Acquisition. Saravanan Vijayakumaran and Tan F. Wong. Wireless Information Networking Group. University of Florida

Transcription

1 A Search Strategy for Ultra-Wideband Signal Acquisition Saravanan Vijayakumaran and Tan F. Wong Wireless Information Networking Group University of Florida Gainesville, Florida , USA Abstract The ultra-wideband channel (UWB) is characterized by the presence of dense multipath and robustness to multipath fading. By taking system performance subsequent to acquisition into account, it was shown recently that there are multiple phases (called the hit set) where a receiver lock can be considered as successful acquisition. In this case, the serial search may no longer be the optimal choice for the sequential search strategy in the acquisition system. In this paper, we consider the problem of finding better search strategies in the set of all search strategies which are permutations of the search space. The large size of the search space and the absence of any exploitable structure make the problem of finding the permutation search strategy which minimizes the mean detection time prohibitively complex. However, if we take the first-order approximation that probabilities of detection of all the hit set phases are equal then there exists a permutation search strategy which minimizes the mean detection time. Since the actual probabilities of detection are not equal, this search strategy although not optimal serves as a useful heuristic solution to an otherwise intractable problem. Furthermore, we see that this search strategy has a simple Jump-by-H structure and improves the mean detection time by a significant amount compared to the serial search. The material in this paper was presented in part at the IEEE Vehicular Technology Conference 004-Fall, Los Angeles, CA, September 004.

2 I. INTRODUCTION In spread spectrum systems, timing synchronization is typically performed in two stages [10], [11]. The first stage achieves coarse synchronization to within a reasonable amount of accuracy in a short time and is known as the acquisition stage. The second stage is known as the tracking stage and is responsible for achieving fine synchronization and maintaining synchronization through clock drifts occurring in the transmitter and the receiver. In the acquisition stage, the timing ambiguity region is usually coarsely quantized and the candidate timing estimates, called phases, are checked in a serial manner. In a multipath channel, the energy corresponding to the true signal phase is spread over several multipath components (MPCs). The primary difference between the acquisition problems in a multipath channel and a channel without multipath is that there is more than one hypothesized phase which can be considered a good estimate of the true signal phase. In a dense multipath environment like the UWB channel [1], the receiver may lock onto a non-line-of-sight (non-los) path and still be able to perform adequately as long as it is able to collect enough energy. Taking post-acquisition performance into account, the existence of multiple such phases (called the hit set) for UWB systems was demonstrated in [], [3]. The analogous observation of the existence of multiple in-phase cells in frequency selective fading channels for direct-sequence spread-spectrum systems was made in [4]-[9]. When there are multiple elements in the hit set, the serial search may no longer be the optimal sequential search strategy. The problem of finding efficient search strategies for ultra-wideband acquisition was first considered in [1]. In [1], the authors assume that the acquisition and false alarm states are the same and compare various search strategies on the basis of the mean stopping time. In this paper, we consider the problem of finding efficient search strategies in the set of all search strategies which are permutations of the search space. Finding the optimal permutation search strategy which minimizes the mean detection time when the search space is large and the probabilities of detection of the hit set elements are arbitrary turns out to be prohibitively complex. However, if we assume the probabilities of detection of all the hit set phases to be

