DE10304170B4 - Method and device for determining the transfer function of discrete or continuous systems with determined signals - Google Patents

Abstract

method to determine an unknown, discrete or continuous, transfer function a technical or biological system if possible low system load, characterized in that for excitement a broadband, determinate test signal is used, which the impulse response of a first, discrete Hilbert transformer equivalent.

Description

The The invention relates to a method and a device for carrying out the Method by which the unknown transfer function H (p) or the frequency response H (f) with p = σ + j2πf of a technical or biological system without unduly high system load, as it occurs with impulse or jump excitation, metrologically can be determined. It can also unknown systems in control applications in the closed loop can be determined.

In 1 are the classical methods for determining the transmission behavior, ie the frequency response H (f) of linear, time-invariant systems.

In the first column are the known test signals x (t), in the second Column the associated Signal spectra (amounts the Fourier transform of the signals) and in the third column the system responses are shown.

The Lines (1) and (2) indicate that the pulse or the Step response by Fourier transform the complete locus (the frequency response) lets determine. In this case, impulse or step excitation place a high load on the system Which of sensitive systems, such as biological Arrangements, can not be easily coped with. on the other hand needed one at unknown (high) corner frequency of the system to be analyzed very broadband excitations; Here the impulse is better suited as the jump.

Line (3) shows the classical measuring technique, ie excitation with a sinusoidal signal. For a fixed frequency f ₀ , a sinusoidal signal having the same frequency, different amplitude and different phase is again obtained after the transient process. The respective amplitudes and phases then represent points of the locus or the frequency response of the system to be analyzed. This method is very time consuming in systems that need to be measured over a broadband range.

In the line (4) is a modern analysis method shown in the excitation is done with a broadband random signal. For details, this is through the lectures and the publications by Prof. dr. Klaus Lehmann and Jochen Lehnert on the occasion of XVI.

international Conference "Science in practice "in Budapest in April 2000 and by the contribution of Prof. Jürgen Weith on the occasion of the XVII. International conference "Science in Practice" in Schweinfurt im May 2001.

The International patent application WO 00/13032 A1 describes the analysis the transfer function an unknown system by means of various process options if possible low system load. However, the described require Method due to the test signals used a high metrological Effort to generate or analyze the test signals.

In spite of the diverse options to signals and the continuous improvements is the invention still the problem behind, an easy to implement process show, with the transfer function of a system without impermissible high system loads can be determined metrologically.

It is therefore an object of the present invention, a device and to identify a method whereby special broadband deterministic, i.e. predetermined test signals are used, with the same quality of the result a significantly lower system load compared to a pulse or jump excitation than in the prior art methods is and their generation and analysis with significantly less effort connected is.

These The object is achieved by a method according to claim 1 or by a Device according to claim 10 solved.

The inventive method and the device is characterized in that the Er a broadband, determinate test signal is used, which corresponds to the impulse response of a first discrete Hilbert transformer.

mit dem Hilbert-Transformationskern im Frequenzbereich –jsgn(f), multipliziert, so erhält man nach Rücktransformation in den Zeitbereich das Signal ν ^(t) die Hilberttransformierte von ν(t). Das Ursprungssignal ν(t) und das hilberttransformierte Signal ν ^(t) sind orthogonal zueinander. Wird jetzt das Signal ν ^(t) erneut einer Hilbert-Transformation unterworfen, so wird das inverse Ursprungssignal abzüglich des Gleichanteiles erhalten.The basis of the inventively preferred, advantageous signal generation for metrological determination of the frequency response H (f) of an unknown system forms the so-called Hilbert transformation of a one-dimensional time signal ν (t). Is the signal spectrum (ie the Fourier transform of the signal) of the original signal with

multiplied by the Hilbert transform kernel in the frequency domain -jsgn (f), we obtain the signal ν ^ (t) after Hilbert transformation into the time domain, the Hilbert transform of ν (t). The original signal ν (t) and the hilbert transformed signal ν ^ (t) are orthogonal to each other. If the signal ν ^ (t) is again subjected to a Hilbert transformation, the inverse original signal minus the DC component is obtained.

The 2 and 3 clarify this fact in graphic representation, where 2 the original signal (top) and the simple hilbert transformed signal (bottom) shows, and 3 the simply hilbert transformed signal (top) and the double hilbert transformed signal (bottom) shows.

