Ultra-Directional Microphones: Part 4. James A. Moorer. Sonic Solutions

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1 Copyright - Ultra-Diretional Mirophones: Part 4 Soni Solutions Abstrat In a series o artiles dating rom the early 97 s, Mihel Gerzon suggested using anellation between two adjaent mirophones to ahieve high diretionality in a limited requeny range. In this paper, we extend this analysis to linear arrays o mirophones by borrowing ertain aspets o phased-array radar. The unique issue that audio has is the requirement that the requeny response be lat over 5 otaves or more. We show that this requirement an be met by the use o multiple olinear arrays, ollowed by a signiiant amount o signal proessing. Bakground Derivation o the Phased-Array Mirophone We start with an array o mirophones plaed at equal distanes along a line. Let d be their separation. Let a plane wave impinge on the array at an angle o q rom the perpendiular to the array. Assume that the plane wave is a sinusoid with a wavelength o l. I n is the number o mirophones, then we an write the response to the plane wave in mirophone k as ollows: () sin( π ( t + sin θ)) λ kd For onveniene, we let the number o mirophones be odd, and we all the enter mirophone number zero. The variable t represents time in seonds. I we sum these signals over all the mirophones and simpliy, we obtain the ollowing: () sin( π t) + λ ( n ) k= os(π kd sin θ ) λ The seond term o the above represents the amplitude o the resulting sum. This is plotted or various values o wavelength in Figure. Note that the maximum response is developed in a diretion perpendiular to the mirophone array. The varying width o the response maximum show that dierent wavelengths will have dierent pikup patterns. We an steer the entire array by applying a simple delay to eah mirophone as ollows: d (3) t k = sin φ where j is the angle where the greatest sensitivity is desired. Unpublished Manusript /7/

2 Copyright - This has the eet o moving the maximum o the response o the array, but it also hanges the width o the enter lobe. Figure 3 shows the eet o steering the array rom -45 to 45. Note that the main response widens a bit as the array is steered away rom the enter. This is beause the eetive mirophone spaing is redued by the osine o the angle. Sine the amplitude term in equation () resembles a Fourier series, we might envision the use o window untions to hange the tradeo between enter lobe width and side lobe suppression. Indeed, it works pretty muh as one might antiipate. Figure 4 shows the eet o hanging the strength o the window. We an see learly the inrease in lobe width with inreasing window strength. So ar, this is all taken diretly rom phased-array radar tehnology. To make this useul or audio, we need to aomplish the ollowing: Produe a uniorm lobe width over all requenies. Ahieve -otave range with lat requeny response (roughly Hz to khz) The reason we want uniorm lobe width is to redue the oloration o the sound in the prinipal diretion o the array. Sine the array depends on anellation and reinorement o the wave ronts, it is neessarily a highly requeny-dependent proess. We need to ollow it with suiient proessing to minimize the requeny dependenies. The basi array exhibits reasonable response over about otaves overing wavelengths rom about.5d and 6d. Wavelengths longer than this produes very wide prinipal lobes, and wavelengths shorter than this produe multiple prinipal lobes. We an take the enter otave o this (in a geometri -mean sense) as the main region o response, whih is rom about.d to about 4.4d. The remainder o the response range will be used to overlap with other arrays that over other otaves. We obtain wide response by having multiple arrays on the same line with the same mirophone in the enter. Figure 5 shows a simpliied diagram with three olinear arrays with spaings at d, d and 4d. To over the ull audio range with equal spatial resolution would require a total o ten arrays. Eah array will ontribute one otave o requeny response to the overall result. The upper and lower hal-otave o eah array will overlap with the adjaent arrays. Controlling the Width o the Prinipal Lobe: The next problem to be addressed is ontrol o the width o the prinipal lobe. As noted above, a window untion an be used to adjust the width o the enter lobe. Sine we need a dierent lobe width at eah dierent requeny, we must ilter the output o eah array with individual ilters that are designed to realize a ertain window untion at eah requeny. The ilters have a urther requirement that they sum properly with the responses o adjaent arrays to produe lat requeny response and uniorm lobe width when summed over all the arrays. Sine window untions always make the lobe wider and never more narrow, we must take the widest lobe width and math all the other widths to this. The widest lobe in the range o interest ours at 6d. By a simple optimization, we an derive values o the beta parameter o the Kaiser-Bessel window that give us the desired window width. Figure 6 shows the result o suh an optimisation. As the wavelength moves rom 6d down to.5d, the beta parameter an be inreased steadily to widen the prinipal lobe. Figure 7 shows the result o applying dierent window untions to the array at dierent wavelengths. Note that at the shortest wavelength, the sideband rejetion starts to rise again, probably due to the eetive shortening o the array. There is nothing partiularly speial about the Kaiser-Bessel window. It is used here simply beause it omes with a single parameter that ontrols the width o the window in a smooth, ontinuous, and monotoni ashion. One ould equally derive an optimum window by a least-squares tehnique. This Unpublished Manusript /7/

