
94 ◾ Lionel R. Watkins
so that the CWT of Equation (3.6) can be written as a convolution integral:
=ψ−
∞
(,)()(
bftbtdt
(3.35b)
Hence, by the properties of the Fourier transform,
F =ω
{(,)}
()
.Wabf
(3.35c)
Because
ψω=ψω
∗
()
ˆ
,
(3.35d)
the CWT is readily calculated from the inverse FFT according to
F
(,){
()
ˆ
}
1
ba
f
(3.35e)
=
ωψ
∞
ˆ
()
ˆ
()
a
fad
ib
(3.35f)
If we intend to implement the CWT using a FFT, then the data will obvi-
ously be sampled, and we should write Equation (3.35f) as [16]
∑
=ωψω ω
−
∗
(,)
ˆ
()
ˆ
()exp( )
1
bafa
f
N
kk k
(3.36a)
where
=
π
≤
−
π
>
2
2
2
2
w
k
N
k
N
k
N
k
N
k
(3.36b)
and we have assumed unit sampling interv ...