ISSN 1064�5624, Doklady Mathematics, 2011, Vol. 84, No. 1, pp. 586–590. © Pleiades Publishing, Ltd., 2011.Original Russian Text © G.A. Leonov, N.V. Kuznetsov, M.V. Yuldashev, R.V. Yuldashev, 2011, published in Doklady Akademii Nauk, 2011, Vol. 439, No. 4, pp. 459–463.

586

Phase locked loops (PLLs) are widely used in radioengineering [10, 15] and computer architectures [1, 2,4–9, 11–13]. Nowadays various software and hard�ware implementations of PPLs are used. An advantageof software implementations is that they are relativelyeasy to create. However, this limits the maximum pos�sible speed of operation, which is due to the internalimplementation of the software code [1, 3]. A short�coming of hardware implementations is that theyrequire a complex nonlinear analysis of PPL models[1, 11]. Below, we address one aspect of this analysis.

To construct an adequate nonlinear mathematicalmodel of PPLs, we have to determine [1] the charac�teristic of a phase detector (PD), i.e., a nonlinear ele�ment whose input is fed with signals from a referenceoscillator and a voltage controlled oscillator (VCO)and whose output contains a low�frequency correctingsignal.

We consider a standard phase detector in the formof a signal multiplier [10, 15]. The approachesdescribed in [7, 12] are used to develop a method forcomputing the phase detector characteristics for vari�ous classes of signals.

Consider the transmission of the product of high�frequency oscillations through a linear filter (Fig. 1).Here, ⊗ is the multiplier, f 1(θ1(t)) and f 2(θ2(t)) arehigh�frequency oscillations (signals produced by thereference and VCOs, respectively) with their productfed as input into the linear filter (low�frequency filter),and g(t) is the output of the filter.

Assume that f 1(θ) and f 2(θ) are bounded 2π�peri�odic piecewise differentiable functions (i.e., functionswith a finite number of jumps that are differentiable ontheir continuity intervals). Then, according to the

Lipschitz criterion [14], the Fourier series corre�sponding to f 1(θ) and f 2(θ) converge to function val�ues at continuity points and to the half�sum of the leftand right limits at discontinuity points. Recall thatfunctions different at a finite number of points are

equivalent in . Therefore, f 1(θ) and f 2(θ) areconsidered with the values at discontinuity pointsindicated by the Lipschitz criterion; i.e.,

(1)

L π– π,[ ]

1

f 1 θ( ) c1 ai1 iθ( )sin bi

1 iθ( )cos+( ),

i 1=

∞

∑+=

f 2 θ( ) c2 ai2 iθ( )sin bi

2 iθ( )cos+( ),

i 1=

∞

∑+=

aip 1

π�� f p x( ) ix( )sin x,d

π–

π

∫=

bip 1

π�� f p x( ) ix( )cos x,d

π–

π

∫=

cp 1π�� f p x( ) x, pd

π–

π

∫ 1 2,{ }, i �.∈ ∈=

Computation of Phase Detector Characteristicsin Synchronization Systems

Corresponding Member of the RAS G. A. Leonova, N. V. Kuznetsova, b, M. V. Yuldasheva, b, and R. V. Yuldasheva, b

Received February 10, 2011

DOI: 10.1134/S1064562411040223

a Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504 Russiae�mail: leonov@math.spbu.ru, nkuznetsov239@gmail.com, renatyv@gmail.com, maratyv@gmail.comb University of Jyväskylä, P.O. Box 35, FI�40014, Mattilanniemi 2 (Agora), Finland

CONTROL THEORY

f1(θ1(t)) g(t)

f 2(θ2(t))

f1(θ1(t))f 2(θ2(t))RO Filter

VCO

Fig. 1. Multiplier and the filter.

