## The Design and Simulations of Phase and Timing

## Tracking Circuits with Pre-assured 2

### nd

## order Digital

## Loop Filter

Chien-Hsing Liao, Fu-Nian Ku, and Fu-Hao Yeh*Program of Information Technology *
*Fooyin University *

*Kaohsiung, Taiwan (ROC)*

Jia-Chin Lin and Mu-King Tsay Department of Communication Engineering,

National Central University, Taiwan (ROC) Taoyuan, Taiwan (ROC)

**Abstract—Phase and timing tracking circuits are always a crucial ****issue in circuit designs, especially in mobile communications and **
**many other applications which will produce phase shift or timing **
**jitter due to relative activity or environments variations. For **
**keeping track of these crucial changing parameters, a **
**synchronization circuit with 2nd order digital loop is generally **
**applied for its simplicity, flexibility, and fast setting time. In this **
**paper, a typical 2nd order digital loop filter with pre-assured **
**stable convergence triangle from Jury criteria is developed, **
**which can then be directly applied in many circuit designs with **
**phase or timing jitter by simply adjusting two digital loop **
**parameters. The convergence property and the effectiveness of **
**this digital loop filter and its applied circuits with derived **
**recursive formulations are well studied and simulated. This **
**simple and direct scheme can easily tell the designers the stable **
**convergence contours under specific circumstances in the whole **
**design stages. **

**Keywords- Phase tracking circuit;Timing tracking ****circuit;pre-assured 2****nd**** order digital loop filter;convergence triangle **

I. INTRODUCTION

The importance of synchronization circuits in modern circuit designs is indisputable and is always a crucial issue in circuit designs, especially in mobile communications and many applications which will produce phase shift or frequency drift due to relative activity, components aging, or environments variations. Therefore, a fast, stable, and flexible tracking loop design is always a necessity. For example, the Doppler effects will be generated while the systems are in relatively mobile conditions. And it will be much more severe, especially when the systems are operating in higher frequency bands, e.g., a few kHz drift will be normal for Ku-band satellite communications. In addition, the inherent frequency variations of crystal oscillator no less than 10 ppm in the whole operating frequency band and temperature range due to components aging and environments are generally happened, which will cause relatively large phase shift or frequency drift if a specific synchronization circuit is not applied. If sophisticated spread spectrum communications, e.g., fast frequency hopping, is considered, a synchronization circuit with even fast and stable convergence performance has to be made for achieving the specified hopping performance.

Conventionally, the most well known synchronization loop design is analog phase lock loop (PLL) with added voltage controlled oscillator (VCO) and loop filter, or with direct digital synthesizer (DDS) [1-5]. But, nowadays, full digital PLL design methods and approaches with digital loop filters are also applied in many related fields for its inherent benefits [4]. The use of higher-order digital loop filter should generally be restricted to filtering purpose, as these designs can be slower than lower-order PLLs and may exhibit a lower phase margin. And if a fast-settling PLL application is desired, e.g., a fast frequency hopping circuit, it is appropriate to select a lower-order digital loop for its simplicity and flexibility [8-10]. The theoretic analysis and design of a PLL circuit is complicated, therefore, many design approaches based on some specific models or simulation tools are taken to evaluate system parameters in a fast manner [14-15] [6-7]. In this paper, a simple and direct scheme based on a pre-assured stable 2nd order digital loop filter for designing a phase and timing tracking circuit is developed, which can tell the designer the convergence area by simply adjusting or selecting two loop parameters in the whole design stages.

The remainder of this paper is organized as follows. In Section II, the basics for a pre-assured stable 2nd order digital loop design will be addressed thoroughly, which include its function block diagram, recursive formulations, transfer function, and convergence characteristics and responses with two simply adjusted parameters. In Section III and IV, two typical application cases with pre-assured stable 2nd order digital loop design, i.e., QPSK phase tracking and Radio frequency tracking, respectively, will be presented. Conclusion is in final Section V.

II. PRE-ASSURED 2ND

ORDER DIGITAL LOOP FILTER

In this section, the function block diagram, recursive formulations, transfer function, and convergence triangle and response for a pure 2nd order digital loop filter design are addressed.

