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Impulse testing techniques for auditoria
Applied Acoustics 17 (1984) 165 181
Impulse Testing Techniques for Auditoria M. Barron Department of Architecture, University of Cambridge, Cambridge (Great Britain) (Received: 15 February, 1983)
S UMMA R Y
Though computer-generated low crest-factor signals can now be produced and processed to derive the impulse response of auditoria, the use o f impulsive signals remains the simplest and, in many cases, a fully viable technique. The advantages o f a uni-polar half-cycle sine wave test signal are discussed and compared with those of the hi-polar single-cycle signal The particular requirements for visual display of the impulse response are reviewed, as are the signal requirements and analysis options for objective measures involving integrated energy. In each case the implications are considered o f interference between reflections which arrive simultaneously.
INTRODUCTION Evaluation of the impulse response o f an auditorium is fundamental to m u c h current work in the area o f objective testing. Since the transmission between two points in a r o o m is linear, it might be expected that techniques applied to other linear systems would automatically be appropriate to r o o m response as well. This is, however, not necessarily the case, due principally to the complexity of r o o m response, including as it does thousands of reflections or, from a modal standpoint, thousands o f modes. There is also a further difference, namely that for auditoria analysis of energy is made on a constant percentage bandwidth basis, generally in octaves or third-octaves (i.e. logarithmic frequency), rather than linear frequency. 165 Applied Acoustics 0003-682X/84/$03-00 ~'i Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
166
M. Barron
The problems of testing with an impulsive signal are well publicised. To achieve adequate signal-to-noise ratio within limits set by linear sound propagation is not easy and for this reason testing with long duration signals generated by computer has been proposed by Schroeder, 1 Berkhout et al. z and Aoshima. 3 In a quiet environment, testing with impulsive signals is nevertheless a viable technique which requires only simple apparatus and no computer. In acoustic models, impulsive spark sources also have advantages, particularly their omni-directional characteristic. Testing with impulsive signals will remain a common technique. This paper reviews the implications of using typical test signals. The results of this review have been used in the selection of test signals for the current Acoustic Survey of British Auditoria.
T H E BASIC R E L A T I O N S H I P S For completeness, but at the risk of repetition, it is appropriate to quote the fundamental mathematical relationships governing transmission between two points in a room. The impulse response, g(t), is a real function of time and is a unique descriptor of this transmission characteristic. The frequency spectrum, G(f), where f is frequency, is related to g(t) by Fourier transformation. G(f) is a complex function. The frequency response, as conventionally measured, is IG(f)l:
G(f) =
f
QC
g(t) exp (-j2nft) dt --
(1)
OC
The signal, p(t), received in a room from a source is equal to the impulse response convoluted with the source signal, s(t): p ( t ) = g ( t ) * s( t)
P(f) = a(f)s(f)
(2)
where P(f) and S ( f ) are the spectra of the received and emitted signals. If s(t) is an even function, then S ( f ) is also a real even function, j = x / / - 1. Schroeder 4 has shown that the traditional reverberant decay, n2(t), produced by switching offa noise signal is related to the impulse response. A noise decay contains random fluctuations due to the random time history immediately prior to switch-off. However, the ensemble average
Impulse testing techniques for auditoria
167
of an infinite number of squared noise decays, (n2(t)), is related to the squared impulse response, g2('r), as follows: (n2(/))
=
g2(z)
dz =
g2(z)
dr -
g2(z)
dz
(3)
The time interval for integration is between t and infinity, which implies measurement in reverse time. The second form above gives an alternative, namely the difference between the total integrated energy and the integral from 0 to t. Equation (3) thus supplies the link between the two traditional methods of measuring reverberation time: with noise and impulse signals. The traditional level recorder will measure instantaneous rms pressure level with an appropriate averaging time to smooth fluctuations. The property of an exponential function, that it is replicated on differentiation or integration, implies that the same decay rate will result for a true exponential decay measured either with noise or an impulsive signal in the traditional manner. The integrated impulse method, based on evaluation
i,?S |
0 Fig. 1.
- - ~ - .
t
(a) Idealised squared impulse response. (b) Corresponding build-up curve (integral from 0 to t) and decay curve (integral from t to or).
168
M. Barton
of the right hand side of eqn. (3), is now well established to determine the detailed nature of sound decay and the subjectively significant early decay, s Equation (3) also indicates the complementary nature of sound buildup and decay. This is illustrated for an idealised impulse characteristic in Fig. 1. When the integrated energy is converted into decibels, only the decay has a linear characteristic.
T W O BASIC TEST SIGNALS The impulse response, g(t), refers to the response to the idealised Dirac delta function, which has a spectrum with equal energy per cycle. In practice, there is a limit to intensity due to non-linear propagation. The energy in a real pulse is proportional to its duration. Extremely short pulses, which would have a wide bandwidth, are unlikely to result in an adequate signal-to-noise ratio. Fortunately, the true impulse response, g(t), is rarely required as such. For the selection of a suitable impulse signal, it is necessary to consider both the temporal signature and frequency spectrum of the signal. Initially, two basic impulse signals will be discussed: the half- and whole-cycle sine wave. The half-cycle sine wave is illustrated in Fig. 2(a). Being derived from a sine wave, sin 2rtfot, of frequency f0, it has a duration 1/2j0. Its amplitude spectrum is given by:
I S ( f ) l - ~(fo2 _ f z ) cos 2~0
(4)
This is plotted in Fig. 3 in a constant percentage bandwidth form. At low frequencies, below fo, its spectrum is identical to a ideal delta function, namely flat on an energy per cycle basis and falling at 3 dB/octave on a constant percentage bandwidth basis. For measurements in octave bands, this signal is usable for centre frequencies ofjo and below. Both the low frequency characteristic and rapid drop off of energy at high frequencies occur for any shape of impulse which is uni-polar (other examples are a triangular or rectangular form). The spectrum for a bi-polar signal, such as a single-cycle pulse, contains no DC component. Figure 2(b) illustrates how a single-cycle impulse is the convolution of a half-cycle sine wave and an odd-impulse pair. The
169
Impulse testing techniques Jbr auditoria
(a)
% ½f,
Fig. 2.
