The Influence of Overlapping Band Filters on Octave Band Decay Curves

Summary This study showed that the overlap of practically-used bandpass ﬁlters can inﬂuence the octave band decay curves, especially if the decays are calculated from a ﬁltered impulse response that has been created from octave band energy responses. Energy from a frequency band with a long reverberation time can leak into a neighbouring band with a shorter reverberation time. This also means that neither octave band decays from a measured response are independent, nor are measured octave band reverberation times.


Introduction
When using energy-based geometrical room acoustic modelling techniques, room acoustical parameters are normally calculated at the centre frequencyo fo ctave bands.This assumes that the energy response in each band only depends on the material properties of the very band.If these results are to be used for auralisations, it is necessary to create atotal full bandwidth pressure impulse response from the octave band energy impulse responses [1,2,3,4].Here, full bandwidth response refers to ar esponse that covers the entire frequencyrange of interest, typically the audible range.The full bandwidth pressure response can be obtained by first creating octave band pressure impulse responses and then summing these.
If the full bandwidth pressure impulse response is refiltered into octave band impulse responses, the decays of these are unlikely to be the same as the decays of the responses from before the summation, because of overlaps between adjacent bands that cause energy leakage.Reverberation times are often calculated from the octave band responses, expecting these to be valid also for the full bandwidth impulse response.This letter demonstrates that theya re not necessarily so.When measured impulse responses are processed with bandpass filters, the decays in the bands are not independent, and simulations assuming independent bands therefore do not correspond to measurements.
This study focuses on decay curves and reverberation times, because the effect of the frequencyleakage is much larger when considering the energy decay than the total energy or the steady state response.
The phenomenon is not limited to energy-based models, because one might use apressure-based model to calculate an impulse response within ao ctave band and determine the decay from this.In this case, it is also possible that the obtained decay will not correspond to one that would be obtained if the impulse response of aw ider frequency range had been calculated and then filtered to the octave band.
To the best knowledge of the authors, this issue of overlapping bands has not yet been sufficiently discussed in this application field.Arelated issue for measurements of narrowband decays, is the influence of the time responses of the filters, which has been studied [5,6].The present study illustrates howthe overlapping bands influences the decays from energy-based models, and investigates this through simple examples.

Full bandwidth impulse response
Af ull bandwidth impulse response from energy-based methods is often found by first determining octave band impulse responses and then taking the sum of those as where p b (t)i sthe impulse response of the octave band b. p b (t)c an for instance be found with an octave band noise signal that is used to fill an energy impulse response.The octave band impulse response will in that case be givenby where w b (t)isthe energy impulse response of the band b, n(t)i saG aussian noise signal, and h b (t)i st he impulse response of the filter of band b. n(t) * h b (t)i st hus ao ctave band noise signal of which the content will be mainly within the cutoff frequencies of the band b,b ut there will be some content outside depending on the sharpness of the filter.The method of Equation (2) is, for example, used to obtain apressure impulse response from acoustical radiosity in the simulation tool PARISM [4].APoisson process with random sign can also be used rather than the Gaussian noise signal [3].
To compare with Equation ( 2),am ethod that does not employbandpass filters is tested.The rational behind this approach is to limit the overlaps of the octave band responses.Fort his, sine functions of random phases are used, and the impulse response within asingle band is then givenby where L is the number of included sines within band b and ϕ l is arandom phase.f l refers to frequencies between the lower and upper cutoff frequencies of band b.W ith this formulation, the overlap of the bands in the creation of the full bandwidth response only comes from the attenuation of the sines due to the decay factor w b (t).Regardless of whether p b,noise or p b,sin is used, there will be an overlap of the bands if the full bandwidth response is refiltered with non-ideal filters.By comparing p b,noise and p b,sin ,i tc an be determined howm uch of the total effect is due to the fact the filters in p b,noise overlap, and how much is due to the overlap of the filters for refiltering.The refiltered octave band response is found as where the subscript RF denotes that it is the refiltered response, and p(t)i sf ound with Equation (1).I nt he following p b,noise,RF refers to a p b,RF using p b,noise in Equation (1),and p b,sin,RF refers to a p b,RF using p b,sin in Equation (1).

Example with geometrical room acoustics
Firstly,the influence of overlapping bands on decay curves is illustrated with an example using the room acoustical simulation tool CARISM [7] (Combined Acoustical Radiosity -I mage Source Method).CARISM is an energy-based combination of acoustical radiosity (AR) and the image source method (ISM), and the results from CARISM are octave band energy impulse responses.The method of Equation ( 2) is applied to obtain apressure impulse response, and the filters used there and in the refiltering are octave bandpass filters constructed from the 7th order high-and low-pass Butterworth filters.The bandpass filters are constructed such that the sum of their frequency responses is flat, and theym eet the requirements of IEC 61260-1 [8].
Octave band decay curves and reverberation times from CARISM can then be obtained with twomethods.Method I: Directly from the octave band energy impulse responses (w b (t)),w here the octave band results are independent of each other.T his is the standard method in CARISM.Method II: By filtering the full bandwidth impulse response that is constructed, Equation (4).
The chosen test room is based on an existing room at the laboratories of the Technical University of Denmark and has dimensions [4.38 × 3.29 × 2.7] m.The calculations are done in the octave bands from 125 Hz to 8k Hz, and the sampling frequencyfor the pressure impulse response is 24 kHz.The scattering coefficients of all surfaces are set to [0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1] for the eight octave bands, respectively.T he absorption coefficient is 0.05 for all surfaces and frequencies.The air absorption is determined according to ISO 9631-1 [9], and since the surface absorption is frequency-independent most of the differences in reverberation times overf requencyw ill be due to the air absorption.The reverberation times T 30 ,calculated with both methods Iand II, are plotted in Figure 1.Differences are seen between the twom ethods at 4a nd 8kHz, and that values obtained with method II are higher than those of method ItoII.
The decay curves for the 4kHz and 8kHz bands from methods Iand II are plotted in the lower part of Figure 1.The 8kHz curveofmethod II follows the 8kHz method I curveinthe very early part, and then the slope changes to be more similar to that of the 4kHz method Icurve.This indicates that energy from the 4kHz band is influencing the 8kHz band in the part of the decay where the energy is lowinthe 8kHz band.It is also observed that the curve of 8kHz band of method II is tending more towards being double-sloped than that of method I.The 4kHz curves of the twomethods are more similar.

