The statistical bandwidth of Butterworth filters

Journal

ofSound

and Vibration

(1987) 115(3), 539-549

THE STATISTICAL BANDWIDTH OF BUTTERWORTH J. L. Commonwealth

DAVY

AND

I.

FILTERS

P. DUNN

Scientific and Industrial Research Organization, Division of Building Research, P.O. Box 56, Highett, Victoria 3190, Australia

(Received

3 June 1986, and in revised form 13 September 1986)

The precision of standard architectural acoustic measurements is a function of the statistical bandwidth of the band pass filters used in the measurements. The International and United States Standards on octave and fractional octave-band filters which specify the band pass filters used in architectural acoustics measurements give the effective bandwidth, but unfortunately not the statistical bandwidth of the filters. Both these Standards are currently being revised and both revisions require the use of Butterworth filter characteristics. In this paper it is shown theoretically that the ratio of statistical bandwidth to effective bandwidth for an nth order Butterworth band pass filter is 2n/(2n - 1). This is verified experimentally for third-octave third-order Butterworth band pass filters. It is also shown experimentally that this formula is approximately correct for some non-Butterworth third-octave third-order band pass filters. Because of the importance of Butterworth filters in the revised Standards, the theory of Butterworth filters is reviewed and the formulae for Butterworth filters given in both revised Standards are derived.

1. INTRODUCTION

Draft revisions of the International Standard [l] and the U.S. Standard [2] on octave and fractional octave-band filters are both based on the use of Butterworth filter characteristics. One of the reasons why the current U.S. Standard [3] is different from the current International Standard [4] is because the U.S. Standards Committee believe that the recommendations contained in the International Standard “strongly discourage filter design approaching the ideal for noise analysis. For example, if the frequency response of a Butterworth filter is centered within the tolerance, the effective bandwidth is 7% greater than it should be”. It would appear from the Draft International Standard that international opinion now agrees with the U.S. views on this point. Both draft revisions contain formulae for the design goal of the filter shape. However, the draft U.S. revision gives only the simpler formula for the transmission loss, while the draft international revision gives only the more complicated formula for the transfer function which is obtained by factorization of the transmission loss. There would appear to be a strong argument for both drafts to give both formulae. The transmission loss expression is much simpler, while the transfer function is needed if the phase response of the filter is also required. Different definitions of filter bandwidth are given on page 228 of reference [5]. Both draft revisions give the analytic expression for the ratio of the effective bandwidth to the half-power bandwidth for a Butterworth band pass filter although they express it as the ratio of the design bandwidth quotient to the reference bandwidth quotient. The effective bandwidth of a filter is required by the Standards to be equal to the nominal filter bandwidth within certain tolerances. The effective bandwidth of a filter is the appropriate bandwidth to use when determining how much white random noise power a filter will pass. However, when one is interested 539 0022-460X/87/120539+ 11 %03.00/O

@ 1987 Academic Press Inc. (London) Limited

540

J.