(hi (a) Half-cycle sine wave. (b) Half-cycle sine wave convoluted with an odd-impulse pair produces a single-cycle sine wave.
I
I
I
I
I
I
I
I
-5 -10
/
dB
/
-15
/1
f
9d%ve//
/
-20 -25
/
I I
/
I I I
/ // i
f"16"" Fig. 3.
f
"/8
i
f
"/4
f
l
=
hi
"/2
f°
2f°
"x
~ave I 4f.
\ 8f. Frequency
Spectra on a constant percentage bandwidth basis of a half-cycle sine wave (continuous line) and a single-cycle sine wave (broken line).
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M. Barton
amplitude spectrum of an odd-impulse pair is: Is(f)l = 2
sin
rtfz
(5)
so the spectrum of the full-cycle sine wave is the product of expressions in eqns. (4) and (5). It contains additional frequencies of zero energy and has an additional 6 dB/octave slope at low frequencies, also shown in Fig. 3. For measurements in octave bands, this signal is only really valid for octave centre frequencies close to fo. Natural impulsive sources tend to produce bi-polar rather than unipolar responses. Spark sources, at least those using a third electrode for a trigger, generally have a spectrum similar to that for a bi-polar signal. The spark energy is a determinant of signal duration and hence the higher frequency limit of measurement. Predictably, signal-to-noise problems have been encountered at low frequencies with spark sources; use of high energy sparks creates the risk of non-linear acoustic propagation. If the signal-to-noise ratio is adequate, however, a very simple , '25
ms,
(a)
(b) Fig. 4.
Acoustic signal from a spark source before (a) and after (b) passing through an integrator.
lrnpulse testing techniques for auditoria
171
technique enables octave measurements to be made with a bi-polar source signal. Comparison of the low-frequency spectra in Fig. 3 suggests the use of a 6 dB/octave bass boost filter. Such a filter is equivalent to an integrator and if a single cycle of a sine wave, sin 2rCfot, is fed into such a filter, the output is (1 - c o s 2rCfot), where 0 < t < 1Ifo. This filter thus produces the response that would be obtained for a uni-polar signal. Figure 4 illustrates the acoustic time signature of a spark source before and after passing through such an integrator. (The integrated signal is not fully uni-polar due to the negative peak in the spark signal being larger; the subsequent decay to zero is due to incidental high-pass filtering to block DC in the analysis system.) Loudspeakers fed with uni-polar signals can distort them severely; both a fiat frequency response and clean impulse response are required. (Perfect reproduction is not possible since loudspeakers cannot radiate zero frequency.) Theoretically, a certain time signature exists which when fed to the loudspeaker will produce the desired acoustic output. A paper by Winter et al. 6 describes this procedure: how with a loudspeaker with an impulse response, h(t), the required signal spectrum, S ( f ) , is calculated as 1 / H ( f ) , where H ( f ) is the Fourier transform of h(t) (see eqns. (1) and (2)). This technique demands some compromises, however, involving some form of frequency band limiting, which is not described.
VISUAL P R E S E N T A T I O N OF I M P U L S E R E S P O N S E S The display of the impulse response in an auditorium on an oscilloscope screen remains a traditional testing technique. Either the pressure or the pressure squared is displayed as a function of time (see Fig. 5). The pressure squared response is currently preferred for two principal reasons: firstly, that subjective response appears to be better represented objectively by measures which relate to energy rather than pressure, and secondly, that in the process of squaring the significant reflections are better separated from low level scattered components. Due to the inevitable temporal proximity of reflections at some locations, interference will occur which can frustrate isolation of individual reflections. To minimise interference, short duration signals are to be preferred. This cannot be taken to the extreme, however, because it is necessary also to consider the spectrum of the signal. With a very short duration signal, a high proportion of the energy will be high
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M. Barton
(a)
(b) Fig. 5.
The pressure (a) and pressure squared (b) impulse responses measured in a concert hall at a source-receiver distance o f 23-2 m.
frequency, which would accentuate reflections off small surfaces, reflections which are unlikely to be significant subjectively. Optimum signal durations for visual responses of auditoria probably lie in the region of 0.3-1 ms. As will be stressed below, comparative assessment of impulse responses is appropriate and ideally the same acoustic source signal should be used for such comparisons. Whatever source signal is used, cases will still exist where simultaneous reflections overlap. The only solution to avoid such interference is to take impulse responses at a series of adjacent microphone positions. The visual impulse response gives a clear indication of the arrival time of early reflections. It can obviously also be used to expose intense echoes. However, the attempt to relate subjective characteristics to the visual impulse response has been largely unsuccessful. The work at G6ttingen in this field during the 1950s is summarised in reference 7. Likewise,
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173
(a)
(b) Fig. 6.
Squared impulse responses, measured in the same concert hall as the example in Fig. 5, at two source-receiver distances: (a) 13-6 m and (b) 30.6 m.
attempts at measures based on reflection statistics 8 have not proved useful. Most objective measures now considered valuable are related to integrated energy, which will be discussed in the following sections. A fundamental problem of interpreting impulse responses is that the visual nature of the response is very much a function of the source-receiver distance. For receiver positions close to the source, the delays of the first reflections tend to be large and reflections are well spread out and at low levels relative to the direct sound. At receiver positions far from the source the first reflection delays are small, the density of reflections is high and levels of the first reflections are close to the direct sound level. This behaviour occurs in all auditorium spaces and is an obvious consequence of the fact that the image positions associated with the first reflections are fixed in space. Figure 6 illustrates this
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M. Barron
Configurofion 2
Configurcttion 6
Configurcltion 4 Fig. 7.