3 3 equal then there exists a permutation search strategy which minimizes the mean detection time. Since the actual probabilities of detection are not equal, this search strategy although not optimal serves as a useful heuristic solution to an otherwise intractable problem. Furthermore, we see that this search strategy has a simple Jump-by-H structure and improves the mean detection time by a significant amount compared to the serial search. The features of the system model of [3] relevant to the problem considered are briefly described in Section II. The mean detection time of an arbitrary permutation search strategy is calculated in Section III and the best permutation search strategy under the assumption of equal probabilities of detection is found in Section IV. We present some numerical results in Section V quantifying the improvement in mean detection time performance followed by some concluding remarks in Section VI. II. SYSTEM MODEL In this section, we briefly describe those aspects of an UWB acquisition system which are relevant to the problem of finding efficient search strategies. A more detailed description can be found in [3] where we analyzed and compared two approaches, namely square-and-integrate (SAI) and integrate-and-square (IAS), for the acquisition of UWB signals which perform equal gain combining (EGC) to utilize the energy in the multipath. It was found that the IAS approach without EGC (i.e. with EGC window size equal to one) was the better strategy suggesting that EGC may not be a good method to utilize the energy in the multipath to improve acquisition performance. In this paper, we consider the IAS acquisition system without EGC which has the structure shown in Fig. 1. The transmitter transmits a periodic signal with period N s T c during the acquisition process, where T c is the UWB pulse duration and N s is a positive integer. We assume that the pull-in range of the tracking loop is T c and hence the acquisition search only needs to search the timing ambiguity region in increments of T c. To simplify the analysis, we assume that the true phase is an integer multiple of T c. It is then reasonable to choose the

4 4 hypothesized phase to always be an integer multiple of T c. The timing ambiguity region is equal to the period of the transmitted signal and hence the search space, which is the set of all hypothesized phases, is given by {0, T c, T c,..., (N s 1)T c }. The received signal is correlated with a locally generated reference signal and the correlator output is squared to generate the decision statistic R( τ;h) where τ = ˆτ τ, the difference between the hypothesized phase ˆτ and the true phase τ of the received signal, and h is a random vector containing the channel taps. The decision statistic R( τ;h) is compared to a threshold γ and the hypothesized phase ˆτ used to generate the reference signal is accepted as an estimate of the true phase of the received signal if the threshold is exceeded. If the threshold is not exceeded, the process is repeated with a new value for the hypothesized phase. A search strategy is then the sequence of hypothesized phases which are checked until the threshold is exceeded. We will find it convenient to represent the search space by S = {1,, 3,..., N s }, where the integer n indexes the hypothesized phase (n 1)T c. As mentioned earlier, there may be multiple phases in a dense multipath environment which can be considered a good estimate of the true phase. A typical paradigm for transceiver design is the achievement of a certain nominal uncoded bit error rate (BER) λ n. Then all those hypothesized phases such that a receiver locked to them achieves an uncoded BER of λ n can be considered a good estimate of the true signal phase. We define the hit set to be the set of such hypothesized phases. For a given true phase τ, let P E ( τ) denote the BER performance of the receiver when it locks to the hypothesized phase ˆτ where τ = ˆτ τ. Let Υ m be the minimum SNR at which the receiver achieves a BER of λ n when it locks to the LOS path, that is, P E (0) λ n when the SNR is Υ n and P E (0) > λ n for all SNRs less than Υ n. Then for an SNR Υ Υ n and true phase τ, the hit set is given by S h = {ˆτ S : P E ( τ) λ n }. (1) In this paper, we assume that a partial Rake (PRake) receiver [13] is employed for demodulation and hit set in this case has been derived in [3]. The hit set S h is typically a block of H consecutive phases in the search space S where two elements i, j are considered to be consecutive if i j

5 5 (mod N s ) = 1 or N s 1. For a particular value for the true phase of the received signal, the position of the first element of the hit set block is p, which is assumed to be equally likely to be any element of S. Given p, the positions of all the hit set elements are completely specified. When p > N s H + 1, the last p N s + H 1 hit set phases wrap around and are represented by the first p N s + H 1 phases of the search space. This is due to the periodicity of the transmitted signal. For a given threshold γ, the average probability of detection is given by P D (γ, τ) = E h [Pr(R( τ;h) > γ)], ˆτ S h, () and the average probability of false alarm is given by P FA (γ, τ) = E h [Pr(R( τ;h) > γ)], ˆτ / S h, (3) where E h [ ] denotes expectation with respect to the distribution of the channel taps. We assume the channel model given in [14] in the evaluation of these average probabilities. In traditional spread spectrum acquisition systems, the decision threshold is chosen such that the probability of false alarm in each of the non-hit set phases is small. Usually, a verification stage in the acquisition system aids in the identification and rejection of false alarm events with high probability. However, for threshold-based UWB acquisition systems it is difficult to build a good verification stage which can distinguish between a detection event and a false alarm [15]. Thus a more appropriate choice of decision threshold is one which restricts the probability that the acquisition process encounters a false alarm to be small. So if P F (γ) is the average probability that the acquisition process ends in a false alarm, then the decision threshold γ d is chosen such that P F (γ) is constrained by a small positive constant δ 1, γ d = argmin P F (γ) δ. (4) γ The calculation of P F (γ) when the acquisition system employs a serial search strategy is given in [3]. Since the serial search results in the largest value of P F (γ) among all permutation search strategies, this choice of threshold gurantees that P F (γ d ) δ for any arbitrary permutation search strategy.

