CC BY
12Journal of Siberian Federal University. Engineering & Technologies 3 (2008 1) 276-286
УДК 621.384 191
Accelerated Phase-lock-loop Frequency
Control Methods of User's Equipment in Perspective Radio
Navigation Systems
Evgeny V. Kuzmin*
Siberian Federal University, 79 Svobodny, Krasnoyarsk, 660041 Russia 1
Received 05.08.2008, received in revised form 10.09.2008, accepted 17.09.2008
This paper investigates noise-immunity of accelerated phase-lock-loop frequency control algorithms of user equipment in perspective ground-based radio navigation systems. Three algorithms of accelerated phase-lock-loop frequency control are suggested and described. Statistic simulations of signal processing in involved system are given.
Key words: Radio navigation, spread-spectrum signal, minimum shift keying, phase-shift discriminator, phase synchronization system, acceleratedphase-locked-loop frequency control, phase-tracing error, statistical modeling, quasi-optimal algorithm.
Introduction
Spread spectrum signals with minimum shift keying (MSK) are widely used in modern radio navigation systems (RNS), e.g.: GEOLOC (France). High accuracy of coordinate measuring in the whole RNS working area requires providing phase shift measurements with root-mean-square (RMS) error ct9 < 3°, when signal-to-noise ratio threshold equals to -40dB (in the band of MSK-signal). That is why, the meaning of phase-lock-loop frequency control pass band equals to 0,1 Hz. Thus, locking time is 600 5, and can grow by a factor of 10 under noise and jamming influence [1].
Recently, researchers have shown an increased interest in Kalman filtering, because it can provide high accuracy of phase tracing measurements. But Kalman filter has a significant disadvantage -computational complexity, therefore, in the foreseeable future it can't be used for preprocessing algorithms.
Due to limits in computational technology, it's necessary to investigate phase tracking algorithms with performance objectives: small values of locking time and RMS error. So, the hypothesis that will be tested is that multistage (several meanings of pass band) phase-lock-loop frequency control algorithms can provide adequate accuracy of phase-tracing measurements and greatly smaller locking time. Consequently, investigation of accelerated phase-lock-loop frequency control algorithms with invariable phase shift accuracy is a topical scientific problem.
* Corresponding author E-mail address: [email protected]
1 © Siberian Federal University. All rights reserved
1. Navigation signal model of perspective RNS
Total realization of received MSK-signal and additive white Gaussian noise (AWGN) can be described as:
y(t) = Re{S(t)exp[ j(2%(f0 ± Fd)t -cp,)]} + ^(t), (1)
here j - imaginary unit; f - carrier frequency; Fd - Doppler frequency shift; cps - starting phase of signal; £,(t) - AWGN; S(t) - complex envelopeof MSK-signal:
S (t ) = D (t )J2F exp [ j9( t)], (2)
t
where Ps - signal's power; D(t) ==±1 - information signal; 9(f) = —Jd(t')dt' - function which
2T 0
N-1
determines angle modulation, d(t) = ^dtrect(t-/T), {dj} - pseudorandom sequence (PRS) of
1=0
N-length, T - one's bit PRS duratio n, rect(t) - square pulse with T duration [2].
2. Phase synchroniz2tion system of M SK-signal receiver
Structural chart of MSK-signal receiver's digit2l phase-lock-loop frequency control system (PLFS) is presented in Fig. 1. Values yi = y(ti) (tj = iAt, i = 0,1,..., At - sampling interval) are incoming observations to digital phase-shift discriminator (DPD), formed by analog-digital converter (ADC).
