Power-based application of frequency-averaged ℓ 1 -norm regularisation technique for the synthesis of accelerating indoor tyre pass-by noise

Summary 1 Pass-by noise contribution analysis is an engineer- 2 ing procedure employed to estimate the contributions 3 from various noise sources on a vehicle to the overall 4 sound pressure level. This can be realised by placing 5 a set of microphones close to the various sources to 6 estimate their source strengths and then synthesising 7 the response at a far-field linear array in the presence 8 of the remaining sources. The results described in this 9 paper rely on measured near-field pressure data close 10 to the tyres of an electric vehicle under accelerating 11 conditions. The number and position of the estimated 12 virtual source strengths used is a compromise be- 13 tween complexity and accuracy, which has previously 14 been addressed mostly empirically. A power-based, 15 frequency-averaged ℓ 1 -norm regularisation technique 16 is investigated to optimise the equivalent source posi- 17 tion and strength for one operating tyre and, subse- 18 quently, the far-field pass-by noise pressure estimates. 19 It is shown that for the tyre under investigation, opti- 20 mising the positions of only 2 equivalent sources over 21 the frequency range of interest gives a good represen- 22 tation of the measured far-field spectra. 23


24
Vehicle pass-by noise can nowadays be measured in- 25 doors with a far-field microphone array and a sta-26 tionary vehicle on a rolling road, according to ISO-27 362:2016 [1]. In this procedure, the various vehi-28 cle noise source contributions can also be quantified, 29 which is described as pass-by noise contribution anal- 30 ysis and is nowadays widely used in automotive NVH. 31 The indoor pass-by noise contributions are esti-32 mated in fully operational conditions by using a set 33 of microphones close to each component and applying 34 the concept of acoustic transfer path analysis (TPA) 35 [2-3]. The sources are discretised into sets of equiv-36 alent sources, which are quantified by performing an 37 inverse method, using the near field measured spec-38 tra and the measured transfer responses between these 39 sources and their respective near-field microphone po- 40 sitions. New sets of acoustic transfer functions are 41 then measured between the source positions and a 42 linear microphone array 7.5 m away from the vehicle, 43 which are then used to synthesise the far-field acoustic 44 pressure and quantify the noise source contributions. 45 Tyre noise has emerged as one of the most impor-46 tant source contributions due to substantial efforts 47 in reducing engine noise and the increasing popular-48 ity of electric vehicles. While sound radiation from 49 tyres has been rigorously studied in the past [4], vari-50 ous studies have been also conducted to estimate tyre 51 noise contribution to pass-by noise in fully operational 52 conditions. A few of these studies deploy numerical 53 means, such as different variations of the Boundary 54 Element Method [5-6], to calculate vehicle pass-by 55 noise. However, despite their good accuracy, they are 56 mostly suited for pass-by noise estimation on the ba-57 sis of the CAD/CAE computer models, early in the 58 design stage. In [7], tyre pass-by noise is calculated 59 experimentally using particle velocity sensors, which 60 can prove impractical at capturing aeroacoustic phe-61 nomena, while, in [8], a separate test-bed diagnosis is 62 needed for tyre noise estimation. 63 In [9], the acoustic transfer path analysis described 64 above was used to estimate tyre noise and was proven 65 very practical due to the small number of microphones 66 needed close to the tyres to capture the near field 67 spectra. This work served as the basis for a few sim-68 ilar studies [10-12] utilising the inverse methodology, 69 while an alternative power-based approach was intro-70 duced in [13] and was shown to produce good results. 71 A very similar concept was also used in [14][15], where 72 the need for regularisation in the inverse method was 73 first introduced. ℓ 2 -norm regularisation was inves-74 tigated more extensively in [16][17], showing that it 75 can improve tyre noise synthesis accuracy by avoid-76 ing source overestimation at low frequencies. Through 77 all of this work, a predefined small set of equivalent 78 tions that are fixed over frequency and thus more use-126 ful for the given problem since the optimised source 127 geometry can then be used for subsequent measure- was previously described in [33][34]. 137 Since the calculation of the SPL for pass-by noise 138 engineering purposes does not require the use of the 139 phase information and in an effort to alleviate the ef-140 fect of the phase-related errors, this paper focuses on 141 the investigation of the frequency-averaged ℓ 1 -norm 142 regularisation, which was formulated in [20], using a 143 power-based formulation, similar to the one presented 144 in [13]. The new power-based, frequency-averaged 145 ℓ 1 -norm regularisation as a means of optimising the 146 equivalent source distribution, which is the novelty 147 of this work, omits the phase information, while the 148 equivalent source strength cross spectra are also not 149 taken into account, in an effort to keep the complexity 150 of the regularisation problem to a minimum. A full 151 vehicle indoor pass-by noise test campaign is also per-152 formed using accelerating tyre noise data and with the 153 vehicle operated under fully operational conditions, in 154 order to assess the performance of the methodology. 155 The accelerating near-field tyre noise data and the 156 corresponding measured transfer responses are used in 157 the convex optimisation problem to estimate the opti-158 mum source geometry and reconstruct the equivalent 159 source distribution. The distribution is then used, 160 together with measured far-field transfer responses, 161 for far-field tyre noise synthesis. The aim of this pa-162 per is therefore to investigate the application of the 163 power-based, frequency-averaged ℓ 1 -norm regularisa-164 tion method to optimise the equivalent source geome-165 try and far-field synthesised spectra for various num-166 bers of sources without assuming prior knowledge of 167 the noise -generating mechanisms of the tyre.

