Evaluation of measurement value and uncertainty of sound pressure level difference obtained by active device noise reduction

Measurement 96 (2017) 67–75

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Evaluation of measurement value and uncertainty of sound pressure level difference obtained by active device noise reduction Józef Wiora ⇑, Stanisław Wrona, Marek Pawelczyk Institute of Automatic Control, Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland

a r t i c l e

i n f o

Article history: Received 29 June 2016 Received in revised form 20 October 2016 Accepted 22 October 2016 Available online 24 October 2016 Keywords: Active noise control Active casing Measurement result evaluation Spatial variability Type A uncertainty decomposition

a b s t r a c t Acoustic insulation of a device from the environment can be enhanced by appropriate control of its casing vibrations. The level of noise reduction obtained in such way is considered as the main point for evaluating the performance of the active control system, hence its appropriate measurement constitutes a vital issue. In this study, measurement uncertainty evaluation of the sound pressure level difference is presented. Three independent components of the Type A evaluated uncertainties are derived. Many sets of conducted experiments allow to estimate the components, while information read from the calibration certificate of the employed sound level meter allow to estimate the Type B components. Prepared uncertainty budgets show that the most contributing source is the location dependency. Recommendations for appropriate performance measurement of an active control system are stated basing on the obtained results. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Some of the most common noise sources in the human environment are devices and machinery. Prolonged exposure to a highlevel noise, as in some industrial environments, can lead to hearing damage. In turn, domestic appliances can also be a source of the noise, causing annoyance and significantly obstructing work or leisure. Passive methods are commonly applied to reduce the excessive device noise, however, they are ineffective for low frequencies and often are inapplicable due to increase of size and weight of the device and its potential overheating. When passive methods are exhausted, alternatively, active control methods can be applied [1,2]. They efficiently complement the passive methods in their weak points—the low-frequency noise and heat transfer problems. The general idea of active noise control is to reduce the unwanted sound, referred to as the primary noise, with an additional sound, referred to as the secondary sound, which should be close in amplitude and opposite in phase to the primary disturbance. But, in the three-dimensional space, it often results in only local zones of quiet. In case of the device noise, global noise reduction is more desired. To obtain this goal, the Active Structural Acoustic Control (ASAC) can be applied, which uses vibrational inputs to reduce the actual noise emission to the environment.

⇑ Corresponding author. E-mail addresses: [email protected] (J. Wiora), (S. Wrona), [email protected] (M. Pawelczyk). http://dx.doi.org/10.1016/j.measurement.2016.10.050 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.

[email protected]

If the device generating noise is surrounded by a thin-walled casing, or if it can be enclosed in an additional casing, control inputs can be applied directly to the structure, and as a whole it can be used as an active barrier enhancing acoustic isolation of the device [3]. Such approach is referred to as the active casing approach, and was further developed by the authors and successfully applied in previous research [4,5]. It results, if appropriately implemented, in global noise reduction instead of local zones of quiet. The level of obtained acoustic reduction is considered as the main point for evaluation of the active control system performance, hence its appropriate measurement and uncertainty evaluation constitute a vital issue. In order to quantify the noise level, an acoustic sound pressure, intensity or power levels are measured. All of them are expressed in log scales, in dB. An impact of the noise on humans is described by an application of the levels modified by frequency weights, often the A-weights. The Sound Pressure Level (SPL) is measured using a measurement device, which consists of a microphone, filters, amplifiers and other electronic components [6]. Sometimes, it is built as a virtual device using a software application working with data acquisition cards [7]. The reduction of noise levels is quantified by conducting two following measurements—in loud and silent environments. If a wall is used to reduce the noise, the level difference (LD) is calculated basing on measurements conducted at both sides of the wall. To calculate the LD provided by the ASAC, the SPLs are read out without and with active control.

