Time-Hopping Correlation Property and Its Effects on THSS-UWB System 107 () ij L Cl N  (11) and max L C N  , (12) where ij . Proof: According to Definition 3, we have 1 0 111 () () () () () () (1) () 000 () () { [( ) ,( ) ] [( ) ,( ) ]} LL LL ij NL ij l LNL jj ii NL NL NL NL ka k k a k kba NL C l Cl hc c b hN c c b           The analyses of the above equation is similar to Theorem 1, and then we can obtain 11 () 2 () 00 () (( ) ) L LN j ij i N k kb NL C l num c b L      and () ij L Cl N  . Also, it is obvious that max L C N  since max () ij CCl . Q.E.D From Theorem 1 and Theorem 2, we can see that TH correlation function averages () ii Cl and () ij Cl are determined by sequences period L and the number of time slots N . When L and N are fixed, both () ii Cl and () ij Cl will be fixed for any TH sequence. In order to explain the conclusions, we give an example. We use linear congruence codes (LCC) (Titlebaum, 1981) and QCC. For LCC sequences, () () () L i P k Cki , where LP , 01kp, 11ip , and P is a prime. In this example, let 5P  and 5N  . Then, we have   5 (1) () {0,1,2,3,4} k C  and   5 (2) () {0,2,4,1,3} k C  . When l is from 1 to 24 (here 1 24NL   ), auto-correlation sidelobes of TH sequence   5 (1) () k C constitute the set {1,0,0,0,0,4,2,0,0,0,0,3,3,0,0,0,0,2,4,0,0,0,0,1}. When l is from 0 to 24, cross- correlation values between   5 (1) ()k C and   5 (2) ()k C constitute the set {1,1,2,0,1,1,1,0,2,1,1,2,1,0,1,1,0,2,2,0,1,1,0,1,2}. Then, the averages of elements in two sets are equal to 5/6 and 1, respectively. The results correspond to 2 1,1 5 () 16 LL Cl NL     and 1,2 () 1 L Cl N  in terms of Theorem 1 and Theorem 2. In addition, for QCC sequences, we have   5 (1) () {0,1,4,4,1} k C  and   5 (2) () {0,2,3,3,2} k C  . When l is from 1 to 24, auto-correlation sidelobes of   5 (1) () k C constitute Novel Applications of the UWB Technologies 108 {0,1,0,1,1,2,1,1,0,1,1,1,1,1,1,0,1,1,2,1,1,0,1,0}. When l is from 0 to 24, cross-correlation values between   5 (1) () k C and   5 (2) () k C constitute the set {1,2,0,2,0,0,2,2,1,1,0,0,2,1,2,0,0,1,2,1,0,2,1,0,2}. Then, the averages of elements in two sets are also equal to 5/6 and 1, respectively. As a result, for any sequence, both of () ii Cl and () ij Cl will be fixed as long as L and N are fixed. Based on Theorem 1 and Theorem 2, the further result can be also obtained. Two corollaries on TH correlation properties are expressed as follows. Corollary 1: For a TH sequences family with period L , we have max max ,1SC . (13) Corollary 2: When the period L and the number of time slots N are fixed, in order to obtain good TH correlation properties, correlation function values () ii Cl and ( ) ij Cl should be close to their averages as possible. In practice, we are also interested in maximal TH correlation function values {()} ij max C l which is the maximum of all correlation function values include cross-correlation function values and auto-correlation sidelobes. Then, the following theorem gives the low bound of { ( )} ij max C l . Theorem 3: For a TH sequences family with period L and family size u N , the average of TH correlation function values can be expressed as 2 (1)2 () (1)2 u u LN L Cl NL N     (14) and then 2 (1)2 {()} (1)2 u ij u LN L max C l NL N     . (15) Proof: For a TH sequences family with period L and family size u N , the number of auto- correlation sidelobes and the number of cross-correlation values are equal to (1) u NNL and (1) 2 uu NN NL  , respectively. Then, the