Fixed Point Iteration Chaos Controlled ZCDPLL

The stable operation of first and second order Zero Crossing Digital Phase Locked Loop (ZCDPLL) is extended by using a Fixed Point Iteration (FPI) method with relaxation. The non-linear components of ZCDPLL such as sampler phase detector and Digital Controlled Oscillator (DCO) lead to unstable and chaotic operation when the filter gains are high. FPI will be used to stabilize the chaotic operation and consequently extend the lock range of the loop. The proposed stabilized loop can work in higher filter gains which are needed for faster signal acquisition.


Introduction
Digital Phase Locked Loop (PLL) has been widely used and for many years in wireless and wired communications subsystems.It is an essential component in clock and carrier recovery, and frequency synthesizer.Digital Phase locked Loops (DPLLs) have better reliability and higher stability compared to analogue counterpart at lower cost and can easily be part of a digital processing equipment [1].The researchers show strong interest in the design of digital PLLs (DPLLs) to solve problems associated with analog DPLLS, such as, sensitivity to DC drift and component inaccuracies and saturation, and their need for initial calibration [1] [2].The sampler type classifies DPLL into two major categories: uniform sampling DPLLs (US-DPLLs) and non-uniform sampling DPLLs (NUS-DPLLs).Different types of NUS-DPLLs have been introduced according to the way to detect the phase difference between locally generated carrier and the input signal to the loop from the sampled signal such as zero crossing DPLL (ZCDPLL) [3] [4] and digital tan-lock loop (DTLL) [5] [6] [7].ZCDPLL is a closed loop system used to follow the zero crossing of the input carrier signal.It consists of a sampler (acting as a phase detector), a digital loop filter and a digital controlled oscillator [8] [9] [10] [11].
The most commonly used DPLL is the Zero Crossing Digital Phase Locked Loop (ZCDPLL).The operation is based on tracking the signal input phase by using non uniform sampling techniques.The sample value is a function of the signal input phase.These values are filtered before they are used back to control the next sampling time by the help of Digital Controlled Oscillator (DCO).The non-linear behaviour of ZCDPLL leads bifurcation instabilities to its path to chaos [12].
A number of methods were proposed for chaos control [13] such as using Pyragas method to broaden the tracking range by extending the stable operation behaviour of ZCDPLL to a larger digital filter gain, which leads to larger input frequency [11].Fixed Point Iteration (FPI) with relaxation will be presented to extend the stable operation range of both first and second order ZCDPLL.The stabilized loops are analyzed and the results are verified using bifurcation theory and a numerical simulation.It is the first time that FPI used to stabilize the chaotic operation of the DPLL.
In Section 2, the conventional first order ZCDPLL operation is described.Section 3 discusses the Fixed Point stabilization algorithm, and in section 4 the second order ZCDPLL is presented, while Section 5 details the operation of the second order ZCDPLL when FPI chaos control is included in the loop.Simulation results are presented in Section 6 and finally conclusions are given in Section 7.

First Order ZCDPLL
Conventional first order ZCDPLL is shown in Figure 1.Let us assume that the input signal ( ) where ( ) n t is Additive White Gaussian Noise (AWGN), 0 φ is initial phase (can be assume zero without loosing generality), ω is the input signal frequency, and n ω is the nominal frequency or DCO free running frequency when no input signal is applied.
The input signal is assumed to be noise free [ ( ) ( ) the sampling instants k t can be represented by Q. Nasir k T is the Digital Controlled Oscillator (DCO) period, which is given by [15]: where ( ) y − is the output of digital filter.The input signal phase can be represented as k θ .The sampled signal input ( ) The phase error k φ is determined by: Then The sampled values k x is passed through a digital filter D(z) to produce the output k y .The digital loop filter can be of zero order (just gain block) or first order (gain and summation blocks).The loop filter output k y can be written as: where K is the zero order filter gain (First Order ZCDPLL), while for first order filter or second order ZCDPLL, the outputs will be: where 1 K and 2 K are the loop filter gains.If a frequency step of a value s w is ap- plied to the ZCDPLL ( s n w w w = − ), then the signal input phase can be expressed as: ( ) where ( ) is normalized frequency step size.Consequently the phase error can be written as ( ) Therefore first order ZCDPLL phase error operation function will be: ( ) The phase error mapping function ( ( ) ) will be: ( ) ( )