Es ist die Impulsantwort eines Systemes, welches die kontinuierliche Hilbert-Transformation ausführt. Diese Tatsache ist seit etwa 150 Jahren bekannt, nur lässt sich ein analoges (kontinuierliches) System mit der geforderten Impulsantwort nicht realisieren. Durch die steigende Verbreitung der digitalen Signalverarbeitung in den letzten 15 Jahren wurde offenbar, dass eine diskrete Realisierung eines Systemes zur Hilbert-Transformation mit sogenannten FIR-Filtern (Finite-Impulse-Response) mit fast beliebiger Genauigkeit möglich ist.The Fourier inverse transform of the Hilbert transform kernel -jsgn (f) is known

It is the impulse response of a system that performs the continuous Hilbert transform. This fact has been known for about 150 years, only an analogue (continuous) system with the required impulse response can not be realized. Due to the increasing popularity of digital signal processing in the last 15 years, it has become clear that a discrete realization of a system for Hilbert transformation with so-called finite impulse response (FIR) filters is possible with almost any accuracy.

With the integer variable k, the impulse response of a discrete Hilbert transformer results

Thus, the infinite gain at kT = 0, which is required for analog implementation, avoided. T is the equidistant sampling clock of the discrete system H _H (z). A shift and windowing of the discrete impulse response leads to a realizable discrete system.

mit N als ganzzahliger Verschiebungsparameter, der zur Realisierung des ersten Systems H_H(z) erforderlich ist. Das in der Kette folgende unbekannte System H(z), dessen Übertragungsfunktion ermittelt werden soll, wird somit mit einer wertmäßig endlichen Folge x(k), deren Hüllkurve mit 1 / k abnimmt, erregt und das Antwort-Signal y(k) ermittelt.First, the preferred inventive concept for determining the unknown transfer function H (z) of a discrete system will be explained. In 4 For example, a discrete Hilbert transform system H _H (z) is excited with a discrete delta pulse γ ₀ (k). The output signal x (k) is of the form

with N as integer displacement parameter required to implement the first system H _H (z). The unknown system H (z) following in the chain, whose transfer function is to be determined, is thus excited with a value-finite sequence x (k) whose envelope decreases by 1 / k, and the response signal y (k) is determined.

A new Hilbert transform of the response signal y (k) with the same system H _H (z) as used to generate the signal of x (k) provides that of the DC component x (k) freed inverse impulse response h ₀ (k) of the unknown system H (z). A discrete Fourier transform of this signal eventually leads to H (z). In this case, the missing constant-size transmission factor H (z = 1) can be obtained either by simple separate measurement or by the interpolation of the acquired spectrum of h ₀ (k).

In the discrete frequency range for z = e ^jΩ with Ω = ω · T as the normalized angular frequency, the basis of the inventive ^{idea becomes} obvious: Each subsystem H _H (e ^jΩ ) which performs a Hilbert transformation has the frequency response H _H (e ^jΩ ) = -jsgn (Ω).

wird am Ausgang der Systemkaskade in 4 die folgende diskrete Fourier-Transformation erhalten: DFT{–h0(k)} = –jsgn(Ω)·H(ejΩ)·[–jsgn(Ω)] = [–jsgn(Ω)]·[–jsgn(Ω)]·H(ejΩ) = –H(ejΩ) With

is at the output of the system cascade in 4 receive the following discrete Fourier transform: DFT {-h 0 (k)} = -jsgn (Ω) · H (e jΩ ) · [-Jsgn (Ω)] = [-jsgn (Ω)] · [-jsgn (Ω)] · H (e jΩ ) = -H (e jΩ )

bestimmbar.This relationship exists because assuming linear-time-invariant (LTI) systems, the order in the cascade order can be reversed. Thus, the discrete Fourier transform DFT {-h ₀ (k)} = -H (e ^jΩ ), and thus the impulse response h is ₀ (k) except for the DC component

determinable.

In 5 Fig. 4 illustrates the magnitude response of a discrete 42nd order Hilbert transformer designed with the REMEZ EXCHANGE algorithm. 6 shows the corresponding finite impulse response, that of the discrete sequence x (k) in 4 for this example corresponds.

Here it is clearly recognizable that instead of a single high pulse at k = 0, the unknown system is excited with a larger finite number of pulses falling in the values of 1 / k, the frequency spectrum, see 5 , broadband is almost constant.

A further advantage of the particularly preferred method of exciting the unknown system H (z) with broadband, finite, long-term determined signals is that the in 4 shown signal x (k) prior to the application of H (z) differentiated or integrated. The respective inverse operation is to be arranged on the receiving side before or after the Hilbert transformation in order to determine the impulse response h ₀ (k) of the system H (z) to be analyzed. The explained facts describe the 7 in graphical representation, being at the top of the already in connection with 4 described way and in the middle and the lower position, the differentiation or integration of x (k) before and after the application to the system H (z) is shown.

