PN Code Tracking Loops

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1 Wireless Information Transmission System Lab. PN Coe Traking Loops Institute of Communiations Engineering National Sun Yat-sen University

2 Introution Coe synhronization is generally arrie out in two steps : Aquisition (oarse synhronization) : To aligns the inoming signal an the loal PN signal to within one hip or less. Traking (fine synhronization) : To bring the ifferene of the two phases to zero. In most situation, the PN aquisition is performe before (or at best onurrently with) the arrier reovery an traking. Note that the PN traking an arrier traking are ontinuous operation after the PN phase has been aquire beause (a) there is relative motion between the transmitter an the reeiver, an (b) imperfet osillators.

3 Introution Coe traking is aomplishe using phase-loke tehniques, very similar to those use for generation of oherent arrier referenes. The prinipal ifferene between the phase-loke loops use for arrier traking an oe traking is in the implementation of the phase isriminator. For arrier traking, the isriminator is often as simple as a multiplier, whereas for moern oe traking loops, several multipliers an usually pairs of filters an envelope etetor will be employe in the phase isriminator. A traking iruit generally onsists of a feebak loop that monitor the error an ajusts the esire signal in suh a way that the error goes to zero. 3

4 Introution The loop banwith will be selete to be a ompromise between a wie banwith, whih failitates traking the ynamis of transmission elay, an a narrow banwith, whih minimizes the traking jitter ue to interferene. A onvenient measure of the traking performane is the synhronization error variane (traking jitter), whih shoul be mae small. There are two ommon methos use for traking : Delay-Lok loop (DLL) traking metho. A traking loop that makes use of two inepenent orrelators. Tau-Dither Loop (TDL) traking metho. A traking loop that time shares a single orrelator. 4

5 Timing jitter Introution A result that is partiularly important to the sprea-spetrum system esigner is the relationship between the traking jitter an the reeive signal-to-noise ratio in the loop banwith. τ τ The traking jitter is efine as δ The rms traking jitter is σ δ = ( f ) H ( jπf ) f where S n" (f) is the power spetrum of the Gaussian noise proess at the input to the loop filter, an H(s) is the loseloop transfer funtion efine : H( s) 5 τ ( s) τ ( s) T S n

6 Optimum Traking of Wieban Signals It has been shown that the optimum traking isriminator for an arbitrary wieban signal reeive with aitive white Gaussian noise (AWGN) is a multiplier that forms the prout of the reeive signal plus noise an the first erivative with respet to time of the reeiver generate replia of the transmitte signal. This isriminator is optimum in that its output is a maximum likelihoo estimate of the phase ifferene between the two wieban signals in an AWGN environment. This means that the output phase error estimate is the most probable phase error, given the available reeive information. 6

7 Optimum Traking of Wieban Signals The optimum traking isriminator for an arbitrary wieban signal reeive with AWGN : 7

8 Optimum Traking of Wieban Signals The reeive signal r(t) = s(t - T ) + n(t) is multiplie by a ifferentiate an elay reeiver-generate replia of s(t). The multiplier output ontains a omponent relate to the elay error ( T T ), where T is the estimate of the transmission elay. This omponent is extrate by the lowpass filter an use to orret the elay of the voltage ontrollable elay line. This traking loop onfiguration is not ommonly use in moern sprea-spetrum systems an the operation of the loop will be esribe for a single speial ase. 8

9 Optimum Traking of Wieban Signals T > T T T < T 9

10 Optimum Traking of Wieban Signals Operation of the oe traking loop for a speial ase: The reeive signal is shown in figure (a). The traking loop is to proue the signal (t- ), shown in Tˆ figure (b), with δ = ( T ˆ ) as small as possible. T T The first erivative of figure (b) with respet to time is shown in figure (). The erivative is a series of impulse funtions. The traking loop multiplier output is shown in figure (). The omponent of the multiplier output is the time average of ( t T ) ( t T ) t whih is (N + 1)/NT. 10

