that 2 In case of n cs > 5 in Tabe-I, we need two roots to generate the code matrix. 3 Sparse recovery agorithms and theories are generay deveoped in the research area caed Compressive sensing.

3 3 contros the RA code detection performance. The factor is caed matrix coherence. We show that the matrix coherence of the underying IUS probem depends on the code matrix. We then suggest a code matrix design procedure that can ensure the optimum vaue of matrix coherence. The simuation resuts ceary demonstrate that we can indeed significanty enhance the performances of the state of the art IUS parameter estimation agorithms [4], [5], [7] by using the code matrices generated by our suggested procedure. In particuar, we can achieve the optimum code detection performance in Tabe-I by using our preferred code matrices. A. Singe RT II. DATA MODEL In this work, we consider the upink data transmission protoco of an LTE-ike system [6]. However, the proposed anaysis can be extended for any LTE systems []. Consider a system with N subcarriers and N g cycic prefixes. Therefore, the ength of an OFDM symbo is N = N + N g. Tota M adjacent subcarriers are reserved for the Physica Random Access Channe (PRACH) [6]. We denote their indices by {j m : m =,2,...,M}. When a downink synchronized RT wants to start communicating via the enodeb, it must choose a coumn of a pre-specified M G random access (RA) code matrix C = [ c (u) c (u2) 2 c (ug) G ] (4) uniformy at random, and send this code via the PRACH during a random access opportunity. Note that if a RA codes are generated from same ZC root then u = u 2 = = u G. We now mention an important property of the ZC sequence which wi be hepfu for deveoping the RA data mode in the foowing section. One can express (2) as: c (u ) k+,+ = Zu (k)e i2πu kn cs/m e iπu n cs(n cs+)/m (5) with 0 k < M [7], [8]. Since the index of m-th random access subcarrier is j m, and the random access subcarriers are adjacent to each other, we have j k+ = j +k. Substituting k = j k+ j in (5) gives c (u ) k+,+ = Zu (k)e i2πu j k+ ncs M e iπu ncs M (ncs+ 2j). (6) Let T be a specific RT which attempts to synchronize with the enodeb at a particuar RA opportunity. In particuar, T chooses the coumn c (u ) from C and transmits it through PRACH to the enodeb. The RT transmits N + 2N g number of time domain channe symbos which are constructed from c. We denote these N + 2N g channe symbos by {w(k)} N+Ng k= N g. To construct the channe symbos, at first cacuate s(q) = N M m= c m, exp{i2πj m q/n}, q = I, (7) where I := {0,,2,...,N }, and c m, denotes the m th component of c. Subsequenty, {w(k)} N+Ng k= N g are constructed as { s(k mod N), Ng k N, w(k) = (8) 0, N k N +N g, where as usua, (k mod N) := k N k/n. For any integer k positive or negative, (k mod N) I. Note that the ast N g symbos in (8) are zero vaued guard symbos [6], [9]. Note that the upink protoco of standard LTE system [] is sighty different than the mode in (7). Standard LTE uses SC-FDMA upink. This requires the ZC sequences first DFT pre-coded and then mapped onto subcarriers by IDFT, i.e., in (7) the c (u ) m, wi be repaced by ĉ(u ) m, where ĉ(u ) m, its the DFT of c (u ) m,. Nevertheess, when the ength of ZC sequence is a prime number then the DFT of ZC sequence is another ZC sequence conjugated and scaed [20], [2]. As a resut, the proposed anaysis for ZC code in OFDMA structure is fuy compiant to the SC-FDMA. Suppose h(p), p I are the upink channe impuse response coefficients between the transmitter T and enodeb. Let {v(k)} N+Ng k= N g be the contribution of T in the symbos received by the enodeb during the IUS opportunity. These are deayed and convouted version of the transmitted symbos: N v(k) = e i2πkǫ/n p=0 N = e i2πkǫ/n p=0 h(p) w(k p d), k I h(p) s{(k p d) mod N}, k I. where deayddepends on the distance between T and enodeb, and ǫ is the carrier frequency offset (CFO). During an upink synchronization period, the CFO is mainy due to Dopper shifts and downink synchronization error, they are assumed to be significanty smaer than the subcarrier frequency spacing. Hence, the impact of CFO on the initia upink synchronization is generay negected [3], [4], [6]. Thus {v(k)} N k=0 is the resut of ength N circuar convoution of h, and s circuary shifted by d paces. The enodeb cacuates N point discrete Fourier transform (DFT) of v to obtain V(n) = N k=0 (9) v(k) exp( i2πkn/n), n I. (0) Let S and H denote the N point DFTs of s and h, respectivey. It is we known [22] that in the DFT domain the circuar convoution in (9) is equivaent to V(n) = H(n)S(n) exp( i2πdn/n). () Note that equation (7) is compacty given in matrix form as [ s(0) s() s(n ) ] = N F Θ c (u ), (2) where (.) and (.) denote transpose and compex conjugate transpose respectivey, Θ is an M N row seector matrix which is constructed by seecting M contiguous rows from an N N identity matrix where the first row of Θ is the j -th

