WO1997008598A1 - Simultaneous, phase-sensitive digital detection process for time-resolved, quasi-simultaneously captured data arrays of a periodically stimulated system - Google Patents

Abstract

A process for measurement and analysis technology supplies by phase-sensitive detection the response of a system that may be periodically stimulated. For that purpose, an external thermodynamic parameter is periodically modified. Data arrays are used instead of the scalar measurement values normally used in the state of the art both for capturing and for processing data. During data processing, the period is subdivided into a determined number of supporting points that represent each a data vector at the corresponding moment in time within the period. The supporting points of at least one period are digitally stored after simultaneous phase-sensitive detection is carried out. By simultaneously processing the measurement data, the number of components of the data array is reduced in time while maintaining the signal/noise ratio at the same level. The phase angle array, which represents for each component the delay of the system response in relation to the stimulation, is an essential element to characterise the kinetics of the stimulated process. Devices for Fourier transformation infrared and X-ray diffraction modulation are disclosed. The process has application in optical spectroscopy, in diffraction methods, in image processing, and in general in measurement techniques which use unidimensional or bidimensional array detectors, or fast (almost simultaneous) serial data capture.

Description

Method for the simultaneous, digital phase-sensitive detection of time-resolved, quasi-simultaneous data arrays of a periodically stimulated system

The present invention relates to a method and its application for the simultaneous, digital phase-sensitive detection of time-resolved, quasi-simultaneous data arrays of a periodically stimulated system according to claim 1, and a device according to claim 16.

Phase-sensitive detection (PSD) is an analytical measurement method that enables the selective and time-resolved detection of the periodic response of a system to a periodic change in external parameters, such as pressure (p), temperature (T), concentration (c), and electric field (E), electrical current density (j), radiation flux density (I) etc. The information content on the dynamics (kinetics) of the stimulated process is in the frequency dependence of the signal amplitudes, as well as the phase shifts of the signals with respect to the external stimulation. The selectivity of the method is given by the fact that only signals with the stimulation frequency ω and their overtones nω are detected with PSD. All other signals are suppressed.

PSD processes have been used in a variety of ways and have been described in various ways (Hs.H. Günthard, Modulation Specroscopy, Ber. Bunsengesellschaft Phys.Chemie, 78, 1110-1115 (1974); Ch. J. Manning and PR Griffiths , Step-Scanning Interferometer with Digital Signal processing, Applied Specroscopy, 47, 1345-1349 (1993)).

In the following, the procedure on a system with I measurands is presented. These measurement data can be, for example, intensities, absorbances or transmittances, with the wave numbers v _ιr v ₂ , ... v, ... v. be, which are combined in an array, eg spectrum, ^¬ . If the system is a periodic, external Exposed to stimulation with the fundamental frequency ω, this frequency occurs with all or only with individual components of this array, depending on the response of the system. The response of the system must be measured for each component (eg wavenumber) at least during one period and to characterize the system a Fourier analysis using Fast Fourier Transformation

(FFT) or by phase sensitive detection (PSD).

In the previous, conventional measuring methods, this process was always carried out point by point, ie the normally continuous registration process (scan) was interrupted for each component (e.g. wave number v _L , or defined mirror position of the interferometer) in order to determine the behavior of the system by PSD ( Günthard (1974)) or FFT (Manning and Griffiths (1993)). In the latter case, it was a step scanning interferometer, in which the I characteristic measurement variables mentioned at the outset were the I components

Represent (intensities) of the interferogram.

In contrast to the previously described methods, the object of the invention is to acquire and process measurement data arrays simultaneously or quasi-simultaneously, i.e. to proceed with the measurement data arrays as if they were individual measurement points. As a result, the measuring time can be considerably shortened without any significant loss of accuracy.

The main area of application of the vectorial analysis method according to the invention is in modulation spectroscopy using Fourier transform and diode array spectrometers.

According to the invention, this object is achieved by means of methods according to the wording of patent claims 1-15 and by means of devices according to the wording of patent claims 16-19. The invention is based on the following definitions of terms. 1. Definitions and symbols

(Vectors are shown in bold letters below).

2.1 DC term, A ₀

Output signal of the PSD (Fig. IA, Pos. 10), which arises when S (t) is averaged over a period. Time-independent part of S (t).

1.2 DC term array, A ₀

Output signal of the PSD (Fig IB, item 20), which arises when the I components of the array S (t) are averaged over a period. Time-independent part of the system response.

2.3 root note, A, ( ^' φi-φps _D , ι)

Output signal of the PSD during demodulation with the stimulation frequency ω (fundamental tone frequency, index 1). The signal is dependent on the phase shift φ, the fundamental component of S (t) (see FIG. 1A), and on the choice of the phase angle of the PSD, φ _PSD _ ,. φj is the system-related phase angle of the system response with the frequency ω (see also FIG. 2A and modulation spectra).

1 . 4 basic onarray, A- Cφj-φ _PSD , ι)

Output signal of the PSD with simultaneous detection with the stimulation frequency ω (fundamental tone frequency, index 1). The signal is dependent on the phase shift φ _{n of} the fundamental component of the i-th data array component of S (t) (see FIG. IB), and on the choice of the phase angle of the PSD, φ _{PSD> 1} . .phi..sub.i is the system ^¬ related phase angle array of the portion of the system response ω with frequency. It consists of the components φ _llr where l≤i≤I (see also Fig. 2B and modulation spectra).

2.5 interval

The interval is the Nth part of a period τ, the kth interval beginning within a period at the time t _k = (kl) Δt (FIG. 2A). 1 . 6 components (array, vector), 1, 1

Each data array S _k (t), for example the time-resolved spectrum at the k-th support point, consists of I components S _ki (t), for example the intensities, absorbances or transmittances at the _wavenumbers V _lf where 1 <i <I .

2.7 measuring cycle

(N + l) support points (data arrays with I components each) recorded synchronously with the external stimulation within a period τ. Because of the periodicity of the process, the first (k = 1) and the last (k = N + 1) support point are identical (FIG. 2B).

1 . 8 support point S _k

A function value or coaddition of M assigned to an interval k, where 1 <k <N (FIG. 1A).

1 . 9 Average support point, S _j ,

Function mean value generated by coaddition of corresponding support points from P periods (FIG. 1A).

2.20 Support point array (support point spectra), S _k Data array assigned to an interval k or coaddition of M data arrays with I components each, where 1 <k <N and 1 <i <I (see FIG. IB).

2.22 Ge / nitte2ter support array, S _j r

By coaddition of corresponding support point arrays from P

Period generated mean (see FIG. 2B).

1 . 12 support earray component, S _kl , _j S ^ i-th component of the k-th support point array, resp. of the averaged kth node array. Beginning of the kth interval.

1 . 13 modulation spectra, A _n (§ _n - $ _PSD , _n )

If the original data array S (t) (FIG. IB) consists of spectra, for example FIG. 5, fundamental and harmonics of the output signals of the PSD, respectively. tion spectra of an FFT as Modula ^¬ refers to the corresponding result (see fundamental, harmonic, ORIGINAL DATA array).

1 714 Multicomponent system response, multicomponent The change in the state of a system manifests itself not only in the change in a single measurement variable (scalar), but also in I measurement variables, which can be combined into a vector (array) with I components and simultaneously or quasi-identically can record experimentally early, e.g. as an intensity, absorption or transmittance spectrum, in which characteristic changes are shown at certain wavelengths, or as an interferogram, in which the corresponding changes result at I defined mirror positions of the interferometer.

1 . 15 Multiparametric stimulation, multi-parametric stimulation of the system by simultaneously changing more than one external parameter.

1 . 1 6 overtone, A „($ _n - $ _{PSDt π} ), A _n

Output signal (Fig. IA, Pos. 10) of the PSD when stimulated with the fundamental frequency ω and detection with the n-fold frequency (nω, (n- l) th overtone). The signal is dependent on the difference between the absolute phase shift φ _n of S (t) with regard to the stimulation and the phase setting φ _{PSD, n} at the PSD.

2.27 Overtone array, A _n (φ _n -φp _SD , J, A _n

Output signal array (Fig. IB, item 20), the PSD when stimulated with the fundamental frequency ω and simultaneous demodulation with the n-fold frequency (nω, (nl) th overtone). The signal is dependent on the difference between the absolute phase shifts φ _{1R of} the individual data array component S, (t) with regard to the stimulation and the phase setting φ _{PSD> n} on the PSD (see also FIG. 2A and modulation spectra).

1 . 18 original data array S (t)

System response to a time-dependent (periodic) external fault. The original data array consists of I components S, (t), (where 1 <i <I), for example the intensities, absorbances or transmittances in the wavenumbers v ,, or. the intensities an interferogram at the corresponding mirror positions of an interferometer. Each of these components shows a time dependency characteristic of the system. In the case of periodic excitation, each array component can be described by a Fourier series in the steady state. The corresponding Fourier coefficients are experimentally accessible through a PSD or FFT.

1 . 19 External parameter, p, T, c, E, j, I,. . .

Thermodynamic parameters such as pressure p, temperature T, concentration c, electric field E, electric current density j, radiation flux density I, etc.

1 .20 period, X

Cycle time of the stimulation or the system response, τ = N-Δt, where Δt is the interval time, see FIG. 2A.

2.22 phase angle, φ "

Phase shift of the system response with the frequency nω with respect to the stimulation. Measure of the delay in the system response due to the dynamic properties of the periodically excited system (eg chemical reaction) with respect to the stimulation. N denotes the (nl) th overtone with the angular frequency nω at which the demodulation was carried out in the PSD (n = l: root). Each interpolation point is therefore assigned as many phase angles φ _n as there are demodulation frequencies.