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The fundamental document regarding evaluation of a measurement result is the Guide to the expression of Uncertainty in Measurement (GUM) [8]. According to this document, each result should include information about the quality of the measurement. The most acceptable way of reporting such information is to present its measurement uncertainty. Nowadays, there are plenty of scientific works on the evaluation of the uncertainty in different areas of engineering. On the other hand, in the field of acoustics, especially during calculation of sound pressure level difference as a result of active noise control, reporting of uncertainties is not a rule, namely it is rather not discussed at all. In the literature on acoustics, discussions about the uncertainties are frequently related with international or national standards [9–15]. Sometimes, uncertainty is treated qualitatively and no estimation is conducted or only standard deviation of data is calculated. Such qualitative approach is presented in the work about sound insulation of dwellings in Denmark [9] or an outlier detection in acoustic noise [16]. There is a group of works regarding the sound level measurements. There is stated that most ISO standards and researches in sound power measurement deal with reproducibility and/or repeatability, omitting the Type B component. This component, however, gives usually a small but non-negligible contribution to the combined uncertainty [17]. In another work, an uncertainty of third-octave sound pressure level in an anechoic chamber was evaluated. Nine uncertainty sources were taken into consideration. The obtained combined standard uncertainties were about 0.4 dB for frequencies between 80 Hz and 3.1 kHz [10]. The SPL may also be measured under water. In such a case, the uncertainty is assessed taking into account resolution and reproducibility of outputs as well as hydrophone calibration uncertainty [18]. Measurements of sound pressure level differences are frequently related with building acoustics. One can find an uncertainty evaluation of normalized and standardized airborne sound insulation between rooms. The presented calculations base on standard deviations obtained in selected positions of the source and receiving rooms, uncertainty of the reverberation time and uncertainty of the equipment setup. No correlation between input quantities were assumed. Expanded uncertainties were determined for frequencies between 100 and 3150 Hz, as well as for a single number quantity, and were lower than 2 dB. [11]. It is concluded in a similar work that the most important source is the reproducibility. Furthermore, uncertainty increases with decrease of frequency that is a result of low frequency resonances occurring in sound transmission loss characteristics [12]. In other work, uncertainties of normalized level differences in building acoustics were investigated. The single number quantities and their uncertainties were obtained in field measurements of walls, floors and façades. Authors emphasize that there are significant variabilities in obtained sound isolation parameters [13]. Working with single number ratings for airborne sound insulation requires taking into consideration the correlation between the sound reduction indices in the 1/3 octave bands [14]. Uncertainty of noise reduction measurements of passive hearing protector were also assessed. Eight sources were analyzed. The uncertainties were similar for all frequency bands and were from 1.5 to 2.0 dB [15]. Many authors conclude that it is difficult to identify all sources of uncertainties [11,12]. In this work, the measurement uncertainty evaluation of the sound pressure level difference obtained by appropriate global control of noise in an active casing is presented. The uncertainty is calculated using the Type A and Type B evaluations. Three independent random variables are analyzed within the Type A evaluation: location dependent, series dependent and common, independent of the formers, whereas information taken from

technical data of the used sound level meter are considered within the Type B evaluation. Such approach allows for building an uncertainty budget with independent contributions, which take into account all important uncertainty sources. Finally, the result of the level difference obtained by active noise reduction is given as a value and expanded uncertainty. Undoubtedly, the approach is a bit complicated and time consuming, however, the inconvenience may be overcome during routine measurements using an appropriate prepared spreadsheet or script. 2. Material and methods 2.1. Laboratory The measurements of sound pressure levels were conducted in a laboratory of the Silesian University of Technology, Gliwice, Poland (Fig. 1). Its cubature is about 40 m3. The laboratory is intentionally full of equipment in order to obtain the environment similar to real working conditions. At each measurement location, the sound pressure level was a result of interference of direct and reflected sound waves. Primary noise was generated by a loudspeaker driven by a power audio amplifier, for which a computer-generated signal was provided. The loudspeaker was placed in an active noisereducing casing, described broadly elsewhere [19]. As the noisecancellation actuators, 21 electrodynamic exciters mounted to the casing walls were used [20]. The signals for the exciters were computed using an adaptive feed-forward ASAC algorithm, described by Wrona and Pawelczyk [21]. Configuration parameters of the algorithm were unchanged during all conducted experiments. 2.2. Measurement instrumentation All sound level measurements were conducted using a certified class 1 SVAN 912AE sound level meter with a type 4166 Brüel & Kjær microphone. During all performed measurements, the meter used the following settings: sound pressure level (SPL), Aweighting filter, RMS linear detector averaged in 1 s period. The sound level measurements were performed in eight randomly chosen locations, marked with pink sheets of paper, chosen to correspond to regular usage, stuck on their bases and labelled with letters ‘A’ to ‘H’ (see Fig. 1).

Fig. 1. A photo of the laboratory. Pink sheets with letters – places of measurements; 1 – the source of sound (loudspeaker in the active casing); 2 – the place labelled as ‘‘A” with the sound level meter.

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J. Wiora et al. / Measurement 96 (2017) 67–75 Table 1 A-weighted SPL of the floor noise obtained from three repetitive measurement series – summary. Values expressed in dB. Location Mean uA ðLp Þ

A

B

C

D

E

F

G

H

33.4 1.2

38.3 1.8

41.0 2.1

34.3 0.7

31.3 0.9

36.3 0.9

33.0 1.0

34.7 0.3

signals consisted of one, three and five harmonics, respectively. Parameters of the signals are gathered in Table 2.

Table 2 Primary noise signals used in experiments. The amplitude is expressed as a fraction of reference voltage of a bipolar DAC. Name

Amplitude

Frequency, Hz

S1

0.70

142

S2

0.25 0.14 0.10

142 155 92

S3

0.15 0.13 0.12 0.11 0.10

133 121 168 155 92

2.3.3. Sound pressure level difference The sound pressure level difference (LD), D, was measured in the following way. Firstly, sound level meter was set at a given location, primary noise was run and the SPL was measured – the first indication L1 was obtained. Next, the adaptive active noise reduction algorithm was switched on. The algorithm needs some time to converge, therefore a few up to a few dozens of seconds was needed to wait. After the time period, the second indication L2 was obtained. The LD was calculated as

D ¼ L1  L2 :

In such a way, the positive sign of the difference means lowering of the signal level.

