errors in real-time room acoustics dereverberation - Wire ...

ERRORS IN REAL-TIME ROOM ACOUSTICS DEREVERBERATIONPANAGIOTIS D. HATZIANTONIOU AND JOHN N. MOURJOPOULOSAudio Group, Wire Communications LaboratoryElectrical and Computer Engineering DepartmentUniversity of Patras, Patras, 265 00 GreeceTel.: +30 261 0 996474Fax: +30 261 0 991855E-mail: mourjop@wcl.ee.upatras.grSignificant measurable audible distortions are generated during realtimeroom acoustics dereverberation, far greater in magnitude thanthose expected from identical simulated (off-line) dereverberationexperiments. Long inverse filters worsen this effect. Possiblemechanisms for such discrepancy between simulated and real-timetests are examined. It is also shown that dereverberation based onComplex Smoothing is immune to such errors.0. INTRODUCTIONThe term dereverberation is used here to define the broad category of digitalsignal processing methods, which attempt to undo (partially or completely) thephysical or perceptual artifacts generated by audio signal reproduction insidereverberant spaces. This term is comparable to other, alternative terms for1

similar methods, such as: “room correction”, “room compensation”,“reverberation reduction”, “room equalization”, etc. Also, the termdereverberation can include similar methods applied to restricted audiobandwidth such as “low-frequency correction”, “modal-equalization”, etc.Given the mixed-phase nature of room response due to room boundaryreflections, this paper will mainly address the inversion (“undoing”) of suchmixed-phase responses, since such methods present the most general anddifficult case of the problem. The conclusions of this work will also apply tothe simpler case of minimum-phase room response inversion, which isusually referred to as “room equalization”.The theoretical aspects of room dereverberation based on room impulseresponse inverse filtering have been introduced more than two decades agofor minimum-phase [1] and for mixed-phase response inversion [2]. Theseearly studies have highlighted a number of theoretical problems, namely, theacausal nature of such inverse filters when it is chosen to compensate for thenon-minimum phase room response (practically realized via the introductionof appropriate delay [2, 3]), the sensitivity of the processing with respect tothe measured source / receiver placement inside the room [4, 5], as well asthe large number of inverse filter coefficients required. For the first of thoseproblems it has been found that in principle, non-minimum phasecompensation is beneficial compared to magnitude (minimum-phase)correction, not only for reducing reverberation energy, but also for phaselinearization which is especially desirable for (anechoic) loudspeakerresponse equalization [6,7]. The disadvantage of such acausal filters, is that2

methods achieve by definition a non-ideal RTF inversion and hence lowerdereverberation performance than theoretically could be achieved by the“ideal” dereverberation, but generally are optimized to minimize audibledegradations due to the previously discussed problems. These authors haverecently produced results for such a method, based on the ComplexSmoothing of room responses [20], and tested it over a large variety ofdifferent spaces [29]. This method is free to a large extend of all of the aboveproblems affecting practical dereverberation, i.e. audible artifacts,displacement sensitivity and measurement variations.During those past studies, the results were largely derived via off-linesimulation tests and the issue of real-time dereverberation has been onlypassingly examined, often found to be highly sensitive to many practicalfactors so that the robustness of these methods and perceived gains aftertheir real-time application could not be easily quantified. An often-discussed,but never formally investigated aspect of these methods relates to the audibledistortions generated during real-time dereverberation which althoughreported to be severe, could not be accounted for by the expected theoreticalerrors such as the previously discussed ones which were measured by offlinesimulations and tests. Such distortions were initially described in earlyreal-time listening dereverberation tests, by Neely and Allen [1] (for minimumphaseresponse inversion) and Mourjopoulos [4] (for mixed-phase responseinversion). It is often felt that these distortions were audible manifestations of4

the previously discussed error artifacts [7-9], although such conclusions werenot derived from formal in-situ measurements of the inversion errors.Here it will be shown that such distortions are manifestations of large scalemeasurable errors generated during practical real-time dereverberation,which are far greater in magnitude than those expected from the theoreticalstudies and which could be evaluated during any identical simulateddereverberation experiment. It will be also shown that such large discrepancybetween simulated and real-time tests is not a result of system non-linearity,but instead, the inversed room path must be practically treated as a “weaklynon-stationary” system [42], that is, a system whose properties will practicallyvary between measurement and test even if source / receiver placement andother physical room parameters have not been intentionally changed. Thisindicates that an “uncertainty” will appear between the responsemeasurement and subsequent inverse filter realizations and hence suchfindings are related to previously published results concerning room responsemeasurement [34, 37, 38]. It will be shown experimentally that those factorsare becoming more critical when long inverse filters are employed whichgenerate larger orders of measurable errors and audible degradations in thedereverberated signals. Possible sources for such errors will be discussedand analytical mechanisms for their generation will be examined. It will bethen also shown that the proposed dereverberation method based onComplex Smoothing of room responses, is immune to such discrepanciesbetween simulated and real-time tests.5

The paper is organized as follows: section 1 establishes the experimental setupand theoretical background for real-time and simulated (off-line) tests;section 2 provides experimental results which show the quantitative andqualitative aspects of differences between the dereverberation performance ineach of the above cases; section 3 will analytically explain the possible natureof such error; section 4 will illustrate why dereverberation based on roomresponse Complex Smoothing is free of such errors; section 5 will providediscussion and conclusions.1. THEORY: DIFFERENCES BETWEEN REAL-TIME AND OFF-LINEDEREVERBERATIONTo investigate any discrepancy between simulated and real-timedereverberation methods, it is necessary at first to define the measurementand test chains for each class of experiments: (a) the simulateddereverberation experiments are conducted within a computer, employingmeasured room responses, subsequently also referred to as “off-line” or“theoretical” tests and, (b) the real-time dereverberation measurements, aresimilarly employing previously measured responses, but are conducted in-situ(inside a room).To compare directly these two types of tests, all relevant parameters weredefined and kept identical in both classes. Nevertheless, given the largenumber of such parameters which potentially can affect the outcome of the6

esults, these parameters are noted in the corresponding Figures (1,3-6) andare also listed in Table 2, in the Appendix.In the above Figures the detailed experimental set-up is shown and allpotentially adjustable and variable parameters are indicated inside squarebrackets. Each module in the experimental chain, this being electrical (analogor digital), electroacoustical or acoustical, is considered to be a linear system,characterized by its corresponding measurable impulse response function, sothat well-established expressions can be employed, as is illustrated below.1.1 Room response measurementFollowing Figure 1, it is:[ s ( )]ste ( t)= DACten(i)s′ ( t)= g ⋅s( t)(ii)srrtetate( t)= s′( t)∗ h ( t)(iii)teL[ s ( t)∗ h ( t)] n ( )( t)= t(iv) (1)ta ta R+tem( t)= r ( t)∗ h ( t)(v)taMr ′ ( t)= g ⋅ r ( t)(vi)tete[ r ( )]r ′ ( n)= ADC t(vii)′te tewhere the symbol ∗ denotes the continuous-time convolution. All othersymbols are listed in Table 2 in the Appendix.7

