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CORRECTING SOUND LEVEL MEASUREMENTS FOR BACKGROUND NOISE

Michael A. Staiano Staiano Engineering, Inc.

Rockville, Maryland

ABSTRACT This paper examines the uncertainty associated with

background noise corrections and their validity for high­background noise measurements, and considers confi­dence bounds for low-background noise situations. A ocrrection scheme is proposed which ocnsists of repeti­tive measurements of the source-signal-with-back­ground and background noise alone then the ocmputa­tion of a signal estimate and confidence interval. The procedure assumes that both the source of interest and background noise are uncorrelated, random processes which are stationary over the duration of the measure­ments. For useful results, the numbers of measure­ments must be selected to provide calculated ocnfidence intervals which acceptably ocntain the prediction error. These requirements are strongly influenced by the var­iability of the measured processes. When background noise is relatively low, the technique is useful primarily for quantifying measurement confidence bounds.

INTRODUCTION Sound levels measured to quantify the noise emissions

of a specific source may be contaminated by background noise. Good measurement practice requires that sound pressure levels be measured both with and without the operation of the source of interest. If the sound pres­sure level with the source is at least 15 dB greater than the background noise alone, the sound pressure levels measured with the source (and background noise) are essentially those due to the source alone-the preferred condition, but one frequently unobtainable in field measurements. If the background sound pressure levels

and those with the source operating differ by 4-15 dB, the sound pressure level due to the source alone may be calculated by applying ocrrections.(ANSI) If the sound pressure level measured with the sound source operat­ing is no more than 3 dB above the background noise, the sound pressure level due to the source of interest is less than or equal to the background sound pressure level. In this circumstance, correction for background noise is discouraged by all measurement standards since 'the correction is large and unreliable .. ." (Hassall)

Unfortunately, in many cases, background noise is within 4 dB of sound levels measured with the desired source-at least in some frequency bands. In these cases, the alternatives are: forego any information quantifying the source emissions, use the sound levels measured with the background noise influence as an­upper-bound source emission approXimation, or pro­ceed with a background noise correction to obtain an estimate of source sound levels. Even when background noise is relatively low, some uncertainty is introduced by the background noise correction. This paper examines the validity of background noise corrections in high­background-noise measurements and the quantification of confidence intervals for low-background-noise situa­tions.

EXISTING BACKGROUND NOISE CORRECTIONS The background noise correction scheme, such as that

provided for in ANSI S1.13, consists of the logarithmic subtraction of the measured background noise sound pressure level from that of the measured total of the background and source sound pressure levels. That is,

Presented at the 1994 International Mechanical Engineering Congress &; Exhibition, of the Winter Annual Meeting

Chicago, illinois - November 6-11, 1994

'. ,....------------ ­\ , \ . -\ \ \ \,

1 -. L_ ~_~ _'________=_~:-------

o ~ 12 '\~

1- N (-:l9'J

FIGURE 1. BACKGROUND NOISE CORRECTION per Equation {3}

{l}

where S is the estimated source-signal sound pressure level, T is the measured signal-with-noise sound pres­sure level, and N is the measured background-noise sound pressure lev'el. With a little manipulation, this expression can be reformulated as:

S =T-K {2}

where K is a background noise correction factOr (dB),

K= -1010g[1- 1O-(T-N)/l~_ {3}

This correction factor is plotted in Figure 1 versus the difference in the sound levels measured with and with­out the source operating, (T - N). As (T - N) goes to odB, K goes to infinity. For most measurement proce­dures, the curve in Figure 1 is truncated at (T - N) < 4 dB.

A background noise correction for small values of (T - N) would still be quite acceptable if the sound lev­els with and without the source could be measured ex­actly. Unfortunately, all measurements have some un­certainty, and this unsureness propagates through the correction procedure. Even where (T - N) > 4, finite errors can be generated due to the uncertainty. The ambiguity entailed by the background-noise correction becomes very large as the difference in the measure­ments becomes small.

SAMPLING OF SOUND LEVELS Data representing a physical phenomenon can be pre~

sented in terms of a finite-length amplitude-time record.

The collection of all of these time-history records de­fines the process describing the phenomenon. The properties of the data can be computed by averaging over this ensemble of records. When the average value at some relative time in each record remains conStant with average values at other relative times, the process is stationary and the properties computed over short time intervals do nOI vary significantly from interval to interval.