Pressure impulse responses measured at fixed source-receiver positions in three stage configurations in the Gulbenkian Great Hall, Lisbon.
b e h a v i o u r with impulse responses m e a s u r e d in a full-size concert hall with source-receiver distances o f 13.6 m and 30.6 m. The question, therefore, is w h e t h e r this obvious transition in the impulse response c o r r e s p o n d s with a subjective response. If one has to generalise, then the answer appears to be no. This is particularly the case with music; in concert halls, experience of listening suggests that a l t h o u g h there are subjective differences between the s o u n d at the front a n d rear,
Impulse testing techniquesfor auditoria
175
there is not an overriding subjective correlate of the inevitable transition between the extremes illustrated in Fig. 6. A more hopeful comparison would appear to be between impulse responses measured at similar source-receiver distances. This obviously restricts the usefulness of the impulse response as an indicator, but with a library of responses recorded in different halls comparisons would be possible. The following is included here to indicate the serious problem of deriving subjective response from the visually displayed impulse response. In the study of the Gulbenkian Great Hall in Lisbon, 9 impulse responses were recorded in the same conditions as those in which subjective tests were conducted; the subjective effects of changes in the stage conditions were investigated for fixed source-receiver positions. Figure 7 shows the (pressure) impulse responses for three stage configurations for which the original labels 2, 6 and 4 have been u s e d . 9 The responses illustrate approximately the first 100 ms after the direct sound. There is a clear increase in the number of early reflections from configuration 2 to 6 and 6 to 4, which corresponds with expectations since the number of reflecting surfaces around the stage was being increased. On the basis of these impulse responses, one might expect an increase in perceived clarity whereas the clarity did not change, rather the opposite: the perceived reverberance, envelopment and intimacy all increased significantly in the transition from configuration 2 to 6 and reverberance and envelopment also increased in the transition from configuration 6 to 4. Configuration 2 was incidentally the most disliked configuration of those tested. (Impulse responses are generally now displayed over the first 200 ms since this includes the early reverberant decay.)
TEST SIGNALS F O R I N T E G R A T E D E N E R G Y MEASUREMENTS The effect of interference
The energy of a transient signal is derived by squaring and integrating. An important consideration in choosing a test signal is whether interference between reflections is likely to occur and if it does what influence will the interference have on the measured results. To determine the effect of interference, it is appropriate to consider two reflections of equal
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M. Barron
amplitude arriving at times t t and t 2 deriving from a source signal s(t) (the reflection amplitude can be considered unity for convenience). The integrated energy is then
f{s(t- tl) +s(t-
t2)} z dt
= fs2(t-tl)dt+
fs2(t-tE)dt+2
fs(l-tl)s(l-t2)dt
(6)
In the absence of interference (i.e. with a signal for which s(t) = 0 for Itl > It2- Ill) the final term is zero. The final term is proportional to the autocorrelation function of the signal at a delay (t z - t 1) and as a fraction of reflection energy without interference it is identical to the value of the normalised autocorrelation function, 7(z), at z = t z - tl. Since the autocorrelation function is zero for delays larger than the signal duration, a short duration signal minimises interference. The relevant autocorrelation function here though is not just that of the signal, but rather that associated with the combined impulse response of the source signal, the transfer function of the microphone and amplifying circuitry but especially the bandpass filter (usually octave or 1/3-octave) in the analysis system. Since we do not wish our chosen test signal to influence the result, this leaves as a criterion for a suitable signal that in the frequency domain its spectrum is fiat over the relevant frequency range. There is relatively little information regarding how our ears treat interfering reflections. What evidence there is suggests that where the acoustic information is available, the ears will treat the reflections as if they were separate. As an illustration of this characteristic, Schultz and Watters~ 0 describe an interesting experiment concerning perception of a viola note, whose fundamental is suppressed due to interference. Finally it is appropriate to discuss the effect of interference for a r a n d o m ordering of reflections. F r o m the argument above it is clear that interference will not affect the integrated energy for a r a n d o m order of reflections if
f
~ 7(r) dr = 0 oC
where 7(z) is the autocorrelation function of the impulse r e s p o n s e , f (t), of the total measuring system. By reference to the Fourier transform relationship between the autocorrelation function and the power
Impulse testing techniques Jbr auditoria
177
spectrum, it can be shown that this condition is satisfied by functions for which ~_ f ( t )
dt
0
i.e. for functions with no DC component. Since bandpass filtering eliminates any DC component, this condition is satisfied and interference for a random ordering of reflections does not influence integrated energy. Test signals with narrow bandwidths In general the requirements for an impulse test signal are as follows: (a) short duration (b) adequate energy (c) reasonably flat spectrum in the measurement range An additional constraint is introduced if no filtering is envisaged in the analysis, namely that the bandwidth be narrow. As narrow band signals, the useful signal envelopes are familiar from window functions for nonstationary signals, see, for example, Section 3.6.1 in reference 11. The Gaussian envelope has been used by Kiirer ~z and others, because it has the virtue of having no side lobes in the frequency domain. Atal e t al. 13 propose the use of a Hamming envelope; that is, a single cycle of a cosine 2 envelope on a small pedestal, which has the property of low level side lobes. In terms of maintaining good temporal resolution within a specified bandwidth these signals are near optimum. The signals clearly have to be radiated from a loudspeaker. In most cases, however, the complication of generating the signal does not appear to be outweighed by advantages associated with not requiring filtering in the analysis system. Test signals when filtering is included in the analysis The requirements for a suitable test signal for this situation have now all been stated. To minimise the effects of interference it is necessary to have a signal with a spectrum reasonably flat over the measuring frequency range. The most appropriate signal appears to be the uni-polar half-cycle sine wave already discussed, which can be used for measurement at or below the frequency, f0, corresponding to the fundamental frequency.
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M. Barton !
!
i
!
!