6 6 The performance of spread-spectrum acquisition systems has typically been characterized by the calculation of mean acquisition time [10], [16]. In mean acquisition time calculations, a false alarm penalty time is assumed which is the dwell time of the verification stage, i.e., the time required by the acquisition system to recover from a false alarm event. Thus mean acquisition time calculations implicitly assume the existence of a verification stage. For UWB signal acquisition systems, if the threshold is set according to (4) the mean detection time is a reasonable metric for system performance. The mean detection time is defined as the average amount of time taken by the acquisition system to end in a detection, conditioned on the nonoccurrence of a false alarm event. The calculation of the mean detection time thus does not require any assumption on the verification stage. III. MEAN DETECTION TIME CALCULATION As mentioned earlier, the problem of finding the optimal permutation search strategy when the probabilities of detection are arbitrary is complex. But if we assume that the probabilities of detection in the hit set elements are equal, we are able to find a suboptimal search strategy which reduces the mean detection time significantly. This permutation search strategy serves as a useful heuristic solution to the otherwise intractable problem. So we proceed to find the permutation search strategy which minimizes the mean detection time under the assumption of equal detection probabilites in all hit set elements. We first calculate the mean detection time when the search strategy is an arbitrary permutation R of the search space. Let P d be the average probability of detection in any hit set element. For a particular initial position p of the hit set in the search space, let the positions of appearance of elements of hit set elements in the sequential search be {t p,i : i = 1,,..., H}. So the first appearance of a hit set element is at t p,1, the second appearance is at t p, and so on. Table I illustrates this for the serial search starting in position 1 of S when N s = 8 and H = 3, where the positions in boldface indicate the presence of a hit set element. The last three columns of the table contain the positions of the first, second and third appearances of a hit set element for a particular value

7 7 of p. Table II shows the positions of appearance of the hit set elements for the permutation search strategy (1, 4, 7,, 5, 8, 3, 6) when N s = 8 and H = 3. Note that the columns indicating the presence of hit set elements in Table II are obtained by permuting the corresponding columns of Table I. Also note that a hit set element appears in every position of the permutation exactly H times where each appearance corresponds to a distinct value of p in S. It is easy to see that this is true for any permutation search strategy and for all values of N s and H. A detection event is defined by the position t p,i where we have a hit and a particular number of misses j of S h. Let T be the dwell time of the correlator. The time taken for a miss event is N s T. The time for a particular detection event defined by (p, i, j) is then T(p, i, j) = t p,i T + jn s T. (5) The probability that there is a hit in position t p,i is given by P h (i) = P d (1 P d ) i 1. The probability of j misses of S h is equal to P j M where P M = (1 P d ) H. The mean detection time conditioned on the fact that the first element of the hit set is in position p of the search space is given by T det (p) = = H H T(p, i, j)p j M P h(i) j=0 [t p,i T + jn s T]P j M P h(i) j=0 The mean detection time is then given by = T H t p,ip h (i) + N H stp M P h(i) 1 P M (1 P M ) = T H t p,ip h (i) 1 P M + N stp M 1 P M. (6) T det = 1 N s T det (p) N s p=1 = T H ( N s p=1 t p,i)p h (i) + N stp M. (7) N s (1 P M ) 1 P M Note that the second term in the right hand side of (7) does not depend on the permutation R. Then any optimization with respect to R can only hope to minimize the first term.