*A. * *Function Block Diagram *

**Fig. 1 shows a typical synchronization function block **
diagram, which is consisted of the synchronization circuit and
2nd* order digital loop filter with two adjustable C1 and C2*

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978-1-4244-8163-7/11/$26.00 ©2011 IEEE 978-1-4244-8164-4/11/$26.00 ©2011 IEEE

*parameters. The synchronization circuit is labeled as f(x)=1 for *
analyzing the pure 2nd order digital loop filter only. The initial
*input signal δxm value is δx0=x0* and the feedback signals

within this function diagram are updated periodically by
*system clocks. The C1 and C2* are the only two parameters

*which we can select to let δxm*converge to zero for a stable

response. In this diagram, two internal small feedback loops
*are built for generating Sm+1, xm+1, ym+1*, and many other

parameters for recursive formulations.

**Fig. 1 The typical synchronization function block diagram with two **
**adjustable C****1**** and C****2**** parameters from 2nd order digital loop filter**

*B. Recursive Formulations *

The basic recursive formulations are derived from (1a) to
(1d), respectively, based on the two internal feedback loops
**from Fig. 1. **
1 2
*m* *m* *m*
*S* + =*C* ⋅δ*x* +*S* **, ** **(1a) **
1 1 1
*m* *m* *m*
*x* + =*C* ⋅δ*x* +*S* +, (1b)
1 1
*m* *m* *m*
*y* + =*y* +*x* + , (1c)
1 0 1 0
*m* *m* *m* *m*
*x* *x* *y* *x* *x* *y*
δ + =δ − + ⇒δ =δ − , (1d)

*where δxm* is the input difference quantity. Moreover, the

*transfer function H(z) of this 2*nd order digital loop filter with
*f(x)*=1 is derived and shown in (2) by wrapping up and taking
z-transform of (1a) to (1d).

### ( )

### (

### )

### (

### )

### (

### )

2 2 0 1 2 1 1 2 1*m*

*z*

*x*

*H z*

*x*

*z*

*C*

*C*

*z*

*C*δ δ − = = + + − ⋅ + − (2)

*C. Convergence Triangle*

The pure 2nd* order digital loop filter will be stable if C1 and *

*C2parameters in the denominator of the H(z) transfer function *

could satisfy the constrained stable conditions from Jury criteria [11-12], which is represented as

1 2 1 2

0<*C* <2 *C* >0 2*C* +*C* >4, (3)

*where these three different curves formed by C1 and C2*

**combinations will generate a convergence triangle. Fig. 2 **
shows this theoretic syable convergence triangle based on (3).

Within this constrained region, the arbitrary combinations
*of C1 and C2* parameters will generate stable system responses

only with the differences of clock cycles required to achieve
**stable conditions. Fig. 3 shows the simulated convergence **
clock cycle contours with only 0.1% oscillating amplitude
variations, i.e., the final amplitude variations to reach stable
*criteria is within 0.1%. If the combinations of C1 and C2* are in

the inner contours, the clock time to reach stable criteria is shorter, i.e., it will reach stable condition more quickly. On the

contrary, in the outer contours, the clock time to reach stable criteria is longer.

*We can choose any combinations of C1 and C2* within this

region to have a specific stable response. For example, within
the inner constrained contour (grey color), the convergence
clock cycles no more than ten (10) are required to reach the
stable criteria of only 0.1% amplitude variations. In the same
manner, for the outer pink contour the convergence clock
cycles required will be increased to no more than twenty (20)
*whenever any combinations of C1 and C2* values within this

contour are selected. Finally, near the edges of this convergence triangle, the convergent clock cycles will be much more than one hundred (100) cycles, which means the convergence speed to reach the stable criteria of 0.1% amplitude variations will be even slower and more sensitive.

Convergence Triangle

*C _{1}*·100

*C*

*2*_{·1}

00

Convergence Index Contour

C1·100

(a) (b)

**Fig. 2 Theoretic stable convergence triangle (C****1**** is 0~2 and C****2**** is 0~4)(a); **

**clock cycle contour within 0.1% amplitude variations for stable criteria(b)**

*D. Convergence Response *

*The δxm***convergence responses curves are shown in Fig. **

**3, where the x-axis represents the number of updating clock ***cycles (m) and the y-axis represents the δxm*amplitudes. The

first combination takes about ten cycles to be stable, and the
second one take at least 50 cycles to be settled down. But
*when C1 and C2* are selected near the edges of convergence

* triangle as shown in Fig. 2 e.g., C1= C2*=0.19 combination,

obviously it will need more clock cycles to converge or even
become unstable if they are outside of this region. Therefore,
*the convergence speed simply depends on the choice of C1* and

*C2. If C1 and C2 *are in the inner part of the convergence

triangle, it will converge more quickly; on the contrary, it will converge more slowly.

