Improved Performance Phase Detector for Multiplicative Second Order PLL Systems Using Deformed Algebra Marconi O. de Almeida, Eduardo T. F. Santos and Jos´e M. Ara´ ujo
‡
Instituto Federal de Educa¸c˜ ao, Ciˆencia e Tecnologia da Bahia Grupo de Pesquisa em Sinais e Sistemas Rua Em´ıdio dos Santos, S/N, Barbalho, 40301-015, Salvador-BA ‡Correspondign author e-mail:
[email protected]
Abstract Phase-locked loops (PLL) is a phase and/or frequency tracking system, widely used in communication and control systems. The sinusoidal multiplicative type PLL still remains a recurrent model, due the fact that its derivation is originated from the maximum likelihood approach. In this note, it is showed as a generalized product, called q-product, which can be used to implement the phase detector and improve some important parameters of the PLL system, as the block linearity and pull-in characteristics. Numerical examples are presented in order to illustrate the proposal. Keywords: PLL, Deformed algebra, Phase detector 1. Introduction PLL is a phase and/or frequency tracking systems used to track the phase of a sinusoidal signal, even under reasonable deviations between the phase of the input reference signal and VCO block. Several applications involving phase synchronization, as communications and frequency measurement systems can be carried out by PLL systems (see [1], [2], [3] and references therein). The most common type of PLL system is the multiplication-based one for the phase detector block. Its derivation is made in a stochastic manner by maximum likelihood technique [4],[5]. This PLL type is known to be almost linear in a small phase displacement sense, however, for larger phase displacements the linear approximation sin x ≈ x is no longer valid. Then, a nonlinear treatment is required in such situation. The deformed algebra Preprint submitted to Journal of Circuits, Systems, and Computers
August 21, 2013
Figure 1: PD maps for conventional and deformed product.
[6] gives flexibility on operations generalization, e.g. the q-product, which generalizes the ordinary multiplication and recovers it in the limit for q → 1. Such generalization is inspired in Tsalliss nonextensive statistical mechanics [7]. In this note, focused in second order PLLs, it is shown phase detector that π π characteristic can be made quasi -linear over the interval − 2 , 2 for phase offset by using a deformed product instead the conventional one at the phase detector stage. Also, simulations were performed in order to compare the modified PLL performance in relation to the conventional based on ordinary multiplication, where maximum rate for ramp reference frequency tracking, pull-in time, and robustness in noise presence are shown better when one uses the q- product phase detector. 2. Preliminaries 2.1. Multiplicative PLL System In so-called multiplicative PLL system, the phase detector block is implemented as a product between an input signal x(t) = A1 sin (ωi t + θi ) and a synthesized signal from the output of a VCO block y(t) = A2 sin (ωo t + θo ). By setting Θi ≡ ωi t + θi and Θo ≡ ωo t + θo , the resulting signal at the output of the PD block is given by: 1 z(t) = x(t) y(t) = A1 A2 {sin[(ωi + ωo )t + θi + θo ] + sin(Θi − Θo )} 2
(1)
The high frequency first term in Eq. (1) can be neglected due the action of the loop filter F (s). Thus, defining Θi − Θo = φ and considering the amplitudes normalized, the PD block can be simplified as: 2
Figure 2: PD maps for conventional and deformed product.