2.3. Plan of experiments 2.3.1. Background noise In order to assess the floor noise level, three series of measurement at each of the eight locations were performed. SPLs were read after few seconds from the time of placing the meter at given location. The lowest measurement range of 70 dB was set. Results of measurements were collected in Table 1. Type A evaluated measurement uncertainties, uA , were calculated according to GUM as the standard deviation of the mean. Investigations of floor noise impact on measured values were conducted using the following relationships. Compensated SPL, Lcmp , taking into account the floor noise level, Lnoise , and the uncorrected indication of SPL, Lind , can be calculated as [6]


 Lcmp ¼ 10 log 10Lind =10  10Lnoise =10 :

2.3.4. Investigation of amplitude impact on LD Linearity of the ASAC was tested by changing the amplitude of the primary noise signal. One series of measurements at location A was performed. Signal S2 was chosen, in which its amplitudes were modified (Table 3). The lowest (70 dB) measurement range of the sound level meter was set. 2.3.5. Frequency bandwidth The tested control algorithm is destined for reduction of a low frequency noise, such as the sound emitted by distribution power transformers. The investigations were limited to determination whether the LD changes significantly with frequency. After preliminary tests, the range was known in which the noise-reduction algorithm works correctly. In the given configuration, the chosen range was 80 up to 198 Hz. Next, the range was split into eight parts, equal to each other in the log scale. One series of measurements for pure-tone signal at location A was conducted. Results are gathered in Table 4. In order to avoid the saturation of the exciters at high displacement, the amplitudes were lowered comparing to other tests. The lowest (70 dB) measurement range of the sound level meter was set.

ð1Þ

After some transformations, the correction bnoise , which should be added to indications, can be expressed as


 bnoise ¼ Lcmp  Lind ¼ 10 log 1  10DLI-N =10 ;

ð2Þ

where DLIN ¼ Lind  Lnoise is a difference between the indication of the sound level meter and the noise level, both expressed in dB. The sensitivity coefficient cðLnoise Þ, describing influence of noise level uncertainty on the correction, defined as

cðLnoise Þ ¼

ð4Þ

@bnoise @Lnoise

2.3.6. Repeatability and location dependence Ten series of measurements was performed in order to assess repeatability and LD variability dependent on location. The S1 signal was emitted and LDs were determined at locations A, B, . . ., H in the way described earlier. The SPLs were acquired after about 15 s from starting ASAC algorithm. The meter range of 90 dB was set. Level differences Djk are calculated for each location j (j ¼ A; B; ; H) and each series k (k ¼ 1; 2; . . . ; K; K ¼ 10) according to Eq. (4). Mean values of SPLs measured without and with ASAC,

ð3Þ

were obtained by Monte-Carlo simulations with 106 trials. 2.3.2. Primary noise Experiments were conducted using three primary noise signals, named as ‘S1’, ‘S2’ and ‘S3’, generated by a computer program. The

Table 3 Impact of amplitude on the level difference. A-weighted SPLs are measured at location A for signal S2 with different amplitudes. The unit-less amplitude ratio g is the amplitude of primary noise signal divided by the amplitude defined in Table 2, respectively for each tone; L1 is the SPL obtained without active control, L2 – with active control, D is the level difference, L1;cmp ; L2;cmp and Dcmp – the same quantities but after the floor noise compensation, calculated using Eq. (1) with Lcmp ¼ 33:4 dB. All values are expressed in dB.

g

L1

L2

D

L1;cmp

L2;cmp

Dcmp

1/1 1/2 1/4 1/8

63.5 57.5 51.4 45.7

46.6 41.0 35.9 34.2

16.9 16.5 15.5 11.5

63.5 57.5 51.3 45.4

46.4 40.4 32.3 26.5

17.1 17.3 19.0 19.0

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Table 4 Impact of frequency on the level difference. A-weighted SPLs are measured at location A for signal S1 with changing amplitude and frequency f. Symbols are the same as in Table 3. f, Hz

g

80 0.1

90 0.2

100 0.2

112 0.2

126 0.2

141 0.2

158 0.2

177 0.2

198 0.2

L1 L2 D

41.3 39.0 2.3

50.4 40.5 9.9

47.0 42.0 5.0

55.6 53.2 2.4

48.0 43.5 4.5

57.8 39.5 18.3

56.2 50.2 6.0

48.0 38.8 9.2

45.2 41.8 3.4

Table 5 Level differences dependent on locations for the pure-tone noise. Symbols: L1;j and L2;j – mean value of A-weighted SPLs at location j without and with active control, respectively, Dj – mean value of the level differences at location j, sbj ðÞ – experimental standard deviation calculated according to Eq. (31). All values are in dB. j