0 5 10 15 20 25 30 35 40 45 50 2 − 1 − 0 1 2 C1= 0.5 C2=0.1 C1= 1.2 C2=1.5 C1= 1.9 C2=0.19 δ(x) Convergence Curves (degree)

m

**Fig. 3 Convergence response curves of 2nd**_{ order digital loop with three }**adjustable C****1**** and C****2**** combinations**

III. QPSK PHASE TRACKING

In this section, the function block diagram, recursive formulations, transfer function, and convergence response for a QPSK phase tracking circuit design are addressed.

*A. Function Block Diagram *

**Fig. 4 shows the function block diagram of a QPSK phase **
tracking circuit consisted of phase rotator, phase discriminator,
and the pre-assured 2nd order digital loop filter. This is a
typical phase tracking system, where the initial phase shift (θ*0*)

and residue angular frequency (ω*r*) from analog I and Q

channel signals are mixed into this tracking system. The aim is,
therefore, to correct the shifted signal phases by this tracking
system. When this tracking loop works, the phase differences
*δθ will cause the phase rotator to rotate, and the two phase *
*rotated signals J and P dependent on δθ values in each clock *
cycle will be input to the phase discriminator for a
discriminated phase output. Thereafter, the two adjustable
*parameters C1 and C2 *inside the pre-assured 2nd order digital

loop filter are taken for a stable output performance according to the specified convergence triangle as aforementioned. The phase rotator can be implemented with SRAM chips or a FPGA chip in a table look-up style.

**Fig. 4 Function block diagram of QPSK phase tracking circuit rotator **

*B. Recursive Formulations *

**The basic recursive formulations based on Fig. 1 and Fig. 4 **
are derived from (3a) to (3d), respectively.

### (

### )

1 2*m*

*p*

*m*

*m*

*S*

_{+}=

*C*⋅

*k f*δθ +

*S*

**,**(3a)

### (

### )

1 1 1*m*

*p*

*m*

*m*

*x*

_{+}=

*C k f*⋅ δθ +

*S*

_{+}

**,**(3b) 1 1

*m*

*m*

*m*

*y*

_{+}=

*y*+

*x*

_{+}

**,**(3c)

### (

0### )

*m*

*rmT*

*ym*δθ = θ +δω − , (3d)

*where kp* is the approximated slope for linear phase

discriminator output response; δθ*m* is the mth phase tracking

*error; f(δθm)* is phase discriminated output represented and

approximated as (4) and (5), respectively [13].

### (

_{m}### )

### (

_{m}### )

sgn### (

### (

_{m}### )

### )

### (

_{m}### )

sgn### (

### (

_{m}### )

### )

*f*δθ =

*J*δθ

*P*δθ −

*P*δθ

*J*δθ (4)

### (

### )

mod , 2 4 2 2*m*

*m*

*m*

*f*π δθ π δθ δθ π π

_{+} = − (5)

**Fig. 5 shows the real phase discriminator characteristic **
curve (red solid) and approximated curve (blue dashed) with
*kp* slope (≒8/1.414π). This is basically a saw-tooth waveform

with π/2 phase cycle. For no loss of generality, if |I |=|Q|=1 is assumed, it will be available for each possible case from (5) when QPSK constellations are considered.

*2*
− −*1.5* −*1* −*0.5* *0* *0.5* *1* *1.5* *2*
*1.5*
−
*1*
−
*0.5*
−
*0*
*0.5*
*1*
*1.5*

Phase Detector Characteristic Curves

θ/ 0.5π

**Fig. 5 Phase discriminator characteristic curves (red solid: real; **
**blue dashed: approximated) **

*C. Convergence Response *

**Fig. 6 shows three different δθ***m*convergence response

*curves by selecting three C1 and C2* combinations for QPSK

phase tracking loop with residue frequency equal to -0.01 clock rate and initial phase of 135 degrees. The first combination takes about twenty cycles to be stable, and the second one take less than 10 cycles to be settled down. The third combination takes about at least 100 cycles to be within the specified 0.1% amplitude variations. The only convergence limitation is to keep the swing within a stable region, and it is always possible.

