WO2021233985A1 - Amplificateur à détection synchrone à réduction du bruit optimale - Google Patents

Amplificateur à détection synchrone à réduction du bruit optimale Download PDF

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Publication number
WO2021233985A1
WO2021233985A1 PCT/EP2021/063265 EP2021063265W WO2021233985A1 WO 2021233985 A1 WO2021233985 A1 WO 2021233985A1 EP 2021063265 W EP2021063265 W EP 2021063265W WO 2021233985 A1 WO2021233985 A1 WO 2021233985A1
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filter
lock
designed
amplifier
filters
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PCT/EP2021/063265
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German (de)
English (en)
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Andreas Wieck
Arne Ludwig
Daniel HÄGELE
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RUHR-UNIVERSITäT BOCHUM
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Publication of WO2021233985A1 publication Critical patent/WO2021233985A1/fr

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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R19/00—Arrangements for measuring currents or voltages or for indicating presence or sign thereof
    • G01R19/0007—Frequency selective voltage or current level measuring

Definitions

  • the invention relates to a lock-in amplifier with optimal noise suppression.
  • a lock-in amplifier is an amplifier for measuring a weak electrical alternating signal, which is connected in a phase-locked manner to a reference signal known in frequency and phase.
  • a lock-in amplifier is a narrow-band bandpass filter and thereby improves the signal-to-noise ratio.
  • Lock-in amplifiers are widely used to measure small signals over a noisy background.
  • the signal is deliberately periodically modulated in time via an external parameter, the modulation being phase-locked to a known reference.
  • the lock-in amplifier determines the amplitude of the signal at the frequency of the reference and its phase relative to the reference. All commercial lock-in amplifiers use a filter function that determines the time interval required to estimate the amplitude and phase of the signal when using polar coordinates, or X and Y of the signal when using Cartesian coordinates .
  • the filter function of a simple RC filter is a decaying exponential function with a time constant.
  • the lock-in output at time t therefore also contains contributions from the signal that were present at the input several time constants before time t.
  • the RC -based filter functions never achieve a constant zero value.
  • An RC filter is a circuit made up of an ohmic resistor R, for the English word “resistor”, and a capacitor C, for the English word “capacitor”. RC filters are implemented digitally in modern lock-in amplifiers.
  • US 2017/153279 A1 relates to a lock-in amplifier containing a clock signal generator which is configured so that it has a first demodulation clock signal and a second Demodulation clock signal with a phase difference of 90 degrees and the same demodulation frequency generated; and a detector which is configured to provide an offset voltage corresponding to an offset of the lock-in amplifier in a first operating mode and a first output voltage and a second output voltage on the basis of an input signal, the first demodulation clock signal and the second demodulation clock signal provides, each of which corresponds to a demodulation frequency component of the input signal in a second operating mode.
  • EIS 2014/218103 A1 relates to a combination of a lock-in amplifier unit and a phase-synchronous processing unit. This combination leads to a multitude of possibilities for signal analysis and processing. These possibilities include (i) the extraction of time domain properties of the input signal, (ii) the extraction of statistical properties of the input signal, (iii) the extraction of frequency domain properties of the input signal and (iv) the preconditioning of the lock-in input signal.
  • DE 102007 015 913 A1 relates to a lock-in amplifier which determines a control variable in order to adapt the integration duration T or time constant.
  • the document has no reference to the optimization of noise suppression by means of a suitably shaped filter.
  • the signal-to-noise ratio is only improved by extending the integration time.
  • the object of the invention is to avoid an unnecessary increase in the measurement time.