Physical data are often conveniently described by meaDS of statistics quamifying the steady and fluctuating components of a time history_ (Bendat and Piersol) The static component is described by the mean, i.e., the average of all values. The dynamic component is de­scribed by the variance, i.e., the mean square variation about the mean. (The standard deviation is the positive square rOOI of the variance.) The process statistics may be estimated from a sample of time histories. For a process x with n independent observations (i.e., sample size), the sample mean is

x=< ~ x./n, i = 1 -+ n {4}I

and the sample variance is

{5}

A st.atistic computed from one sample will seldom ex­actly equal the parameter of interesL Therefore, confi­dence intervals are used to describe how clo~e the value of a st21tistic is likely to be to the value of the parameter and itS probability of being that close. Where the mean value, J.I.. ,of a process x is desired, it is estimated from the sample mean, x, and expressed as being contained within a range. x±d II:. with a chance it is not. In other words: J.I.. lies wi!hi~ x ±daJ2 wilh (1 - a)% confidence. ["(l-a)%~ will be used to symbolize confidence in per­cent, more strictly written as "100(1 - a)%."J In situa­tions where the process standard deviation, ax, is un­known (essentially aU cases of interest), the sample standard deviation, s , and Student'S t distribution are used to define the corffidence bands, provided rhe process is Gaussian, as follows (Dixon and Massey)

- (dt) Yo - f) ,v,x + t n sin . :s J.I.. :s x + t ,,(d s 10 {6}cr, ~ X 1-crl" X

where tidf) is the value of the Student's-t distribution for df degrees of freedom (df = n - 1) and ¢ probability. Since the t-distribution is symmetrical, t (dt) = -I (df). a k defineda~ove, the confidence inten-'al describes a range about the sample mean, x ±d

cr12, in .whiCh the

process mean is expected to be found. ThiS protects against positive and negative extremes larger and small­

2

OQ r- 70 --------------------,;

iii Z 0' I ... ~ ~

~

s§

I ~

50

'..1I

'0-" -.

..

a Signal, S; Noise, N; and Total, T b. Signal Est., Sm; Upper Bnd., Se+; Lower Snd., Se-FIGURE 2. EFFECT OF MEASUREMENT UNCERTAINTY

hypothetical case of T known exactly and (T-N) known with =:3 dB uncertainty er than the specified value-considering the symmetri­cal, two-sided sha pe of the sample mean dis tribution. In many Situations, concern may be limited to the mean value being DOt more than or not less than some magni­tude, i.e., one-sided tests establishing only upper or lower bounds, respectively. For example, if an upper­bound limit is desired, then the appropriate confidence interval is}J. :5: X+ d

l.

Many proxcesses reSUlt in sound level distributions which are non-Gaussian. However, the Central Limit Theorem in statistics allows the sampling distribution of the mean of any random variable to be assumed normal provided the sample size is sufficiently large. The as­sumption of normality is reasonable in many cases for n > 4 and quite accurate in most cases for n > 10. (Ben­dat and Piersol)

PROPOSED CORRECTION SCHEME Many acoustical measurement situations involve ran­

clam processes, such as in environmental or community noise evaluations. The statistical techniques described above are readily applicable to these requirements. In building or industrial noise measurements, noise sources are often highly periodic-for example, the blade-passage frequency of a fan, the hum of an electri­cal motor, or the firing fate of a diesel engine. For these evalua{jons, deterministic techniques are most suitable. However, if the sources are operated under fluctuating loading conditions or are combined with numerous other noise sources in an uncorrelated man­ner, the assumption of a random process may become applicable. Even such apparently steady sinusoidal noise sources as a motor-pump set or fan may exhibit low-magnitude-but significant-noise emission fluctu­ations such as due to electrical-power and pump-load

variations for the pump set or inflow turbulence to the fan. Furthermore, as noise propagation distances in­crease, steady source noise emissions will become modi­fied by propagation conditions. In environmental measurements, large amplitude fluctuations may be in­trOduced by varying atmospheric conditions and turbu­lence even over moderate propagation distances (e.g., fan blade-passage noise has been observed by the author to fluctuate over a 9-dB range at a 500-ft distance).

