Patent Application: US-48111403-A

Abstract:
the present invention relates to a method for designing a modal equalizer for a low frequency audible range , typically for frequencies below 200 hz for a predetermined space and location therein , in which method modes to be equalized are determined at least by center frequency and decay rate of each mode , creating a discrete - time description of the determined modes , and determining equalizer filter coefficients on the basis of the discrete - time description of the determined modes , and forming the equalizer by means of a digital filter by defining the filter coefficients on the basis of the properties of the modes . in accordance with the invention following method steps are used : forming a mode detection function ), defining the highest peak angular frequency ω p of the function ), making ar analysis of predefined order for solving the poles of the mode representing the highest peak angular frequency , selecting the pole closest to point e jωp on the unit circle , determining the decay time of the mode and if is over a predetermined level , designing a modal correction filter for this mode , and repeating this procedure until no decay time exceeds over the predetermined level , and each individual filter obtained in previous stages of the method is combined to a cascaded filter .

Description:
the problems in resolving very closely positioned modes and mode groups was the reason in this study to experiment with methods that have better control over frequency resolution . several ideas are available for improvement , including frequency warping [ 22 ] and frequency selective modeling such as selective linear prediction [ 15 ], multiband ar / arma techniques [ 16 ], and many other high - resolution signal modeling methods . frequency warping is a convenient technique when either lowest or highest frequencies require enhanced frequency resolution . this approach can be extended to kautz filters that exhibit interesting properties of generalized frequency resolution control [ 23 ]. frequency selective modeling has been applied for example in linear prediction of speech . in a simple case a target response can be low - pass filtered and decimated in order to model the low - frequency part of the response . a range of higher frequencies can be modulated down and decimated before similar modeling . actually any subband of a given frequency range can be modeled this way , and finally the resulting parameters ( poles and zeros ) can be mapped back to the corresponding original frequency range . this approach is called here modeling by frequency zooming . it resembles also the multiband techniques used in [ 16 , 17 , 24 ]. the fz - arma ( or fz - ar ) analysis starts by modulating ( heterodyning ) the desired frequency range down to the neighborhood of zero frequency [ 25 , 26 , 27 ] by where ω m = 2 πf m / f s , f m is the modulation frequency , and f s is the sample rate . in the z - domain this means clockwise rotation of poles z i by angle ω m , i . e ., ω i , rot = ω i − ω m = arg ( z i )− ω m ( 5 ) but retaining the pole radius . the next step to increase frequency resolution is to limit the frequency range by decimating , i . e ., lowpass filtering and down - sampling the rotated response by a zooming factor k zoom to obtain a new sampling rate f s , zoom = f s / k zoom . this implies mapping to a new z - domain where poles are scaled by the rule together mappings ( 5 ) and ( 6 ) yield new poles . eqs . ( 5 ) and ( 6 ) characterize how the z - domain properties of a given response are changed through modulation and decimation , but the estimated pole - zero pattern of an arma model will be obtained only in the next step . { circumflex over ( z )} i =| z i | k zoom e j ( arg ( z i )− ω m ) k zoom ( 7 ) now it is possible to apply any ar or arma modeling to the modulated and decimated response obtained from h ( n ). notice that this new signal is complex - valued due to the one - sided modulation operation . this approach resembles multirate and subband techniques . the advantage gained by frequency zooming is that in the zoomed subband the order of ( arma ) analysis can be reduced by increasing the zooming factor k zoom and , consequently , the solution of poles and zeros as roots of denominator and numerator polynomials of the model function eq . ( 1 ) is simplified . additionally this means that a different resolution can be used in each subband , for example based on knowledge about the modal complexity in each subband . after solving the poles and zeros within a zoomed subband , they must be remapped to the full band . this means inverse scaling the radii of poles as well as rotating them counter - clockwise , i . e ., z i ={ circumflex over ( z )} i ( 1 / k zoom ) e jω m ( 8 ) due to the one - sided down - modulation in ( 4 ), each pole z i must be used as a complex conjugate pair