Content uploaded by Csaba Huszty
Author content
All content in this area was uploaded by Csaba Huszty on Aug 24, 2016
Content may be subject to copyright.
Performance comparison of monoexponential and
multiexponential decay function estimation methods
Csaba Huszty 1
1ENTEL Engineering Research and Consulting Ltd. Inspired Acoustics division
H-1025 Budapest, Sz´
epv¨
olgyi ´
ut 32., Hungary
ABSTRACT
Various decay function and decay time estimation methods show different performance properties in terms
of accuracy, reliability, convenience in use and ease of implementation. In this paper a systematic way for
evaluating and comparing decay function estimation methods is presented. The method is based on decay
models. Commonly used monoexponential a multiexponential decay time estimation methods are compared
and a new multi-component non-linear method for multiexponential decay function estimation is briefly
introduced and compared to the Bayesian method. Model functions are used to evaluate the current and
newly introduced methods.
Keywords: Multiexponential, Decay function, Room acoustics
1. INTRODUCTION
Reverberation time is still one of the most sought-after room acoustic parameter. The decay function
that is most often used in practice to determine the reverberation time has certain properties that limit the
accuracy of the estimation. First these issues will be identified and then new reverberation time estimation
methods will be proposed and compared systematically to the current methods published in the literature.
Some already published methods were intentionally omitted as they are providing similar results to the ones
examined here.
2. MODELS
2.1. RIR model and functions
The empirical definition of the reverberation time corresponds to a 60 dB decay of the sound energy
density1, which has an exponential decay envelope in a diffuse space. The multiexponential RIR envelope
function with Ncomponents is expressed as the sum of monoexponentials with different time constants. It
1csaba@inspiredacoustics.com
has 2Nnumber of parameters to estimate: mixing constants of A1..ANand monoexponential equivalent
reveberation time values of R1..RN.
h(t)m=
N
X
i=1
Ai·exp −kt
Ri(1)
With i= 1 the monoexponential RIR envelope function is obtained. kis a constant supporting to fulfill a
60 dB decrease at h(t=R1)for the monoexponential case, therefore k= 3 ln 10 assuming a normalized
amplitude of A1= 1. A decay process that can be considered multiexponential or ’bended’ has multiple time
constants and equivalent reverberation times. Consequently, some reverberation time estimation methods
assuming a monoexponential model will yield different decay time values not necessarily corresponding to
the internal time constants. Also note that the reverberation time in its original definition indicates a temporal
support required by the process to decay 60 dB but does not indicate the internal structure of such process.
2.2. Energy decay function
The energy decay function2(EDF) and its graphically plotted representation, the energy decay curve,
EDC, allows a highly reproducible method for obtaining decay functions using only a single measurement3,4 .
With energy normalization it can be expressed as
D(t) = 10 log10 R∞
th2(τ)dτ
R∞
0h2(τ)dτ != 10 log10 1−Rt
0h2(τ)dτ
R∞
0h2(τ)dτ !.(2)
The second formulation uses sound buildup, which can be efficiently implemented in computer calculations5.
Due to energy normalization the EDC starts at 0dB and monotonically decreases for finite length data or
square integrable data vanishing in time. Assuming a monoexponential decay D(t)d=−60
Rt, so the energy
decay function – on the decibel scale – is a straight, decreasing line. In practical applications, the energy
decay function shows two types of bias: a cutoff bias caused by the upper limit of integration (ULI) of length
L, and a bias due to additive stationary background noise (BN). Graphically, these two bias effects transform
the EDC towards the opposite directions and under special conditions their effects are balanced, often causing
confusion in the evaluation process.
2.2.1. Cutoff bias
In general, the cutoff bias causes a time-dependent decreasing shift compared to the bias-free energy
decay function. For the special case of a monoexponential decay, the cutoff bias can be shown in the expres-
sion:
D(t)L= 10 log10 1−1−exp −2kt
R
1−exp −2kL
R!.(3)
The bias is the error term exp −2kL
Rin the denominator. This cutoff bias is often not corrected in decay
time estimations, despite that its effect is not negligible. The reason for applying no correction in practice is
shown in the error term: an a priori knowledge of the reverberation time would be required for the appropriate
correction unless there are other ’blind’ methods available.