1 sin φ (2) 2 This approach is a classical implementation, since it derivation is made in a stochastic point view, by means of the maximum likelihood technique [4,5]. Figure 1 depicts the simplified representation of the sinusoidal PLL system. f (φ) =
2.2. Deformed or q-Product Inspired in Tsalliss nonextensive statistical mechanics, deformed product, also known as q-product, is given by [6],[8] : 1
x ⊗q y = sign(x)sign(y)[|x|1−q + |y|1−q − 1]+1−q
(3)
in which q ∈ R, sign(•) stands to the algebraic sign of the argument, and [A]+ ≡ max(A, 0). Such operation may then be considered a generalization of the conventional product where q is the generalization parameter. It can be verified that limq→1 x⊗q y= xy, that is, the conventional product is recovered. Such operation can then be considered a generalization of the conventional product where q is the generalization parameter. 3. Characteristic of the modified PLL By deforming the PD through the q-product, a numerical study is then performed for investigate the effect of q parameter in the map f (φ). Figure 2 a and b illustrates such map in the intervals − π2 < φ < π2 and 0 < φ < 2π. 3
Figure 3: Standard deviation of residuals for linear regression of f (φ) × φ
It can be noticed that the map is almost linear in the first case for q = 50, and an almost triangular function can be observed in the later. In the Figure 3, the residual standard deviation of a linear regression in the points of map f (φ) makes clear that for values of the q parameter greater than 30, no improvements were made in the sense of the map linearity. 4. Simulations results 4.1. Example #1 In this case, a numerical noise free example is performed in two different scenarios of the loop filter √ topology. By setting the PLL key parameters as: (2)
ωn = 2π × 3rad s−1 ; ζ = 2 , and defining a high gain for the VCO block 0.0741s+1 for the passive Ko = 2π ×1000s−1 , the loop filter is given by F (s) = 17.6893s+1 0.0741s+1 lead-lag case and F (s) = 17.6098s for PI design. For the lead-lag filter, an initial frequency off-set ∆ωn = 2π × 30rad s−1 is applied. In the Figure 4 one can see the better performance of the modified PD when compared with the conventional multiplicative one. The pull-in times are shown on table 4
Figure 4: The pull-in process for the conventional and deformed PD. Table 1: Performance comparison between conventional vs. deformed PD.
PD Pull-in time (s) Conventional 3.6 Defromed (q = 50) 2.9
Maximum reference rate (rad s−2 ) 344.3 422.8
1. In the PI loop filter simulation, a frequency ramp reference with rate ˙ n = 344.3rad s−2 is applied, and the conventional multiplier PLL fails in ∆ω locking this reference, while the PLL with deformed PD locks in, as can be seen on Figure 5. In Table 1, it is also shown the maximum rate for ramp references at the given frequency. 4.2. Example #2 This example is for illustrate the performance improvement of the qdetector under severe noise in the input phase. The PLL parameters where borrowed from [1], and are: Ko = 250s−1 ; F (s) = 0.0976s+1 . A ramp ref0.5945s −2 erence at frequency is applied with rate of 190rad s . It was found that 5
Figure 5: Performance comparison between conventional and deformed PD under frequency ramp reference.
Figure 6: Lock-in performance for PLL of Example #2 under severe noise at input.
the conventional multiplicative PD no longer locks the phase with a noise of SN R = 11dB at input. In the other hand, the deformed PD exhibits more robustness and destabilize only for a noise with SN R = 5.5dB, i.e., a 100% more severe level of noise. The Figure 6 shows the performance of both detectors when SN R = 11dB.
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Concluding Remarks A deformed product PD is proposed in this paper in substitution of conventional multiplier PD for multiplicative PLL systems. A numerical analysis of the representative input-output map is made in order to compare conventional and deformed PDs. Simulation results show that the deformed one gives superior performance in lock-in process. One potential use for this proposed PD is an all-digital PLL system implemented in a DSP or microcontroller application. Acknowledgments This work was supported by FAPESB and PRPGI-IFBA. References [1] R. Best, Phase Locked Loops 6/e : Design, Simulation, and Applications: Design, Simulation, and Applications, McGraw-Hill Professional Publishing, 2007. [2] G. Bianchi, Phase-Locked Loop Synthesizer Simulation, McGraw-Hill electronic engineering series, McGraw-Hill, 2005. [3] S. Kameche, M. Feham, M. Kameche, Optimizing pll performance levels, Microwaves and RF 51 (2012). [4] G. Eynard, C. Laot, Extended linear phase detector characteristic of a software pll, pp. 62–67. [5] B. Farhang-Boroujeny, Signal Processing Techniques for Software Radios, Lulu.com, 2011. [6] E. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A: Statistical Mechanics and its Applications 340 (2004) 95–101. [7] C. Tsallis, Possible generalization of boltzmann-gibbs statistics, Journal of Statistical Physics 52 (1988) 479–487. [8] R. Pessoa, E. Borges, Generalising the logistic map through the qproduct, Journal of Physics: Conference Series 285 (2011). 7
