3D Time-of-flight distance measurement with custom - Universität ...

90 CHAPTER 4
4.2 Noise limitation of range accuracy
In Chapter 3 we discussed the fact that the performance of solid state imagers is
limited by noise and that there are several different noise sources in both CCD
sensors and photodiode arrays. The essential ones are electronic and optical shot
noise, thermal noise, reset noise, 1/f noise and quantization noise. All of these
noise sources can be reduced or eliminated by different signal processing
techniques or cooling, except photon shot noise. Therefore, we first only investigate
the influence of shot noise on the ranging accuracy.
Quantum noise as final limitation
Shot noise describes the statistical Poisson-distributed nature of the arrival process
of photons and the generation process of electron-hole pairs. The standard
deviation of shot noise is equal to the square root of the number of photons (optical
shot noise) or photogenerated charge carriers (electronic shot noise). In the
following the required number of photoelectrons per sampling point and pixel to
achieve a given range resolution is derived.
As introduced in Chapter 2, the phase can be calculated from the sampling points
with the following equation (four-tap approach):
Following the rules of error propagation
∆ ϕ
=
2
⎛ ∂ϕ
⎞ 2
⎜ ⋅ ∆
A ⎟
⎝ ∂ 0 ⎠
⎛ A − ⎞
ϕ =
⎜ 1 A
atan 3
⎟ . Equation 4.8
⎝ A0
− A2
⎠
2
⎛ ∂ϕ
⎞
⎜ A ⎟
⎝ ∂ 1 ⎠
⎛ ∂ϕ
2
⎞
⎜ A ⎟
⎝ ∂ 2 ⎠
⎛ ∂ϕ
2
⎞
⎜ A ⎟
⎝ ∂ 3 ⎠
2
2
2
( A ) + ⎜ ⎟ ⋅ ∆ ( A ) + ⎜ ⎟ ⋅ ∆ ( A ) + ⎜ ⎟ ⋅ ∆ ( A )
0
1
2
3
Equation 4.9
and considering that each of the integrated sampling points A0..A3 shows a
standard deviation of ∆ A i = Ai
, one obtains the quantum noise limited phase
error ∆ϕ:
∆ ϕ
=
2
2
2
2
⎛ ∂ϕ
⎞ ⎛ ∂ϕ
⎞ ⎛ ∂ϕ
⎞ ⎛ ∂ϕ
⎞
⎜ A0
A1
⋅ A2
+ ⋅ A3
A ⎟ ⋅ + ⎜
0
A ⎟ ⋅ + ⎜
1
A ⎟
⎜
2
A ⎟
⎝ ∂ ⎠ ⎝ ∂ ⎠ ⎝ ∂ ⎠ ⎝ ∂ 3 ⎠
. Equation 4.10