The effect of the cutoff bias was simulated for the noise-free monoexponential decay, and results suggest
that it can be treated as negligible only if the R/L reverberation-to-length ratio is comparable to or less than
about 0.5. If the energy decay function is calculated without cutoff bias correction for a monoexponential
decay, then 0.47 < R/L < 0.87 is required in order to assure that the first 90% of the decay function is
reliable within 1dB, and 0.45 < R/L < 1.3to ensure reliability within 2dB.
2.2.2. Stationary background noise bias
Let us denote the decay function as D(t) = 10 log10 (d(t)) where d(t)=1−1
ERt
0h2(τ)dτ and
where Eis the total energy of the noisy RIR sample. Assume that the RIR is noisy and consists of h(t) =
p(t) + n(t)where p(t)is the noiseless component and n(t)is a given representation of a white Gaussian
process. Since Rt
0h2(τ)dτ =Rt
0(p+n)2(τ)dτ =Rt
0p2(τ)dτ +Rt
0n2(τ)dτ + 2 Rt
0p(τ)n(τ)dτ let us
denote the last term as X(B−A)$RB
Ap(τ)n(τ)dτ. In case of a sound build-up expression let us use
X(t) = Rt
0p(τ)n(τ)dτ, and for a backward integrated formulation X(L−t). Assuming t > R, the term
X(t)converges to a constant value because as tbecomes large, the noise-free impulse response vanishes
(p(t)→0), and consequently the terms in the integration also vanish, yielding a finite limt→∞ X(t) = C∈
R. The actual value of X(t)depends on the concrete representation of the noise and RIR samples. Since
σ2=1
tRt
0n2(τ)dτ is the second moment or variance of the noise, or also its mean square, or squared RMS
value, or its autocorrelation function’s t= 0 point, this value is constant assuming an adequately large time
lag t. Therefore, combining the above yields
d(t)=1−1
EZt
0
p2(τ)dτ −σ2
E·t−2X(t)
E=1
EZL
t
p2(τ)dτ +σ2
E·(L−t) + 2X(L−t)
E(4)
where the first representation of (4) is the decay function in the sound build-up formulation and the second
is the backward integrated formulation. Considering the above, the bias in the energy decay function due to
additive stationary background noise contains two main effects; the first, a decreasing line (before taking the
dB scale) whose slope is dependent on the noise variance and the total energy of the noisy signal, and the
second due to the time-dependent function X(t), which introduces the integrated ’dependency information’
between the noise and the noise-free RIR up to a given finite length. This term cannot be assumed to be
constantly zero for any t. In fact, in the sound build-up formulation (4), a considerable portion of energy
is contained in this last term. If the formulation is a backward integrated one, the third term is evaluated
between the integration limits t..L, and as tincreases, the energy in the term decreases with time. This
allows neglecting this term in the simplified decay function (as discussed later) assuming a considerably high
SNR.
2.3. Decay function models
2.3.1. Simplified decay function
Consider the noisy finite length backward integrated decay function and assume that the third term
X(L−t)is zero for any decay function. Such assumption yields the simplified decay function in a general
case, expressed as d(t)≈c1·F(t)+ cN·(L−t), where c1=1
Eand cN=σ2
Eare constants for the noise-free
energy decay F(t)and for the noise, respectively, and Eis the total energy.
2.3.2. Simplified infinite monoexponential decay function with finite noise
Assume that there is no cutoff error (an infinite integration length is used) and that the energy in the EDF
contains only the energy of the noise-free decay and no noise energy. Then the first term of the simplified
decay function becomes
F(t) = 1
Ep
lim
L→∞ ZL
t
p2(τ)dτ = exp −2k
Rt(5)
which approximation also appears in the literature5:
d(t)≈c1·exp (−c2t) + cN·(L−t)(6)
where c2=2k
R.
2.3.3. Simplified finite monoexponential decay function with finite noise
For a finite length monoexponential decay, the decay function becomes
F(t) = 1
Ep,L ZL
t
p2(τ)dτ =exp −2k
Rt
1−exp −2k
RL−exp −2k
RL
1−exp −2k
RL(7)
leading to
d(t)≈c1·exp (−c2L)−exp (−c2t)
exp (−c2L)−1+cN·(L−t)(8)
where c2=2k
Rand k= 3 ln 10 in a RIR model with normalized amplitude.