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

COMPUTATION OF PHASE DETECTOR CHARACTERISTICS 587

The properties of Fourier coefficients for piecewisedifferentiable functions [14] imply the estimates

(2)

The input ξ(t) and the output σ(t) of the linear filterare related by the formula

(3)

where α0(t) is an exponentially decaying functiondepending linearly on the initial state of the filter at t = 0and γ(t) is the impulsive transition function of the lin�ear filter (α0(t) and γ(t) are hereafter assumed to be dif�ferentiable functions with bounded derivatives). Then,according to (3), the function g(t) has the form

(4)

Let θ1(t) and θ2(t) be given by

where ω1(t) and ω2(t) are positive differentiable func�tions, while ψ1 and ψ2 are constants. Based on theassumptions about ω1(t) and ω2(t) made above, ωp(t)can be treated as the frequencies; θp(t), as the phases;and ψp, as the initial phase shifts of the reference oscil�lator and VCO at times corresponding to transientprocesses.

Let us formulate the high�frequency properties ofthe oscillators. On a fixed time interval [0, T], whichcan be divided into small subintervals of the form[τ, τ + δ], we have

(5)

where C is a constant independent of δ or τ. Withoutloss of generality, we assume that the boundedness ofthe derivative of γ(t) implies the similar relation

(6)

Suppose that there exists a number R such that

(7)

Assume that the frequency difference is uniformlybounded:

(8)

Here, C1 is independent of δ.It follows from (6) and (7) that, at short time inter�

vals, ωp(t) is “almost a constant,” and its value is suffi�ciently large.

Now consider the 2π�periodic function

aip O 1

i��⎝ ⎠

⎛ ⎞ , bip O 1

i��⎝ ⎠

⎛ ⎞ , p 1 2,{ }.∈= =

σ t( ) α0 t( ) γ t τ–( )ξ τ( ) τ,d

0

t

∫+=

g t( ) α0 t( ) γ t τ–( )f 1 θ1 τ( )( )f 2 θ2 τ( )( ) τ.d

0

t

∫+=

θ1 t( ) ω1 t( )t ψ1, θ2 t( )+ ω2 t( )t ψ2

,+= =

ωp τ( ) ωp t( )– Cδ, p 1 2,{ }∈≤

t∀ τ τ δ+,[ ], τ∀ 0 T δ–,[ ],∈ ∈

γ τ( ) γ t( )– Cδ t∀ τ τ δ+,[ ],∈≤

τ∀ 0 T δ–,[ ].∈

ωp τ( ) R 0, p 1 2,{ }, τ∀ 0 T,[ ],∈ ∈>≥

R R δ( ) O 1

δ2����⎝ ⎠

⎛ ⎞ , R CT.>= =

ω1 τ( ) ω2 τ( )– C1 τ∀ 0 T,[ ].∈≤

(9)

It follows from (2) that this series uniformly converges,while ϕ(θ) is continuous and bounded on �.

Consider the block diagram shown in Fig. 2. Here,PD is a nonlinear phase detector with the outputϕ(θ1(t) – θ2(t)) and G(t) is the output of the filter,which, according to (3), is given by

(10)

Theorem 1. If conditions (5)–(8) hold, then, in thesame initial state of the filter, we have

(11)

Proof. Let t ∈ [0, T]. Consider the difference

(12)

Let K ∈ � be the smallest positive integer such that

[0, T] ⊂ kδ, (k + 1)δ]. Let m ∈ � be such that

t ∈ [mδ, (m + 1)δ]. Without loss of generality, assume

that (m + 1)δ ≤ T. Clearly, m = m(δ) = O . In what

follows, let k ≤ m. The continuity condition impliesthat γ(t) is bounded on [0, T]. Moreover, f 1(θ), f 2(θ),and ϕ(θ) are bounded on �. Then

(13)

It follows that (12) can be represented as

ϕ θ( ) c1c2=

+ 12�� al

1al2 bl

1bl2+( ) lθ( )cos al

1bl2 bl

1al2–( ) lθ( )sin+( ).

l 1=

∞

∑

G t( ) α0 t( ) γ t τ–( )ϕ θ1 τ( ) θ2 τ( )–( ).