1 .22 phase array, phase vector, phase angle array, φ "

Array of phase shifts (phase array difference) between

System response with the frequency nω and the stimulation. φ _n is composed of components I φ _nl and is a measure of the rule by the Dynami ^¬ properties of the excited system periodically (for example chemical reaction) induced delay of the system response. φ _n can vary component by component (φ _ni ) / n denotes the (nl) th overtone with the angular frequency nω at which the demodulation was carried out in the PSD (n = l: fundamental). Each interpolation point is therefore assigned as many phase vectors φ _n as there are demodulation frequencies. 1 . 23 phase resolution Δφ

Is determined by the number of support points N: Δφ = 360 '/ N. The phase resolution can be refined approximately arbitrarily by interpolation.

1 . 24 phase angle of the PSD, PSD phase angle, φ _{PSD, n}

Phase angle of the PSD switching function freely selectable by the experimenter based on the use of external stimulation. In general, the detection frequency (reference frequency) is n times the stimulation (fundamental) frequency (n = 1,2,3, .... N / 2). For n = 1 there is the fundamental and for n> 1 the overtones. The corresponding phase angles are, φ _{PSD; 1} , Φ _PSD . ₂ / Φ _PSD . ₃ '/ΦPSD.Π

(see Fig. 1). The frequency is limited by the sampling theorem n≤N / 2.

1 . 25 Quasi-simultaneous, guasi-simul tan

If a sequential acquisition of the components of an original data array (eg spectrum) takes place in a sampling time τ _{s that} is much shorter than the duration of a period τ, that is, τ _s <τ, the data acquisition of the components becomes quasi-identical timely and the data processing referred to as quasi-simultaneous. This means that the time that elapses between the acquisition of the first and the last (I-th) component of the array is neglected.

1 . 26 sampling time, sampling time, X,

Time to acquire a data array, e.g. of a spectrum.

2.27 stimulation period, x = 2π / ω

Time after which the periodic process repeats (Fig. 2A).

1 . 28 system

Object, e.g. chemical reaction mixture that can be influenced by changing external parameters.

1 . 29 Systematic word (periodic)

(Periodic) reaction of the system to a (periodic) change in an external parameter (see also multicomponent). 1 . 30 Stationary condition

If a system that is in thermodynamic equilibrium is exposed to a periodic external stimulation, after a settling time t _E (see there) it reaches a state that can be described component by component by Fourier series and is called stationary. See system response and Fig. IB, item 20.

2.32 Settling time t _E , relaxation times τ _f

If a system that is in thermodynamic equilibrium is exposed to a sudden change in an external parameter (see there), the system relaxes to the new state of equilibrium. This process is practically complete after a time that corresponds to three times the longest relaxation time of the system, t _E = 3τ _r ^BΛX _/ . τ _r ^πιax is determined by the kinetics of the stimulated process. The data acquisition in modulation experiments should only take place t _E time units after switching on the periodic, external stimulation (see stationary state).

2.33 symbols

The following symbols are defined as follows:

τ Length of a stimulation period τ _s Time for a scan τ _r Vector of the relaxation times of the stimulated system τ _r ^min Shortest relaxation time of the stimulated system τ _r ^max Longest relaxation time of the stimulated system

Ξ (t) stimulation (excitation) function (periodic)

Ξ_, _ax Maximum value of the stimulation (excitation) function exp exponential function to the base 10 i index of the i th component of the data array, for example the

Corresponds to wavenumber v _L or the i-th reference point (mirror position) of an imterferogram

I Number of vector components per data array k kth interval within a period

M Number of accumulations per support point in data acquisition solution n multiplier for the detection frequency n-ω; n = 1 gives the fundamental, n = 2 the 1st overtone, etc. Because of the sampling

Theorems must be n <N / 2. N Number of intervals into which a period is divided

(see FIG. 2B) P number of accumulated measurement cycles (periods), (see FIG. 2A) q qth measurement cycle, where 1 <q <P (see FIG. 2A) Ro reference spectrum (background spectrum), single-channel spectrum without Sample. Vector (array) with I components. t _E settling time of the periodic process, t _E = 3τ _r ^max

The invention is described in more detail below with reference to FIGS. 1-12. Show it:

Fig. IA Schematic representation of a known method for conventional modulation spectroscopy

IB Schematic representation of the method according to the invention for modulation spectroscopy

2A shows a section of the course of the harmonic stimulation and the ith component of the data array of the system response in the stationary state

2B stationary behavior of the i-th vector component of a data array after averaging over P periods

3 Examples of periodic functions for multifrequency stimulation (MFS)

4 time-resolved Fourier transform infrared (FTIR)

Intensity spectra as array support points (original data arrays)

Fig. 5 Principle of demodulation with a rectangular switching function (see equation (6)). Phase differences between signal and PSD switching function: A. φ-φ _PSD = 0 °, B: φ-φ _PSD = 180 °, C: φ-φ _PSD = 67. 5 °

Firg. 6 Non-modulated, time-independent portion of the support array

7 time-resolved absorption spectra of the sample. Periodic system response

8 shows the phase-resolved fundamental modulation spectra (n = 1), with N = 16 intervals and Δφ = 22.5 ° phase resolution

9 representation of the phase-resolved 1st harmonic modulation spectra (n = 2), with N = 16 intervals Δφ = 45 ° phase resolution

10 Device for the rapid acquisition of original data arrays in FT devices with the aid of a controlled frequency multiplication unit

11 device for carrying out an FTIR temperature (T) modulation experiment

Fig. 12 Device for performing an XRD moisture (c) modulation experiment

1A shows a schematic illustration of a known method for modulation spectroscopy. A one-component measuring device 1, which can be, for example, an FTIR spectrometer or a diode array spectrometer, contains a sample 2 or a system which is to be analyzed. For this purpose, the sample 2 is periodically excited or stimulated via a stimulation and reference unit 3 with the frequency ω, while the demodulation by the PSD at the frequencies ω, 2ω, ... nω,

(where n≤N / 2) and the corresponding phases φ _{PSD (1} , Φ _PSD , ₂ ^ Φ _PSD . _Γ , is controlled by reference unit 3. Periodic stimulation via an external parameter (p, T, c, E, j, I, etc.) triggers periodic reactions in the system Data acquisition unit 4, which is connected to the one-component measuring device 1 via lines 5, becomes a measuring point analog (Hs.H. Günthard, Modulation Spectroscopy, Ber. Bunsengesellschaft Phys.Chemie, 78, 1110-1115 (1974) or digital ( Ch. J. Manning and PR Griffiths, Step-Scanning Interferometer with Digital Signal processing, Applied Spectroscopy, 47, 1345-1349 (1993). The data acquisition unit 4 thus contains the time course of a one-component signal as a system response to a periodic, external one A typical course of the signal S = S (t) is indicated in Fig. 1A in a phase-sensitive detector (PSD) 6, which on the one hand via lines 7 to the data acquisition unit 4 and on the other hand via a line 8 is connected to the stimulation / reference unit 3, is phase-sensitive rectified in a known manner (lock-in amplifier, quadrature demodulator), the PSD via line 8 Reference signal S _ref with respect to the frequencies ω, 2ω, ..., nω and the phases φ _{PSD> 1} , ΦPSD.2 / ••• / Φ _P SD.Π is supplied.

In the analog or digital PSD 6, the system response S (t) is demodulated in accordance with the reference signal S _ref . The system is characterized by the amplitudes of the DC term, the fundamental (ω), the overtones (2ω ... nω) and the corresponding phase shifts with respect to the excitation (φ ,, φ ₂ , ..., φ _n ). With a constant phase angle φ _{PSD, n} / ^{m with} n = 1, 2, ..., ≤N / 2, between the stimulation / reference unit 3 and the switching function of the phase-sensitive detector 6, the signal S ( t) one data point per time of the time-independent background absorption A ₀ (DC component), the basic tone A _j and the overtones A ₂ , ..., A _{n are} recorded. The signals which are selectively rectified with respect to the frequencies ω, 2ω ... nω are fed via a line 9 to an output unit 10, in which they are represented, for example, as a function of the phase φ _{PSD n} , specifically as signals Aj (φ ₁ -φ _{PSD <} j), A ₂ (φ ₂ -φ _{PSD 2} ), ..., A _n (φ _n -φ _{PSD> n} ), where φ _{PSD, n} = nφ _{PΞD <1} = nφ _PSD .

In the output unit 10 thus are the outputs of the PSD function of the freely chosen on the reference unit phases ^¬ angle φ _PSD and the typical system, absolute Phasenverschiebun ^¬ gen φ, (root), φ _2, ..., φ _n (overtones) for Grouting. The corresponding components have the following designation: A _Q for the DC component, Aj for the fundamental with the frequency ω and A ₂ , ..-. , A _n for the overtones with the corresponding frequencies 2ω ..., nω.

IB shows a schematic representation of the method according to the invention for a simultaneous, multicomponent modulation spectroscopy. Sample 2, stimulation and reference unit 3 and line 8 with the reference signal S _ref correspond to FIG. 1A. A multicomponent measuring device 11 or array measuring device, which can be, for example, an FTIR spectrometer or a diode array spectrometer, contains a sample 2 or a system which is to be analyzed. For this purpose, the sample 2 is periodically excited or stimulated by means of a stimulation and reference unit 3 with the frequency ω. Such periodic stimulation via an external parameter (p, T, c, E, j, I, etc.) triggers periodic reactions in the system. In a data acquisition unit 14, which is connected to the multi-component measuring device 11 via lines 15, instead of the known point-by-point data acquisition, PSD and registration, entire data vectors S (t) are acquired with the vector components Si = Si (t) (i = 1, 2, ..., I). In the data acquisition unit 14, the time profile of a multi-component signal is thus recorded as a system response to a periodic, external stimulation in the stationary state. Each component S _x (t) corresponds to a signal according to FIG. 1A. The entirety of the components (i = 1, 2, ..., I) form an array S (t). They characterize the system by the amplitudes of the DC term, the fundamental (ω), the overtones (2ω ... nω) and the phase shifts with respect to the excitation (φ _u , φ ₂₁ , ..., φ _nι ).