Following the well-established direct approach for ideal inversion of themeasured response h (n)[3, 33] (for a specific source/receiver position), an“inverse filter”h i(n) with length Li(samples) has to be introduced, such thata new response d i(n)will be produced, according to the followingexpression:d ( n)= h(n)⊗ h ( n)≅ δ ( n),n = 0,1, K,L(2)iidwhere δ (n)is the ideal impulse response, L L + L −1(samples) is thed= ilength of the deconvolved response and the symbol ⊗ denotes the discretetimelinear convolution. The inverse filter response h i(n)was calculated hereto have the best (least-square) approximation for the ideal response δ (n)[3].Other inversion strategies (e.g. via direct DFT inversion) may also producecomparable results, noting potential sources of practical errors [9].The test chain for such inversion is shown in Figure 3, where it can beobserved that all processing is carried-out within a simulated computerenvironment. The majority of results presented in the dereverberationliterature were derived by such off-line tests.1.2.2 Real-time room response inversionIn this case, the source excitation signal is pre-filtered in real-time by theresponse inverse filter so that after acoustic reproduction in the room, themeasured response at the microphone would produce the inverted9

(deconvolved) room impulse response, as is shown in Figure 4 and it isdescribed below:sfte( n)= s ( n)⊗ h ( n)(i)te[ s ( )]isfte ( t)= DACften(ii)s′ ( t)= g ⋅s( t)(iii)srrfteftafte( t)= s′( t)∗ h ( t)(iv) (3)fteL[ s ( t)∗ h ( t)] n ( )( t)= t(v)fta fta R+ftei( t)= r ( t)∗ h ( t)(vi)ftaMr ′ ( t)= g ⋅ r ( t)(vii)ftefte[ ( )]r ′ ( n)= ADC r t(viii)′fte fteAll symbols are listed in Table2 in the Appendix.The recorded dereverberated excitation signal defined by eq. (3viii) isappropriately processed (see previous case) yielding the measured real-timeinverted discrete-time system (loudspeaker/room/microphone) responsed ˆ ( n),n = 0,1,K . This response can be then analyzed as in Figure 3.i,L d1.2.3 Error criteria in room response dereverberation.The dereverberation performance can be monitored via the time domaindifference error e (n)between the desired off-line ( d i(n)) or real-time ( ˆ ( n )inversed response:di )e( n)δ ( n)− d ( n),n = 0,1, K,L − 1(4a)eˆ( n)=idδ ( n)− dˆ( n),n = 0,1, K,L − 1(4b)=id10

The term “dereverberation error” will be used from now on to describe theabove error sequences.Any discrepancy between real-time and off-line inversion as was described inthe previous sections, is defined here as the difference error e d(n), betweenthe two corresponding inverted responses (equations (2) and (3)), i.e.:e ( n)dˆ( n)− d ( n),n = 0,1, K , L − 1(5)d=iidThe term “dereverberation discrepancy error” will be then used form now onto describe the above error sequence. Equivalently in the discrete frequencydomain:E) = Dˆ( ω ) − D ( ω ) , ω 2π k L , k = 0,1 KL− 1(6)d( ωki k i kk=ddwhere d i(n), D ω ) and dˆ i( n), D ˆ ( ) are Discrete Fourier Transform pairs.i(kiω kFor the objective analysis of the dereverberation performance in the timedomain, the off-line dereverberation error energy J (dB), and the real-timedereverberation error energy Ĵ (dB) can be derived from eq. (4), definedrespectively as:L⎧⎫= ⎨ ⋅ ∑ − 11 d2J 10log10 [ e(n)] ⎬7(a)⎩Ldn=0 ⎭L⎧⎫= ⎨ ⋅ ∑ − 11d2J ˆ 10log10 [ eˆ(n)] ⎬7(b)⎩Ldn=0 ⎭The dereverberation discrepancy error energy Jd(dB) is defined, then, as:J d= J ˆ − J(8)11

In the frequency domain, the standard deviation V of the DFT modulus isused as an objective criterion of the spectral flatness of the off-line or realtimeinverted response, defined respectively as:V⎧⎪ 1⎨Ld⎪⎩−1k = 02L −1⎡1⎤ ⎫⎪⎢10logDi( ω k ) − 10logDi( ω k ) ⎥ ⎬(9a)⎣Ldk = 0⎦ ⎪⎭L d d= ∑ ∑0.5Vˆ⎪⎧ 1⎨Ld⎪⎩L d −1L d −1k = 02⎡ˆ 110log ˆ⎤ ⎪⎫⎢10logDi( ωk) −Di( ωk) ⎥ ⎬(9b)⎣Ldk = 0⎦ ⎪⎭= ∑ ∑0.5The term “dereverberation spectral deviation” will be used from now todescribe this function.Any discrepancy between off-line and real-time dereverberation spectraldeviation can be then defined as:V d= V ˆ −V(10)The term “dereverberation discrepancy spectral deviation” will be used todescribe the above function.1.3 Dereverberation of audio signalsWhen a typical audio segment substitutes the source excitation signal, thensubjective evaluation of dereverberation can be carried-out. As with theprevious analysis, such tests can be carried-out under simulated (“theoretical”or “off-line”) conditions, or via real-time reproduction and audition.12

1.3.1 Off-line audio dereverberationAs is shown in Figure 5, a simulated audio dereverberation test can becarried-out within a computer using a measured room response, so that theprocessed signal can be auditioned, typically via headphones. In such a case,it is:r t( n)= s(n)⊗ h(n)(i)s ( n)= r ( n)⊗ h ( n)(ii)it[ s ( )]isi ( t)= DACin(iii) (11)s′ ( t)= g ⋅ s ( t)(iv)iis′ ( t)= s′( t)∗ h ( t)(v)ihipAll symbols are listed in Table 2 in the Appendix.1.3.2 Real-time audio dereverberationIn this case, the source audio signal is pre-filtered in real-time by theresponse inverse so that after acoustic reproduction in the room, the audiosignal at the listener / microphone position would in theory be devoid of roomacoustic distortions, as is shown in Figure 6 and it is described below:s ( n)= s(n)∗ h ( n)(i)fi[ s ( )]sf ( t)= DACfn(ii)s′ ( t)= g ⋅ s ( t)(iii) (12)srffaf( t)= s′( t)∗ h ( t)(iv)fL[ s ( t)∗ h ( t)] n ( )( t)= t(v)fa fa R+i13

All symbols are listed in Table 2 in the Appendix.Clearly this experiment is appropriate for realistic subjective evaluation of anydereverberation method.1.3.3 Response Spectral Complex Smoothing and dereverberationAs was noted in the introduction, many different methods have beenproposed as alternative to the previously described “ideal inversion”. Forsome time now [14, 20] room response pre-conditioning via spectralsmoothing has been proposed, so that a modified perceptually compliantversion of this response can be derived which is appropriate fordereverberation [29].According to this approach, the discrete-time measured room impulseresponse h (n)can be transformed into a smoothed response h cs(n)by theapplication of the Complex Smoothing operation on the discrete-frequencyresponse Η ω ) [20]. Specifically, a spectral smoothing window function( kWsm (m,ωk) operates on the complex discrete-frequency response H( ωk)according to the following convolution expression:HL= ∑ − 1 cs(ωk) H(ωk− ω ) ⋅Wsm(m,ωl)l=0l(13)In case that the spectral smoothing window function W m,ω ) has the formsm(kof a rectangular (spectral) window function, the above expression can besimplified as:14