L. DAVY

AND

I. P. DUNN

in calculating the variance of the random noise power passed by a filter, the statistical bandwidth of the filter is the bandwidth that should be used in the calculations. Neither the present Standards nor the proposed revisions provide or require information on this characteristic, because in the past the difference between statistical bandwidth and effective bandwidth has not been widely recognized. Because of their interest in the uncertainty of room acoustical measurements, the authors have long been interested in the statistical bandwidth of filters. However, experience shows that the amount of work involved in experimentally determining either the effective bandwidth or statistical bandwidth of a band pass filter is large. It is not just a matter of obtaining the frequency response plot of the filter on a graphic level recorder and checking that the filter shape lies within the tolerances marked on a transparent overlay. The transmission loss of the filter has to be digitized (at a large number of frequencies) using a graphic level recorder chart or a voltmeter. Then these digitized values have to be used to calculate numerical approximations to the integrals involved in the bandwidth definitions. The first determination of effective and statistical bandwidth that the authors made [6] was the manual measurement and calculation of a 1 kHz third-octave filter. This filter was found to have an effective bandwidth 5% greater than its nominal bandwidth and a statistical bandwidth 30% greater than its nominal bandwidth. These measurements were made by measuring at 21 frequencies spaced l/24 of an octave apart. When one considers that a third-octave filter is typically only one of 30 in a filter set, the task of measuring the effective and statistical bandwidth of all of them seems daunting. This filter was manufactured as a third-order band pass Butterworth filter whose lower 3 dB cut-off frequency was equal to 0.8945 times the centre frequency of the filter. We used a computer program to calculate the bandwidth quotients and centre frequencies of the three first-order sections of a filter of this type. We then used these bandwidth quotients and centre frequencies to obtain a formula for the transmission loss of the filter. This formula was then numerically integrated to show that the effective bandwidth was equal to the nominal bandwidth and that the statistical bandwidth was 20% greater than the nominal bandwidth. The numerical integrations were carried out from 0.8 times the midband frequency to 1.25 times the midband frequency in steps of 0.01 times the midband frequency. The other fact we knew at this time was that the statistical bandwidth is always greater than or equal to the effective bandwidth for any filter. The checking of filter shapes and the determination of effective and statistical bandwidths was a task which obviously called for a frequency synthesizer which could be computer controlled. It is still not a task to be undertaken lightly. Our computer program took 4 hours and 28 minutes to check the 30 third-octave filters in the filter set. The recent appearance of the draft revisions of the filter Standards, with their emphasis on Butterworth filters as the design goal, renewed our interest in such filters and their statistical bandwidths. We have been able to show analytically that the ratio of statistical bandwidth to effective bandwidth for a Butterworth band pass filter of order n is 2n/(2n - 1). Thus, for a third-order Butterworth band pass filter the statistical bandwidth is 20% greater than the effective bandwidth, which agrees with our earlier numerical calculation. Although this 20% difference leads only to a 10% difference in the formulae for standard deviation, this is still significant when comparing theory and experiment. For example, see reference [7] where theory and experiment when averaged across the frequency range 100 Hz to 5 kHz agree to within 10%. In this paper the theory of Butterworth band pass filters is reviewed and the derivation of the ratio of their statistical bandwidth to their effective bandwidth is presented, together with some experimental measurements of the bandwidth of actual filters.

STATISTICAL

BANDWIDTH

OF

541

FILTERS

2. BUTTERWORTH FILTERS Let s be the Laplace transform variable, w the radian frequency and j the square root of minus one. If H(s) is the transfer function of a filter, then the frequency response of the filter is HCjw) and the magnitude squared of the frequency response is (a list of symbols is given in the Appendix) GCjw) = 1HCjw) I*= HCjw)fi(jo)

= HCjw)H(-jw).

(2.1)

A transfer function must be expressible as a ratio of two polynomials in s with real coefficients. Hence GCjw) must be expressible as the ratio of two polynomials in w. Also one sees from equation (2.1) that G(jw) must be an even function of w. A low pass Butterworth filter of order n is defined by G(jo) = l/L(w*),

(2.2)

where L(x) is a polynomial of order n in x, and dkL(x)/dxk I._,, = 0

for kc n - 1.

(2.3)

Note that the numerator of G(jw) is 1 and GCjw) is an even function of w so the denominator must be a polynomial in o*. Equation (2.3) says that as many derivatives of L(x) as possible must be zero when x is zero. This is the reason why Butterworth filters are also known as maximally flat filters. If Bmxrn, Wl=O

L(x) = i

(2.4)

then condition (2.3) implies that B,=O

for Lyman-1.

(2.5)

If one now normalizes the transfer function by requiring that 1H’(O) ( = G(0) = 1,

(2.6)

B,,= 1.

(2.7)

then

Thus one has GCjo) = l/( 1 + B,p2”).

(2.8)

One can further normalize this expression by requiring that the half-power frequency occurs at a frequency of 1 radian per second: i.e., IH(jl) I’= G(j1) = l/2.

cut-off (2.9)

This gives B, = 1,

and

G(jw) = l/(1 SW’“).