2.3.4. Simplified finite double exponential decay function with finite noise
For a finite length double exponential decay of
p(t) = Aexp −k
R1
t+Bexp −k
R2
t=Aexp (−at) + Bexp (−bt).(9)
The noise-free decay function becomes
F(t) =
4ABab
exp(t(a+b)) −4ABab
exp(L(a+b)) +A2ba+A2b2
exp 2at −A2ba+A2b2
exp(2aL)+B2a2+B2ab
exp(2bt)−B2a2+B2ab
exp(2bL)
A2b2+A2ba +B2ab +B2a2+ 4ABab −4AB ab
exp(L(a+b)) −A2ba+A2b2
exp(2aL)−B2a2+B2ab
exp(2bL)
(10)
and the simplified decay function with noise can be obtained by substituting F(t)into the simplified decay
function formula. Triple and quadruple decay functions are also possible to evaluate analytically, but they are
rather long to display – yet they are highly applicable in a non-linear regression method in practice.
3. METHODS
3.1. Monoexponential estimation methods
3.1.1. Truncation and subtraction
The truncation method3,5,6 is a method for BN bias mitigation exploiting that the ’knee point’ of the
decay function (corresponding to the SNR) is defined by the upper integration length: an appropriately trun-
cated noisy decay will show a nearly linear decay, apparently as if the decay would continue below the noise
level. For this method an adequately small level of noise and long measurement is required, and the truncation
itself can also cause additional estimation issues. By exploiting the fact that the optimum truncation point
on the energy decay curve (EDC) and on the energy time curve (ETC) is the same point in time, an iterative
truncation method can also be developed7. However, in order to find the true optimum truncation point, both
the reverberation time Rand the SNR (expressed in dB) must be known a priori unless an analytically proper
blind method is used.
Another possible way to mitigate the effects of BN is the subtraction method8, which relies on the fact
that a nearly linear decay function can be obtained if the RMS of the noise is deducted from the squared
impulse response in the calculation of the EDC. The subtraction method does not correct for the cutoff bias.
3.1.2. Extrapolation
The currently standardized4,9 extrapolation method assumes a monoexponential model due to diffusity
of the late part of the decay. The logarithmic representation of a monoexponential EDF is a straight line
and allows a regression-based fit and determination of the reverberation time accordingly. In such cases the
reverberation time can be calculated either by directly reading two preceding points where noise still does
not dominate, for example using a −5dB reference point and an ndB decay point or by using a least-
square linear regression in that interval, as standardized4. The extrapolation method mitigates the effect
of background noise most trivially by avoiding using decay level ranges contaminated by noise. Usage of
the starting reference point as −5dB is limited by source-receiver distance9which can otherwise result in
reverberation time underestimation. As of the time of writing, a rule of thumb can be found in the literature3
so at least 10 dB of headroom is required for a given lower reference level. The extrapolation method yields
different reverberation times at different extrapolation and reference levels or ranges if the measured decay is
non-exponential, indicating the mistmatch of a monoexponential model to the empirical data.
3.1.3. Nonlinear regression (NLS)
The nonlinear iterative regression, or nonlinear least squares analysis (NLS)10 was first reported to be
applied to backward integrated decay functions in 19955, and later, a similar method was proposed using a
scaled exponential decay model11. The nonlinear regression method applied to backward integrated decay
functions assumes a monoexponential decay and uses a simplified decay model which takes into account the
finite length of integration in the noise-term, but approximates the reverberation time as if it was without
cutoff error and without assuming noise energy in that term. The NLS method presented in the literature
using matrix inversion. It reportedly may have convergence issues in improper initial conditions 5and may
find a local solution instead of a global11. If the noise estimate can be done accurately, requiring sufficient
measurement length, the results of the NLS are reportedly ”virtually equivalent” to that of the subtraction
method5, however, this statement is questioned based on the findings presented in this paper.
In the present work we compare the previously published method with others and propose using the
model of Eq.(7) and mitigate the convergence issues and matrix inversion problems by using a trust-region-
reflective optimization approach.