8 8 IV. THE JUMP-BY-H PERMUTATION SEARCH STRATEGY We want to minimize g(s) = H s ip h (i), where s = (s H, s H 1,...,s 1 ) and s i = N s p=1 t p,i, over all permutations of S. Note that s H > s H 1 >... > s 1. By the fact that H s i = N s(n s+1)h and that P h (i) is a decreasing function of i for all P d, we have the following result. Lemma 1: g(s) is Schur-concave [17] on A = {s = (s H,...,s 1 ) : s i = N s p=1 t p,i, i = 1,,..., H, for some permutation R of S}. Proof: Let D = {(x 1,...,x n ) : x 1 x n }. Note that A is a subset of D. For all s D and k = H, H 1,...,, g(s H,...,s k+1, s k + ɛ, s k 1 ɛ, s k,...,s 1 ) = H s i P h (i) ɛ[p h (k 1) P h (k)], (8) which is decreasing in ɛ since P h (k 1) P h (k). The Schur-concavity of g(s) on D follows from Lemma 3.A. in [17]. Since A is a subset of D, g(s) is Schur-concave on A. Thus g(s) is minimized if s is the maximal vector of A. If x y, i.e. if x is majorized by y for some x,y A, then and k x i k y i, k = 1,..., H 1, (9) H x i = H y i. (10) where x i and y i are the (H i + 1)th components in x and y respectively. Lemma : Let r k be not greater than the minimum value of k s i over all permutations of S for k = 1,..., H 1 and r H = Ns(Ns+1)H. If r i+ r i+1 r i+1 r i for i = 0,...,H, then the vector q = (r H r H 1, r H 1 r H,...,r 3 r, r r 1, r 1 ), (11) majorizes all the vectors in A. Proof: By hypothesis, we have q 1 q... q H and H q i = Ns(Ns+1)H, where q i is the (H i + 1)th component of q. Let s A and let s i be its (H i + 1)th component. Since

9 9 s 1 s... s H, the sum of the k smallest components of s is k s i r k = (r k r k 1 ) + (r k 1 r k ) (r r 1 ) + r 1 = k q i (1) for k = 1,,..., H 1. Furthermore, H s i = Ns(Ns+1)H. Thus q majorizes s and since the choice of s was arbitrary, q majorizes all the vectors in A. We now proceed by finding one particular set of r k s which satisfy the conditions of Lemma and then exhibit a permutation R of S whose corresponding vector x A is equal to the vector q defined by these r k s. This vector x then majorizes all the vectors in A. Hence the permutation search strategy R minimizes the mean detection time. Theorem 1: The minimum value of k s i over all permutations of S is N k(n k +1)H +(N s k N k H)(N k + 1) for k = 1,...,H, where N k = Nsk. These minima are all simultaneously H achieved by a permutation R of S given by R i = (i 1)H (mod N s ) + i 1 ( Ns d ) + 1, (13) where R i is the element in its ith position and d is the greatest common divisor (GCD) of N s and H. Thus the search strategy R is optimal in the set of permutation search strategies. Proof: First, we note that it is not entirely obvious but easy to show that (13) does indeed define a permutation. Suppose R i and R j are equal for some integers i, j such that i = l( Ns d )+n i and j = m( Ns d ) + n j where 1 n i, n j Ns d and 0 l, m d 1. If l = m, then R i R j = 0 if and only if (i j)h (mod N s ) = 0. Since n i n j ( Ns ) 1, this implies i = j. Now d suppose (without loss of generality) that l > m. Then R i R j = [(l m)( N s d ) + n i n j ]H (mod N s ) + l m = (n i n j )H (mod N s ) + l m, (14) which is not equal to zero since the first term is a multiple of H and the second term l m is not greater than d 1 < H. Thus the R i are distinct for i = 1,..., N s and it then follows by the pigeonhole principle that R is a permutation of S.