Implementation of a background noise correction re­qUires repetitive measurements of .he source signal with background noise and of the background noise alone to increase confidence in the measured magnitUdes of the sound pressure levels Which define T and N and to de­termine the variability associated with their measure­ment. From the variabilities (and numbers of meas­urements), confidence interval bounds can be defined._ The source-signal magnitude at any instant, S., is the "decibel difference" in the T. and N. amplitudes, per Equation {l}. Over time, il c!.an be r~presented by the level of the ariLhmeric mean of "the antilogarithm differ­ences (i.e., the level of the expected SOurce sound pres­sures-squared). Confidence intervals can bederived from the measurement means and standard deviations-assuming both the source of interest and the background noise are uncorre13ted, random proc­esses. The difficulty that arises is {hat T and N cannot bath be measured simultaneously. However, jf the source and background are stationary over the time pe­riod spanned by the measurements, T and N can be measured separately and the source magnitude, $, com­puted.

The measurement errors for T and N are such that they resull in upper and lower bound estimates of the signal which-when expressed in decibels-are asym­

3

TABLE 1. GAUSSIAN RANDOM NUMBERS TEST RESULTS

n 11EAN -ERROR- -c. I. for a - -ERR ~ C. L for a OK SIN Mean SD 95.0 99.0 99.5 95.0 99.0 99.5 TRlS (dB) (dB) (dB) (dB) (dB) (dB) (%) (%) (%) (%)

6 10.1 0.5 2.0 2.0 2.9 3.3 75 91 94 100 30 10.0 0.0 0.9 0.9 1.2 1.4 81 91 91 100

100 10.1 0.1 0.5 0.5 0.7 0.8 91 94 94 100

6 0.0 -0.4 2.9 3.3 4.6 5.1 80 87 97 94 30 -0.1 0.1 1.1 1.5 2.1 2.3 91 97 97 100

100 0.1 -0.1 0.6 0.9 1.2 1.3 94 97 100 100

6 -9.8 5.1 5.9 6.5 8.2 8.8 36 55 59 69 30 -9.7 -0.6 6.7 7.2 8.4 8.7 67 67 71 75

100 -10.0 0.5 5.9 4.7 5.6 5.9 64 89 89 88

metric, i.e., the upper-bound eSlimate is more tightly constrained than the lower bound. This is illustrated in Figures 2 which shows the two-sided, upper- and lower­bound estimates (Se+ and S .' respectively) of a signal for various magnitudes of signal-tO-noise ratio. Thus, signal estimates may be obtained for (T - N) differences less than 4 dB with relatively confined upper bounds.

Correction Procedure. A correction procedure is proposed consisting of the following steps: 1. Perform n discrete measurementS of source sig­

nal with background noise, T., and of back· ground noise alone, N., (for a total of 2n meas­urements). )

2. Arbitrarily pair T. and N. measurementS, (T.,J J 1

N.).I

3. COmpute the sound-pressures-squared (or each measuremenl, p 2 = lO(XiIlO). {7}

4. (11culate the ~ressures.squared differences, 8. = p1j2 - PN. . {8} ~ I I .

5. Kepe.at Steps 2-4 for the n measurement pairs and determine the mean (8) and standard devia­tion (5,s) over the measurements.

6. Estimate the signal magnitude as S =: 10 log (8). {9}

7. Calculate the signal estimate upper bound as Sl-a: = 10 log (8 + k1-o-sS) {l0}

where k} = t} (df)/nY2 .",ith t (dC) the value 1

of Student's-t d'istribution for--<>"d[ degrees of freedom and l-a probability (i.e., confidence).

LImitations. Fundamental to this approaCh is the as­sumption that not only are the signal and noise uocone· lated, but that the measurements of the background noise with and without the source signal are uncorrelat­

ed. In most situations of in terest, this premise is rea­sonable since the au!Ocorre)ation function for random noise goes to zero rapidly with increasing intra­measurement time delay. (Bendat and Piersol) Even narrow frequency bands of random noise may be ana­lyzed provided the sampling interv'al between the meas· urements is sufficiently long.

A required condition is the stability of the background noise magnitude during the measurementS. Stationarity of the background sound pressure levels can be tested by performing background measurements before and after the with-source measurementS. If necessary, be­fore and after background sound levels can be averaged, or the computation aborted if excessive drift is ob­served.