4 4 row of the identity matrix, and the N N DFT matrix F is defined eement-wise as [F] k,m = exp{ i2π(k )(m )/N}. where [F] k,m is the eement of F at its k-th row and m-th coumn. Thereby, the DFT of (2) is given by [ S(0) S() S(N ) ] = Θ c (u ). (3) Using the reations in (6) and (3), and denoting φ + = πu n cs (n cs + 2j )/M, we can express () as V(j m ) = Z u (m)h(j m )e i2π N jm(u ( )n cs N M +d) e iφ. (4) for m =,2, M. Note that φ is known for a. After cacuating V, the enodeb forms the vector v = [ V(j ) V(j 2 ) V(j M ) ], (5) From the theory of DFT, it is we known that H(n)e i2πdn/n is the DFT of h circuary shifted by d paces [22], which is written compacty as [ H(0) H()e i2πd/n H(N )e i2πd(n )/N ] = Fh (d), (6) where h (d) is the circuary shifted version of the channe impuse response by d paces expressed as a vector: h (d) := [h(n d) h(n ) h(0) h() h(n d )]. (7) Hence by (6) and (7), we have [ H(j )e i2π N jd H(j 2 )e i2π N j2d H(j M )e i2π N jmd ] = ΘFh (d). (8) Hence (4) and (5) impy that v = E h (d), (9) E = diag(z u )diag(p ) ΘF. (20) h (d) = e iφ h (d) (2) p = [e i2π N j(u ( )n cs N M ),,e i2π N jm(u ( )n cs N M ) ] (22) where diag(p ) denotes a diagona matrix where the vector p is its diagona entry. Typicay, we know a number P, known as the maximum channe order [3], [4], such that h(k) = 0 for k P. In addition, the ce radius gives an upper bound D on d. Thus, by construction of h (d), ony first D +P of its rows are non-zero. Hence it is fine to truncate h (d) to a D+P dimensiona vector, and thus it is enough to work with ony first D +P coumns of E. By denoting N = P +D, we can rewrite (9) as v = Ẽ h (d) ( : N ), (23) where, Ẽ = diag(z u )diag(p )ΘF(:, : N ) here F(:, : N ) denotes the submatrix of F formed by taking its first N coumns and h (d) ( : N ) denotes the sub-vector of h (d) consisting of its first N components. Note that Ẽ is known to the enodeb for any. However, h (d) is unknown. In fact, the enodeb knows neither the vaues of d, nor, nor the channe impuse response. B. Mutipe RTs Let Ñ be the number of terminas transmitting code c (u ). Note that the vaue of Ñ can be arger than one. However, Ñ > impies that mutipe RTs wi coide by seecting the same RA code in a particuar IUS opportunity. To prevent the coision and maintain Ñ, different scheduing approaches have been considered by the Third Generation Partnership Project []. Thus, in the foowing, we assume Ñ {0,}. Suppose d be the deay of the RT transmitting c (u ). Let h = { h (d ), Ñ =, 0, Ñ = 0. (24) Then by the principe of superposition and using (23) the data vector y received by the enodeb at the PRACH subchannes is given by y = Ax+e, (25) x := [ h ( : N ) h 2 ( : N ) h G ( : N ) ], A = [ Ẽ Ẽ 2 Ẽ G ]. where e is the contribution of noise. Since Ẽ is known for any, A is aso known. The power received by the enodeb corresponding to the code c (u ) is given by [3, eq. (5)]: M m= Γ = P p=0 h (p)e i2πpjm/n 2 2. (26) M where z p denotes the p norm: z p = ( t z(t) p ) /p. C. IUS parameter estimation probem Given y, the enodeb needs to i) find the set L = { : Γ 0}; and ii) for every L find Γ and d. Reca that the first d components of h (d) are zero, see (7). Hence by construction of h in (24), the index of the first nonzero component of h is +d. This observation can be used to find d from an estimate of h. III. IUS PARAMETER ESTIMATION AS A SPARSE SIGNAL RECOVERY PROBLEM By construction, A C M G.N in (25) is a known matrix. On the other hand, x and e are unknowns and we have to obtain an estimate of x to resove the IUS probem. Typicay, the tota number of RTs K = G =Ñ G, impying Ñ = 0 (and therefore h = 0) for a vast majority of the vaues {,2,...,G}. This makes x very sparse, motivating a sparse recovery approach for soving the IUS probem. Suppose we obtain a sparse estimate x of x by appying a sparse recovery agorithm on (25). From the estimate x, the enodeb can extract the IUS information as foows. Partition x into G sub-vectors: x = [ h h 2 h G ], where each h is of ength N. Then we decare L ony if h 2 0 and the index of the first nonzero component of h eads to an estimate of d.