For these cases shows 8th the test signal for H (z) resulting from differentiation of x (k) during 9 which lists the test signal resulting from the integration of x (k).

Of the Advantage of differentiated or integrated test signals in that differentiated test signals increase the excitation spectrum for higher frequencies In contrast, integrated test signals reduce the excitation spectrum for higher frequencies have as a consequence.

Finally it will be related to 10 described how the analysis of continuous (analogous) systems H (p) can be carried out with the new inventively advantageous determinate finite length test signals.

The finally long test function x (k) does not need explicitly by a first Hilbert transformation system to be produced, but you can the known and predictable Store values in a read-only memory and this by a microcontroller call one after the other. Even the differentiated or integrated Test signals can - once calculated - in stored in a read-only memory.

Using a clock generator, the controller causes the values stored in the read-only memory (eg ROM, EPROM, etc.) to be output once or periodically with selectable pauses in succession to a digital-to-analog converter (DAC). The downstream Sample & Hold circuit (S & H) generates the value-discrete, time-continuous test signal, which serves as excitation of the unknown system H (p) (see 11 ).

The output of this system is brought into the discrete form via an analog-to-digital converter (ADC), subjected to the digital Hilbert transformation and further evaluated either directly, integrated or differentiated as -h ₀ (k). Any subsequent discrete Fourier transform directly provides the frequency response of the unknown system H (p).

The integration or differentiation on the evaluation page in 11 In each case represents the inverse operation to the rule used for the generation of the test signal.

Claims (18)

Method for determining an unknown, discrete or continuous, transfer function of a technical or biological system with the least possible system load, characterized in that a broadband, determined test signal is used for excitation, which corresponds to the impulse response of a first, discrete Hilbert transformer. Method according to claim 1, characterized in that that the response of the unknown system when excited according to claim 1 is subjected to a second Hilbert transform, whereby directly gives the negative impulse response of the unknown system. Method according to claim 1 or 2, characterized that the broadband, determined test signal of a digital / analog conversion with subsequent sample & hold Circle is subjected. Method according to one of claims 2 or 3, characterized that the reaction of the unknown system of an analog / digital conversion and the second Hilbert transform in discrete Area is made. Method according to one of claims 1 to 4, characterized that the broadband, determinate test signal in the form of a sequence is generated by discrete signal values. Method according to one of claims 1 to 5, characterized that the broadband, determinate test signal is differentiated, predominantly high-frequency frequency components for the analysis of the unknown Systems prefer. Method according to one of claims 1 to 5, characterized that the broadband, determined test signal is integrated, predominantly low-frequency frequency components for the analysis of the unknown system to prefer. Process according to claims 6 or 7, characterized that for easy system analysis each for differentiation or to integrate the test signal inverse operation before or after the second Hilbert transform. Method according to one of the preceding claims, characterized characterized in that the generated test signal once, periodically repeated or periodically repeated with intermediate defined Breaks is generated. Device for detecting an unknown, discrete or continuous transfer function a technical or biological system if possible low system load, the device means for generating a test signal, means for energizing the system with the test signal and means for detecting the response of the system thereby in that the means for generating a test signal generate a broadband, determinate test signal, which is the Impulse response of a first, discrete Hilbert transformer corresponds. Device according to claim 10, characterized in that that the means for detecting the response of the system is a device which subjects the sensed response to a second Hilbert transform, which directly causes the negative impulse response of the unknown system results. Device according to claim 10 or 11, characterized in that the means for generating the broadband, determinate test signal include a digital / analog conversion device with replicating Sample & Hold circuit, which is subjected to the test signal. Device according to one of claims 11 or 12, characterized that the means for detecting the response of the system is a device for analog / digital conversion of the detected reaction, as well as a means to discrete the second Hilbert transform Make area. Device according to one of claims 10 to 13, characterized in that the means for generating the broadband, determinate test signal include means for passing the test signal as a result of discrete ones Generate signal values. Device according to one of claims 10 to 14, characterized that means for generating a broadband, deterministic Test signal comprise a differentiation device to predominantly high-frequency Frequency shares for to prefer the analysis of the system. Device according to one of claims 10 to 14, characterized that means for generating a broadband, deterministic Test signal include an integration device, mainly low-frequency Frequency shares for to prefer the analysis of the system. Device according to claims 15 or 16, characterized to simplify the system analysis, the means for detecting the reaction of the system comprise a device, each of which for differentiation or integration of the test signal inverse Operation before or after the second Hilbert transform. Device according to one of claims 10 to 17, characterized that means for generating a broadband, deterministic Test signal the test signal once, periodically repeated or periodically repeated with intervening defined pauses produce.