643

ISSN 1064–5624, Doklady Mathematics, 2008, Vol. 78, No. 1, pp. 643–645. © Pleiades Publishing, Ltd., 2008.Original Russian Text © G.A. Leonov, 2008, published in Doklady Akademii Nauk, 2008, Vol. 421, No. 5, pp. 611–613.

Based on special research techniques for high-fre-quency oscillations, a method is described for comput-ing the phase detector characteristics of phase lockedloops for clock synchronization.

Clock synchronization in television and radioloca-tion [1–6] and, then, in circuit engineering [7–15] hasbeen an important task over the last fifty years. Ade-quate mathematical models of clock synchronizationsystems are required for their effective analysis andsynthesis. For this purpose, methods for computingphase detector characteristics have to be developed.Such methods are discussed in this paper.

As a rule, phase locked loops (PLLs) in high-fre-quency harmonic generators have phase detectors withharmonic characteristics [1–6]. For pulse generators,the computation of phase detector characteristics is amore complicated task.

Consider a

2

π

-periodic differentiable function

g

(

x

)

that has exactly two extrema on

[0, 2

π

]

(

g

–

<

g

+

) andpossesses the following property: for any number

α ∈

(

g

–

,

g

+

)

, the equation

g

(

x

) = –

α

has exactly two roots

Define the function

if

g

(

x

) < –

α

on (

β

1

(

α

),

β

2

(

α

)

) and

if

g

(

x

) > –

α

on (

β

1

(

α

),

β

2

(

α

)

), and

a

<

b

,

ω

.

Assume that

ω

is much greater than

a

,

b

,

α

,

and

π

.

0 β1 α( ) β2 α( ) 2π.< < <

F α( ) 1β2 α( ) β1 α( )–

π----------------------------------–=

F α( ) 1β2 α( ) β1 α( )–

π----------------------------------–⎝ ⎠

⎛ ⎞–=

Lemma 1.

It is true that

(1)

Lemma 1 follows from the definition of

F

(

α

)

.Consider a product of high-frequency pulse oscilla-

tions passing through a linear filter (Fig. 1). Here,

⊗

isthe multiplier,

(2)

(3)

A

j

> 0

and

ψ

j

(

j

= 1, 2

) are constants,

γ

(

t

)

is the pulsetransition function of the linear filter, and

α

(

t

)

is anexponentially decaying function that depends linearlyon the initial state of the filter at the time

t

= 0.The high-frequency property of the generators can

be reformulated as follows. Consider a long fixed timeinterval [0,

T

] divided into short subintervals of theform [

τ

,

τ

+

δ

],

τ ∈

[0,

T

], where

(4)

(5)

(6)

Assume that

δ

is sufficiently small with respect to thefixed numbers

T

,

C

,

and

C

1

,

while

R

is sufficiently largewith respect to

δ

:

R

–1

=

O

(

δ

2

)

.

α g ωt( )+[ ] tdsgn

a

b

∫ F α( ) b a–( ) O1ω----⎝ ⎠

⎛ ⎞ .+=

f j t( ) A j ω j t( )t ψ j+( ),sinsgn=

g t( ) α t( ) γ t τ–( ) f 1 τ( ) f 2 τ( ) τ,d

0

t

∫+=

γ t( ) γ τ( )– Cδ, ω j t( ) ω j τ( )– Cδ,≤ ≤t∀ τ τ δ+,[ ], τ∀ 0 T,[ ];∈ ∈

ω1 τ( ) ω2 τ( )– C1, τ∀ 0 T,[ ];∈≤

ω j τ( ) R, τ∀ 0 T,[ ].∈≥

CONTROL THEORY

Computation of Phase Detector Characteristics in Phase Locked Loops for Clock Synchronization

Corresponding Member of the RAS G. A. LeonovReceived February 7, 2008

DOI: 10.1134/S1064562408040443

Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, St. Petersburg, 198504 Russiae-mail: leonov@math.spbu.ru

f1(t) g(t)

f2(t)

Filter

Fig. 1. Multiplier and the filter.

644

DOKLADY MATHEMATICS Vol. 78 No. 1 2008

LEONOV

The last assumption means that, on the short inter-vals [τ, τ + δ], γ(t) and ωj(t) are nearly constants, whilefj(t) are fast oscillating functions. Clearly, these condi-tions hold for high-frequency oscillations.

Consider the 2π-periodic function ϕ(θ) defined as

(7)

and the flow chart in Fig. 2, where θj(t) = ωj(t)t + ψj, PDis a nonlinear phase detector with the output ϕ(θ1(t) –θ2(t)), and

(8)

Theorem 1. If conditions (4)–(6) are satisfied, then,for the same initial states of the filter, it holds that

(9)

where D is a number independent of δ.

ϕ θ( )A1A2 1 2θ

π------+⎝ ⎠

⎛ ⎞ if θ π– 0,[ ]∈

A1A2 1 2θπ

------–⎝ ⎠⎛ ⎞ if θ 0 π,[ ]∈

⎩⎪⎪⎨⎪⎪⎧

=

G t( ) α t( ) γ t τ–( )ϕ θ1 τ( ) θ2 τ( )–( ) τ.d

0

t

∫+=

G t( ) g t( )– Dδ, t∀ 0 T,[ ],∈≤

Proof. Obviously,

Here, m is a number such that t ∈ [mδ, (m + 1)δ].Using Lemma 1, we obtain the estimate

which proves Theorem 1.Consider a flow chart of a typical phase locked loop

[1–6] (Fig. 3). Here, RG is a reference generator, TG isa tuned generator, and ⊗ is a signal multiplier.