EVALUATION OF PROCEDURE The prOced-ure was tested by simulating its implemen-­

tation in a number of uials. Each tri.f1l consisted of the repetitive sampling of noise and signal-with-noise six or more times (to evaluate the effect of sample size) and (he computation of the expected signal value and upper­bound estimate (for various confidence intervals). For each trial, the signal-estimation error was determined by subtracting the actual signal magnitude from that ex­pected using the correction procedure; and the actual magnitude was compared to the upper-bound estimates. This process was repeated for 32 trials using random numbers and eight trials using experimental measure· ments.

The results of the trials were examined with respect (0

four considerations: Accuracy is defined in terms of the mean error experienced in eslimating a kno'W"l1 signaL Consistency of Ihe estimates is quantified by the standard deviation of the errors.

4

TABLE 2. RAYLEIGH-LIKE RANDOM NUMBERS TEST RESULTS

n MEAN ---ERROR--- ---C.I. for a --- -ERR ~ C.I. for a OK SiN Mean SO 95.0 99.0 99.5 95.0 99.0 99.5 TRLS

(dB) (dB) (dB) (dB) (dB) (dB) (%) (%) (%) (%)

6 9.8 0.3 0.9 1.1 1.7 2.0 78 97 97 100 30 9.9 -0.2 0.5 0.5 0.7 0.8 97 100 100 100

100 10.0 -0.0 0.2 0.3 0.4 0.4 84 97 97 100

6 0.2 -0_2 1.3 1.7 2.5 2.9 84 94 97 100 30 -0.0 0.2 0.5 0.8 1.1 1.2 84 91 94 100

100 -0.0 -0.0 0.4 0.5 0.6 0.7 91 97 97 100

6 -9.7 1.6 5.6 6.2 7.8 8.4 54 62 77 81 30 -9.8 -0.0 3.1 4.4 5.5 5.8 77 91 95 69

100 -10.0 -0.8 2.9 3.4 4.2 4.5 84 87 94 _00

• Assurance that the estimate upper-bound con­tains the actual magnitude determines the va~

lidity of the procedure. It is quantified by the fraction of the trials for which the confidence interval is greater than or equal to the error, Success is considered to exist when the mean of the signal~with-noisemeasurements is greater than or equal to the mean of the noise-alone measurements. (As signal-to-noise ratio de­creases, the probability increases of a 0) < 0 dB. l! 0 < adB, then the signal cannot be estimated and the measurement trial fails.)

In evaluating the correction method, the procedure re­sultS were considered useful if the assurance and success were at least 90%.

Random Numbers Tests Gaussian Distribution. The correction procedure

was tested using normally distributed, random numbers. The random numbers were used to create sound-Ievel­like magnitudes by means of a constant piUS the prodUCt of a dispersion faCtor and a random function. The dis­persion factor yielded a standard deviation of approxi­mately 3 dB, in other WOrds a difference between 10th­and 90th-percentile sound levels (L\O-L

90) of about

8 dB. [Experience with environmenta nOlse measure­ments shows that L -L varies between about 4­

10 9010 dBA with 8 dBA typical. Near steadily operating fa­cilities (e.g-, power or process plants), this range is like­ly to be much smaller.] From such random n lJ mbers, values for T and N were generated. The constants were selected to yield desired nominal signal-to-noise ratios (S!l'l"). TestS were simulated to correspond to 6, 30, and 100 repetitive measurements with nominal signal-tO­noise ratios of +10 dB, 0 dB, and -10 dB. For each number of measurements, errors and 95.0,99.0 and

99.5% confidence inlerv'al bounds were computed over 32 trials.

The resultS, summarized in Table 1, are: Accuracy-mean error magnitudes typically ~

0.6 dB; Consistency-standard deviation of errors < 3.0 dB for nominal SIN 2: adB and 6-7 dB for SIN ::::: -10 dB; Assurance~onfidence interval contains error > 90% of time for nominal SIN 2: 0 dB and confidence 1-0: 2: 99% (except SIN ::::: 0 dB and n = 6), and confidence interval contains error about 90% of time for SIN =:; -10 dB with 1-0: 2:

99% and n =100; and • Success-~ 94% successful trials for nominal

SiN 2: 0 dB and about 90% success rate for SIN <>:: -10 dB.