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In Section 2, the power-based methodology used 169 for the tyre noise synthesis including the frequency-170 averaged ℓ 1 -norm regularisation is covered, while, in 171 Section 3, the measurement set-ups are presented. 172 The corresponding results are discussed in Section 4, 173 while the conclusions drawn are covered in Section 5. 174 2 Formulation 175 2.1 Power-based, frequency-averaged 176 ℓ 1 -norm regularisation 177 As a first step, the pressure is assumed to be mea-178 sured with a near-field array of M microphones. The 179 accelerating tyre noise has a time dependence that 180 can be regarded as random with non-stationary sta-181 tistical properties. For the purposes of the frequency-182 domain formulation used in this paper, the power 183 spectral densities are estimated by averaging the mod-184 ulus squared spectra over multiple overlapping short 185 segments of data in the time-domain, where the pres-186 sure signals are locally assumed stationary [35]. As-187 suming x time segments, for each segment of period 188 T r , the raw periodogram is formed as where X Tri (f ) denotes the Fourier transform at a cer-190 tain frequency f . The power spectral density at the 191 m-th microphone for the given frequency is then de-192 fined by pnear,m (f ). (2) These spectra will in practice be contaminated by where S pnear is a vector of S pnear (f ) at the M near- Parameter α is a scalar, which modulates the max-232 imum value of ∥S Q ∥ 1,2 and, thus, sets the ceiling for 233 the sum of the power spectral densities of the source 234 strength vectors over frequency. By assigning a low 235 value to α, the problem is constrained to be solved 236 with a number of sources smaller than the original set, 237 thus introducing sparsity. A common sparsity profile 238 for the various frequency lines is introduced, which 239 translates to a number of non-zero equivalent source 240 positions in the source vector, which are fixed over 241 frequency [36]. The lower the value of α, the more it 242 will enforce sparsity in the solution and thus a smaller 243 number of non-zero equivalent sources will be used 244 in the linear problem. This constrained optimisation 245 can, therefore, estimate source strength power vectors 246 S q which share the same sparsity pattern with re-247 spect to the source positions and minimise the sum of 248 the least-squares linear problem over frequency. The 249 SPGL1 toolbox is used to solve the problem [37-38].

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After the use of the frequency-averaged ℓ 1 -norm 251 regularisation technique, the problem is reduced to 252 a determined or overdetermined one with a reduced 253 vector of source power spectra S ′ q of size L f × 1 and a 254 reduced modulus-squared transfer response function 255 The inverse problem is then solved using the squared 257 modulus pseudo-inverse |G ′ near | 2 + formed between 258 the non-zero source components and the near-field mi-259 crophones 2.2 Far-field pressure synthesis

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The power spectral density of the sound pressure in 262 the linear far-field array is estimated by where |G ′ far | 2 is the modulus-squared transfer re-264 sponse matrix connecting the far-field microphones 265 to the non-zero equivalent source positions, and, in 266 practice, should also be measured. The sound pres-267 sure level of the estimatedSPL t is energetically added 268 to those corresponding to the remaining sources on 269 the vehicle which are not under investigation, and 270 are quantified by assuming a fixed small number of 271 sources close to each source and performing the same 272 power-based inverse methodology. Assuming Z re-273 maining sources, the final synthesised far-field re-274 sponseSPL ov at each microphone is the sum of the 275 tyre noise contribution and the remaining synthesised 276 noise source contributions which is compared to SPL ov , the response measured 278 directly using a far-field microphone array.  The method described in Section 2 was validated on 317 a full vehicle indoor pass-by noise measurement cam-318 paign. The vehicle used was a SimRod rear-wheel 319 drive electric car developed for Siemens by Kyburz, 320 Switzerland, as a test version of the commercial eRod 321 vehicle. Tyres of type Pirelli 195/50R15 with radius 322 28.8 cm were used. The experiment was split into two 323 measurement sessions which took place in the semi 324 anechoic chamber at Siemens Digital Industries Soft-325 ware [39]. In the first instance, the left front (LF) tyre was se-329 lected as the tyre under investigation when only noise 330 from the two front tyres was emitted. The vehicle was 331 fixed so that its front tyres were placed on top of the 332 rolling road with a smooth road surface, which was 333 controlled by one motor for both tyres. A schematic 334 of the measurement set up is given in Figure 2. 18 microphones were placed in a linear array 4.8 m 336 away from the left side of the car, as shown in Figure 337 3(a). The microphone height was set to 1 m, while 338 the microphone spacing was set to 0.9 m with a 15.3 339 m total array length. A near-field circular array of 16 340 microphones with a radius of 29 cm was placed 15 cm 341 away from the left front tyre, as shown in Figure 3(b). 342 The microphone number was chosen to be double the 343 number of the maximum equivalent sources assumed 344 within the investigation. 4 microphones were placed 345 close to the right front (RF) tyre, 2 on the leading 346 edge and 2 on the trailing edge of the tyre.

