Figure 7: Costas loop.
4 COSTAS LOOP
Consider now a block diagram of the Costas loop
(Fig. 7)
Here all denotations are the same as in Fig. 1, 90
o
– is a quadrature component. As before, we consider
here the case of the high-frequency harmonic and im-
pulse signals f
j
(t).
However together with the assumption that condi-
tions (2) and (4) are valid we assume also that (3) is
satisﬁed for the signal of the type (1) and the relation
|ω
1
(τ) − 2ω
2
(τ)| ≤ C
1
, ∀τ ∈ [0,T ], (23)
is valid for the signal of the type (5).
Applying the similar approach we can obtain dif-
ferential equation for the Costas loop, where
˙z = Az + bΨ(σ)
˙
σ = c
∗
z + ρΨ(σ).
(24)
Here A is a constant n ×n-matrix, b and c are con-
stant n-vectors, ρ is a number, Ψ(σ) is a 2π-periodic
function, satisfying the following relations
ρ = −2aL, W (p) = (2L)
−1
c
∗
(A − pI)
−1
b,
ψ(σ) =
1
8
A
2
1
A
2
2
sinσ −
ω
1
(0) − ω
2
(0)
L(a +W (0))
,
σ = 2θ
1
− 2θ
2
(in the case (1);
ψ(σ) = P(σ) −
ω
1
(0) − 2ω
2
(0)
2L(a +W (0))
,
P(σ) =
−
2
A
2
1
A
2
2
µ
1
+
2σ
π
¶
,
σ
∈
[
0
,
π
]
− 2A
2
1
A
2
2
µ
1 −
2σ
π
¶
, σ ∈ [−π, 0]
σ = θ
1
− 2θ
2
(in the case (5).
From the above equations it follows that for
deterministic (when the noise is lacking) description
of the Costas loops the conventional introduction
of additional ﬁlters turns out unnecessary. Here a
central ﬁlter plays their role.
REFERENCES
Abramovitch, D., 2002. Phase-Locked Loope A control
Centric. Tutorial, in the Proceedings of the 2002 ACC.
Aleksenko, A., 2004. Digital engineering, Unimedstyle.
Moscow. (in Russian)
Best Ronald E., 2003. Phase-Lock Loops: Design, Simula-
tion and Application, McGraw Hill, 5
ed
.
Egan, W.F., 2000. Frequency Synthesis by Phase Lock ,(2nd
ed.), John Wiley and Sons, 2
ed
.
Gardner F., 2005. Phase–lock techniques, John Wiley &
Sons, New York, 2
ed
.
Kroupa, V., 2003. Phase Lock Loops and Frequency Syn-
thesis, John Wiley & Sons.
Kung, S., 1988. VLSI Array Processors, Prentice Hall. New
York.
Lapsley, P., Bier, J., Shoham, A., Lee, E., 1997. DSP Pro-
cessor Fundamentals Architecture and Features, IEE
Press. New York.
Leonov, G., Reitmann, V., Smirnova, V., 1992. Nonlocal
Methods for Pendulum-Like Feedback Systems, Teub-
ner Verlagsgesselschaft. Stuttgart; Leipzig.
Leonov, G., Ponomarenko, D., Smirnova, V., 1996.
Frequency-Domain Methods for Nonlinear Analysis.
Theory and Applications, World Scientiﬁc. Singapore.
Leonov, G., Seledzhi, S., 2002. Phase locked loops in array
processors, Nevsky dialekt. St.Petersburg. (in Rus-
sian)
Lindsey, W., 1972, Sinchronization systems in communica-
tion and ontrol, Prentice-Hall. New Jersey.
Lindsey, W., Chie, C., 1981. A Survey of Digital Phase
Locked Loops. Proceedings of the IEEE.
Razavi, B., 2003. Phase-Locking in High-Performance Sys-
tems: From Devices to Architectures, John Wiley &
Sons.
Smith, S., 1999. The Scientist and Engineers Guide to Dig-
ital Dignal Processing, California Technical Publish-
ing. San Diego.
Solonina, A., Ulahovich, D., Jakovlev, L., 2000. The Mo-
torola Digital Signal Processors. BHV, St. Petersburg.
(in Russian)
Ugrumov, E., 2000. Digital engineering, BHV,
St.Petersburg. (in Russian)
Viterbi, A., 1966. Principles of coherent communications,
McGraw-Hill. New York.
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