3.1.4. Fourier transformation based reverberation time estimation
In this section a new method of reverberation time calculation is briefly introduced. When a one-sided
monoexponential decay is Fourier transformed, the ratio of its real and imaginary parts yields the unknown
decay constant12 from which the reverberation time can be calculated. It can be seen analytically that the
backward integrated decay function can also be applied in this method in a similar way. The proposed
method estimates the reverberation time by taking the ratio of the imaginary and real parts of the Fourier
transform of the finite length energy decay function D(ω)L, at low ωvalues, as follows:
R=−23 ln 10
ω
Im D(ω)L
Re D(ω)L
.(11)
One of the most significant merits of this method is that transforming the simplified infinite and finite decay
function with finite noise as proposed both yield the same results at ω > 0. This means that the method
inherently includes compensation for the cutoff and background noise bias. Additional merits include a very
fast and convenient calculation of the reverberation time as well as its relatively good accuracy under certain
conditions, as well be shown later.
3.1.5. Forward integration based reverberation time estimation
The author previously proposed a method to calculate the reveberation time without using regression
or backward integration based on moments of the RIR and forward integration13 . By using a smoothed
representation of the RIR h(t, p).
=|h(t)|p, the reverberation time is obtained by
R(p) = kp ·R∞
0t·h(t, p)dt
R∞
0h(t, p)dt (12)
The first term kp of (12) assures a p-independent result for the pure exponential decay, while the second term
can be interpreted as the generalized gravity point of the smoothed RIR (a generalized center time). The word
generalized is used because not h(t)directly, but its p-th power is involved. For a monoexponential decay
the result is a single value without a dependency on p.
3.2. Multiexponential estimation methods
3.2.1. Ambiguity of multiexponential decomposition
The main problem of multiexponential analysis is due to the non-orthogonality of exponentials along the
real axis,10,14. Although the inverse Laplace transform provides the solution of the multiexponential analysis
problem, the measured data is not available analytically and even if it were, it would require the solution of
an integral equation in the complex plane that is also ill-posed, meaning that the solution may not exist or be
unique10. The same is true for smoothed decay functions by backward integration. Therefore it cannot be
expected that decay parameters (decay constants and amplitudes) found by such methods are unique solutions
of the exponential decomposition problem even if a given threshold of error is fulfilled.
Examples can be easily found where the multiexponential analysis can in itself be ambiguous, with
results sometimes difficult to interpret, even if the applicability of a multiexponential model to particular
measurement data is already known. For example, estimating a noisy double decay can lead to different
values if only little information on the second decay is visible. If the model order is unknown, the princi-
ple of parsimony (Ockham’s razor15) is applied, namely that the simplest model with the least number of
components is preferable among all possibilities. This, however, might not be explaining the underlying phe-
nomena. For example, a noisy triple decay with parameters R= [1,0.1,0.3],A= [0.1,0.5,1] and SNR= 40
dB can be approximated by a 2-component decay (using NLS) of R= [0.259,0.942],A= [1,0.921] and
SNR = 39.97 dB with a goodness-of-fit of r2= 0.9998 and almost no visually identifiable difference. The
principle of parsimony would suggest a two-component model for this case, while the generated signal was
of three components.
3.2.2. Bayesian inference
The application of the Bayesian inference, using a maximum likelihood estimation and the Bayes In-
formation Criterion (BIC) to determine the number of components16–18, is particularly useful for identifying
coupled spaces. The method uses a simplified multiexponential decay function which must be undersampled
in order to guarantee success in the numerical evaluation of the Gamma function used in the BIC. At low
frequencies, due to the fixed smoothing features of the energy decay function, the undersampling must be
moderate, whereas at high frequencies a greater degree of undersampling can be used. The Bayesian method
generally requires an exhaustive search in the full parameter space, therefore in a general case merits of in-
troducing the probability framework are questionable. Slice sampling reducing the estimation time can be
applied if certain probability assumptions can be made with respect to the decay. The Bayesian inference
supports determining the model order without a-priori information on the model, however, as test presented
here showed, the determination sometimes should be ’guided’ by properly set range parameters, and failure
can be expected even in a noise-free case. In our implementation of the Bayesian inference made according to
the literature, the results were highly dependent on the ranges of parameter search and the exhaustive search
performed rather slowly. More optimum implementations may be possible but were not available at this time
of writing.