5 5 A. Performance of sparse recovery agorithm for resoving the IUS probem To observe the IUS parameter estimation performance by using a sparse recovery agorithm, we appy a popuar approach caed east absoute shrinkage and seection operator (Lasso) [23], [24]. Using the Lasso paradigm, we need to sove the foowing optimization probem: x = argmin z λ z + 2 Az y 2 2 (27) where the vaue of λ > 0 depends on the noise eve. The typica resuts of IUS parameter estimation by using the Lasso with simiar setup in Tabe-I has been demonstrated in Tabe-II. Simiar to the state of the art IUS agorithms, the performance of Lasso aso depends on the code matrix. In fact, when SMUD and SRMD perform we, Lasso aso performs we. On the other hand, for the seection of code matrices eading to performance deterioration of the SMUD and SRMD, Lasso aso shows very cear deterioration of performance. This observation inspires us to investigate the dependency of agorithms performance on the code matrix by using the compressive sensing theory 4. B. The coherence parameter In genera, the performance of sparse recovery agorithms depends on some properties of the matrix A. Two types of metric are commony used to characterize the properties of a matrix: i) restricted isometry property (RIP) [25], [26] and (ii) mutua coherence [24], [27]. A matrix satisfying the restricted isometry property wi approximatey preserve the ength of a signas up to a certain sparsity, thereby, provides performance guaranty of sparse recovery agorithms. However, evauating the RIP of a matrix is a computationay hard probem in genera [28]. On the other hand, computing mutua coherence of a matrix is easy. In this respect, the coherence based resuts of performance guaranty of sparse recovery agorithms are appeaing since they can be evauated easiy for any arbitrary matrix. In this work, we seek the performance guarantee of sparse recovery agorithms based on the mutua coherence. The mutua coherence µ(a) of A is defined as [24] µ(a) = max i j [A] i [A] j [A] i 2 [A] j 2, (28) where [A] i denotes the i-th coumn of A and [A] i is its compex conjugate transpose. In particuar, the coherence is defined as the maximum absoute vaue of the crosscorreations between the normaized coumns of A. When the coherence is sma, the coumns ook very different from each other, which makes them easy to distinguish. Thereby, it is used as a measure of the abiity of sparse recovery agorithms to correcty identify the true representation of a sparse signa [29]. The theoretica resuts in [24], [27] show that the performance of a sparse recovery agorithm can be improved by minimizing µ(a). We vaidate this theoretica resut for IUS appication in Tabe-II where we ist the vaues 4 Sparse recovery agorithms and theories are generay deveoped in the research area caed Compressive sensing. of µ(a) for different ZC root u and n cs. By comparing those vaues of µ(a) with the P s of Lasso, we see that the performance of Lasso increases with decreasing the vaue of µ(a). Thereby, we can improve the performance of Lasso by minimizing the vaue of µ(a). In the foowing section, we show that the vaue of µ(a) can be controed by propery designing the code matrix. IV. SINGLE ROOT CODE MATRIX DESIGN In this section, we derive an expression for µ(a) by assuming that a codes in the code matrix C are generated from the same ZC root, i.e., we assume that u = ũ; {,2, G} in (4). Consequenty, we propose a code matrix design procedure that can ensure the optimum vaue of µ(a). In the next section, we extend the code matrix design procedure for mutipe roots. A. Coherence of A for singe root The foowing emma gives the coherence property of A. Lemma : Assume that u = ũ; {,2, G} in (4). Furthermore, π 2 M N/2. Then it hods that µ(a) = sinc(g (ũ) M/N) sinc(g (ũ) /N), (29) where sinc(x) = sin(πx)/(πx) and g (ũ) = min{,ζ(ũ)}. Here we define {( ) } ncs ũn ζ(ũ) := min m M ( m)+(p k) mod N p,k N (30) with m, {,2, G}. Proof: See Appendix-A. In a typica LTE system [6] the vaues of M = 839 and N = 644. Therefore, the assumption of Lemma- i.e., π 2 M N/2 hods in practice. Under the assumptions of Lemma-, the vaue of µ(a) is the smaest when g (ũ) =. If the vaue of ζ(ũ) becomes smaer than one then it wi increase the vaue of matrix coherence. The code design procedure proposed here wi aim to maintain ζ(ũ). In this foowing sections, we first demonstrate the conventiona procedure