in order to obtain real - valued filters . the final step is to combine poles and zeros obtained from different subbands into a full model . this is a non - trivial task but on the other hand , partitioning of the whole problem and recombining again brings increased flexibility . it is advantageous to pick poles and zeros only within the central part of each ( overlapping ) subband to avoid problems at the boundaries of subbands due to band - limitation . in the investigations of fz - arma below the frequency - zooming method of solving the arma coefficients is the steiglitz - mcbride method . notice also that the filter orders n and p refer to real - valued filters with complex conjugate pairs constructed from one - sided zeros and poles obtained from the above method . thus the orders are twice the numbers of zeros and poles from the above procedures , respectively . in this section , the performance of the fz - arma analysis method is illustrated through synthetic signals . in particular we are interested in investigating the modeling capability when dealing with signals exhibiting beating and / or two - stage decay in their envelopes . simple signals featuring these characteristics can be obtained by s ⁡ ( n ) = ∑ k = 1 m ⁢ a k ⁢ ⅇ - n / τ k ⁢ sin ⁡ ( 2 ⁢ π ⁢ ⁢ n ⁢ ⁢ f k f s + θ k ) ( 9 ) where m is the number of modal frequencies present in s ( n ), τ k are the decay time constants , f k the modal frequencies , f s the sampling frequency , and θ k the initial phases of the modes . let us start with case ( a ) in which the amplitude envelope of the signal consisting of two modes shows beating . the parameters used to generate the signal as well as those adopted in the fz - arma modeling are given in table 1 . the target responses in sine and cosine phase , their fz - arma envelopes , and resynthesized versions are shown in fig3 . the envelopes are obtained from the complex decimated signals . each resynthesized response is computed as the impulse response of the filter that is obtained by combining the complex conjugate poles and zeros from the fz - arma analysis . notice that the band - limitation in frequency focusing produces compensating zeros ( and poles ) that show artifacts in the reconstructed impulse response . in a typical case of focusing on a narrow resonance band the reconstructed impulse response has a high - amplitude impulse as its first sample ( see also [ 27 ]). this has been removed in the resynthesized response simulations of this patent application . this removal has only marginal effect on the spectral properties of the response . in the simulation results of case ( a ) in fig3 , an arma ( 4 , 4 ) model suffices to represent properly the envelope decays in subplots ( c ) and ( d ), while the initial phase characteristics of the resynthesized signal in ( f ) deviate form ( b ). note that it is almost impossible to distinguish between the dashed and solid lines in the subplots ( c ) and ( d ) of fig3 . in case ( b ) we verify the fz - arma modeling of a two - mode response for which the amplitude envelope exhibits a two - stage decay . the parameters used to generate this signal , as well as those of fz - arma modeling , are summarized in table 2 , and the results of modeling are shown in fig4 . the slower decaying mode is modeled properly although its initial level is 10 db below the stronger one . this ability of two - stage decay analysis can work down to − 30 db in a clean synthetic case . in simulation case ( c ), in order to verify the fz - arma modeling when dealing with noisy signals , we contaminate the impulse responses shown in the plots ( a ) and ( b ) of fig3 with zero - mean additive white gaussian noise . in this example the variance of the noise is chosen to produce a signal - to - noise ratio ( snr ) of − 5 db in the beginning of the signal . of course , the local snr decreases towards the end of the signal . the results are displayed in fig5 , which follows the same structure as the previous figures . now , by looking at subplots ( c ) and ( d ) of fig5 it can be seen that the envelopes of the modeled signals ( solid lines ) differ substantially from those of the noisy signals ( dashed lines ). moreover , the resynthesized signals based on the computed models , shown in subplots ( e ) and ( f ), are free of visible noise and follow closely their corresponding clean versions , which are depicted in the subplots ( a ) and ( b ) of fig3 . the highly successful result of reducing the additive noise in simulation case ( c ) can be understood when considering the frequency zooming to a narrow band around the modal frequencies of interest , whereby snr is improved by the zooming ratio , i . e ., by 10 log 10 200 = 23 db in this case . the low - order arma ( 4 , 4 ) modeling further reduces the influence of noise due to good correlation with the modal signals only . a primary assumption when applying