3 Copyright - would allow us to ine tune the response at any given requeny by adjusting the tradeo between mathing the enter lobe to the prototype response (whih is the response at the longest wavelength, 6d) to the o-axis response. We note in Figure 6 that the o-axis peaks get greater as the wavelength gets longer. This is to be expeted, sine smaller values o Beta allow the sidelobes to inrease in amplitude. We an deine a window untion, w, then deine a weighting untion at eah angle as p. We may then desribe an objetive untion as ollows: k i (4) M ( n ) F = pi Di i= k= w k kd os(π sin θ λ i ) where D i represents the desired response. In our ase, we might produe a desired response by windowing the response at the maximum wavelength o 6d. Using this as the prototype response, we an math this as losely as we like by hoosing the weighting untion, p i, and inding the window untion oeiients, w k, that minimize F in equation (4). Sine the response o the array is linear with respet to any given window oeiient, equation (4) represents a linear least-squares problem. The normal equations an be ormed and solved by any number o methods, suh as singular-value deomposition [***re]. One might hoose, or instane, p to math the desired response as well as possible over the entire untion. One might hoose = i p over the main lobe and = i p to ore the response to math the desired response as well as possible at the main lobe and less well outside the main lobe. Sine the Kaiser-Bessel window is relatively simple, we will use this in the remainder o this disussion with the understanding that any suitable window that allows mathing o the prinipal lobes an be used. Implementing the Frequeny-Dependent Window: To implement a window untion that varies with requeny, we must implement a ilter or eah mirophone that has the desired gain at eah wavelength. This gain is determined by the value o the Kaiser-Bessel window or that mirophone at the value o beta indiated by the urve o Figure 6. The resulting window untion is, in at, a amily o window untions, sine the window untion will be dierent or eah dierent requeny. We might represent this as w (λ) or the weighting o mirophone k at a wavelength o λ. Figure 7 shows a plot o our dierent mirophone oeiients as untions o wavelength. These represent the ilters that must be realized to produe equal main lobe widths over the requeny range o interest. There are many ways to alulate the ilter oeiients [***res MLellan, Dzeky, et], so this aspet need not be disussed any urther here. Sine a ilter will respond over the entire range, we do need to speiy the urves outside o the range shown in Figure 7. It is suiient to just extend the urves to zero requeny and the Nyquist rate by simply dupliating the values at the end points shown in Figure 7. That is, the response o the ilter at wavelengths greater than 6d an have the same response at a wavelength o 6d, and wavelengths shorter than.5d an have the same response as at a wavelength o.5d. These values are somewhat arbitrary but are suiient to produe a working design. Note that window untions are symmetri. This means that or an array o n mirophone, only ( n ) windowing ilters need be implemented. Mirophones on eah side o the enter mirophone may be summed beore iltering, thus eliminating the need or a number o ilters. Overlapping the Arrays: i k Unpublished Manusript 3 /7/