3.2.3. Multicomponent NLS
The idea of the NLS can be applied to multiple decays by exchanging the monoexponential model
functions to multicomponent ones as presented above. Although the resulting formulas can become long, the
complexity of estimation is practically unchanged. An algorithm using a trust-region reflective approach19
with multicomponent model functions was implemented in MATLAB R
. The resulting method is easy to
apply and yields useful results without the need of using the probability framework, while the principle of
parsimony remains applicable based on goodness-of-fit measures and estimated mixing amplitudes.
4. RESULTS
A systematic comparison of mono and multiexponential decay time estimation methods was conducted
based on simulations. In each simulation a decay envelope model was created, noise was added, and the
resulting RIR was used as the input of each estimation method. The estimated parameters were compared
to the reference parameters and the error was calculated based on their differences. The simulation was
conducted in the parameter space of different RL ratios (corresponding to the cutoff bias) and at different
SNR values (corresponding to the noise bias). Empirical noise bias reduction methods requiring two or more
measurements (such as Hirata’s method20 ) were excluded in the simulation as they require two RIRs in the
same position as opposed to other methods requiring only one.
4.1. Evaluation of monoexponential estimation
The following methods were compared: truncation method, subctraction method, extrapolation method
(standardized as ISO 3382), non-linear regression method according to the literature5, non-linear regres-
sion method according to the author, Fourier-based decay time estimation method according to the author,
forward-integrated method according to the author.
4.1.1. Performance at middle and higher frequencies in the presence of noise
At middle and higher frequencies the RIR envelope looks often smooth and the model decay can be
considered as similar to measured RIRs. Results are summarized in Fig. 1. The best and most reliable results
were obtained by using the enhanced nonlinear regression method implementing the trust region reflective
approach, allowing the calculation of decay times reliably even at 0 dB SNR and at a truncated response
(the error was within 5% if 0.05 < R/L < 2regardless of SNR). The Fourier method outperformed the
subtraction, truncation, extrapolation, and the original iterative nonlinear regression methods for cases when
the cutoff bias is large (at large RL ratios), or otherwise stated, when the measurement length is short for a
given reverberation time. These traditionally used methods seem to become inaccurate at approximately 12
dB of SNR, and the Fourier method becomes inaccurate if a low SNR and a short reverberation time compared
to the length happen simultaneously. In such cases, the truncation and subtraction methods performed better
than the Fourier and the traditional nonlinear regression methods, but the NLS method may perform better
than the Fourier method if the initial conditions allow its convergence. In the practical measurement ranges
reported in the literature5the Fourier method performs similarly, but is simpler to apply and implement.
4.1.2. Performance in case of a model mismatch
In case the RIR is not monoexponential but a monoexponential estimation method is used, the estimation
model does not match the input data model, so reliable estimations cannot be expected. This is often the
case in practical measurements or when no information is available about the decay process and only visual
feedback (i.e., plotting the EDC) is possible. We examined the similarity of noise-free multiexponential and
noise-free monoexponential RIRs using a simulation. 40,000 double decay RIRs were generated for each
case and compared to monoexponentials created using the largest time constant of the corresponding double
decay, being a likely estimation for the full process. The estimation accuracy is expressed by the goodness-
of-fit r2. Results showed that in order to distinguish between double decays and single decays effectively,
the double decay must have its shorter time constant component mixed very strongly, producing a clearly
noticeable drop in the decay curve. Otherwise, distinguishing by any method including visual feedback
would not be easy or possible. If the extrapolation method is used for estimating a (noise-free) double-decay
process, different extrapolation ranges will yield different reverberation time values, as already known. If the
extrapolation ranges are chosen to be large, differentiation between a single and double decay process will
become more difficult in the R2/R1/L < 1region. The nonlinear regression method tends to be an effective
candidate to indicate a model mismatch by using goodness-of-fit measures in many cases. At this point, a
reliable method for checking the validity of the applied model, applicable to all estimation types, is yet to
be found and defined. From a signal processing viewpoint, a finite Gaussian background noise sample is not
really distinguishable from very long time constants of multi-component decay processes (but practically,
this issue can be mitigated). Even in a noise-free case an ’indication’ of model difference is the current
best practice. For example, in the extrapolation method different time constants are obtained which can be
used to indicate a model mismatch. In the Fourier-based method a larger dependency of ωwill appear, and
in the forward integration method p-dependency will be significant. In the nonlinear regression method the
goodness of fit is a possible candidate for such indication.