(2.10, 2.11)

To convert from this normalized low pass filter to a band pass filter with midband frequency w, and half-power bandwidth B, one replaces w on the right side of equation (2.11) with (w -&lw)lB

= Qtwlwn -wn/~),

(2.12)

where

Q=wnlB

(2.13)

542

J. L. DAVY

AND

1. P. DUNN

is the bandwidth quotient of the band pass filter. Thus the magnitude squared of the frequency response of a band pass Butterworth filter of order n is G(jw) = 1H&o) I2= l/[ 1 + Q2”(~/o,

- w,/w)“‘].

(2.14)

The maximum value of this function is HZ,,, = 1H(jw,,,)I’=

1.

(2.15)

For complete generality the right side of expression (2.14) would have to be multiplied by the maximum value of the actual filter response. One can avoid the need to do this by regarding G(jw) as the ratio of the actual value to the maximum value. The half-power bandwidth and midband frequency of this filter can now be determined, to show that they are actually B and w, as claimed. The half-power frequencies occur when - w,/w)~” = (w - w;/w)~“/ B2” = 1.

Q2’$+,,,

(2.16)

Since the half-power frequencies must be real, this gives (w-w;/w)/B=il,

(2.17)

since w, and B are real. Solving this quadratic equation gives o=*(B/2)~[wZ,+(B/2)2]“2=w,{~1/(2Q)~[l+1/(2Q)2]”2}.

(2.18)

The half-power bandwidth is the modulus of the difference between the largest and smallest positive half-power frequencies. It is easily seen that the half-power bandwidth is B. The midband frequency is the geometric mean of the largest and smallest positive half-power frequencies. Thus it is equal to

Wdt+W2)21"2 The effective bandwidth

-(B/2)}{[wZ,+(B/2)2]“2+(B/2)}}“2=w,.

(2.19)

of the filter is given by ~~/[~+Q’“(w/~,-w~/w)~~]~w l/[ 1 + Q2”(x - l/x)‘“]

=W,

In Appendix A of the Butterworth band pass system, the white noise The ratio of the effective Putting

(2.20)

dx.

Draft U.S. Standard [2], it is stated that “By transforming the transmission-loss equation to the equivalent low pass coordinate power in the passband may be calculated directly by integration. bandwidth to the 3-dB bandwidth is given by (r/2n)/sin (7r/2n)“. 22=x-l/x,

(2.21)

one obtains x=z*(z2+1)“2

and dx=[l*z/(z’+l)“*]dz.

(2.22, 2.23)

Since the integration here is between 0 and co only the positive square root is used. Hence the effective bandwidth is given by OD B,=w, (2.24) {1/[1+(2Qz)2”]}dz+w, m {z/{[l+ (ZQz)‘“](z’+ 1)“‘)) dz. I -0j I -CC The second integrand is an odd function so its integral from -co to +OOis zero, and the first integrand is an even function. Therefore B, = 2w, =

W(l,[1+(2Qz)2”]}dz= I B[7r/[2n)]/sin [7r/(2n)J,

B

I0

iT[l/(l+y2n)]dy (2.25)

STATISTICAL

BANDWIDTH

543

OF FILTERS

where integral number 3.241.2 of Gradshteyn and Ryzhik [8] has been used. This result agrees with the statement quoted above from the Draft U.S. Standard. One can now evaluate the following integral: 12= OW]H(ju)l’do. I Applying the same transformations bandwidth yields I1 = B

(2.26)

to this integral as used\ in evaluating the effective

+Y”‘)~] dy = B[(2n - 1)r/(4n2)]/sin

OD[l/(1

Jo

where integral number 3.241.5 of Gradshteyn The statistical bandwidth of the filter is

[m/(2n)],

(2.27)

and Ryzhik [8] has been used.

B.=[~~lHUw)12d~]2/lg.lHUW)14dy=B:l~2

(2.28)

since H2,,, equals 1. Hence the statistical bandwidth is B, = B[7r/(2n - l)]/sin

[7r/(2n)],

(2.29)

and the ratio of the statistical bandwidth to the effective bandwidth is B,/B,

- 1).