  • a lock-in amplifier having a filter, the filter being designed to filter out a signal from a measurement signal by means of a parameter-dependent filter function that is limited to a time interval.
  • the choice of the parameter allows the user to interpolate between a square filter and a sine filter and thus obtain optimal suppression for noise situations that lie between the extreme cases of broadband or narrowband noise.
  • a measurement signal can advantageously be measured without delay by a filter of the lock-in amplifier according to the invention.
  • the basic idea of the invention is therefore to use filters which, after some time, assume the constant value zero. Such filters are referred to as finite impulse response (FIR) filters.
  • FIR finite impulse response
  • the lock-in amplifier according to the invention makes it possible to completely avoid contamination by a preceding signal.
  • the lock-in amplifier with the filter according to the invention is the best possible compromise between filters that offer very good suppression of white background noise and represent filters with a narrow frequency response. A spectrally narrow filter suppressed strong interfering signals at frequencies close to the signal frequency. A good frequency resolution with good noise suppression is achieved.
  • the filter is designed to let its function for a white background noise in the measurement signal become a rectangle. In this way, a large part of the white background noise can advantageously be filtered out of the recorded measurement signal.
  • the filter is designed to let its function for noise in the measurement signal close to a reference frequency become the sine function.
  • a narrow-band filter can thus advantageously be used in a targeted manner at frequencies close to the reference frequency.
  • the filter is implemented by a parameter-dependent soft rectangle filter.
  • a soft rectangle filter is a filter with a finite impulse response (FIR filter). It can be provided that the filter function of the filter has a shape of a rectangle with rounded edges, a rounding of the edges decreasing with an increase in a proportion of background noise, and the rounding of the edges increasing with a proximity of a noise close to the reference frequency.
  • the Soft Rectangle filter can also be selected with a narrower band (parameter alpha> 0) in order to spectrally separate the interference signal - Rectangle filter for a given suppression of white noise always a better RMS narrow band than a Tukey filter and all other conceivable FIR filters ch achieved an optimization method, reference being made to equation (41) of this descriptive text with regard to an exemplary embodiment of the optimization. In the case of the soft rectangle filter, this is therefore carried out in accordance with this Example of the calculation of the filter function using matrix calculations.
  • An analytical formula for the Soft-Rectangle-Filter allows a quick and direct calculation without using matrix calculations, whereby reference is made to the descriptive text.
  • the filter which is designed to filter out a signal from the measurement signal by means of the parameter-dependent filter function, which is limited to the time interval, with one or more others different from the filter Filtering is connected in series.
  • the filter which is designed to filter out a signal from the measurement signal by means of the parameter-dependent filter function, which is limited to the time interval, is also designed to turn its function for a white background noise in the measurement signal into a rectangle .
  • a filter can be connected in series upstream of this filter, which filter is designed to reduce the amount of data of the measurement signal to be processed.
  • the upstream filter can be designed to reduce fluctuations in the measurement data.
  • This filter is designed as a kind of “decimation filter”.
  • the filter following the decimation filter which is, for example, the soft rectangle filter, can be implemented with a slower algorithm and with less memory expenditure.
  • the filter which is designed to filter out a signal from the measurement signal by means of the parameter-dependent filter function, which is limited to the time interval, is also designed, its function for a white background noise in the measurement signal to form a rectangle to be let.
  • a filter can be connected in series upstream of this filter, which filter is designed to reduce the amount of data of the measurement signal to be processed.