of code matrix design. We show that the procedure may not achieve the minimum vaue of coherence of A. Finay, we demonstrate an efficient code matrix design procedure. B. Conventiona procedure of generating RA code matrix (CRA) The conventiona procedure of generating RA code sequences has been described in [], [4]. For the given specification of an LTE system, we need to compute the smaest integer for n cs that satisfies (3). For a given specification of the LTE system, we take ˆn cs as the smaest integer satisfying (3). Set n cs = ˆn cs and compute G = M/n cs which is the maximum number of codes that can be generated from a singe root. We then choose an arbitrary non-zero positive

6 6 TABLE II IUS USER DETECTION PROBABILITIES BY LASSO AND MUTUAL COHERENCE OF A FOR DIFFERENT VALUES OF OF u AND n cs. SNR=0dB, TOTAL IUS USERSK = 3 AND P s DENOTES THE PROBABILITY OF SUCCESSFULLY DETECTING THE USERS. ZC root u = ZC root u = 2 n cs = n cs = 3 7 P s µ(a) integer for the root ũ such that ũ < M []. Subsequenty, we generate G number of RA codes {c } G = by using (2). However, this procedure may produce a code matrix A with bad vaue of µ(a). Such exampes are readiy constructed. Consider the LTE system with N = 644,N g = 768 and M = 839 and OFDM symbo samping interva is 30 ns. The ce radius γ =.5 km and N = 05. Using (3), we obtain a ower bound ˆn cs = 3. Suppose we want to generate tota 50 codes. By using those vaues, we construct a code matrix C by appying the above procedure where we set ũ =. We found that ζ() = 0.99 and consequenty µ(a) = C. Coherence based code generation (CCG) Reca that the minimum vaue of µ(a) in (29) can be obtained by makingζ(ũ), i.e., for everyp,k {,2, N } and,m {,2, G} with m, we have to satisfy {( ) ncs ũn M ( m)+(p k) } mod N. (3) The foowing proposition gives a sufficient condition to fufi the requirement in (3). Proposition : If N n csũn M N N G, (32) then (3) hods for every p,k {,2, N } and,m {,2, G} with m. Proof: It is sufficient to show that if (32) hods then n cs ũn M ( m)+(p k) N (33) for any p,k {,2, N } and,m {,2, G} with m. Suppose that (32) hods. Then using reverse triange inequaity [30], we see that n cs ũn M ( m)+(p k) n cs ũn M ( m) (p k) n cs ũn M (N ) (34) where the ast inequaity foows from the first inequaity of (32). TABLE III COHERENCE BASED CODE GENERATION (CCG) FOR SINGLE ZC ROOT Input: The vaue of ˆn cs and a ZC root ũ. Initiaization: Set C is empty.. Find the smaest positive integer n cs such that MN ˆn cs n cs and satisfies ower bound of (32) i.e., N ncsũ. 2. Compute G using (36): G = + M(N N ) Nn. csũ 3. For = : G 4. Generate c (ũ) using (2). 5. Update C = [C c (s) ]. 6. End for. 7. Output: Code matrix C. On the other hand, triange inequaity impies n cs ũn M ( m)+(p k) n cs ũn M ( m) + (p k) n cs ũn(g ) +(N ) M N N N + = N (35) where we use the second inequaity of (32). Combining (34) and (35) we get (33). Remark : Note that for some given vaues of n cs and ũ, eq. (32) gives an upper bound on the maximum number of codes that can be generated i.e., G + M(N N ). (36) Nn cs ũ Tabe-III shows the proposed CCG agorithm. Here ˆn cs denotes the vaue of ower bound of n cs that satisfies (3). Unike the CRA agorithm (Section-IV-B), we do not use the vaue of ˆn cs directy for code generation. Instead, we use ˆn cs to choose an appropriate vaue of n cs in Step- which satisfies the ower bound in (32). Givenn cs, we use (36) to compute the vaue of G in Step-2. Subsequenty, the agorithm generates G number of codes using Step-3 to Step-6. V. MULTIPLE ROOT CODE MATRIX DESIGN In practice, the code matrixcin (4) can be generated by using mutipe ZC roots. We denote U as the set of a roots that has been used to generate C i.e. u U; {,2, G}. Let J u be the set of a coumn indices of C that are generated from the root u. To give an exampe, assume that C has been generated from the two roots {, 2}, where the first 3 coumns of C are generated using root number. Hence, U := {,2}