Extending the Stable Operation of First Order ZCDPLL
Various methods and techniques were used to control the instability of chaotic operation of control loop such as Ott-Grebogi-Yorke (OGY) or Pyragas [14].In this paper the Fixed Point Iteration (FPI) with relaxation is used to extend the stable operation of the ZCDPLL.FPI was used the first time by Babylonian (2000 B.C) to estimate the square root.The original version was used for finding a as: ( ) Then Hillam [16] proposed FPI with relaxation for fixed point stability as follow: ( ) p is fractional constant which control the amount of feedback.This algorithm can't be used when (8,9).Let us apply the above FPI with relaxation to stabilize ZCDPLL operation.Then the operation Equation (12) should become as: ( ) The system will be stable when 1 k φ′ < .This condition of the derivative of Equation (15) will be: ( ) The stable operation phase error ( * k φ ) was found to be at . The values of the constant (p) which can stabilize ZCDPLL operation is determined from Equation ( 16).The values are The stabilized first order ZCDPLL using FPI with relaxation is shown in Figure 2.

Second Order ZCDPLL Operation
The first order filter transfer function

( )
D z of the second order ZCDPLL can be written as: ( ) ( ) ( ) To express Equation ( 16) in time domain: ( ) Then the operation Equations ( 10) is given by [17] ( ) If we assume , then the second order ZCDPLL operation equation can be written as ( ) ( ) To guarantee the stable operation of the loop, then inequality should be satisfied [17]

Extending the Stable Operation of Second Order ZCDPLL
The proposed FPI with relaxation for fixed point stability is applied for second order as well and the new operation equation can be written as: The system state vector is defined as around the stable operating point ( ) In order to have eigen values of ( ) ( ) ( ) Using Jury stability test [18], the roots of the polynomial ( ) F λ defined in (25) are within a unit circle, or the eigen values are less than 1, if ( ) ( ) Then p should satisfies the following: ( ) ( ) this leads that 0.5 p < − .Jury stability test ap- plied on ( ) F λ the absolute value of the constant term of the equation should be less than 1.This leads to ( )

System Performance
The first and second order conventional and FPI chaos controlled ZCDPLL is simulated by using MATLAB.The input signal is assumed to be ( ) ( )

Conclusion
This paper proposes a Fixed Point Iteration (FPI) with relaxation to control the chaotic operation of the ZCDPLL.The analytic expressions for the stable operation for both conventional and FPI chaos control first and second order ZCPLL are found and confirmed by simulation.It is found that the lock range of the FPI chaos controlled loop is larger than that of the conventional loop for both orders.The validity of the results is conformed through numerical simulations.It is also found that careful selection of chaos control parameters is needed to ensure that the loop is still working in stable operation.This extended operation of the ZCDPLL leads to larger lock range.The larger values of filter gains of FPI chaos controlled will automatically decrease the input signal acquisition time.
simulation, the first 100 samples of the DCO period ( DCO T ) values are discarded to allow the loop to stabilize.The next 100,000 samples are collected and recorded to generate bifurcation plot.The bifurcation plot maps the DCO T values versus the filter gain(s).It will be used to compare the operation ranges of the conventional and FPI chaos controlled ZCDPLL.First order ZCDPLL is subjected to a frequency step of 1.3 Hz.Then DCO T values are recorded for three cases.The first case shown in Figure 3(a) is for conventional ZCDPLL.The conventional loop bifurcates at 1 0.32 K = .DCO period jitter is used to show the amount of deviation of the period compared to the input signal period (jitter Q. Nasir

Figure 4 .
Figure 4. Second order FPI-ZCDPLL performance for different values of p and 1.1 r = .

Figure 5 .
Figure 5. Second order FPI-ZCDPLL performance for different values of p and 1.6 r = .

Figure 6 .
Figure 6.Second order FPI-ZCDPLL performance for different values of p and 2.1 r = .