Examples with exponential decays
The energy impulse responses of Eqs. ( 2) and ( 3) are then chosen to decay exponentially.W et hus set w b (t): = e − δ b t ,r ather than determining w b (t)t hrough as imulation as in Section 3. δ b is the exponential decay constant, from which the reverberation time can be found by T = 3l n(10)/δ.The initial filters used here are the same as in the example of Figure 1.As in the previous example, the 4kHz and 8kHz octave bands are used.Firstly,t he frequencyr esponses of the twob ands are regarded with both reverberation times set to 0.9 s.The spacing of the frequencies f l in Equation ( 3) is 1Hz.The realisations of p b,noise and p b,sin were repeated 200 times, because each realisation will be slightly different due to the random noise in p b,noise and the random phases in p b,sin .The frequencyresponses are calculated by the Fourier transform of the impulse responses and the means of the squared magnitudes of the 200 realisations of frequencyr esponses are plotted in Figure 2. The overlap of the p b,sin frequencyresponses is very small (0.085% of the total energy), which makes good sense.The overlap of the frequencyr esponses of p b,sin,RF is then much larger (4.3% of the total energy).The frequencyresponses of p b,noise overlap much already,s oi ti sb arely increased for p b,noise,RF (from 4.3% to 4.4% of the total energy).
Ad i ff erence in the reverberation times between the 4kHz and 8kHz octave bands is then introduced.Theyare set to 1.7 and 0.9 s, respectively,which are taken from the example of Figure 1.
The mean decay curves from 200 realisations of p b,noise and p b,noise,RF are shown in Figure 3. p b,sin and p b,sin,RF are left out of this figure because theya re very similar.T he decay curveo fp b,noise,RF in the 8kHz band is very much influenced by the 4kHz band.It is not single-sloped and  follows the one of 8kHz p b for the very first part of the decay,but in the later part the slope approaches that of the 4kHz p b,noise decay.T he p b,noise and p b,noise,RF decays for the 4kHz band coincide, which confirms that the leakage between bands mostly influences the band with the shorter reverberation time.
The mean reverberation time (T 30 )a nd mean early decay time (EDT)o ft he 200 realisations were also calculated, and the relative differences between those from p b,noise,RF and p b,sin,RF ,a nd those from p b,sin and p b,noise are calculated as  I, it is seen that the difference is obviously largest for the 8kHz band, and that it is the reverberation time that is most influenced.Forthe 8kHz bands, the changes in reverberation times are above the just noticeable difference, which is stated as 5% in ISO 3382-1 [10].Forthe 8kHz early decay times, it is only ΔEDT p b,sin that is belowthe just noticeable difference.Forthe 4kHz values there are also small changes, all belowthe just noticeable difference.But since the differences are consistently reductions, theyc annot be random and must stem from the leakage.The changes are generally smaller for p b,sin,RF ,but still large enough to showthat the overlap of the refiltering filters can create anoticeable difference.
In order to test the influence of the filter design, higher order Butterworth filters are tested.The bandpass filters are then created from the 9th order filters rather than the 7th order.T he differences in the EDT and T 30 with these filters are shown on the right side of Table I.The differences caused by the leakage are smaller when the filters are sharper,but it is still only the 4kHz ΔEDT p b,sin that is belowthe just noticeable difference.
When choosing and designing filters, their computational cost and stability should be considered.Fort he present Butterworth filters, the highest possible order for stable filters is 7, if the 125 Hz octave band should be included.Moreover, when filtering to obtain decays in octave bands possible, ringing of the filters in the time domain should also be considered, because it can influence the decays [6].Ringing in the time domain tends to increase when the filter are sharper in the frequencydomain.

Concluding remarks
When creating full bandwidth pressure impulse responses from octave band energy responses, the overlaps of the applied bandpass filters influence the decays of the octave bands.The effect can be important when looking at decays, even when the leakage in energy is marginal.If an octave band has aneighbouring band with aslower decay than itself, leakage from the slowly decaying band will makeits decay slower when calculated from the full bandwidth impulse response.The shape of the decay curve will furthermore tend to be double-sloped.Even if the construction of the full bandwidth response is done such that there is hardly anyo verlap between the octave band responses, the overlaps of the filters used for refiltering the full bandwidth response are big enough for spillover between the bands to influence the reverberation times.This indicates that the same will be true when filtering and processing am easured impulse response, which means that the assumption of independent bands in simulations is an approximation and may lead to noticeable errors.Finally,the design of the bandpass filters has an influence on leakage, and sharper filters naturally reduce the effect.But even with sharper filters than required in IEC 61260-1 [8], the influence on the reverberation time of non-ideal filtering is found to be higher than the JND.

5 )
ΔT 30,p b = T 30,p b,RF − T 30,p b /T 30,p b • 100% (ΔEDT p b = EDT pb,RF − EDT p b /EDT p b • 100%, where the T 30 and EDT values are means of the 200 realisations.In Table

Table I .
Percentage differences in T 30 and EDT from p b to p b,RF .Note that the JNDis5%.