10 10 There are N s k number of t p,i s in k s i, where each t p,i is in the set {1,,..., N s } with the restriction that each distinct value of t p,i appears at most H times. We can obtain a lower bound of k s i by assigning the smallest values in {1,..., N s } to the t p,i s such that each value is assigned H times. Then the elements in the set {1,..., N k } are each assigned H times and N k + 1 is assigned N s k N k H times where N k = Nsk. Thus we have H for k = 1,...,H. k s i 1 H + + N k H + (N k + 1) (N s k N k H) = N k(n k + 1)H Let r k be equal to the lower bound obtained in (15), i.e., + (N s k N k H)(N k + 1), (15) r k = N k(n k + 1)H + (N s k N k H)(N k + 1) (16) for k = 1,...,H. Then r H = Ns(Ns+1)H and r k+1 r k = (N k + 1) (H N s k + N k H) + (N k + ) H + + N k+1 H +(N s (k + 1) N k+1 H) (N k+1 + 1), for k = 0, 1,...,H 1. For each k {0, 1,..., H 1}, r k+1 r k is a sum of N s terms belonging to the set {N k + 1,...,N k+1 + 1} with each distinct value appearing at most H times. Since N k+1 N k + M, r k+ r k+1 r k+1 r k for i = 0, 1,..., H. Thus the r k s satisfy the conditions in Lemma and consequently q = (r H r H 1, r H 1 r H,..., r r 1, r 1 ), (17) majorizes all the vectors in A. In the appendix, we show that the vector x A corresponding to the permutation R defined in (13) in fact equals q. It is easy to see that R has a Jump-by-H structure. R consists of d consecutive blocks each containing Ns d for 1 i d. elements, where the ith block consists of the elements (i, H+i,...,(Ns d 1)H+i)

11 11 V. NUMERICAL RESULTS In order to compare the mean detection time performance of the heuristic permutation search strategy with the serial search strategy, we chose the following values for the system parameters: Size of the search space N s = 9696, T c = ns, dwell time T = N s T c, and SNR = 7,10 db. The hit set was obtained under the assumption that the PRake receiver has 5 fingers and the nominal uncoded BER requirement is λ n = The threshold was set according to (4) with δ = Table III shows the mean detection times for the serial search and heuristic search strategies. The mean detection time of the serial search strategy does not change much with increase in SNR even though the size of the hit set increases significantly. This is because the mean detection time is dominated by the time spent by the acquisition system in evaluating and rejecting the non-hit set phases before it reaches the hit set. The heuristic permutation strategy provides an improvement of more than 70% in the mean detection time compared to the serial search. VI. CONCLUSIONS We began with the observation that the serial search may no longer be the optimal search strategy when the hit set consists of multiple phases which is the case for the dense UWB channel. We provided a heuristic suboptimal solution to the generally intractable problem of finding the permutation search strategy which minimizes the mean detection time by assuming that the detection probabilities of all hit set elements are equal. We also found that the heuristic search strategy has a simple Jump-by-H structure and hence it can be generated easily obviating the need to store the whole permutation. APPENDIX A. Proof that q (defined in (17)) is the vector in the set A corresponding to the permutation R. We first show that k s i is equal to r k defined in (16), for k = 1,,..., H, for all possible values of d = GCD(N s, H).

12 1 Case 1: d = H Let N s = MH for some positive integer M. Then the ith position in the permutation R is given by i 1 R i = (i 1)H (mod N s ) + + 1, (18) M for i = 1,,..., N s. It is easy to see that the permutation consists of H consecutive blocks each having M elements where the kth block can be written as (k, H + k, H + k,...,(m 1)H + k), (19) for k = 1,,..., H. Since any two positions in a block are at least H 1 phases apart in S p, for any position p of the first element of the hit set there is exactly one position in the block where a hit set element appears. Thus, for a particular value of p S p, the ith appearance of a hit set element is in the ith block, i.e., (i 1)M+1 t p,i im for i {1,,..., H}. Furthermore, a hit set element appears in every position of a block exactly H where each appearance corresponds to a distinct value of p in S p. Then we have and N s s i = t p,i = ((i 1)M + 1) H + ((i 1)M + ) H + + im H, (0) p=1 k s i = 1 H + H + + km H = = N k(n k + 1)H km(km + 1)H + (N s k N k H)(N k + 1), (1) where the last equality follows from the fact that N k = Mk and N s = N k H = MHk. Case : d = 1 In this case, the ith position in the permutation is given by R i = (i 1)H (mod N s ) + 1, () for i = 1,,..., N s.