347
The car was not in operation and, instead, the 348 rolling road was driven to excite the front tyres, there- Hz, was used [40]. The source was moved to the var- Spectrogram of the near-field pressure signals for 1 m/s 2 acceleration patches, the distance is 12 to 15 cm. For the right 376 front tyre, 2 equivalent sources were assumed in the 377 middle of the contact patch, one at each edge of the 378 tyre, as shown in Figure 5(b). Assuming that the 379 centre of the tyre is a point with coordinates (0, 0, 380 0), Table 1 gives the detailed coordinates of the 12 381 candidate equivalent sources in cm for the tyre under 382 investigation. The FRFs were measured between the 12 source 384 positions and both the far-field linear array and the 385 near-field circular array, while, for the other tyre, the 386 transfer responses were then measured between the 387 2 source positions and both the far-field linear array 388 and the 4 microphones placed close to the tyre. Each   In these measurements, the car was driven while 423 the rolling road was operated in a minimal resistance 424 mode, which means that no manual exterior force was 425 exercised at the rolling road to counter against the 426 movement of the tyres. The pressure spectrum was 427 measured at all the various microphones from 5 to 60 428 km/h with an acceleration of 1 m/s 2 . The volume 429 velocity source was then used to measure the trans-430 fer responses between the 12 equivalent source posi-431 tions along the right rear tyre circumference and both 432 the far-field linear and the near-field circular array. 433 The same positioning as in Section 3.1 was used for 434 the equivalent sources of the tyre under investigation. 435 Corresponding FRFs were also measured for the two 436 equivalent sources, one at the centre of the contact 437 patch of each edge of the left rear tyre, and for the 438 one equivalent source close to the gearbox. Fully cou-439 pled matrices were, therefore, calculated between the 440 equivalent source positions and their corresponding 441 far-field and near-field microphones at each frequency 442 line over the frequency range of interest (100 to 10,000 443 Hz). An overview of the equivalent source positions 444 assumed in this measurement set up is given in Figure 445 9. In Figure 10 The next step in the analysis was the utilisation of 514 the equivalent source strengths to synthesise the far-515 field response of the left and right tyre and add them 516 energetically. The final synthesised response was then 517 compared to the one measured directly.

518
In Figure 12, the synthesised SPL in 1/3 octave 519 bands at far-field microphone No. 11 is given over the 520 frequency range of interest for the left front tyre using 521 the regularisation technique with various numbers of 522 equivalent sources and for the right front tyre using 523 two predefined sources. The two responses were ener-524 getically added and the final synthesised response was 525 compared to the one measured directly at the far-field 526 linear array. The results for microphone No. 11 were 527 chosen as it is the one which is subject to the highest 528 acoustic response for the 1 m/s 2 case. The directly 529 measured spectra reveal a dominant region between 530 400 Hz and 1.5 kHz, although a small dip is identi-531 fied around 800 Hz. The synthesised response for the 532 right front tyre is the same for all the various cases 533 as it is not dependent on the ℓ 1 -norm regularisation 534 investigation.

535
For the 1-source case, the accuracy is very ac-536 ceptable up to approximately 1.5 kHz, while using 537 2 sources extends the accuracy up to approximately 538 3 kHz. This is a substantial improvement compared 539 to the accuracy achieved taking the coherence and 540 phase information into account in [20] and is due to 541 the ability of the power-based method to create larger 542 patches on the tyre with an average source strength. 543 For a higher number of sources, the reconstruction ac-544 curacy improves even at higher frequencies. However, 545 this improvement is not critical since the response 546 from the tyre above 3 kHz is attenuated significantly 547 compared to the dominant response between 400 Hz 548 and 1.5 kHz. This is in line with the results in Fig. 549 10, where it was seen that no significant improvement 550 should be expected when using more than 2 equiv-551 alent sources. The overestimation of the response at 552 low frequencies, which was seen in [20], is limited since 553 the phase information, which is the most important 554 reason for the creation of errors during the transfer 555 matrix inversion, is not taken into account. However, 556 it is seen that the overestimation worsens by increas-557 ing number of equivalent sources, which could be due 558 to the fact that, at low frequencies, the tyre geometry 559 dimensions are small in comparison to the wavelength 560 in air and, thus, a one source solution gives the best 561 results.