number of all correlation function values without auto-correlation peak should be equal to (1) (1) 2 uu u NN NNL NL   . According to the proof of Theorem 1, the sum of auto-correlation sidelobes for every TH sequence is equal to 2 LL  . Then, the sum of auto-correlation sidelobes for TH sequence family is equal to 2 () u NL L  . Similarly, the sum of cross-correlation values for TH sequence family is equal to 2 (1) 2 uu NN L  . Then, the sum of all correlation function values without auto-correlation peak should be equal to 22 (1) () 2 uu u NN NL L L   . In terms of the above analyses, we can obtain that Time-Hopping Correlation Property and Its Effects on THSS-UWB System 109 22 2 (1) () (1)2 2 () (1) (1)2 (1) 2 uu u u uu u u NN NL L L LN L Cl NN NL N NNL NL         . Also, it is obvious that {()} () ij max C l C l and 2 (1)2 {()} (1)2 u ij u LN L max C l NL N     . Q.E.D According to Theorem 3, TH correlation function average ()Cl is determined by three parameters of period L , the number of time slots N and family size M . When L , N and M are fixed, ()Cl is fixed for any TH sequence family. 4. Improvement of TH correlation properties In this section, we will provide a method that improves the correlation properties of TH sequences. Before the corresponding analyses, the maximum TH correlation function values are further analyzed according to Definition 3. We give Theorem 4 as follows. Theorem 4: For TH sequences with period L, the upper bound can be given by 1 () () max max () () 0 ,2([(),()]) LL L j i NN ka k k SC maxhc c b      . (16) Proof: According to the equation (4), we have 11 () () () () () () (1) () 00 () [( ) ,( ) ] [( ) ,( ) ] LL LL LL jj ii i j NL NL NL NL ka k k a k kk Cl hc c b hN c c b        . We first discuss the first part of ( ) ij Cl, namely 1 () () () () 0 [( ) ,( ) ] LL L j i NL NL ka k k hc c b      . Note that it operates modulo NL. When it operates modulo N, the possibility of collisions between () () () L i N ka c  and () () () L j N k cb is larger than that of collisions between () () () L i NL ka c  and () () () L j NL k cb . Then, we have 11 () () () () () () () () 00 [( ) ,( ) ] [( ) ,( ) ] LL LL LL jj ii NL NL N N ka k ka k kk hc c b hc c b         . Similarly, the second part of ( ) ij Cl satisfies 11 () () () () (1) () (1) () 00 [( ) ,( ) ] [( ) ,( ) ] LL LL LL jj ii NL NL N N ka k ka k kk hN c c b hN c c b       1 () () (1) () 0 [( ) ,( ) ] LL L j i NN ka k k hc c b      . When the shift l is from 0 to NL – 1, it is obvious that 11 () () () () () () (1) () 00 ([( ),( )]) ([( ),( )]) LL LL LL jj ii NN NN ka k k a k kk max h c c b max h N c c b        . Novel Applications of the UWB Technologies 110 Therefore, we have 11 () () () () max max () () (1) () 00 1 () () () () 0 ,([(),()][( ),()]) 2([( ),( )]) LL LL LL LL jj ii NL NL NL NL ka k k a k kk L j i NN ka k k SC maxhc c b hNc c b max h c c b           Q.E.D Based on Theorem 4, we can obtain another theorem which indicates that the correlation properties of TH sequences will be improved when the number of TH time slot satisfies N  2N h + 1. Theorem 5: Let   () () L i k c and   () () L j k c denote two TH sequences with period L , respectively. When 21 h NN , we have 1 () () () () 0 1 () () (1) () 0 [( ) ,( ) ], 0 1 () [( ) ,( ) ], 1 LL LL L j i NL NL h ka k k ij L j i NL NL h ka k k hc c b b N Cl hN c c b N b N                      and 1 () () max max () () 0 ,([(),()]) LL L j i NN ka k k SC maxhc c b      . (17) Proof: According to the equation (4), we