4 Copyright - As noted above, eah array overs about two otaves. We will separate this into the main region rom about.d to about 4.4d, and the overlap regions whih onstitute the remainder o the ull two otave range. At the extremes o the requeny range, there is no overlap, so the highest array will over up to.5d r and the lowest array will over down to 6 d, where d j represents the mirophone spaing o array j. Using 4 khz as the highest requeny or whih overage is desired, we an set the spaing o the mirophones in the highest requeny array as about m. From this, we an derive the ollowing: Mirophone Spaing Low Frequeny High Frequeny m 8 Hz 67 Hz m 4 Hz 8 Hz 4 m Hz 4 Hz 8 m Hz Hz 6 m 5 Hz Hz 3 m 5 Hz 5 Hz 64 m 5 Hz 5 Hz.8 m 6.5 Hz 5 Hz.56 m. Hz 6.5 Hz These requenies are not exat, they have been rounded to onvenient boundaries or larity. Note again that the highest requeny array extends rom.5d to 4.4d, and the lowest requeny band extends rom.d to 6d. All the others extend rom.d to 4.4d. This shows that the entire requeny range may be aptured by 9 ollinear arrays. I desired, the larger arrays at lower requenies may be eliminated. The only eet o this is that the pikup will not be highly diretional at low requenies due to the widening o the prinipal lobe o the array response. Note again that steering the array away rom angle zero (straight ahead) does have the eet o widening the prinipal lobes, sine it lowers the eetive distane between the mirophones. This table was omputed at angle zero. We might hoose the table based on a dierent angle. To be as onsistent as possible, we should ompute a dierent set o requeny-dependent window untions or eah desired pikup angle so that the prinipal lobe width would be onstant over the entire steering range o the array, whih is rom -45 to 45. For many appliations, however, it is aeptable to allow the width o the prinipal lobe to hange, as long as other properties o the array are preserved, suh as overall requeny response latness, and mathing o the prinipal lobes among the arrays to prevent oloration o the sound in the prinipal lobe. In addition to the iltering desribed above to apply the requeny-dependent window untion to eah mirophone in eah array, there is a ilter that must be applied to the total response rom a given array so that eah array ontributes to the overall response mainly in its prinipal requeny region. We also require that the sum o the responses aross all the arrays be lat over the audible range. We may express this by Unpublished Manusript 4 /7/

5 Copyright - onsidering the impulse response o eah array, then stating onditions on these responses whih represent the design goals. We may say or onveniene that the impulse response o eah array will be symmetri. This is not stritly neessary, but it guarantees that there will be no phase variane rom one array to the next. I we represent the impulse response o ilter i by h is, then we may state the onditions or latness o overall requeny response as ollows: (5), s = his = i, s This is neessary and suiient to guarantee peretly lat requeny response. In general, this ondition will not be met exatly. All we require is that the deviation rom identity be suiiently small so it is not heard as a oloration o the sound. To ompute the overlap ilters, we irst reate an ideal prototype ilter that is onstruted so that it overlaps peretly. We then ompute approximations to the prototype ilter using standard approximation tehniques [***res Parks MLellan, et]. Although we need to onstrut a separate prototype ilter or eah band, there are some similarities that make the proess simpler. We an separate the ilters into the two at the extremes o requeny, and all the rest. For the ilters that are not at the extremes, we an require that they are idential, exept that eah band spans twie the requeny o the previous band. I we say that a partiular requeny band goes rom to, then we may deine a ilter as ollows: (6) (4 3) (7) ( 3) (8) (8 3) (9) ( ϑ ) ( H( ϑ) = ( ϑ) /( ) ) ϑ < ϑ < <= ϑ < ϑ Figure shows a plot o this untion or the requeny band -4 Hz. As noted, the ilter extends down to 333 Hz and up to 5333 Hz. It will peretly overlap the ilters in the next higher and next lower requeny bands, and the sum o these overlapping ilters is exatly one by onstrution. This is only one way that prototype ilters may be hosen. There are any number o prototype ilters that have this property. At the extremes o requeny, we simply allow the ilter to stay at unity gain on one side or the other. Using the deinitions above, we may deine the ilters or the extremes as ollows: () H( ϑ) = ( ϑ) ( ) ϑ < ϑ < < ϑ Unpublished Manusript 5 /7/