7 7 and J := {,2,3}. Hence, in (4) we have u = ; =,2,3 and u = 2; = 4, G. Thus, in this case C = [ c () c () 2 c () 3 c (2) 4 c (2) G ]. Let B u be the matrix which is constructed by concatenating the matricesẽ; J u. For exampe, if the first three coumns of C are generated from the root u then J u := {,2,3} and B u = [Ẽ Ẽ 2 Ẽ 3 ]. In this way, we can view A as a concatenation of matrices B u ;u U. To derive an expression for µ(a), we require the foowing definition. For any two matrices U D, we extend the definition of mutua coherence in (28) to bock coherence as ˆµ(U,D) = max j,k [U] j [D] k [U] j 2 [D] k 2. (37) where [D] k denotes the k-th coumn of D. Therefore µ(a) = max{max µ(b p), max ˆµ(B u,b u2 )}. p U u u2; u,u2 U (38) In the foowing, we deveop a code design procedure so that we can maintain µ(a) to its minimum vaue i.e., µ(a) = sinc(m/n) sinc(/n). (39) We assume that the foowing condition hods for any p U µ(b p ) = sinc(m/n) sinc(/n). (40) In particuar, we can satisfy (40) by constructing B p using the CCG method in Section-IV-C. Next, we see that max ˆµ(B u,b u2 ) = max ˆµ(Ẽ,Ẽm). u u2; u,u2 U J u; m J u2 u u2; u,u2 U (4) It is difficut to deveop an exact expression for (4). Nevertheess, we provide an upper bound for the vaue of (4). By using the definition of bock coherence in (37), we get ˆµ(Ẽ,Ẽm) = max k,p N M [Ẽ] k [Ẽm] p. (42) Next we give an upper bound on the right hand side of (42). Proposition 2: For any k,p {,2, N } with J u ; m J u2 ; u,u2 U and u u2, it hods that M [Ẽ] k [Ẽm] p = M { M exp i π M (u2 u) n=0 where, ϑ(u,u2,,m,p,k) = 2 + (u2 u) {n+ϑ(u,u2,,m,p,k)} 2} (43) {n cs ( u2(m ) ) u( ) +(p k) M N }. (44) Proof: See Appendix-C. Lemma 2: Suppose the foowing conditions hod for some k,p {,2, N } with J u ; m J u2 and u u2: C. M is an odd prime integer. C2. (u2 u) is an even integer. C3. ϑ(u,u2,,m,p,k) is integer vaued. Then the foowing reation hods M [Ẽ] k[ẽm] p =. (45) M Proof: Under conditions C-C3, the sequence z(n) = exp ( i π M (u2 u){n+ϑ(u,u2,,m,p,k)}2) (46) with n = 0,,2, M satisfies M n=0 z(n) = M [3]. Thereby, (45) hods. Remark 2: Under conditions C-C3 of Lemma-2, the sequencez(n) (defined in (45)) is known as the Frank-Zadoff- Chu (FZC) sequence [7], [8], [3]. Note that in LTE system M = 839. Thus, the first condition of the Lemma-2 hods. Furthermore, we can fufi the second condition by appropriatey choosing the ZC root sequences. However, the third condition may not hod for a k,p {,2, N }. We observe that M n=0 z(n) M, whenever ϑ(u, u2,, m, p, k) is approximatey an integer. In practice, the vaue of (43) remains cose to / M for any feasibe vaue of ϑ(u, u2,, m, p, k) defined in (44). Furthermore, for a typica LTE system (M = 839,N = 644) Hence, M sinc(m/n) sinc(/n) max ˆµ(Ẽ,Ẽm) J u; m J u2 u u2; u,u2 U. sinc(m/n) sinc(/n) (47) hods with high probabiity, and thus we shoud be abe to find roots u and u2 satisfying (39). Nevertheess, to be more precise, we deveop a systematic procedure to choose the ZC roots so that (39) hods. The root seection probem can be described formay in the foowing way. Suppose, tota t number of coumns of C have aready been generated from the set of roots U such that (39) is satisfied. Given U and J u ;u U we need to choose another root u2 and generate additiona coumns of C so that (39) hods. Note that for some given vaues of u2 and n cs satisfying the ower bound of (32), we can generate maximum G u2 = M(N N) N n csu2 + number of coumns of C (see (36)). Denote J u2 = {t+,t+2 t+g u2 ]. Since the coumns are generated by satisfying (32), we have µ(b r ) = sinc(m/n) sinc(/n).

8 8 TABLE IV COHERENCE BASED CODE GENERATION (CCG) FOR MULTIPLE ZC ROOT Input: The vaue of G and the vaue of ˆn cs. Initiaization: Set C is empty, U is empty, u = and t = 0. repeat. Find the smaest positive integer n cs such that MN ˆn cs n cs which satisfies ower bound of (32) i.e., N ncsu. 2. Compute G u using (36): G u = + M(N N ) Nn. csu 3. Set J u := {t+,t+2 t+g u}. 4. If the condition in (48) does not satisfy then goto Step For = : G u 6. Generate c (u) using (2). 7. Update C = [C c (u) ]. 8. t = t+. 9. If t = G then goto Step End for.. Set U := {U, u}. 2. u = u+. Continue to Step. 3. Output: Code matrix C. Then according to Proposition-2, it is sufficient to check M ( max p,k {,2, N } M exp i π M (u2 u) m J u2, J u; u U n=0 {n+ϑ(u,u2,,m,p,k)} 2) sinc(m/n) sinc(/n). (48) Based on the idea, the mutipe root code matrix generation procedure has been described in Tabe-IV. Suppose that we want to generate tota G number of codes. The agorithm initiates with the ZC root u =. Given the vaue of u, and ˆn cs as the ower bound of n cs, we compute the vaues of n cs in Step and G u in Step-2. In Step-4, we check that whether the condition in (48) is satisfied. If the condition is satisfied then we concatenate G u number of codes for C in