On Solving Ambiguity Resolution with Robust Chinese Remainder Theorem for Multiple Numbers

Transcription

1 1 On Solving Ambiguity Resolution with Robust Chinese Remainder Theorem for Multiple Numbers Hanshen Xiao and Guoqiang Xiao arxiv: v1 [cs.it] 9 Jun 018 Abstract Chinese Remainder Theorem (CRT) is a powerful approach to solve ambiguity resolution related problems such as undersampling frequency estimation and phase unwrapping which are widely applied in localization. Recently, the deterministic robust CRT for multiple numbers (RCRTMN) was proposed, which can reconstruct multiple integers with unknown relationship of residue correspondence via generalized CRT and achieves robustness to bounded errors simultaneously. Naturally, RCRTMN sheds light on CRT-based estimation for multiple objectives. In this paper, two open problems arising that how to introduce statistical methods into RCRTMN and deal with arbitrary errors introduced in residues are solved. We propose the extended version of RCRTMN assisted with Maximum Likelihood Estimation (MLE), which can tolerate unrestricted errors and bring considerable improvement in robustness. Index Terms Frequency Ambiguity Resolution, Phase Ambiguity Resolution, Robust Chinese Remainder Theorem, Maximum Likelihood Estimation, Remainder Errors. I. INTRODUCTION Localization of nodes [3], [5] and frequency estimation [4] are two fundamental problems in sensor networks. Due to the restriction on precise synchronization and hardware resources such as high-rate analog to digital converters (ADC), the phase detection based ranging methods and sub-nyquist sampling are two important approaches used in these kinds of applications. Especially, the radio interferometric positioning system (RIPS) [13], which receives considerable attraction recently, is also based on the idea to measure the phase of the interference signals generated by two transmitters. However, all of above-mentioned methods are confronted with the ambiguity resolution problems. To be formal, let m l, l = 1,,..., L, denote a group of moduli selected and X i, i = 1,..., N, denote multiple numbers. In the model of undersampling frequency estimation [4], [1], [11], [14] {X i } represent the frequencies to be estimated and the moduli {m l } stand for the sampling frequency used. For a complex waveform f(t) = Hanshen Xiao is with CSAIL and the EECS Department, MIT, Cambridge, USA. hsxiao@mit.edu. Guoqiang Xiao is with the College of Computer and Information Science, Southwest University, Chongqing, China. gqxiao@swu.edu.cn.