Reference signals of carrier frequency cos i (t) = cos(2n(f0 ± Fd (t))ti) and sini..(t) = cos(2n(f0 ± Fd(t))ti) c;ome into supporting inputs of DPD. These signals are formed by digital synthesizer (DS) and based on Doppler frequency shift estimation Fd(it) in each t-period of filtering. Reference signals Qt = sin9.. and I, = cos9., which are synchronous with quadrature components of MSK-signal, are formed by delay lock system. Quadrature components of bandwidth compressing signal (after MSK-detection) are formed by summarizing of multiplications of quadrature components of realization (1) and reference signals Ii, Qt and integration on intervals t e[kT ,(k + 1)T)], t = 0,1,..., (T) = 40ms - MSK-signal's period). Time of one cycle radio-range beacon transmition equals Tc = 25T). Error sigtal which is proportional to phase mismatch forms in compliance with quasi-optimal algorithm [3]:
Zd (tr) = sign (Z1 (t )>2 (t) = D (t)Z2 (t), (3)
where sign (x) - sign function, D(2) ^^s^i^ation ol" information signal D(t) on t-period of filtering, z1 (t) and z2 (t) - quadrature components of correlation, computed on interval t e [+1)T). Error signal Zd (t) comes into digital filter (DF). Output signal ofDF used to control signals cos i. (t) and sin <ii (t) frequencies. When there is no noise, discriminaTion characteristic can be described as
Zd (cp) = -1-tt/ i^ign (coscp)sin cp
Structural chart of the DPD is presented in Fig. 2, where x - multiplier; + - adder; £ - adder accumulator (digital integrator), which interrogated in ttT) moments, t = 0,1, . . .; M = T) / At - integer.
Normalized discrimination (curves 1, 2) and fluctuation (curve 3) characteristics of DPD are presented in Fig. 3. At that, curve 1 corresponds with no-noise case, and curves 2, 3 present discrimination and fluctuation characteristics respectively. Curves 2, 3 are the statistical simulation
Fig. 1
Fig. 2
Z,a2 (cp)
Fig. 4
data then signal-to-noise ratio equals to -40dB. Length of using PRS N = 214 -1 = 16383. Number of statistical examinations equals to 104.
The model of PLFS is presented in Fig. 4, where Zd (9) - discrimination characteristic of DPD; Tt - time constant of integrator; K = K Kc - instantaneous element, taking account of transfer constants of digital filter K and digital synthesizer Kc; the meaning of another designation are clear without comments.
Doppler frequency shift on k-period of filtering is estimated in compliance with the following algorithm:
Fd (k) = K^Zd (k)+x(k -1)+(k - l)j. (4)
Discriminator nonlinearity in case of using quasi-continuous analyzing method for digital synchronization systems is taking into account by it parameters, which depend on signal-to-noise ratio [4].
3. Accelerated phase synchronization target setting
In phase navigation systems RMS error of coordinate measuring (in meters) can be approximately determined as
^ ACT9' (5)
where - wave-length, A - geometric quotient, ap - RMS error of phase-shift measurements [5]. In steady-state regime phase-tracking error dispersion value can be determined by using quasi-continuous analyzing method for digital synchronization systems [6]:
< = 2a2TpF9, (6)
here a;; - phase fluctuation dispersion, which can be calculated as
a2
„2 =
kd
_ „2
a2 = t' (7)
where ad =ad (0) - fluctuation characteristic for algorithm (3) of phase mismatch failing; kd =dZd («p)/^«^ - discrimination characteristic slope for algorithm (3), line from the top means statistical estimation. Noise pass band of PLFS can be written as
Fp=2-j] k (>) X (8)
2n 0
where K (jra) - complex transfer coefficient of PLFS.
Using (5) it can be shown that in case of r= 1,5 (rho-rho navigation), = 150m for attainment of coordinate measuring accuracy with RMS ac < 2 m needed RMS error of phase-shift measurements value is ap<3,3 -0,053rad. Further, using results [3] for ad and k], when signal-to-noise ratio threshold equals to -40 dB, and using equation (6) let's compute required noise pass band of PLFS for MSK-signal receiver:
F9<-0L< °,°532 - 0,1 Hz. (9)
9 -a2T 2 • 0,364 -0,04 w
rad
20 k/103
F, Hz
0,2
b
Fig. 5
0,1--
20 kj 103
Thus, PLFS must provide RMS error of phase-shift measurements value ct9 < 0,05 rad in case of noise pass band value Ftf < 0,1 Hz.
Functional dependences of phase-tracking and frequency estimation error average values from discrete time k in digital PLFS are presented in Fig. 5, a and 5, b respectively. Computational approach conditions are equal to discriminator modeling, except number of statistical examinations - 102.
Presented functional dependences are correspondent to noise pass band value F9 = 0,1 Hz, user's top speed equals Vmax = 100km/h (peak level of Doppler frequency shift |Fdmax| = 0,2Hz) and capture probability Pc ^ 1.
Analysis of statistic simulation data of digital PLFS (Fig. 5) shows that average locking time has intolerable level for perspective RNS for special users - t«15 • 103 • Tp = 600 s.
