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

OPTICAL TOF RANGE MEASUREMENT 41
Ssq,
int(
f)
=
+
δ
⎧
1 ⎪
⎨...
2 ⎪
⎩
+
δ
j7ϕ
ϕ
( ω + 7ω
) ⋅ e 0
j5
δ(
ω + 5ω
) ⋅ e 0 δ(
ω + 3ω
)
jϕ
( ω + ω ) ⋅ e 0 δ(
ω − ω )
−j3ϕ
− ϕ
( ω − 3ω
) ⋅ e 0
j5
δ(
ω − 5ω
) ⋅ e 0 δ(
ω − 7ω
)
0
3π
3 ⋅
2
+
0
7π
7 ⋅
2
δ
+
0
π
1⋅
2
+
+
0
5π
5 ⋅
2
0 +
5π
5 ⋅
2
−jϕ
⋅ e 0
0
π
1⋅
2
+
−j7ϕ
0 ⋅ e
7π
7 ⋅
2
j3ϕ
0 ⋅ e
3π
3 ⋅
2
0
0
⎫
⎪
+ ... ⎬
⎪
⎭
Equation 2.31
Now we sample with fsamp=4f0 (ω0=2πf) and observe aliasing, as in the previously
examined case. The spectrum at f=f0 S sq,
int, samp(
f = f0
) , looks like this:
⎪⎧
j11ϕ
1 e 0
Ssq,
int, samp(
f = f0
) = ⎨...
+
π ⎪⎩
121
+
−jϕ
− ϕ
e 0 j5
e 0
+
1 25
j7ϕ
e 0
49
+
+
−j9ϕ
e 0
81
j3ϕ
e 0
9
We can split this into amplitude Asq,f0 and phase ϕsq,f0:
A
sq,
f
0
1
= ⋅
π
⎛ cos
⎜...
+
⎝ 9
⎛ sin
+ ⎜...
+
⎝ 9
⎛ sin
... 0
⎜ +
ϕ atan 9
sq,
f0
= ⎜
⎜ cos
... 0
⎜ +
⎝ 9
+
+
−j13ϕ
e 0
169
( 3ϕ
) cos(
− ϕ ) cos(
− 5ϕ
)
0
( 3ϕ
) sin(
− ϕ ) sin(
− 5ϕ
)
0
+
+
1
1
0
0
+
+
25
25
( 3ϕ
) sin(
− ϕ ) sin(
− 5ϕ
)
+
1
( 3ϕ
) cos(
− ϕ ) cos(
− 5ϕ
)
+
1
0
0
+
+
0
0
⎞
+ ... ⎟
⎠
⎞
+ ... ⎟
⎠
⎪⎫
+ ... ⎬
⎪⎭
2
2
0 ⎞
+ ... ⎟
25 ⎟
0 + ...
⎟
25 ⎠
Equation 2.32
Equation 2.33
Equation 2.34