From these testS, the correction procedure was deter­mined to provide useful estimates even for very poor signal-to-noise ratios (i.e., -10 dB). For high signal-to­noise ratios (about +10 dB), even six measurementS are adequate; for poor signal-to-noise (about 0 dB), at least 30 measurements are necessary; and, for very poor sig­nal-to-noise ratios, 100 measurementS and 99% confi­dence intervals are needed to obtain marginally accept­able consistency and assurance.

Rayleigh-Like Distribution. Environmental sound levels often do not distribute normally but·exhibit a Rayleigh distribution. Such a distribution is positively skewed with 3 finite lower bound but an asymptotic upper bound. A rna thema I ica I expression was de­veloped to generate random numbers eXhibiting a Ray­leigh-like distribution. When random numbers with sound-Level-like magnitudes were created with the se­lection ora suitable constant, the resultant distributions

5

"

'" ~ ~ ~ .,

~~~~~~I I'.

-----,

' ' I

~~~~~V~I, ~ ~

~

---I

,""

1'1.TJ':t?~~

, ,I ., 10 11 'I: u ,. .,

,~'" "''' 1;:­-- -,­a. 15-min time history b. 2-min segment of "l.vhite noise" excitation

FIGURE 3. EXPERMENTAL MEASUREMENTS-SPEAKER-EXCITATION TIME HISTORIES

TABLE 3. SAMPLING ERROR IN ABSENCE OF BACKGROUND NOISE (over eight trials)

n ----ERROR (dB) for SAMPLING TYPE--­-----P~riodic---- -----Random----­

mean SO mean SD

6 1.8 3.8 10 1.4 2.7 30 0.1 1.9 56 0.2 1.1

-0.1 1.6 -0.2 1.5 -0.0 0.5 +0.2 0.4

yielded standard deviations of about 1.6 dB (if normally distributed, L -L :::: 4 dB). As for the Gaussian raD­

10 90dam numbers, values ofT and N were generated to );eJd nominal +10 dB, 0 dB, and ·10 dB signal-to-noise ratios for tests with 6,30, and 100 repetitive measurements.

The results, summarized in Table 2, are: • Accuracy-mean error magnitudes typically :s

0.8 dB; • Consistency-standard deviation of errors gen­

erally less than about 3 dB; Assurance-> 90% for nominal SIN ~ 0 dB and greater than about 90% for SiN <:::: ·10 dB with l-a 2: 99% and n ;:: 30; and Success-100% for all but SiN =:: ·10 dB and n 530.

The results from these tests again showed the proce. dure to pro\-;de useful estimates. The accuracy 'with the non-Gaussian variables was about the same as with the Ga ussian-distribu ted variables. The consis tency of the results was greater and the confidence intervals smaller with the Rayleigh-like distribution-probably as a result of tbe reduced variabilit:y of the test parameters. In

spite of the tighter confidence intervals, the assurance remained high. For even poor signal-la-noise ratios (approx. adB), as few as six measurements were suffi· cient; while, for very poor signal-la-noise, 100 meas­urements and 99% confidence intervals appear to be necessary.

Experim entaI Measurements Further analyses of the correction procedure were de­

signed based upon the results of the random numbers tests. These tests were performed indoors and consisted of two loudspeakers and a centrally placed microphone. Speaker A was excited with filtered random noise (Standard deviation, approx. 0.7 dB; L ·L approx.

10 90,

2 dB), as shown in Figure 3a. The excitatlon inuoduced­to Speaker B, as also shown in Figure 3a, was one of two approximately IS-min. recordings of traffic noise made at about 100 ft from the edge of a very busy freeway (standard deviation, approx. 2.3 dB; L -L

9Q, approx..