  • the filter which is designed to filter out a signal from the measurement signal by means of the parameter-dependent filter function, which is limited to the time interval, and that the filter is implemented by a parameter-dependent soft rectangle filter .
  • a filter can be connected in series upstream of this filter, which filter is designed to reduce the amount of data of the measurement signal to be processed. The above remarks on the filter, which reduces fluctuations in the measurement data, apply analogously to this example.
  • the calculation effort for the filtering can be reduced.
  • the examples described above relate to two filters connected in series. It is conceivable that further filters are connected in series for this purpose.
  • the filter is designed with a digital approximating implementation of the soft rectangle filter by means of an iterative algorithm, reference being made to the text of the description.
  • a computation time when applying the soft-rectangle filter to the measured values can advantageously be considerably shortened compared to a normal calculation using a convolution sum.
  • a series connection of several filters for example a filter to reduce the amount of data in front of a soft rectangle filter, is no longer necessary in order to keep the calculation effort low.
  • the digital approximating realization of the soft rectangle filter by the iterative algorithm is based on an analytical formula for the Soft-Rectangle Filter is based.
  • ß (beta) is gain and - ß (beta) is attenuation.
  • the formula can be interpreted as the sum of three complex-valued filters, whereby the complex-valued filters can all be implemented individually using an efficient iterative algorithm analogous to a sliding discrete fourier transform, sliding DFT.
  • sliding DFT reference is made to the statements from the publication by Jacobsen, E. and Lyons, R. on The Sliding DFT in IEEE SIGNAL PROCESSING MAGAZINE (March 2003), pages 74 to 80. Equation 7 from this publication corresponds to equation (44).
  • the lock-in amplifier is designed to monitor a background noise spectrum and to adapt an integration time to a signal-to-noise ratio (SNR) to be achieved.
  • SNR signal-to-noise ratio
  • the lock-in amplifier is designed to output an error value for each measured value. This can be used to assess the quality of the method and the measurement data, for example. It should be noted that the RC filter according to the prior art contains a statistical error as well as a systematic error which depends on the previous measured values. The method according to the invention, however, only generates statistical errors.
  • the lock-in amplifier is designed to output an amplitude and / or a phase and / or coordinates X and Y of at least one higher harmonic.
  • the higher harmonics can thus be recorded together with the “fundamental harmonics” and no measurements have to be repeated for this purpose. It should be made clear that outputting at least one harmonic is expressly outputting a Large number of harmonics as required, ie depending on the experiment, expressly included.
  • the lock-in amplifier can be set to a mode that mimics a lock-in amplifier with one or more serial RC filters.
  • the lock-in amplifier is designed to measure with the RC filters at the same time as with the filter according to the invention or a filter according to a modified embodiment or a combination thereof.
  • the lock-in amplifier is designed to output an amplitude, phase or coordinates X and Y like a lock-in amplifier according to one of the preceding modified embodiments or according to the invention and configured is to simultaneously output amplitude, phase or coordinates X and Y like a lock-in amplifier with one or more RC filters.
  • the RC filters described above can be those with several filter stages.
  • 1 schematically shows a time response function of one, two, three and four sequentially used RC filters
  • Fig. 3 schematically profile curves for an RMS bandwidth of a Tukey
  • Fig. 1 shows schematically a time response function of one, two, three and four sequentially used RC filters.
  • a weighting of the measurement signal in the past based on various RC filter stages is shown.
  • the past relates to a time before the time of the measurement, which is indicated in the graph in FIG. 1 at point “0” on the x-axis.
  • the graph in FIG. 1 goes up to 10 time constants in the past in order to illustrate how much signal from the past is also measured in the various filter stages.