A

B

C

D

E

F

G

H

L1;j sbj ðL1 Þ

70.52

75.53

73.06

68.76

73.34

65.07

66.38

77.19

0.18

0.24

0.30

0.70

0.21

0.35

0.45

0.14

L2;j sbj ðL2 Þ

47.01

52.10

48.64

48.63

50.22

41.91

49.02

54.53

0.37

0.54

0.75

0.39

0.41

0.50

0.32

0.36

Dj sbj ðDÞ

23.51

23.46

24.42

20.13

23.12

23.16

17.36

22.66

0.27

0.48

0.72

0.76

0.28

0.56

0.36

0.43

Table 6 Level differences dependent on series for the pure-tone noise. L1;k and L2;k – mean value of A-weighted SPLs in series k without and with active control, respectively, Dk – mean value of the level differences in series k, sck ðÞ – experimental standard deviation calculated according to Eq. (35). All values are in dB. k

1

2

3

4

5

6

7

8

9

10

L1;k sck ðL1 Þ

71.5

71.3

71.2

71.4

71.2

71.1

71.2

71.1

71.2

71.1

3.6

3.6

3.6

3.5

3.6

3.9

3.5

3.6

3.4

2.6

L2;k sck ðL2 Þ

48.9

49.1

49.2

49.3

49.1

49.1

48.8

49.0

48.8

48.9

2.7

2.7

2.9

3.2

3.2

3.1

2.8

3.1

3.0

3.1

Dk sck ðDÞ

22.58

22.24

22.04

22.10

22.11

22.08

22.35

22.13

22.45

22.21

2.4

2.4

2.2

2.1

2.5

2.4

2.5

2.5

2.4

2.3

Table 7 Level differences dependent on locations for the multi-tone noises. Symbols are the same as in Table 5. All values are in dB. j

A

B

C

D

E

F

G

H

14.93

20.40

19.10

14.27

17.07

15.53

16.77

22.67

0.19

0.24

0.43

0.29

0.09

0.31

0.21

0.12

7.53

9.20

16.40

5.03

11.67

8.07

12.47

9.60

0.05

0.22

0.37

0.26

0.38

0.17

0.05

0.08

Signal S2 Dj sbj ðDÞ Signal S3 Dj sbj ðDÞ

noted as L1;j and L2;j , respectively, are calculated as the average value from all (K) indications at given location, whereas mean values of LDs, Dj , as the average value from all level differences at given location:

L1;j ¼

K 1X L1;jk ; K k¼1

L2;j ¼

K 1X L2;jk ; K k¼1

Dj ¼

K 1X Djk : K k¼1

ð5Þ

The Type A evaluated uncertainties, uA ðÞ, are calculated over all respective data obtained at given location. Values obtained during the experiments are gathered in Table 5. The same indications obtained in the experiments are applied to assess dispersions in each measurement series. Similar means are calculated as

L1;k

J 1X ¼ L1;jk ; J l¼A

L2;k

J 1X ¼ L2;jk ; J l¼A

J 1X Dk ¼ Djk : J l¼A

Results of the calculations are presented in Table 6.

ð6Þ

Similar investigations are performed for multi-tone signals S2 and S3. Number of series is limited to three. LDs and their Type A uncertainty components are calculated in the same way as for the mono-tone signal. The second indications (L2 ) were read after 20 and 30 s, counting from starting ASAC algorithm, for signals S2 and S3, respectively. Intermediate results of the tests are gathered in Tables 7,8.

Table 8 Level differences dependent on series for the multi-tone noises. Symbols are the same as in Table 6. All values are in dB. k

1

2

3

17.6

17.6

17.6

2.7

2.9

2.6

10.0

10.0

10.0

3.4

3.4

3.0

Signal S2 Dk sck ðDÞ Signal S3 Dk sck ðDÞ

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J. Wiora et al. / Measurement 96 (2017) 67–75

1 1 xk ¼ l þ pffiffi n þ pffiffi i þ j: J J

3. Uncertainty analysis of the level difference The procedure of determining the LD is described in Section 2.3.3. According to it, the same measurement instrument indicates two quantities: L1 – the sound pressure level obtained without active reduction and L2 – with the reduction. For the reason, the two quantities are strongly correlated. It means, that many uncertainty sources, which are value independent, are cancelled during calculation of the LD: D ¼ L1  L2 . In order to assess the uncertainty of the LD, the following sources enumerated bellow are taken into consideration.

ð14Þ

The average over all measurements leads to

x ¼ Mean xj ¼ Mean xk ; j

ð15Þ

k

which transforms to

1 1 1 x ¼ l þ pffiffiffiffiffi n þ pffiffi i þ pffiffiffiffi j: JK J K

ð16Þ

Let the random variable a be defined as

a ¼ x  x:

3.1. Repeatability

ð17Þ

After substitutions of Eqs. (7) and (16), one obtains: The bellow-presented derivations base on fundamental information included in GUM [8] and the textbook regarding random variables, expectations and variances [22, Chapter IX]. Let a random variable x be notated as a bold symbol, EðxÞ be the expectation of the random variable, VarðxÞ be the variance and P Meanj ðxÞ ¼ 1I i xi be the mean (average) over all i ¼ 1 . . . I. Experimentally obtained values may differ between each other due to errors. Their variability is described by repeatability which may be decomposed into three independent components, as it is shown in this section. Thanks to it, a further evaluation of combined uncertainties is possible. Otherwise, without the decomposition, during calculations of uncertainties of mean values, correlation coefficients, often very difficult to determine, should be taken into consideration. Common disturbances, such as thermal noise and discretisation of measurement instrument, influence on each measurement. They can be described by a random variable n, which satisfies EðnÞ ¼ 0 and VarðnÞ ¼ r2n . There are also some errors, which are location dependent, due to properties of sound waves. Let they be described by a random variable i, which satisfies EðiÞ ¼ 0 and VarðiÞ ¼ r2i . The third component is a random variable j; EðjÞ ¼ 0 and VarðjÞ ¼ r2j , which describes the changing conditions between measurement series. Combining the information, one can write:

x ¼ l þ n þ i þ j;

ð7Þ

where l is the mean that would result from an infinitive number of measurements. The random variabilities n; i and j are uncorrected. The value obtained at jth location and in kth series is

xjk ¼ l þ njk þ ij þ jk :

ð8Þ

The expectation of a sum of random variables is the sum of the expectations, therefore,

EðxÞ ¼ Eðl þ n þ i þ jÞ ¼ l:

ð9Þ

Similar relationships is valid for variances:

VarðxÞ ¼ Varðl þ n þ i þ jÞ ¼ r2n þ r2i þ r2j :

ð10Þ

1 1 1 a ¼ l þ n þ i þ j  l  pffiffiffiffiffi n  pffiffi i  pffiffiffiffi j: JK J K

The sum of two random variables makes that their variances are added. From that:

a¼

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi K 1 JK  1 J1 pffiffiffiffiffi n þ pffiffi i þ pffiffiffiffi j: K JK J

EðaÞ ¼ 0: Let the variance

r2a

ð20Þ

r be defined as 2 a


 ¼ VarðaÞ ¼ E ða  EðaÞÞ2 :

JK  1 2 J  1 2 K  1 2 n þ i þ j þ ðw1 ni þ w2 nj þ w3 ijÞ: JK J K

a2 ¼

r2a ¼

JK  1 2 J  1 2 K  1 2 rn þ ri þ rj þ 0: JK J K

The estimate of the variance Eq. (17) is

s2a ¼

1 1 xj ¼ l þ pffiffiffiffi n þ i þ pffiffiffiffi j: K K

2 1 XX xjk  x : JK j k

j

equals to

j

ð24Þ

Let the random variables bj be defined for all locations j as

bj ¼ x  xj :

ð25Þ

ð26Þ

and after reductions:

ð12Þ

Similarly, it is possible to calculate the mean in each series k:

xk ¼ l þ Mean njk þ Mean ij þ jk

ð23Þ

r2a of the random variable defined in

1 1 bj ¼ l þ n þ i þ j  l  pffiffiffiffi n  i  pffiffiffiffi j K K

which can be expressed as

ð22Þ

The sum in the parenthesis on the right site of the above equation is a sum of products of uncorrelated random variables with weights w. The expectations of the products are products of expectations. Because the expectations of the mean zero random variables are zero, the expression in the parenthesis is also zero. Furthermore,   E n2 ¼ r2n ,. . . From that, the following relationships is obtained:

xj ¼ l þ Mean njk þ ij þ Mean jk ;

ð11Þ

ð21Þ

The square of the random variable a from Eq. (19) is

Using Eqs. (7) and (12), one obtains

k

ð19Þ

The expectation of a sum of mean zero random variabilities is also zero:

It is possible to calculate the mean of all measurements conducted at jth location. The value is k

ð18Þ

ð13Þ

pffiffiffiffiffiffiffiffiffiffiffiffi K 1 bj ¼ pffiffiffiffi ðn þ jÞ: K

ð27Þ

The expectation of bj is

  E bj ¼ 0;

square of the random variable is

ð28Þ

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J. Wiora et al. / Measurement 96 (2017) 67–75

2

bj ¼

 K 1 2 n þ 2nj þ j2 K

ð29Þ

and the variance is

r2b


 K  1  2 ¼ E bj ¼ r2n þ 0 þ r2j : K

The estimate of

s2bj ¼

r2b is s2b . It is possible to estimate for every location

2 1 X xjk  xj : K k

ð31Þ

2 1X 2 1 XX sbj ¼ xjk  xj : J j JK j k

ð32Þ

In the same way, let c j be defined as

c k ¼ xk  xk :


 J  1 2 rn þ r2i J

ð34Þ

and its estimate calculated from

s2ck ¼

2 1 X xjk  xk J j

ð35Þ

2 1 XX xjk  xk : JK k j

ð36Þ

is

s2c ¼

Combining above presented derivations, it is possible to calculate the estimates of the elementary variances r2n ; r2i and r2j from the set of Eqs. (23), (30) and (34):

8 s2 ¼ JK1 s2n þ J1 s2i þ K1 s2j > > K JK J < a 2 K1 2 K1 2 sb ¼ K sn þ 0 þ K sj > > : s2 ¼ J1 s2 þ J1 s2 þ 0 c n i J J

ð37Þ

where estimates of variances s2a ; s2b and s2c are calculated from Eqs. (24), (32) and (36), respectively. The Type A uncertainty is the square root of the variance of the mean expressed in Eq. (16):

uA ðxÞ ¼ sðxÞ ¼

ð41Þ

3.2. Type B components In addition to the above mentioned a posteriori evaluated uncertainties, known as the Type A evaluated components, there are also these ones, which can be quantified a priori, known as the Type B components. While estimates of the former may be lowered by increasing the number of experiments, estimates of the latter are difficult to change. To lower it, the so-called combined measurement method may be applied, but the method requires some modifications in order to adopt it to the properties of sound [23]. Bellow, the most important Type B components are enumerated and their estimates assessed.