2 N i=1 A ie πjxit sampled in frequency m l, the undersampled waveform becomes x ml [n] = N i=1 A ie πjxin m l, n Z. The spectrum of x ml [n] can be obtained via an m l -point Discrete Fourier Transform, i.e., DF T ml (x ml [n])[k] = N i=1 A i1(k X i ml ), where the residue sets, {r il = X i ml i = 1,,..., N}, l = 1,,..., L, can be read from the peaks on spectrum respectively, though the correspondence between X i and r il is unknown. Here X i ml denotes the residue of X i modulo m l and 1 is the indicator function. Similarly, in a localization system [3], [5], {X i } stand for the distances and {m l } denote the wavelengths, respectively. In a nutshell, addressing the ambiguity problems is equivalent to recover X i with the residue sets, {r il }, which is a generalized Chinese Remainder Theorem (CRT) problem. It is well known that CRT describes a closed-form relationship between an integer and its residues modulo given pairwise co-prime moduli. However, even if very small errors are introduced in the residues, it may result in an incredibly large deviation in reconstruction with conventional CRT. In the presence of error il in each residue, which is almost inevitable in practice, the problem turns to estimate X i with X i, which is reconstructed by erroneous residues sets, R l = { r il = X i + il ml i = 1,,..., N}, l = 1,,..., L. To address the problem arising from error sensibility, Robust CRT (RCRT) is formally proposed and studied during last decade. Ideally, RCRT is expected to achieve a reconstruction deviation proportional to the errors in residues. The studies in this area have been elaborated in [10]. Certainly, any improvement over the error bound or the dynamic range max i X i in RCRT will lead to more robust and efficient estimation schemes in many applications. In this paper, trodding the line of research in [9], [6], [3], we initiate the study on RCRT for multiple numbers (RCRTMN) with tolerance of arbitrary errors. Besides presenting the specific algorithms, we show how to sharply reduce the complexity and introduce MLE for further improvement. A. Remainder Codes for Hamming-weight Errors II. PRELIMINARIES Error correction coding is a well-studied field, where most research has concentrated on errors measured with the Hamming weight. Classic remainder code under such scenario was formally proposed during the 1960s. The first polynomial time error correction scheme was constituted by Goldreich et al. [1] based on LLL lattice reduction and further improved by Guruswam et al. in []. We conclude their results as the following lemma. Lemma 1: Given L co-prime integers, M 1, M,..., M L, which are in an ascending order, the residue vector of an integer X within [0, K l=1 M l), K L, is expressed as x = (x 1, x,..., x L ) = ( X M1, X M,..., X ML ). If there exist λ many coordinates that are erroneous in ˆx = (ˆx 1, ˆx,..., ˆx L ), i.e., there exist λ many indexes, l {1,,..., L}, such that coordinate-wise x l ˆx l and λ L K, then X can be uniquely recovered in polynomial time from the residue vector with errors. Before we proceed, we have to stress the fact that when the number of erroneous residues, λ, is no bigger than L K, then x, the error-free residue vector of X, can be uniquely recovered. However, another noteworthy feature is that, when λ > L K, we can still use similar scheme to implement error correction, though it is not guaranteed that there exists a unique code, of which the hamming distance to ˆx is no bigger than λ. It is clear that x is one of candidates when λ L K, i.e., X is possible to be recovered but may not be distinguished due to multiple possible solutions when L K < λ L K. In coding theory, to find all possible codes within a fixed distance

3 3 away from the erroneous vector ˆx is called list decoding. In [], Guruswam et al. proved that there still exists polynomial time list decoding scheme of remainder code when λ < L KL. For general λ, the corresponding results can be refereed in [7]. We will use the above results in the following proof. B. Framework of conventional RCRTMN with bounded errors In the case of bounded errors, assume δ > max il il. Denote M = {m 1, m,..., m L } as the moduli selected, where m l = M l. Throughout the paper, we always assume that M l are co-prime and = 4Nδ. For the erroneous residues, denote the residue sets as R l = { r il = X + il ml i = 1,,..., N}, l = 1,,..., L. Let r c il = r il, which are termed as common reminders. We arrange the set of common remainders { r il c } in an ascending order represented as R = {γ 1, γ,..., γ κ }, where i = 1,,..., N, l = 1,,..., L, and κ NL. It is not hard to observe that the main difficulty to construct RCRTMN arises from the absence of the correspondence between { r il } and X i, and interference from introduced errors. We first review the techniques in RCRT from the point of macroscopic view. To achieve robustness, the folding number Xi plays a key role. It is noted that X i M l = X i ml X i Ml = r il ri c Ml (1) Unfortunately, we can not trivially replace r il and r c i by r il and r c il in (1) due to presence of errors. For example, if there exist some l 1 and l such that r c i + il 1 < and 0 r c i + il <, we have r il 1 r c il 1 Ml1 = Xi +1 M l1 while r il r c il Ml = Xi M l. However, things are different if the order of index l is known such that {r c i + il} are sorted incrementally, as illustrated in Fig.1, where the order is l 1, l, l 3. Under this situation, combined with the information of r c il = rc i + il on a circle modulo illustrated in Fig., we can recover the relative position of r c i + il represented in the axis in Fig.1, if max il il min il il <. In this example, relative positions of r c i + il on axis is determined since it can be inferred that either < r c 1 + il1 < 0 and 0 r c 1 + ilj <, or 0 r c 1 + il1 < and r c 1 + ilj <, j =, 3. Anyway, the problem raised by residue inconsistence of Xi in (1) is solved naturally. The following lemma is refined from [8]. Lemma : If the error il introduced in each residue satisfies max il il < δ = 4N, there exists j 0 {1,,..., κ 1} such that γ j0+1 γ j0 > δ or j 0 = κ such that γ 1 γ κ + > δ. In addition, for each i {1,,..., N}, the order of l of {ˆr c il } defined in () and (3) in Algorithm 1 below is exactly the same as that of {rc i + il} when both sets {ˆr c il } and {rc i + il} are arranged in an ascending order. 1 We define γ (l)i = r c il = rc i + il, i {1,,..., N}, where (l) i is the index such that il is the l th smallest error introduced in the residues of X i, illustrated in Fig. 3. Corollary 1: In Lemma, j 0 should be (L) i0 and j κ should be (1) i1 for some i 1, i 0 {1,,..., N}. Proof. Revisit the definition of {ˆr il c } in () and (3), which is merely a shift on { rc il }. The ascending order of {ˆr c il } corresponds to that of il for each i based on Lemma. Thus i0l 0 corresponding to γ j0 or r c i 0l 0 should be the maximum for some i 0 {1,,..., N}, since it corresponds to the largest element of {ˆr il c }. As the first 1 Said another way, there exist an empty interval ranging from γ j0 to γ j0 +1 κ on the circle modulo with distance at least δ clockwise. We can cut the circle at γ j0 and stretch it to be a line, where the relative location of ˆr c il on the line is the same as that of rc i + il on the axis in Fig.1.

