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7
)
sin(
)
(
)
2
cos(
2
1
)
cos(
2
1
)
sin(
)
(
)
sin(
)
sin(
)
(
)
(
t
A
t
B
t
A
A
A
A
t
A
t
B
t
t
A
A
t
S
t
S
S
R
R
I
R
I
R
R
I
R
I
psd
In the time region, the output of the PSD consists three parts:
The first part is a DC signal. If A
I
, A and the phase difference (φ − δ) between the input signal and the reference
signal are constants, this part is a DC signal.
The second part is the frequency-doubled reference AC signal.
The third part is the result of multiplication of the noise and the reference. Because the sine signal is periodic and
there is no relevance between the noise signal and the reference signal. The integral of this part is zero.
In the frequency region, we can redraw these three parts:
The first part is at 0Hz, which is known as the DC component of one signal.
The second part is at 2fref Hz.
The third part is a random signal at all the frequencies, such as white noise. The frequency spectrum of white noise
does not change after any frequency drifts.
To sum up, the LPF output is defines as:
)
cos(
2
1
R
I
Output
A
A
S
Although we can determine the amplitude the amplitude of the input signal through adjusting the phase difference
)
(
, the accuracy is unsteady and insecure. In order to solve this problem, the dual phase lock-in amplifier was
invented, see Fig.3.
Fig.3 Dual Phase Lock-in Amplifier diagram
Now, we define the phase difference
, the LPF0 output
0
Output
S
X
and LPF1 output
1
Output
S
Y
. Then
we calculate the amplitude R which is independent of
θ
:
R
I
A
Y
X
A
R
)
(
2
2
2
2
