Abstract
In this paper, we discuss how to partially determine the Fourier transform
F
(
z
)
=
∫
-
1
1
f
(
t
)
e
i
z
t
𝑑
t
,
z
∈
ℂ
,
F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C},
given the data
|
F
(
z
)
|
{\lvert F(z)\rvert}
or
arg
F
(
z
)
{\arg F(z)}
for
z
∈
ℝ
{z\in\mathbb{R}}
. Initially, we assume
[
-
1
,
1
]
{[-1,1]}
to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of
F
(
z
)
{F(z)}
. Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.
Quantitative phase retrieval of a complex-valued object using variable function orders in the fractional Fourier domain