  • the curve with the reference symbol “A” corresponds to a time response function of a used RC filter of 6 dB.
  • the curve with the reference symbol “B” corresponds to a time response function of two sequentially used RC filters of 6 dB each, ie. H.
  • the course curve with the reference symbol “C” corresponds to a time response function of three sequentially used RC filters of 6 dB each, ie. H. a total of 18 dB.
  • the curve with the reference symbol “D” corresponds to a time response function of four sequentially used RC filters of 6 dB each, ie. H. a total of 24 dB.
  • a simple analog lock-in amplifier implements a low-pass filter with a capacitor that is charged by the input signal x (t) via a resistor.
  • the filter output y (t) satisfies the differential equation
  • each filter function is one.
  • the Standford Research manual for the SR830 lock-in amplifier recommends a waiting time for the 6dB filter (12) for the measurement so that a trustworthy value of y (t) can be set. In general, a reasonable waiting time depends on the type of filter.
  • y (t) be an estimate for the average value xflj.
  • the average value of y (t) is (13) which means that y (t) is an unbiased estimator of ⁇ x (t)> for any filter function with areas one.
  • the variance ⁇ y 2 (Sigma_y ⁇ 2) of y (t) is a measure of the measurement error. The variance can best be interpreted in the frequency domain
  • ⁇ N (Alpha N) is crucial for comparing different types of filters.
  • ⁇ N (Alpha_N) is used in the context of this application as "noise-time Bandwidth product ".
  • the smaller ⁇ N (Alpha_N) is for a filter to be characterized, the stronger the suppression of white background noise.
  • the value of ( ⁇ N only depends on the ratio of Tw and Tu, since ⁇ enbw (Omega_ ⁇ enbw ⁇ ) scaled with Tu -1.
  • Table 1 Properties of filters (RC, Butterworth, and Time Rectangle).
  • the waiting time would be 2,303 / 1,570, i.e. H. about 1.47 times longer than a 24dB filter of a better fock-in amplifier, whereby users also suffer from poorer sideband suppression by the 6dB RC filter compared to the 24dB RC filter.
  • a family of filters is derived which contains the window with the lowest noise time bandwidth and the window with the lowest RMS bandwidth.
  • g (t) g (t)
  • the RMS bandwidth is given by
  • a small noise product I I 2 requires a small k (kappa), which is obtained for a large integral F 1 (30)
  • R is an NxN matrix that is filled with 1 in each entry.
  • the long-range multipliers method is used to include the constraint in a single minimization problem
  • the time courses of the function of the soft square filter are standardized here.
  • the appearance of a sine window is advantageous because this has the best possible RMS frequency bandwidth of an FIR filter (compare Starosielec and Hägele in Signal Processing 2014).
  • the noise-time bandwidth product (XN (Alpha N) gives 1.0 for the rectangle and Pi ⁇ 2/8, which corresponds to about 1.234, for the sine filter.
  • ⁇ N (Alpha_N) means better suppression of white Noise.
  • ⁇ N (Alpha_N) should not be confused with the parameter ⁇ (Alpha) of the soft rectangle filter.
  • ⁇ (alpha) between 0 and 1 there is a smooth transition from the sharp rectangle to the sine curve.
  • the actual setting of ⁇ (alpha) in a future lock-in amplifier is based on the character of the background noise.
  • a white background is best suppressed by the rectangular shape.
  • a spectrally narrow one Noise near the reference frequency is best suppressed by the spectrally narrower sinusoidal filter.
  • FIG. 3 shows schematically progressive curves for the RMS bandwidth (sigma omega in units of (1 / T_W)) of the Tukey window filter and the soft rectangle filter plotted over the various values of their parameters alpha N, which the suppression characterize the white noise (white noise suppression parameter).
  • FIG. 3 serves to compare the performance of the Tukey filters with the Soft Rectangle filters according to the invention.
  • a comparison of Tukey filters and soft rectangle filters is shown in FIG.
  • the behavior of the Tukey filters for different alpha N is shown in dashed lines and the corresponding curve is provided with the reference symbol “J”.
  • the behavior of the soft rectangle filters for different alpha N is shown with a solid line and the corresponding curve is provided with the reference symbol “K”.
  • the soft rectangle filter includes the cosine window (cosine filter) for one parameter.
  • the Tukey filter includes the Hann window (cosine square window) for one parameter.