!
125
250
500
lk
i
0.
-5
dB -10
-20 63
2k
Frequency (Hz) Fig. 8. Constant percentage bandwidth spectra of half-cycle sine wave pulses with the same peak voltage at fundamental frequencies of (a) I kHz and (b) 125 Hz.
SOres
I_
SO+fo Fig. 9.
--I
Impulse response with delta impulse (a) and half-cycle sine wave signal (b) for direct sound and a 50 ms delay reflection.
Impulse testing techniques for auditoria
179
The requirement of adequate signal-to-noise ratio generally implies the use of more than one test signal to cover the whole frequency range. In particular, the higher background noise levels in buildings at low frequencies necessitate this. To cover the frequency range in the four octaves 125 Hz to 1 kHz, our current survey 'effectively' uses half-cycle sine pulses at centre frequencies 125 Hz and 1 kHz. The relevant spectra (omitting upper frequency behaviour for clarity) are shown in Fig. 8. Due to its longer duration, the 125 Hz half-cycle signal (with the same peak voltage) contains more energy and offers a 16 dB improvement at 125 Hz over the 1 kHz signal. (The word 'effective' has been used above since the dodecahedron loudspeaker being used has a 6 dB/octave frequency characteristic rising in the bass, so the signal being fed to the loudspeaker is in fact a single-cycle sine wave to provide a reasonably fiat frequency spectrum for the acoustic signal.)
Temporal smearing due to finite duration signals The true impulse response refers to a signal pulse of infinitely short duration, the finite duration of actual test signals implies a temporal smearing of the room response. This has to be taken account of when measurements are made which involve temporal components of the impulse response. For the purposes of this discussion it is most convenient to consider the implications for measurement of the energy during the first so many milliseconds after the direct sound, such as would be required for measurement of the 50 ms energy fraction originally proposed by Thiele. 14 Figure 9 shows the impulse response for direct sound and a 50 ms reflection, as well as the received signal for a half-cycle sine wave test signal of total duration 2t o ms. It is clear in this case that the appropriate sampling time is 50 + t o ms. In general, the additional sampling time, to, is that for half the total integrated energy of the combined impulse response of the test signal and filtering during analysis. For example, a typical analogue third-octave filter has a half-integrated energy time of six cycles at the centre frequency. These additional sampling times are far from insubstantial at low frequencies. An alternative technique is to gate the received signal prior to filtering during analysis. In this case the relevant half-integrated energy time is that of the signal alone, which may allow it to be ignored if it is short relative to the sampling time. The example discussed above and illustrated in Fig. 9 effectively refers to gating prior to filtering.
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M. Barron CONCLUSIONS
The true impulse response between a source a n d a receiver in a r o o m is a u n i q u e descriptor o f transmission between them. F r o m the impulse response can be derived the frequency response, the reverberation time, etc. The direct m e a s u r e m e n t o f the impulse response is c o m p l i c a t e d by the need to p r o d u c e a short d u r a t i o n pulse of suffÉcient intensity. T h e discussion here has indicated the suitable forms o f signal for this p u r p o s e a n d the need for simple c o m p e n s a t i o n for signals of a f o r m similar to a single cycle o f a sine wave. Signals f r o m spark sources and explosives are often o f this f o r m a n d for this reason tend to be deficient in low frequency energy. L o u d s p e a k e r sources have the a d v a n t a g e o f m u c h better reproducibility a n d obvious flexibility of signal. The provision o f an intense signal is n o t as difficult as it m i g h t seem, as m o s t l o u d s p e a k e r units will w i t h s t a n d impulsive inputs with peak voltages well in excess of their c o n t i n u o u s rating.
REFERENCES 1. M. R. Schroeder, Integrated impulse method measuring sound decay without using impulses, Journal oj the Acoustical Society of America, 66 (1979), pp. 497-500. 2. A. J. Berkhout, D. de Vries and M. M. Boone, A new method to acquire impulse responses in concert halls, Journal oj the Acoustical Society oJ America, 68 (1980), pp. 179-83. 3. N. Aoshima, Computer-generated pulse signal applied for sound measurement, Journal of the Acoustical Society o/America, 69 (t981), pp. 1484-8. 4. M. R. Schroeder, New method of measuring reverberation time, Journal of the Acoustical Society o[' America, 37 (1965), pp. 409-12. 5. B.S. Atal, M. R. Schroeder and G. M. Sessler, Subjective reverberation time and its relation to sound decay, 5th International Congress on Acoustics, Liege (1965), G 32. 6. T. G. Winter, J. Pereira and J. Bee Bednar, On driving a transducer to produce pulses shorter than the natural period of the transducer, Ultrasonics, 13 (1975), pp. 110-12. 7. E.G. Richardson and E. Meyer, Technicalaspects ojsound, Vol. lit, Elsevier Publishing Company, New York, 1962. 8. L. B, Preizer, Statistics of high level reflections in auditoriums, Soeiet Physics Acoustics, 11 (1966), pp. 407-11. 9. M. Barron, The Gulbenkian Great Hall, Lisbon, If: an acoustic study of a
Impulse testing techniques for auditoria
10. 11. 12. 13. 14.
181
concert hall with variable stage, Journal of Sound and Vibration, 59 (1978), pp. 481-502. T. J. Schultz and B. G. Watters, Perception of music heard by interfering paths, Journal of the Acoustical Society of America, 36 (1964), pp. 897-902. R. B. Randall, Application orB and K equipment to frequency analysis, Briiel and Kjaer, Naerum, 1977. R. K/irer, Untersuchungen zur Auswertung yon Impulsmessungen in der Raumakustik, Dissertation, Technischen Universit~it, Berlin, 1972. B. S. Atal, M. R. Schroeder, G. M. Sessler and J. E. West, Evaluation of acoustic enclosures by means of digital computers, Journal of the Acoustical Society of America, 40 (1966), pp. 428-33. R. Thiele, Richtungsverteilung und Zeitfolge der Schallriickwiirfe in R~iumen, Acustica, 3 (1953), pp. 291-302.
