In this section we discuss the computer simulation of the inversion process and we describe how such properly designed simulations are used in the results presented in the following two chapters. More specifically, we show that when the inversion process is simulated by convolving a properly chosen model of the plant matrix C with the model of the inverse matrix H under evaluation, the outcome is indeed in good agreement with the measured results of the inversion process. On the other hand, this is not the case when the strictly anechoic model of the plant matrix is used for the convolution.
The comparison of the measured inversion results with the two different cases of simulation results is plotted in figure 3-8. The same model of the inverse matrix H is used in all three cases depicted in the figure. This is H190, i.e. the inverse computed by substituting the plant model C190 in the method of §2.3. In this instance the regularisation is set to the value β=0, the inverse length is set to 4000 coefficients and the modelling delay Δ is set to 2000 samples.
With blue line in figure 3-8 we plot the inversion results measured by means of the experimental arrangement of figure 3-2 (with the notional switches in the upper, dashed grey line position). This is then the actual, directly measured, performance of the inversion against which the simulation process should be compared. With red line in the figure we plot the results obtained when the model of the plant containing the measured impulse responses cij(n) with the sample index n in the range 1 ≤n≤ 300041 is convolved with the properly normalised (see p. 63) chosen model of the inverse H190. Finally, with green line we plot the results obtained with the strictly anechoic model of the plant C190 is convolved with H190. We note that this latter method is the one typically used when computer simulations are used for the evaluation of the inversion process. In such cases, a model of the plant that corresponds to the strictly anechoic part of the plant responses is used for the determination of the inverse and this inverse is then convolved with the same anechoic model of the plant. Obviously
then, as is confirmed with the green line results of figure 3-8, when no regularisation is used and the length of the inverse is chosen to be sufficiently high, it is straightforward to (erroneously) obtain perfect inversion results.
102 103 104 −120 −110 −100 −90 −80 −70 −60 −50 −40
Left l/s Left ear
Frequency [Hz] dB 102 103 104 −120 −110 −100 −90 −80 −70 −60 −50 −40
Left l/s Right ear
Frequency [Hz]
dB
Figure 3-8: Measured inversion results for H190 (inverse computed using C190 as the plant model,β=0, inverse length set to 4000 coefficients) (blue line). Simulation results obtained by convolving the 3000-coefficients model of the plant with H190 (red line). Simulation results obtained by convolving C190 with H190 (green line).figure_14p1(21,pappos)
On the other hand, the comparison of the measured results (blue line) with the results obtained when the full-length model of the plant is used for the convolution (red line) can be seen in the figure 3-8 to validate this properly designed simulation process. As can be seen in the time-domain plot of figure 3-9, the simulated and measured impulse responses are identical above the measurement noise-floor. A small deviation of the simulation results from the measured results in figure 3-8 is due to the fact that, for the non-regularised case chosen here, the response level is rather close to the noise-floor. Hence the difference below the noise-floor between the simulation and the measurement becomes significant. In regularised cases where the level of the response rises higher above the noise-floor, the agreement between measurement and simulation is virtually perfect as is seen in the following example.
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Left l/s Left ear
Sample number dB 0 1000 2000 3000 4000 −160 −140 −120 −100 −80 −60 −40
Left l/s Right ear
Sample number
dB
Figure 3-9: Measured inversion results for H190 (inverse computed using C190 as the plant model,β=0, inverse length set to 4000 coefficients) (blue line). Simulation results obtained by convolving the 3000-coefficients model of the plant with H190 (red line). Simulation results obtained by convolving C190 with H190 (green line).
figure_14p2(21,pappos)
A second case of comparison between measurement and simulation is presented in figures 3-10 and 3-11 in the time-domain and frequency-domain respectively. In this case, the plant contains a strong reflection42 but the inverse is computed on the basis of a model of the plant that does not contain this reflection. In this case, the regularisation is set to the value β=10-4, the total length of the inverse to 2000 coefficients and the modelling delay Δ to 1000 samples. Again with blue line we plot the directly measured results and with red line the results obtained when the inversion process is simulated by convolving the full-length model of the plant with the chosen inverse model.
42 This is the case with the wall reflection described in §5.2. A detailed description of that experimental arrangement that contains strong early reflections is given in that section. In this instance we are only interested in the comparison between the measured and the properly simulated results.