have 11 () () () () () () (1) () 00 () [( ) ,( ) ] [( ) ,( ) ] LL LL LL jj ii i j NL NL NL NL ka k k a k kk Cl hc c b hN c c b      . When 01 h bN , we have () () 0( ) 2 1 L j NL h k cb N   since () () 0 L j h k cN . Similarly, we also have () (1) ()21 L i NL h ka Nc N   when 21 h NN. As a result, it is obvious that 1 () () (1) () 0 [( ) ,( ) ] 0 LL L j i NL NL ka k k hN c c b      . Then, 1 () () () () 0 () [( ) ,( ) ] LL L j i i j NL NL ka k k Cl hc c b      when 01 h bN . When 1 h NbN , we have () () () 1 L j NL h k cbN   since () () 0 L j h k cN. Combining the result with () () 0( ) L i NL h ka cN  , we can obtain 1 () () () () 0 [( ) ,( ) ] 0 LL L j i NL NL ka k k hc c b       . Hence, 1 () () (1) () 0 () [( ) ,( ) ] LL L j i i j NL NL ka k k Cl hN c c b       when 1 h NbN   . Time-Hopping Correlation Property and Its Effects on THSS-UWB System 111 According to Theorem 4, we have 1 () () max max () () 0 ,([(),()]) NL L j i LN ka k k SC maxhc c b      . Q.E.D. To show how Theorem 5 works, we give a simple example using QCC sequences, where 11p  and 11L  . Fig. 6 and Fig. 7 show the distributions of correlation function values of QCC sequences when 11N  and 21N  , respectively. By comparing two figures, we can see that the maximum TH correlation function values are deceased to a half of original values. -60 -40 -20 0 20 40 60 0 5 10 15 Correlation Function Values shift -60 -40 -20 0 20 40 60 0 1 2 3 4 Correlation Function Values shift (a) (b) Fig. 6. The distribution of correlation function values of QCC sequences, where 11N  . (a). ACF of   11 (2) ()k c ; (b). CCF between   11 (3) ()k c and   11 (5) ()k c Novel Applications of the UWB Technologies 112 -100 -50 0 50 100 0 5 10 15 Correlation Function Values shift -100 -50 0 50 100 0 0.5 1 1.5 2 Correlation Function Values shift (a) (b) Fig. 7. The distribution of correlation function values of QCC sequences, where 21N  . (a). ACF of   11 (2) ()k c ; (b). CCF between   11 (3) ()k c and   11 (5) ()k c 5. TH sequences with ZCZ In this section, we begin with the definition of ZCZ of TH sequences to understand how ZCZ works. We then construct a class of TH sequences with ZCZ and prove the correlation properties of such TH sequences when the shifts between ZCZ TH sequences are in the range of ZCZ. 5.1 Definition of ZCZ of TH sequences According to Definition 3 on TH period correlation function, we can define the ZCZ of TH sequences as follows. Definition 5: Let C ij (l) denotes TH periodic correlation function between two TH sequences   () () L i k c and   () () L j k c with period L , and then ZCZ of TH sequences can be expressed as Time-Hopping Correlation Property and Its Effects on THSS-UWB System 113 ,0 () 0, 0 | | 2 ii A CZ Ll Cl Z l         (18) and () 0, 0 || , 2 CCZ ij Z Cl l i j    , (19) where A CZ Z and CCZ Z denote TH zero auto-correlation zone (ZACZ) width and TH zero cross-correlation zone (ZCCZ) width, respectively. According to definition 5, both of CCF and ACF sidelobes are equal to zero when the shifts between TH sequences are in the range of CZ Z , where {,} CZ ACZ CCZ ZminZZ  . Then, orthogonal communications can be realized when the approximate chip synchronization is held between users in whole system. 