Step-5 to Step-0. In Step-, we add the active root u with the set U. VI. SIMULATION RESULTS We simuate an LTE system simiar to [6] where N = 644, and cycic prefix ength N g = 768. The subcarrier frequency spacing is.25 khz, and the samping interva T s = 30 ns. The PRACH consist of M = 839 adjacent subcarriers. The wireess ce radius is.3 km, which corresponds to D = 70. The wireess channes are modeed according to a mixed channe mode specified by the ITU IMT-2000 standards: Ped- A, Ped-B, and Veh-A. For each RT, the simuator seects one of the above channe modes uniformy at random. The mobie speed varies in the interva [0,5] m/s for Ped-A, Ped- B channes, and [5,20] m/s for Veh-A. The channe impuse response of the RTs have a maximum order of P max = 35 taps [6]. Simiar to [3], [4], we assume that enodeb has an approximate knowedge about P max, and we set N = P max +D in (23). The RA codes are the Zadoff-Chu (ZC) sequences of ength 839. The ower bound of n cs for the ZC sequence is P s P s P s SMUD: u=, n cs SMUD: u=, n cs =3 SMUD: u=, n cs =5 SMUD: u=, n cs =7 SMUD: u=2, n cs SMUD: u=3, n cs (a) SMUD [4] SRMD: u=, n cs SRMD: u=, n cs =3 SRMD: u=, n cs =5 SRMD: u=, n cs =7 SRMD: u=2, n cs SRMD: u=3, n cs (b) SRMD [5] Lasso: u=, n cs Lasso: u=, n cs =3 Lasso: u=, n cs =5 Lasso: u=, n cs =7 Lasso: u=2, n cs Lasso: u=3, n cs (c) Lasso Fig.. Code detection performance as a function of number of users for different vaues of ZC root u and n cs. Signa SNR=0 db. The CRA matrices are generated with n cs {,3} and u {,2,3}. The CCG matrices are generated using n cs {5,7} and u =.

9 9 cacuated by using (3), and we get n cs. The number G of avaiabe RA codes in the matrix C for contention based random access is 50. Reca that, at a particuar random access opportunity, the set of active RA code indices is L. Let ˆL be the set of code indices detected by an agorithm. The probabiity that L = ˆL, denoted by P s, is used to quantify the merit of the agorithm [32]. The signa to noise ratio (SNR) is defined as SNR = 0og 0 (σh 2/σ2 e), where σh 2 is the variance of a channe tap, and σe 2 is the variance of a component of e, respectivey [3]. For the Lasso agorithm in (27), we set λ = 8σe(+α)n(G.N 2 ) [27], where α = 4. The foowing resuts are based on 200 independent Monte- Caro simuations. In Figure-, we compare the code detection performance of different agorithms with different code matrices generated by the CRA and CCG methods. For the LTE configuration under consideration, the conventiona code matrix generation procedure i.e., CRA method (see Section-IV-B) suggests using n cs = and we can set the vaue of ZC root arbitrariy. Hence, we choose three different vaues of ZC root i.e., u {,2,3} to iustrate performance of the IUS agorithms. In contrast, the CCG method, described in Section-IV-C, suggests usingu = andn cs 5. Thus, we use two different vaues of n cs {5,7}. Note that we appy the mutipe root CCG method (Tabe-IV) for code generation whenever n cs 7. As can be seen in Figure-, the code detection probabiity of any particuar agorithm remains amost simiar for any vaue of n cs {5,7} with u =. However, a arger vaue of n cs decreases the vaue of G u (see (36)), that is the maximum number of codes that can be generated from the singe root u with the CCG agorithm. Hence, we sha use n cs = 5 for CCG agorithm in the foowing experiments. As can be seen, a agorithms exhibit their optimum performances with the CCG code matrices. For exampe, the code detection probabiities of SMUD agorithm in Figure-(a) are 0.92 and 0.8 with the CCG matrix (n cs = 5) for 2 and 0 users respectivey. In contrast, SMUD exhibits different types of performance for three different vaues of u with the CRA matrices. The code detection probabiities remain 0.6, 0.35 and 0.78 with 2 users for n cs = and u =,2 and 3 respectivey. As expected the performance deteriorates with the increase in the number of users. With 0 users, the P s are 0.7,0.0 and 0.54 receptivey for n cs = and u =,2 and 3. The SRMD agorithm in Figure-(b) aso shows simiar performance with the code matrices. We have stated in Section-III that the code detection probem can be soved by using a sparse recovery agorithm. The resuts in Figure-(c) justify our caim. Simiar to the state of the art IUS agorithms, the Lasso can detect RA codes efficienty. To iustrate the robustness of the IUS agorithms with the CCG matrix in noisy