  • the comparison shows the RMS (root mean square) bandwidth (narrow band) for a given alpha N of the suppression of white noise. A small bandwidth and a small alpha_N are cheaper here.
  • the soft rectangle filter always behaves better than the Tukey filter for various parameters of the filter.
  • the Tukey (tapered cosine) window or filter is a finite filter which, depending on the parameters, interpolates between a rectangular filter and a cosine square filter. For a given integration time and suppression of white noise, it always has a poorer RMS bandwidth than the soft-rectangle filter.
  • FIG. 4 shows a time profile for a lock-in output (in English: “lock-in output”) of a 24 dB RC filter compared with a time profile for a lock-in output of a soft rectangle Filter shown.
  • lock-in exit is after a switching process in the experiment.
  • the behavior of the soft rectangle filter over time is shown with a solid line and the corresponding curve is provided with the reference symbol “M”.
  • the time behavior of the RC filter with 24 dB is shown with a dashed line and the corresponding curve is provided with the reference symbol “L”.
  • the soft rectangle filter is 1.52 times faster than a 24 dB RC filter with the same noise level s (sigma) in the output. It is even 2.4 times faster than a 6dB RC filter and 3.2 times faster than a fourth-order Butterworth filter.
  • TN denotes the time constant of the RC filter.
  • the soft rectangle filter can be implemented with a convolution algorithm or with a faster iteration algorithm.
  • the lock-in amplifier is designed with a digital approximating implementation of the soft rectangle filter using an iterative algorithm.
  • One possible embodiment for such an approximating implementation of the soft rectangle filter using an iterative algorithm is described below.
  • the soft rectangle filter can be implemented with a convolution algorithm or with a faster iteration algorithm.
  • the computational effort for the following iteration algorithm is independent of N and therefore very efficient.
  • the parameter ⁇ (beta) relates to the steepness of the rise of the soft-rectangle filter at early times (x close to 0) or to the steepness of the fall of the soft-rectangle filter at late times (x close to 1).
  • the normalization factor A is chosen so that (45) is satisfied.
  • the practical calculation of soft-rectangle filters as a function of N and beta can therefore be carried out quickly using equations (43) and (44) without using equation (40).
  • beta 0 the sine filter follows, for increasing beta the limit case of the rectangular filter is reached.
  • the three summands in the argument of the real part of the right-hand side of equation (44) can be interpreted as three complex-valued (FIR) filters: a square-wave filter, a complex (FIR) frequency filter for frequency Pi and attenuation beta (in units of 1 / T_W) and a complex (FIR) frequency filter for frequency -Pi and gain beta. All three sub-filters have the advantage that they can be implemented with an iterative algorithm.
  • the speed of iterative algorithms does not decrease with the filter length N.
  • the memory requirement for the intermediate storage of measured values s [n] or xi [n] increases linearly with N.
  • the usual convolution algorithms become slower by the factor N, the memory requirement also increases linearly with N.
  • FIR filter A filter with a finite impulse response (“finite impulse response”, or “FIR” for short), in short: FIR filter was used in a self-made lock-in amplifier by Qin et al. (Jianhuan Qin, Zhiming Huang, Yujian Ge, Fun How, and Junhao Chu, “Tandem demodulation lock-in amplifier based on digital processor for dualmodulated specroscopy,” Rev. Scientific Instruments, vol. 80, pp. 033112, 2009) used.
  • the Blackman window was used as the filter, which is by far not optimal with regard to the noise-time bandwidth product (XN (Alpha_N).
  • a Gaussian FIR filter can be used by the user in the lock-in amplifiers SR860 and SR865 from Stanford Research. However, its use is limited to time constants below 3s. Here, too, the Gaussian filter is not optimal with regard to ( ⁇ N (Alpha_N).
  • the invention can be applied in a variety of areas, such as general spectroscopy applications (microwaves, infrared, visible, UV, xuv, ...) Medical technology
  • AD temporal response function of one, two, three and four sequentially used RC filters (6dB, 12dB, 18dB, 24dB) EI curve of an amplitude of a function of a soft square filter for a value of a