5.2 Construction of ZCZ TH sequences The principle of construction of ZCZ TH sequences can be depicted in Fig. 8, where     (1) (1) () () LL kk ce and     (2) (2) () () 1 LL eh CCZ kk cNZe respectively denote two TH sequences, and   () () L i k e is any existing TH sequence satisfying () () 0 L i eh k eN. Fig. 8. The principle of construction of ZCZ TH sequences According to Fig. 8, (1) (1) (0) (0) 2 LL ce   , (1) (1) (1) (1) 0 LL ce   , (2) (0) 1 L e  , 3 eh N  and 6 CCZ Z  . Then, we have (2) (2) (0) (0) 1316111 LL eh CCZ cN Ze  . In terms of such principle, a class of ZCZ TH sequences can be constructed as follows. Construction of ZCZ TH Sequences: For the given ZCZ width CZ Z which is determined by THSS-UWB systems, a novel ZCZ TH sequence   () () L i k c can be expressed as () () () () (1)( 1 ) LL ii eh CZ kk ciN Ze   . (20) The widths of ZACZ and ZCCZ satisfy CCZ c Z T      (1)   f ACZ eh c TZ N T (1) eh c NT  112 0 Novel Applications of the UWB Technologies 114 (1) 1 1 ACZ u eh CZ eh eh ZNN ZN NN   and CCZ CZ ZZ , where (1) ueh CZ NN N Z  and () () 0 L i eh k eN. Based on Definition 3, correlation properties of the constructed ZCZ TH sequences can be proved as follows. Proof: (1). We first consider the case of ij  , namely CCF. Let the synchronization error  of a THSS-UWB system satisfy || 2 CZ c Z T    when the approximate chip synchronization is held in the whole system. Correspondingly, the shift between two TH sequences is equal to laNb   , where 0 1aL   and 0 CZ bZ . The evaluation of ( ) ij Cl will be carried out in two steps on the basis of its two components. i. According to the equation (20), the first part of ( ) ij Cl can be expressed as 11 () () () () () () () () 00 [( ) ,( ) ] [(( )( 1 ) ( )) ,( ) ] LL LL LL jj i i NL NL eh CZ NL NL ka k ka k kk hc c b h i j N Z e e b          , where () () () () LL j i eh eh ka k Ne e N     since () () ()() 0, LL j i eh ka k eeN  . Then, it is obvious that () () () () (( )( 1 ) ( )) 1 LL j i eh CZ NL CZ ka k ijN Z e e Z     when ij . If ij  , we will have (1) 1 u Nij . Then, () () () () (1)( 1 ) ()( 1 )( ) (1) (1 ) 0 LL j i uehCZeh ehCZ ka k eh CZ eh CZ NN ZNijN Ze e NZN Z               We can further obtain that () () () () ( ) () ( ) () (( )( 1 ) ( )) |( )( 1 ) ( )| (1)( 1 ) (1)(1)(1) (1) ( 1)(1 ) 1 LL LL jj ii eh CZ NL eh CZ ka k ka k uehCZeh ueh CZ u eh CZ eh uehuu CZ CZ ijNZee NLijNZee NL N N Z N NN Z L N N Z N NL N NLN Z Z             As a result, when ij  , we have () () () () (( )( 1 ) ( )) 1 LL j i eh CZ NL CZ ka k ijN Z e e Z    . Due to 0 CZ bZ , we can obtain that Time-Hopping Correlation Property and Its Effects on THSS-UWB System 115 11 () () () () () () () () 00 [( ) ,( ) ] [(( )( 1 ) ( )) ,( ) ] 0 LL LL LL jj i i NL NL eh CZ NL NL ka k ka k kk hc c b h i j N Z e e b           ii. The second part of ( ) ij Cl can be expressed as 1 () () (1) () 0 1 () () (1) () 0 [( ) ,( ) ] [(( )( 1 ) ( )) ,( ) ] LL LL L j i NL NL ka k k L j i uehCZ NLNL ka k k hN c c b hN ijN Z e e b           Similarly, when ij  , we can obtain that () () (1) () (( )( 1 ) ( )) 1 LL j i uehCZ NLCZ ka k NijN Z e e Z         . Due to 0 CZ bZ , we can obtain that 1 () () (1) () 0 [( ) ,( ) ] 0 LL L j i NL NL ka k k hN c c b      . In terms of the above analyses, the CCF values of the constructed ZCZ TH sequences are equal to zero when the shifts are in range of CCZ Z , namely ( ) 0 ij Cl  when 0 2 CCZ Z l and ij . (2). Secondly, we consider the case of ij  , namely ACF. For an approximately synchronized THSS-UWB system, when multipath delay is in the range of ACZ c ZT , the shift of TH sequence   () () L i k c is correspondingly equal to laNb, where 0a  and 0 A CZ bZ   . Similar to ( ) ij Cl, the evaluation of () ii Cl will be carried out in two steps. i. According to