Impulse Testing Techniques for Auditoria M. Barron Department of Architecture, University of Cambridge, Cambridge (Great Britain) (Received: 15 February, 1983)
S UMMA R Y
Though computer-generated low crest-factor signals can now be produced and processed to derive the impulse response of auditoria, the use o f impulsive signals remains the simplest and, in many cases, a fully viable technique. The advantages o f a uni-polar half-cycle sine wave test signal are discussed and compared with those of the hi-polar single-cycle signal The particular requirements for visual display of the impulse response are reviewed, as are the signal requirements and analysis options for objective measures involving integrated energy. In each case the implications are considered o f interference between reflections which arrive simultaneously.
INTRODUCTION Evaluation of the impulse response o f an auditorium is fundamental to m u c h current work in the area o f objective testing. Since the transmission between two points in a r o o m is linear, it might be expected that techniques applied to other linear systems would automatically be appropriate to r o o m response as well. This is, however, not necessarily the case, due principally to the complexity of r o o m response, including as it does thousands of reflections or, from a modal standpoint, thousands o f modes. There is also a further difference, namely that for auditoria analysis of energy is made on a constant percentage bandwidth basis, generally in octaves or third-octaves (i.e. logarithmic frequency), rather than linear frequency. 165 Applied Acoustics 0003-682X/84/$03-00 ~'i Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
166
M. Barron
The problems of testing with an impulsive signal are well publicised. To achieve adequate signal-to-noise ratio within limits set by linear sound propagation is not easy and for this reason testing with long duration signals generated by computer has been proposed by Schroeder, 1 Berkhout et al. z and Aoshima. 3 In a quiet environment, testing with impulsive signals is nevertheless a viable technique which requires only simple apparatus and no computer. In acoustic models, impulsive spark sources also have advantages, particularly their omni-directional characteristic. Testing with impulsive signals will remain a common technique. This paper reviews the implications of using typical test signals. The results of this review have been used in the selection of test signals for the current Acoustic Survey of British Auditoria.
T H E BASIC R E L A T I O N S H I P S For completeness, but at the risk of repetition, it is appropriate to quote the fundamental mathematical relationships governing transmission between two points in a room. The impulse response, g(t), is a real function of time and is a unique descriptor of this transmission characteristic. The frequency spectrum, G(f), where f is frequency, is related to g(t) by Fourier transformation. G(f) is a complex function. The frequency response, as conventionally measured, is IG(f)l:
G(f) =
f
QC
g(t) exp (-j2nft) dt --
(1)
OC
The signal, p(t), received in a room from a source is equal to the impulse response convoluted with the source signal, s(t): p ( t ) = g ( t ) * s( t)
P(f) = a(f)s(f)
(2)
where P(f) and S ( f ) are the spectra of the received and emitted signals. If s(t) is an even function, then S ( f ) is also a real even function, j = x / / - 1. Schroeder 4 has shown that the traditional reverberant decay, n2(t), produced by switching offa noise signal is related to the impulse response. A noise decay contains random fluctuations due to the random time history immediately prior to switch-off. However, the ensemble average
Impulse testing techniques for auditoria
167
of an infinite number of squared noise decays, (n2(t)), is related to the squared impulse response, g2('r), as follows: (n2(/))
=
g2(z)
dz =
g2(z)
dr -
g2(z)
dz
(3)
The time interval for integration is between t and infinity, which implies measurement in reverse time. The second form above gives an alternative, namely the difference between the total integrated energy and the integral from 0 to t. Equation (3) thus supplies the link between the two traditional methods of measuring reverberation time: with noise and impulse signals. The traditional level recorder will measure instantaneous rms pressure level with an appropriate averaging time to smooth fluctuations. The property of an exponential function, that it is replicated on differentiation or integration, implies that the same decay rate will result for a true exponential decay measured either with noise or an impulsive signal in the traditional manner. The integrated impulse method, based on evaluation
i,?S |
0 Fig. 1.
- - ~ - .
t
(a) Idealised squared impulse response. (b) Corresponding build-up curve (integral from 0 to t) and decay curve (integral from t to or).
168
M. Barton
of the right hand side of eqn. (3), is now well established to determine the detailed nature of sound decay and the subjectively significant early decay, s Equation (3) also indicates the complementary nature of sound buildup and decay. This is illustrated for an idealised impulse characteristic in Fig. 1. When the integrated energy is converted into decibels, only the decay has a linear characteristic.
T W O BASIC TEST SIGNALS The impulse response, g(t), refers to the response to the idealised Dirac delta function, which has a spectrum with equal energy per cycle. In practice, there is a limit to intensity due to non-linear propagation. The energy in a real pulse is proportional to its duration. Extremely short pulses, which would have a wide bandwidth, are unlikely to result in an adequate signal-to-noise ratio. Fortunately, the true impulse response, g(t), is rarely required as such. For the selection of a suitable impulse signal, it is necessary to consider both the temporal signature and frequency spectrum of the signal. Initially, two basic impulse signals will be discussed: the half- and whole-cycle sine wave. The half-cycle sine wave is illustrated in Fig. 2(a). Being derived from a sine wave, sin 2rtfot, of frequency f0, it has a duration 1/2j0. Its amplitude spectrum is given by:
I S ( f ) l - ~(fo2 _ f z ) cos 2~0
(4)
This is plotted in Fig. 3 in a constant percentage bandwidth form. At low frequencies, below fo, its spectrum is identical to a ideal delta function, namely flat on an energy per cycle basis and falling at 3 dB/octave on a constant percentage bandwidth basis. For measurements in octave bands, this signal is usable for centre frequencies ofjo and below. Both the low frequency characteristic and rapid drop off of energy at high frequencies occur for any shape of impulse which is uni-polar (other examples are a triangular or rectangular form). The spectrum for a bi-polar signal, such as a single-cycle pulse, contains no DC component. Figure 2(b) illustrates how a single-cycle impulse is the convolution of a half-cycle sine wave and an odd-impulse pair. The
169
Impulse testing techniques Jbr auditoria
(a)
% ½f,
Fig. 2.