6 Copyright - () ϑ < H( ϑ) = ( ϑ ) ( ) ϑ < ϑ We are being somewhat areless with the notation, in that the above ormulas all use the same symbols or the important requenies (,, and ), but we intend them to apply just to the partiular band o interest. As noted above, or the band rom to 4 Hz, would be 333 Hz, and would be 5333 Hz. For other bands, these requenies would be saled appropriately to represent the requeny range o the partiular band. As an example, in the lowest band as shown in the table above, would be Hz, and would be Hz. Equation () represents the lowest ilter, whih extends down to zero requeny. Having deined a suitable set o prototype ilters or overlapping the mirophone arrays, we may ompute ilter oeiients that approximate these ilters to any degree o auray. I the ilters are all o zero-phase, then they will sum to an approximation o an impulse, desribed by Equation (5). This is by onstrution. Sine the sum o all the prototype ilters is unity, the resulting impulse response must be a simple impulse. Consequently, the sum o a series o ilters that approximate the prototype ilters will naturally be an approximation to an impulse. O ourse, i the ilters are not o zero-phase design, they will not neessarily sum to an impulse. We should point out that as we steer the array so that the prinipal lobe is at a non-zero angle, the eetive shortening o the mirophone spaing by the ator o os(θ ) indiates that all the ilters, both the windowing ilters and the overlapping ilters, should be reomputed using a mirophone spaing o. Additionally, we an adjust the Beta parameter o the Kaiser-Bessel window (or whatever d os(θ ) window untion is used) so that the width o the prinipal lobes remains onstant over the usable steering range o -45 to 45. There has been an impliit deision in the above to implement the requeny-dependent window untion and the overlapping ilter using FIR, or inite impulse-response ilters. This is not stritly neessary, but it allows us to use linear-phase ilters. A linear-phase ilter has an inherent delay in the signal path. I all the ilters have the same number o multiplies, then they will all exhibit the same delay, and they may be summed. I the ilters do not have the same number o multiplies, then we will have to equalize the delays beore summing the results o the windowing ilters. We an oset these delays by ombining them with the delays neessary or steering the array (Equation (3)). I some mirophones end up with negative delays, then all the mirophones must be delayed to assure ausality. About Diretional Mirophones: So ar, we have not disussed the diretional harateristis o the individual mirophones in the array. This disussion is peretly aurate i the mirophones are omni-diretional. Some modiiations to the exposition will have to be made to show the eet o diretional mirophones, suh as the pressuregradient type. Figure shows a shemati representation o a pressure-gradient mirophone. There are two diaphragms that are used to generate a voltage. These may then be weighted and summed to produe a diretional pikup. This kind o mirophone has the ollowing angular response: () C + ( C)os( θ ) The response straight ahead (zero angle) is exatly one. The response to the rear is (C-). For a ardioid pattern, C is set to one-hal, so the response to the rear is exatly zero. Other values o C produe dierent patterns. Unpublished Manusript 6 /7/