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  • General Physics & Mathematics (AREA)
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Abstract

L'invention concerne un amplificateur à détection synchrone comprenant un filtre, ce filtre étant conçu pour filtrer un signal à partir d'un signal de mesure au moyen d'une fonction de filtre dépendant de paramètres, laquelle est limitée à un intervalle de temps.
PCT/EP2021/063265 2020-05-19 2021-05-19 Amplificateur à détection synchrone à réduction du bruit optimale WO2021233985A1 (fr)

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DE102020113558.0 2020-05-19
DE102020113558.0A DE102020113558B4 (de) 2020-05-19 2020-05-19 Lock-In-Verstärker mit optimaler Rauschunterdrückung

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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE102007015913A1 (de) 2007-04-02 2008-11-13 Siemens Ag Lock-In-Verstärker und Verfahren zum Filtern eines Messsignals mittels eines solchen Verstärkers
US20140218103A1 (en) 2013-02-04 2014-08-07 Sadik Hafizovic Lock-in amplifier with phase-synchronous processing
US20170153279A1 (en) 2015-11-27 2017-06-01 Samsung Electronics Co., Ltd. Lock-in amplifier, integrated circuit and portable measurement device including the same

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE102007015913A1 (de) 2007-04-02 2008-11-13 Siemens Ag Lock-In-Verstärker und Verfahren zum Filtern eines Messsignals mittels eines solchen Verstärkers
EP2130295A1 (fr) * 2007-04-02 2009-12-09 Siemens Aktiengesellschaft Amplificateur synchrone et procédé de filtrage d'un signal de mesure au moyen d'un tel amplificateur
US20140218103A1 (en) 2013-02-04 2014-08-07 Sadik Hafizovic Lock-in amplifier with phase-synchronous processing
US20170153279A1 (en) 2015-11-27 2017-06-01 Samsung Electronics Co., Ltd. Lock-in amplifier, integrated circuit and portable measurement device including the same

Non-Patent Citations (7)

* Cited by examiner, † Cited by third party
Title
AYAT M.: "Design of Multiple Modulated Frequency Lock-In Amplifier for Tapping-Mode Atomic Force Microscopy Systems", IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, vol. 65, no. 10, 2016, pages 2284 - 2292, XP011622680, DOI: 10.1109/TIM.2016.2579438
AYAT MEHDI ET AL: "Design of Multiple Modulated Frequency Lock-In Amplifier for Tapping-Mode Atomic Force Microscopy Systems", IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, IEEE SERVICE CENTER, PISCATAWAY, NJ, US, vol. 65, no. 10, 1 October 2016 (2016-10-01), pages 2284 - 2292, XP011622680, ISSN: 0018-9456, [retrieved on 20160912], DOI: 10.1109/TIM.2016.2579438 *
ERIC JACOBSENRICHARD LYONS: "The sliding DFT", IEEE SIGNAL PROCESSING MAGAZINE, vol. March, 2003, pages 74
JIANHUAN QINZHIMING HUANGYUJIAN GEFUN HOWJUNHAO CHU: "Tandem demodulation lock-in amplifier based on digital processor for dualmodulated spectroscopy", REV. SCIENTIFIC INSTRUMENTS, vol. 80, 2009, pages 033112
MAXIMILIAN HOFMANNRUDOLF DR. BIERLTHOMAS RÜCK: "Implementation of a dual-phase lock-in amplifier on a tms320c5515 digital signal processor", EUROPEAN DSP EDUCATION AND RESEARCH CONFERENCE (EDERC, 2012, pages 20 - 24
QIN JIANHUAN ET AL: "Tandem demodulation lock-in amplifier based on digital signal processor for dual-modulated spectroscopy", REVIEW OF SCIENTIFIC INSTRUMENTS, AIP, MELVILLE, NY, US, vol. 80, no. 3, 25 March 2009 (2009-03-25), pages 33112 - 33112, XP012128140, ISSN: 0034-6748, DOI: 10.1063/1.3098948 *
S. STAROSIELECD. HÄGELE: "Discrete-time windows with minimal RMS bandwidth for given RMS temporal width", SIGNAL PROCESSING (ELSEVIER, vol. 102, 2014, pages 240

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