4 4 Fig. 1. Positions of r c i + il on the axis Fig.. Positions of r c il = rc i + il on the circle modulo. element of {ˆr c il } sorted incrementally, γ j 0+1 κ or r c i 1l 1 corresponding to i1l 1, should be the smallest for some i 1 {1,,..., N}. Substituting the notations above, j 0 = (L) i0 and j = (1) i1. Done. The rest work is to apply Generalized CRT for multiple numbers (GCRTMN) reconstruction in the error-free case [8] on q il = r il ˆr c il Ml to uniquely recover q i for each i, where q i Ml = q il. The reconstruction of q i also naturally determines the correspondence between X i and r il c. To deal with bounded errors, no redundant moduli are required. Finally, we conclude the algorithms in [8] as follows with K moduli. III. RCRTMN WITH ARBITRARY ERRORS We divide the construction of RCRTMN for arbitrary errors into two parts, the estimation of folding number Xi and the common residues X i, respectively. Throughout the rest of the paper we consider a system formed by L moduli, where (L K) moduli are redundant. We merely assume that K l=1 M l is big enough, where the specific lower bound of K l=1 M l given X i to apply GCRTMN on q il to uniquely recover q i can be referred in [8]. Based on Lemma 1, we can correct up to L K errors. Following the notations given before, for each γ j, j {1,,..., κ}, let I j denote the interval (γ j, γ j + δ], if γ j + δ < ; otherwise, I j = (γ j, ) [0, γ j + δ ]. For each γ j = r c il, it is assigned with a label [l], the index of the residue set it belongs to. In the following, we divide the index set {1,,..., L} into two parts G and B: R l, l G, are called Good residue sets, in which the errors are bounded by δ; while R l, l B, are called Bad residue sets, in which the errors can be arbitrary and unbounded. Throughout the paper, we always assume B L K. Let R j denote the set of labels in I j. For example, in Fig.4, R (1)i1 for the interval I (1)i1 satisfied and the label of γ j is not in R j. is {[l ], [l 3]}. Let N denote the index set for those j such that, for I j, R j L K is

