Local Polynomial Phase Modeling and Estimation ◾ 207
simplicity, if we assume a slowly varying amplitude and ignore the eect
of noise, we have
Γ+Γ−≈α+α +α +α
+α+α
()()()exp[ {2
2( 3)}]
2
01 2
2
3
3
23
2
ym ym ay jy
ym
ii ii
ii
(6.37)
In Equation (6.37), the right-hand side comprises a complex signal whose
phase has two components: (1) the term 2(α
i0
+ α
i1
y + α
i2
y
2
+ α
i3
y
3
), which
is invariant to m and is a constant for a given y, and (2) the term 2(α
i2
+
3α
i3
y) m
2
, which is quadratic in m, and the corresponding quadratic coef-
cient is essentially the IFR.
Hence, the CPF in Equation (6.35) can now be expressed as
∑
Ω≈ α+α+α+α
+α+α ×−Ω
≈α+α +α +α
−Ω−
CP(, )()exp[{2( )
2( 3)}] exp[ ]
()exp[ 2( )]
exp[ {IFR()}]
2
01 2
2
3
3
23
2
01 2
2
3
3
2
yayj yy y
ym jm
ay jyyy ...