Hcs1(ωk) = H ωl 2m+ 1( )(14)k∑ + ml=k −mThe discrete variable m (samples) is defined as a function of the discretefrequencyindex k , allowing a variable degree of spectral averaging for eachfrequency typically employing fractional octave or other non-uniformfrequency smoothing profiles. In this way Complex Smoothing may beconsidered as a “generalized” form of the more traditional fractional octaveanalysis (e.g. via 1/3 octave filter banks), employed for at least half a centuryin room acoustics and audio engineering. Following such an approach, m (k)can be considered as a half-bandwidth function that expresses thedependence of the desirable spectral averaging on frequency. It must benoted that the smoothing operation is practically meaningful when thesmoothing index m(k)is defined for the range of values 0 < m(k)≤ L / 4 . Theabove operation allows mapping of the Complex Smoothed room frequencyresponse into a corresponding smoothed room impulse response. Typicalresults of a smoothed version of the room response (shown originally inFigure 1), are shown in Figure 7.The proposed dereverberation scheme is based on such pre-processing ofroom responses [29]. After the application of this operation on the measuredroom response h(n) , a reduced-order smoothed responseh cs(n) with aTransfer Function Hcs(ωk) is derived. Then, an inverse filter (n)evaluated, which inverts the Complex Smoothed response, i.e.:h cs iis15

dDcs ics i(n) = hcs(k) = H(n) ⊗ hcs(ω ) ⋅ Hkcs i(n) ≅ δ(n)cs i(ω ) = 1k(15)In theory such an inverse filter responseh cs i(n) will achieve a compromisedresult when employed on the original measured room response h(n) ,according to the following expression:d ′(n)= h(n) ⊗ hiD ′(k)= H(ω ) ⋅ Hikcs i(n)cs i(ωk)(16)Such compromised, but nevertheless perceptually more beneficialperformance can be observed on the (off-line) dereverberation results ofFigure 8. This dereverberation method was found to produce objective andsubjective improvement enhancing all known acoustic parameters related tomeasured responses, irrespective of the room size, without introducingaudible distortions (see [29] and off-line audio demonstrations inhttp://www.wcl.ee.upatras.gr/audiogroup/Equalization/index.html).2. RESULTS: ERROR IN REAL-TIME DEREVERBERATIONTests were carried-out for all types of experiments described in section 1.Real-time tests were conducted in a professional laboratory room withdimensions and acoustics close to those recommended by the IEC 268-13standard for loudspeaker evaluation, i.e. L: 7,15m X W: 4,60m X H: 2,90m.Reverberation time and ambient noise data for this room are listed in theAppendix. For the room response measurements, a small 2-way closed-boxloudspeaker (Yamaha NS-10) was located at the center of the room, at 3 mfrom an omnidirectional microphone and at a height of 1.7 m above the floor.16

The excitation signal was a maximum-length sequence (MLS) reproducedand averaged 16 times to reduce the effects of additive background noise.The original room response (Figure 1) was derived from calculations basedon double-precision floating-point arithmetic. The overall system responsewas obtained to a length ofL = 512Ksamples. The real-time inverted systemresponses were also obtained using the above method. Pre-filtering wasapplied on such excitation sequences by using the appropriate inverse filters,having different length L iranging from 1K to 512K samples. The measuredreal-time inverted responses were trimmed to an analysis lengthL L + L −1 (samples). Off-line inverted responses were obtained by lineard=iconvolution between the original room and the inverse filter response. Fromthe results of those tests, the following conclusions were derived:(i) In all cases the dereverberation error (as defined by eq. (4)) wassignificantly larger during real-time inversion than during the identicalparameteroff-line inversion. Typical results are shown in Figure 9, where thetime and frequency domain response inversion results are shown for filterlengths ofL = 1K(short filter), (Figure 9(a), (b)) and L = 512K(long filter),ii(Figure 9(c), (d)). As can be observed, the real-time inversion errors were upto 20 dB higher especially for low frequencies, resulting to low-frequencyringing in the time domain. In detailed examination, it can be observed thatduring real-time tests the compensating poles of the inverse filter do not fullycancel the RTF zeros (dips), hence generating additional mismatched ringing17

poles. Such effects were found to be significantly smaller during the off-linetests.(ii) As a result of this, informal listening tests carried-out with audio materialwere less distorted during off-line inversion (equations (11)), than during realtimeinversion (equations (12)). For the case of off-line inversion, theexpected from the diagrams overall flattening of the spectrum was alsoobserved (heard) resulting to reduced coloration, but some low-amplitudepost-ringing together with just audible pre-ringing were also noticeable. Thisfinding is in agreement with past references [4, 8]. However, during real-timetests, the audio material was significantly more degraded with additional andmore prominent harsh pre- and post- ringing resonances. Typical audioexamples are available in the electronic address:http://www.wcl.ee.upatras.gr/audiogroup/Equalization/index.html(iii) By increasing inverse filter length L i (samples), as is well-known [3, 33],the theoretical dereverberation error can be reduced in both time andfrequency domain, albeit with the expected increase of the low-level pre-echolength corresponding to the partial deconvolution of the now-lengthier acausalroom response error component. Again, this finding is in agreement with pastreferences [2, 3]. However, rather unexpectedly and not previously noted inthe literature, the real-time inversion results were significantly worse forlonger inverse filters (see Figures 9(c) and 9(d)). As can be observed in thosefigures, the previously observed low-frequency errors were now combinedwith large errors spread to most frequency ranges, generating broadband18

approximately, not dramatically increasing thereafter. This difference betweenreal-time and off-line error (dereverberation discrepancy error energy,equation (8)) can be more clearly observed in Figure 11(a), where thisunexpected increase in real-time error is clearly shown. Significantly,Complex Smoothing based dereverberation error (results of correspondingtests using such smoothed responses are also plotted in these figures)remains largely independent of filters length or test class. Similar trends canbe also observed in the frequency domain where dereverberation spectraldeviation (see equation (9)) is increasing for longer filters for real-time tests,whereas, it remains independent of it for Complex Smoothed dereverberation(Figures 10(b) and 11(b)).These findings, believed not to have been previously reported, at firstconsideration cannot be accommodated within the existing theoreticalanalysis of the room response inversion problem. Hence, subsequentsections of this paper will examine in more detail the factors affecting thisdiscrepancy and will propose possible mechanisms responsible for thegeneration of such errors.3. ERROR ANALYSIS FOR REAL-TIME DEREVERBERATIONIn order to interpret those results it is suggested here that real-time is differentto off-line dereverberation possibly due to the instantaneously time-varyingnoise amplified by the inverse filter compensating poles. These narrowbandwidth (high-Q) large-gain poles, in a frequency domain sense are20

attempting to compensate for high-Q RTF zeros, where signals to noise ratio(SNR), as well as spectral resolution appear to be extremely criticalparameters, susceptible to variations between response measurement andtest conditions. Such mismatch between response measurement anddereverberation test noise level at those critical regions will severely degradeperformance due to any instantaneous small variations. In a time domainsense, these dereverberation mismatched components can be very lengthydue to ringing of mismatched poles, (including pre-ringing for the case ofmixed-phase inverse), so that the distortion is finally manifested as modulatednarrow-band noise. Practical tests maybe also affected by otherenvironmental as well as equipment-related factors which may generate smallshifts in the signal frequency. Such mechanisms may be due to roomtemperature variations between response measurement and dereverberationtest or digital audio equipment jitter.These two potentially dominant error mechanisms, namely the effect ofbackground noise and of spectral shift, will be analyzed below.3.1 Effect of background noiseAs it was shown in previous sections during practical tests the measuredimpulse response will always contain a level of background noise. Hence, themeasured discrete-time room response h (n), having a discrete-frequency21