5 5 Algorithm 1 RCRTMN with bounded errors in [8] Input. Moduli: {m 1 = M 1, m = M,..., m K = M K }. Residue Sets: R l = { r il i = 1,,..., N}, l = 1,,..., K. Step 1. Calculate the common residues γ j = r il, j = 1,,..., κ, arranged in an ascending order. Step. Find out j 0 {1,,..., κ 1} such that γ j0+1 γ j0 > δ or j 0 = κ such that γ 1 γ κ + > δ Step 3. When j 0 κ, for each i and l, if r c il > γ j 0, define else if r c il γ j 0, When j 0 = κ, ˆr c il = r il for each i and l. Step 4. Let ˆr c il = r il. () ˆr c il = r il. (3) q il = r il ˆr il c Ml (4) and apply GCRTMN on q il to recover q i, i = 1,,..., N, q il = q i Ml, and correspondence between X i and r il. Output. Xi = q i + [ K l=1 ˆrc il K ], X i X i < δ, as the estimation of X i, where [ ] is the round operation of R. A. Folding Number Estimation In the presence of arbitrary errors, Lemma and Corollary 1 both are no longer tenable since there may exist an r c il, l B, between rc i 0l 0 and r c i 1l 1 on the circle modulo. Neverless, if there does exits an empty I j, then j here can certainly be such j 0 in Lemma and with the same definition on ˆr c il and q il, the problem is easy to solve. In order to find two successive elements in { r il c, l G } with distance at least δ over the circle, we will construct such an interval without any r c il. To this end, for an I t, t N, we remove all residues, γ j, where γ j = r c il, l R t. Let {γ j } denote the left κ many common residues in an ascending order. Since the label of γ t is not within R t, γ t will be kept and denote γ t as γ t in {γ j }. Let G = G R t and B = B R t, where R t = {1,,..., L}/R t. Since the number of residue sets removed, R t, is upper bounded by L K, thus G G R t K. We will show in the following that we can always find such an I t on the circle and consequently, by applying Lemma on the rest residues, q i can be uniquely recovered similar to Algorithm 1 with list decoding of errors in Hamming weights up to L K. Lemma 3: For an I t, t N, let γ t = γ t after residues are removed. In the case of t κ, let ˆr il c = rc il, when r il c = r il > γ t0 ; let ˆr il c = rc il, when ˆrc il γ t 0. In the case of t = κ, let ˆr il c = rc il. Then for each i {1,,..., N} and l G, the relative location of ˆr c il is exactly the same as that of rc i + il on axis. Proof. With the notations above, now we can find two successive γ t and γ t +1 κ such that γ t +1 κ γ t + 1(t = κ ) > δ, where 1(t = κ ) = 1 iff t = κ, otherwise it equals 0. We search two elements r c il, l G, say γ α and γ β, closest

6 6 Fig. 3. An example for positions of r c i + il on the axis where N = and L = 3. Fig. 4. Continuing example for positions of r c il (γ j) on the circle modulo. to γ t counterclockwise and to γ t +1 κ clockwise, respectively. Then ˆr c il, l G, defined in Lemma 3 is the same as those obtained in () and (3) when j 0 = α. Furthermore, the clockwise distance between γ α and γ β is at least δ. Thus based on Lemma, for l G, the claim holds and the relative location of r c i + il on axis is determined. Done. With Lemma 3, we can similarly define q il = r il ˆr c il Ml. After removing all residue sets with same labels in I t, t N, there exist at most L K many erroneous q il, l B, which are not the residue of q i. Here q i is the same notation as Algorithm 1 and satisfies q il = q i Ml for l G. However, in the worst case that all labels of γ j in I t are from G and B = R t = L K, after residue sets removed, it is reduced to a system with L L K moduli, where B = B = L K and G = L L K. As there are merely L K L K redundant moduli left, based on Lemma 1, the number of errors exceeds the unique correction capability (L K L K )/. When we apply list decoding [7], [] to correct up to L K errors in each step of GCRTMN [8] on q il, it is not guaranteed that q i can be uniquely recovered from q il. Nevertheless, q i should be in the decoding list since G K and B L K. On the other hand, based on Lemma, there exists γ j0 = r c il, l G, such that I j 0 does not contain any r c il, l G. Therefore, j 0 N and the labels of elements in I j0 must be all from B if they exist. Assuming that R j0 is τ, then the number of residue sets or moduli left is L τ and B = L K τ. Therefore the error correction capacity is L τ K, which is no less than L K τ. Thus we enumerate the operation on each I t, t N, with list decoding based error correction until q i can be distinguished with the unique solution. We formally conclude the scheme as follows. In section II, we give the notation (j) i. In the rest of the paper, let γ (j)i = r c il only for l G. It is clear that (j) i N for j = 1,,..., K 1, since γ (K)i, γ (K+1)i,...γ (K+ L K ) i are all within I (j)i, i.e., there exist at least ( L K + 1) labels in such I (j)i for j = 1,,..., K 1. Thus N (L K + 1)N and the complexity of