0

t

∫+=

G t( ) g t( )– C2δ t∀ 0 T,[ ].∈≤

g t( ) G t( )–

= γ t s–( ) f 1 θ1 s( )( )f 2 θ2 s( )( ) ϕ θ1 s( ) θ2 s( )–( )–[ ] s.d

0

t

∫

[k 0=

K

∪

1δ��⎝ ⎠

⎛ ⎞

γ t s–( )f1 θ1 s( )( )f2 θ2 s( )( ) sd

t

m 1+( )δ

∫ O δ( ),=

γ t s–( )ϕ θ1 s( ) θ2 s( )–( ) sd

t

m 1+( )δ

∫ O δ( ).=

θ1(t) G(t)

θ2(t)

ϕ(θ1(t)) − θ2(t))RO

VCO

FilterPD

Fig. 2. Phase detector and the filter.

588

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

LEONOV et al.

(14)Conditions (6) imply that, on each of the intervals

[kδ, (k + 1)δ], we have

(15)

which holds uniformly in t and O(δ) is independentof k. Then, using (14), (15), and the boundedness off 1(θ), f 2(θ), and ϕ(θ), we obtain

(16)

Define

(17)

Since ϕ(θ) is continuous and bounded on �, it is truethat

(18)

By assumption, f 1(θ) and f 2(θ) are bounded on �.If f 1(θ) and f 2(θ) are additionally continuous on �,then an estimate similar to (18) also holds for f 1(θ1(s))and f 2(θ2(s)).

Let us derive an estimate similar to (18) in the casewhen f 1(θ) and f 2(θ) have discontinuity points.

Since conditions (7) and (5) hold and a ωp(s) is acontinuous function, we can introduce a set Wk (theunion of sufficiently small neighborhoods of the dis�

continuity points of f p(θp(s)) and f p( (s)), p ∈ {1, 2},s ∈ [kδ, (k + 1)δ]) such that

(19)

Combining this with the fact that f p(θ) is piecewisecontinuous and bounded, we obtain

(20)

Then (16) can be rewritten as

g t( ) G t( )– γ t s–( )

kδ

k 1+( )δ

∫k 0=

m

∑=

× f 1 θ1 s( )( )f 2 θ2 s( )( ) ϕ θ1 s( ) θ2 s( )–( )–[ ]ds O δ( ).+

γ t s–( ) γ t kδ–( ) O δ( ), t s,>+=

s kδ k 1+( )δ,[ ],∈

g t( ) G t( )– γ t kδ–( )k 0=

m

∑=

× f 1 θ1 s( )( ) f 2 θ2 s( )( ) ϕ θ1 s( ) θ2 s( )–( )–[ ] sd

kδ

k 1+( )δ

∫ O δ( ).+

θkp s( ) ωp kδ( )s ψp

, p 1 2,{ }.∈+=

ϕ θ1 s( ) θ2 s( )–( ) ϕ θk1 s( ) θk

2 s( )–( )– sd

kδ

k 1+( )δ

∫

= O δ2( ).

θkp

ds

Wk

∫ O δ2( ).=

f 1 θ1 s( )( )f 2 θ2 s( )( ) sd

kδ k 1+( )δ,[ ]

∫

= f 1 θk1 s( )( )f 2 θk

2 s( )( ) sd

kδ k 1+( )δ,[ ]\Wk

∫ O δ2( ).+

(21)

By the Jordan test for the uniform convergence ofFourier series [14], on each of the intervals free of dis�continuity points, the Fourier series of f 1(θ) and f 2(θ)converge uniformly. Then there exists a number M =M(δ) > 0 such that the remainders of the series of f 1(θ)and f 2(θ) do not exceed δ outside the neighborhoodsof the discontinuity points. A similar assertion holds forϕ(θ). Then the boundedness of f 1(θ) and f 2(θ) implies

(22)