fz - arma or any lti system modeling is that the frequencies of modes do not change within the duration of the analyzed segment . if this requirement cannot be satisfied , e . g ., in strongly plucked string instrument tones having initial pitch shifting [ 28 ], the envelope behavior of the target signal can still be modeled . a straightforward way , if the frequency trajectory of the pitch shift is know , is to resample the signal so that the shift is eliminated . another way is to apply fz - arma modeling but adopting higher orders for the numerator and denominator so that this can capture the effect of frequency shift . alternatively , one can compute an arma model for the envelope of a modulated and decimated signal ( fz - env - arma ). in that way the envelope behavior can be approximated with a lower model order . simulation case ( d ), an example that compares the standard fz - arma modeling against the fz - env - arma , is shown in fig6 . the test signal plotted in subplot ( a ) is a version of that plotted in fig3 , but now the frequencies of the modes start 50 hz above the values indicated in table 1 and then converge exponentially with a time constant of 100 ms to the nominal values . the plots on the left column show the original and modeled envelopes for different fz - env - arma model orders . the plots on the right column do the same but employing fz - arma models . to resynthesize a changing pitch signal based on the fz - env - arma computed model , it is necessary to estimate its pitch behavior . then , after obtaining a model for the amplitude envelope , a frequency modulation corresponding to the original frequency shift should be employed during synthesis . for a direct fz - arma modeling this is not needed if the estimation is capable of capturing the given behavior of the shifting modal frequencies . it can be verified from fig6 that , in constrast to what happens to the fz - arma modeling , increasing the model order in the fz - env - arma does not essentially help to improve the model fit , since the inherent phase relations of the original signal have been lost in the computation of the envelope that is used as a target . nevertheless , for low - order modeling , fz - env - arma yields better envelope fit than the equal order fz - arma modeling . if the response of a target system is highly complex in mode density , such as a room response at medium to high frequences , a detailed modal description may not be feasible or even desired . in such cases the envelope behavior can be represented simply by fitting a lower order model to the decaying envelope in a desired frequency range by fz - env - arma techniques . this can be useful in decay time estimation . there exist many methods , however , that are better motivated for example in reverberation time rt 60 estimation [ 13 ]. simulation case ( e ) in fig7 depicts the decay envelope of an example room response for the octave band 1 - 2 khz and a related envelope curve fitting by low - order fz - arma modeling . another form of non - lti behavior are nonlinearities . a small degree of nonlinearity in a system can be accepted , and even quite severe deviation from linear can be tolerated if we accept the fact that the parameters are then signal - dependent , for example dependent on the level of a signal . the choice of the fz - arma parameters , i . e ., ω m , k zoom , and the arma orders n and p , depends on several factors . considering first the zoom factor , it can be said that the larger k zoom is , the higher the frequency resolution . this favors cases in which the modes are densely distributed in frequency . on the other hand , high values for k zoom imply a more demanding signal decimation procedure and less samples available for modeling in the decimated signal . a possible strategy is to define a minimum f s , zoom beforehand and then derive k zoom . for instance , the criterion may be based on the number of samples that remain in the decimated signal . another natural choice when there are relatively isolated modes or mode groups is to select the frequency range of focusing to cover such a group and its vicinity until neighboring modes or groups start to have influence . it is recommended to to choose the range of focus such that resonance peaks are not placed at the ends of the subband . as a rule of thumb , a suitable choice is to set f m = f r − f s , zoom / 4 , which brings the resonance frequency f r in the middlepoint of the decimated frequency range . the order of an arma model will be dependent on the number of modes associated with each resonance group . experiments on two - mode resonances reveal that adopting an arma ( 4 , 4 ) model in general yields satisfactory results for such cases . better modeling accuracy can be achieved by increasing the order , although the result may not be physically interpretable for a two - mode case . high - order analysis also rises the probability of ending up with an unstable model . note that in fz - arma the modulation frequency f m must correspond to the lower edge of the focusing range and the zooming factor k zoom , in relation to sampling rate , determines the zooming bandwith . ar / arma modeling has many applications in modern audio signal processing . linear and time - invariant models can be applied for example in room acoustics , sound synthesis , and audio reproduction . based on the previous theoretical overview and examples , in this section we will take examples from these domains to study the feasibility of the methods , particularly of the fz - arma technique , in several audio applications . a challenging application for ar / arma modeling is to find compact but perceptually valid approximations for measured ( or computed ) room impulse responses [ 12 ]. this is needed for example in modal analysis of rooms at low frequencies , artificial reverb designs , or equalization of loudspeaker - room responses . as case study ( f ), an analysis of low - frequency modal behavior of a room impulse response is carried out using different ar and fz - arma methods . the room has approximate dimensions of 5 . 5 × 6 . 5 × 2 . 7 m 3 . fig8 ( a ) describes the time - frequency behavior ( cumulative decay spectrum ) for frequencies below 220 hz as computed from a measured room impulse response . the room shows particularly intense and long modal decays around 45 hz . a straightforward ar modeling of the room impulse response below 220 hz using linear prediction yields fairly accurate results when the all - pole filter order p is about 100 or above for the low - frequency range . fig8 ( b ) shows the model response decay plot when p = 80 . the original sample rate of 44100 hz was decimated by a factor of 110 before ar modeling . a comparison with fig8 ( a ) reveals that the decay times of prominent modes are quite well modeled but many weaker modes are too short or too damped due to insufficient model order . direct arma modeling by steiglitz - mcbride method yields a better time domain fit with a given denominator order than the corresponding ar model . for example using numerator order of n = 30 and denominator p = 100 worked fairly well for the room response above , although in many cases the steiglitz - mcbride algorithm gives an unstable result already with such moderate filter orders . fz - arma is a powerful method for accurate modeling of modal behavior in a limited frequency range . fig9 depicts modeling results of the prominent modal region 33 . . . 58 hz in the response of fig8 ( a ). the region includes three major modes at frequencies of 37 , 46 , and 55 hz . fig9 shows the decay envelope of the modal region for the original signal ( dashed line ) and as a result of applying the steiglitz - mcbride method of different orders ( solid line ). increasing the filter order improves the envelope fit , but finally it may start to model the background noise envelope . the pole - zero plots on the right hand side indicate that for an order of p = 6 the poles correspond to the three modes , while for higher orders there are extra poles and it is not easy to associate them with the modes . at higher frequencies , above the critical frequency ( schroeder frequency ) [ 29 ] of the room , the modal behavior is diffuse , i . e ., the modal density is high and modes overlap in frequency . full audio range ar and arma modeling is difficult , if not impossible . however , it is possible to apply the fz - arma analysis to a narrow frequency band of a reverberant response . fig1 describes a fitting to the room response studied within a critical band at 1 khz ( 920 . . . 1080 hz ) by different model orders . with the highest model order p = 60 , envelope fitting is good for the first 250 ms and for about a 40 db dynamic range . equalization of a loudspeaker response or a loudspeaker - room reproduction chain means correcting the system response closer to desired perceptual or technical criteria . ma and arma modeling have been reported in the literature in several forms for loudspeaker and in - situ frequency response equalization , both in on - line and off - line formulations [ 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 17 , 40 , 41 , 42 , 43 , 23 ] equalization of the free - field magnitude response ( possibly including the phase response ) of a loudspeaker by dsp can be carried out using many known techniques . for highest quality loudspeakers there is hardly any need to improve its free - field response , but the loudspeaker - room combination may benefit greatly from proper equalization . the combined task of loudspeaker and room equalization is also demanding since it is essentially a problem of finding a perceptually optimal time - frequency equalization , instead of simple flattening of the magnitude spectrum and / or linearization