A typical signal curve of the signals S = S (t), which are recorded in the data acquisition unit 14, is shown schematically in three dimensions in FIG. 1B.

In a digital multi-component phase-sensitive detector (PSD) 16, which is connected on the one hand via lines 17 to the data acquisition unit 14 and on the other hand via a line 8 to the stimulation and reference unit 3, the arrays are rectified in a phase-sensitive manner (Lock-m amplifier ), the reference signal S _{ref being} fed to the PSD 16 via the line 8. The digital multi-component phase-sensitive detector (PSD) is used for the simultaneous demodulation of the multi-component system response in accordance with the reference signal S _ref , or. of the algorithms described. In contrast to the one-component PSD, the multi-component PSD works with entire signal vectors and is therefore more efficient (by a factor of one) in the number of vector components.

If the phase angle φ _PSD between the stimulation / reference unit 3 and the phase-sensitive detector 16 is kept constant, a data vector of the time-independent background absorption (DC component) A _Q , the fundamental tone, is generated for each operation in the PSD of the signal S (t) A and the overtones A ₂ , ..., A "registered. The signals selectively rectified with respect to the frequencies ω, 2ω ... nω are fed via lines 19 to an output unit 20, in which they are represented as a function of the phases φ _PSDιn , specifically as signal _vectors A _Q , A- (φ _j —φ _{PSD # 1} ), A ₂ (φ ₂ -φ _{PSD 2} ),

•

The multi-component output signals of the PSD are thus in the output unit 20 as a function of the difference in the freely selectable phase angle φpso. _n ^and d of the system-typical phase vectors φ _n are available. The corresponding output signals are: A _Q for the DC components, A, for all fundamental tones (frequency ω) and A ₂ , ..., A ,, for all overtones (frequencies 2ω ..., nω).

Note: The processing of entire data vectors S (t) leads to a time saving in the amount of the number of components of the data vector, which proves to be particularly advantageous and which will be explained in more detail later.

2. Forms of stimulation

Due to the waveform of the stimulation, two types of stimulation can be differentiated. The harmonic stimulation, in which only one frequency is used and which is referred to below as monofrequency stimulation or single frequency stimulation (SFS), and the multifrequency Stimulation or Multiple Frequencies Stimulation (MFS), in which two or more frequencies are included in the excitation waveform. Since the excitation is periodic and can therefore be represented by a Fourier series, the MFS must be the fundamental tone frequency and the corresponding harmonic frequencies.

2.1. Monofrequency stimulation (SFS)

With the SFS, the excitation is purely harmonious, i.e. as a function of type

Ξ (t) = (Ξ, " _ax / 2) [1 + sin (ωt - φ _s )], (1)

where φ _{s represents} any phase angle (φ _s = 0 'results in a sinus stimulation and φ _s = 90 "in a cosine stimulation). From the analytical point of view, SFS has the advantage of simplicity and clarity ¬ tion frequency arise, this is a clear indication that nonlinear processes are running in the system, which in turn allows important conclusions to be drawn about the dynamics (kinetics) of the stimulated process.

≤ ω ≤ —— (2)

2τ ^B ; now

X r

However, SFS has the disadvantage that only information about a frequency ω and its overtones nω with n = 2.3,..., N / 2 are obtained per measurement. This means a relatively large expenditure of time, since measurements at as many frequencies as possible are very helpful or even necessary for a kinetic analysis of the stimulated process (Gunthard (1974)). The range of suitable stimulation ^frequencies is described by equation (2), where τ _r ^min and τ _r ^{max mean} the shortest and the longest relaxation time of the system.

2.2. Multi-frequency stimulation (MFS)

MFS always applies when a periodic function not reι ^r is harmonious. In addition to the fundamental frequency ω there is at least one frequency (overtone) nω with n> 2. As a consequence, the system is stimulated not only with the basic frequency, but also with multiples thereof. The system responses are delayed in accordance with the characteristic relaxation times τ _r ^raln <τ _r <τ _r ^max by the phase angle φ _r and in parallel the corresponding amplitudes are damped. The frequency dependency of both effects depends significantly on the reaction scheme on which the stimulated process is based. MFS can therefore simultaneously provide information about the periodic behavior of a system at several frequencies, which means that the measurement effort involved in elucidating a reaction mechanism can be significantly reduced.

2.2.1. Experimental MFS

In certain cases, the generation of a sinusoidal stimulation waveform can be so complex that a harmonic excitation (see above) is intentionally dispensed with. This inevitably leads to an MFS. In this case it will be necessary to carry out a Fourier analysis of the excitation waveform in order to be able to carry out significant kinetic analyzes.

2.2.2. Targeted MFS

Since MFS can be used for simultaneous excitation at different frequencies, this technology has considerable time-saving potential. In the following, stimulation waveforms are discussed, with the help of which targeted multi-frequency information about fundamental and overtones of the system response, i.e. can be collected through the mechanism of the stimulated process.

2.2.2.1 Symmetrical Rectangular Stimulation: Odd Multiples of the Fundamental Tone Frequency This waveform occurs when the stimulation starts ^suddenly and is switched off again ^suddenly after half a period. This waveform with maximum value Ξ „, _ax can be represented by the following ^¬ de Fourier series (3): Ξ (t) = <Ξ ,. _ax / 2) {l + 4 / π ∑ l / (2σ + l) - sin [(2σ + l) (ωt-φ.)]} (3) σ-o

φ _s = 0 means that the switch-on process takes place at time t = 0, as shown in FIG. 3, curve 24. If φ _s > 0, stimulation is delayed by the angle φ _s . The function (3) results in a simultaneous excitation with the frequencies ω, 3ω, 5ü), ie with the odd multiples of the

Fundamental frequency. As a consequence, the linear behavior of the system can be determined simultaneously at these frequencies. At the same time, it can be clearly clarified whether the system generates harmonics of the frequencies 2ω, 4ω, 6ω, ... through non-linearities (e.g. through reactions that are not of the 1st order).

2.2.2.2 Positive half-wave of a sinus stimulation: Even multiples of the fundamental tone frequency

This waveform consists of the positive half wave of a sine function. The second half wave (half period) is zero. The Fourier series (4) represents:

Ξ (t) = Ξ _^ .fl / π + 1/2 sin (ωt-φ.) +

- 2 / π ∑ l / [(2σ) ² -l] cos [2σ (ωt-φ _s )]) (4)

Oil

For φ _s = 0, stimulation begins at the beginning of the period, ie zero in the second half of the period, as shown in FIG. 3, curve 25. Function (4) gives simultaneous excitation with the frequencies ω, 2ω, 4ω, ie with the even multiples of the fundamental tone frequency. As a consequence, the linear behavior of the system can be examined simultaneously at these frequencies. At the same time, it can be clearly clarified whether the system generates harmonics of the frequencies 3ω, 5ω, 7ω through nonlinearities (eg through reactions that are not of the 1st order).

2.2.2.3 Non-symmetrical rectangular stimulation: broadband stimulation with possible amplitude adjustment

3, curve 26 represents a non-symmetrical rectangular stimulation that is centered in the middle of the period. It is represented by the Fourier series (5). Ξ (t) = Ξ _max {Δτ / τ + 2 / π ∑ {(-l) ^σ / σ} • sin (σπΔτ / τ) • sin (σωt)} (5) σ-l

The function (5) contains all overtones of the stimulation frequency ω, provided the scaling factor is sin (σπΔτ / τ) ≠ O, i.e. Δτ / τ ≠ n, with n = l, 2,3, which e.g. in the case of Δτ = τ / 2 for all straight lines

Values of σ applies. In this case, symmetrical rectangular stimulation occurs. Equation (5) merges into equation (3) with φ _s = - π / 2. In the case of non-symmetrical excitation, ie Δτ ≠ τ / 2, the amplitude pattern of the excitation frequencies can be influenced in a targeted manner by choosing the stimulation duration Δτ. As an example, the 1st overtone ((5 = 2) of (5) should reach maximum intensity. This applies if the scaling factor sin (2πΔτ / τ) = 1, ie Δτ / τ = l / 4. From equation ( 5) it follows that with this choice the amplitude of the fundamental tone is 0.200-Ξ ^, while the maximized 1st overtone has an amplitude of O.Sδδ-Ξ.- _a ,,.

2.2.2.4 General, periodic forms of stimulation

There are any number of periodic forms of suggestion, of which, however, only a limited number can be implemented with reasonable, experimental effort. The largest selection is available for electrical stimulation. Example 2.2.2.3 is intended to show that excitation waveforms can be assembled in order to generate certain frequency and amplitude patterns. The analysis of the response of a system to ready-made forms of stimulation can contribute significantly to elucidating the mechanism of the stimulated response.

2.2.2.5 Multi-parameter stimulation

By simultaneous stimulation of the system by changing two or more external parameters with waveforms, as described in sections 2.2.2.1 to 2.2.2.4, but with a different frequency for each parameter, followed by appropriate signal processing and PSD, respectively. FFT, more detailed information about the system can be obtained. 3. Data acquisition

Frg. 2A shows a section of a signal curve of the system response 22 of the i-th component of the data array with harmonic stimulation 21.

In the case of harmonic stimulation 21, the reference zero crossing can be recognized at the beginning of the period.

The system response 22 here consists of the i-th component of the data array S ^ S.ft) with fundamental (ω) and 1st overtone (2ω) components. The sluggishness of the system in response to the external stimulation leads to phase shifts φ _n (ω) and φ ₁₂ (2ω).