106 dB). (As can be seen from inspection of the white­noise time history, that excitation was neither perfectly stationary throughout the measurements nor truly ran· dom. Closer examination of a 2-min. interval in Figure 3b shows the excitation in fact to be quasi-random wilh a periodicity of about 4/min.) The gains of the loud­speaker amplifiers were ~elected such that.nominal +10-dB, Q·dB, and -10-dB signal-to-noise ratios were obtained. All of the speaker outputs were measured 3t I-sec intervals over (!bou{ 15 min. for a total of approx· imately 900 meas urements in terms of i-sec equivalenr sound level, L> (1 s), values. A, A + B an.d B were measured for e1h test condition with A and B arbiWlri· Iy defined as signal or noise, Graphical methods can be used to show both excitations to be very nearly Gausian,

6

TABLE 4. PERIODICALLY SAMPLED MEASUREMENTS TEST RESULTS

n MEAN -ERROR- ---e. 1. for a - -ERR .:: C. I. for a OK SiN Mean SD 95.0 99.0 99.5 95.0 99.0 99.5 TRLS

(dB) (dB) (dB) (dB) (dB) (dB) (%) (%) (%) (%)

6 11. 6 1.8 3.2 1.2 1.8 2.1 38 38 38 100 10 12.0 1.3 2.6 1.0 1.5 1.7 50 63 63 100 30 12.6 0.0 1.9 0.6 0.9 1.0 75 75 75 100 56 12.7 0.2 1.1 0.5 0.8 0_8 75 75 75 100

6 1.2 1.5 3.8 1.3 1.9 2.2 38 38 63 100 10 1.6 1.4 2.7 1.1 1.5 1.7 50 63 63 100 30 2.3 0.1 2.2 0.7 0.9 1.0 63 63 63 100 56 2.4 0.3 1.3 0.5 0.7 0.8 63 63 63 100

6 -8.8 2.3 3_9 3.0 4.2 4_7 63 75 75 100 10 -8.5 1.4 2.7 2.7 3.6 3.9 63 63 88 100 30 -7.8 0.5 2.3 1.7 2.2 2.4 63 75 75 100 56 -7.7 0.7 1.2 1.2 1.6 1.8 63 75 75 100

The measured data were sampled such that effeCtively trials of 6, 10,30, and 56 L (1 s) measurements were obtained for each sourcelb~kgroundnoise condition. The sampling was done both periodically (every 2 sec) and randomly. For each number of measurements, er­rors and 95.0, 99.0 and 99.5% con fidence interval bounds over 8 trials were computed.

(The two traffic noise recordings were employed such that one recording consistently was added to the white­noise and the other recording consistently represemed the excitation-alone, "actual" -magnitude measuremem. [The separate recordings were used to preclude possible false favorable results that may have arisen from com­paring an excitation to itself.] The error associated v.ith the sampling alone, i.e., aside from the presence of background noise, can be determined by estimating the magnitude of one recording from the other using the correction procedure with infinitesimal background noise. The results of this comparison are given in Ta­ble 3. With periodic sampling, the error is relatively large [> 1 dB] for n :s 10; with random sampling the error is small [~ 0.2 dB] for all n.)

The resultant tests had actual signal-lo-noise ratios ranging from -9 to + 13 dB, relatively high compared to

the nominal magnitudes. The differences between the nominal and actual SIN ratios are the result of the man­ual setting of amplifier gains.

Periodic Sampling. The results, summarized in Ta­ble 4, are:

Accuracy-mean error magnitudes =:: 1.4 dB for n ~ 10 and about 2 dB for n = 6; Consiscency-standard deviation of errors <

3 dB for n ~ 10 and 3-4 dB for n = 6; Assurance-<:onfidence interval con tains error :S 75% of lhe time for nearly all cases; and Success-all tria15 were successful.

While the accuracy, consistency and success of these results are quite good, the assurance performance is unacceptable since at least 2.5% of the trials resulted in errors exceeding the confidence interval. Funhermore, the assurance did not improve significantly as the num­bers of measurementS increased from 6 to 56. The poor performance of the correCtion procedure may be the result of twO factOrs-the periodicity exhibited by the '\vhite noise" excita tio nand the possibility that the highway noise samples taken at 2-s intervals may be at... least panially correlated. In fact. comparison of Tables 3 and 4 shows the periodically sampled results to be strongly influenced by sampling error. Therefore, at least for quasi-random and/or highway-noise-like proc­esses such as those tested, a modified sampling scheme is necessary. random sampling of the stored los-interval Leq values. sampling with intervals significamly greater than 2 s, and/or greatly increased numbers of measure­ments.