11 Optimum Traking of Wieban Signals When Tˆ < T an T - Tˆ <T, all of the impulses at the multiplier output are negative an the omponent is -(N + 1)/NT. Optimum elay isriminator output omponent for m- sequene baseban traking loop: 11

12 Then the elay line input will be positive an that this will erease T as require to rive elay error to zero. When T < T an T T < T, all of the impulses at the multiplier output are negative an the omponent is (N + 1)/NT. When T T T, there is an equal number of positive an negative multiplier output impulses, an the level is zero. Whenever T T < T, a voltage exists whih pushes the elay in the orret iretion. T Optimum Traking of Wieban Signals 1

13 Coherent Delay-Lok Traking Loop os ( π ft+ θ ) ˆb( t) () + n( t) s t ( t + τ + τ ) ( + τ) τ < T t ( t + τ τ ) v 1 (t) Low Pass Filter PN Sequene Generator VCO y(t) u 1 (t) Loop Filter z(t) - + ( π ft+ θ ) v (t) os From arrier reovery system. Low Pass Filter Delay-lok traking loop with oherent arrier emoulation. 13 ( ) u (t) ˆb t From reeiver output.

14 Coherent Delay-Lok Traking Loop We assume that traking is initiate after the aquisition iruit has brought the phase ifferene to within ± T. We further assume without loss of generality that the signal ( ) ( ) ( π θ ) st ( ) = P t b t os ft + at the input of the reeiver, i.e., the phase of (t) is zero. The loal PN generator generates (t+τ), where τ < T. In aition, it also proues avane an elaye version of PN signal: (t+τ+τ ) an (t+τ-τ ) for a fixeτ. Then the avane an elaye (early an late) PN signals are mixe (multiplie) with the inoming signal an the arrier os( π f t + θ ) from the arrier reovery iruit. 14

15 Coherent Delay-Lok Traking Loop The ifferene of two esprea signals is an error signal whih is feebak to ajust the phase of the PN generator. The outputs from the mixers are ( ) ( ) ( ) υ τ τ π θ 1() t = P t t + + b t os( ft+ ) P = t t + + b t + f t + υ τ τ π θ () ( τ τ ) ()[ 1 os(4π θ) ] () ( ) () () t = P t t + b t os( ft + ) P = t t + b t + f t + () ( τ τ ) ()[ 1 os(4π θ) ] The lowpass filters have banwith wie enough to pass b(t), but narrow enough that it average (lowpass filters) the (t)(t+τ+ τ ) omponent. 15

16 Coherent Delay-Lok Traking Loop Therefore, after lowpass filtering an mixing with b (t), the signals are Pbt () bt () t P u1() t t (') t (' ) t' bt () btr () ( ) NT + τ + τ = τ + τ t NT Pbt () bt () t P u( t) t ( ') t ( ' ) t' bt ( ) btr ( ) ( ) NT + τ τ = τ τ t NT where R (τ) is the normalize autoorrelation funtion of the PN signal, whih has a perio of NT : δ 1, if δ - int < T for some integer i R ( δ ) T 0, otherwise 16

17 Coherent Delay-Lok Traking Loop Assume that b( t) = b( t), so that b( t) b( t) = 1. The ifferene between u 1 (t) an u (t) is P ( τ ) z() t noise= 0= u( t) u1( t) = [ R ( τ τ ) R ( τ + τ )] where bt ( ) is an estimate of b() t. Estimation of b(t) has some oasional errors, so the loop filter, whih is a lowpass filter with a small banwith, averages out the prout b ( t) b( t). Sine the probability of error in a properly working reeiver is small, the estimate b ( t) equals b( t) most of the time an the average of b ( t) b( t) is nearly 1. The funtion (τ) is alle the elay isriminator harateristi, an it is also a perioi funtion with perio NT. 17

18 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = 0.5. Delay Disriminator Charateristi (Tau / T = 0.5) PN Coe Perio = 31 Correlation On Time Late Early S-Curve=Late-Early [ T ] 18