13 13 We first calculate s 1 = N s p=1 t p,1. Thus we need to enumerate the positions of the first appearance of a hit set element in the permutation for all p S p. Consider the first N positions of the permutation, namely R 1, R,...,R N1 +1. Note that the locations of these positions in the search space are such that any two consecutive positions are located H 1 phases apart and any block of H consecutive phases in the search space S p contains one of these positions. We claim that for all p S p, the first appearance of a hit set element occurs in one of these positions, i.e., t p,1 N Suppose this is false. Then there exists a p S p such that when the first element of the hit set block is in p, none of the H consecutive hit set phases appear in R 1, R,...,R N1 +1. This implies that there is a block of H consecutive phases in S p which does not contain any of R 1, R,..., R N1 +1, which is a contradiction. Any two positions in the first N 1 positions of the permutation, namely R 1, R,...,R N1, are at least H 1 phases apart in the search space. Thus for a particular value of p, a hit set element can appear only in one of these positions. So every appearance of a hit set element in these N 1 positions is a first appearance and corresponds to a distinct p S p. Since a hit set element appears exactly H times in every R i, there are exactly H first appearances in each one of R 1, R,...,R N1 and there cannot be any more appearances. This accounts for the first appearance of a hit set element corresponding to N 1 H distinct p s in S p. By the fact that t p,1 N 1 +1 for all p S p, the remaining N s N 1 H appearances have to occur in R N1 +1. Thus s 1 = N s p=1 t p,1 = 1 H + H + + N 1 H + (N 1 + 1) (N s N 1 H) = N 1(N 1 + 1)H + (N s N 1 H)(N 1 + 1). (3) The remaining H (N s N 1 H) appearances of a hit set element in R N1 +1 are second appearances of a hit set element since these correspond to p s in S p for which the first appearance of a hit set element is not in R N1 +1 but in one of R 1, R,...,R N1. In order to calculate s = N s p=1 t p,, we consider the positions R N1 +, R N1 +3,...,R N +1. The locations of these positions in the search space are such that any two consecutive positions are located H 1 phases apart and any block of H consecutive phases in the search space

14 14 S p contains one of these positions. Hence by the argument in the previous paragraph, a hit set element appears in one of these positions for all p S p. Since the first appearance of a hit set element occurs in R 1, R,...,R N1 +1 for all p S p, each appearance of a hit set element in R N1 +, R N1 +3,...,R N +1 is either the second or third 1 appearance of a hit set element. Thus we have t p, N + 1 for all p S p. Furthermore, any two positions in the positions R N1 +1, R N1 +,...,R N are at least H 1 phases apart in the search space. Thus for a particular value of p, a hit set element can appear only in one of these positions. So each of the H appearances of a hit set element in R N1 +, R N1 +3,...,R N is a second appearance of a hit set element and corresponds to a distinct p S p. None of these appearances is a third appearance of a hit set element because the only way this can happen is that for a particular p in S p the second appearance occurs in R N1 +1 and for that same p a hit set element appears in R N1 +, R N1 +3,...,R N. We already know that H (N s N 1 H) second appearances of a hit set element occur in R N1 +1. So far we have enumerated H (N s N 1 H) + (N N 1 1)H (= N H N s ) second appearances of a hit set element in R N1 +1, R N1 +,...,R N corresponding to distinct p s in S p and there cannot be more any more appearances in these positions. Since t p, N + 1, the remaining N s (N H N s ) second appearances have to occur in R N +1. Thus N s s = t p, = (N 1 + 1) (H (N s N 1 H)) + (N 1 + ) H + (N 1 + 3) H + p=1 + N H + (N + 1) (N s N H) (4) and using (3) we have s i = N s t p,i = 1 H + H + + N 1 H + (N + 1) (N s N H) p=1 = N (N + 1)H + (N s N H)(N + 1). (5) The value s k = N s p=1 t p,k for k > can be calculated using arguments very similar to those 1 Due to the fact that some second appearances occur in R N1 +1.