(hi (a) Half-cycle sine wave. (b) Half-cycle sine wave convoluted with an odd-impulse pair produces a single-cycle sine wave.
I
I
I
I
I
I
I
I
-5 -10
/
dB
/
-15
/1
f
9d%ve//
/
-20 -25
/
I I
/
I I I
/ // i
f"16"" Fig. 3.
f
"/8
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l
=
hi
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f°
2f°
"x
~ave I 4f.
\ 8f. Frequency
Spectra on a constant percentage bandwidth basis of a half-cycle sine wave (continuous line) and a single-cycle sine wave (broken line).
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amplitude spectrum of an odd-impulse pair is: Is(f)l = 2
sin
rtfz
(5)
so the spectrum of the full-cycle sine wave is the product of expressions in eqns. (4) and (5). It contains additional frequencies of zero energy and has an additional 6 dB/octave slope at low frequencies, also shown in Fig. 3. For measurements in octave bands, this signal is only really valid for octave centre frequencies close to fo. Natural impulsive sources tend to produce bi-polar rather than unipolar responses. Spark sources, at least those using a third electrode for a trigger, generally have a spectrum similar to that for a bi-polar signal. The spark energy is a determinant of signal duration and hence the higher frequency limit of measurement. Predictably, signal-to-noise problems have been encountered at low frequencies with spark sources; use of high energy sparks creates the risk of non-linear acoustic propagation. If the signal-to-noise ratio is adequate, however, a very simple , '25
ms,
(a)
(b) Fig. 4.
Acoustic signal from a spark source before (a) and after (b) passing through an integrator.
lrnpulse testing techniques for auditoria
171
technique enables octave measurements to be made with a bi-polar source signal. Comparison of the low-frequency spectra in Fig. 3 suggests the use of a 6 dB/octave bass boost filter. Such a filter is equivalent to an integrator and if a single cycle of a sine wave, sin 2rCfot, is fed into such a filter, the output is (1 - c o s 2rCfot), where 0 < t < 1Ifo. This filter thus produces the response that would be obtained for a uni-polar signal. Figure 4 illustrates the acoustic time signature of a spark source before and after passing through such an integrator. (The integrated signal is not fully uni-polar due to the negative peak in the spark signal being larger; the subsequent decay to zero is due to incidental high-pass filtering to block DC in the analysis system.) Loudspeakers fed with uni-polar signals can distort them severely; both a fiat frequency response and clean impulse response are required. (Perfect reproduction is not possible since loudspeakers cannot radiate zero frequency.) Theoretically, a certain time signature exists which when fed to the loudspeaker will produce the desired acoustic output. A paper by Winter et al. 6 describes this procedure: how with a loudspeaker with an impulse response, h(t), the required signal spectrum, S ( f ) , is calculated as 1 / H ( f ) , where H ( f ) is the Fourier transform of h(t) (see eqns. (1) and (2)). This technique demands some compromises, however, involving some form of frequency band limiting, which is not described.
VISUAL P R E S E N T A T I O N OF I M P U L S E R E S P O N S E S The display of the impulse response in an auditorium on an oscilloscope screen remains a traditional testing technique. Either the pressure or the pressure squared is displayed as a function of time (see Fig. 5). The pressure squared response is currently preferred for two principal reasons: firstly, that subjective response appears to be better represented objectively by measures which relate to energy rather than pressure, and secondly, that in the process of squaring the significant reflections are better separated from low level scattered components. Due to the inevitable temporal proximity of reflections at some locations, interference will occur which can frustrate isolation of individual reflections. To minimise interference, short duration signals are to be preferred. This cannot be taken to the extreme, however, because it is necessary also to consider the spectrum of the signal. With a very short duration signal, a high proportion of the energy will be high
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(a)
(b) Fig. 5.
The pressure (a) and pressure squared (b) impulse responses measured in a concert hall at a source-receiver distance o f 23-2 m.
frequency, which would accentuate reflections off small surfaces, reflections which are unlikely to be significant subjectively. Optimum signal durations for visual responses of auditoria probably lie in the region of 0.3-1 ms. As will be stressed below, comparative assessment of impulse responses is appropriate and ideally the same acoustic source signal should be used for such comparisons. Whatever source signal is used, cases will still exist where simultaneous reflections overlap. The only solution to avoid such interference is to take impulse responses at a series of adjacent microphone positions. The visual impulse response gives a clear indication of the arrival time of early reflections. It can obviously also be used to expose intense echoes. However, the attempt to relate subjective characteristics to the visual impulse response has been largely unsuccessful. The work at G6ttingen in this field during the 1950s is summarised in reference 7. Likewise,
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(a)
(b) Fig. 6.
Squared impulse responses, measured in the same concert hall as the example in Fig. 5, at two source-receiver distances: (a) 13-6 m and (b) 30.6 m.
attempts at measures based on reflection statistics 8 have not proved useful. Most objective measures now considered valuable are related to integrated energy, which will be discussed in the following sections. A fundamental problem of interpreting impulse responses is that the visual nature of the response is very much a function of the source-receiver distance. For receiver positions close to the source, the delays of the first reflections tend to be large and reflections are well spread out and at low levels relative to the direct sound. At receiver positions far from the source the first reflection delays are small, the density of reflections is high and levels of the first reflections are close to the direct sound level. This behaviour occurs in all auditorium spaces and is an obvious consequence of the fact that the image positions associated with the first reflections are fixed in space. Figure 6 illustrates this
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Configurofion 2
Configurcttion 6
Configurcltion 4 Fig. 7.
