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

N ′ (t, φ(t)), is a zero-mean, rapidly fluctuating disturbance which needs to be filtered out
by the PLL. One of the major attractions of a PLL is that it can cope with a significant
amount of noise. Note that N ′ (t, φ(t)) is a dimensionless quantity that can be viewed
as an angular phase disturbance which replaces the additive bandpass noise n(t) in the
equivalent base-band model (Meyer and Ascheid [15, pp. 106-107]).
Now consider the effects of filtering the signal g(t) with the loop filter response function
f(t), shown in Figure 3.1. When g(t) is passed through the time response function of the
loop filter f(t), the output is given by the convolution,
f(t) ∗ g(t) =
∫ t
0
f(t − τ)g(τ)dτ,
where f(t) = 0 on the interval (−∞ < t < 0). Since convolution integrals can be complicated,
the effects of filtering g(t) are more easily analyzed in the Laplace-frequency domain.
The analysis of a linear system is performed conveniently by the use of Laplace transform
techniques because it transforms linear ordinary differential equations into algebraic equations.
The unilateral Laplace transform of the loop filter f(t) is the filter-transfer function
F(s), defined by
F(s) =
∫ ∞
0
e −st f(t)dt,
where s = α+iω is the Laplace transform variable. Since the system is linear, the output
of the filter is a product in the Laplace domain and one has the following Laplace transform
pair:
f(t) ∗ g(t) ←→ F(s)G(s),
where G(s) is the Laplace transform of the signal g(t), provided it exists.
The purpose of the loop filter F(s) is to smooth the dynamic-error signal g(t) and to
reduce the amount of noise that enters the VCO [16]. Furthermore, F(s) is a filter element
of the closed-loop transfer function H(s) which controls what is called the “loop-noise”
bandwidth B L . Both H(s) and B L are defined later in Section 3.5 of this Chapter. The
most common type of loop filters used in applications are first and second-order filters. A
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