of phase . there seems to be quite a common misunderstanding that just flattening the response , at least at low frequencies where it might be technically possible , would be an ideal solution . a better strategy is to improve the balance of overall acoustical parameters , particularly of the reverberation time . as discussed in [ 44 ], this can be done by controlling the decay times of individual modes at low frequencies , typically below 200 hz , to match the reverberation time at mid frequencies . this is called modal equalization . it may be followed by a traditional correction of the envelope of magnitude response . the need for such active correction of room acoustics is particularly prominent around 100 hz even in spaces designed for listening purposes , such as audio monitoring rooms [ 45 ]. in [ 44 ] we proposed a method for modal equalization . in the present patent application we suggest another technique to realize modal equalization , optionally combined with magnitude envelope correction . the general framework of modal equalization has been discussed in detail in the previous paper . a brief description of the procedure is : 1 . measure the combined loudspeaker plus room impulse response in the listening position of interest . any modern technique for reliable response measurement can be applied . 2 . analyze the average reverberation time rt 60 at mid frequencies , for example between 500 hz . . . 2 khz . 3 . determine an upper limit of modal decay time as a function of frequency for the low - frequency range , typically below 200 hz . this value can be allowed to grow slightly toward lowest frequencies [ 46 , 47 ], for example linearly by 0 . 2 s when the frequency decreases from 300 hz to 50 hz . 4 . find the modes that need equalization , i . e ., those that have a longer decay time than the upper limit defined above . if the magnitude level of a mode is so low that its tail remains below a given level , it does not need modal equalization even when its decay time is longer than the upper limit . estimate modal parameter values for these modes , particularly the modal frequency and the decay time constant , and compute the angles and radii of the corresponding poles . 5 . design a correction filter for each mode requiring equalization so that the filter shortens the decay time to meet the upper limit criteria specified in step 3 . this means canceling the estimated pole pair , which represents a mode with a long decay time , by a zero pair , and replacing it with a new pole pair having the desired decay time . this can be done with an iir filter [ 44 ] h c ⁡ ( z ) = ( 1 - r ⁢ ⁢ ⅇ jθ ⁢ z - 1 ) ⁢ ( 1 - r ⁢ ⁢ ⅇ - jθ ⁢ z - 1 ) ( 1 - r c ⁢ ⁢ ⅇ jθ c ⁢ z - 1 ) ⁢ ( 1 - r c ⁢ ⁢ ⅇ - jθ c ⁢ z - 1 ) ( 10 ) where r and r c are the ( complex conjugate ) pole radii of the original decay and the corrected decay , respectively , and θ and θ c are the corresponding pole angles . 6 . compute steps 4 and 5 either in a batch mode , i . e ., in parallel for each mode to be equalized , or iteratively so that modes are equalized one by one , starting from the most prominent one and returning to step 4 , to be applied to the result of the previous equalization . the process is terminated when all remaining modes meet the decay time criteria or when a preset upper limit of correctable modes has been reached . 7 . traditional magnitude equalization can be applied to the result of modal equalization , if needed , by any method or technique appropriate . in this context we are only interested in step 4 as a part of batch or iterative analysis . all other steps follow the general scheme described in [ 44 ], where the mode search and the decay time estimation were based on a time - frequency representation and fitting of a logarithmic decay plus background noise model using nonlinear optimization . while the previously proposed method is found robust for modes that are separated well enough , strongly overlapping or multiple modes with closely similar frequency are an inherent difficulty of that method . since ar / arma models search for a global optimum and don &# 39 ; t try to separate modes , they are potentially a better alternative in such cases . in the equalization cases below , mode finding and parameter estimation are carried out iteratively in the following way : 1 . compute a function that can be used robustly to find the most prominent modes and their frequencies . this can be done in different ways , for example directly by ar or arma analysis and finding the poles with largest radii . because the selection of proper model order can be problematic , we have first applied here a separate mode detection function g ( ω )=√{ square root over (| h ( ω )| max ( 0 , d ( arg ( h ( ω ))))}{ square root over (| h ( ω )| max ( 0 , d ( arg ( h ( ω ))))} ( 11 ) where h ( ω ) is the fourier transform of the measured response , ω is the normalized angular frequency ( angle in the z - plane ), and d is a differentiation operator ( in the frequency domain ). an example of g ( ω ) function is plotted in fig1 ( d ). positive peaks indicate strong modes that may need decay time equalization . note that this function combines both the magnitude level and the decay time ( through phase derivative ) information . this is one possible detection function , and there can be other functions for this purpose . 