= 2n/(2n

(2.30)

For a third-order Butterworth band pass filter this ratio is equal to 1a2 which agrees with our earlier numerical calculation described in the introduction. Appendix A of the draft revision of the International Standard gives the normalized transfer function of a Butterworth band pass filter of order n. Appendix B of the draft revision of the U.S. Standard gives the normalized transfer function of a Butterworth band pass filter of order 3. The normalized transfer function is needed if one wishes to calculate the phase or group delay response of the filter. It can now be shown how the normalized transfer function is obtained from the magnitude squared of the frequency response. First one factorizes the magnitude squared of the frequency response for a normalized low pass Butterworth filter of order n. Rutting s equal to jw in equation (2.11) gives G(s) = H(s)H(-s)

= l/[l+

(-l)‘?‘“].

(2.31)

For the filter to be stable the poles of H(s) mist lie in the left half of the complex plane. Thus one has to find the poles of G(s) which lie in the left half of the complex plane. If s is a pole of G(s), then s 2n =

-1 =

exp b(2k+

1=

expCj2kr)

l)~]

if n is even if n is odd I ’

(2.32)

where k is an arbitrary integer. Thus the poles of G(s) are S= where k=O,...

exp b(2k+ 1 exp (jkr/n)

l)r/(2n)]

if n is even if n is odd I ’

(2.33)

,2n - 1. If n is odd the left-hand poles are

s=-1

and

-exp (*jkr/n)

for k = 1,. . . , (n - 1)/2.

(2.34)

544

J. L. DAVY

AND

I. P. DUNN

If n is even the left-hand poles are s=-exp[*j(2k+l)fl/2n]

for k=O,...,n/2-1.

(2.35)

The pole at s equals -1 produces the term (s + 1). The complex conjugate pole pairs of the form s = -exp (+jnR) the factorization to produce terms of the form [s+exp

(j&)][r+exp

(-j&)1

can be combined in

= s2+2s cos (rR)+l.

(2.36)

Thus for a normalized low pass Butter-worth filter of order n the transfer function is (n-l)/2 H(s) = 1 (s+l) fl [.s2+2scos(rrk/n)+1] (2.37) !+=I /I i if n is odd, and n/Z-l H(s)

=

1

{s2+2s cos [(2k+ l)r/(2n)]+

n

I

l}

(2.38)

k=O

is n is even. To convert from this normalized low pass filter to a band pass filter with midband frequency w, and half-power bandwidth B one replaces s on the right side of equations (2.37) and (2.38) with (s+f&/r)/B=

Q(s/o*+W,/S).

(2.39)

Hence for a normalized band pass Butterworth filter of order n the transfer function is (n-1)/2

hJs)l

H(r) = I/ [l+ Q(s/%l+ { +2Q(s/w,

l-I Cl+ Q2Wwn +%Js)2 k=l

+w,Js)

cos (?rk/n)]

(2.40)

I

if n is odd, and n/2-1 H(s)=1

k!.

I

{1+Q2(s/w,+W,/s)2+2Q(s/w,+w,/s)cos[(2k+l)~/(2n)l} (2.41)

if n is even. The frequency response of a normalized band pass Butterworth obtained by putting s equal to jo. This gives

A

H(jc.0) = 1

(n-1)/2

[l +jQ(w/o,

+_G’Q(~/w,

filter of order n is

-

- o,/w)

%/~)I

,rI, [I- Q2Wwn - ~m/~)2 (2.42)

cos (rk/n)]

if n is odd, and n/2--L H(jw)=

1 /

k!. {l- Q’(w/o,

-W,/W)‘+j2Q(w/wm

-w,/w)

cos [(2k+ l)n/(2n)]} (2.43)

if n is even. Equations (2.42) and (2.43) agree with equations (Al) and (A2) of Appendix A of the draft revision of the International Standard [ 11. If equation (2.42) is evaluated for n = 3, the result agrees with equation (Bl) of the draft revision of the U.S. Standard [2], provided the inverse of the right side of equation (Bl) is used.