Pressure impulse responses measured at fixed source-receiver positions in three stage configurations in the Gulbenkian Great Hall, Lisbon.
b e h a v i o u r with impulse responses m e a s u r e d in a full-size concert hall with source-receiver distances o f 13.6 m and 30.6 m. The question, therefore, is w h e t h e r this obvious transition in the impulse response c o r r e s p o n d s with a subjective response. If one has to generalise, then the answer appears to be no. This is particularly the case with music; in concert halls, experience of listening suggests that a l t h o u g h there are subjective differences between the s o u n d at the front a n d rear,
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there is not an overriding subjective correlate of the inevitable transition between the extremes illustrated in Fig. 6. A more hopeful comparison would appear to be between impulse responses measured at similar source-receiver distances. This obviously restricts the usefulness of the impulse response as an indicator, but with a library of responses recorded in different halls comparisons would be possible. The following is included here to indicate the serious problem of deriving subjective response from the visually displayed impulse response. In the study of the Gulbenkian Great Hall in Lisbon, 9 impulse responses were recorded in the same conditions as those in which subjective tests were conducted; the subjective effects of changes in the stage conditions were investigated for fixed source-receiver positions. Figure 7 shows the (pressure) impulse responses for three stage configurations for which the original labels 2, 6 and 4 have been u s e d . 9 The responses illustrate approximately the first 100 ms after the direct sound. There is a clear increase in the number of early reflections from configuration 2 to 6 and 6 to 4, which corresponds with expectations since the number of reflecting surfaces around the stage was being increased. On the basis of these impulse responses, one might expect an increase in perceived clarity whereas the clarity did not change, rather the opposite: the perceived reverberance, envelopment and intimacy all increased significantly in the transition from configuration 2 to 6 and reverberance and envelopment also increased in the transition from configuration 6 to 4. Configuration 2 was incidentally the most disliked configuration of those tested. (Impulse responses are generally now displayed over the first 200 ms since this includes the early reverberant decay.)
TEST SIGNALS F O R I N T E G R A T E D E N E R G Y MEASUREMENTS The effect of interference
The energy of a transient signal is derived by squaring and integrating. An important consideration in choosing a test signal is whether interference between reflections is likely to occur and if it does what influence will the interference have on the measured results. To determine the effect of interference, it is appropriate to consider two reflections of equal
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amplitude arriving at times t t and t 2 deriving from a source signal s(t) (the reflection amplitude can be considered unity for convenience). The integrated energy is then
f{s(t- tl) +s(t-
t2)} z dt
= fs2(t-tl)dt+
fs2(t-tE)dt+2
fs(l-tl)s(l-t2)dt
(6)
In the absence of interference (i.e. with a signal for which s(t) = 0 for Itl > It2- Ill) the final term is zero. The final term is proportional to the autocorrelation function of the signal at a delay (t z - t 1) and as a fraction of reflection energy without interference it is identical to the value of the normalised autocorrelation function, 7(z), at z = t z - tl. Since the autocorrelation function is zero for delays larger than the signal duration, a short duration signal minimises interference. The relevant autocorrelation function here though is not just that of the signal, but rather that associated with the combined impulse response of the source signal, the transfer function of the microphone and amplifying circuitry but especially the bandpass filter (usually octave or 1/3-octave) in the analysis system. Since we do not wish our chosen test signal to influence the result, this leaves as a criterion for a suitable signal that in the frequency domain its spectrum is fiat over the relevant frequency range. There is relatively little information regarding how our ears treat interfering reflections. What evidence there is suggests that where the acoustic information is available, the ears will treat the reflections as if they were separate. As an illustration of this characteristic, Schultz and Watters~ 0 describe an interesting experiment concerning perception of a viola note, whose fundamental is suppressed due to interference. Finally it is appropriate to discuss the effect of interference for a r a n d o m ordering of reflections. F r o m the argument above it is clear that interference will not affect the integrated energy for a r a n d o m order of reflections if
f
~ 7(r) dr = 0 oC
where 7(z) is the autocorrelation function of the impulse r e s p o n s e , f (t), of the total measuring system. By reference to the Fourier transform relationship between the autocorrelation function and the power
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spectrum, it can be shown that this condition is satisfied by functions for which ~_ f ( t )
dt
0
i.e. for functions with no DC component. Since bandpass filtering eliminates any DC component, this condition is satisfied and interference for a random ordering of reflections does not influence integrated energy. Test signals with narrow bandwidths In general the requirements for an impulse test signal are as follows: (a) short duration (b) adequate energy (c) reasonably flat spectrum in the measurement range An additional constraint is introduced if no filtering is envisaged in the analysis, namely that the bandwidth be narrow. As narrow band signals, the useful signal envelopes are familiar from window functions for nonstationary signals, see, for example, Section 3.6.1 in reference 11. The Gaussian envelope has been used by Kiirer ~z and others, because it has the virtue of having no side lobes in the frequency domain. Atal e t al. 13 propose the use of a Hamming envelope; that is, a single cycle of a cosine 2 envelope on a small pedestal, which has the property of low level side lobes. In terms of maintaining good temporal resolution within a specified bandwidth these signals are near optimum. The signals clearly have to be radiated from a loudspeaker. In most cases, however, the complication of generating the signal does not appear to be outweighed by advantages associated with not requiring filtering in the analysis system. Test signals when filtering is included in the analysis The requirements for a suitable test signal for this situation have now all been stated. To minimise the effects of interference it is necessary to have a signal with a spectrum reasonably flat over the measuring frequency range. The most appropriate signal appears to be the uni-polar half-cycle sine wave already discussed, which can be used for measurement at or below the frequency, f0, corresponding to the fundamental frequency.
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!
i
!
!
!
125
250
500
lk
i
0.