19 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = 0.5. Delay Disriminator Charateristi (Tau / T = 0.50) PN Coe Perio = 31 Correlation On Time Late Early S-Curve=Late-Early [ T ] 19

20 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 0.75) PN Coe Perio = 31 Correlation On Time Late Early S-Curve=Late-Early [ T ] 0

21 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 1.00) PN Coe Perio = 31 Correlation On Time Late Early S-Curve=Late-Early [ T ] 1

22 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 0.5) PN Coe Perio = 4095 Correlation On Time Late Early S-Curve=Late-Early [ T ]

23 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = 0.5. Delay Disriminator Charateristi (Tau / T = 0.50) PN Coe Perio = 4095 Correlation On Time Late Early S-Curve=Late-Early [ T ] 3

24 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 0.75) PN Coe Perio = 4095 Correlation On Time Late Early S-Curve=Late-Early [ T ] 4

25 Coherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 1.00) PN Coe Perio = 4095 Correlation On Time Late Early S-Curve=Late-Early [ T ] 5

26 Coherent Delay-Lok Traking Loop For the purpose of isussion, suppose that y(t)=z(t). The signal y(t) ontrols the voltage-ontrolle lok (VCC). If y(t)=0, the VCC an PN generator nee no ajustment. When y(t) 0, the phaseτof the loal PN generator is ajuste appropriately (by elaying or avaning the phase, epening on whether y(t) is positive or negative). The phaseτof the PN generator must be inrease when y(t) >0 an erease when y(t) <0. 6

27 Nonoherent Delay-Lok Traking Loop () + n( t) PN Sequene t ( + τ) τ < T Generator t ( + τ τ ) s t ( t+ τ + τ ) w 1 (t) w (t) BPF BW=B@f v 1 (t) BPF BW=B@f v (t) square law envelope etetor VCO ( ) LPF y(t) Loop Filter ( ) LPF z(t) square law envelope etetor u 1 (t) - + u (t) Delay-lok traking loop with nonoherent emoulation. 7

28 Nonoherent Delay-Lok Traking Loop There are two ifferene between oherent an nonoherent DLL : A square-law envelop etetor is use to eliminate the arrier. Aoring to the square-law etetor, we on't nee b (t). Sine the signal s(t) an the noise n(t) is the same as above, the signal at the input of the upper banpass filter is w1 ( t) = P( t) ( t + τ + τ ) b( t) os(πf t + θ ) + noise Let the banpass filters have a passban entere at ±f an a banwith of B Hz an unity gain. The banwith is wie enough to pass b(t), but narrow enough that the PN signal gets average out. 8

29 Nonoherent Delay-Lok Traking Loop With these, we have υ1 ( t) = PR ( τ + τ ) b( t) os(πf t + θ ) + where the noise is a banpass noise. The signal υ ( t 1 ) is passe through the square-law envelop etetor, it is square an lowpass filtere : u1 ( t) = PR ( τ + τ ) + noise where the noise is now a lowpass noise. Similarly, u ( t) = PR ( τ τ ) + noise Therefore, the input to the loop filter is zt () = u() t u1() t = P R( τ τ) R( τ + τ) + noise 9 noise

30 Nonoherent Delay-Lok Traking Loop The elay isriminator harateristi is the signal omponent of z(t): ( τ) z( t) = P R ( τ τ ) R ( τ + τ ) with ( τ ) noise= 0 0 for NT / < τ T τ τ + τ P 1 +, for T τ < τ T + τ T 4τ τ P 1 +, for T + τ < τ τ T T 4τ τ = P 1, for τ < τ τ T T 4τ τ P 1, for τ < τ T τ T T τ τ P 1, for T-τ < τ T + τ T 0 for T + τ < τ < NT / 30

31 Nonoherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 0.5) PN Coe Perio = 4095 Correlation On Time Late Early S-Curve=Late-Early [ T ] 31

32 Nonoherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 0.45) PN Coe Perio = Correlation On Time Late Early S-Curve=Late-Early [ T ] 3

33 Nonoherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 0.75) PN Coe Perio = Correlation On Time Late Early S-Curve=Late-Early [ T ] 33

