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

A use of standard trigonometric identities enables one to simplify the expression by expanding
and collecting like terms, to yield
g(t) = K [
m
A 2 m 2 (t)sin(2φ(t)) + 2Am(t)n x (t)sin(2φ(t))
2
]
+ [x 2 (t) − n 2 y(t)]sin(2φ(t)) + 2[Am(t) + n x (t)]n y (t)cos(2φ(t)) .
(3.18)
The expression, given in (3.18), can be linearized on the assumption of a sufficiently small
phase error, i.e., |φ(t)| ≤ 30 0 = π/6 (Lindsey [16, pg. 131]). Thus, if the phase-locked loop
is capable of reducing the phase error to a small value, then one can use the first term in
the Taylor series expansion for sin(2φ(t)) and cos(2φ(t)) which are given by
sin(2φ(t)) ≈ 2φ(t)
cos(2φ(t)) ≈ 1.
(3.19a)
(3.19b)
A substitution of (3.19) into (3.18) yields the linear approximation for g(t), given by
g(t) = K [
m
2A 2 m 2 (t)φ(t) + 4Am(t)n x (t)φ(t) + 2[n 2
2
x(t) − n 2 y(t)]φ(t)
]
+ 2[Am(t) + n x (t)]n y (t)
]
= K m
[A 2 m 2 (t)φ(t) + N(t, φ(t)) , (3.20)
where N(t, φ(t)) is defined by
N(t, φ(t)) = 2Am(t)n x (t)φ(t) + [n 2 x(t) − n 2 y(t)]φ(t) + [Am(t) + n x (t)]n y (t). (3.21)
In this study it is assumed that the data pulses are unit amplitude, equal width, and
are non-overlapping. Furthermore, the input signal-to-noise ratio is relatively high and
the low-pass filters are wide enough so that the data modulation m(t) incurs relatively
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