STATISTICAL

BANDWIDTH

OF

FILTERS

545

The transmission loss or attenuation A of the filter relative to the minimum transmission loss or attenuation (which for Butterworth filters is the same as the midband frequency transmission loss or attenuation) is - 10 log,,, [ G(jw)]

A = 10 log,, [max G(jw)]

= 10 log,,[l+

- w,,,/ti j”‘],

Q’“(o/w,

(2.44) where equations (2.14) and (2.15) have been used. Equation (2.44) agrees with equation (10) of the draft revision of the U.S. Standard. The relative attenuation can also be calculated from equations (2.42) and (2.43), with A= -1010g10(H(jw)~2=2010g,o

(2.45)

(l/IH(jw)().

This second method is the one suggested by the draft revision of the International Standard [I], despite the fact that it is more complicated than equation (2.44). The phase $J is given by 4 =arctan

{Imag [H(jw)]/Real

(2.46)

[H(jw)]}.

The group delay of the filter is the derivative of the phase 4 with respect to the frequency w. Butterworth band pass filters of order n will normally be implemented as a cascade of n first-order band pass filters. The midband frequencies and bandwidth quotients of these first-order band pass filters are calculated starting from the complex conjugate pole pairs of the low pass Butterworth filter of order n given by equations (2.34) or (2.35). If one lets sk and Sk be a complex conjugate pole pair, then the transfer function of the low pass filter contains a factor of the form

I/[(s-&)(S-%)I. Converting to a band of the form

(2.39) one obtains

a factor

&l[Q(~/% + %2/s) - %I1 =(W,/Q)2S2/{[S2-(O,/Q)SLS+OZm][S’-(Om/Q)SkS+Wzm]}.

(2.48)

l/{[Q(S/%

pass filter by replacing

(2.47)

+6&/s)

s with formula

-

The poles of this factor are xk=&+,,{[~~/(2Q)]*{-l+[~k/(2Q)]2}”Z} Both these pairs of complex

conjugate s/[s*-2s

These factors can now be compared

poles combine

and

xk.

to each produce

factors of the form

Real(x,)+l~,1~].

to the transfer

(2.49)

(2.50)

function

of a first order band pass filter,

(w,/Q)sl[s2+(w,/Q)s+wbl,

(2.51)

to define the midband frequency w, and bandwidth quotient of the first order band pass filter. When n is odd the pole at -1 produces a low pass filter factor of the form s + 1. Converting to a band pass filter by replacing s with formula (2.39) produces a factor of the form

~/[Q(~/~m+~m/~)+~l=(~,/Q)~/[~~+(~,/Q)~+~~l, which is a cascade filter section as the original filter.

with the same midband

frequency

and bandwidth

(2.52) quotient

J. L. DAVY

546

For a third-octave Butterworth one-third octave is given by

band

AND

1. P. DUNN

pass filter of order 3 the effective

Be = (2”h-2m”h)~, The half-power

bandwidth

bandwidth

=0*23156w,.

(2.53)

is

B= Be sin[~/(2n)]/[~/(2n)]=3B,/~=0.22113w, and the bandwidth

quotient

(2.54)

is Q = w,/ B = 4.52229.

Using equation

(2.18) one finds that the positive 0.895530,

The half-power frequencies of

of

frequencies

should

half-power

and

or 3 dB down frequencies

l.l1666w,.

be compared

2-Ww m = 0.8909Ow,

(2.55)

and

are (2.56)

to the nominal

third-octave

2’16w, = l.l2246w,.

band

edge

(2.57)

The individual first-order band pass filters needed to form this filter can be calculated as described above to have the following midband frequencies and bandwidth quotients: Bandwidth quotient 9.0861 4.5223 9.0861.

Midband frequency 0.90869w, WWl 1.10048w, The statistical

bandwidth

is B, = 2nBe/(2n

The transfer

function

HCjw) = 1/{[1

(2.58)

- 1) = 1.2B, = 0.27787w,.

(2.59)

of the filter is

+jQ(wlw, -wJw)I[l-

Q2(w/wm-~,/w)‘+jQ(~/w,

-wJw)l~, (2.60)

and the magnitude

squared

of the frequency

GCjw) = 1H(jw) I’= l/[l

response

is

+ Q6(w/w,

3. COMPUTER-CONTROLLED

FILTER

- w,/w)~].