15 15 used in calculating s. Finally, we get for k = 1,,..., H. k s i = N k(n k + 1)H + (N s k N k H)(N k + 1). (6) Case 3: 1 < d < H Let M = Ns d. Then the ith position in the permutation R is given by i 1 R i = (i 1)H (mod N s ) + + 1, (7) M for i = 1,,..., N s. As in Case 1, the permutation consists of d consecutive blocks each having M elements where the kth block can be written as (R M(k 1)+1, R M(k 1)+,..., R Mk ) = (k, H + k, H + k,..., (M 1)H + k), (8) for k = 1,,..., d. The ith position in the first block (R 1, R,...,R M ) is given by R i = (i 1)H (mod N s ) + 1, (9) for i = 1,,...,M. Note that the structure of the permutation in the first block is the same as the structure of the permutation in Case, i.e., any two consecutive positions are located H 1 phases apart in the search space. Let K = H. Then N d K = Ns = M. Since in Case, the Hd value of s k depended only on the relative locations of the positions R 1, R,..., R Nk +1, we have k s i = N k(n k + 1)H + (N s k N k H)(N k + 1), (30) for k < K. This argument cannot be extended for the case when k = K because the positions 1,,..., R NK +1 go beyond the first block and hence their relative positions are not as in Case. Nevertheless, the positions R NK 1 +1, R NK 1 +,...,R NK are such that any two positions are at least H 1 phases apart in the search space and any block of H consecutive phases contains exactly one of these positions. So each of the H appearances of a hit set element in R NK 1 +, R NK 1 +3,...,R NK is a Kth appearance of a hit element and corresponds to a distinct p in S p. We know from Case that a Kth appearance occurs H (N s (K 1) N K 1 H) times

16 16 in R NK This accounts for H (N s (K 1) N K 1 H) + (N K N K 1 1)H (= N s ) Kth appearances of a hit set element. Since these Kth appearances all correspond to distinct p s in S p, this accounts for all possible Kth appearances and hence we have K s i = N K(N K + 1)H = N K(N K + 1)H where the second equality follows from the fact that N s = Ns d + (N s N K H)(N K + 1), (31) = MH = N KH. Thus the first K appearances of a hit set element occur in the first block. Since the relative locations of the positions in any of the d blocks are the same as those in the first block, the kth K appearances of a hit set element occur in the kth block. Thus kk s i = Mk(Mk + 1)H, (3) for k = 1,,..., d. Furthermore, the ith appearance of a hit set element in the kth block occurs in the same positions within the block as the ith appearance of a hit set element in the first block and the ith appearance of a hit set element in the kth block is the ((k 1)K + i)th appearance overall of a hit set element in the permutation. Thus we have (k 1)K+i l=(k 1)K+1 s l = im(k 1)HN s + i s l, (33) for k = 1,,..., d. Note that for j = (k 1)K + i, we have Ns(k 1)H Ns ((k 1)K + i) + N d s i N j = = = M(k 1) + N s = M(k 1)+N i. (34) H H H Thus for j = (k 1)K + i, from (3) and (33) we have j s l = l=1 = = (k 1)K l=1 s l + (k 1)K+i l=(k 1)K+1 s l M(k 1)(M(k 1) + 1)H M(k 1)(M(k 1) + 1)H +(N s N i H)(N i + 1) = N j(n j + 1)H l=1 + im(k 1)HN s + i l=1 + im(k 1)HN s + N i(n i + 1)H + (N s N j H)(N j + 1) (35) s l