-5
dB -10
-20 63
2k
Frequency (Hz) Fig. 8. Constant percentage bandwidth spectra of half-cycle sine wave pulses with the same peak voltage at fundamental frequencies of (a) I kHz and (b) 125 Hz.
SOres
I_
SO+fo Fig. 9.
--I
Impulse response with delta impulse (a) and half-cycle sine wave signal (b) for direct sound and a 50 ms delay reflection.
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The requirement of adequate signal-to-noise ratio generally implies the use of more than one test signal to cover the whole frequency range. In particular, the higher background noise levels in buildings at low frequencies necessitate this. To cover the frequency range in the four octaves 125 Hz to 1 kHz, our current survey 'effectively' uses half-cycle sine pulses at centre frequencies 125 Hz and 1 kHz. The relevant spectra (omitting upper frequency behaviour for clarity) are shown in Fig. 8. Due to its longer duration, the 125 Hz half-cycle signal (with the same peak voltage) contains more energy and offers a 16 dB improvement at 125 Hz over the 1 kHz signal. (The word 'effective' has been used above since the dodecahedron loudspeaker being used has a 6 dB/octave frequency characteristic rising in the bass, so the signal being fed to the loudspeaker is in fact a single-cycle sine wave to provide a reasonably fiat frequency spectrum for the acoustic signal.)
Temporal smearing due to finite duration signals The true impulse response refers to a signal pulse of infinitely short duration, the finite duration of actual test signals implies a temporal smearing of the room response. This has to be taken account of when measurements are made which involve temporal components of the impulse response. For the purposes of this discussion it is most convenient to consider the implications for measurement of the energy during the first so many milliseconds after the direct sound, such as would be required for measurement of the 50 ms energy fraction originally proposed by Thiele. 14 Figure 9 shows the impulse response for direct sound and a 50 ms reflection, as well as the received signal for a half-cycle sine wave test signal of total duration 2t o ms. It is clear in this case that the appropriate sampling time is 50 + t o ms. In general, the additional sampling time, to, is that for half the total integrated energy of the combined impulse response of the test signal and filtering during analysis. For example, a typical analogue third-octave filter has a half-integrated energy time of six cycles at the centre frequency. These additional sampling times are far from insubstantial at low frequencies. An alternative technique is to gate the received signal prior to filtering during analysis. In this case the relevant half-integrated energy time is that of the signal alone, which may allow it to be ignored if it is short relative to the sampling time. The example discussed above and illustrated in Fig. 9 effectively refers to gating prior to filtering.
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M. Barron CONCLUSIONS
The true impulse response between a source a n d a receiver in a r o o m is a u n i q u e descriptor o f transmission between them. F r o m the impulse response can be derived the frequency response, the reverberation time, etc. The direct m e a s u r e m e n t o f the impulse response is c o m p l i c a t e d by the need to p r o d u c e a short d u r a t i o n pulse of suffÉcient intensity. T h e discussion here has indicated the suitable forms o f signal for this p u r p o s e a n d the need for simple c o m p e n s a t i o n for signals of a f o r m similar to a single cycle o f a sine wave. Signals f r o m spark sources and explosives are often o f this f o r m a n d for this reason tend to be deficient in low frequency energy. L o u d s p e a k e r sources have the a d v a n t a g e o f m u c h better reproducibility a n d obvious flexibility of signal. The provision o f an intense signal is n o t as difficult as it m i g h t seem, as m o s t l o u d s p e a k e r units will w i t h s t a n d impulsive inputs with peak voltages well in excess of their c o n t i n u o u s rating.
REFERENCES 1. M. R. Schroeder, Integrated impulse method measuring sound decay without using impulses, Journal oj the Acoustical Society of America, 66 (1979), pp. 497-500. 2. A. J. Berkhout, D. de Vries and M. M. Boone, A new method to acquire impulse responses in concert halls, Journal oj the Acoustical Society oJ America, 68 (1980), pp. 179-83. 3. N. Aoshima, Computer-generated pulse signal applied for sound measurement, Journal of the Acoustical Society o/America, 69 (t981), pp. 1484-8. 4. M. R. Schroeder, New method of measuring reverberation time, Journal of the Acoustical Society o[' America, 37 (1965), pp. 409-12. 5. B.S. Atal, M. R. Schroeder and G. M. Sessler, Subjective reverberation time and its relation to sound decay, 5th International Congress on Acoustics, Liege (1965), G 32. 6. T. G. Winter, J. Pereira and J. Bee Bednar, On driving a transducer to produce pulses shorter than the natural period of the transducer, Ultrasonics, 13 (1975), pp. 110-12. 7. E.G. Richardson and E. Meyer, Technicalaspects ojsound, Vol. lit, Elsevier Publishing Company, New York, 1962. 8. L. B, Preizer, Statistics of high level reflections in auditoriums, Soeiet Physics Acoustics, 11 (1966), pp. 407-11. 9. M. Barron, The Gulbenkian Great Hall, Lisbon, If: an acoustic study of a
Impulse testing techniques for auditoria
10. 11. 12. 13. 14.
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concert hall with variable stage, Journal of Sound and Vibration, 59 (1978), pp. 481-502. T. J. Schultz and B. G. Watters, Perception of music heard by interfering paths, Journal of the Acoustical Society of America, 36 (1964), pp. 897-902. R. B. Randall, Application orB and K equipment to frequency analysis, Briiel and Kjaer, Naerum, 1977. R. K/irer, Untersuchungen zur Auswertung yon Impulsmessungen in der Raumakustik, Dissertation, Technischen Universit~it, Berlin, 1972. B. S. Atal, M. R. Schroeder, G. M. Sessler and J. E. West, Evaluation of acoustic enclosures by means of digital computers, Journal of the Acoustical Society of America, 40 (1966), pp. 428-33. R. Thiele, Richtungsverteilung und Zeitfolge der Schallriickwiirfe in R~iumen, Acustica, 3 (1953), pp. 291-302.