Reverberation chambers show transient behaviour when excited with a pulsed signal. The field intensities can in this case be significantly higher than in steady state, which implies that a transient field can exceed predefined limits and render test results uncertain. Effects of excessive field intensities of short duration may get covered and not be observable in a statistical analysis of the field characteristics. In order to ensure that the signal reaches steady state, the duration of the pulse used to excite the chamber needs to be longer than the time constant of the chamber. Initial computations have shown that the pulse width should be about twice as long as the time constant of the chamber to ensure that steady state is reached. The signal is sampled in the time domain with a sampling frequency according to the Nyquist theorem. The bandwidth of the input signal is determined using spectral analysis. For a fixed stirrer position, the reverberation chamber, wires, connectors, and antennas can jointly be considered as a linear time-invariant system. In this article, a procedure will be presented to extract characteristic signal properties such as rise-time, transient overshoot and the mean value in steady state from the system response. The signal properties are determined by first computing the envelope of the sampled data using a Hilbert transform. Subsequent noise reduction is achieved applying a Savitzky–Golay filter. The point where steady state is reached is then computed from the slope of the envelope by utilising a cumulative histogram. The spectral analysis is not suitable to examine the transient behaviour and determine the time constants of the system. These constants are computed applying the method of Prony, which is based on the estimation of a number of parameters in a sum of exponential functions.

An alternative to the Prony Method is the Time-Domain Vector-Fit method. In contrast to the first mentioned variant, it is now also possible to determine the transfer function of the overall RC system. Differences and advantages of the methods will be discussed.

Reverberation Chambers are commonly used for immunity and emission tests, where an important class of immunity tests is performed with pulsed modulated sinusoidal signals. A reverberation chamber (RC) is characterised by its quality factor

Pulsed signal.

The duration of the pulse is denoted

A large number of publications have already dealt with transient behaviour in RCs

The possible overshoot can best be observed by measurements in time-domain. Artz has shown in his work that strong overshoot is accompanied by a low amplitude in the steady state and vice versa.

In contrast to the work of

The focus of this work is on signal analysis which depends on the stirrer position. First, in Sects.

An algorithm that extracts signal characteristics such as the amplitude of the overshoot when switching on and off as well as the amplitude at steady state is presented in Sect.

The parameters thus obtained are required for further signal analysis.

The chamber response is recorded with an oscilloscope for each stirrer position and is therefore time-discrete. This time series of measurements is interpolated to a function that is continuous over time. The function depends on the location within the test volume, the time and the stirrer position.

In Sect.

An alternative is the Time-Domain Vector-Fit method

A typical RC which can be regarded as rectangular cavity of width

Reverberation chamber with stirrer and antennas.

Resonant cavities possess a linear time-invariant system and can therefore be considered as a linear time-invariant (LTI) system

Reverberation chamber as linear time-invariant system.

The system response

To alleviate this problem an approximation of the mode density

A key performance indicator of a RC is the quality factor

A theoretical value for

A RC can electrically be viewed as a network of RLC circuits, see

The signal displayed in Fig.

Spectrum

To generalise the processing of the signals, it is beneficial to decompose the signal into its carrier and envelope parts. In a first step the analytic signal

The Hilbert transform is defined as

The integral representation of the Hilbert transform in Eq. (

In contrast to the Fourier Transform the Hilbert Transform does not change the domain,

The Fourier transform of the kernel

Expressing the spectrum of

Evaluating Eq. (

The analytic signal

The transition from a time continuous signal

As noted in Sect.

Figure

Measurement setup and signal path.

A generator and an amplifier delivers the signal

The carrier frequency

Spectrum of the input signal

The input signal excites several modes simultaneously which can lead to high field intensities in the chamber.

Figure

Switching on the input signal, measured at the port of the signal generator.

Switching off the input signal, measured at the port of the signal generator.