equation (20), the first part of () ii Cl can be expressed as 11 () () () () 00 [(( )( 1 ) ( )) ,( ) ] [(0) ,( ) ] LL LL ii eh CZ NL NL NL NL kk kk hiiN Z e e b h b      . Due to 0 ACZ bZ , we have 11 () () () () 00 [( ),( )] [(0),()]0 LL LL ii NL NL NL NL ka k kk hc c b h b       . ii. The second part of () ii Cl can be expressed as 1 () () (1) () 0 1 () () (1) () 0 [(( )( 1 ) ( )) ,( ) ] [( ( 1 ) ( )) ,( ) ] LL LL L ii uehCZ NLNL kk k L ii ueh CZ NL NL kk k hN iiN Z e e b hN N Z e e b               Due to () () (1) () LL ii eh eh kk Ne e N     and (1) ueh CZ NN N Z  , we can obtain that eh NN () () (1) () (1)( ) LL ii ueh CZ eh kk NN Z e e N N   . Also, since 01 SCZ eh bZ NN  , then, Novel Applications of the UWB Technologies 116 11 () () () () (1) () (1) () 00 [( ) ,( ) ] [( ( 1 ) ( )) ,( ) ] 0 LL LL LL j i ii NL NL u eh CZ NL NL ka k k k kk hN c c b hN N Z e e b        According to the above analyses, the ACF sidelobes of the constructed ZCZ TH sequences are equal to zero when the shifts are in range of A CZ Z . Q.E.D. 6. Effects of TH correlation properties on MAI in THSS-UWB systems By transforming the signal model of THSS-UWB communication systems, we obtain expressions for the relation of MAI values and TH correlation function values in this section,. 6.1 Binary model of TH sequences According to the equation (1), we can see that only one pulse is transmitted to each user within any frame time f T , i. e. One-Pulse-Per-Frame structure (Erseghe, 2002b; Scholtz et al, 2001). The pulse position is decided by TH sequence   () () L i k c , namely () () . L i c k cT . For more easiness to understand, the structure is depicted in Fig. 9, where elements of a TH sequence are binary ones. Fig. 9. The hopping format of pulses in PPM We assume that “1” denotes the time slot where a pulse is modulated, and the other time slots in frame time f T are “0”. As a result, the binary TH sequence   () () NL i n a can be obtained. The sequence   () () NL i n a corresponds to   () () L i k c and its period is equal to NL . According to the above analyses, the equation (1) may be transformed as () () () () [/( )] () ( ) NL s ii i c nnNN n St a wtnT d       , (21) () () NL i n a 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 f T c T () (3) 1 L i c  () (2) 3 L i c  () (1) 2 L i c  [...]... period correlation function in Definition 4 In the equation (26), the first part is the signal that we desire The second part represents the MAI that the other users make to user 1, and the last part is the interference made by noise We are interested in the second part, which will be analyzed in the following The analysis of THSS -UWB MAI is similar to the performance evaluation for DSSS multiple-access... Description of our model with first stage synchronization The basic idea behind TDT is to find the maximum of square correlation between pairs of successive symbol-long segments These symbol-long segments are called “dirty templates” because: i) they are noisy, ii) they are distorted by the unknown channel, and iii) they are representing estimate offset of subject to the unknown offset Then, we will... (20), the estimation of delay τ0 is made possible due to the presence of the term εA(τ ) - εB(τ ) Unfortunately for the estimator x (M; τ), this term exists only if the transmitted sequence presents an alternating sign between the symbols s(m − 2) and s(m) Thus, for the synchronization in NDA mode, the performances of this approach are affected by the sign of