Using the formulas for the product of sines and cosinesyields

(23)

g t( ) G t( )– γ t kδ–( )k 0=

m

∑=

× f 1 θk1 s( )( )f 2 θk

2 s( )( ) ϕ θk1 s( ) θk

2 s( )–( )–[ ] sd

kδ k 1+( )δ,[ ]\Wk

∫

+ O δ( ) γ t kδ–( )k 0=

m

∑=

× c1 ai1 iθk

1 s( )( )sin

i 1=

∞

∑ bi1 iθk

1 s( )( )cos+ +⎝ ⎠⎜ ⎟⎛ ⎞

kδ k 1+( )δ,[ ]\Wk

∫

× c2 aj2 jθk

2 s( )( )sin

j 1=

∞

∑ bj2 jθk

2 s( )( )cos+ +⎝ ⎠⎜ ⎟⎛ ⎞

∫ – ϕ θk1 s( ) θk

2 s( )–( ) ds O δ( ).+

g t( ) G t( )– γ t kδ–( )k 0=

m

∑=

× c1c2 c1

M���� aj

2 jθk2 s( )( )sin(

⎩⎨⎧

j 1=

M

∑i 1=

M

∑+

kδ k 1+( )δ,[ ]\Wε k,

∫

+ bj2 jθk

2 s( )( ))cos c2

M���� ai

1 iθk1 s( )( )sin bi

1 iθk1 s( )( )cos+( )+

+ ai1 iθk

1 s( )( )sin bi1 iθk

1 s( )( )cos+( )

��× aj2 jθk

2 s( )( )sin bj2 jθk

2 s( )( )cos+( )⎭⎬⎫

∫ – ϕ θk1 s( ) θk

2 s( )–( ) ds O δ( ).+

ai1 iθ1( )sin bi

1 iθ1( )cos+( ) aj2 jθ2( )sin bj

2 jθ2( )cos+( )

= 12�� ai

1aj2 bi

1bj2+( ) iθ1 jθ2–( )cos(

+ ai1bj

2 bi1aj

2–( ) iθ1 jθ2–( ) ) 12�� bi

1bj2 ai

1aj2–( )(+sin

× iθ1 jθ2+( )cos ai1bj

2 bi1aj

2+( ) iθ1 jθ2+( ) ).sin+

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

COMPUTATION OF PHASE DETECTOR CHARACTERISTICS 589

Define

(24)

Then

(25)

It follows from (7) that

(26)

Taking into account (26), (7), and (2), we obtain theestimate

(27)

Now we consider the term containing cos(i (s) +

j (s)) in μi, j(s). According to (7), iω1(kδ) + jω2(kδ) ≥(i + j)R. Then it follows from (26) that

(28)

Conditions (2) imply

(29)

μi j, s( ) 12�� ai

1aj2 bi

1bj2+( ) iθk

1 s( ) jθk2 s( )–( )cos(=

+ ai1bj

2 bi1aj

2–( ) iθk1 s( ) jθk

2 s( )–( ) )sin

+ 12�� bi

1bj2 ai

1aj2–( ) iθk

1 s( ) jθk2 s( )+( )cos(

+ ai1bj

2 bi1aj

2+( ) iθk1 s( ) jθk

2 s( )+( )sin ).

g t( ) G t( )– γ t kδ–( )k 0=

m

∑=

× c1c2 c1

M���� aj

2 jθk2 s( )( )sin(

⎩⎨⎧

j 1=

M

∑i 1=

M

∑+

kδ k 1+( )δ,[ ]\Wk

∫

+ bj2 jθk

2 s( )( ))cos c2

M���� ai

1 iθk1 s( )( )sin bi

1 iθk1 s( )( )cos+( )+

+ μi j, s( )⎭⎬⎫

ϕ θk1 s( ) θk

2 s( )–( )– ds O δ( ).+

Rs ψ+( )cos sd

kδ

k 1+( )δ

∫ O δ2( ).=

1M���� aj

p jθkp s( )( )sin

kδ k 1+( )δ,[ ]

∫j 1=

M

∑i 1=

M

∑ bjp jθk

p s( )( )dscos+

= O δ2( ) O 1

j2��⎝ ⎠

⎛ ⎞

j 1=

M

∑ O δ2( ), p 1 2,{ }.∈=

θk1

θk2

i ω1 kδ( )s ψ1+( ) j ω2 kδ( )s ψ2+( )+( )cos sd

kδ k 1+( )δ,[ ]\Wk

∫

= O δ2( )O 1i j+�������⎝ ⎠

⎛ ⎞ .

bi1bj

2 ai1aj

2–2

���������������������

kδ k 1+( )δ,[ ]\Wk

∫j 1=

M

∑i 1=

M

∑

× i ω1 kδ( )s ψ1+( ) j ω2 kδ( )s ψ2+( )+( )dscos

= O δ2( ) O 1ij i j+( )��������������⎝ ⎠

⎛ ⎞ .

j 1=

M

∑i 1=

M

∑

Since the series converges, we see that

expression (29) is O(δ2). An equality similar to (29)

holds for the term sin(i (s) + j (s)).