The stimulation period τ = 2π / ω was divided into N equidistant time intervals Δt, which led to N + l support points. Because of the periodicity of the process, the first and the last support point (N + 1) are the same as soon as the steady state is reached. At least one measurement data array is assigned to each interval (1 <k <N), representative of the state of the system at the time t = qτ + (k- l) Δt assigned to the corresponding interval. The acquisition of the data arrays begins at the beginning of each new interval. After the first period in the steady state has elapsed, this process can be repeated for any number of P periods (1 <q <P), with data arrays being averaged during or after the data acquisition over each M within the interval Δt, resulting in the corresponding interpolation point array arises. The shortest period is thus determined by the sampling time, or sampling time, τ _s , for acquiring a data array and by the number of intervals, N. It is τ _mιn = N- τ _s ; however, the duration of a period is not restricted upwards, since τ = NM-τ _s with M = 1, 2, ... can be selected.

3.1. Version with internal trigger

3.1.1. Scanning time τ _c as a unit of time

Each time a period with the duration τ has elapsed, the measuring apparatus generates a control signal that enables the next measuring cycle to be stimulated without delay. By issuing a second Control signal, respectively. by switching off the first, after the time Δτ (see FIG. 3), the stimulation (disturbance) of the system is canceled again. In general, this step takes place after half a period, ie Δτ = τ / 2. These control signals must be available without delay so that the system remains stationary. The period is minimal τ _min = N- τ _s , but it can be extended as desired by adding a number of M data arrays per support point. This gives for the duration of an interval Δt = M- τ _s , resp. for the period τ = N * Δt = M- τ _mln = MN-τ _s . Since the time resolution corresponds to the interval duration Δt in this method, the corresponding phase resolution results in Δφ = 360 ° / N. The mean time assigned to the kth interpolation point in the qth measurement cycle is thus t _k = q- τ + (kl) -Δt with 1 <q <P and 1 <k <N. This approximation can produce a very good signal / Noise ratio can be achieved with a minimum of time. This procedure is always permissible if the interval duration is small compared to the shortest relaxation time of the stimulated system: Δt <τ _r ^min . To improve the signal-to-noise ratio, averaging is generally carried out over P measurement cycles (periods) in the steady state (FIG. 2B).

3.1.2. Internal timer

A timer of the data acquisition system determines the interval duration Δt and triggers the external stimulation process after each period τ = N-Δt. As described in section 3.1.1, the stimulation is canceled after Δτ. If the steady state is reached, a new measuring cycle is stimulated at times (ql) -NΔt, with (1 <q <P). At the beginning of each interval, the data acquisition for the corresponding support point is started (FIG. 2A), the number M of the additions having to be selected such that Δt> M-τ _s . With this method, the phase resolution can be improved to the value Δφ = (360 ° / N) ^■ (M-τ _s / Δt) compared to that described in section 1.1, however at the expense of the signal / noise ratio with the same measurement time. To improve the signal-to-noise ratio, averaging is generally carried out over P measurement cycles in the steady state (FIG. 2B). 3.1.3. Time between two successive sampling points of the interferogram as a time measurement unit (rapid scan method) This experimentally more complex method is used when the kinetics of the stimulated process is too fast to be detected with the time resolution τ _{s of} the scanner (see 3.1 .1.) .

In FT optical spectroscopy, i.a. for the exact determination of the relative position of the movable mirror, the interferogram, item 50, FIG. 10, of a monochromatic light source with the wavelength λ is used. This is recorded parallel to the spectroscopic interferogram (original data array) in the same interferometer and corresponds to a cosine wave with zero crossings at the mirror positions x = (2m + l) λ / 4, where m = 0, 1, 2, .... These zero crossings are used as sampling points for the interferogram support points.

If these zeros are also used as stimulation points of the system, in that the signal is fed via a line, item 51, to the stimulation and reference unit, item 3, then I stimulations per scan take place at I interpolation points. Depending on the selected scan speed, the time resolution can be improved by several powers of ten. In this method, the period τ corresponds to the time between two successive sampling or stimulation points. In order to be able to carry out a simultaneous, digital PSD, the period must still be divided into N intervals of the duration Δt = τ / N, the intensity of the light used for the spectroscopy being measured at each interpolation point S _xl , and the i-th interferogram point is assigned in the kth support point. 2: i = 1, 2, ..., I and k = 1, 2, ..., N, as in the other methods already described. This means that after each scan N, exactly time-resolved interferograms S _{k are available} as original data arrays, which can be added P times to improve the signal-to-noise ratio.

Technically, this process can be implemented using a PLL-controlled oscillator, item 52. This control unit receives a signal, item 51, with the frequency ω ₀ and generates the N times the frequency Ncύ _t ,. Cύ _b can be, for example, the above-mentioned cosine-shaped interferogram of the monochromatic light source, the zero crossings of which determine the time when the sample is stimulated. With the aid of the zero crossings of the output signal of the frequency multiplication unit (frequency NCύ _f c), the time-resolved data acquisition of the N interferogram points S _ki to the respective ith reference points is triggered via a line, item 53, see FIG. 10.

3.2. Version with external trigger

The interval duration Δt is determined by an external timer. After each period τ = N-Δt, the external stimulation process is triggered again and canceled after Δτ. The support points are recorded as in section 3.1.2. described.

4. Data processing: phase sensitive detection (PSD)

2B schematically shows the signal 23 averaged over P periods, which represents the periodic behavior of the i-th vector component of a data array (e.g. data point of an interferogram, or intensity, absorbance, or transmittance of a spectrum at the wave number vj at the N support points ( here N = 16). This averaging process can improve the signal / noise ratio by a factor of 1 / "VP.

The averaged l-th vector component Si is given by (N + l) -Stutz ^¬ filters (here N = 16) at a distance .DELTA.t (see Fig. 2B) stored digitally, wherein the support points k = l and k = are identical 17th

4 shows, as an exemplary embodiment, the support points (data arrays) 27 of a measurement cycle averaged according to FIG. 2B with N = 16 intervals, recorded as Fourier transform infrared (FTIR) intensity spectra (single-channel spectra).

Each array S _k consists of 1 = 3450 components corresponding to the wave number range 600 cm ^"1 to 4000 ^{cm" 1} cm with about 1 ^"distance. At certain, the system characteristic waves for pay ^¬ indicate the corresponding components of S _kl the in Figure 2B behavior shown schematically, but which is still masked by the dominating, non-modulated background 42 (FIG. 6). In each interval k to Δt = 16.5 s, M = 12 spectra were added and assigned to the k-th support point. N = 16 intervals form a period with the duration τ = 4.4 min. The averaging was carried out over P = 10 periods in the steady state. There is therefore a phase shift of Δφ = 22.5 ° between the support points. The numbering (k) of the spectra 27 (FIG. 4) gives the assignment to the corresponding reference points, the 1st and 17th spectrum being identical because of the periodicity of the process. The phase shift of the kth interpolation point with respect to the 1st interpolation point (beginning of the measuring cycle) is thus φ _0k = (kl) -Δφ.

After a transformation, which will be described in detail later, a phase-sensitive detection (PSD) can now be used to split each component Si (t), with i = 1,2, ..., I, of the original data array S (t) into a constant one , non-modulated part of the data arrays A _0i (mean over a period), as well as in the corresponding fundamental A _u (φ _n -φ _PSDj ,), with angular frequency ω, and the overtones A ^ (φ _nl -φ _{PSD , n} ) with the angular frequencies n-ω, where n = 2,3, ..., N / 2. The corresponding I components for the DC term or for the frequencies ω, 2ω, ..., nω then result in the output signals A "(φ _n -φ _{PSD n} ) of the PSD shown in FIG. IB. Spectroscopic experiments still require certain data transformations, which will be discussed later. In the following, demodulation methods (switching functions) are discussed, with the aid of which original data arrays, as shown in FIG. 4, can be broken down into the output signals 20 of the PSD just mentioned.

4.1. Demodulation with a rectangular function

4.1.1. Root note

The N support points (arrays) (Fig. 2B) are with a rectangular switching function (6)

R (t) 4 / π ∑ (i / 2σ + i) sin [(2σ + l) (ωt-φ _{PSD / 1} )]; 6) σ = 0 multiplied. R (t) takes the value +1 during one half-period and -1 during the following half-period. After that, the mean must be formed over a full period. This switching function represents a Fourier series, the result still having to be scaled by the factor π / 2 in order to ensure a 1: 1 amplitude transmission. φ _PSDιl is the phase _setting that can be selected by the experimenter at the reference unit of the PSD. φ _{PSD / 1} = 0 means that the switching function R (t) runs in phase with the stimulation; φ _PSD, ι ≠ 0 results in a corresponding shift of the PSD switching point with respect to the start of stimulation. Due to the data acquisition method described in Section 2, φ _PSD-1 can assume the discrete values φ _{PSD / 1} = (kl) -Δφ, where 1 <k ≤ N.

If the phase resolution .DELTA..phi.

4.1.2. Overtones

The overtones with angular frequencies 2ω, 3ω,, nω result analogously, the corresponding periods being τ / 2, τ / 3,, τ / n, with n <N / 2 (sampling theorem). The original data can therefore be evaluated 2, 3, and n times to ensure that

Improve signal / noise ratio. The phase resolution is reduced accordingly, unless new interpolation points are approximated. The number of interpolation points N should therefore be chosen such that N / 2, N / 3,, N / n are integers, which can also be achieved approximately by interpolation (preferably trigonometric).