7 Copyright - The eet o using a pressure-gradient mirophone in this array is that the o-angle response will be multiplied by the diretional pattern desribed by Equation (). The eet would be that, or instane, the plot shown in Figure 3 would also show an amplitude dierene as the prinipal lobe was steered rom let to right. All the urves in Figure 3 would be multiplied by Equation (). Note that we an easily normalize the peak amplitude o the prinipal lobes in Figure 3 by simply orreting or the expeted attenuation due to the diretional harateristis o the mirophones. As Gerzon noted in his seminal work in this domain [***re], it is also possible to take the voltages rom the anterior and posterior diaphragms separately, thus produing two separate eeds rom eah mirophone. These an then be ombined later to produe diretional harateristis. For instane, we might weight the anterior diaphragm by one-hal and the posterior diaphragm by minus one-hal and sum them to produe a orward-aing ardioid pikup, with % rejetion o sounds oming rom diretly behind. Alternately, we might weight the posterior diaphragm with one-hal and the anterior diaphragm with minus one-hal to produe a rear-aing ardioid pikup with % rejetion o sounds oming rom diretly in ront. In this manner, using a single array o pressure-gradient mirophones, we an mix the eeds o the diaphragms dierently so that the same mirophone array may be used or sounds in ront o the array and behind the array with equal angular resolution and idential idelity (requeny-response). O ourse, the iltering shown in Figure 9 would have to be dupliated or the rear-aing array. Curvature o the Waveront With phased-array radar, there is always the expliit assumption that the inoming wave is a plane wave. With the phased-array mirophone, the plane wave assumption may be used when the sound soures are suiiently distant rom the mirophone itsel. I this is not the ase, the waveront will be urved. We an orret or this urvature, but we need to know the loation o the sound soure to make this orretion. I the plane-wave approximation an be made, then we need not know the distane between the sound soure and the array. To orret or the urvature o the waveront, we need to apply a orretion to the amplitude and to the arrival time. The amplitude orretion is needed to oset the r attenuation the waveront experienes. The orretion to the arrival time is neessary sine the urvature will have the eet o delaying the oenter parts o the waveront. We an quantize this as ollows: Let q and r be the angle and distane rom the sound soure to the enter mirophone o the array. We an then desribe the amplitude and time delay ompensation as ollows: (3) P n r = n r = os θ + sin θ n r n r (4) n = = { r os θ + ( r sin θ nd) r } d r where rn represents the distane rom the sound soure to mirophone n. The eed rom mirophone n should be multiplied by P n and should be advaned by n seonds. Sine this orretion is speii to the partiular loation o the sound soure, we would expet that The rejetion o the o-axis sound would be aeted. Indeed, we will experiene more leakage rom oaxis sounds when this kind o orretion is applied. Further Sharpening o the Response Unpublished Manusript 7 /7/

8 Copyright - Note that when the sound soure onsists o a number o disrete soures at known angles and possibly known distanes, then the response in a partiular diretion an be enhaned by subtrating o the signals rom the known diretions. O ourse, the delays aross the varying angles must be equalized beore a signal rom one angle an be subtrated rom a signal rom another angle. We might think o this as a kind o analog to the lateral inhibition ound in optial reeptors in the retina o the eye. Mirophone Mismath So ar in this exposition, we have operated under the impliit assumption that the mirophones were idential. This is, o ourse, not a valid assumption: there will be some mismath. We should examine the eet o the mismath and see what this requires o the mirophones. We an obtain a worst-ase bound on the error in the array by taking the seond term o Equation (), applying a window untion, assuming that the osine term is always unity, and assuming that the mirophone error is a uniorm ator o e. This gives us the ollowing upper bound: (5) ( n = + ) M ε w k = w k The window untion is normalized so that the above sum (aross all the points o the window untion) is unity, so the error is bounded by the individual mirophone error. We an take e to represent the expeted value o the error. Some mirophones will exhibit somewhat more error and some will exhibit somewhat less. A mean deviation o db then will produe error in the resulting pikup pattern that is about 8 db down. The error we are talking about is a distortion o the pikup pattern itsel, as shown in Figures, 3, and 4. This is not so important or the prinipal lobe, but it will make a signiiant dierene in the sideband suppression, sine in some ases, the error will be o the same order o magnitude as the sideband amplitude itsel. We an expet that the atual sideband rejetion will be several db less than the theoretial values with a db variation among the mirophones. O ourse, better mathing will allow us to ahieve more sideband rejetion. Eets o Room Reverberation on the Array So ar we have disussed sounds oming rom point soures that are in ront o (or behind) the array. What happens when we have room reverberation, whih an ome rom any diretion? We may (somewhat artiiially) divide room reverberation into three epohs: the diret sound, the early reletions, and everything else. The diret sound and the early reletions an all be treated as point soures o sound. The array an be steered to pik up eah one o these soures separately (or not, depending on the goals o the reording). The late reverberation an be onsidered to be omnidiretional, and will thus aet the array uniormly regardless o the steering diretion. O ourse, non-uniorm reletions, suh as slap ehos, will appear as speular reletions and thus will appear as point soures to the array. Extension to 3 Dimensions To extend the phased-array mirophone to three dimensions, we must irst extend it to two dimensions. This an be done by extending the array as shown in Figure. This shows a regular -dimensional array o mirophones that is apable o steering plus or minus 45º in the horizontal diretion and plus or minus 45º in the vertial diretion. Note that or some appliations, it may not be neessary to have the same resolution in the vertial diretion as in the horizontal diretion. Figure 3 shows an array with higher resolution in the horizontal diretion than in the vertial diretion. A single -dimensional array an only be steered aross about a 9º range in the orward diretion and a 9º range in the reverse diretion. To allow steering through the ull 36º range, we need to use two arrays at Unpublished Manusript 8 /7/