ð33Þ

After some transformations, the following variance is obtained:

r2c ¼ E c2j ¼

rffiffiffiffiffiffiffiffiffi 1 2 s : uA;j ðxÞ ¼ K j

ð30Þ

The pooled estimate of variance based on J series of independent observations is the mean of variances:

s2b ¼

and the series-dependent component:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 1 2 s þ s þ s : JK n J i K j

ð38Þ

From that, the uncertainty may be expressed as the root sum square of three components, as the common component:

sffiffiffiffiffiffiffiffiffiffi 1 2 s ; uA;n ðxÞ ¼ JK n

ð39Þ

the location-dependent component:

sffiffiffiffiffiffiffiffi 1 2 s uA;i ðxÞ ¼ J i

ð40Þ

3.2.1. Readings at calibration frequency The SPL meter reading error of 0.09 dB shown in the calibration certificate propagates to both L1 and L2 values in the same way. By subtracting the two SPLs in order to obtain LD, the errors are cancelled. 3.2.2. Frequency characters According to the calibration certificate, the determination error for A-weighted characteristic is below 0.2 dB for frequencies less than 1 kHz. The activation of ASAC algorithm does not change significantly the spectrum of the signal. Therefore, the error keeps the same values for both SPLs and is cancelled during the LD measurement. 3.2.3. Linearity According to the calibration certificate, the SPL meter has excellent linearity. In the measurement span of 60 dB, there was no difference between predicted value and meter reading. The standard uncertainty of the determination is 0.1 dB. The two readings, needed for calculation of LD, are different and independent, therepffiffiffi fore, the uncertainty component should be 2 times higher and is 0.14 dB. 3.2.4. Self generated noise This level is determined in the certificate as 17.2 dB. The indications obtained during LD measurements are at least 20 dB higher. According to information included in Table 9 and discussion presentation in Section 4.1, the impact on SPL for those indications is negligible. For this reason, the uncertainty source should be omitted. 3.2.5. Resolution Each indication arises by quantisation of a measured quantity. The resolution of the SPL meter is 0.1 dB, therefore, the displayed values are rounded and have rectangular distribution with limits of 0:05 dB. Hence, according to the GUM, the uncertainty pffiffiffiffiffiffi component of the SPL is 0.1 dB= 12. The roundings are independent in each measurement, therefore by subtracting two SPLs,

Table 9 Corrections of SPL indications accounting the floor noise. Calculations were performed using Eqs. (1)–(3). The floor noise level, Lnoise , is 33.4 dB – location A. Symbols: Lind – the indicated SPL; DLIN ¼ Lind  Lnoise – the difference between the indication and the noise level; bnoise ¼ Lcmp  Lind – the correction of the indication; cðLnoise Þ ¼ @b=@Lnoise – the sensitivity coefficient. All values are in dB.