7 7 Algorithm RCRTMN for arbitary errors Input. Moduli: {m 1 = M 1, m = M,..., m L = M L }. Residue Sets: R l = { r il i = 1,,..., N}, l = 1,,..., L. Step 1. Calculate the common residues γ j = r il, j = 1,,..., κ, arranged in an ascending order. Step. For each t N, do the following steps. Step 3. Delete all the residues with the same labels in I t. Step 4. For the rest κ many residues γ j, γ t = γ t. Case 1: t κ. When r c il > γ t, define ˆr c il = rc il ; Otherwise, ˆr c il = rc il. Case : t = κ. Let ˆr c il = rc il. Step 5. Calculate q il = r il ˆr c il Ml and apply Generalized CRT with list decoding based error corrections on each step for q il to obtain q i, i = 1,,..., N. Step 6. For each error correction step, if the solution is unique, output Xi = q i + [ L Rt l=1 ˆr c il L R t ]. Algorithm is upper bounded by (L K + 1)N times using GCRTMN to recover integers. In the following, we proceed to present further optimization to reduce the complexity of Algorithm to N times accessing GCRTMN. Let P denote the index set for those j such that the number of labels in I j is no less than K + L K. ζ and ζ in P are called consecutive index if γ ζ I ζ or γ ζ I ζ. Theorem 1: P can be divided into at most N disjoint subsets, within which the index are consecutive. Moreover, in Step of algorithm, j only needs to enumerate the element in N, which is clockwise closet to the first element of a subset. Proof. Clearly, for each i {1,,..., N}, (1) i P. We claim that for each subset in P, at least one (1) i should be within it. If the claim is true, then the number of such subsets is upper bounded by N. Assume there exists a ζ P, ζ (1) i, which is not successive to (1) i for any i. Since I ζ contains at least K + L K labels, which is much bigger than L K, it must contain some labels from G. Supposing r c i l I ζ, l G, then γ ζ I (1)i or γ (1)i I ζ, which leads to a contradiction. Next we prove the rest half of the theorem. Recalling Corollary 1 and Lemma 3, there exists j 0 N and clearly j 0 P such that γ j0 = γ ( G )i0. Moreover, after removing all r c il, l R j 0, there exists γ (1)i1, which is closest to γ j0 counterclockwise for all r c il, l G. Therefore, γ j0 is counterclockwise before the first element, denoted by γ ζ, of the consecutive subset containing γ (1)i1. Anyway, γ ζ is lying in the interval ranging from γ j0 to γ (1)i1 clockwise. In Algorithm, when we set t = j 0, q il can be uniquely recovered. On the other hand, it is clear that ˆr c il defined in Step 4 of Algorithm keeps the same for all l G when we set either t = ζ or t = j 0 in Step of Algorithm. Done. Based on Theorem 1, the complexity of Algorithm is reduced to N times accessing GCRTMN. Especially when N = 1, the complexity in [9] is L K times higher than that of ours. Moreover, the analysis above is based on reconstruction of multiple integers, but it can be generalized trivially to the real number case [6]. B. Maximum Likelihood Estimation Based Common Residue Estimation In Algorithm, we briefly give an estimation of ˆr i c = X i q i by the average of ˆr il c, while it is not the maximumlikelihood estimation (MLE). The residue errors, { il }, are random variables and may have different variances due

