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

OPTICAL TOF RANGE MEASUREMENT 27
c(
τ)
= ϕsg(
τ)
=
1
= lim
T'→∞
T'
=
a
⋅ cos
2
[ 1+
a ⋅ cos( ϖτ − ϕ)
] ⊗ [ cos( ϖτ)
]
+
−
T'
2
∫
[ 1+
a ⋅ cos( ϖt
− ϕ)
] ⋅ [ cos( ϖt
+ ϖτ)
]
T'
2
( ϕ + ϖτ)
dt
Equation 2.8
We evaluate this function for different phases ϖτ. Choosing ϖτ0=0°, ϖτ1=90°,
ϖτ2=180° and ϖτ3=270° allows us to recalculate the phase ϕ and amplitude a of the
received optical signal s(t). Considering that the received signal is mostly
superimposed on a background image, we must add an offset K to the correlation
function to obtain the real measured values C(τ)=c(τ)+K:
a
C(
τ0
) = c(
τ0
) + K = ⋅ cos(
ϕ)
+ K
2
a
C(
τ1)
= c(
τ1)
+ K = − ⋅ sin()
ϕ + K
2
a
C(
τ2
) = c(
τ2
) + K = − ⋅ cos(
ϕ)
+ K
2
a
C(
τ3
) = c(
τ3
) + K = ⋅ sin()
ϕ + K
2
Equation 2.9
With this evaluation of the correlation function at four selected points we can
determine the phase ϕ and amplitude a of s(t):
⎛ C( τ ) − C( τ ) ⎞
ϕ = atan ⎜ 3 1 ⎟
⎜
⎟
⎝ C( τ0
) − C( τ2
) ⎠
2
[ C(
τ ) − C(
τ ) ] + [ C(
τ ) − C(
τ ) ]
3
1
0
Equation 2.10
a = Equation 2.11
2
Sampling a sinusoidal signal
Another, slightly different approach is to sample the modulated signal
synchronously. Sampling always means to convolve the input signal with a
sampling function. In that sense, if one chooses the same sampling function as the
function to sample we get the same conditions as described above. Before
sampling, the input signal must be bandwidth limited in order to avoid aliasing
effects. Assuming that we sample a periodic signal we can use the equations of the
DFT (Discrete Fourier Transform) in order to calculate both amplitude and phase of
the base frequency and harmonics contained in the signal. Using N sampling points
2
2