The discrete Fourier Transform (DFT) of the actual input signal is computed employing the Fast Fourier Transform (FFT) algorithm.
Figure

Spectrum of the input signal

These frequency components will excite modes in the RC, because of the LTI nature of the system. To filter these unwanted frequency components from the chamber response a 4th-order Butterworth band-pass filter is used, with a bandwidth of 200 MHz.

The spectra of the idealised and of the actual signal are symmetrical with respect to the carrier frequency in displayed frequency range.

Figure

Chamber response to the actual input signal, dashed lines mark the trigger points.

The spectra of the actual input signal and the chamber response differ which implies that transfer function of the RC is frequency dependent. The peak value at the carrier frequency is decreased by 25 dB corresponding to the insertion loss of the chamber.

Figure

Comparison of chamber responses in time- and frequency-domain.

In Fig.

Chamber response

Chamber response

Figure

Spectra of chamber response, recorded with single shot

Figure

Spectra of the averaged chamber response

It is now possible to downsample the complex envelope of the chamber response, because the upper frequency limit

The signal

In this section an algorithm is presented that extracts signal properties like rise time and overshoot amplitude. Table

The averaged downsampled envelope of the input signal

In order to extract the signal properties of interest the signal is partitioned into four parts

rise phase,

quasi-steady state (QSS),

decay phase,

off phase.

smoothing of the envelope,

clipping,

computation time-derivative,

deriving threshold for detecting the quasi-steady state,

identification of start and end points.

In the following section the single steps for the determination of the quasi-steady state are presented.

Signal smoothing is accomplished by employing a Savitzky-Golay filter, see

The quasi-steady state has to be within the boundary of the trigger time and the switch off time, see Fig.

The envelope of the chamber response in grey, the smoothed signal

An ideal QSS is defined by a constant amplitude over a certain time period. The derivative for this time period is zero. In practice a deviation around zero is expected and an alteration limit needs to be defined, see Fig.

Time-derivative

The time-derivative is calculated with a forward-difference quotient

In order to determine the maximum deviation around zero for the QSS a cumulative histogram is used, see Fig.

Histogram with 100 bins of equal width, the cumulative percentage and the threshold

In the final step of the algorithm the start and end points of the QSS are determined. The QSS is defined by coherent regions where every value has to be smaller or equal than the threshold. These regions are

The optimal outcome is that only one coherent segment results from the process. The identification of the QSS is obvious.

If more than one coherent segment have been found, the longest segment is chosen as the actual QSS.

Where

The previously determined steady-state value

The envelope of the chamber response in black, the smoothed signal

Due to the oscillating nature of the measured signals, the system responses are interpreted as oscillating systems. Based on this assumption it is possible to determine time constants for the rise and fall of each signal. A simple damped harmonic oscillation is

Computation of the unknowns in Eq. (

Extracted signal properties.

The described algorithm is now applied to a series of measurements of

The distribution of incidences of selected properties are shown in Figs.

Histogram of the QSS mean value

In Table

Population mean

The standard deviation indicates a wide spread around the mean value of the

Figure

Histogram of the QSS mean value

The spread of the peak values in the Rise and Decay phase around their mean value is smaller than for the QSS mean value. In the histogram this can be seen by an accumulation of values around the mean value in Fig.

Histogram of the peak values in the Decay phase

The relationship between the overshoot and the QSS mean value is shown in Fig.

Plot of the signals overshoot

The results of the algorithm are the foundation for further investigations.

With Prony's Method a decomposition of a signal into a sum of weighted complex exponentials can be performed

Prony's Method is sensitive to noise

The signal

The unknown factors appear linear as pre-factors

The core of Prony's method is the assumption that the sum of the exponential functions satisfies a

Rise phase of signal

The data set for the Decay phase includes

Decay phase of signal

Figure

Conjugate complex poles of

An alternative interpolation method is the TD-VF where pole locations within a given frequency band are selected.

Prony's method represents the measured chamber response as a sum of exponentials. This sum is the result of a convolution of a transfer-function and an excitation. Purely employing Prony's method makes it difficult to
extract information about underlying characteristics of the signal transfer. An alternative method is Time-Domain Vector-Fit (TD-VF), which is
based on the Vector-Fit (VF) algorithm

Vector-Fit offers the possibility to approximate a transfer-function to a measured response. In the following sub-sections are VF and TD-VF briefly summarised. Afterwards an application to measured data is presented.