Random Sampling. Random sampling of the stOred L values was evaluated with the same numbers of m~surements. The results, summarized in Table 5, are:

Accuracy-mean error magnitudes :s 1.2 dB for all cases (and :s 0.5 dB for n ~ 30); Cons iSlency-s ta nda rd devia tio n 0 f er ro rs :s 1.2 dB for nominal SiN ~ 0 dB, and for SIN :::: -10 dB with n ~ 30, but about 3 dB for SIN =::

-10 dB with n :S 10;

7

TABLE 5. RANDOMLY SAMPLED MEASUREMENTS TEST RESULTS

n MEAN ---£RROR--- ---e. I. for a --- -ERR ~ C.l. for a OK S!N Mean SO 95.0 99.0 99.5 95.0 99.0 99.5 TRlS (dB) (dB) (dB) (dB) (dB) (dB) (%) (%) (%) (%)

5 12.2 0.7 1.4 1.5 2.3 2.6 75 88 100 100 10 13.0 -0.1 1.6 1.1 1.6 1.8 75 100 100 100 30 12.9 0.1 0.6 0.7 LO 1.1 88 88 88 100 56 12.7 -0.1 0.3 0.5 0.7 0.7 100 100 100 100

6 1.8 1.2 1.4 1.6 2.4 2.7 50 88 88 100 10 2.5 0.1 1.4 1.1 1.6 1.8 63 100 100 100 30 2.3 0.2 0.7 0.7 1.0 1.1 75 88 88 100 56 2.3 0.1 0.4 0.6 0.8 0.9 88 88 88 100

6 -8.0 0.7 3.5 3.3 4.6 5.1 63 63 75 100 10 -7.3 0.9 2.7 2.2 3.0 3.3 63 75 75 100 30 -7.6 0.5 1.5 1.7 2.3 2.5 75 75 75 100 56 -7.7 0.5 0.5 1.2 1.6 1.7 75 100 100 100

• Assurance--amfidence interval contains error ~ 88% of the time for nominal SIN 2': 0 dB with I-a 2': 99%, and 100% of the time for SIN =:<

-10 dB with n =56 and l-a 2': 99%. Success-all trials were successful.

Randomized sampling greatly improved the assurance of the prOcedure such thai the correction method again produced useful estimates, given sufficient numbers of measurements. With I-a 2': 99%, as few as six meas­urements gave good results for SIN > 0 dB. However, for very poor signal·to·noise conditions, a substantial number of measurements are required, n 2': 56.

Discussion. The experimental lestS were performed using measurements of 1-s equivaleol sound levels. Equivalent sound levels of other durations could have been used-or for that mauer other noise descriptors-provided the "total" and "noise" measure­ments are of a common descriptOr and duration and the measurement results can be input validly into Equation {I}. Certainly meeting the sampling require· ments for some descriptors, e.g" L (l hr), would be laborious and other prerequisites~~Chas stationarity of the variables--difficult to meet.

The favorable results of the experimental tests may have been enhanced by the normally distributed charac~

ler of the constituent processes. However, as indicated by the random number leSts, good resulls can be ob­tained with non-Gaussian parameters. The usefulness of this procedure for strongly skewed and/or multi· modal distributions must be demonstrated. However, it is speculated that useful resulls will be obtainable given sufficient sample size and suitable sampling technique.

CONCLUSIONS Corrections for High-Noise Measurements. The

correction procedure was found to give useful results for poor (SIN =:< 0 dB) and very poor (SIN "':: -10 dB) signal-tO-noise ratios provided the numbers of meas­urements are adequate. In Table 6, the minimum re­quired numbers of measurements observed in tbe fOUf

testS of the procedure are summarized. For all evaluat· ed parameter distributions, variabilities, and other tirne­history Characteristics-acceptable results (I.e.) with assurance and success equaling or exceeding 90%) were obtained with some measurement scheme. The re­quired number of measurements decreased with de­creasing parameter variabillty-to as few as 6 [or poor~ signal-la-noise and as few as 30 for very poor signal·tO· noise-even for non-Gaussian parameter distributions. Using wider confidence intervals (i.e., higher confidence estimates), reduces the minimum Dumber of measure· ments required; however, a more precise interval may be worth the eXira measurements (e.g., for Gaussian numbers and SIN :::: 0 dB-as I-a goes from 95.0 ...... 99.5%, n . = 30 ->0 6 but the confidence interval goes

IDln . ~

from 1.5 - 5.1 dB). For some requirements, acceptable results may be obtained only wjth a suitable sampling scheme, such as random sampling.

