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

46 CHAPTER 2
2.2.3 Influence of system non-linearities
Generally, the transfer characteristic of all components involved in the analogous
output, ranging from the on chip output amplifier to the digitization, is not exactly
linear. In this section we describe the influence of quadratic non-linearities on the
measured phase using the 4-tap algorithm.
From the four acquired sampling points A0..A3 the phase ϕ is calculated using the
following equation:
⎛ A ⎞
⎜ 0 − A 2
ϕ = arctan
⎟
Equation 2.35
⎝ A1
− A 3 ⎠
Considering linear and quadratic distortions in the transfer function leads to the
measurement of a⋅Ai + b⋅Ai 2 rather than Ai. For the measured phase ϕmeas this
means:
( ) ( )
( ) ( ) ⎟⎟
2 2
A
⎞
0 − A 2 + b ⋅ A 0 − A 2
2 2
A − A + b ⋅ A − A
⎛
⎜ a ⋅
ϕmeas
= arctan
⎜
Equation 2.36
⎝ a ⋅ 1 3 1 3 ⎠
This can be rewritten as:
( ) ( ) ( )
( ) ( ) ( ) ⎟ ⎟⎟⎟
⎛
b
⎞
⎜ A 0 − A 2 + ⋅ A 0 − A 2 ⋅ A 0 + A 2
ϕ = ⎜
meas arctan
a
⎜
b
Equation 2.37
⎜ A1
− A 3 + ⋅ A1
− A 3 ⋅ A1
+ A 3
⎝
a
⎠
As can be seen from Figure 2.7, the sum “A0+A2” and “A1+A3” equal two times the
offset B. This is because cos(x) = -cos(x+180°). So we obtain:
⎛
⎜
ϕ
⎜
meas = arctan
⎜
⎝
( A − A ) + ⋅ ( A − A )
( A − A ) + ⋅ ( A − A )
⎛ A ⎞
⎜ 0 − A 2
= arctan
⎟ = ϕ
⎝ A1
− A 3 ⎠
0
1
2
3
2 ⋅ B ⋅ b
a
2 ⋅ B ⋅ b
a
0
1
2
3
⎞ ⎛ ⎛ 2 ⋅ B ⋅ b ⎞
⎟ ⎜ ⎜1+
⎟ ⋅
⎟ ⎜ ⎝ a ⎠
= arctan
⎟ ⎜
⎛ 2 ⋅ B ⋅ b ⎞
⎟ ⎜ ⎜1+
⎟ ⋅
⎠ ⎝ ⎝ a ⎠
( A − A )
0
( A − A )
1
2
3
⎞
⎟
⎟
⎟
⎟
⎠
Equation 2.38
This means that the 4-tap algorithm is not sensitive to linear or quadratic distortions
in the analogous transfer characteristic.