17 17 Thus for all possible values of d we have shown that k s i = N k(n k + 1)H + (N s N k H)(N k + 1) = r k, (36) for k = 1,,..., H. Then the vector in the set A corresponding to the permutation R is given by (s H, s H 1,..., s 1 ) = (r H r H 1, r H 1 r H,...,r r 1, r 1 ) = q. (37) REFERENCES [1] M. Z. Win and R. A. Scholtz, On the robustness of ultra-wide bandwidth signals in dense multipath environments, IEEE Commun.Lett., vol., pp , Feb [] S. Vijayakumaran and T. F. Wong, Equal gain combining for acquisition of UWB signals, in Proc. IEEE Military Communications Conf. (MILCOM 03), Boston, MA, Oct 003. [3], On equal gain combining for acquisition of time-hopping ultra-wideband signals, IEEE Trans. Commun., Jun. 003, submitted for publication. Revised Oct Revised Feb [4] B. B. Ibrahim and A. H. Aghvami, Direct sequence spread spectrum matched filter acquisition in frequency-selective Rayleigh fading channels, IEEE J. Select. Areas Commun., vol. 1, pp , Jun [5] R. R. Rick and L. B. Milstein, Parallel acquisition in mobile DS-CDMA systems, IEEE Trans. Commun., vol. 45, pp , Nov [6], Optimal decision strategies for acquisition of spread-spectrum signals in frequency-selective fading channels, IEEE Trans. Commun., vol. 46, pp , May [7] L.-L. Yang and L. Hanzo, Serial acquisition of DS-CDMA signals in multipath fading mobile channels, IEEE Trans. Veh. Technol., vol. 50, no., pp , Mar [8] O.-S. Shin and K. B. Lee, Utilization of multipaths for spread-spectrum code acquisition in frequency-selective Rayleigh fading channels, IEEE Trans. Commun., vol. 49, pp , Apr [9] L.-L. Yang and L. Hanzo, Serial acquisition performance of single-carrier and multicarrier DS-CDMA over Nakagami-m fading channels, IEEE Trans. Wireless Commun., vol. 1, no. 4, pp , Oct. 00. [10] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications. Englewood Cliffs, NJ: Prentice Hall, [11] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications: Volume III. Computer Science Press, MD Rockville, USA, [1] E. A. Homier and R. A. Scholtz, Rapid acquisition of ultra-wideband signals in the dense multipath channel, in Proc. 00 IEEE Conf. Ultra Wideband Sys. Tech., Baltimore, MD, 00, pp [13] D. Cassioli, M. Z. Win, F. Vatalaro, and A. F. Molisch, Performance of low-complexity RAKE reception in a realistic UWB channel, in Proc. IEEE Intl. Conf. on Commun., vol., 00, pp

18 18 [14] D. Cassioli, M. Z. Win, and A. F. Molisch, The ultra-wide bandwidth indoor channel: From statistical model to simulations, IEEE J. Select. Areas Commun., vol. 0, pp , Aug [15] S. Vijayakumaran, T. F. Wong, and S. Aedudodla, On the asymptotic performance of threshold-based acquisition systems in fading multipath channels, June 004, Submitted to IEEE Transactions on Information Theory. [16] A. Polydoros and C. Weber, A unified approach to serial search spread spectrum code acquisition: Part I. general theory, IEEE Trans. Commun., vol. 3, pp , May [17] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications. New York: Academic Press, 1979.

19 19 p t p,1 t p, t p, TABLE I SERIAL SEARCH FOR N s = 8 AND H = 3. p t p,1 t p, t p, TABLE II PERMUTATION SEARCH (1, 4,7,,5, 8,3, 6) FOR N s = 8 AND H = 3. SNR Hit set size Serial Search MDT Heuristic Search MDT 7 db s s 10 db s s TABLE III MEAN DETECTION TIME (MDT) VALUES FOR THE SERIAL SEARCH AND HUERISTIC SEARCH STRATEGIES.

20 0 r(t) Correlator Squaring Operation R( τ ;h) Is R( τ ;h) > γ? Yes s(t τ ) No Reference Signal Generator τ Clock Control Fig. 1. Block diagram of the acquisition system.