4. Digital PLFS statistical simulation
Progress in locking time decrease can be realized by varying of PLFS noise pass band. Thus, using "wide" noise pass band Ftfw = 0,5Hz on the first time stage and "narrow" Ftfn = 0,1 Hz on the second time stage, it is possible to attain benefit in synchronization time.
Digital PLFS statistical simulation results, namely: phase 9 and frequency F tracking errors average meanings (a, c), and RMS phase ct9 and frequency aF tracking errors (b, d) are presented in Fig. 6 and in Fig. 7. All curves are functional dependences on discrete time k.
Curves 1, 2, and 3 are signifying Doppler frequency shifts: 0; 0,02; 0,2 Hz respectively. Noise pass bands are described by discrete time step functions (10). Function F9 (k) describes noise pass band for Fig. 6, and F;(k) for Fig. 7.
a
a
10,5--
0-
-0,5-1-
c, rad
0,75
7,5
-^-►k/102 10
Fig. 6
5
2
b
F, Hz A
0,4
0,2 -
0 -0,1
a f , Hz 0,2
0,1-
0
d
2,5
7,5
7,5
10
k/102
k/102
10
Fig. 6 (сontinue)
5
c
5
10,5
0
0,5 j -1
2,5
7,5
10
k/ 102
1,5
0,75
12,5
25
37,5
50
^ k/102
Fig. 7
5
a
b
■ k/102
15 20
CTf , Hz
-1
0,2-1
0,01-
MjlVvy
0,0075-
0,005
h-1
19,5 19,75 20
d
5 10
15 20
■k/102
Fig. 7 (сontinue)
c
2
0
Fw = 0,5 Hz,0 < k < 300, Fm = 0,1 Hz, k > 300,
(10)
Fw = 0,5Hz,0 < k < 300, K(k) = 1 = 0,1 Hz, 300 < k < 1000,
F = 0,02Hz,k > 1000.
It becomes clear from Fig. 6, 7 that using multistage phase-lock-loop frequency control algorithms for MSK-signals receivers, average locking time can be significantly decreased (in comparison with autonomous algorithm F9 = 0,1 Hz) to t ~ 1000 • Tp = 40 s, with phase tracking RMS error desired value (ct9 = 0,05rad) in case of using function F'(k). Using function F9(k), it can be shown that phase tracking RMS error desired value is provided in time equal to 40 s. Also, using function FF(k~) it is possible to achieve a9 = 0,03 rad in 120 s and in steady-state regime a9 = 0,02 rad (k> 200 s).
Number of statistical examinations for Fig. 6 and Fig. 7 equals to 103. In all examinations there are no tracking losses. Described two- and three-stage phase-lock-loop frequency control algorithms with discrete time step functions (10) can be used in MSK-signal receivers of perspective frequency-limited RNS.
In present paper multistage phase-lock-loop frequency control algorithms of perspective RNS user's equipment are suggested. Statistical simulation was used to prove that a two-stage phase-lockloop frequency control algorithm, using function F'(k), has gain in synchronization time equal to 560 s (in comparison with autonomous algorithm) and provides steady-state RMS error values a9 < 3° and ctf < 0,03Hz. It was also stated that a three-stage phase-lock-loop frequency control algorithm has two benefits: first, gain in synchronization time is not less than 560 s; second, RMS error values in steady-state regime (k > 200 s) is ct9 < 1,1 ° and aF < 0,01 Hz - better than required.
This article contains specific results which can be used in digital phase synchronization systems of user's equipment for perspective RNS with spread-spectrum MSK-signals. The investigated algorithms of accelerated phase synchronization can be easily realized on the basis of field programmable gate array technology (FPGA).
The author of this paper would like to thank the following people: prof. V. N. Bondarenko, for scientific help and scientific training; associate prof. O. A. Almabekova, for language support in this article's preparation; prof. V. I. Kokorin, for his practical advice and scientific help; associate prof. V. A. Vyahirev, for his support, advice and help.
These investigations are financially supported with grants № 08-08-00849-a of Russian Foundation for Basic Research (RFBR) and № 18G041 of Krasnoyarsk Regional Scientific Foundation (in Russian).
1. Kuzmin E.V. Complex phase locked-loop system of spread-spectrum signal receiver / E.V. Kuzmin, V.N. Bondarenko // Modern problems of radio electronics, 2007. - P. 31 - 34. (in Russian).
Conclusions
Acknowledgements
References
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