M
ATHEMATICAL
DESCRIPTION
A digital LockIn realises its filtering by certain mathematical procedures.
Think, Y(t), the signal to be analised, is dependent on a value X(t). For
instance: we measure a photovoltage 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
ω
ref
and detect with a narrow band
filter at this frequency.
Thus, 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
=
1
∫
− /
2
/
2
Y
X
t
e
−
i m
t
dt
Manual Anfatec PCI-Lockin Amplifier AMU5.0 – Rev. 1.01
Page 11 (34)
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
Figure 5: Output signal of a LockIn amplifier for a single frequency input signal of 100
kHz and 20 mVrms amplitude at the input. The time constants have been kept constant.