RTF H(ω k) with ω = 2πkL,k = 0,1, K L , can be considered as the sum ofk,the noiseless system response h o(n)and of the background noise (n), i.e.:h( n)= h ( n)n ( n), n = 0,1,K,L(17)o+mand in the discrete frequency domain:H ω ) = H ( ω ) + N ( ω )(18)(k o k m kwhere h o(n), Ho( ωk) and n m(n), N m(n)are Discrete Fourier Transform pairs.Following the ideal dereverberation discussed earlier, the inverse filter withresponseh i(n) of length Liwill be introduced, so that the theoretical (off-line)inverted system response d i(n)will be:ii[ ho( n)+ nm(n)] ⊗ hi( n),n = 0,1, Ldd ( n)= h(n)⊗ h ( n)=K,(19)and, equivalently, in the frequency domain:D (ωik[ ) N (ω )] ⋅ H (ω ))= H (ωok+ (20)mkikwhere ωk= 2πk L , k = 0,1 KLd−1and Ld= L + Li(samples) is the length ofthe now inverted system response.dn mDuring real-time equalization, provided all parameters remain the same, it canbe assumed that the noiseless system response h o(n)also remainsunchanged, but the measured background noise n m(n)it is possible to havedifferent per sample values (although it may retain the same longer-termstatistics as for the previously measured response noise) and hence a newnoise sequence n i(n)is present in the equalized system response dˆ i( n).Then, according to equations (19) and (20):22

i[ ho( n)+ ni( n)] ⊗ hi( n),n = 0,1, K Ldd ˆ ( n)= ,(21)Dˆ( ωik[ H ( ω ) + N ( ω )] ⋅ H ( ω )) = (22)okiCombination of equations (20) and (22) yields:kikH (ω ) N (ω )Dˆ o k+i ki(ωk) =⋅ Di(ωk)(23)H (ω ) + N (ω )okmkFrom the above expression it is obvious that the real-time inverted roomresponse differs from the off-line one mainly due to the effect of thebackground noise. Combination of equations (6), (20) and (22) yields:[ N (ω ) − N (ω )] ⋅ H (ω )E (ω ) = . (24)dkikmkikEquivalently, in the discrete-time domain, the dereverberation discrepancyerror function e d(n)will be:[ n ( n)− n ( n)] ⊗ h ( )e ( n)= n . (25)dimiGiven that nm ( n)and ni( n)may have different per sample values, thedifference ni ( n)− nm( n)is a non-zero sequence. Hence, a filtered (by theinverse system response) noise sequence will be generated during the realtimeresponse inversion. In contrast, the off-line equalized response will onlygenerate the significantly lower theoretical equalization error.From a physical point of view the above analysis leads to the conclusion thatin spectral bins where the power of the measured system response is lowrelative to the acoustic noise floor, then the effect of any inverse will largelydepend on the noise properties, being likely that any small (instantaneous)noise variations to generate mismatch error which will be significantlyamplified by the now-mismatched compensating poles. This conclusion can23

partially explain the results described in Section 2. However, the interdependenceof noise and inverse filter length may be further explained via anidealized example where a room response null will be represented via asimple notch filter. This example will show how such a noise amplificationmechanism becomes more dominant for longer inverse filters. To create sucha null at a specific frequency f0(Hz), a pair of complex-conjugate zerosz ,e1 2± jω0= are introduced on the unit circle at a normalized angularfrequency ω 0, whereω0R ( ω k) for such a notch system will be:jω − jω−( ) ( e e ) ( e )jω −− ⋅ − eR ωk0 k 0 jω= 1k0= 2π f fs. Then the discrete frequency response1 (26)In order to simulate the realistic measurement conditions, a spectralcomponent for the background noise must be added, so that:Rjω jω− jω − jω( ω ) = ( − e e ) ⋅ ( − e e ) Ν ( ω )k0 − k 0 k1 1+(27)The ideal inverse filter response Ri( ω k) for the above system will be:Ri( ω )kjω jω− jω jω( 1−e e ) ⋅ ( 1−e e ) N ( ω )mk1(28)−= 0 k 0 − k+such that, the off-line equalized notch response will be:ik( ) ⋅ R ( ω )D ( ω ) = R ω(29)kikAssuming now that this ideal inverse filter is applied to the original notchsystem response under noisy test conditions and recalling the previousexpression for the dereverberation discrepancy error (equation (24)), suchfiltering yields:[ N ( ω ) − N ( ω )] ⋅ R ( ω )mEd( ωk) =i k m k i k, ωk= 2πk L , k = 0,1 KLd−1(30)kd24

The discrete inverse filter response Ri( ω k) is maximized when evaluatednear the null angular frequency ω 0. Clearly, as the bin spacing of the DiscreteFourier Transform (DFT) approaches zero, then the maximum absolute valueof the inverse filter response Ri( ω k) approaches infinity. Hence the worstcase for generating such error will occur when the differenceN ω ) − N ( ω ) is large (i.e. noise differs between response measurementi(k m kand equalization test) and simultaneously, the frequency spacing is very fine,i.e. when the inverse filter length approaches infinity for a given roomresponse length and sampling frequency.Figure 12 (a) shows the magnitude response of the idealized notch system forDFT length of 1K, 8K and 64K samples hence resulting to a differentfrequency bin separation. For illustration purposes, a typical spectrum ofacoustic noise is also drawn. Figure 12(b) shows the response of the idealinverse filters (see equation (28)) generated for different filter length valuesLi, which compensates for the noisy notch system. As can be observed, byincreasing filter length, the gain and mismatch of the compensating inversefilter also increases due to the combined affect of noise and increasedspectral resolution. This has the effect of increasing mismatch spectraldistortion in the deconvolution result, shown in Figure 13(a).3.2 Effect of spectral shift25