environment, we present Figure-2, where we evauate the code detection performance of IUS agorithms in reativey ower SNR. As can be seen, the agorithms sti perform better with the CCG code matrix. We examine the miss-detection probabiity P md of different IUS agorithms for different code matrices in Figure-3. Here miss-detection occurs when an agorithm incudes the index of an inactive code into ˆL. As can be seen the miss-detection P s SMUD: u=3, n cs SRMD: u=3, n cs Lasso: u=3, n cs SMUD: u=, n cs =5 SRMD: u=, n cs =5 Lasso: u=, n cs = Fig. 2. Code detection performance by different agorithms at SNR=5 db. The CRA matrix is generated with n cs = and u = 3. The CCG matrix is generated using n cs = 5 and u =. probabiity of a IUS agorithms are higher with the CRA matrix compared to the CCG matrix. For exampe, with 6 users the miss-detection probabiity of Lasso are 0.08 and 0.03 for CRA and CCG code matrices respectivey. The missdetection probabiity increases with increasing the number of active users. With 9 users, the miss-detection probabiity of Lasso are 0.23 and 0. respectivey for CRA and CCG code matrices. Both SMUD and SRMD agorithms aso perform simiary. Miss detection Probabiity SMUD: u=3, n cs SRMD: u=3, n cs Lasso: u=3, n cs SMUD: u=, n cs =5 SRMD: u=, n cs =5 Lasso: u=, n cs = Fig. 3. Miss-detection probabiity by different IUS agorithms at SNR=0 db. The CRA matrix is generated with n cs = and u = 3. The CCG matrix is generated using n cs = 5 and u =. In Figure-4 we pot the mean squared error (MSE) associated with the estimate of the power Γ and timing offset as a function of K for different code matrices. As expected, the MSE of power estimate by different agorithms is better with the CCG code matrix compared to the CRA matrix. It is interesting to note that the power estimation performance of the Lasso is poor for CRA matrix, however, it performs the best compared to other agorithms when we use the CCG matrix. Figure-4 (b) aso exhibits that we can enhance

10 0 MSE of power estimate SMUD: u=, n cs SRMD: u=, n cs Lasso: u=, n cs SMUD: u=, n =5 cs SRMD: u=, n cs =5 Lasso: u=, n =5 cs MSE of timing estimate SMUD: u=, n cs SRMD: u=, n cs Lasso: u=, n cs SMUD: u=, n =5 cs SRMD: u=, n =5 cs Lasso: u=, n =5 cs (a) Fig. 4. MSE of initia upink synchronization parameter estimations. (a) MSE of estimated channe power versus the number of active RTs, (b) MSE of estimated timing offset versus the number of active RTs. SNR=0 db. The CRA matrix is generated with n cs = and u =. The CCG matrix is generated using n cs = 5 and u =. (b) the timing estimate performance of different agorithms by appying the proposed code matrix. VII. CONCLUSION In this work, we study the dependency of performance of some state-of-the-art initia upink synchronization agorithms on the random access (RA) code matrix which has been generated from the Zadoff-Chu (ZC) sequences. We observe that the agorithms can not perform equay for a ZC sequences. To identify the efficient ZC sequences, we appy a theory of compressive sensing. At first we deveop a data mode of the received signa at enodeb over the PRACH. The data mode aows us pose the IUS probem as a sparse signa representation probem on an overcompete matrix. Consequenty, we appy a sparse recovery agorithm for resoving the IUS probem. The compressive sensing theory says that the performance of sparse recovery agorithm depends on a property of the overcompete matrix caed mutua coherence where a smaer vaue of the mutua coherence ensures better performance of the agorithm. We show that the vaue of mutua coherence can be controed by propery designing the RA code matrix. We then deveop a systematic procedure of code matrix design which ensures the optimum vaue of coherence. The empirica resuts show that if we design the code matrix by using the proposed method then it aso increases the performance of state of the art IUS agorithms. In particuar, compared to conventiona code matrix, the IUS agorithms with the proposed code matrix show significant performance improvements in the mutiuser code detection, timing offset and channe power estimations. APPENDIX A PROOF OF THE LEMMA- We begin the proof by describing some important properties of the sinc function. Define S(r) = sinc(rm/n) sinc(r/n). (49) For any vaue of r <, S(r) = S(N r), (50) S(r) = S(r mod N). (5) We have shown in Appendix-B that for π 2 M N/2, the S(r) aso satisfies the foowing properties: ) S(r) is a monotonicay decreasing function for r [0,], 2) for r [,N ), the maximizer of S(r) is unique and is given by arg max S(r) =. (52) r<n We are now ready to prove Lemma-. For any two matrices B D, we extend the definition of mutua coherence in (28) to bock coherence as ˆµ(B,D) = max j,k [B] j [D] k [B] j 2 [D] k 2. (53) By using the definitions of coherences in (28) and (53), it can be verified that µ(a) = max{µ(ẽ),max m ˆµ(Ẽm,Ẽ)}. (54) Since a codes of C are generated from the singe ZC root ũ, by using (20) we have ( ( [Ẽm] q,k = Zũ(q)exp i2πjq (m ) n )) cs ũn +(k ) N M (55)

