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

38 CHAPTER 2
obtain positive coefficients and can easily deduce the amplitude and phase
spectrum. We obtain:
Sint(
f)
=
a
2
2
with a =
π
{
+
− jϕ
( ( ) e 0
jϕ
δ ω − ω
( ) e 0
0 ⋅ + δ ω + ω0
⋅ )
j
( ( 3 ) e
( π−3ϕ0
) j
( 3 ) e
( 3ϕ
⋅ δ ω − ω
0 −π)
0 ⋅ + δ ω + ω0
⋅ )
1
3
⎫
⎬
⎭
Equation 2.25
Sampling with fsamp=4f0 (ω0=2πf) leads to aliasing. For this simple function
(Equation 2.22 and Equation 2.23) the spectrum Sint,samp at f=f0, which is the place
we are interested in, is only superimposed by the -3f0 component of the actual
signal spectrum displaced to 4f0 by convolution with the sampling spectrum:
1 j(
3ϕ
−π)
⎞
+ ⋅ e ⎟⎠
3
1 ⎛ − jϕ
S = = ⋅ ⎜ 0
0
int, samp(
f f0
) e
π ⎝
We can split this into amplitude Af0 and phase ϕf0:
( ) ( ) ( ) ( ) 2
2
1
⎞ ⎛ 1
⎞
− ϕ + cos 3ϕ
− π + sin − ϕ + sin 3ϕ
− π
1 ⎛
Af 0 = ⋅ ⎜cos
0
0 ⎟ ⎜ 0
0 ⎟
π ⎝
3
⎠ ⎝ 3
⎠
ϕ
f0
⎛
⎜ sin
= atan ⎜
⎜
cos
⎝
( − ϕ ) + sin(
3ϕ
− π)
0
( ) ( )⎟ ⎟⎟⎟
− ϕ + cos 3ϕ
− π
0
1
3
1
3
0
0
⎞
⎠
Equation 2.26
Equation 2.27
Equation 2.28
With the knowledge of this relationship, the real signal phase ϕ0 can be deduced
from the aliasing disturbed measured phase ϕf0. In practical applications one could
efficiently solve this by means of a look up table (LUT). Figure 2.12 illustrates the
influence of natural sampling and aliasing on the amplitude and phase spectrum for
the example given above. Figure 2.14 d shows the relation of the real phase to the
measured phase and the corresponding error. Using the signal of Equation 2.22
rather than a pure sine would result in a maximum phase error of ±19.5° if no LUT
correction were performed.