3.2 M
ATHEMATICAL
DESCRIPTION
A digital LockIn realises it's filtering by certain mathematical procedures.
Think, Y(t), the signal to be analysed, is dependent on a value X(t). For instance: we measure a
photo voltage U(t) which depends on the light intensity given by an LED I(t). If the background
light intensity is much too high to detect small intensity changes of the LED light, one can
modulate the light intensity I(t) at a certain frequency
ref
and detect with a narrow band filter at
this frequency.
This X(t) is modulated as X(t) = X
0
+X
1
cos(
w
ref
t). Then, Y(X(t) can be developed into a Taylor-
Series:
Equ.(1)
For the value of Y at the time t, all measurement values Y(t) detected in the period
are used.
This period
t
should be a whole-numbered multiple of 2
pw
ref
. Due to the modulation of X(t) at
w
ref
,
modulations of Y(t) at the frequencies m
w
ref
with m = 1,2,... are expected. These modulations
equal the Fourier components Y(m
w
ref
) with m = 1,2,... of the input signal Y(t), which are given
as:
Y
(
m
ω)=
1
τ
∫
−τ /
2
τ/
2
Y
(
X
(
t
))
e
−
i m
ω
t
dt
and together with Equ. 1
Y
m
ref
=
∑
k
=
0
∞
Y
k
X
0
X
1
k
k !
K
m
k
with
K
m
k
=
1
∫
−/
2
/
2
cos
k
ref
t
e
−
i m
ref
t
dt
.
The factors K
m
k
get zero for k to infinity. For small k, the K
m
k
remain too big to be neglected. In
order to neglect n-th order parts, the n-th derivative of Y(t) to X(t) has to be negligible.
The first 10 (k = 1 ... 10) coefficients K
m
k
for the first 4 harmonics m = 1 ... 4 are given in the
following table:
k =
m
1
2
3
4
5
6
7
8
9
10
1
1
2
3
8
5
16
35
128
63
256
2
1
4
1
4
15
64
7
32
105
512
3
1
8
5
32
21
128
21
128
4
1
16
3
32
7
64
15
128
Manual Anfatec PCI-Lockin Amplifier AMU2.4 – Rev. 1.10 dated 30/09/20
Page 12 (70)
Y
X
0
X
1
cos
ref
t
=
∑
k
=
0
∞
Y
k
X
0
⋅
X
1
k
k !
⋅
cos
k
ref
t
⋅
Y
k
X
=
d
k
Y
dX
k
∣
X
0
k
∈
N