Results across far-field array 563
The far-field pressure synthesis accuracy with the con-  In this section, a more challenging synthesis concept 611 was tested using data from fully operational measure-612 ment conditions, as presented in Section 3.2.

613
In Figure 15, a comparison between the measured 614 Figure 15: SPL in 1/3 octave bands at far-field microphone 11 over the frequency range for the measurements described in Section 3.1 and 3.2. spectra at far field microphone No. 11 is shown for 615 the two measurement set-ups described in Sections 616 3.1 and 3.2. The spectra measured in the fully opera-617 tional case are higher throughout the frequency range 618 of interest, with the exception of a region around 300-619 400 Hz and at 1.7 kHz where the responses are very 620 similar, although a smaller increase at low frequen-621 cies is seen. A similar pattern is seen in terms of the 622 dominant frequencies of the spectra, as the range be-623 tween 500 Hz and 1.5 kHz is the strongest with the 624 exception of a dip at 800 Hz for both set-ups.

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The power-based, frequency-averaged ℓ 1 -norm reg-626 ularisation was used to decide the number and po-627 sition of the equivalent sources for the right rear 628 tyre under the presence of noise emitted also by the 629 left rear tyre and the gearbox. Different synthesised 630 curves for the right rear tyre were obtained for dif-631 ferent numbers of equivalent sources and these were 632 added energetically to the synthesised curves using 2 633 fixed sources for the left rear tyre and 1 fixed source 634 for the gearbox. The overall synthesised spectra were 635 then compared to the one measured directly. The 636 analysis was again done for a car driven from 5 to 60 637 km/h with a 1 m/s 2 acceleration using 1, 2, 4 and 8 638 equivalent sources for the right rear tyre. It is impor-639 tant to add that the source number and positioning 640 for the remaining sources were chosen by experience 641 and could be a possible source of error in the estima-642 tion.

643
By performing the ℓ 1 -norm regularisation in the 644 power-based formulation it was shown that the set 645 of regularisation parameters where the error decreases 646 substantially translates to the use of 2 sources whereas 647 the use of a higher number of sources was not shown 648 to further improve the reconstruction accuracy. Once 649 again, in order to choose the equivalent source posi-650 tions, the regularisation parameter that minimises the 651 least-squares error over frequency was chosen within 652 the range of the parameter which results in the use of 653 a chosen number of non-zero sources. This is done for 654 the cases of 1, 2, 4 and 8 equivalent sources.

655
The fixed positions chosen using the ℓ 1 -norm regu-656 larisation are given in Figure 16. Further research would also be needed to assess the 760 performance of the method when the radiation con-761 ditions vary substantially as the frequency-averaged 762 source selection, which is performed in this method-763 ology, could potentially lead to poor reconstruction 764 accuracy. was investigated for one tyre obtaining corresponding 806 synthesised responses, which were added energetically 807 to the ones acquired by the remaining sources on the 808 car, and the overall response was compared to the one 809 measured directly. 810 For the first set-up, where only tyre noise was 811 present, using fixed source positioning gave an accept-812 able level of synthesis accuracy, with the best accuracy 813 being achieved when using 2 sources for the left front 814 tyre. The ability of the power-based approach to cre-815 ate larger effective surface patches represented by an 816 average source strength and to avoid the phase errors 817 which have a significant effect on the inversion pro-818 cess, gives good synthesis accuracy by using only one 819 source at each edge of the tyre, which is a substan-820 tial improvement compared to the approach where the 821 phase information and source coherence were taken 822 into account, as presented in [20] for steady-speed 823 measured data.

824
For the second set-up, where tyre and gearbox noise 825 were present, similar conclusions were drawn regard-826 ing the optimum number of equivalent sources needed 827 for the power-based approach. The 1-source far-field 828 estimate was seen to have a smaller deviation from 829 the directly measured spectra compared to the first 830 set-up, while slight larger overestimations of the final 831 response were seen when using 8 sources. With the 832 aim of minimising the number of sources and target-833 ing the dominant region of 400 Hz to 1.5 kHz, the 834 2 -source case was selected as the optimum one, al-835 though results of similar accuracy were also seen in 836 the 4 -source case. 837 tional.