Theorem 1 implies that, for pulse generators withoutput signals (2), this flow chart can be asymptotically(for high-frequency generators) replaced at the level offrequency and phase relations by the flow chart shownin Fig. 4 [15]. Here, PD is a phase detector with charac-teristic (7).

Thus, the asymptotic analysis of high-frequencypulse oscillations (Lemma 1 and Theorem 1) was usedto calculate the phase detector characteristic (7).

Now, we describe the scheme for calculating thephase detector characteristics for a pulse PLL with asquare-law generator [3, 6]. For this purpose, we againconsider the flow chart shown in Fig. 1, where

Next, we consider the flow chart in Fig. 2, where PD isa unit with the characteristic F(θ) = 2A1ϕ(θ). In thiscase, Theorem 1 remains valid and it is proved follow-ing the same scheme as in the previous case.

To conclude, we note that the transient time in PLLsfor present-day processors has to be no longer than 10 s,while the clock frequency reaches 10 GHz. Choosing

g t( ) α t( )– γ t s–( )A1A2 ω1 s( )((cos[sgn

0

t

∫=

– ω2 s( ) )s ψ1 ψ2–+ ) ω1 s( ) ω2 s( )+( )s(cos–

+ ψ1 ψ1+ ) ]ds

= A1A2 γ t kδ–( ) ω1 kδ( )((cos[sgn

kδ

k 1+( )δ

∫k 0=

m

∑– ω2 kδ( ) )kδ ψ1 ψ2 )– ω1 kδ( ) ω2 kδ( )+( )s(cos–+

∫ + ψ1 ψ2 ) ]ds O δ2( )+ + ,

t 0 T,[ ].∈

g t( ) α t( ) A1A2 γ t kδ–( )ϕ θ1 kδ( )(k 0=

m

∑⎝⎜⎛

+=

∫ – θ2 kδ( ) )δ⎠⎟⎞

O δ( )+ G t( ) O δ( ),+=

f 1 t( ) A12

1 ω1 t( )t ψ1+( )sinsgn+( )2,=

f 2 t( ) A2 ω2 t( )t ψ2+( ).sinsgn=

FilterPDθ1(t)

θ2(t)

G(t)

Filter

TG

RG

Filter

TG

RG PD

θ2(t)

θ1(t)

Fig. 2. Phase detector and the filter.

Fig. 3. PLL with a multiplier.

Fig. 4. PLL with a phase detector.

DOKLADY MATHEMATICS Vol. 78 No. 1 2008

COMPUTATION OF PHASE DETECTOR CHARACTERISTICS 645

δ = 10–4 (i.e., dividing each second into one thousandtime intervals), we obtain a practically acceptable con-dition for the asymptotic computation of phase detectorcharacteristics as described above:

REFERENCES1. A. J. Viterbi, Principles of Coherent Communications

(McGraw-Hill, New York, 1966; Sovetskoe Radio, Mos-cow, 1970).

2. S. C. Gupta, Proc. IEEE 63, 291–306 (1975).3. W. C. Lindsey, Synchronization Systems in Communica-

tion and Control (Prentice-Hall, Englewood Cliffs, N.J.,1972; Sovetskoe Radio, Moscow, 1978).

4. V. V. Shakhgil’dyan and A. A. Lyakhovkin, PhaseLocked Loops (Svyaz’, Moscow, 1972) [in Russian].

5. W. C. Lindsey and C. M. Chie, Proc. IEEE 69, 671–685(1981).

6. B. Sklar, Digital Communications: Fundamentals andApplications (Prentice Hall, Englewood Cliffs, N.J.,2000; Williams, Moscow, 2004).

7. E. P. Ugryumov, Digital Circuit Engineering (BHV,St. Petersburg, 2000) [in Russian].

8. G. A. Leonov and S. M. Seledzhi, Phase Locked Loopsin Array Processors (Nevskii Dialekt, St. Petersburg,2002) [in Russian].

9. P. Lapsley, J. Bier, A. Shoham, and E. A. Lee, DSP Pro-cessor Fundamentals: Architectures and Features(IEEE, New York, 1997).

10. S. W. Smith, The Scientist and Engineers Guide to Dig-ital Signal Processing (California Technical Publishing,San Diego, 1999).

11. A. Solonina, D. Ulakhovich, and L. Yakovlev, MotorolaDigital Signal Processors (BHV, St. Petersburg, 2000)[in Russian].

12. A. Solonina, D. Ulakhovich, and L. Yakovlev, Algo-rithms and Digital Signal Processors (BHV, St. Peters-burg, 2001) [in Russian].

13. G. A. Leonov and S. M. Seledzhi, Avtom. Telemekh.,No. 3, 11–19 (2005).

14. G. A. Leonov and S. M. Seledzhi, Int. J. Innovative Com-put. Inform. Control 1 (4), 1–11 (2005).

15. G. A. Leonov, Avtom. Telemekh., No. 10, 47–55 (2006).

ω 1–10

10–10

2– δ2( ) O δ2( ).= = =