2 . find the highest peak position ω p of the detection function g ( ω ) as the best candidate for modal equalization . run ar analysis of predefined order ( here we applied orders 50 . . . 70 for the frequency range below 220 hz ) on the minimum phase version of the target response to find poles and select the pole closest to the point e jωp on the unit circle . this pole and its complex conjugate now represent the most prominent mode . 3 . if the decay time of the mode is below the upper limit allowed and the value of g ( ω p ) is below a threshold experimentally determined , go to finalize the process in step 4 . if not , design a second - order modal correction filter of type eq . ( 10 ) to change the modal decay time to a desired value below the upper limit . apply this to the response to be equalized and use the result when going back to iterate from step 1 . 4 . finally , collect the correction filters into a cascaded filter which is now the modal equalizer for the system . a simulated modal equalization , case ( i ), is illustrated in fig1 . a loudspeaker impulse response is filtered to add five simulated modes at frequencies 50 , 55 , 100 , 130 , and 180 hz , with 60 db modal decay times of 1 . 4 , 0 . 8 , 1 . 0 , 0 . 8 , and 0 . 7 seconds , respectively . the cumulative decay spectrum of this synthetic response is shown in fig1 ( a ). fig1 ( b ) proves that the effect of the modes can be cancelled out almost perfectly , leaving the loudspeaker response only , by moving the pole radii to correspond to a very short decay time ( about 60 ms ) using the procedure described above . in fig1 ( c ) the result of modal equalization depicted is more appropriate for real room conditions . the decay time of each mode is equalized to 250 ms . the two nearby modes partly overlapping , at 50 and 55 hz , do not cause any difficulties , and the modal equalization works almost perfectly . in case ( j ) of fig1 the most prominent single mode at 46 hz is equalized by shortening the decay time from a value above 1 second to about 300 ms using the algorithm described above and limiting the search for modes to only one . in fig1 ( b ) the originally problematic mode decays now clearly faster . furthermore , the equalized response up to 80 hz has much smoother shape since the modal equalization also affects the magnitude spectrum . however , the decay times of some other modes remain long . multi - mode equalization of the same room , case ( k ), is shown in fig1 . the room is the same as the one analyzed in case ( f ), fig8 . the procedure described above is iterated 100 times , yelding 100 second - order correction filter sections , to shorten the mode decay times . the cumulative decay spectrum of the resulting equalized response is illustrated in fig1 ( b ). the target value for equalized modal decay time ( 60 db ) has been 150 ms . in this case the result is not as perfect as in the synthetic or single mode case . there is about 10 db of fast decay in the beginning , as shown by backward integrated plots in fig1 ( c ), and thereafter the decay rate follows the original one . although the ideal shortening of the decay time is not achieved precisely , it already makes sound reproduction in the room more balanced in the terms of reverberation . furthermore , the equalization procedure can be improved by careful adjustment of the details . the final step , i . e ., smooth envelope equalization of the magnitude response , is not discussed here since many known techniques could be applied to equalize the magnitude response . an interesting choice is , however , to integrate the magnitude equalization phase together with the ar / arma modal equalization process . in accordance with fig1 a in one typical implementation of the invention the system comprises a listening room 1 , which is rather small in relation to the wavelengths to be modified . typically the room 1 is a monitoring room close to a recording studio . typical dimensions for this kind of a room are 6 × 6 × 3 m 3 ( width × length × height ). in other words the present invention is most suitable for small rooms . it is not very effective in churches and concert halls . the aim of the invention is to design an equalizer 5 for compensating resonance modes in vicinity of a predefined listening position 2 . type i implementation modifies the audio signal fed into the primary loudspeaker 3 to compensate for room modes . the total transfer