# FRF measurements subject to missing data: quantification of noise, nonlinear distortion, and time-varying effects

Quantifying the level of nonlinear distortions and time-varying effects in frequency response function (FRF) measurements is a first step towards the selection of an appropriate parametric model structure. In this project we tackle this problem in the presence of missing data, which is an important issue in large-scale low-cost wireless sensor networks. The proposed method is based on one experiment with a special class of periodic excitation signals.

In many scientific disciplines parametric models are identified from experiments on dynamical systems either to get insight into complex physical phenomena, or for computer aided design, fault detection/monitoring, prediction and control [1]. Choosing the appropriate model structure of the parametric model is one of the most difficult steps in the identification procedure. A nonparametric procedure allowing the user/scientist to decide which type of dynamical model structure is most appropriate for a particular application would be very helpful in this respect.

A first step towards the solution of this difficult problem is presented in [2]: via an experiment with a random phase multisine, which is a special periodic signal, the time-varying frequency response function, the noise level, the level of nonlinear distortions and the time-varying effects are estimated nonparametrically. It allows one to decide which of the following dynamical models is best suited to describe the measurements: linear time-invariant (LTI), linear time-varying (LTV), nonlinear time-invariant (NLTI), or nonlinear time-varying (NLTV). In addition, it gives insight into the complexity of LTI and LTV dynamics via the best linear time-invariant and best linear time-varying approximations, respectively.

In this project we extend the results of [2] to measurements subject to missing data. The proposed approach is based on a non-trivial combination of the algorithm for estimating the time-varying frequency response function using random phase multisines [2] and the missing data algorithm for frequency response functions using periodic excitations [3].

The whole procedure is illustrated of the following time-varying bandpass filter.

*Figure 1: Transient response (33 mVrms) of a second order time-varying bandpass filter to two periods of a random phase multisine (97 mVrms). *

It can be seen that the estimates (black lines), their noise (green lines), and the total (red lines) variances coincide, except for the total variance of the 20% clipping above 20 kHz. From the right column of Fig. 2 it follows that the error of the missing data estimates (gray lines) are of the level of their predicted total uncertainty (noise and nonlinear distortion). The maximal error (first frequency excluded) on G0 and G1 is, respectively, -56.8 dB and -65.0 dB for the 20% clipping, and -58.2 dB and -63.1 dB for the 30% random pattern.

*Figure 2; Estimated direct model – Measurement example. Left column: the frequency response functions (3 coinciding black lines) and their noise (green lines) and total (red lines) variances. Dark green and dark red: no missing data, medium green and medium red: 30% randomly missing data, and light green and light red: 20% clipping. Right column: Magnitude of the difference between the estimated frequency response functions using all data and the missing data sets (grey lines), and total variance of the missing data estimates (red lines). Dark grey and medium red: 30% randomly missing data, and light grey and light red: 20% clipping.*

It can be seen that the estimates (black lines), their noise (green lines), and the total (red lines) variances coincide, except for the total variance of the 20% clipping above 20 kHz. From the right column of Figure 2 it follows that the error of the missing data estimates (gray lines) are of the level of their predicted total uncertainty (noise and nonlinear distortion). The maximal error (first frequency excluded) on and is, respectively, -56.8 dB and -65.0 dB for the 20% clipping, and -58.2 dB and -63.1 dB for the 30% random pattern.

Figure 3 compares the estimated and the true missing data. It follows that the ratio of the rms value of the estimation error and the rms value of the missing samples is about for both missing data patterns. This estimation error is mainly due to the nonlinear distortions.

*Figure 3. Comparison of the estimated and the true missing data. Top row (only one 1 out of 8 missing samples are shown): true (black x) and estimated (red o) samples – Measurement example. Bottom row (all missing samples are shown): difference between the true and estimated samples (red lines) and the predicted standard deviation (black lines). Left column: 30% randomly missing data (32 mVrms) with an estimation error of 0.14 mVrms. Right column: 20% clipping (58 mVrms) with an estimation error of 0.24 mVrms.*

References

- L. Ljung, System Identification: Theory for the User, second edition. Upper Saddle River, New Jersey (US): Prentice-Hall, 1999.
- R. Pintelon, E. Louarroudi, and J. Lataire, “Time-variant frequency response function measurement on weakly nonlinear, arbitrarily time-varying systems excited by periodic inputs,” IEEE Trans. Instrum. and Meas., vol. 64, no. 10, pp. 2829-2837, Oct. 2015.
- D. Ugryumova, R. Pintelon, and G. Vandersteen, “Frequency response matrix estimation from partially missing data - for periodic inputs,” IEEE Trans. Instrum. Meas., vol. 64, no. 12, pp. 3615–3628, Dec. 2015.