Recalling the example response in the previous section, let assume now thatfor some reason (such as room temperature changes or due to equipmentrelatedfactors such as digital audio jitter), the original simplified notchresponse R( ωk) (see equation (26)) has slightly changed, so that there is asmall shift of the pair of the complex-conjugate zeros along the frequency axis(a shift of the angular frequency on the unit circle), i.e. the new null angularfrequency is now ω~0= ω 0± dω, where d ω denotes a small shift. Then, the~“shifted” response R( ω k)~R ωwill be:jω~-jωk − jω( ) ( e e ) ( ek)jω ~00 −= 1−⋅ 1−ek(31)The ideal inverse filter response R ω ) for the noiseless response R ω ) isdefined as:i(k(kRi( ω )k=1jω0 − jωk − jω( e e ) ( ek)jω 0 −1−⋅ 1−e(32)such that, the ideal off-line equalized notch response will be:Di( ωkk i k) = R(ω ) ⋅ R ( ω )(33)~Thus the filtering of R(ω ) by the inverse filter R ω ) will now produce animperfectly equalized frequency response Dˆ i( ω k):Dˆ(ωkki(k~) = R(ω ) ⋅ R ( ω )(34)kikThe dereverberation discrepancy error )will be:Ed(kE ω between Dˆ ( ) and D ( ω )~) = Dˆ( ω ) − D ( ω ) = R(ω ) ⋅ R ( ω ) - R(ω ) ⋅ R ( ω ) =d( ωki k i kk i kk i k~( R ( ωk) − R(ωk)) ⋅ Ri( ωk== )iω kik26

( − 2cos( ~ -jωk-j2ωk-jωk-j2ωk1 ω ) ⋅e+ e −1+2cos( ω ) ⋅e−e) ⋅R( ω == )0 0i k( −~ -jωkcos( ω ) cos( ω )) ⋅ e ⋅ R ( ω ⇔= ⋅)20 0i k-j( ω ± dω/ 2) ⋅ sin( ± dω/ 2) ⋅ eω⋅ R ( )kEd( ωk) = 4 ⋅ sin0 iωk(35)E ω isBy observing equation (35), it is obvious that the error function ( )zero-valued when the spectral shift d ω is zero. However, even small valuesof the spectral shift ωd will result ( )E ω to approximate infinity for values ofdthe discrete angular frequency ω kapproaching the null frequency ω 0. Besides,it can be easily concluded that by decreasing the discrete frequency spacing(increasing inverse filter length for given sample rate), the error functiond( )E ω becomes more sensitive to smaller angular frequency shifts aroundkthe original angular frequency ω 0.kdkFigure 13(b) shows the deconvolution result for different filter lengths, appliedto the notch system response whose notch frequency was shifted by 0.1%prior to deconvolution. Clearly, the longer filters generate more severedistortion after inverse filtering.4. IMPROVING REAL-TIME DEREVERBERATION VIA RESPONSESMOOTHINGIn order to overcome the previously discussed practical problems, it becomesclear that any robust room acoustics dereverberation / equalization schememust employ techniques that avoid compensation for sharp response dips,having also controlled resolution in an inverse spectral sense, i.e. employing27

a moderate length and order for the inverse filter for any given sampling rate.At the same time, perceptually-significant spectral resolution must not becompromised by such inverse filters, ideally in order to be able to tackle lowfrequencynarrow-Q room resonances [21, 27, 28, 41]). The real-timeapplication of such filter should be able to improve direct / reverberant signalratios without adding any unwanted artifacts; hence a mixed-phase inversefilter must be employed. These requirements can be met by ComplexSmoothing dereverberation as it was described in section 1.3.3. Suchprocedure corrects gross spectral effects due to early room reflections withoutattempting to compensate for many of the original narrow-bandwidth spectraldips, and on the other hand in the time domain (being primarily a mixedphasecompensation scheme), shapes more power in the direct and earlyreflection path sounds and less power in some of the later reverberantcomponents. As was previously noted, the method allows user-definedvariable frequency resolution/smoothing so that full-bandwidthdereverberation can be achieved; however in such a case low-frequencyresolution may be compromised. To illustrate the immunity of this method tothe previously discussed problems, let consider the effect of the smoothingoperation on the notch filter example response. Equation (27), defining thenotch filter frequency response under realistic (noisy) condition, is now writtenas:R( ω )(− jωjkek − ωkω ) = 1 − 2cos( ω )2⋅ + e +0Ν(36)m kApplication of the Complex Smoothing operation to the above response, asdefined by eq (14), yields:28

Rcs( ωk) = 1−2cos( ω) ⋅ α⋅ e+ α⋅ e+ N( ω− jωk−2jωk0 12mcs k)(37)where,ααN12( 2m(k)+ 1)[ π( 2m(k)+ 1)/ L]1 sin= ⋅, (38)sin( π / L)( 2m(k)+ 1)[ π( 2m(k)+ 1)/ L]1 sin 2= , (39)sin(2π/ L)mcsk + m(k )1( ωk) = ∑N( ωl)(40)2m(k)+ 1ml=k −m(k )being the smoothed background noise spectrum.The ideal inverse filter ( )RcsiR ω for the above system will be:csik1( ω ) = k(41)− jωk2 jωk1 − 2 cos( ω ) ⋅ α ⋅ e + α ⋅ e−+ N ( ω )0 12mcs kThus the filtering of R( ωk) by the inverse filter Rcsi( ωk) will now produce an′ :imperfectly equalized frequency response D ( ω )D′ ω ) = R(ω ) ⋅ R ( ω )(42)i(kk csi kFrom equations (38)-(41) it is evident that the inverse filter response Rcsi( ω k)is controlled by the parameters α 1and α 2so that when evaluated near thenull angular frequency ω 0, then the inverse filter response will be always finitein amplitude and always smaller than the original (ideal) inverse R ω ) . Toiki(killustrate this let evaluate this response for the case when the smoothingwindow function m (k)is a 1/3 octave half-bandwidth function of the discretefrequency index k , choosing also a large value of the inverse filter lengthL = 512K samples. Then, for a notch angular frequency ω0 = 0. 1425 radsi29

(corresponding to f0= 1 kHz and = 44. 1equation (38) is evaluated as:ωlim{ R ( ω )}f kHz), lim { m(k)} = 2734sω k →ω0{ N ( ω )}samples= 1-5-5− 8.5394 ⋅10+ 3.7934 ⋅10⋅ i lim(43)csi kk → ω0+ω →ωThe corresponding ideal inverse filter response defined by equation (28), nearthe notch angular frequency ω0 = 0. 1425ωlim{ R ( ω )}will be:= 1-6-7− 1.2910 ⋅10+ 1.8520 ⋅10⋅ i lim(44)i kk → ω0+kkω →ω0mcs{ N ( ω )}As can be observed, recalling the error functions defined by equations (30)and (35), the inverse filter for the Complex Smoothed response will alwaysamplify to a lesser degree any noise, so that any distortion due to mismatchor frequency shifting will be reduced. Figure 14(a) shows the magnitudespectrum for the above example for the noisy original and the smoothednotch response. Then, after inversion with Complex-Smoothed filters ofdifferent length no mismatch artifacts are generated, as is shown in Figure14(b).0mkk5. DISCUSSION AND CONCLUSIONSAlthough the principles of room acoustics dereverberation have been knownfor at least 20 years, the robust application of such methods to real-life caseswas often unsuccessful, since their performance was degraded byundesirable processing artifacts. Since that time it was found that audiblereal-time dereverberation benefits were lower than the off-line (simulated)30