the transmitted symbols To increase the chances... computationally complex as they need high sampling rates In (Djapic et al, 2006), a blind synchronization algorithm that takes advantage of the shift invariance structure in the frequency domain is proposed An accurate signal processing model for a Transmitreference UWB (TR -UWB) system is given in (Dang et al, 2006) The model considers the 124 Novel Applications of the UWB Technologies channel correlation... equivalent to the output signal-to2 N1 noise (SNR) ratio that one might observe in single link experiments Then, the BER can be given by Pe  1 2  I exp(  x 2 / 2)dx SIR (32) 120 Novel Applications of the UWB Technologies According to the equations (30)-(33), we can see that the interference and the BER are determined by C max when A1 , Ai , N s , Tc and ISNR are specific For the construction of TH sequences... degradation of UWB radios due to mistiming (Tian & Giannakis, 20 05) The complexity of which is accentuated in UWB owing to the fact that information bearing waveforms are impulse-like and have low amplitude In addition, compared to narrowband systems, the difficulty of timing UWB signals is increased further by the dense multipath channel that remains unknown at the synchronization step These reasons... making integration between the received signal and its replica shifted by Tf on a window of width Tcorr τ being the estimate delay deducted after the first synchronization floor and the width integration window 132 Novel Applications of the UWB Technologies value’s T will be given in Section 5 This principle is illustrated in Fig.3 We can write the integration window output for the nth step nδ as follows... mode 5. 2 TH-PPM UWB system in single-user links In addition to other parameters described previously, the time shift associated with binary PPM is δ = 1ns Then, the performance is tested for various values of M In Fig 8, we first test the MSE of NDA and DA TDT algorithms for UWB TH-PPM systems We note that increasing the duration of the observation interval M leads to improved Fine Synchronization in UWB. .. represents the propagation delay of the channel Then, the received waveform is given by r(t) = √ε ∑ s(k)p t − kT − τ , − τ + η(t) (3) where , is arbitrary reference at the receiver representing the delay relative to the arrival moment of the first pulse, ( ) is the additive noise and ( ) denotes the received symbol waveform as p (t) = ∑ p(t − iT −c (i)T ) ∗ g(t + τ ) (4) where * indicates the convolution... define the timing offset as ∆ ≔ , − Let us suppose that ∆ is in the range of [0, Ts) and we will show in the rest of this paper that this assumption will not affect the timing synchronization 2.2 TH-PPM UWB system model for single-user links With PPM modulation (Durisi & Benedetto, 2003; Di Renzo et al, 20 05) , the transmitted signal in single-user links is described by the following model 126 Novel Applications . transmitter 1, the demodulation output of the k th bit is Novel Applications of the UWB Technologies 118 (1) (1) () () sc sc kN NT k kNNT TrtVtdt      . ( 25) Then, the received. Definition 4. In the equation (26), the first part is the signal that we desire. The second part represents the MAI that the other users make to user 1, and the last part is the interference. -50 0 50 100 0 5 10 15 Correlation Function Values shift -100 -50 0 50 100 0 0 .5 1 1 .5 2 Correlation Function Values shift (a) (b) Fig. 7. The distribution of correlation function values of