DLIN Lind

5.0 38.4

6.0 39.4

7.0 40.4

9.0 42.4

12.0 45.4

16.0 49.4

20.0 53.4

bnoise cðLnoise Þ

1.7 0.52

1.3 0.37

1.0 0.27

0.6 0.15

0.3 0.070

0.1 0.027

0.0 0.011

J. Wiora et al. / Measurement 96 (2017) 67–75

pffiffiffi the uncertainty is 2 times higher. Finally, the component is pffiffiffi 0.1 dB= 6 ¼ 0:040 dB. 4. Results 4.1. The floor noise The SPL of the floor noise in the laboratory varied from 31.3 dB at location E up to 41.0 dB at location C. Evaluation of measurement uncertainty of the floor noise is not within the scope of this work. The Type A component is relatively high and is up to 2.1 dB. The most important noise sources are computer fans. Table 9 presents corrections of indications, calculated for location A, and sensitivity coefficients describing how uncertainty of the noise level propagates to uncertainty of the correction. If the indication is only about 5 dB higher than the floor noise level, the correction is significant and is about 1.7 dB. For this case, the uncertainty of the noise level propagates to the corrected SPL with high coefficient equal about 0.5. It means that for weak measurement trueness of the noise level, the trueness of the correction is also weak. However, with the increase in indications, the corrections and sensitivity coefficients decreases and the trueness improves. If the indicated SPL is about 54 dB (20 dB higher than the noise), the correction is less than resolution of the meter. 4.2. Investigation of amplitude impact on LD Data obtained during investigations regarding an influence of the primary noise signal amplitude on the level difference are included in Table 3. When the amplitude of the signal decreases, there is seen a decrease in the level difference D. It is due to the lowering of the SPL, especially L2 , down to the floor noise level. After compensating the floor noise according to Eq. (1), the compensated level difference Dcmp keeps nearly constant. The test does not allow to say about the quality of the obtained values – only one measurement series was performed. Additionally, the floor noise level has also weak trueness which propagates to Dcmp for low amplitudes. Therefore, the obtained values allow only to state that the level difference is nearly independent on amplitude in the tested range after the floor noise compensation. 4.3. Frequency bandwidth The ASAC algorithm is optimized for a particular frequency range. A test was performed to say how frequency in the given range influences the level difference. Data included in Table 3 evidences that the LD is significantly higher for 141 Hz than for other frequencies. The noise level compensation is not needed because the obtained characteristic is only indicative. 4.4. Repeatability for the pure-tone noise The measurement data concerning the pure-tone investigations are included in Table 5 for determination of location dependence and in Table 6 – for series dependence. It is seen that the mean SPL without ASAC, L1;j , varies significantly with location. The minimal value is 65.07 dB at location F, whereas maximal – 77.19 dB at H. It makes a span of 12.12 dB. It also means, that the sound wave interferences in the laboratory are significant. With the ASAC, the span keeps comparable and is 12.62 dB between the same locations. The variations in SPLs do not propagate into the LDs, which are between 17.36 dB at location G up to 24.42 at location C – it makes a span of 7.06 dB. Existing variability in the LD is justified by the mechanism of ASAC: The exciters applied to the casing walls are reducing or changing the vibration distribution to limit the

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noise emission to the environment. However, the sound field is measured with a limited number of error microphones, only at particular locations. Hence, depending on the primary noise wavelength, the acquired sensor signal provides only an approximate information representing the overall continuous sound field. For low frequencies, it is enough to provide global noise reduction (instead of local zones of quiet), but the achieved noise reduction may vary dependent on location. The interferences in the laboratory and actuators placement also affect the obtained LDs. Undoubtedly, the SPL is location dependent, therefore the estimated standard deviation of SPL may be higher than uncertainty of LD, as it is evidenced in Table 5. It results from the fact that the meter was placed at given location and two following measurements were performed: without and with ASAC. In the next series, the microphone of the meter might be situated at a bit different position, therefore the SPL might have another value due to interferences but the noise reduction has kept nearly unchanged. Therefore, the Type A uncertainty component of LD should be calculated directly from LDs obtained in each measurement. Calculations from variability analysis of L1 and L2 and uncertainty propagations with accounting correlations may lead to worse estimates [24]. The standard deviations are the highest at locations D and C, which are the nearest to the casing, whereas are the lowest at locations A and E which are far (see Fig. 1). It probably means that the variability of LD is higher near the casing than in the distance. The maximal standard deviation is 0.76 dB at location D. Conclusions drawn from Table 6 are similar to those for repeatability analysis at given location. Here, the standard deviations of SPLs are significantly higher than of LDs for all series. Variabilities of the mean SPLs and LDs in the following series, seen in Table 6, are much lower and the span is about 0.5 dB. Also the standard deviations are nearly constant. It means that the conditions during measurements do not change in significant way. On the other hand, the standard deviations are huge, up to 2.5 dB, due to the location dependences. Analysis of all measurement data, performed according to the theory presented in Section 3.1, lead to the following estimates: s2n ðDÞ ¼ 0:30 dB2; s2i ðDÞ ¼ 5:4 dB2; s2j ðDÞ ¼ 0:006 dB2. The small negative variance s2j ðDÞ in comparison with the other components means that the series do not influence the uncertainty. From that, the following estimates of standard deviation are: sn ðDÞ ¼ 0:55 dB and si ðDÞ ¼ 2:3 dB, which lead to the common Type A uncertainty qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s2n ðDÞ ¼ 0:061 dB and to the locationcomponent uA;n ðDÞ ¼ 810 qffiffiffiffiffiffiffi dependent component uA;i ðDÞ ¼ 18 s2i ¼ 0:82 dB. The mean LD calculated from all 80 measurements is D ¼ 22:23 dB. 4.5. Repeatability for the multi-tone noises Intermediate results for a three-tone signal S2 and a five-tone signal S3 are gathered in Tables 7,8. Dependent on location, the LD varies from 14.27 dB up to 22.67 dB giving 8.40 dB measurement span for signal S2 and from 5.03 dB up to 16.40 dB giving 11.37 dB measurement span for signal S3. This means that with increase in harmonics in a signal, the span of LD also increases. Despite that the standard deviations are calculated basing only on three measurements, their values do not differ significantly from those for pure-tone signal. There are no significant differences between results obtained in series – both means and standard deviations are nearly the same. The estimated standard uncertainties show that there is no series dependency, the common uncertainty components are 0.068 dB and 0.062 dB, respectively for signal S2 and S3, and the locationdependent components are 1.03 dB and 1.23 dB, respectively. It is

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J. Wiora et al. / Measurement 96 (2017) 67–75