measured performance of such methods. However, this discrepancy was notbeen previously formally investigated and it was often related to thecompromised inverse filter design caused by the inadequate real-timeprocessing power available at the time for such tests. This discrepancybetween real-time and off-line dereverberation performance has beenmeasured and analyzed here, illustrating some novel aspects, not previouslydiscussed in earlier publications which were largely based on simulated tests.It is found here that by keeping all parameters identical and unchangedbetween room response measurement and dereverberation test, themismatch errors were significantly higher for the case of real-time tests thanfor the corresponding off-line tests. It was also found that by increasinginverse filter length (i.e. the number of filter coefficients), real-timedereverberation performance was further degraded; furthermore thediscrepancy between real-time and off-line performance increased since forsuch lengthier filters the theoretical off-line mismatch error decreases. It isalso known that allowing for more inverse filter coefficients, ringing poles aremainly implemented which attempt to compensate for the original RTF zeros.Given that the perceptual benefits from such compensation of narrow RTFzeros are questionable, it is becoming obvious that such filters will usuallyproduce a negative effect during practical dereverberation performance.Specifically, it was found that under practical real-time conditions such “high-Q” ringing poles are extremely sensitive to small variations in system (room)properties which can possibly change between response measurement and31

dereverberation test even if source / receiver positions and equipmentsettings remain identical, so that any such test will suffer from significantmismatch error. Hence, from this point of view the room maybe considered asa “weakly non-stationary system”. Although the exact nature of such systemvariation mechanisms could not be fully identified, in subsequent sections ofthis paper it was proposed and analytically discussed that possible factors arethe varying environmental acoustic noise and/or also small drifts in signalfrequency caused by environmental or equipment-related factors. It wasillustrated that such variations affect proportionally more the low-SNR binsassociated with high-resolution discrete-frequency RTF zeros. Due to theabove factors, it has been shown that the “uncertainty” betweenmeasurement and test is increasing for increasing the discrete-time inversefilter length and spectral resolution, so that under such conditions the couplingbetween the discrete and continuous-time acoustic domain seems tobreakdown. It must be also noted here that averaged MLS room responsemeasurements may themselves generate results which may presentinconsistent and biased properties at low-SNR regions (such as responsetails), due to such time-dependent variations in room noise and air currents,which can cause variable phase modulations in the received signal.In order to further assess and discuss the findings of these tests, it isnecessary to note that the known distortions (artifacts) generated bydereverberation methods can be broadly grouped into: (a) “mismatch errors”which up to now have been mainly associated with source / receiverdisplacements with respect to the original RTF measured positions, or by32

insufficient inverse filter performance (typically due to length restrictions) and,(b) pre-echoes, mainly due to the effects of acausal components of theinverse filters employed to tackle the non-minimum phase RTF.For the first of these distortions, it has been shown here that such errors willbe present during ideal real-time dereverberation, irrespective of processingpower (inverse filter length) and source / receiver position coincidence. Theseerrors would obviously increase if dereverberation is carried-out outside theRTF measurement area, as has been repeatedly shown by past studies, butin the first place would always be generated due to the sensitivity of thecompensating poles to the ambient noise appearing around the low-SNR RTFzeros. Such errors would become more prominent when inverse filter lengthis increased even if under similar conditions the theoretical (off-line)dereverberation performance would always seems to improve.Over the past years, most of the proposed dereverberation methods haveattempted to avoid compensating for sharp RTF dips. The findings of thiswork also indicate that given that mismatch error will be generated at all“high-Q” RTF regions, any robust dereverberation method must be based onmoderate-resolution inverse filters, i.e. having a “smoothed” spectral profilewith respect to the measured RTF and hence rather short length (in theregion of 2K samples for 44.1 KHz sampling rate). Ideally, a carefulfrequency-varying smoothing profile would not sacrifice significant resolutionat low frequencies, so that “narrow Q” room resonances can be alsocorrected.33

It has been also shown that this desirable form of spectral compensation canbe achieved by inverse filters derived from a Complex Smoothed RTF havingfrequency-depended profile, which also results to shorter filters, immune tothe undesirable noise sensitivity associated with the longer inverse filters.Such filters can be mixed-phase in which case it has been previously shownthat they can remove some of the unwanted reflection energy, improvingdirect /reverberant signal ratios and other similar room acoustic metrics up toa limit when pre-echo artifacts become perceptually detrimental. Furthermore,Complex Smoothing-based dereverberation appears to be beneficial acrossthe complete audio frequency range.Given that the Complex Smoothing method appears to be largely immune tothe first class of errors and hence to present a viable solution, the secondclass of dereverberation artifacts, namely the pre-echoes, appears at presentto be the most challenging aspect of the problem for removing the timedispersedlate reflection energy. Currently, it appears that it is practicallyimpossible to fully recover the anechoic source signal via real-time anddistortion-free processing. However this should not be a realistic engineeringtarget for dereverberation methods, since they should aim at improvingaudible impression without removing all evidence of the specific listeningspace’s reverberance. Provided that sophisticated and robust perceptualmodels can evolve which could clearly outline the complex interactionsbetween the multiple parameters affecting the perception of reverberation34

(e.g. see [39]), then a better compromise between dereverberationperformance and perceived effect could be achieved by such future methods.6. ACKNOWLEDGEMENTThe authors express their gratitude to the anonymous reviewer for pointingout the potential biasing of averaged MLS measurements due to room noisevariations.7. REFERENCES[1] S.T.Neely and J.B.Allen, “Invertibility of a Room Impulse Response”, J.Acoust. Soc. Am., Vol. 66, pp.165-169, (1979).[2] J. Mourjopoulos, P. M. Clarkson, J .K. Hammond, "A Comparative studyof Least-Squares and Homomorphic Techniques for the Inversion ofMixed-Phase Signals", Proc. IEEE ICASSP’82, pp.1858-1861, (1982).[3] P.Clarkson, J. Mourjopoulos, J. Hammond, “Spectral, Phase, andTransient Equalisation for Audio Systems”, J. Audio Eng. Soc, Vol. 33,pp. 127-132, (1985).[4] J. Mourjopoulos, “On the Variation and Invertibility of Room ImpulseResponse Functions”, Journal of Sound and Vibration, Vol. 102, pp.217-228, (1985).[5] B.D. Radlovic.; R.C. Williamson; R.A. Kennedy, "Equalization in anAcoustic Reverberant Environment: Robustness Results", IEEE Trans.Speech and Audio Processing, Vol. 8, no. 3, pp. 311-319, (2000).35

