Phase Estimation in Optical Interferometry

Uncertainty in Phase Measurements ◾ 323

Replacing the covariance terms by Equation (9.19), we can formally

write these sums using the convolutions





∑

∑

φ= ∗+ ∗

∂

∂













−∗

∂

∂

















































=⋅−φ =ℑ

Γ

−φ

()

1

()

()

() (,)sin()[()]

2

222

22

2

2

2

2

u

SM N

huSh

S

g

h

S

g

ug

hH eh

l

ll

l

i

pfpp

(9.67)

9.7.3 Uncertainty from Uncorrelated Influences

When contributions that are uncorrelated among pixels but have identical

variance such as electron noise are inserted into Equation (9.67), this leads to



()

φ= ∗=

ΓΓ

(, )

1()

{1}

1

{[](0)}

2

2

2

22

2

2

22

uuncorr

S

uS

MN

h

S

uSFT h

(9.68)

where we have used the fact that convolution of a function with the unit func-

tion is proportional to ...