9 Copyright - right angles as shown in Figure 4. Note that or this to work, eah array would have to be aoustially transparent, so that o-axis sounds will easily pass through it to reah the other array. To extend the array to three dimensions, we take two -dimensional arrays shown in Figure 4 and plae another array in the horizontal plane to over the vertial diretion. In this manner, we may ahieve pikup in any diretion. Constrution o the Array There is a wide range o ways to implement the array, depending on the goals o the implementation. The most straightorward way to onstrut the array would be to simply onnet wires to eah transduer in the array and run all the wires to the required proessing hardware. O ourse, there must be preproessing or eah transduer in the orm o a mirophone preampliier and an A/D onverter. Figure 5 shows the proessing neessary or eah mirophone in the array. O ourse, dierent tehnology an aet the elements in the igure. For instane, the use o eletret or other mirophone tehnology may render the preampliier unneessary. Similarly, it is possible to ombine the mirophone preampliier (i any) with the irst stage o the A/D onverter. In any ase, the result o the preproessing is a sequene o digital audio samples. Sine a large array may ontain hundreds o mirophones, running individual wires rom eah mirophone to the required pre-proessing and subsequent proessing may be undesirable. With modern tehnology, high-levels o integration are possible. Both analog and digital iruitry an be put into the same pakage, i not the same substrate. See, or instane, US patent 5,5,799 [Paul et al]. It is possible to produe a very ompat realization o the preampliier and D/A onverter. It is even possible to ombine the mirophone preampliier with the irst stage o the D/A onverter or even a more ompat realization. Suh iruitry an be on the order o the same size as the mirophone apsule or even smaller. Figure 7 shows the idea o inluding the preproessing and A/D onversion in the same physial loation as the mirophone apsule itsel. In addition, some kind o data multiple xing iruit is inluded with eah mirophone so that the outputs o multiple mirophones may be ombined into a single wire. A wide range o multiplexing tehnology may be used, ranging rom simple time-domain or requeny-domain multiplexing [US patent 4,9,536, Hoque] to omputer-type network tehnology, suh as Ethernet [Metal and Boggs, 976]. The end result o this multiplexing is that the data rom the entire array is available in a small number o ables, or even a single able, in a manner suh that the samples rom eah individual mirophone may be separated or the required spatial proessing as shown in Figure 9. Figure 8 shows the logial extension o this train o logi to the mirophone array abri. Here, we show power being ed to eah transduer/proessor/multiplexor node via alternating vertial positive and negative supply wires. One o the wires an also be used as the network medium itsel by AC-oupling the data bak onto the wire. Similarly, lok distribution to the individual A/D onverters may be aomplished by plaing the lok itsel on one o the supply wires. By use o requeny-domain multiplexing, the data an be plaed on the wire in requeny bands that are well above the lok requeny. Note that the entire array ould just as easily be wireless (exept or the supply rails). Eah node ould simply broadast a low-power RF signal that ould be reeived and demultiplexed or urther proessing. Eah node would have to have some unique ID in the orm o a network address, a dediated requeny, a dediated time slot, or any other way o identiying the node so that the samples may be reovered and related bak to the original array position o the node. It is lear that any medium o transmission ould be used to onvey the data rom the array to the proessing elements. For instane, eah node ould emit digital data as light on wavelengths that people an not see. The data ould be multiplexed either by the wavelength o the individual lights, or by time so that only one node transmitted data at a time. Unpublished Manusript 9 /7/