4.1.3. Meaning of rectangle demodulation

That in Sections 4.1.1. and 4.1.2. The demodulation method described with square-wave switching functions has the great advantage of simplicity, since it mainly requires addition and subtraction operations. It is therefore usually possible to use commercial array acquisition systems (eg diode array spectrometers, CCD methods, array detectors in X-ray diffractometers), as well as with, without additional device modification. fast registering devices (quasi-array detection, for example conventional Fourier transform (FT) spectrometer, rapid-scan FT spectrometer and rapid-scan dispersion spectrometer) are used, since the software provided by the device manufacturer is macro-programmed in the is generally sufficient to perform rectangular demodulations.

A disadvantage of square wave demodulation is, however, that due to the odd overtones present in the switching function R (t) (see Eq. (6)), corresponding overtones of the signal ((2σ + l) ω), which are caused by the system's orharmonicity caused by anharmonic stimulation (see also MFS) are demodulated into the fundamental tone array A, (φj-φp _sD i). However, this disadvantage is hardly significant if no exact analysis of the dynamics (kinetics) of the stimulated process is required, ie if an exact phase / amplitude frequency analysis does not have to be carried out. Because of the general importance and the simple feasibility, the demodulation of the fundamental tone of a periodic signal 28 (FIG. 4) will be discussed in more detail here. The rectangular switching function 29 has its positive edge (start of demodulation) at k = 3, ie that the phase angle Φ _PSD . _I ⁼ 67.5 '. Since the demodulation is carried out by multiplying the signal array 28 by the switching function 29, the signal 28 between the support points k = 3 and k = ll remains unchanged (multiplication by +1), while in the remaining half-period (k = ll to k = 3) the nodes are inverted (multiplication by -1). The DC component 31 of the demodulated signal 32 becomes maximally positive if the freely selectable PSD phase angle φ _{PSD, n>} ^and the phase shift φ _n caused by the system dynamics are the same size, 30, FIG. 5A, ie φ _n -φ _PSD . _n ⁼ 0 ^* • Accordingly, the DC component 34 of the demodulated signal 35 becomes maximally negative if φ _n -φ _PSD-n = ± 180 °, ie if there is a half phase shift 33 between the signal and demodulation, see FIG. 5B. The DC component 37 of the demodulated signal 38 disappears if there is a phase difference of φ _n -φ _{PSD, n} ^{= ±} 90 ° between the signal 28 and the switching function 36, FIG. 5C. Modulation signals can therefore have a positive or negative sign. In addition, they can disappear completely at a phase difference of ± 90 °. 5D shows the more general case φ _PSD, - ⁼ 0 ° / ie stimulation and PSD switching function start at the same time, which in concrete example 39 leads to a phase difference φ _n -φ _PSD, ι ⁼ ~ 67-.5 ". In this case, the demodulated signal 41 assumes an intermediate amplitude A _n 40, A _n ^max > -A _n > A _n ^min . The phase can now be changed step by step, in the specific case here with a resolution of Δφ = 22.5 ^* (360 ^* / N), resulting in phase-resolved modulation spectra (see FIG. 8 and 9) This signal conversion produces a pulsating DC voltage 32, 35, 41, from which the DC voltage component 31, 34, 40 is averaged out by integration over a whole period. The demodulation of the nth overtone is carried out analogously Signal 28 is multiplied by a square-wave switching function of frequency nω, since the period of the switching function is n times shorter than the period of the signal, the phase resolution is now Δφ = n-22.5 ^* polation possible.

4.2. Single Frequency Demodulation (SFD)

4.2.1. Root note

The N support points (arrays) (Fig. 2B) are component by component with the harmonic switching function (7)

R (t) = sin (ωt - φ _{PSD (1} ) (7)

multiplied. Applied to the kth interpolation point array, the multiplier (8) results for each component S _{kl of} the interpolation point array S _k

R _κ = sm [(kl) -Δφ - φ _{PSD> 1} ] = sιn [{(k-1) ^• 360 ° / N} - φp _SD .,] (8)

As in the case of rectangular demodulation, the phase angle of the PSD assumes the discrete values φ _{PSD / 1} = (kl) -360 ° / N, that is to say the switching function is also averaged over the averaged here

Reference point arrays with k = 1,2,, N are pushed (see Fig. 2B and Fig. 5A-5D) to the phase-resolved fundamental tone arrays (e.g. To _obtain modulation spectra) A- (φi-φp _sD,, ). The averaging takes place analogously to the rectangular demodulation, with a harmonic factor of 2 having to be taken into account in the harmonic demodulation so that the signal amplitudes are transmitted 1: 1.

4.2.2. Overtones

The overtones with the angular frequencies 2ω, 3ω,, nω result analogously, the corresponding periods being τ / 2, τ / 3,, τ / n and can therefore be evaluated 2, 3,, n times in the original data array. However, the phase resolution is reduced accordingly. The number of interpolation points N should therefore be chosen such that N / 2, N / 3,, N / n are integers, which is also approximate

Interpolation (preferably trigonometric) can be achieved.

4.2.3. Importance of harmonic demodulation

If precise information about the amplitudes and phases of a system response is required, only harmonic demodulation (SFD) can be considered, since any other periodic demodulation waveform that contains harmonics also converts all corresponding harmonic frequencies of the system response into a DC voltage would. The proportion of the fundamental can then no longer be distinguished directly from the harmonic components. This is particularly true if special forms of suggestion, as described in Section 2.2.2. Multi-frequency stimulation (MFS) have been described. SFD enables a clear Fourier analysis of the system response. By comparing the Fourier coefficients of the system response with those of the stimulation, an analysis of the behavior of the system with respect to linear and non-linear signal transfer is possible. Particularly noteworthy is the high information content per time expenditure when combining MFS / SFD with digital PSD. Since the time-resolved interpolation point arrays are stored digitally as output data for the PSD, an SFD can be carried out on this data material at the fundamental frequency ω and the overtones nω with any phase resolution. The number of overtones that can be evaluated is only limited by the sampling theorem n≤N / 2. The Experimental phase resolution is also determined by the number of intervals N per period τ, however, it can be refined by interpolation as desired (see section 3, data acquisition)

4.3. Fourier transform demodulation

Fast Fourier Transformation (FFT), or modified algorithms applied to the N reference point arrays of a wave train also lead to the desired results, namely DC term, fundamental and overtone arrays (FIG. IB), whereby, as in the PSD method, the maximum evaluable number of overtones is limited by the sampling theorem n <N / 2. The disadvantage of the FFT compared to rectangular or harmonic demodulation is the greater computing effort and memory requirement; Boundary conditions that are not always met by commercial device computers.

4.4. Multi-parameter demodulation

All in sections 4.1. until 4.3. The demodulation methods described can also be used for multiparameter stimulation. Double stimulation of the system by simultaneous external change of any two parameters, generally referred to as P _x and P _{2, is} considered as an example. The different frequencies were used to differentiate the corresponding system responses

used. Parts of the system that only respond to parameter P ₁ react with the frequency

and their overtones, while those that only respond to P _{2 have} only the frequency ω in the system response and their overtones. System parts that respond to both parameters react as usual with double modulation with the sums and differences of the corresponding frequencies, namely:

and the same applies to the overtones and to all other possible linear combinations of the two fundamental frequencies. In order to obtain output signals corresponding to FIG. 1A, item 10, or FIG. IB, item 20, double demodulation with a switching function of the frequency (ύ _} and ω _{2 is} required. 5. Application example in FTIR modulation spectroscopy

5th-1st Node arrays

The intensity spectra 27 (FIG. 4), which lie between an initial and end wave number, v _{lt and} . v _τ , recorded in a time τ _s per scan and possibly accumulated M times, correspond to the support points (data arrays) according to FIGS. 2A and 2B. The number of array components is determined by the measuring range and the spectral resolution of the spectrometer. 4 is the result of a time-resolved data acquisition with N = 16 intervals

(Δφ = 22.5 ° phase resolution). The individual interpolation point spectra each correspond to the mean value of M = 12 accumulated individual spectra, according to FIG. 2B. They were created in response to an external temperature stimulation in response to a thin layer of the polypeptide poly-L-lysine HBr spread on an ATR plate. The data was recorded in the infrared range using a commercial FTIR spectrometer. To stimulate the sample, the temperature was modulated approximately sinusoidally according to equation (1) with the amplitude Ξ _max / 2 = ΔT / 2 = 2 ° C around the mean value T = 28 ° C, which results in a periodic T modulation between 26 ^* C and 30 ^* C leads. The average relative humidity was 80%. P = 5 periods were averaged (see Figs. 2A and 2B). The stimulation period was x = 4.4 min. Each of the 16 averaged support spectra

(Arrays) S _k consisted of I = 3450 array components (approx. 1 data point per cm ^"1 ).

5.2. Forming of the support points in spectroscopic experiments

At this point it should be pointed out that in the following the

Expression 'exp' is used for the base 10 exponential function.

According to the Lambert-Beer law, the i-th component of the k-th support point can be described as follows:

S _k i = R _c i "exp (-εiC _k d) (9) R _0i is the intensity 42 (FIG. 6) at the point v _if without a sample in the beam _path (reference intensity, background). e is the molar absorption coefficient for the wave number v _x and c _{k is} the concentration to the k-th spectrum of interpolation points, ie at the time (k-1) ^• Δt.