Confidence Intervals for Low·Noise Measure­ments. For low-noise condilions (i.e., SIN :::: +10 dB), these results are primarily useful in showing the need for sampling and confidence intervals since existing standards do nOt address the variability of the measured parameters and the sampling requirements for them-accepting a single measurement of each as ade·

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TABLE 6. OBSERVED MINIMUM REQUIRED NUMBER OF MEASUREMENTS for acceptable accuracy, consistency, assurance, and success

TEST CONFIDENCE -REQUIRED n. for SiN (dB)- -e.!. (dB) for n, and SiN (dB)­mIn mln(l-a)% +10 0 -10 +10 0

Gaussian Random Numbers 95.0 100 30 0.5 1.5 (LI0 - L90 = 8 dB) 99.0 6 30 >100 2.9 2.1 =5.6

99.5 6 6 >100 3.3 5.1 =5.9 .,Rayleigh-Like Random Numbers 95.0 30+ 100 0.5 0.5 (LIO - L90 = 4 dB) 99.0 6 6 100 1.7 2.5 4.2

99.5 6 6 100 2.0 2.9 4.5 ., ., ., ., .,*Periodically Sampled Measmts. 95.0 ., .,

(LID - L90 = 2 and 6 dB) 99.0 .,99.5

Randomly Sampled Measmts. 95.0 56 .,

0.5 (LID - L90 = 2 and 6 dB) 99.0 10" 10" 56 1.6 1.6 1.6

99.5 6" 10" 56 2.6 1.8 1-7

~

no acceptable number '" poorer performance obtained with greater n

TABLE 7. CONFIDENCE OF SIGNAL ESTIMATES WITH LOW BACKGROUND NOISE SIN::::: +10 dB, n = 6 measurements

--------------T£ST------------ ----ERROR (dB)- CONFIDENCE INTERVAL (dB) for a Mean S.D. 95.0 99_0 99.5

Gaussian Random Numbers 0.5 2.0 2.9 3.3 Rayleigh-Like Random Numbers 0.3 0.9 1.7 2.0 Periodically Sampled Measmts. 1.8 3.2 Randomly Sampled Measmts. 0.7 1.4 2.6

., assurance inadequate quate. Appropriate confidence intervals for the low background noise tests with six measurements aIe sum­marized in Table 7. For six measurements and an ade­quate sampling technique (e.g., random sampling if nec­essary), the prediction error was about 0.5 dB with up­per-bound confidence intervals about 3 dB. A lO-dB SIN ratio is well within that permitted by all measure­ment standards. Oearly, even in these favorable condi­tions, multiple measurements are desirable.

RECOMMENDAnONS The correction procedure described above appears to

give useful results for a number of realistic measure­ment conditions. The procedure should be used in all measurements with background noise to gain more ex­perience with its use so that sampling requirementS can be specified for measurement conditions not tested in­this paper. Ultimately, the proposed seven-step correc­tion procedure should be incorporated into field meas­urement standards.

More tests are necessary with different parameter

variations especially those which are highly variable and/or otherwise ill-behaved, e.g., bi-modal distr..ibu­tions. In addition, sampling guidelines for Ihe imple­mentation of this procedure need to be defined furlher: the required numbers of measurements, appropriate sampling intervals or schemes (i.e., random \IS. periodic sampling), and the required confidence values appro­priate for various measurements.

REFERENCES American National Standards Institute, American Na­

tional Standard Methods for the Measurement of Sound Pressure Levels, ANSI S 1.13-1971 (R 1986).

Hassall, J.R, "Noise Measurement Techniques: Chapter 9, Handbook of Acoustical Measurement in Noise Control-Third Edition, (Ed. eM. Harris), McGraw Hill, Inc., 1991.

Bendat, J.S. and AG. Piersol, Random Data: Analvsis and Measurement Procedure, WiJey-Imerscience, 1971.

Dixon, W.J. and F.J. Massey, Jr., Introduction to S13­

tistical Analvsis (Fourth Ed.), McGraw-Hili, Inc., 1983.

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