Table 10 Uncertainty budget of sound pressure level difference calculated for three signals S1– S3. Values are expressed in dB. Sourcensignal

S1

S2

S3

Repeatability Common Location-dependent Linearity Resolution

0.06 0.82 0.14 0.04

0.07 1.0 0.14 0.04

0.06 1.2 0.14 0.04

Mean value Combined uncertainty Expanded uncertainty

22.2 0.84 1.7

17.6 1.0 2.1

10.0 1.2 2.5

also seen that with increase in harmonics, the LD decreases, beginning from about 22 dB for pure-ton signal down to 10 for five-tone, and its uncertainty components increase. The mean LD calculated from all 24 measurements for signal S2 is D ¼ 17:59 dB and for signal S3 is D ¼ 10:00 dB. 4.6. Uncertainty budget The uncertainty budget is presented in Table 10. According to the above presented discussion, only four sources are taken into considerations: common and location-dependent repeatability components, linearity and resolution of the SPL meter. The combined uncertainty is calculated as a root sum squared standard uncertainties of the sources and is composed from several independent sources so it is allowed to assume that its distribution tends to Gaussian. The expanded uncertainty is the combined uncertainty multiplied by two providing an approximately 95% level of confidence. It is easily seen that the dominated source is the location-dependent repeatability. The results of the sound pressure level difference measurements are as follows:  for signal S1

DS1 ¼ ð22:2  1:7Þ dB;

ð42Þ

 for signal S2

DS2 ¼ ð17:6  2:1Þ dB;

ð43Þ

 for signal S3

DS3 ¼ ð10:0  2:5Þ dB;

ð44Þ

all with a 95% confidence.

5. Discussion It is obvious that more measurements may give more information. However, in the stage of experiment planing, a minimal set of test should be chosen to obtain requested data. After conducting the planed experiments and elaboration of the data, some remarks are formulated. The better time-weightings of the SPL meter would be Slow instead of Linear. According to the manual of the meter, when using the Linear setting, the outcome is calculated as subsequently linearly averaged elementary results over 1 s period, which are the 5 ms RMS values. On the other hand, the Slow time-weighting performs a low-pass filter with 1 s time-constant. When the ASAC is switched on, some a few Hz fluctuations in SPL are observed. In such situation, the low-pass filtering gives more stable indications than windowing. Amplitude tests should be conducted in a broader span. The performed tests include only four measurement points in the span of 18 dB. The L2 reaches the noise level for the fourth measurement

and lowering the SPL below the noise level is aimless. However, the louder signals should be generated, which would give more information about the linearity of the system. The level difference is strongly dependent on frequency. It is due to a complex frequency response of the casing structure itself, characterized by multiple resonances and antiresonances in the considered frequency range. The nine tests performed in the selected band show that the LD varies from 2.3 up to 18.3 dB. Undoubtedly, more tests may give more information about the shape of the frequency characteristic. The repeatability tests show that the series dependency makes irrelevant contribution. Therefore, in the future measurements, it is better to increase the number of locations in exchange for the number of series. Such approach would allow for lowering the location-dependent estimate of uncertainty. In order to obtain information about the level difference, which is the most valuable for hearing protection, the measurement locations should be chosen in such placements in which ears of workers may be found. Therefore, in the future works, measurements conducted on the level of a floor should be avoided. 6. Conclusions The measurement and uncertainty evaluation of the noise reduction levels were presented in this work. The global noise reduction was obtained utilizing the original active casing approach. As the primary noise, tonal and multi-tonal signals were used. They were generated by a loudspeaker placed inside the active casing. For the purpose of active control, 21 electrodynamic exciters mounted to the casing walls were used. The measurements of sound pressure levels were conducted in a university laboratory arranged to resemble real working conditions. Three Type A uncertainty components were distinguished: location dependent, series dependent and common (independent on the two former). A way of estimating the components was presented. Investigations of floor noise influences on the measured noise reduction levels were conducted. Linearity of the ASAC system was tested by changing the amplitude of the primary noise signal. Also, the frequency dependency were evaluated. Ten series of measurements were performed for a pure tone signal and three series of three-tone and five-tone signals. Each series consisted of measurements conducted at eight randomly chosen locations. Many sources of uncertainties were identified and taken into considerations. Basing on the data, uncertainty budgets were built. The most contributing source, above 97.6%, was the Type A evaluated location-dependent component, and the second was the Type B evaluated linearity of SPL meter. Other sources had a negligible impact. The obtained results were presented and discussed in order to formulate a set of recommendations for performance measurement of an active control system. Acknowledgment This work was supported by the National Science Centre, Poland (Grant No. DEC-2012/07/B/ST7/01408). References [1] D. Bismor, K. Czyz, Z. Ogonowski, Review and comparison of variable step-size LMS algorithms, Int. J. Acoust. Vib. 21 (1) (2016) 24–39, http://dx.doi.org/ 10.20855/ijav.2016.21.1392. [2] K. Mazur, M. Pawelczyk, Virtual microphone control for a light-weight active noise-reducing casing, in: Proceedings of 23th International Congress on Sound and Vibration, 2016.

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