Note: For the sake of simplicity, the Lambert-Beer law was formulated in equation (9) and in the following as if there were only one stimulable species in the system, the concentration of which is c (t) and which is spectroscopic due to the molar absorption coefficient e, an array with I components. In the case of ρ reacting species, which generally applies, c and ε have to pass through the sums

+ c ₂ + .... + c ₀ , resp. ε = ∑ε _p = ε ₁ + ε ₂ + .... ε _{0 can be} replaced. This simplification, however, has no restrictive effect on the scope of the invention, but only enables the mathematical formulations to be made simpler and more transparent. In the case of a spectroscopic modulation experiment, the question generally arises as to the behavior of the concentrations of the reaction participants, ie according to the respective phase and amplitude, or. according to the Fourier coefficients used to describe the behavior over time. The Lambert-Beer law (9) shows that the relationship between the measured quantity S _ki and the concentration c _{k is} not linear. If the concentration c (t) changes periodically due to external stimulation, S (t) is also a periodic function and can therefore be described with a Fourier series. The corresponding Fourierko ^¬ modified Bessel functions of entire order are effective. These can be calculated from the Fourier coefficients of c (t) using the Jacobi-Anger theorem (E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves, Dover Publication

Inc., New York (1945) S). Conversely, from the experimentally accessible Fourier coefficients of S (t), the Fourier coefficients of c (t) sought can be calculated. However, a conversion of the measurement data according to Jacobi-Anger is essential, which would not always be possible with the hardware and software equipment of the computers of commercial spectrometers. However, an exception is the special case that the Amplitudes of the concentration modulation are very small, which enables the linearization of equation (9)

Since this case does not generally apply, it is advisable in spectroscopic applications to convert the intensity reference point spectra S _k before the PSD into absorption spectra a _{k in} order to achieve a direct proportionality to the concentrations. It follows for the i-th component of the k-th support point

-log [S _ki ] = -log.RoJ + EiC _k d = -log.RoJ + a _ki (11)

a _kl is the i-th component of the k-th absorption support array. For the entire array, this results from GI. (11)

a _κ = -log [T _k ] = -log [S _k / Ro] = εc _k d (12)

Instead of the original data arrays (intensity spectra) S _k (cf. Fig. 4 and GI. (9)) now occur according to GI. (12) Absorbance spectra a _k as support point arrays. These only depend on the condition of the sample. In the limit case c _k = 0, the absorbance a _k disappears accordingly. The index k stands for the number of the kth support point, ie k = 1, 2, ... N + 1. Since concentrations are fundamentally always positive, in a modulation experiment the concentration c _{k of} a reaction participant at the support point k always has to move periodically around the mean value c ₀ formed over an entire period, which no longer depends on the time. For this reason, c _{k can be broken down} into a time-independent (DC term) and a time-dependent (modulated) concentration component:

c _k = c ₀ + c _k ^mod (13)

From (12) and (13) there follows a DC and a modulated component for the corresponding absorbance: a _k = ε - d (c ₀ + c _k ^mod ) = a ₀ + a _k ^mod (14)

The array of DC components 44, a ₀ , is shown in FIG. 6, while the 17 (N + l) time-resolved absorption spectra a _k ^{mod are shown} in FIG. 7. The index k stands for the kth support point. The spectrum 45 in FIG. 7 forms, for example, the 7th support point of the period.

Application of the Lambert-Beer's law in the form (9), respectively. (12) on equation (14) yields for the transmittance array:

T _k = S _k / Ro = exp (-a _k ) = exp (-εc ₀ d) ^• exp (-εc _k ^mod d) (15)

Dissolution of GI. (15) after the support point array 27 (FIG. 4) yields:

S _k = RQ- exp (-εc ₀ d) ^• exp (-εc _k ^mod d) = S ₀ - exp (-εc _k ^mod d) (16)

= S ₀ -exp (-a _k ^mod ) with

S ₀ = Ro-exp (-εc ₀ d) = Ro-exp (-a ₀ ) (17)

as a time-independent, non-modulated portion 43 (FIG. 6) of all interpolation point arrays k = 1, 2, .... N (FIG. 4, S _k ). R _Q is the time-independent intensity spectrum 42 (FIG. 6), which arises in a measurement without a sample. R _Q is generally referred to as the reference spectrum or background spectrum. From (17) the following results for the mean value of the absorbance 44 of the stimulated sample (FIG. 6):

a ₀ = -log (T ₀ ) = -log (S ₀ / Ro) (18)

Division of (16) by (17) gives the kth time-resolved transmittance support array T _k ^mod . The corresponding absorbance support point array, which is directly proportional to the concentration, is obtained by logarithmization (for example 45, FIG. 7, k = 7).

a _k ^mod = -log T _k ^mod = ε c _x ^mod d (1 9)

These time-resolved spectra contain the relevant information about the dynamics of those generated by external stimulation Concentration modulation c _k ^mod .

The above-described way of determining a _k ^mod using equations (12) - (19) shows the formal relationship between the relevant variables of a spectroscopic modulation experiment. In the specific case, however, a _k ^{mod can be determined} more directly from the intensity reference point arrays S _k (original data arrays) 27 (FIG. 4). Since S _{0 is} the mean of S _k over a period according to (16) and (17), it follows from (19):

_a * ₀ ^d ₌ _ι ₀ g (s _k / js _k dt) = -log (S _k / S ₀ ) (20)

period

The experimentally relevant equation (20) requires numerical integration, which, however, is generally based on elementary arithmetic operations. can be carried out with the standard software of commercial spectrometer computers without any problems.

5.3. Phase sensitive detection (PSD)

The reference point arrays used for the PSD are shown in FIG. 7. They were calculated according to equation (20). The

Demodulation was carried out using a rectangular switching function 29 (FIG.

5) as described by equation (6). The fundamental (ω) and the first overtone (2ω) were detected. The

Procedure is shown schematically in Fig. 5A-5D and in

Section 4.1. described.

In the specific case (Fig. 4, 6-9), the number of support points

N = 16, which corresponds to a phase resolution Δφ = 22.5 °.

The signal arrays appearing in the output unit 20, FIG. 1B are determined as follows:

The DC term A _Q results from averaging (eg through numerical integration and scaling) via the absorbance

Support points a _{k of} an entire period. According to equation (14), this value corresponds to the DC term a _{0 of} the absorbance of the sample

44, Fig. 6.

The basic tone array Aj (φ, -φ _PSD, ι) is obtained by forming the

Average of the product S (t) -R (t) over a period and scaling so that a 1: 1 absorbance-amplitude ratio arises (see section 4.1. and Fig. 5). Starting from the absorption support point spectra a _k ^mod (FIG. 7), the following calculation scheme results for Ajtφi-0 ^* ) 46, FIG. 8: averaging over the support point arrays k = 1-9 from FIG. 7 and subtraction of the corresponding mean value over the support point arrays 9-17, the first and the 17th array being identical because of the periodicity. The fundamental tone array A ₁ (φ ₁ -22.5 ^* ) 47, Fig. 8, to the PSD phase angle

Φ _PSD . _I ⁼ 22.5 ° results from analog processing of the support point arrays 2 to 10, respectively. 10 to 2 of FIG. 7. In general, the fundamental tone arrays A, (φ _x - (k-1) ^• Δφ) for the phases φ _PSD, ι = (kl) -Δφ are obtained by averaging over the support point arrays k to ( k + N / 2) and by subtracting the correspondingly formed average over the support point arrays (k + N / 2) to (k + N). The periodicity of the steady-state process has to be taken into account, i.e. as soon as a node number exceeds the value of N, subtract N from it. In addition, the procedure only has to be carried out up to k = (N / 2) -l, since also due to the periodicity Aι (φ, -0 °)

applies, ie after half a period the function values repeat with an opposite sign, cf. also FIG. 5B. The same applies to the rectangular demodulation of the 1st overtone. The fundamental frequency of R (t) GI. (6) is now 2ω, i.e. the experimental phase change from support point to support point is now Δφ = 45 '. The demodulation in the PSD phase φ _{PSD, 2} = 2- Φ _{PSD, I} ⁼ 0 ^* is carried out by adding the array mean values over the quarter periods formed from the support points k = 1 to k = 5, respectively. k = 9 to k = 13. From this array, the mean values over the ranges k = 5 to k = 9, respectively. k = 13 to k = 17 subtracted, where k = 17 again corresponds to k = l. The result, A ₂ (φ ₂ -0 °), corresponds to modulation spectrum 48 in FIG. 9. The modulation spectrum 49 in FIG. 9, A ₂ (φ ₂ -45 °), is obtained by shifting the rectangular switching function by one Interval length, see also Fig. 5. In general, the phase-resolved spectra for the first overtone, A ₂ (φ ₂ -2 (k-1) ^• Δφ), for the phases φ _{PSD, 2} = 2 (kl) -Δφ are obtained Addition of the mean values, formed via the support point arrays K Ois (k + N / 4), respectively. (k + N / 2) to (k + 3N / 4), and by subtracting the corresponding mean values from the support plate arrays (k + N / 4) to (k + N / 2) and (k + 3N / 4) to (k + N). The periodicity of the steady-state process must be taken into account, ie as soon as a node number exceeds N, the value N must be subtracted from it. In addition, the procedure only has to be carried out up to k = (N / 2) -l, since also because of the periodicity A ₂ (φ ₂ -0 °) = - A ₂ (φ ₂ -180 °), ie after one half a period, the function values repeat with the opposite sign, see FIG. 5B. Since there was no 2ω component in the temperature stimulation, the existence of the first overtone in the modulation spectra clearly indicates non-linearities of the system. In the specific case, this generally means that the reactions triggered by the change in temperature include those that do not have first-order kinetics. These include, for example, 2nd order chemical reactions, but also more complex phenomena such as cooperative and allosteric effects. The latter play an important role in changing the conformation of macromolecules.

5.4. Overview of the relevant evaluation steps in the simultaneous, digital phase-sensitive detection of the time-resolved original data arrays S _k based on the spectroscopic application example

• RQ: Before the modulation experiment begins, the reference spectrum Ro (FIG. 6, item 42) is measured.