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[15] R. Wilson, “Equalization of Loudspeaker Drive Units Considering BothOn- and Off-Axis Responses”, J. Audio Eng. Soc., Vol. 39, Number 3pp. 127, (1991).[16] F.Asano, Y.Suzuki, and T. Sone, “Sound Equalization Using DerivativeConstraints”, Acustica, Vol.2, pp.311-320, (1996).[17] Y.Haneda, S.Makino, Y.Kaneda, “Multiple-Point Equalisation of RoomTransfer Functions by Using Common Acoustical Poles”, IEEE Trans.On Speech and Audio Processing, Vol. 5, No 4, (1997).[18] J.N. Mourjopoulos, "Digital Equalization of Room Acoustics", J. AudioEng. Soc., Vol. 42, No 11, pp. 884-900, (1994).[19] M. Karjalainen, E.Piirilä, A. Järvinen, J. Huopaniemi, “Comparison ofLoudspeaker Equalisation Methods Based on DSP Techniques”, J.Audio Eng. Soc, Vol. 47, No 1/2, pp.14 -31, (1999).[20] P. Hatziantoniou and J. Mourjopoulos, “Generalised Fractional-OctaveSmoothing of Audio and Acoustic Responses”, J. Audio Eng. Soc., Vol.48, No 4, pp. 259-280, (2000).[21] A. Mäkivirta, P. Antsalo, M. Karjalainen, V. Välimäki, “Low-FrequencyModal Equalization of Loudspeaker-Room Responses”, Proc. of theAES111th Conv., preprint 5480, (2001).[22] O.Kirkeby and P.A.Nelson, “Digital Filter Design for Inversion Problemsin Sound Reproduction”, J.Audio Eng.Soc., Vol. 47, pp.583-595, (1999).[23] O.Kirkeby, P.Rubak, and A.Farina, “Analysis of ill-conditioning of multichanneldeconvolution problems”, Proc. IEEE Workshop onApplications of Signal Processing to Audio and Acoustics, pp. 155-158,(1999).37

[24] C. Kyriakakis, S. Bharitkar, P. Hilmes, “Robustness of Multiple ListenerEqualization with Magnitude Response Averaging”, Proc. AES 113thConv., preprint 5669, (2002).[25] L.G.Johansen and P.Rubak, “Listening test results from a new digitalloudspeaker/room correction system”, Proc. of the 110 thAES Conv.,preprint 5323, (2001).[26] A. Azzali, A. Bellini, E. Carpanoni, M. Romagnoli, and A. Farina,“AQTtool an automatic tool for design and synthesis of psychoacousticEqualizers”, Proc. of the 114th Conv., preprint 5835, (2003).[27] R.J. Wilson, M. D. Capp and J. R. Stuart, “The Loudspeaker-RoomInterface - Controlling Excitation of Room Modes”, Proc. of the 23rdAES Conf., Copenhagen, (2003).[28] J.A. Pedersen, “Adaptive Bass Control - The ABC Room AdaptationSystem”, Proc. of the AES 23th Conf., Copenhagen, (2003).[29] P.Hatziantoniou and J.Mourjopoulos, “Results for Room AcousticsEqualisation Based on Smoothed Responses”, Proc. of the 114th AESConv., preprint 5779, (2003).[30] D. Rife and J. Vanderkooy, “Transfer-Function Measurement withMaximum-Length Sequences”, J. Audio Eng. Soc., Vol. 37, No 6, pp.419-443, (1989).[31] S. Müller, P. Massarani, “Transfer-Function Measurement withSweeps”, J. Audio Eng. Soc., Vol. 49, No 6, pp. 443, (2001).[32] A. Farina, “Simultaneous measurement of impulse response anddistortion with a swept-sine technique“, Proc. of the 108th AES Conv.,(2000).38

[33] P.M. Clarkson, Optimal and adapive signal processing, (CRC Press,ISBN 0-8493-8609-8), pp. 89-131, (1993).[34] A. Farina, G. Cibelli, A. Bellini, “AQT - A New Objective Measurement ofThe Acoustical Quality of Sound Reproduction in Small Compartments”,Proc. of the 110th AES Conv., preprint 5283, (2001).[35] S. Spors, A. Kunz and R. Rabenstein, “An approach to listening roomcompensation with wave field synthesis”, Proc. of the 24th AES Int.Conf., Banff, CA, (2003).[36] A. Härmä, M. Karjalainen, L. Savioja, V. Välimäki, U. K. Laine, J.Huopaniemi, “Frequency-Warped Signal Processing for AudioApplications”, J. Audio Eng. Soc., Vol. 48, No 11, pp. 1011, (2000).[37] A.Lundeby, T.E. Vigran, H.Bietz, and M. Vorländer “Uncertainties ofMeasurements in Room Acoustics”, Acustica, Vol. 81, pp.344-355,(1995).[38] M. Karjalainen, P. Antsalo, A. Mäkivirta, T. Peltonen, and V. Välimäki,''Estimation of Modal Decay Parameters from Noisy ResponseMeasurements,'' Proc. of the 110th AES Conv., preprint 5290, (2001).[39] J.Buchholz, J.Mourjopoulos, J.Blauert, “Room Masking: Understandingand Modelling the Masking of Reflections in Rooms”, Proc. of the AES110th Conv., preprint 5312, (2001).[40] L.G. Johansen, “Correcting Room Acoustics Using Digital SignalProcessing”, Ph.D. Thesis, Aalborg University, Denmark, (2003).[41] M. Tyril, J.A. Pedersen, P. Rubak, “Digital Filters for Low-FrequencyEqualization”, J. Audio Eng. Soc., Vol. 49, No 1, pp. 36, (2001).39

[42] L.J. Ziomek, Fundamentals of Acoustic Field Theory and Space-TimeSignal Processing, CRC Press, ISBN 0-8493-9455-4, pp. 651-662,(1995).[43] G. Cibelli, E. Ugolotti, A. Bellini, A. Farina, C. Morandi, “ExperimentalValidation of Loudspeaker Equalization Inside Car Cockpits”, Proc. ofthe 106th AES Conv., preprint 4898, (1999)[44] A. Bellini, A. Farina, G. Cibelli, E. Ugolotti, “Experimental Validation ofEqualizing Filters for Car Cockpits Designed with Warping Techniques”,Proc. of the 109th AES Conv., preprint 5278, (2000)[45] M. Karjalainen, P. A. A. Esquef, P. Antsalo, A. Makivirta, V. Valimaki,“AR/ARMA Analysis and Modeling of Modes in Resonant andReverberant Systems”, Proc. of the 112th AES Conv., preprint 5590,(2002).APPENDIX31,5Hz63Hz125Hz250Hz500Hz1kHz2kHz4kHz8kHzAVERAGEL p(dB-SPL)40 38 32 31 25 22 21 20 20 38RT (s) − − 0,468 0,396 0,349 0,342 0,424 0,371 0,371 0,368Table 140

f s (Hz) discrete-time signal sampling frequencyADC[ ], DAC[ ]signal analog to digital & digital to analog conversionL (samples) discrete-time signal lengthL i (samples) discrete-time inverse filter lengthL d (samples) discrete-time inversion lengthQ (bit) discrete-time signal quantization resolutiong (dB) amplification gainx s , x rsource and receiver position coordinates inside the roomθ ( 0 ), φ ( 0 ) source horizontal and vertical angle (with respect to acousticaxis)Τ (degrees in Celsius)h(n)ambient room temperaturemeasured discrete-time room-path, loudspeaker & microphoneimpulse responseh L (t)h R (t)h M (t)h P (t)n m (t)n i (t)loudspeaker impulse responseroom-path impulse responsemicrophone impulse responseheadphone impulse responseambient room acoustic noise during response measurementambient room acoustic noise during dereverberation testd i(n)discrete-time room-path, loudspeaker & microphone impulseresponse, post-filtered by response inversedˆ i( n)discrete-time room-path, loudspeaker & microphone impulseresponse, using pre-filtering by response inverses(n), s(t)discrete and continuous time audio test signals41

s f (n), s f (t), s’ f (t)s te (n), s te (t), s’ te (t)s fte (n), s fte (t), s’ fte (t)r t (n)s i (n), s i (t), s’ i (t), s’ ih (t)s ta (t)s fta (t)audio signals, pre-filtered by response inversetest excitation signalstest excitation signal, pre-filtered by response inverseaudio signal, reverberated by measured responsereverberated audio signals, post-filtered by response inverseacoustic test excitation signal, after loudspeakeracoustic test excitation signal, after loudspeaker, pre-filtered byresponse inverses fa (t)acoustic audio signal, after loudspeaker, pre-filtered byresponse inverser ta (t)acoustic test excitation signal response, after loudspeaker,including room-path distortionsr fa (t)acoustic audio signal, after loudspeaker, including room-pathdistortions, pre-filtered by response inverser te (t), r’ te (t), r’ te (n)r fta (t), r fte (t), r’ fte (t), r’ fte (n)received test excitation response signalsreceived test excitation response signals, pre-filtered byresponse inverseTable 242