The VF method approximates a rational function to frequency domain data using an iterative least squares technique. The transfer-function in frequency domain is

Similar to the VF in frequency domain the TD-VF solves a linearized
least-squares problem in time domain based on a known excitation

In the case of conjugate complex poles Eq. (

If the deviation between approximated response

The TD-VF is applied to two measurements for two different but fixed stirrer positions and a fixed antenna position within the chamber. Before TD-VF is used a low-pass filter is applied to reduce noise and distortions, the procedure is described in Sect.

A common difficulty is the prior knowledge of the number of required poles. So far there is no general procedure available that can solve this problem, apart from iterative solutions. It is possible to locate regions in frequency domain within which poles ought to occur, these regions are in between two local minima. This method is only suitable for frequency responses

The envelope of the excitation signal is depicted in Fig.

Excitation function.

In order to determine the transfer-function from Eq. (

The poles have to be stable, which means that the real part has to be negative. If a pole has a positive real part the stability condition

Poles outside of the frequency range (Eq.

Each pole has to fulfill

Selection of a curve segment and resulting segment that is approximated with TD-VF.

Approximated function

Both data sets consist of

In Fig.

Piece-wise approximated

Both approximations fit well to the original curve. A majority of the deviations appear at segment boundaries.

The TD-VF has been applied on a single measured signal. The signal belongs to a fixed position within the RC-chamber where the stirrer is at rest. The reconstructed signal

A multi-resolution in frequency and time gives the possibility to computationally partition the response signals. An automated TD-VF evaluation of signals, each measured at a fixed but different stirrer position, could then construct the transfer-functions systematically.

A system theoretical approach has been presented to explain the transient behaviour of a reverberation chamber excited with a pulsed signal. A spectral analysis on a pulsed signal and on the chamber response has been employed.

The spectrum of the chamber response shows that resonant frequencies in the neighbourhood of the carrier frequency are excited. Chamber responses of different stirrer positions provide nearly identical spectra. Differences between the chamber responses are more visible in the time domain.

An algorithm for the extraction of defined parameters was developed. Distinction of the individual responses is automatically possible on the basis of these parameters. The parameters include, for example, the duration and amplitude of the steady state as well as the time constants for the rise and decay. The results of the algorithm constitute the basis for further investigations, in particular the identification of the steady state is essential.

For further analysis of the chamber the data were approximated by a continuous function. This approximation is based on a superposition of damped oscillations for which the Prony interpolation method was used. The practical implementation is realised by a piecewise approximation of the data whose segmentation requires the knowledge of the steady state.

A similar approach allows to construct the transfer function of the overall RC system from the measurements with known excitation. This method is called Time-Domain Vector-Fit. The transfer function also represents an oscillating system. Similar to the Prony Method, the data set is segmented and approximated piecewise.

However, the approximations are based on linear time-invariant systems, i.e. the stirrer position must be kept constant for a measurement. Further investigations include the simulation of the chamber behaviour based on the approximated transfer functions as a function of the stirrer position. The transfer function of the chamber can be improved by knowing the transfer functions of the individual system components such as antennas, cables and connectors.

Furthermore, a comparison between the two interpolation methods is particularly useful with regard to the parameters obtained, such as the pole points.

To estimate the time constants for the Rise and the Decay phase from a damped oscillation (Eq.

Properties.

All data presented in this article are available from the corresponding author upon request.

FO implemented the TD-VF and the computation of the time-constants. KP conducted the measurements and implemented the Prony Method. FO and KP prepared the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Kleinheubacher Berichte 2019”. It is a result of the Kleinheubacher Berichte 2019, Miltenberg, Germany, 23–25 September 2019.

This open-access publication was funded by the Technische Universität Dresden (TUD).

This paper was edited by Frank Gronwald and reviewed by two anonymous referees.