8 8 to different sampling frequencies in practice. In the following, it is assumed that, for a given i, { il } are in wrapped normal distribution with mean 0 and a variance σ l for l = 1,,..., L, separately. In [6], a generic framework on MLE based RCRT for one integer is proposed. Following the idea, we proceed to introduce MLE in our scenario for multiple integers. Assume that after recovering q i, the correspondence between q i and q il for each l {1,,..., L}/R t = R t is determined, which further yields the correspondence between X i and r il. Therefore, the left work is to estimate each X i separately. According to [6], the MLE of r c i is MLE( r c i ) = arg min 0 x< l R t 1 σ l d ( r c il, x) (5) where d (X, Y ) = min z Z X Y + z, i.e., the minimal distance between the residues of X and Y over the circle modulo. In [6], it proved that there are R t candidates which can be the optimal solution of (5). In the following, we will derive a simpler closed-form MLE of ˆr c i in our case. With the assumption of il given at the start of Section III, since the relative position of elements in {ˆr c il } is proved to be the same as that in {rc i + il}, for any l 1, l G, ˆr c il 1 ˆr c il il1 + il δ. In the following, we assume K > L, i.e., the number of redundant moduli is smaller than that of information moduli. It is noted that G L L K K. There exists min l G ˆr c il = ˆrc il 0 [, ], such that at least K out of L elements in {ˆr il c } are within [ˆrc il 0, ˆr il c 0 + δ]. However, if there exists ˆr il c, which does not belong to any interval [y, y + δ], y [, ], which contains at least K many ˆr c il, then clearly l G. After removing such ˆr c il, denote the set of label l of the rest residues as H, we will show that: Lemma 4: max l1,l H ˆr c il 1 ˆr c il 4δ Proof. Assuming the smallest element in {ˆr c il }, l H, is just ˆrc il 1, then there are at least K elements ˆr c il, l H, within [ˆr il c 1, ˆr il c 1 + δ]. Otherwise, there is no interval [y, y + δ], y ˆr il c 1, that contains both ˆr il c 1 and at least other K 1 many ˆr c il. On the other hand, the number of rest ˆrc il, l H, beyond [ˆrc il 1, ˆr c il 1 + δ], is at most L K, which is smaller than K. Hence, the rest residues should all be within [ˆr c il 1 + δ, ˆr c il 1 + 4δ]. Done. From the above lemma, it also indicates that the reconstruction error is upper bounded by 3δ. Especially, when N = 1, i.e., = 4δ, after removing all ˆr il c with the same label as those in an I t of length δ, the rest ˆr il c are all within an interval no bigger than δ = δ. Substituting = 4Nδ, max l1,l H ˆr c il 1 ˆr c il N N. Moreover, noticing that d (X, Y ), therefore, ˆrc il 1 ˆr c il = d (ˆr c il 1, ˆr c il ) = d ( r c il 1, r c il ), when for any l 1, l H, as ( r il c ˆrc il ). Therefore referring to (5), the MLE of ˆrc il, l H, can be expressed as MLE(ˆr i c 1 ) = arg min σ (ˆrc il x) (6) l min ˆr il c x max ˆrc il l H It is easy to get the conclusion that the right hand of (6) is minimized when x = MLE(ˆr c i ) = ˆr c il l H σ l l H 1 σ l ˆr il c l H σ l l H 1 σ l With the closed form proposed, the complexity of determining MLE(ˆr i c ) is reduced to O(1). and thus (7)

9 9 Fig. 5. Performance simulation comparison when N = Fig. 6. Performance simulation comparison when N = 3 IV. SIMULATION RESULTS We have shown that how to generalize conventional CRT to solve the ambiguity resolution problems. The most ideal estimation we can expect is that the reconstruction error is linear to the residue error, since when m l > X i, the samples r il should be of X i itself. In the following simulation, a robust estimation is defined as that X i X i 3 4N are satisfied for i = 1,,..., N. We assume that il are independent and identically distributed and follow a normal distribution N(0, σ ) where SNR = 0 log 10 σ. In the simulation, we set N =, K = 4, L = 6 and N = 3, K = 6, L = 10 respectively where SNR is ranged from 60dB to 10dB. The results are shown in Fig 5 and 6, which verify that the proposed scheme bring considerable improvement in strengthening the robustness. V. CONCLUSION In this paper, the first robust Chinese Remainder Theorem tolerating arbitrary errors for multiple numbers has been proposed. Various optimizations have been developed to both reduce the computational complexity and improve the robustness performance to further widen applications of RCRTMN.

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