10 Copyright - Hybrid shemes are also possible. That is, lusters o some number o nodes in a partiular area ould be multiplexed together with, say, iber-opti ables used to relay the data rom eah luster bak to the spatial proessing equipment. Relation to Sound-Field Theory In so-alled sound ield theory [***res Gerzon], we expand the sound pressure wave about the listener in a series o spherial harmonis [***res Hobson, MaRoberts]. This is not an artiiial onstrut. It alls diretly out o the solution to Laplae s equation in spherial oordinates [***Morse & Feshbah]. To the extent that air is linear, sound waves will obey Laplae s equation, and thus the sound ield around a listener an always be represented as a sum o spherial harmonis. This sum is not neessarily inite. I the sound soure is a true point soure, then the sum will not be inite. It an be approximated by a inite sum. As is typial with this kind o expansion, applying a window untion an help smooth out the overshoot ( Gibbs-type phenomena) inherent in trunating an ininite series. The point o making this expansion is that it gives a rational basis or trying to rereate the reording environment at the time o playbak. The idea is that i we an rereate the spherial harmoni expansion o the sound ield about the listener, then we have rereated the waveorm at one point in spae. This assertion is not ontroversial: it is a tautology. What an be argued is how many spherial harmonis are neessary to do a good job o reonstruting the sound ield. I have no partiular wisdom to oer on this point exept that more is better. The problem with atually doing this is two-old: irst is that we need at least one speaker or eah harmoni that we wish to reprodue, and seond is that modern mirophones are only apable o irst-order diretional patterns, as noted in Equation (). The point o the phased-array mirophone is that it is possible to use this diretionality to diretly measure the higher-order harmonis o the sound ield around the enter mirophone o the array. By using more and more mirophones in the array, the diretional pattern an be made arbitrarily narrow. Consequently we an reover any number o terms o the spherial harmoni expansion about the enter mirophone by inreasing the number o mirophones. Reerenes: [$] Paul, J.D.; Clayton, M.D.; Agnello, A.M. Digital Output Transduer, US Patent 5,5,799, issued September [$] Hoque, T.I. Digital Audio Transmission or Use in Studio, Stage, or Field Appliations US Patent 4,9,536, issued May 99. [$] Metale, R.M., and Boggs, D.R. Ethernet: Distributed Paket Swithing or Loal Computer Networks, Communiations o the ACM, Volume 9, Number 7, pp395-44, July 976 Unpublished Manusript /7/

11 Copyright - q d Figure A linear array o mirophones with a spaing o d. We assume a plane wave impinges on the array at an angle q rom the perpendiular to the array Figure Amplitude o the response o the sum o all the eeds rom the mirophone array with hanging angle o inidene. Eah urve represents a dierent wavelength rom.5d (narrowest) to 6d (widest). Unpublished Manusript /7/

12 Copyright Figure 3 This shows the eet o steering the array by adding a simple delay to eah mirophone. The wavelength o the test signal was set to a onstant.5d. Note the widening o the prinipal lobe as we steer the array away rom diretly in ront. This is due to the eetive narrowing o the mirophone spaing by a ator o os(θ ) Figure 4 this shows the eet o using a window untion to hange the tradeo between enter lobe width and side lobe suppression. The window was the Kaiser-Bessel window with the b parameter varying between.5 and 5.5. Unpublished Manusript /7/

13 Copyright - q d Figure 5 Three overlapping arrays sharing enter mirophones. The arrays have spaings o d, d, and 4d. To attain ull requeny response over the audio range with equal spatial resolution at all requenies, a total o at least ten olinear arrays would be required Figure 6: Plot o Beta parameter to Kaiser-Bessel window or values o wavelength in multiples o the mirophone spaing. These values o Beta equalizes the main lobe widths or the given wavelength. This urve appears to be largely independent o the number o mirophones in the array. Unpublished Manusript 3 /7/

