Decoding Error-Correction Codes Utilizing Bit-Error Probability ...

the VCO angular frequency becomes 2πf c +K 0 e(t), where K 0 is the VCO gain factor with
dimension s −1 V −1 , where V denotes volts and s, seconds. The constant K 0 multiplies
e(t) so that the dimension of the phase estimate ̂Φ(t) is radians. It is a well known fact
that frequency is the time derivative of phase. Thus, the control law of the VCO is given
by
d̂Φ(t)
dt
∫ t
= 2πf c + K 0 f(t − τ)g(τ)dτ. (3.25)
0
A differentiation of ̂Φ(t), given in (3.11) and a comparison with (3.25), shows that the
frequency deviation of the VCO from the quiescent frequency is governed by
d̂θ(t)
dt
∫ t
= K 0 f(t − τ)g(τ)dτ. (3.26)
0
A substitution of g(t), given in (3.22), into (3.26) shows that the frequency deviation of
the VCO from the quiescent frequency is given by
d̂θ(t)
dt
[∫ t
= K 0 K D f(t − τ)φ(τ)dτ +
0
∫ t
0
]
f(t − τ)N ′ (τ, φ(τ))dτ . (3.27)
Equation (3.27) can be rewritten in terms of the phase error, φ(t) = θ(t) − ̂θ(t), as follows:
[∫
d
t
dt [θ(t) − φ(t)] = K 0K D f(t − τ)φ(τ)dτ +
0
∫ t
0
]
f(t − τ)N ′ (τ, φ(τ))dτ .
A slight rearrangement of this relation yields the linear first-order stochastic integro-differential
equation for the phase error, given by
dφ(t)
dt
= dθ(t)
dt
[∫ t
− K 0 K D f(t − τ)φ(τ)dτ +
0
∫ t
0
]
f(t − τ)N ′ (τ, φ(τ))dτ , (3.28)
where N ′ (t, φ(t)) is the phase-noise process, or disturbance given in (3.24).
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