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

40 CHAPTER 2
Aliasing phase error for a square wave using 4-tap DFT-detection.
The deduction of the aliasing-caused phase error, which occurs if we use a square
wave as modulation signal, detected with the 4-tap approach, works in the same
way as the previous deduction. The only difference is that the square wave has an
infinite frequency spectrum, which makes the notation more complex. Therefore,
we will simplify wherever possible. A square wave ssq(t) can be spilt into a sum of
odd harmonics of the base frequency with decreasing amplitude for increasing
frequency:
ssq(
t)
= cos
1
− ⋅cos
7
1
3
( ω t − ϕ ) − ⋅ cos(
3ω
t − 3ϕ
) + ⋅cos(
5ω
t − 5ϕ
)
0
1
9
1
5
( 7ω
t − 7ϕ
) + ⋅cos(
9ω
t − 9ϕ
) −...
0
0
0
In the Fourier domain this corresponds to the spectrum Ssq(f):
Ssq(
f)
=
−
δ
0
0
0
j7ϕ
ϕ
( ω + 7ω
) ⋅ e 0
j5
δ(
ω + 5ω
) ⋅ e 0 δ(
ω + 3ω
)
+
δ
jϕ
( ω + ω ) ⋅ e 0 δ(
ω − ω )
−j3ϕ
− ϕ
( ω − 3ω
) ⋅ e 0
j5
δ(
ω − 5ω
) ⋅ e 0 δ(
ω − 7ω
)
0
3
+
0
1
0
5
+
0
−
0
−jϕ
0 ⋅ e
1
0
−j7ϕ
0 ⋅ e
7
Equation 2.29
⎪⎧
j3ϕ
1 δ
⋅ e
⎨...
− 0 + 0 − 0
2 ⎪⎩
7
5
3
0
0
0
⎪⎫
+ ... ⎬
⎪⎭
Equation 2.30
Again, considering the integration of the sampling points, we multiply Ssq(f) with
sinc(π⋅f⋅T/2). This time multiplication with the sinc function eliminates all negative
signs: