Desig of Cross-talk Cacellatio Networks by usig Fast Decovolutio Ole Kirkeby, Per Rubak, Philip A. Nelso*, ad Agelo Faria # Departmet of Commuicatio Techology, Aalborg Uiversity, Fr. Bajers Vej 7, 922 Aalborg Ø, Demark *Istitute of Soud & Vibratio Research, Uiversity of Southampto, Highfield, SO17 1BJ, UK # Departmet of Idustrial Egieerig, Uiversity of Parma, Via delle Scieze, 431 Parma, Italy http://www.isvr.soto.ac.uk/fdag/vap/ Biaural material must be passed through a cross-talk cacellatio etwork before it ca be played back over two loudspeakers. Such a etwork works well oly if it is capable of providig a sigificat boost of low frequecies. The fast decovolutio method usig frequecy-depedet regularisatio is suitable for desigig a matrix of log fiite impulse respose filters that have the ecessary dyamic rage.
Itroductio Biaural material, such as a dummy-head recordig, is geerally iteded for playback over headphoes [1]. I order to achieve the equivalet effect whe such material is played back over two loudspeakers, a cross-talk cacellatio etwork must be used to compesate for the cross-talk (the soud that is reproduced at the right ear by the left loudspeaker, ad vice versa) ad the headrelated trasfer fuctios (HRTFs) associated with a real listeer [2]-[8]. I practice, a cross-talk cacellatio etwork ca be implemeted by a two-by-two matrix of digital filters. Ufortuately, though, efficiet cross-talk cacellatio at low frequecies is possible oly if each elemet of the cross-talk cacellatio etwork is capable of providig a sigificat boost of those frequecies [8]. This is because the differece betwee the direct path HRTF ad the cross-talk path HRTF is very small at low frequecies, ad so oe eds up havig to ivert a almost sigular two-by-two matrix. This problem, which is usually referred to as ill-coditioig, at low frequecies is particularly severe whe the two loudspeakers are positioed close together, as is the case for the stereo dipole where the loudspeakers spa oly te degrees as see by the listeer [9]. I practice, it is advatageous to use frequecy-depedet regularisatio to atteuate peaks selectively. Eve though a strog boost of low frequecies is ecessary for efficiet cross-talk cacellatio, a strog boost of high frequecies is geerally udesirable. It is particularly importat to be aware of this problem whe workig with HRTFs that are measured digitally. The aalogue ati-aliasig filters i the data acquisitio equipmet cause the spectrum of the measured trasfer fuctios to cotai oly very little eergy at high frequecies, ad if oe attempts to ivert such a trasfer fuctio, the solutio will ievitably boost frequecies just below the Nyquist frequecy [1]. The fast decovolutio method [11], [12], which is based o the Fast Fourier Trasform, ca be used to desig a matrix of causal fiite impulse respose filters whose performace is optimized at a large umber of discrete frequecies. The method is very efficiet for both sigle-chael decovolutio, which ca be used for loudspeaker equalisatio, ad multi-chael decovolutio, which ca be used to desig cross-talk cacellatio etworks. Fast decovolutio essetially provides a quick way to solve, i the least squares sese, a liear equatio system whose coefficiets, right had side, ad ukows are z-trasforms of stable digital filters. Frequecydepedet regularisatio is used to prevet sharp peaks i the magitude respose of the optimal filters. A modelig delay [13, Example 7.2.2] is used to esure that the cross-talk cacellatio etwork performs well ot oly i terms of amplitude, but also i terms of phase. The algorithm assumes that it is feasible to use log optimal filters, ad it works well oly whe two regularisatio parameters, a shape factor ad a gai factor, are set appropriately. I practice, the values of the two regularisatio parameters are most easily determied by trial-ad-error experimets. 1 Cross-talk cacellatio etworks 1.1 Priciples ad solutio The geometry of the problem is show i Fig. 1. Two loudspeakers are positioed symmetrically i frot of a sigle listeer. The loudspeakers spa a agle of as see from the positio of the listeer. Whe the system is operatig at a sigle frequecy, we ca use complex otatio to 2
Kirkeby et al, Cross-talk cacellatio etworks describe the variables. Thus, U 1 ad U 2 are two biaural sigals, recorded or sythesized, V 1 ad V 2 are the iputs to the two loudspeakers, ad W 1 ad W 2 are the soud pressures geerated at the listeer s ears (ote that the variables read alphabetically U, V, W, this will make the otatio easier to remember). There are four trasfer paths from the loudspeakers to the listeer s ears, but oly two of them are differet: the direct path C 1 ad the cross-talk path C 2. Similarly, oly two of the four elemets of the cross-talk cacellatio etwork are differet: the diagoal elemet H 1, ad the off-diagoal elemet H 2. From ispectio of Fig. 1 it is easily verified that where ad where C v = w (1a) C = C v w 1 C2 = V 1 = W 1,, C C V W, (1b) 2 1 2 2 H u = v (2a) H = H u 1 H2 = U 1, H H U. (2b) 2 1 2 A ideal cross-talk cacellatio etwork reproduces U 1 at the listeer s left ear (W 1 =U 1 ) regardless of the value of U 2, ad U 2 at the listeer s right ear (W 2 =U 2 ) regardless of the value of U 1. It is straightforward to show that this is achieved whe the H-matrix i Eq. 2a is the iverse of the C-matrix i Eq. 1a. Cosequetly H1 1 2 H = 2 C1 C 2 2 1.2 Ill-coditioig C1 C. (3) 2 It is see from Eq. 3 that whe the differece betwee C 1 ad C 2 is small, H 1 ad H 2 become very large ad almost exactly out of phase. This is a problem particularly at very low frequecies sice the direct path C 1 ad the cross-talk path C 2 are almost equal, regardless of the loudspeaker spa, whe the wavelegth is very log. At Hz, the phase of C 1 ad C 2 is the same, ad it is oly because the spherical atteuatio associated with the cross-talk path C 2 is greater tha the spherical atteuatio associated with the direct path C 1 that the matrix C is ot exactly sigular. Cosequetly, the closer the two sources are to the listeer, the easier it is to implemet the crosstalk cacellatio etwork. A distace i the rage betwee.5m ad 1m is a good choice, eve if the listeer sits further away. Near-field effects start to play a role whe the distace to the source becomes less tha.5m [14]. I practice, it is ot importat that the desig- ad the implemetio distace are the same, it is more importat that the desig- ad implemetatio loudspeaker spa are the same. 3
It is fortuate that whe two biaural sigals U 1 ad U 2 are passed through a cross-talk cacellatio etwork, the dyamic rage of the outputs V 1 ad V 2 of the etwork is geerally sigificatly smaller tha the dyamic rage of H 1 ad H 2 [9], [15]. Whe oe is dealig with measured HRTFs, the ill-coditioig at low frequecies is made eve worse by the poor radiatio efficiecy of the loudspeaker. Cosequetly, a cross-talk cacellatio etwork that also has to compesate for the soud reproductio chai must be implemeted with care i order to avoid overloadig the loudspeakers ad amplifiers, as well as saturatig the digital sigal processig equipmet. 2 FIR filter desig usig fast decovolutio The idea cetral to our filter desig algorithm [11], [12], is to miimise, i the frequecy domai, a quadratic cost fuctio of the type J = E + βv where E is a measure of the performace error e ad V is a measure of the effort v. The positive real umber is a regularizatio parameter that determies how much weight to assig to the effort term. As is icreased from zero to ifiity, the solutio chages gradually from miimizig E oly to miimizig V oly. By makig the regularizatio frequecy-depedet, we ca cotrol the time respose of the optimal filters i quite a profoud way. However, istead of specifyig as a fuctio of frequecy it is advategous to build the frequecy-depedece ito V. 2.1 Frequecy-depedet regularisatio It is coveiet to cosider the regularizatio to be the product of two compoets: a gai factor ad a shape factor B(z) [1], [12]. The gai factor is a small positive umber, ad the shape factor B(z) is the z-trasform of a digital filter that amplifies the frequecies that we do ot wat to see boosted by the cross-talk cacellatio etwork. Frequecies that are suppressed by B(z) are ot affected by the regularizatio. Although it is the frequecy respose, ad ot the time respose, of B(z) that is importat, we prefer to desig B(z) i the time domai. The phase respose of B(z) is irrelevat sice H(z) is determied by miimizig a eergy quatity. 2.2 Ideal optimal filters It is possible to derive a aalytical expressio for a matrix H(z) of ideal optimal filters [11], [12]. We fid T [ 1 1 ] 1 T 1 H( z) = C ( z ) C( z) + β B( z ) B( z) I C ( z ) z m (5) The compoet z m implemets a modelig delay of m samples. It is see that whe is zero, or B(z) is zero, the H(z) is C(z) -1 z -m, as expected. 2.3 The fast decovolutio algorithm The fast decovolutio method works by samplig Eq. 5, which gives H(z) as a cotiuous fuctio of frequecy, at N h poits. Sice the method uses Fast Fourier Trasforms (FFTs), N h must be a power of two. The implemetatio of the method is straightforward i practice. FFTs (4) 4
are used to get i ad out of the frequecy domai, ad the system is iverted for each frequecy i tur. Sice usig the FFT effectively meas that we are operatig with periodic sequeces, a cyclic shift of the iverse FFTs of the optimal frequecy resposes is used to implemet a modelig delay. If a FFT is used to sample the frequecy respose of H(z) at N h poits without icludig the phase cotributio from the modelig delay, the the value of H(k) at those frequecies is give by H [ β ] 1 H H( k) = C ( k) C( k) + B ( k) B( k) I C ( k) where k deotes the k th frequecy lie; that is, the frequecy correspodig to the complex umber exp(i2k/n h ). The superscript H deotes the Hermitia operator that trasposes ad cojugates its argumet, the superscript * deotes complex cojugatio of its scalar argumet. I order to calculate the impulse resposes of a matrix of causal filters the followig steps are ecessary. 1. Calculate B(k) ad C(k) by takig N h -poit FFTs of each of their elemets 2. For each of the N h values of k, calculate H(k) from Eq. 6 3. Calculate oe period of h() by takig N h -poit iverse FFTs of the elemets of H(k) 4. Implemet the modelig delay by a cyclic shift of m samples of each elemet of h() The exact value of m is ot critical; a value of N h /2 is likely to work well i all but a few cases. 2.4 Determiig the regularizatio gai- ad shape factors Sice the purpose of the regularizatio is to impose a subjective costrait o the solutio, it is very difficult to come up with a reliable black box routie that ca set the gai factor ad the shape factor B(z) simultaeously. For audio-related problems, though, the geeric fuctio show i Fig. 2 ofte works very well. As a fuctio of frequecy, the magitude B of B(z) has a lowfrequecy asymptotic value B L, ad a high-frequecy asymptotic value B H (subscript H is for high, ad should ot be cofused with the optimal filters H 1 ad H 2 ). I the mid-frequecy regio, B is oe. B L ad B H are usually much greater tha oe. The frequecies f L1, f L2, f H1, ad f H2 defie the two trasitio bads. Whe the samplig frequecy is high, for example 44.1kHz, it is sometimes advatageous to desig B o a double-logarithmic scale sice this is a good approximatio to the way the ear perceives soud. Oce B(z) is kow, there are plety of methods oe ca use to determie automatically. Sice the mai udesirable feature of the solutio is likely to be sharp peaks i the magitude respose, oe ca try to adjust such that a certai maximum value is ot exceeded, or such that the peak-to-rms ratio is well-behaved withi certai frequecy bads. It is up to the user to specify a criterio that is appropriate for the applicatio at had. 3 Two cross-talk cacellatio etworks for the stereo dipole Whe the two loudspeakers spa oly te degrees as see by the listeer, we refer to the loudspeaker arragemet as a stereo dipole [9]. We will ow use the fast decovolutio method to desig two differet cross-talk cacellatio etworks for this loudspeaker arragemet. The samplig frequecy is 44.1kHz i both cases. The first etwork is based o a pair of HRTFs (6) 5
calculated from a aalytical rigid sphere model [14], [16]. The sphere model ca be used to geerate results i the frequecy domai. These results are the widowed so that a pair of digital time resposes ca be calculated. The secod etwork is based o a pair of HRTFs measured o KEMAR dummy-head [17]. These HRTFs cotai little eergy at the extreme eds of the frequecy rage, ad they are therefore more difficult to deal with tha the modeled HRTFs. 3.1 HRTFs derived from a rigid sphere model The sphere is assumed to have a radius of 9cm, ad the ears ot quite at opposite positios, but rather they are pushed back te degrees so that they are at 1 degrees relative to straight frot [14]. This geometry esures a good match to the true iteraural time differece (although it has bee suggested that a radius of 7cm is better for ear-frotal sources, see [18] for details). Fig. 3 shows the impulse resposes of a) C 1 (z), ad b) C 2 (z) whe the distace from the two sources to the cetre of the listeer s head is 1m. Sice we do ot have direct access to a time domai expressio for the scattered field, the simulated time resposes are calculated by a iverse Fourier trasform of the sampled frequecy respose (see [16] for details). The frequecy resposes have bee widowed i order to esure that the time resposes are of relatively short duratio. The widowig i the frequecy domai is equivalet to covolutio with a so-called digital Haig pulse give by the time sequece {,.5, 1,.5, }. Thus, C 1 (z) ad C 2 (z) are essetially low-pass filtered versios of the true trasfer fuctios, ad this must be compesated for by also low-pass filterig the optimal filters H 1 (z) ad H 2 (z) (this is equivalet to solvig a equatio system whose left ad right had sides have bee multiplied by the same umber). Formally, this is doe by settig the diagoal elemets of a so-called target matrix A(z) equal to the Haig pulse (see [11] for details). Note that C 1 (z) ad C 2 (z) are quite similar because the two loudspeakers are very close together. Fig. 4 shows a) the impulse respose ad b) the magitude respose of the shape factor B(z). This filter is a gradual high-pass filter whose magitude respose icreases from.1 to 1 as the frequecy icreases from.6f Nyq to.9f Nyq. Fig. 5 shows the magitude resposes of a) H 1 (z) ad b) H 2 (z) calculated with frequecydepedet regularisatio (solid lies) ad with o regularisatio (dashed lies). The shape factor B(z) is that show i Fig. 4, ad the gai factor is.5. It is see that the regularisatio has take out the peak just below the Nyquist frequecy ( 22kHz), ad that the respose at high frequecies rolls of getly. Note that eve though the magitude resposes of H 1 (z) ad H 2 (z) are very similar, their phase resposes are completely differet [15]. Fig. 6 shows the two differet impulse resposes, a) H 1 (z) ad b) H 2 (z). Each impulse respose cotais 124 coefficiets, ad they correspod to the magitude resposes show with the solid lies i Fig. 5. Note that both cotai a compoet that decays away very slowly i forward time. This compoet is resposible for the required boost of low frequecies. 3.2 HRTFs measured o KEMAR dummy-head Fig. 7 is equivalet to Fig. 3. It shows the impulse resposes of a) the direct path C 1 (z), ad b) the cross-talk path C 2 (z) whe the two HRTFs are measured o a KEMAR dummy-head i a aechoic chamber (this HRTF data is available o the iteret [17]). Sice the data is ot 6
equalised for the loudspeaker respose, the two impulse resposes do ot cotai much eergy at very high, or very low, frequecies. Fig. 8 is equivalet to Fig. 4. It shows a) the impulse respose ad b) the magitude respose of the shape factor B(z). This filter has the same type of gradual high-pass characteristic as the filter show i Fig. 4, but i additio it allows eergy at frequecies betwee.3f Nyq ad.4f Nyq to pass through. This is doe i order to atteuate a peak that would otherwise appear just below.4f Nyq as show i Fig. 9. Fig. 9 is equivalet to Fig. 5. It shows the magitude resposes of a) H 1 (z) ad b) H 2 (z) calculated with frequecy-depedet regularisatio (solid lies) ad with o regularisatio (dashed lies). The shape factor B(z) is that show i Fig. 8, ad the gai factor is.5. It is see that the regularisatio has take out the peak at approximately.35f Nyq ad also filtered out the uacceptable boost of the frequecies just below f Nyq. Note the cosiderable dyamic rage of the magitude resposes of H 1 (z) ad H 2 (z). The value at DC is more tha 5dB higher tha the value at.1f Nyq. This happes because the filters ow have to compesate for the loudspeaker as well as the cross-talk. Fig. 1 is equivalet to Fig. 6. It shows the two differet impulse resposes, a) H 1 (z) ad b) H 2 (z). Each impulse respose cotais 248 coefficiets, ad they correspod to the magitude resposes show with the solid lies i Fig. 9. Note that the low-frequecy compoet ow decays away i backward time. Had a modelig delay ot bee used, this compoet would be o-causal ad therefore urealisable. It is the o-miimum phase characteristics of the loudspeaker at low frequecies that causes this dramatic differece betwee the results based o a aalytical sphere model ad the results based o the measuremets o a dummy-head. 4 Coclusios Efficiet cross-talk cacellatio over a wide frequecy rage is possible oly whe each elemet of the cross-talk cacellatio etwork is capable of a very powerful boost of low frequecies. If the etwork also has to compesate for the respose of the loudspeaker, the required boost is eve greater. I additio, the o-mimimum phase behaviour that is typical of electro-acoustic trasducers at the extreme eds of the frequecy rage makes it ecessary to use a modelig delay i order to be able to equalise the phase respose as well as the magitude respose. The fast decovolutio method is very suitable for desigig log fiite impulse respose filters that have a large dyamic rage. Frequecy-depedet regularisatio provides a coveiet way to cotrol the power output from the filters, ad the regularisatio ca be used to optimize the subjective performace of the system as well as prevet overloadig of the amplifiers ad loudspeakers. Fially, it is importat to keep i mid that eve though it is computatioally feasible to ivert very log impulse resposes with the fast decovolutio method, a accurate decovolutio of a impulse respose that cotais a lot of detail does ot ecessarily lead to good subjective results. It is ofte better to ivert oly the system s most essetial characteristics. I practice, this usually helps to avoid excessive colouratio of the reproduced soud. 7
5 Refereces [1] H. Møller, C.B. Jese, D. Hammershøi, ad M.F. Sørese, Evaluatio of artificial heads i listeig tests, preseted at the 12d Audio Egieerig Society Covetio i Muich, March 22-25, 1997. AES preprit 444-A1 [2] P. Damaske, Head-related two-chael stereophoy with loudspeaker reproductio, J. Acoust. Soc. Am. 5, 119-1115 (1971) [3] D.H. Cooper ad J.L. Bauck, Prospects for trasaural recordig, J. Audio Eg. Soc. 37 (1/2), 3-19 (1989) [4] D. Griesiger, Equalizatio ad spatial equalizatio of dummy-head recordigs for loudspeaker reproductio, J. Audio Eg. Soc. 37 (1/2), 2-29 (1989) [5] H. Møller, Reproductio of artificial head-recordigs through loudspeakers, J. Audio Eg. Soc. 37 (1/2), 3-33 (1989) [6] P.A. Nelso, H. Hamada, ad S.J. Elliott, Adaptive iverse filters for stereophoic soud reproductio, IEEE Trasactios o Sigal Processig, 4 (7), 1621-1632 (1992) [7] J. Bauck ad D.H. Cooper, Geeralized trasaural stereo ad applicatios, J. Audio Eg. Soc. 44 (9), 683-75 (1996) [8] O. Kirkeby, P. Rubak, L.G. Johase, ad P.A. Nelso, Implemetatio of cross-talk cacellatio etworks usig warped FIR filters, preseted at the 16 th Iteratioal Covetio of the Audio Egieerig Society, Rovaiemi, Filad, April 1-12, 1999 [9] O. Kirkeby, P.A. Nelso, H. Hamada, The stereo dipole - a virtual source imagig system usig two closely spaced loudspeakers, J. Audio Eg. Soc. 46 (5), 387-395 (1998) [1] O. Kirkeby, P.A. Nelso, ad H.Hamada, Digital filter desig for virtual source imagig systems, preseted at the 14th covetio of the Audio Egieerig Society i Amsterdam, May 16-19, 1998. AES preprit 4688 P1-3. Submitted to J. Audio Eg. Soc [11] O. Kirkeby, P.A. Nelso, H. Hamada, ad F. Ordua-Bustamate, Fast decovolutio of multichael systems usig regularizatio, IEEE Tras. Speech ad Audio Processig, 6 (2), 189-194 (1998) [12] O. Kirkeby, P. Rubak, ad