14 Copyright Figure 7: Lobe widths ater normalization by adjusting the Beta parameter o the Kaiser-Bessel window. The wavelengths span the range rom.5d to 6d. Note that the sideband gain inreases at the ends o the requeny range due to the windowing. This is using 5 mirophones in a single array Figure 8: Typial windowing gain urves or our mirophones in a 9-mirophone array at various values o wavelength (in multiples o d). These represent partiular points o the Kaiser-Bessel window as the Beta parameter is swept as shown in Figure 6. The upper urve represents the enter mirophone, and the enter point o the window untion. Unpublished Manusript 4 /7/

15 Copyright - WINDOWING FILTER OVERLAP FILTER Σ Σ Σ OTHER ARRAYS Figure 9: Complete diagram o proessing or overlapped mirophone arrays. Eah mirophone goes to a ilter that imple ments the requeny-dependent window and the steering delay. Eah windowed array is then iltered so that the arrays overlap properly to produe an overall lat response. One windowing ilter is shown or eah mirophone or larity. Sine the window untions are symmetri, pairs o mirophones equidistant rom the enter mirophone would be summed, then iltered by a single requeny-dependent window ilter. I it is desired to simultaneously reeive signals rom dierent diretions (that is, with the array steered to dierent angles), then separate proessing would have to be supplied or eah desired angle. O ourse, the diret mirophone eeds ould be stored and proessed to extrat signals at dierent angles at a later time. Unpublished Manusript 5 /7/

16 Copyright Figure : One kind o prototype ilter overing the band rom Hz to 4 Hz. For proper overlap, the ilter extends into the adjaent bands rom 333 Hz to 5333 Hz. The ilter or the next higher or lower requeny band may be obtained simply by relabeling the requeny axis with either twie the requenies or hal the requenies. O ourse, this ilter design is not unique. There are many suitable hoies or the overlap ilter. POSTERIOR DIAPHRAGM ANTERIOR DIAPHRAGM NEUTRAL CAPSULE Figure : Diagram o a pressure-gradient ondenser mirophone. Typially, the interior apsule is held at ground, and the variations o apaitane between the diaphragms and the apsule generate a voltage. To obtain diretional harateristis, the voltages o the anterior and posterior diaphragms may be weighted and subtrated. This produes the amiliar diretional patterns, suh as ardioid, hyperardioid, and so on. Unpublished Manusript 6 /7/

17 Copyright - Figure : Regular -dimensional array with equal resolution in horizontal and vertial diretions. Unpublished Manusript 7 /7/

18 Copyright - Figure 3: -dimensional mirophone array showing unequal resolution in vertial and horizontal diretions. Unpublished Manusript 8 /7/

19 Copyright - Figure 4: Two -dimensional arrays plaed at right angles. Sine eah array is apable o steering aross an angle o 9 in the orward diretion and 9 in the bakward diretion, two arrays plaed at right angles an over all diretions. Unpublished Manusript 9 /7/

20 Copyright - PREAMPLIFIER A/D CONVERTER FROM OTHER MICROPHONES TO PROCESSING ARRAY Figure 5: A diret implementation o the array has a wire rom eah mirophone in the array to the required preproessing (mirophone preampliier and A/D onverter) and subsequent proessing (Figure 9). PREAMPLIFIER A/D CONVERTER NETWORK INTERFACE INTEGRATED PRE-PROCESSOR FROM OTHER MICROPHONES TO PROCESSING ARRAY Figure 6: Mirophone apsule with integrated pre-proessing. In this oniguration, miniaturized preampliier and A/D stages are integrated with some kind o multiplexing (network) interae. This interae an use requeny or time-domain multiplexing, or may interae to a general-purpose network, suh as Ethernet [**re]. Unpublished Manusript /7/

21 Copyright Figure 7: Eah oval above represents a omplete transduer, preproessor, and network interae as shown in Figure 6. This igure shows how the array may be powered by a vertial array o alternating positive and negative supplies. One rail (e.g. the positive wires) may also serve as the medium or the network (or additional wires may be used or the network interae). Unpublished Manusript /7/