(2.61)

CHECKING

The purchase of a Hewlett-Packard type 3325A Synthesizer/Function Generator enabled us to consider computer-controlled measurements on filters. The filter set that we wished to check was the GenRad type 1925 One-third Octave Multifilter. This multifilter is connected to a GenRad type 1926 Multichannel Rms Detector to form a GenRad type 1921 Real-time Analyzer. For filter sets other than the one which forms part of our real-time analyzer a Hewlett-Packard type 3478A multimeter was used as the measuring device instead of the real-time analyzer. Three Briiel and Kjaer type 1612 third-octave filter sets were tested in this way. The results of two series of measurements of effective bandwidth are shown in Figure 1. The integrals needed for the bandwidth calculations were performed over a 0.3 decade frequency range, with measurements being made in steps of 0.01 of a decade. The choice of 0.3 decade for the frequency range was the result of experimentation; any lesser value risks introducing systematic bias into measurements of bandwidth. The selection of an interval width of 0.01 decade was deemed sufficient sampling within the frequency range.

STATISTICAL

0.8

t

0

1 a a ’ a

BANDWIDTH

’ 8

’ 4

b.0315 0.063 0.125 0.25 0.5

OF

’ I

I

547

FILTERS

’ 3 2

’ i I ’ L / 16' ’ 8

4

Frequency (kHz )

Figure 1. Ratio of measured effective bandwidth to nominal bandwidth for a set of 30 filters. 0, 1981 measurements; 0, 1985 measurements.

as a function

of frequency

in hertz

These measurements were performed on our GenRad Multifilter. The first series was conducted in 1981 and the second in 1985. As stated in the introduction this series of measurements took 4 hours and 28 minutes. The effective bandwidths have been divided by the nominal third-octave bandwidths. It is these ratios that are plotted in Figure 1. The mean of the 1981 measurements was 0.970 with a standard deviation of O-033. The 1985 measurements gave a mean of O-964 and a standard deviation of O-035. All measurements were within 10% of the nominal third-octave bandwidth as demanded by the current standards [3,4]. The draft revision of the U.S. Standard [2] demands that the effective bandwidth error be less than 41 millibel for type 2 and 3 filters. Not surprisingly, this corresponds to an effective bandwidth error of 10%. The 10 millibel error allowed for type 0 and the 25 millibel error allowed for type 1 filters correspond respectively to 2.3% and 6% bandwidth errors. About 20% of these filters would not satisfy this type 1 requirement and more than half would not satisfy this type 0 requirement. The ratios of statistical bandwidth to nominal bandwidth for these two series of measurements are shown in Figure 2. The mean and standard deviation for the 1981 measurements were 1.185 and 0.026, while the 1985 measurements gave 1.192 and O-028. The mean ratios of statistical to effective bandwidth for the two series of measurements were 1.222 and 1.237. Since these filters are supposed to be third-order Butterworth filters, equation (2.30) states that this ratio should be l-2. The program also checked the filter shapes for compliance with the limits in the International Standard [4]. This checking was performed at the measurement points in the range of integration and not over the full frequency range demanded in the standard. Two filters failed marginally in the 1981 test and these two filters and another failed marginally in the 1985 test. All three filters failed only at one measurement point which was the one near the lower third-octave band edge frequency. The transmission loss was just slightly too great at this measurement point. The three Briiel and Kjaer type 1612 third-octave filter sets that were tested were known to contain filters that had failed. They have not been used for measurement purposes in

548

J. L. DAVY

I.10

AND

10 11 11 11 1 ” 0.0315 0.063 0.125 0,25

1 ’ 11

0.5

1. P. DUNN

’ 11 I

““I 2

1’1 4

6

” 16

Frequency (kHz1

Figure 2. Ratio of measured statistical bandwidth to nominal bandwidth as a function of frequency in hertz for a set of 30 filters. 0, 1981 measurements; 0, 1985 measurements.