A. Faria, Fast decovolutio usig frequecy-depedet regularizatio, to be submitted to IEEE Tras. Speech ad Audio Processig [13] R.A. Roberts ad C.T. Mullis, Digital Sigal Processig, Addiso-Wesley, 1987 [14] R.O. Duda ad W.L. Martes, Rage depedece of the respose of a spherical head model, J. Acoust. Soc. Am. 14 (5), 348-358 (1998) [15] O. Kirkeby ad P.A. Nelso, Virtual source imagig usig the stereo dipole, preseted at the 13rd AES Covetio, New York, USA, September 26-29, 1997. AES preprit 4574-J1 [16] O. Kirkeby, P.A. Nelso, ad H. Hamada, Local soud field reproductio usig two closely spaced loudspeakers, J. Acoust. Soc. Am., 14 (4), 1973-1981 (1998) [17] B. Garder ad K. Marti, HRTF Measuremets of a KEMAR Dummy-Head Microphoe, MIT Media Lab, available o the World Wide Web at http://soud.media.mit.edu/kemar.html [18] K.B. Rasmusse ad P.M. Juhl, The effect of head shape o spectral stereo theory, J. Audio Eg. Soc., 41 (3), 135-141 (1993) 8
% Kirkeby et al, Cross-talk cacellatio etworks! Fig. 1. The variables ad the parameters used to defie a cross-talk cacellatio etwork. Note that because of the symmetry there are oly two differet electro-acoustic trasfer fuctios, C 1 ad C 2, ad the etwork cotais oly two differet filters, H 1 ad H 2 MLN JLK 234 56./ 1 789;:<= & '()* (+,- >?@BACDFEGHI Fig. 2. A suggested magitude respose fuctio for the shape factor B(z). This type of frequecydepedet regularisatio esures that the cross-talk cacellatio etwork does ot boost very low, ad very high, frequecies excessively "$#
1.2 a) c1 1.8.6.4.2.2 1 2 3 4 5 6 1.2 b) c2 1.8.6.4.2.2 1 2 3 4 5 6 Fig. 3. The impulse resposes of a) the direct path C 1 ad b) the cross-talk path C 2 as defied i Fig. 1 whe the listeer s head is modeled as a rigid sphere, ad the samplig frequecy is 44.1kHz.3 a) b.2.1.1.2 5 1 15 2 25 3 1 b) B.8.6.4.2.1.2.3.4.5.6.7.8.9 1 Normalised frequecy Fig. 4. The properties of the shape factor B(z) used to desig a cross-talk cacellatio etwork based o the impulse resposes show i Fig. 3. a) the impulse respose of B(z), ad b) its magitude respose
3 a) H1 2 1 db 1 2.1.2.3.4.5.6.7.8.9 1 Normalised frequecy 3 b) H2 2 1 db 1 2.1.2.3.4.5.6.7.8.9 1 Normalised frequecy Fig. 5. The magitude resposes of a) H 1 (z) ad b) H 2 (z) calculated with frequecy-depedet regularisatio (solid lies) ad with o regularisatio (dashed lies). The shape factor B(z) is that show i Fig. 4. Note that the regularisatio has take out the peak just below the Nyquist frequecy.6.4.2.2.4.6.6.4.2.2.4.6 a) h1 1 2 3 4 5 6 7 8 9 1 b) h2 1 2 3 4 5 6 7 8 9 1 Fig. 6. The impulse resposes of the two filters a) H 1 (z) ad b) H 2 (z) whose magitude resposes are show with the solid lies i Fig. 5. Each impulse respose cotais 124 coefficiets
1 a) c1.5.5 1 2 4 6 8 1 12 1 b) c2.5.5 1 2 4 6 8 1 12 Fig. 7. The impulse resposes of a) the direct path C 1 (z), ad b) the cross-talk path C 2 (z) whe the two HRTFs are measured o a KEMAR dummy-head i a aechoic chamber at a samplig frequecy of 44.1kHz. The data is ot equalised for the loudspeaker respose.5 a) b.5 5 1 15 2 25 3 1 b) B.8.6.4.2.1.2.3.4.5.6.7.8.9 1 Normalised frequecy Fig. 8. The properties of the shape factor B(z) used to desig a cross-talk cacellatio etwork based o the impulse resposes show i Fig. 7. a) the impulse respose of B(z), ad b) its magitude respose
db db 5 4 3 2 1 1 2 a) H1 3.1.2.3.4.5.6.7.8.9 1 Normalised frequecy 5 4 3 2 1 1 2 b) H2 3.1.2.3.4.5.6.7.8.9 1 Normalised frequecy Fig. 9. The magitude resposes of a) H 1 (z) ad b) H 2 (z) calculated with frequecy-depedet regularisatio (solid lies) ad with o regularisatio (dashed lies). The shape factor B(z) is that show i Fig. 8. Note that the regularisatio has take out the peak at approximately.35f Nyq ad also filtered out the uacceptable boost of the frequecies just below f Nyq.6.4.2.2.4.6.6.4.2.2.4.6 a) h1 2 4 6 8 1 12 14 16 18 2 b) h2 2 4 6 8 1 12 14 16 18 2 Fig. 1. The two impulse resposes, a) H 1 (z) ad b) H 2 (z) whose magitude resposes are show with the solid lies i Fig. 9. Each impulse respose cotais 248 coefficiets. Note that the low-frequecy compoet ow decays away i backward time