11 where [Ẽm] q,k is the eement of Ẽm at its q-th row and k-th coumn. Now, using (28) and (55) we get µ(ẽ) = max k,p N M [Ẽ] k[ẽ] p k p = M [Ẽ] [Ẽ] 2 = M exp( i2πq/n) = S(). (56) M q=0 Furthermore, for any m using (53) we get ˆµ(Ẽm,Ẽ) = max k,p N M [Ẽm] k [Ẽ] p. (57) Now using the expression of [Ẽm] q,k in (55), for any k,p {,2, N } with m, we have M [Ẽm] k[ẽ] p = M ( ( )) i2πq ncs ũn exp M N M ( m)+(p k) q=0 ( ) ncs ũn = S M ( m)+(p k) (( ) ) ncs ũn = S M ( m)+(p k) mod N (58) where in (58) we use the property (5). Denote ( ) ncs ũn ξ(m,,p,k) = M ( m)+(p k) mod N. We see that ξ(m,,p,k) = N ξ(,m,k,p). Hence, using (50) we can express (58) as M [Ẽm] k [Ẽ] p = S(min{ξ(m,,p,k),ξ(,m,k,p)}). (59) Note that for any vaues of m,, p and k we have min{ξ(m,,p,k),ξ(,m,k,p)} [0,N/2]. Reca that S(r) is a monotonicay decreasing function for 0 r. Thus, whenever min{ξ(m,,p,k),ξ(,m,k,p)}, we have S (min{ξ(m,,p,k),ξ(,m,k,p)}) = S(min{,ξ(m,,p,k),ξ(,m,k,p)}) (60) On the other hand, whenever min{ξ(m,, p, k), ξ(, m, k, p)} >, the property of S(r) in (52) impies that S (min{ξ(m,,p,k),ξ(,m,k,p)}) < S(min{,ξ(m,,p,k),ξ(,m,k,p)}) (6) Now combining (60) and (6), for any vaue of min{ξ(m,,p,k),ξ(,m,k,p)} [0,N/2] we can express (59) as M [Ẽm] k[ẽ] p S(min{,ξ(m,,p,k),ξ(,m,k,p)}). (62) Consequenty using (57), for any m we have ˆµ(Ẽm,Ẽ) ( { }) S min, min {ξ(m,,p,k),ξ(,m,k,p)} k,p N (63) where the equaity hods ony if min k,p {ξ(m,,p,k),ξ(,m,k,p)}. As a resut, for any,m {,2, G} max ˆµ(Ẽ,Ẽm) S(min{,ζ(ũ)}) (64) m ikewise, the equaity hods ony if ζ(ũ). Now combining (56) and (64) with (54), we obtain (29). APPENDIX B PROOF OF THE RELATION IN (52) By construction, S(r) obeys the properties of Fejér kerne [33]. The maximum of S(r) occurs at r = 0. The zeros of S(r) are ocated at the non-zero mutipes of N/M. Hence, there are (M ) number of zeros of S(r) on r [0,N ). There exists ony one oca maximum point between every two consecutive zeros. Consequenty, S(r) wi have (M 2) number of oca maxima on r [0,N ). Suppose that π 2 M N/2. For r [0,N/M], the ony zero of S(r) occurs at r = N/M and the maximum point is at r = 0. Hence, S(r) is a monotonicay decreasing function for 0 r N/M. Here, N/M 2. Hence arg max S(r) =. r [,N/M] Since S(N r) = S(r), we concude that arg max r [,N/M] [N N/M,N ) S(r) =. We emphasize that, in above, N is not incuded in the domain over which the maximum vaue is sought. Let r n is the unique oca maximum point of S(r) ocated in the interva r n (nn/m,(n + )N/M). It is we known that S(r ) S(r n );n {,2, M 2} [33]. As a resut, for r (N/M,N N/M) the maximum vaue of S(r) is S(r ). Hence, to prove the reation (52), it is sufficient to show that S() S(r > i.e., ) sinc(m/n) sinc(r /N) >. (65) sinc(r M/N) sinc(/n) Note that r (N/M,2N/M). By differentiating sinc(rm/n) with respect to r and equating to zero, it can be verified that the oca maximum point of sinc(rm/n) for r (N/M,2N/M) occurs at [34] Hence, r = [ 3/2π 2 3π ] N πm =.4325N M. sinc(r M/N) sinc( rm/n) = /π. (66)

12 2 The power series expansion of sinc(x) is [35] sinc(x) = ( ) n (πx)2n (2n+)!. (67) n=0 Now using (67), we have sinc(r /N) = (πr /N) 2 + t n (68) 3! where, t n = ( ) n (πr /N) 2n (2n+)!. In particuar, if πr /N < 48 then for a non-zero positive integer k i.e., for k Z >0 we have t 2k +t 2k+ = (πr /N) 4k (4k +)! = (πr /N) 4k (4k +)! (πr /N) 4k (4k +)! where in the ast inequaity we set k =. Hence, n=2 t n = k= (t 2k+t 2k+ ) 0. Since r (N/M,2N/M) and M π 2, hence πr /N < 2π/M 2. Now from (68), we get sinc(r /N) > (πr /N) 2 > 2/3. (70) 6 Since N/M 2, hence πm/n π/2. Then using a simiar procedure of (68), we get sinc(m/n) > (πm)2 6N 2 π2 24. (7) Now combining (66), (70), (7) and the fact sinc(/n), we obtain the bound in (65). 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