the last 10 years. They contain passive filters and an output amplifier in contrast to the active filters of the GenRad 1925 third-octave multifilter. They are third-order filters but not Butter-worth filters. Each filter set contains 33 third-octave filters from 25 Hz to 40 kHz inclusive. The three filter sets had five, two and eight filters respectively which failed the filter shape requirements in the International Standard [4]. The filter sets also had four, two and two filters respectively whose effective bandwidth differed by more than the 10% allowed from the nominal third-octave bandwidth. Since some of the filters in these sets had failed completely it was necessary to reject outliers before averaging the effective and statistical bandwidths. The procedure used for rejection of outliers was in accordance with ASTM E178-80 [9]. After outliers were rejected the averages of the ratios of effective bandwidth to nominal bandwidth for the three filter sets were 0,999, 1.014 and 1.018 with standard deviations 0,038, 0.037 and 0.032 respectively. The averages of the ratios of statistical bandwidth to nominal bandwidth were 1*189,1-197 and l-194 with standard deviations of O-017,0*028 and O-019 respectively. These values are close to the values of 1 and 1.2 that we would predict for a third-octave third-order Butterworth band pass filter. Of course, it is again noted that these filters are not designed as Butterworth filters. 4. CONCLUSIONS

Since the filters used in acoustics are designed to have their effective bandwidth equal to their nominal bandwidth, this paper enables the statistical bandwidth of the filters to be calculated. This then enables the use of formulae to calculate the precision of room acoustics measurements. In the past the statistical bandwidth has been approximated by the nominal bandwidth. It has been shown here that the ratio of statistical bandwidth to effective bandwidth for an nth-order Butterworth band pass filter is 2n/(2n - 1). This formula has been verified for third-order filters by experimental measurements on a set of 30 third-octave filters. The formula also agrees reasonably well with measurements on three other third-octave

STATISTICAL

BANDWIDTH

OF

FILTERS

549

third-order band pass filter sets, although these filters are not designed to be Butterworth filters.

REFERENCES 1. INTERNATIONAL ELECTROTECHNICAL COMMISSION 1985 Document-29C (Secretariat) 55. Draft-Octave and fractional octave-band filters (Revision of IEC 225). $2. AMERICAN NATIONAL STANDARDS INSTITUTE 1985 ANSZSl.ll-198X (revision of ANSI Sl.ll-1966 R1976). Specification for octave-band and fractional-octave-band analog and digital filters. Eighth Draft. 3. AMERICAN NATIONAL STANDARDS INSTITUTE 1966 ANSZSl.ll-1966 R( 1976). Specification for octave, half-octave, and third-octave band filter sets. 4. INTERNATIONAL ELECTROTECHNICAL COMMISSION 1966 ZEC 225-1966. Octave, halfoctave and third-octave band filters intended for the analysis of sounds and vibrations. 5. J. S. BENDAT and A. G. PIERSOL 1971 Random Data: Analysis and Measurement Procedures. New York: John Wiley and Sons. 6. J. L. DAVY, I. P. DUNN and P. DUBOUT 1979 Acusticn 43, 12-25. The variance of decay rates in reverberation rooms. 7. J. L. DAVY 1986 Journal ofSound and Vibration 107,361-373. The ensemble variance of random noise in a reverberation room. 8. I. S. GRADSHTEYN and I. M. RYZHIK 1965 Table oflntegrals, Series, and Products. New York: Academic Press, fourth edition. 9. AMERICAN SOCIETY FOR TESTING MATERIALS 1980 ASTM E178-80. Dealing with outliers.

APPENDIX: A B B, B, B, G H H zzmax j k L ; R S X xk Z 4 w w,,,

LIST OF SYMBOLS

relative transmission loss half-power bandwidth effective bandwidth (also known as noise bandwidth) mth coefficient of polynomial statistical bandwidth magnitude squared of frequency response transfer function maximum value of modulus of frequency response value of integral square root of minus one integer polynomial order of filter bandwidth quotient variable Laplace transform variable variable pole variable phase response radian frequency midband radian frequency

t Note added in proof Reference [2] has now been published as AMERICAN NATIONAL STANDARDS INSTITUTE 1986 ANSI 53.11-1986. Specification for octave-band and fractional-octave-band analog and digital filters. The authors of this paper have not seen a copy and do not know if there are any changes from reference

121.