function from the primary radiator to the listening position represented in z - domain is where g ( z ) is the transfer function of the primary radiator from the electrical input to acoustical output and h m ( z )= b ( z )/ a ( z ) is the transfer function of the path from the primary radiator to the listening position . the primary radiator has essentially flat magnitude response and small delay in our frequency band of interest , or the primary radiator can be equalized by conventional means and can therefore be neglected in the following discussion , we now design a pole - zero filter h c ( z ) having zero pairs at the identified pole locations of the modal resonances in h m ( z ). this cancels out existing room 1 response pole pairs in a ( z ) replacing them with new pole pairs a ′( z ) producing the desired decay time in the modified transfer function h ′ m ( z ) h m ′ ⁡ ( z ) = h c ⁡ ( z ) ⁢ h m ⁡ ( z ) = a ⁡ ( z ) a ′ ⁡ ( z ) ⁢ b ⁡ ( z ) a ⁡ ( z ) = b ⁡ ( z ) a ′ ⁡ ( z ) ( 14 ) the new pole pair a ′( z ) is chosen on the same resonant frequency but closer to the origin , thereby effecting a resonance with a decreased q value . in this way the modal resonance poles have been moved toward the origin , and the q value of the mode has been decreased . the sensitivity of this approach will be discussed later with example designs . in accordance with fig1 b , type ii method uses a secondary loudspeaker 4 at appropriate position in the room 1 to radiate sound that interacts with the sound field produced by the primary speakers 3 . both speakers 1 and 4 are assumed to be similar in the following treatment , but this is not required for practical implementations . the transfer function for the primary radiator 3 is h m ( z ) and for the secondary radiator 4 h 1 ( z ). the acoustical summation in the room produces a modified frequency response h ′ m ( z ) with the desired decay characteristics this leads to a correcting filter h c ( z ) where h m ( z ) and h ′ m ( z ) differ by modified pole radii note that if the primary and secondary radiators are the same source , equation 16 reduces into a parallel formulation of a cascaded correction filter equivalent to the type i method presented above h ′ m ( z )= h m ( z )( 1 + h c ( z )) ( 19 ) a necessary but not sufficient condition for a solution to exist is that the secondary radiator can produce sound level at the listening location in frequencies where the primary radiator can , within the frequency band of interest at low frequencies where the size of a radiator becomes small relative to the wavelength it is possible for a radiator to be located such that there is a frequency where the radiator does not couple well into the room . at such frequencies the condition of equation 20 may not be fulfilled , and a secondary radiator placed in such location will not be able to affect modal equalization at that frequency . because of this it may be advantageous to have multiple secondary radiators in the room . in the case of multiple secondary radiators , equation 16 is modified into form h m ′ ⁡ ( z ) = h m ⁡ ( z ) + ∑ n ⁢ h c , n ⁡ ( z ) ⁢ h l , n ⁡ ( z ) ( 21 ) after the decay times of individual modes have been equalized in this way , the magnitude response of the resulting system may be corrected to achieve flat overall response . this correction can be implemented with any of the magnitude response equalization methods . in this patent application we will discuss identification and parametrization of modes and review some case examples of applying the proposed modal equalization to various synthetic and real rooms , mainly using the first modal equalization method proposed above . the use of one or more secondary radiators will be left to future study . in this patent application we have studied the modeling of acoustic and audio system responses that exhibit resonant and reverberant properties . particularly the ar and arma modeling techniques are investigated to obtain efficient all - pole or pole - zero filters . such modeling , if accurate enough and computationally inexpensive , finds applications in solving many audio - oriented problems . the first part of the patent application is a non - theoretical overview of ar and arma modeling methods to demonstrate their inherent properties and limitations . a specific interest of this study has been the modeling methods that can yield good temporal match to a given target response and high frequency resolution , often at the same time . based on earlier studies , primarily on applying prony &# 39 ; s method to subbands , we show that frequency - zooming arma ( fz - arma ) based on the steiglitz - mcbride iteration is a powerful technique for high - resolution modeling in subbands . simulation examples demonstrate the ability of this approach to model complex modal and reverberant behaviors . 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