S ₀ : Averaging (by numerical integration using the Simpson method over the full period, with scaling) of the function S (t) (FIG. IB, item 14), which is carried out by supporting point arrays S _k, (FIG. 4 , Item 27) is determined. The interpolation point arrays S _{k were} treated like individual points. This resulted in the time-independent spectrum S ₀ (FIG. 6, item 47).

• a ₀ : The DC term a ₀ , (Fig. 6, item 44) was calculated according to equation (18).

A _k ^m ° ^{d: The} modulated, time-dependent portion (FIG. 7, item 45) of the absorption support points of the sample was calculated in accordance with equation (20).

• A _Q : The DC term of the absorption spectrum of the system response corresponds to a _c (see above). Aj: phase-resolved fundamental spectra which correspond to the fundamental part of the modulated absorbance of the system response. They were determined by phase-sensitive detection at the fundamental tone frequency ω using a square-wave switching function (6). See section 5.3. above.

A ₂ : Phase-resolved 1st harmonic spectra which correspond to the proportion of the first harmonic of the modulated absorbance of the system response. They were determined by phase-sensitive detection at the frequency 2ω (1st overtone) with a square-wave switching function (6). See section 5.3. above.

5.5. Evaluation of the data at the level of the interferograms

Fourier transform devices deliver interferograms as original data arrays S _k . With a Fourier transformation, these must be converted into interpretable spectra.

In the application example described in 5.1. until 5.4. is described in detail, this conversion was carried out with the original data arrays, i.e. carried out immediately after data collection. In principle, however, this conversion can take place at any point in the data analysis steps described in the application example, since there is a clear connection between corresponding arithmetic operations in the Fourier range and in the time range. For computational reasons, a multiple change between the two areas is also conceivable.

6. Process variants in optical spectroscopy

The following method variants are possible in optical spectroscopy in order to arrive at the desired results AQ, A _lr A _{2 ,,} A ^ according to FIG. IB, item 20:

6.1. Time of performing the phase-sensitive detection (PSD) 6. 1 . 1 on-line PSD

The original data arrays are acquired as shown in FIG. 2A. The interpolation point arrays S _{k of} a period q are, however, not continuously accumulated, but are stored and demodulated after this period, ie during the period q + 1, according to one of the methods described in the two sections below. However, the results A _Q , A _α , A ₂ ,, A “are continuously accumulated, ie with the

Averages of the (q-1) preceding periods averaged. The same procedure is used with the original data arrays recorded in the period (q + 1) to P. The on-line PSD has the advantage that information about the expected results is already available during the experiment.

6.1.2. Off-line PSD

Three methods should be mentioned here:

- Storage of the original data arrays immediately after being recorded in a storage medium (RAM, hard disk, etc.). The entire evaluation is carried out off-line, using one of the demodulation methods described below.

- Storage of the original data arrays immediately after being recorded in a storage medium (RAM, hard disk, etc.). The evaluation is carried out off-line by reading stored data from the buffer during data acquisition and storage and processing it using one of the demodulation methods described below.

- Coaddition of the original data arrays immediately after detection. This corresponds to the illustration in FIGS. 2A and 2B and the application example. The further evaluation takes place according to one of the demodulation methods described below.

6.2. Demodulation process

6.2.1. Procedure without logarithmizing the original data arrays:

(Jacobi-Anger Transformation (JAT)) The time-independent arrays R ^ and S ₀ are determined as in the application example and converted into the DC term array AQ. The determination of A ,, A ₂ ,, A ,, can be carried out in three different ways Types occur, but in principle the order:

S _k => PSD => JAT must be observed. At any of the three possible locations, a division must be made by S ₀ , i.e. by S _k , by the PSD or by the JAT.

6.2.2. Method with logarithmicization of the original data arrays: The time-independent arrays R _Q and S ₀ are determined as in the application example and converted into the DC term array A _Q.

A _u A ₂ ,, A. can be determined in two ways, but in which the original data must always be logarithmized:

- With calculation of the transmittance (see Eq. (15) and (19)): The original data arrays are divided by S ₀ , then logarithmic and fed to the PSD. This path was labeled in the application example.

- Without calculating the transmittance:

The original data arrays are directly logarithmic and then fed to the PSD.

7. Devices for performing the method

11 shows a device for carrying out a temperature (T) modulation experiment.

An FTIR spectrometer or a diode array spectrometer serves as the analytical instrument (60).

The system (2) or the sample is located on both reflection surfaces of an ATR (attenuated total reflection) crystal, of which only half of the ATR plate (70) is shown. The incident into the ATR plate (70) beam of light (71) originates from an interferometer (64) or diode array spectrometer ^¬ Spec. After one or more total reflections in the ATR plate (70), in which the system absorbs light a _k (see equation (14)), the emerging light beam (72) is guided onto a detector (61) or detector array, in which the optical signal is converted into an electrical signal. The analytical instrument (60) exists - in addition to the interferometer (64) or monochromator - from the detector (61) and a computer unit (65), which in turn from a data acquisition

(! 4), a data processing (62), a phase-sensitive detection (16), a control and monitoring unit (63) and an output unit (20). The control and monitoring unit (63) is connected via lines (73) to the interferometer (64) or monochromator, via which all signals necessary for the operation of the analytical instrument are carried, e.g. those for controlling the mirror position of an interferometer. The detector (61) is connected via lines (74) to the data acquisition (14) of the computer unit (65), via which the original data arrays S (t) are fed to the data acquisition (14). A stimulation / reference unit (3) is provided for the external stimulation of the system (2), which is indicated by the arrow

(75) is shown. The external stimulation takes place here as temperature stimulation with the help of two circulation thermostats

(not shown), one of which maintains the circulating fluid at a temperature T and the other at a temperature T ₂ . The heat transfer to the system or to the cuvette takes place via a metallic heat exchanger plate, which is connected to the thermostats via hose connections. The control of the reference / reference unit (3) takes place via lines (76) which connect the former to the computer unit (65). The PSD

(16) receives the reference information digitally, which is processed within the computer unit (65).

If a monochromator and a diode array detector are used instead of the interferometer, the monochromator can be arranged in the beam path either in front of the system (2) or before the sample, or after the system (2) or after the sample .

FIG. 12 shows a device for carrying out an XRD X-ray diffraction-moisture (c) modulation experiment in a schematic representation.

The system (2), or the sample, is a crystal and is located on a goniometer attachment (90). An x-ray source (84 '), which is connected via line 86 to the control unit (84) of an x-ray diffractometer (80), is arranged to the system (2) in such a way that the x-ray (x-ray) (91) hits the crystal hits and creates a diffraction pattern. The diffracted X-rays (92) hit a two-dimensional matrix detector (81) which is part of the X-ray diffractometer (80). At the same time, the crystal is influenced from the outside by a periodic change in a parameter, for example by changing the air humidity, which is done by the stimulation / reference unit (3) and is indicated schematically by the arrow (95). This also modulates reflections from crystal regions that are influenced by the external modulation. The signals which are time-resolved on the matrix detector (81) are fed to a simultaneous, digital PSD. The remaining reference numerals in FIG. 12 correspond to those in FIG. 11 and have been described there.

8. Applications

8.1. General applications

Analysis of the dynamic and static behavior of any system which can be influenced by a periodic change in an external thermodynamic parameter (stimulation) and whose periodic response (reaction) is recorded simultaneously or quasi-simultaneously with a multi-component measuring device and digitally stored as original data arrays becomes. Phase-resolved modulation data arrays are also the basic data for 2D correlation analyzes (2D spectroscopy). If the stimulated process runs too quickly to carry out a time-resolved acquisition of the original data arrays, the new method still provides very precise static information about the state of the system when one or off stimulation. This enables quasi-drift-free differential measurements, which has proven to be a significant advantage over previous methods. 8.2. Special applications

8.-2.1. Optical spectroscopy

The task of optical spectroscopy includes the identification and the determination of the concentration and structure of chemical and biochemical substances. This also enables the determination of reaction schemes for processes that are triggered by a periodic, external stimulation.

8.2.1.1 Pressure (p) modulation

The equilibrium of a chemical reaction can be influenced via the pressure p, provided the total volume of the reactants involved changes during the reaction. A periodic p-modulation can therefore trigger periodic reactions, the maximum oscillation between the states caused by the lower, respectively. upper limit pressure are determined, p-modulation can e.g. to investigate reaction mechanisms of homogeneous and heterogeneous processes in the following areas:

• Environmental analysis: Homogeneous and heterogeneous reaction of air pollutants with each other and with catalyst surfaces.

• Chemical process engineering: Analysis of mechanisms in homogeneous and heterogeneous reactions with gases.

• Pharmaceutical research: Investigation of pharmaceutical biomembrane interactions, as well as pharmaceutical receptor interactions in the areas of drug design and drug delivery.

• Chemistry / biochemistry / biology: analysis of reaction mechanisms, conformation analysis.

• Biotechnology: interaction of gases with cell cultures and cell colonies.

8.2.1.2 Tempera t ur (T) module

The equilibrium of a chemical reaction can be influenced via the temperature T, provided that the reaction produces a heat (heat of reaction). A periodic T modulation can therefore trigger periodic reactions, whose maximum oscillation between the states, which are determined by the lower, respectively. upper limit temperature are determined. T-modulation can be used, for example, to investigate the reaction mechanisms of homogeneous and heterogeneous processes in the following areas:

• Chemical process engineering: Analysis of mechanisms in homogeneous and heterogeneous reactions. Temperature-critical processes (security technology).

• Pharmaceutical research: Investigation of pharmaceutical biomembrane interactions, as well as pharmaceutical receptor interactions in the areas of drug design and drug delivery. Temperature and interaction-related phase changes in biomembranes. Correlation of the data with the effectiveness of the pharmaceuticals.

• Chemistry / biochemistry / biology: analysis of reaction mechanisms, conformation analysis.