4.2. Evaluation of multiexponential methods
The following methods were compared: Bayesian inference and multicomponent NLS. The parameter
estimation consists of the tasks of 1) finding model orders and 2) finding model parameters. As shown
before, there due to the ambiguity of exponential decomposition in the real space, always finding the proper
and unique parameters reliably solely from the real decay function seems unfeasible.
4.2.1. Noise-free double decay
In case of a noise-free double decay the Bayesian inference misidentified the model order for numerous
times especially when the two time constants were similar, despite it was allowed to choose only from order
1or 2. A visual verification of the decay function might be similarly confusing underlining the ambiguity
of exponential decomposition. In this case the Bayesian inference did not perform much better than the
monoexponential NLS applied to the double decay (the ’model mismatch’ case) in the R2/R1/L < 1region.
The multicomponent NLS performed better: when checking the obtained reverberation times it was found
that the NLS identified the parameters correctly except for one region where any two time constants were
mixed in a similar ratio. In this case the method identified one time constant correctly but mixed incorrectly,
and the other time constant incorrectly but mixed correctly. In other words another possible combination of
exponentials that fit correctly was found.
4.2.2. Noisy double decay
In addition to noisy double decays, simulations of triple decays were also conducted, with and without
noise, but they are omitted from this paper because of the complexities of graphically presenting the results.
The Bayesian inference was tested on noisy double decays and the model order was allowed to be 2at most
(Fig.3c,3d). An arbitrary selected amplitude ratio of A= [1; 0.25] was chosen to allow plotting the results. In
this case the Bayesian method found parameters with a good fit at SNR>40 dB and R2/R1/L < 2. Below
this SNR, the estimation did not provide a good fit because of incorrect estimations of the SNR. A high value
of goodness-of-fit does not necessarily mean that the estimated parameters were correct, but suggests that a
matching combination of exponentials along with the estimated noise level was correctly found. From the
results it can be seen that the multicomponent NLS method performs consistently for SNR>8dB for the
selected mixing amplitudes (Fig.3f).
5. CONCLUSIONS
In this paper we presented various reverberation time estimation models and evaluated their perfor-
mance. A general conclusion can be drawn as follows. Among the various methods of monoexponential pa-
rameter estimation, the improved nonlinear regression method is the most reliable. Similarly reliable methods
are the forward integration and Fourier-transform based methods, with the additional merit of easier imple-
mentation. Previously published methods such as the truncation, iterative truncation and subtraction methods
did not perform well and are not recommended to use. Due to the opposite effects of the two biases occur-
ring in the energy decay function, the extrapolation method can produce misleading results in some cases at
low SNR and moderately high reverberation time compared to the available length. For the multicomponent
case the Bayesian method can be used to suggest a model order, but in many cases misidentification can be
expected especially in situations where visual verification is also difficult. When examining the estimated
parameters in a known model order, the proposed multicomponent nonlinear regression significantly outper-
formed the Bayesian search both in simplicity, ease of implementation and speed. The above mentioned
methods were also applied to measured RIRs and among them, the multicomponent nonlinear regression
seems to be most practical providing best matching, but the Fourier and forward integration based methods
were also found to be conveniently applicable. These results, due to length restrictions of this paper, will be
presented in a future work.
6. REFERENCES
[1] L. L. Beranek, Acoustics (Acoustical Society of America, Cambridge, MA) (1993).
[2] M. R. Schroeder, “New method of measuring reverberation time”, The Journal of the Acoustical Society
of America 37, 409–412 (1965).
[3] M. Vorl¨
ander and H. Bietz, “Comparison of methods for measuring reverberation time”, Acustica 80,
205–215 (1994).
[4] ISO, “3382-2:2008 - Acoustics – Measurement of room acoustic parameters – Part 2: Reverberation
time in ordinary rooms”, International Organization for Standardization, Geneva, Switzerland (2008).
[5] N. Xiang, “Evaluation of reverberation times using a nonlinear regression approach”, The Journal of the
Acoustical Society of America 98, 2112–2121 (1995).