*0* *10* *20* *30* *40* *50* *60* *70* *80* *90* *100*
*120*
*140*
*160*
*180*
*200*
*220*
*240*
C1= 0.5 C2=0.1
C1= 1.2 C2=0.85
C1= 1.9 C2=0.15

δ(θ) Convergence Curves (degree)

m

**Fig. 6 Convergence response curves for QPSK phase tracking circuit **
**with three adjustable C****1**** and C****2**** combinations **

IV. TIMING TRACKING WITH EARLY-LATE DETECTOR

*A. Function Block Diagram *

**Fig. 7 shows the function block diagram of an early-late **
timing tracking loop consisted of direct digital synthesizer
(DDS), A/D, early-late timing detector, and the proposed 2nd
order digital loop. This is a typical timing tracking system,
where the phase shift and residue angular frequency from I
*and Q channels (J and P) are mixed into this tracking system. *
The early-late timing detector is just to adjust and extract the
timing position within one chip. ω*s *is angular sampling

*frequency. J and P are two phase rotated signals dependent on *
δω* _{s}T* values in each clock cycle. To extract and align the
sampling position to be in the middle of each chip for the
filtered (raised-cosine filter) and jittered spread spectrum
signals with 210－1 chip length, at least three sampling clocks
per chip are taken in this application case for effective Nyquist
sampling. Moreover, averaging sampled and random data
should be taken for more smooth results.

**Fig. 7 Function block diagram of early-late timing tracking circuit**

*B. Recursive Formulations *

**The basic recursive formulations based on Fig. 7 are **
derived from (6a) to (6d), respectively. There exist nonlinear
*relation between the (m+1)th sampling period sm+1 * and

*sampling period Ts* and can be approximated as shown in (6c)

*if xm+1Ts* <<1**is assumed. **

### (

### )

1 2*m*

*t*

*m*

*m*

*S*

_{+}=

*C*⋅

*k g*δ

*T*+

*S*

**,**(6a)

### (

### )

1 1 1*m*

*t*

*m*

*m*

*x*+ =

*C k g*⋅ δ

*T*+

*S*+

**,**(6b)

### (

### )

1 1 1 1 1 1*m*

*s*

*m*

*s*

*m*

*s*

*s*

*T*

*x*

*T*

*x*

*T*+ + + = + ⋅ −

**,**(6c) 1 3

*m*

*m*

*m*

*T*

_{+}=

*T*+

*M s*⋅

**,**(6d)

*where kt* is the loop coefficients dependent on the averaged

*sampled number M and sampling period Ts, and g(δTm+1)* is the

timing tracking circuit output function with the larger sampled
*J or P channel signal value. *

*C. Convergence Response *

* Fig. 9 shows the δtm* convergence response curves for the

*early-late timing tracking loop with M=300 and initial time *
*position set as 1.35Ts*. The more sampled data averaged, i.e.,

*M* is larger, the smoother the convergence of timing position.
*The y-axis is the relative position to sampling period Ts; the *
x-axis is number of averaged sampled group. It is always
*possible to select C1 and C2* combinations for a stable timing

tracking position. 0 10 20 30 40 50 60 70 80 90 100 1 1.5 2 2.5 3 C1= 1.5 C2=0.1 C1= 1.0 C2=0.2 C1= 0.5 C2=0.3 δ(t) Convergence Curves m

**Fig. 9 Convergence response curve of early-late timing tracking circuit **
**with three adjustable C****1**** and C****2**** combinations (M=300; T****0**** =1.35T****s****)**

V. CONCLUSION

In this paper, a pre-assured 2nd order digital loop filter with easily adjustable parameters for predicted stable convergence characteristics and responses is proposed. Moreover, the general recursive formulations and derived convergence triangle for the filter itself and its applications in phase and timing variations are also presented. This simple and direct scheme can tell the designers the convergence regions by simply adjusting or selecting these parameter combinations in the whole design stages.

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