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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.

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 [

A number of methods were proposed for chaos control [

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.

Conventional first order ZCDPLL is shown in

where

the sampling instants

where

The phase error

Then

The sampled values

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

where

Therefore first order ZCDPLL phase error operation function will be:

The phase error mapping function (

Various methods and techniques were used to control the instability of chaotic operation of control loop such as Ott-Grebogi-Yorke (OGY) or Pyragas [

Then Hillam [

p is fractional constant which control the amount of feedback. This algorithm can’t be used when

The system will be stable when

where

The stabilized first order ZCDPLL using FPI with relaxation is shown in

The first order filter transfer function

Then

To express Equation (16) in time domain:

Then the operation Equations (10) is given by [

If we assume

To guarantee the stable operation of the loop, then inequality should be satisfied [

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 [

Since

plied on

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

quency of DCO is

First order ZCDPLL is subjected to a frequency step of 1.3 Hz. Then

=

controlled first order ZCDPLL bifurcates

There are two filter parameters in the second order loop (

that the FPI chaos controlled loop has extended stable operation when the filter parameter (

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.

Nasir, Q. (2016) Fixed Point Iteration Chaos Controlled ZCDPLL. Int. J. Communications, Network and System Sciences, 9, 535-544. http://dx.doi.org/10.4236/ijcns.2016.911042