Thus, we find from (25) that

(30)

Note that the terms with indices i = j in (30) sum to

ϕ( (s) – (s)) up to O(δ).

Consider the terms with i < j containing cos (simi�lar relations hold for terms involving sin with indicesi > j). In an analogous manner to (28), we have

(31)

The convergence of the series

(32)

implies that (31) is O(δ2).

Taking into account the last argument yields

(33)

as required.

Let us give examples of computing the characteris�tic of a phase detector (multiplier) with the use of for�mula (9) for basic types of signals, namely, sinusoidaland impulsive.

Corollary 1.

(34)

1ij i j+( )��������������

j 1=

∞

∑i 1=

∞

∑

θk1 θk

2

g t( ) G t( )– γ t kδ–( )k 0=

m

∑=

×ai

1aj2 bi

1bj2+

2���������������������� iθk

1 s( ) jθk2 s( )–( )cos

⎩⎨⎧

j 1=

M

∑i 1=

M

∑kδ k 1+( )δ,[ ]\Wk

∫

+ai

1bj2 bi

1aj2–

2��������������������� iθk

1 s( ) jθk2 s( )–( )sin

⎭⎬⎫

– ϕ θk1 s( ) θk

2 s( )–( ) ds O δ( ).+

θk1 θk

2

ai1aj

2 bi1bj

2+2

���������������������� i ω1 kδ( )s ψ1+( )(cos[

kδ k 1+( )δ,[ ]\Wk

∫j 1=

i 1–

∑i 2=

M

∑

– j ω2 kδ( )s ψ2+( ))]ds O δ2( ) O 1ij i j–( )��������������⎝ ⎠

⎛ ⎞ .

j 1=

i 1–

∑i 2=

M

∑=

1ij i j–( )��������������

j 1=

i 1–

∑i 2=

∞

∑

g t( ) G t( )– O δ( ),=

f 1 t( ) A1 θ1 t( )( ), f 2 t( )sin A2 θ2 t( )( ),sin= =

ϕ θ1 θ2–( ) A1A2

2��������� θ1 θ2–( ).cos=

590

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

LEONOV et al.

Corollary 2.

(35)

It is well known [14] that the resulting characteris�tic ϕ(·) coincides with the function plotted in Fig. 3.

Corollary 3.

(36)

f 1 t( ) A1 θ1 t( )( )sinsgn=

= 4A1

����� 1

2n 1+������������ 2n 1+( ) ω1 t( )t ψ1+( )( ),sin

n 0=

∞

∑

f 2 t( ) A2 θ2 t( )( )sinsgn=

= 4A2

����� 1

2n 1+������������ 2n 1+( ) ω2 t( )t ψ2+( )( ),sin

n 0=

∞

∑

ϕ θ1 θ2–( )

= 8A1A2

π2������������ 1

2n 1+( )2������������������ 2n 1+( ) θ1 θ2–( )( ).cos

n 0=

∞

∑

f 1 t( ) A1 θ1 t( )( ),sin=

f 2 t( ) A2 θ2 t( )( )sinsgn=

= 4A2

����� 1

2n 1+������������ (2n 1+( ) ω2 t( )t ψ2+( )),sin

n 0=

∞

∑

ϕ θ1 θ2–( ) 2A1A2

π������������ θ1 θ2–( ).cos=

ACKNOWLEDGMENTS

This work was supported by the Ministry for Edu�cation and Science of the Russian Federation and bythe Academy of Finland.

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A1A2

−A1A2

0

−2π −π 0 π 2π θ

Fig. 3. Plot of ϕ(θ).