• Biotechnology: growth dynamics of cell cultures and cell colonies.

• Liquid-crystal (LC) technology: analysis of phase changes and the interaction between LC and electrode.

8.2.1.3 Concentration (c) modulation

At constant pressure and temperature, the periodic change in the concentration of a reactant causes a periodic course of all reactions of the system in which the externally c-modulated reactant is involved. The maximum concentration amplitudes of the reactants move between the equilibrium values that the upper, respectively. correspond to the lower concentration of the externally modulated reactant. c-modulation can e.g. to investigate the reaction mechanisms of homogeneous and heterogeneous processes in the following areas:

• Environmental analysis: Homogeneous and heterogeneous reaction of air pollutants with each other and with catalyst surfaces.

• Chemical process engineering: Analysis of mechanisms in homogeneous and heterogeneous reactions. • Pharmaceutical research: Investigation of pharmaceutical-biomembrane interactions, as well as pharmaceutical-receptor interactions in the areas of drug design and drug delivery. interaction-related phase transformations in biomembranes, peptides and proteins. Correlation of the data with the activity of the pharmaceuticals.

• Chemistry / biochemistry / biology: analysis of reaction mechanisms, conformation analysis.

• Biotechnology: interaction of active substances with cell cultures and cell colonies.

8.2.1.4 Electrical field (E) modulation

The equilibrium of chemical reactions can be influenced via the electric field, provided that a change in the state of charge and / or total dipole moment occurs between the educts and products during the reaction. In addition, molecules with electrical charges and / or dipole moments can be aligned and / or shifted in the electrical field. A periodic E-field modulation can therefore trigger or influence various types of periodic reactions. E-field modulation can e.g. to investigate the reaction mechanisms of homogeneous and heterogeneous processes in the following areas:

• Pharmaceutical research: Investigation of pharmaceutical-biomembrane interaction, as well as pharmaceutical-receptor interaction under E-field conditions, as they exist on native systems. The results are important in the areas of drug design and drug delivery.

• Chemistry / biochemistry / biology: analysis of reaction mechanisms, in particular structural changes in membranes, peptides and proteins caused by the E field.

• Liquid crystal (LC) technology: Analysis of the dynamics of the LC reorientation in the E field, as well as the interaction between the LC and the electrode.

8.2.1.5 Electrical current density (j) modulation j modulation plays a role in clarifying the mechanism of electrochemical reactions at the electrode interface, as well as in the development of conductive polymers.

8.-2.1.6 Radiation flux density (I) modulation

I-modulation plays a role in clarifying the mechanism of photochemical reactions, as well as in the development of photosensitive elements (e.g. detectors, information storage, energy storage).

8.2.1.7 Polarization (POL) modulation

POL modulation plays a role in the dynamic and static investigation of molecular order states.

8.2.2. Effects created by X-rays

X-ray diffraction (XRD): Modulated diffraction patterns (data arrays) of crystals can e.g. generated by T and c modulation. In the latter e.g. humidity can be modulated to investigate the influence of water of hydration on the molecular structure.

X-Ray Photoelectron Spectroscopy (XPS, ESCA) X-Ray Fluorescence Spectroscopy (XRF)

8.2.3. CW nuclear magnetic resonance spectroscopy (NMR)

By modulating external parameters, the periodic system response can be modulated onto the radio frequency (RF) field when using a continuous wave (CW) NMR spectrometer. Demodulation of the RF signal generates the modulated system response _r, which can be subjected to a simultaneous, digital PSD.

8.2.4. Electron spin resonance spectroscopy (ESR)

The periodic system response can be modulated onto the ESR signal (~ 100 kHz) by modulating external parameters. The demodulation of this signal generates the modulated system response, which can be subjected to a simultaneous, digital PSD.

8.2.5. Mossbauer spectroscopy

The modulation of external parameters can be used to modulate the quadrupole splitting, for example of iron in hamoglo am to generate in Mössbauer spectra (data arrays).

8.-2.6. Digital image acquisition and processing (CCD) CCD detection of optical effects is a fast, simultaneous acquisition of original data arrays. External parameter modulation applied to the system shown therefore leads to modulated portions in the original data arrays, which can then be subjected to simultaneous, digital phase-sensitive detection.

Claims

Claims:

1.- Method for the quasi-simultaneous analysis and characterization of the multicomponent, periodic system response of a system, characterized in that the system is periodically excited with a stimulation by changing at least one external parameter, thereby causing a periodic system response, whereby at least a physical quantity of the system is changed such that the change in the physical quantity of the system is detected by the system response registered simultaneously or quasi-simultaneously in the form of an original data array S (t) in N equidistant interpolation point arrays S _k with 1 <k <N digitally recorded as data per period, stored at least once and processed at any time before a digital, phase-sensitive detection (PSD), that the data prepared in this way are subjected to the digital, phase-sensitive detection, with the support point arrays is treated as w are scalar measurement variables, which leads to a quasi-simultaneous determination of phases and amplitudes, or to determine the Fourier coefficients of the I components S _kl with 1 <i <I, and that the system response is characterized via the phases and amplitudes.

2. The method according to claim 1, characterized in that in a first processing step the original data array S (t) is averaged point by point by coaddition immediately after the acquisition over a given number of periods, and the mean values thus calculated as averaged original data arrays S_ (t) are stored so that the averaged original data array S_ (t) is logarithmized and then fed to phase-sensitive detection ^' (PSD).

3. The method according to claim 1, characterized in that in a first processing step the original data array S (t) is averaged point by point by coaddition immediately after the acquisition over a given number of periods, and the mean values thus calculated as averaged original data arrays £ 3 (t) are stored so that the averaged original data array ∑5 (t) is divided by the mean value S ₀ and logarithmized and then fed to the phase-sensitive detection (PSD).

4. The method according to claim 1, characterized in that the original data array S (t) in a first processing step is averaged point-by-point by coaddition immediately after detection over a given number of periods, and the mean values thus calculated as averaged original data arrays S_ (t) be stored that the averaged original data array ∑3 (t) is fed to the phase-sensitive detection (PSD), that a Jacobi-Anger transformation (JAT) is then carried out, and that the division by the mean S ₀ at any point, respectively . before or after the PSD or after the JAT.

5. The method according to claim 1, characterized in that in a first processing step the original data array S (t) is averaged point-by-point by coaddition immediately after detection over a given number of periods, and the mean values thus calculated as averaged original data arrays S_ (t) are stored in a second processing step, the stored mean values S_ (t) are integrated and scaled over a period, and that each averaged support array S ^ is divided by this mean value S ^, with weak system absorption after elimination of the DC -Proportion of support point arrays a _k arise, which are fed to the phase-sensitive detection.

6. The method according to claim 1, characterized in that the original data array S (t) in the subsequent period, the data ^¬ processing up to the output signals A _k of the PSD is carried out, and thus at the end of each period, an updated system ^¬ response is available .

7. The method according to claim 1, characterized in that the original data array S (t) is acquired over at least one period and is only processed after the data acquisition has been completed.

8. The method according to claim 1, characterized in that the original data array S (t) is acquired and stored over at least one period and that the data processing begins during the data acquisition.

9. The method according to any one of claims 1 to 8, characterized gekenn¬ characterized in that the data acquisition, processing and evaluation at least partially takes place at the level of the interferograms, and that the Fourier transformation is carried out at a later, arbitrary time.

10. The method according to any one of claims 1 to 9, characterized gekenn¬ characterized in that the stimulation is carried out by a single-frequency stimulation (SFS) or by a multi-frequency stimulation (MFS).

11. The method according to claim 10, characterized in that the single-frequency stimulation (SFS) is followed by a single-frequency demodulation (SFD) or a multi-frequency demodulation (MFD) or that the multi-frequency stimulation (MFS) a single-frequency demodulation (SFD) or a multi-frequency demodulation (MFD) follows.

12. The method according to any one of claims 1 to 11, characterized gekenn¬ characterized in that the interpolation points are subjected to an interpolation and thereby an improvement in the phase resolution and the signal / noise ratio is achieved.

13. Application of the method according to one of claims 1 to 12 in FT modulation spectroscopy of fast processes with the aid of the PLL technology (phase locked loop).

14. Application of the method according to one of claims 1 to 12 in modulation experiments in optical spectroscopy, CW nuclear magnetic resonance spectroscopy (NMR), electron resonance spectroscopy (ESR) and Mössbauer spectroscopy.

15. Application of the method according to one of claims 1 to 12 in X-ray diffraction (XRD) and in digital image processing (CCD).

16. Device for carrying out the method according to one of claims 1 to 15, consisting of a system (2) with a multicomponent, periodic system response, a stimulation / reference unit (3) and an analytical instrument

(60), characterized in that a system is present which can be externally stimulated periodically via a stimulation / reference unit (3), an analytical instrument (60) with means for characterizing the physical quantities of the system (2), and a computer unit (65) are provided with means for data acquisition (14), for data processing (62), for PSD (16) and means for output (20) of the measurement results, and that the stimulation / reference unit (3) with the computer unit (65) is connected via control lines (76).

17. The apparatus according to claim 16, characterized in that an interferometer (64) and a detector (61) are provided as a means for characterizing the physical quantities of the system (2).

18. The apparatus according to claim 16, characterized in that a monochromator and a diode array detector are provided as a means for characterizing the physical quantities of the system (2), the monochromator being arranged in the beam path either before or after the system (2) is.

19. The apparatus according to claim 16, characterized in that the analytical instrument is an X-ray diffractometer (80) and as a means for characterizing the physical quantities of the system (2) an X-ray source (84 '), a control unit (84) with line (86) , an X-ray beam (91), the diffracted X-ray beams (92) and a matrix detector (81) are provided.