Synchronous detection is used to detect and measure very low-level signals in the presence of significant noise. A defining characteristic of this measurement approach is the use of a periodic probe signal to excite the system under test. This is followed by mixing of the reference signal and its phase-quadrature with the measured signal. Standard analog to digital converters are employed, usually with the mixing and filtering performed digitally. Most practical high-resolution analog to digital converters employ oversampled sigma-delta modulation and are incorporated as a separate functional block. This paper derives a processing algorithm that combines the oversampled analog to digital conversion with signal mixing into one functional block. There are several important advantages of this approach. The computational complexity of the lock-in amplifier is substantially reduced, with no loss of accuracy. Moreover, the requirement for high-resolution analog-to-digital conversion is relaxed; it is replaced with low-resolution high-rate sampling, which is typically much easier to realize in practice. Experimental results are presented to demonstrate the correctness of the technique as determined via theory and simulation.

1.
P. A.
Probst
and
A.
Jaquier
, “
Multiple-channel digital lock-in amplifier with PPM resolution
,”
Rev. Sci. Instrum.
65
,
747
750
(
1994
).
2.
A.
Restelli
,
R.
Abbiati
, and
A.
Geraci
, “
Digital field programmable gate array-based lock-in amplifier for high-performance photon counting applications
,”
Rev. Sci. Instrum.
76
,
093112
(
2005
).
3.
P.
Jain
,
P.
Priya
,
T. V. S.
Ram
,
K. S.
Parikh
, and
T. N.
Bandi
, “
Digital lock-in amplifier for space rubidium atomic clock
,”
Rev. Sci. Instrum.
92
,
124705
(
2021
).
4.
E. A.
Tholén
,
D.
Platz
,
D.
Forchheimer
,
V.
Schuler
,
M. O.
Tholén
,
C.
Hutter
, and
D. B.
Haviland
, “
Note: The intermodulation lockin analyzer
,”
Rev. Sci. Instrum.
82
,
026109
(
2011
).
5.
A. W.
Barnard
,
E.
Mikheev
,
J.
Finney
,
H. S.
Hiller
, and
D.
Goldhaber-Gordon
, “
Feedback lock-in: A versatile multi-terminal measurement system for electrical transport devices
,”
Rev. Sci. Instrum.
94
,
013902
(
2023
).
6.
G. A.
Stimpson
,
M. S.
Skilbeck
,
R. L.
Patel
,
B. L.
Green
, and
G. W.
Morley
, “
An open-source high-frequency lock-in amplifier
,”
Rev. Sci. Instrum.
90
,
094701
(
2019
).
7.
D.
Uhl
,
L.
Bruder
, and
F.
Stienkemeier
, “
A flexible and scalable, fully software-based lock-in amplifier for nonlinear spectroscopy
,”
Rev. Sci. Instrum.
92
,
083101
(
2021
).
8.
G.
Enemali
,
R.
Zhang
,
H.
McCann
, and
C.
Liu
, “
Cost-effective quasi-parallel sensing instrumentation for industrial chemical species tomography
,”
IEEE Trans. Ind. Electron.
69
,
2107
2116
(
2022
).
9.
F.
Paz
and
M.
Ordonez
, “
High-performance solar MPPT using switching ripple identification based on a lock-in amplifier
,”
IEEE Trans. Ind. Electron.
63
,
3595
3604
(
2016
).
10.
M. N.
Ashraf
,
R. A.
Khan
, and
W.
Choi
, “
A novel selective harmonic compensation method for single-phase grid-connected inverters
,”
IEEE Trans. Ind. Electron.
68
,
4848
4858
(
2021
).
11.
B.
Akin
,
U.
Orguner
,
H. A.
Toliyat
, and
M.
Rayner
, “
Phase-sensitive detection of motor fault signatures in the presence of noise
,”
IEEE Trans. Ind. Electron.
55
,
2539
2550
(
2008
).
12.
R.
Gray
, “
Oversampled sigma-delta modulation
,”
IEEE Trans. Commun.
35
,
481
489
(
1987
).
13.
E.
Nunzi
,
P.
Carbone
, and
D.
Petri
, “
Estimation of the in-band delta-sigma noise power based on windowed data
,”
IEEE Trans. Instrum. Meas.
55
,
2221
2226
(
2006
).
14.
R.
Schrier
and
M.
Snelgrove
, “
Bandpass sigma delta modulation
,”
Electron. Lett.
25
,
1560
1561
(
1989
).
15.
G.
Li
,
S.
Zhang
,
M.
Zhou
,
Y.
Li
, and
L.
Lin
, “
A method to remove odd harmonic interferences in square wave reference digital lock-in amplifier
,”
Rev. Sci. Instrum.
84
,
025115
(
2013
).
16.
S.
Zhang
,
G.
Li
,
L.
Lin
, and
J.
Zhao
, “
Optimization of a digital lock-in algorithm with a square-wave reference for frequency-divided multi-channel sensor signal detection
,”
Rev. Sci. Instrum.
87
,
085102
(
2016
).
17.
I.
Miranda
and
A.
Lima
, “
Impulsive sound detection directly in sigma-delta domain
,”
Arch. Acoust.
42
,
255
(
2017
).
18.
W. N.
Cheung
, “
Correlation measurement by delta-sigma modulation
,” in
IEEE Transactions on Industrial Electronics and Control Instrumentation
(
IEEE
,
1979
), Vol. IECI-26, pp.
88
92
.
19.
J.
Juni
, “
Implementing adaptive dsp algorithms using oversampled sigma delta strategies
,” in
IET Conference Proceedings
(
IET
,
1995
), Vol.
9–9
(
1
).
20.
P. P.
Sotiriadis
and
N.
Stamatopoulos
, “
All-digital frequency synthesis based on single-bit Nyquist-rate sinewave quantization with IID random dithering
,” in
2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC)
(
IEEE
,
2013
), pp.
213
216
.
21.
J.
Leis
, “
Computationally efficient synchronous demodulation using sigma-delta approach
,” in
2020 14th International Conference on Signal Processing and Communication Systems (ICSPCS) (ICSPCS’2020)
(
IEEE
,
Adelaide, Australia
,
2020
) (p
ublished
).
22.
S.
Engelberg
, “
Implementing a ΣΔ DAC in fixed-point arithmetic
,” in
Streamlining Digital Signal Processing
, edited by
R. G.
Lyons
(
Wiley
,
2012
), Chap. 7.
You do not currently have access to this content.