34 Nonoherent Delay-Lok Traking Loop Delay Disriminator Charateristi with τ T = Delay Disriminator Charateristi (Tau / T = 1.00) PN Coe Perio = 4095 Correlation On Time Late Early S-Curve=Late-Early [ T ] 34

35 Nonoherent Delay-Lok Traking Loop One problem with the elay lok loop is that the two branhes in this loop must have mathe harateristis (gain balane). Otherwise, the loop will be loke to a non-zero point, sayτ 1. One metho to solve this question is by using the tau-ither traking loop. Other methos may be foun, for example, in: Rihar A. Yost, an Robert W. Boy, A Moifie PN Coe Traking Loop: Its Performane Analysis an Comparative Evaluation," IEEE Trans. On Communiations, Vol. COM-30, No. 5, pp , May

36 ( ) + n( t) s t Tau-Dither Traking Loop Tau-Dither traking loop (or time-share early-late traking loop) uses only orrelator branh. w(t) q 1 (t) q (t) f square law envelope etetor v(t) Square Wave Gen. ( ) LPF q(t) u(t) ( t+ τ + τ ) t ( + τ τ ) PN Sequene Generator VCO y(t) Loop Filter z(t) 36

37 Tau-Dither Traking Loop The tau-ither loop eliminates the problem of mathing the harateristis of the two orrelator branhes. However, the signal power in the tau-ither loop is 3 B smaller than that in the elay-lok loop an the traking jitter is larger in the tau-ither loop. The ithering is ontrolle by two pulse signals q 1 (t) an q (t), whih alternately let the avane an elaye versions of the PN signal through at a rate of F D =1/T D, where T D»T. The effet of orrelation with early an late versions of the PN signal is ahieve by ithering bak an forth between these early an late signals. The square wave q(t) also ontrols the signal into the loop filter, in synhronism with the phase ithering. 37

38 Tau-Dither Traking Loop 1 Dithering Loop Swithing Funtions q(t) 0-1 T D 1.5 t 1 q 1 (t) T D t 1 q (t) T D t 38

39 Tau-Dither Traking Loop The signal w(t) is, alternately, equal to w 1 (t)an w (t), where w1 ( t) = s( t) ( t + τ + τ ) + noise w ( t) = s( t) ( t + τ τ ) + noise Using q 1 (t) an q (t), we an write w ( t) = q1 ( t) w1 ( t) + q ( t) w ( t) + noise An the signal u(t) alternates between u 1 (t) an u (t), where u t = PR ( τ + τ ) + noise u t = PR ( τ τ ) + noise () () 1 Then, we have zt () = qtut () () = qt () q() tu() t + q() tu() t [ ] 1 1 = q ( t) u ( t) q ( t) u ( t) 1 1 = P q1( t) R ( τ + τ) q( t) R ( τ τ) + noise 39

40 40 If the loop filter has a narrow banwith, ompare to f D, the signal q 1 (t) an q (t) are effetively average, yieling the elay isriminator harateristi of whih is half of that for the elay-lok loop. Therefore, the signal power of the tau-ither loop is 3 B smaller than that of the elay-lok loop. Tau-Dither Traking Loop [ ] [ ] { } [ ] ) ( ) ( ) ( ) ( Ave. ) ( ) ( Ave. ) ( 1 R R P R t q R t q P τ τ τ τ τ τ τ τ τ + = + =

41 Referenes Viterbi, A. J., Priniples of Coherent Communiation, MGraw- Hill, New York, Heinrih Meyr, Mar Moenelaey, an Stefan A. Fehtel, Digital Communiation Reeivers -- Synhronization, Channel Estimation, an Signal Proessing, John Wiley & Sons In., Roger L. Peterson, Roger E. Ziemer, an Davi E. Borth, Introution to Sprea-Spetrum Communiations, Appenix A, Prentie Hall In., John G. Proakis, Digital Communiations, fourth eition, Chapter 6, MGraw-Hill,

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