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−80 −60 −40
Left l/s Left ear
sample number dB 0 1000 2000 3000 4000 −100 −80 −60 −40
Left l/s Right ear
sample number
dB
Figure 3-10: Comparison of measured (blue line) and simulated (red line) inversion results. The regularisation is set to β=10-4 and the inverse length is to set to 2000 coefficients. The inverse is computed using the anechoic model of the plant. figure_26p2
As discussed in further detail in chapter 4, when regularisation is introduced, the level of the inversion rises higher above the noise-floor. This can be seen when the results of figure 3-10 are compared with those of figure 3-9. In that case then, the disagreement between the measured and simulated results below the noise-floor becomes less significant. Hence the agreement between measurement and simulation can be seen to be virtually perfect also in the frequency-domain results of figure 3-11. Concluding then, we see that the two comparison cases presented in this section confirm the validity of the simulation of the inversion process when this is implemented by convolving the 3000-samples-long measured model of the plant with the chosen model of the inverse. More specifically, in that case the simulation results are virtually identical with the directly measured results above the measurement noise-floor in the time-domain. As long as the level of the inversion is reasonably higher than the noise-floor, the simulation results are also identical with the measured results in the frequency-domain.
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Left l/s Left ear
Frequency [Hz] dB 102 103 104 −70 −60 −50 −40 −30 −20 −10
Left l/s Right ear
Frequency [Hz]
dB
Figure 3-11: Comparison of measured (blue line) and simulated (red line) inversion results. The regularisation is set to β=10-4 and the inverse length is to set to 2000 coefficients. The inverse is computed using the anechoic model of the plant. figure_26p1
3.6
Conclusions
In this chapter we described the basics of the inverse filtering objective evaluation procedure that is used in the following two chapters of the thesis. A detailed justification for the specific choice of evaluation procedure was given as well as a description of the experimental arrangement that used for the acquisition of the presented results. The measured model of the plant matrix C was presented that is inverted in a number of different implementation cases in the next chapters. More specifically, it was shown that even in controlled anechoic conditions the strictly anechoic part of the plant model is followed by a series of reflections of low energy and that decay slowly to the level of the noise floor. When substituted into equation (2-20) that describes the inverse matrix H, these different models of the plant matrix C should result to more or less varying models of the inverse. The examination and the comparison of the effectiveness of these inverse models are presented in the next chapter. Finally, it was demonstrated that the use of the strictly anechoic part of the measured model of the plant for the simulation of the inversion process is highly
inaccurate. Conversely, when the full-length model of the plant is used for that simulation, the results were shown to be practically identical to the directly measured inversion results. This provides us with a versatile simulation method that is extensively used in the analysis of the following two chapters but also validates the measurement procedure that was described above.
4
Objective evaluation of inverse filtering –
Inversion efficacy under ideal conditions
4.1
Introduction
In this chapter we present the results obtained when the inverse matrix H is realised with a number of different FIR models. The inversion is implemented under “ideal” conditions as all the assumptions listed on p. 55 regarding the perfect knowledge of the plant are satisfied and, furthermore, the condition of “transparency” of the audio reproduction chain is circumvented by the fact that the transducers used for the measurement of the effectiveness of the audio control are the same as those used for the measurement of the plant. Hence any possible deviations from the flat spectrum and linear phase are taken into account in the inversion.
The only remaining sources of error in the inversion are then the imperfections inherent in the design of the inverse matrix H and the presence of non-linear behaviour in the audio reproduction chain. As was already seen above, the non-linear behaviour in the reproduction chain appears in the measurement as a constant noise- floor of the measured impulse responses which for the -50dB power amplification setting appears at approximately -100dB below the [-1,1] full-scale range of our measurement apparatus43 (see figure 3-4). This is of course far from the -144dBFS noise-floor of the 24-bit quantisation used here. Nevertheless, it exceeds the -96dBFS noise-floor of the 16-bit arithmetic of the standard PCM format that is commonly used in existing home audio installations (e.g. in Compact Disc equipment) and is thus the basic implementation platform of the virtual acoustic imaging system design under examination.
43 Henceforward we denote the dB value relative to the maximum absolute signal value of 1 by the symbol dBFS (dB Full Scale).
As regards the imperfections introduced by the design parameters of the inverse, the parameter that is commonly identified as that primarily impeding the effectiveness of the inversion is the required length of the FIR filters in the inverse matrix H. However, the results and analysis presented in the following show that the influence of the inverse filters’ length on the actual effectiveness of the inversion is in turn determined by two other parameters. These are the length of the responses in the model of the plant matrix C that is used for the determination of the inverse (see the discussion in §3.4) and the amount of regularisation introduced in the inversion. It is shown in the following that the chosen values of these two parameters affect the expected perceived quality of the inversion as much as that of the inverse filters’ length and that for different values of these parameters the required length of the inverse filters varies dramatically. The exact relationships between these three parameters and their combined influence on the effectiveness of the imposed control are the objective of this chapter.