Table captionsTable 1: Sound Pressure Level (L p ) and Reverberation Time (RT) versusfrequency for the laboratory room employed for the real-time tests.Table 2: List of symbols employed for room response measurement, inversionanalysis and audio signal dereverberation.Figure captionsFigure 1: Room impulse response measurement.Figure 2: Measurements corresponding to the test laboratory room: (a)impulse response (energy), (b) magnitude spectrum level (in dB-SPL),together with background acoustic noise.Figure 3: Off-line room impulse response inversion.Figure 4: Real-time room impulse response inversion.Figure 5: Off-line audio signal dereverberation.Figure 6: Real-time audio signal dereverberation.Figure 7: Comparison of Complex Smoothed vs original room response: (a)time domain (energy), (b) frequency domain (magnitude spectrum).43

Figure 8: Typical off-line dereverberation results, using Complex Smoothing:(a) time domain (energy) (b) frequency domain (magnitude spectrum).Figure 9: Results for real-time and off-line room response inversion, asfunction of different inverse filter lengths: (a) and (b) inverted impulseresponse (energy) and magnitude spectrum for 1K filter length, (c) and (d)inverted impulse response (energy) and magnitude spectrum for 512K filterlength. Note the expected increased pre-ringing in (c) compared to (a), due tothe long acausal filter component.Figure 10: Time and frequency domain results for real-time and off-linedereverberation tests for ideal and Complex Smoothing-based inversion, asfunction of inverse filter length: (a) dereverberation error energy (see equation(7)), (b) dereverberation spectral deviation (see equation (9)).Figure 11: Time and frequency domain results for real-time vs off-linedereverberation discrepancy error for ideal and Complex Smoothed basedinversion as a function of inverse filter length: (a) dereverberation discrepancyerror energy (see equation (8)), (b) dereverberation discrepancy spectraldeviation (see equation (10)).Figure 12: Example of a notch system response at the frequency of 1kHz: (a)discrete-frequency magnitude spectrum of the noiseless system as function ofdifferent DFT length L , (b) inverse filter magnitude spectrum compensating44

for the noisy notch system response as function of different inverse filterlength Li.Figure 13: Mismatched equalized magnitude response for the notch system ofFigure 12, as function of different inverse filter length L i: (a) noisy systeminverse filtering (b) inverse filtering after 0.1 % shifting of the notch frequency.Figure 14: Example of a Complex Smoothing-based notch response inversion(notch frequency of 1kHz): (a) original (noisy) and Complex Smoothedspectra, (b) result of inversion by using Complex Smoothed inverse filters ofdifferent length Li.45

P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”Figure 146

P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”0Impulse Response (dB)-20-40-60-80-100-1200 200 400 600 800 1000Time (msec)Figure 2(a)Magnitude Response (dB-SPL)70605040302010Room ResponseBackground Noise010 100 1k 10klog Frequency (Hz)Figure 2(b)47

P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”Figure 3Figure 4P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”48

Figure 5Figure 6P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”49

0Time Energy (dB)-20-40-60-80OriginalComplex Smoothed0 5 10 15 20Time (msec)Figure 7(a)Magnitude (dB)3020100-10-20Original-30Complex Smoothed-4010 100 1k 10klog Frequency (Hz)Figure 7(b)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”50

Time Energy (dB)0-20-40-60-80OriginalDereverberated100 105 110 115 120Time (msec)Figure 8(a)3020Magnitude (dB)100-10-20-30OriginalDereverberated-4010 100 1k 10klog Frequency (Hz)Figure 8(b)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”51

0Impulse Response (dB)-20-40-60-80-100-120Real-TimeOff-Line0 200 400 600 800 1000Time (msec)Figure 9(a)Magnitude Response (dB)40200-20-40Real-TimeOff-Line10 100 1k 10klog Frequency (Hz)Figure 9(b)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”52

Impulse Response (dB)0-20-40-60-80-100-120Real-TimeOff-Line9600 9800 10000 10200 10400Time (msec)Figure 9(c)Magnitude Response (dB)40200-20-40Real-TimeOff-Line10 100 1k 10klog Frequency (Hz)Figure 9(d)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”53

-50DereverberationError Energy (dB)-60-70-80-90-100Ideal (Off-Line)Ideal (Real-Time)Complex Smoothed (Off-Line)Complex Smoothed (Real-Time)2K 8K 32K 128K 512Klog2 Filter Length (samples)Figure 10(a)DereverberationSpectral Deviation (dB)43210Ideal (Off-Line)Ideal (Real-Time)Complex Smoothed (Off-Line)Complex Smoothed (Real-Time)2K 8K 32K 128K 512Klog2 Filter Length (samples)Figure 10(b)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”54

Dereverberation DiscrepancyError Energy (dB)50 IdealComplex Smoothed4030201002K 8K 32K 128K 512Klog2 Filter Length (samples)Figure 11(a)Dereverberation DiscrepancySpectral Deviation (dB)4 IdealComplex Smoothed32102K 8K 32K 128K 512Klog2 Filter Length (samples)Figure 11(b)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”55

Magnitude (dB)80400-40L=1KL=8KL=64KBackground Noise-80900 950 1000 1050 1100Frequency (Hz)Figure 12(a)Magnitude (dB)80400-40Noisless NotchBackground NoiseL i=1KL i=8KL i=64K-80900 950 1000 1050 1100Frequency (Hz)Figure 12(b)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”56

Magnitude (dB)200-20-40-60L i=1KL i=8KL i=64K-80Background Noise Notch Response900 950 1000 1050 1100Frequency (Hz)Figure 13(a)20Magnitude (dB)0-20-40-60-80L i=1KL i=8KL i=64KNotch Response900 950 1000 1050 1100Frequency (Hz)Figure 13(b)P.D. Hatziantoniou and J.N. Mourjopoulos, “Errors in Real-Time Room AcousticsDereverberation”57

200Original (noisy)SmoothedMagnitude (dB)-20-40-60-80900 950 1000 1050 1100Frequency (Hz)Figure 14(a)200Magnitude (dB)-20-40-60Original (noisy)L i=1KL i=8KL i=64K-80900 950 1000 1050 1100Frequency (Hz)Figure 14(b)58