[6] L. Faiget, R. Ruiz, and C. Legros, “The true duration of the impulse response used to estimate reverber-
ation time”, in Proc. International Conference on Acoustics, Speech, and Signal Processing (ICASSP)
96, volume II., 913–916 (IEEE, Atlanta) (1996).
[7] A. Lundeby, T. E. Vigran, H. Bietz, and M. Vorl¨
ander, “Uncertainties of measurements in room acous-
tics”, Acustica 81, 344–355 (1995).
[8] W. T. Chu, “Comparison of reverberation measurements using Schroeder’s impulse method and decay-
curve averaging method”, The Journal of the Acoustical Society of America 63, 1444–1450 (1978).
[9] ISO, “3382-1:2009 - Acoustics – Measurement of room acoustic parameters – Part 1: Performance
spaces”, International Organization for Standardization, Geneva, Switzerland (2009).
[10] A. A. Istratov and O. F. Vyvenko, “Exponential analysis in physical phenomena”, Review of Scientific
Instruments 70, 1233–1257 (1999).
[11] M. Karjalainen, P. Antsalo, A. Makivirta, T. Peltonen, and V. Valimaki, “Estimation of modal decay
parameters from noisy response measurements”, Journal of the Audio Engineering Society 50, 867–878
(2002).
[12] P. D. Kirchner, W. J. Schaff, G. N. Maracas, L. F. Eastman, T. I. Chappell, and C. M. Ransom, “The
analysis of exponential and nonexponential transients in deep level transient spectroscopy”, Journal of
Applied Physics 52, 6462–6470 (1981).
[13] C. Huszty and S. Sakamoto, “Application of calculating the reverberation time from room impulse re-
sponses without using regression”, in Proc. Forum Acusticum 2011, Paper 450, 1929—1934 (European
Acoustics Association, Aalborg, Denmark) (2011).
[14] C. L´
anczos, Applied analysis (Dover Publications) (1988).
[15] J. C. Santamarina and D. Fratta, Discrete signals and inverse problems (John Wiley & Sons, Ltd.)
(2005).
[16] N. Xiang and P. M. Goggans, “Evaluation of decay times in coupled spaces: Bayesian parameter esti-
mation”, The Journal of the Acoustical Society of America 110, 1415–1424 (2001).
[17] N. Xiang and P. M. Goggans, “Evaluation of decay times in coupled spaces: Bayesian decay model
selection”, The Journal of the Acoustical Society of America 113, 2685–2697 (2003).
[18] N. Xiang, P. Goggans, T. Jasa, and P. Robinson, “Bayesian characterization of multiple-slope sound
energy decays in coupled-volume systems”, The Journal of the Acoustical Society of America 129,
741–752 (2011).
[19] T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to
bounds”, SIAM Journal on Optimization 6, 418 (1996).
[20] Y. Hirata, “A method of eliminating noise in power responses”, Journal of Sound and Vibration 82,
593–595 (1982).
(a) Extrapolation, T20 (b) Extrapolation, T30
(c) Subtraction (last 10%), T30 (d) Iterative truncation, T30
(e) Nonlinear regression (as in the literature), T30 (f) Fourier-based method
(g) Nonlinear regression using the trust region reflective
approach
(h) Forward integration (mean at a range of p, where
p∈[0.05,0.7] in 0.05 steps)
Fig. 1 – Comparison of decay time estimation methods using a decay envelope and an alternating noise
model. The signal length was L= 1 s with fs= 48000 Hz. Colors correspond to estimation error
in percentage.
(a) NLS according to the literature 5,r2(b) NLS as proposed in this paper, r2
Fig. 2 – Double decays estimated by using a mismatched monoexponential model using non-linear regression
(NLS). Goodness-of-fit r2is plotted for each RIR. Blue regions correspond to double decays not
distinguishable by this method from monoexponential decays. Light gray ranges indicate r2≤0.75.
(a) Bayesian inference: estimated order (noise-free) (b) Bayesian inference: goodness-of-fit r2(noise-free)
(c) Bayesian inference: estimated order (noisy) (d) Bayesian inference: r2(noisy) A= [1; 0.25]
(e) 2-component NLS: goodness-of-fit r2(noise-free) (f) 2-component NLS: r2(noisy) A= [1; 0.25]
Fig. 3 – Double decay estimated using a two component exponential model. Gray regions indicate goodness-
of-fit r2≤0.75.
