Position Independent Stereo Sound Reproduction

Transcription

1 Nat.Lab. Tehnial Note 2000/002 Date of issue: 06/2000 Position Independent Stereo Sound Reprodution Part I: State-of-the-art on PI-stereo Josep A. Ródenas and Ronald M. Aarts Company Restrited Koninklijke Philips Eletronis N.V. 2000

2 2000/002 Company Restrited Authors address data: Josep A. Ródenas WY82; Ronald M. Aarts WY81; Koninklijke Philips Eletronis N.V All rights are reserved. Reprodution in whole or in part is prohibited without the written onsent of the opyright owner. ii Koninklijke Philips Eletronis N.V. 2000

3 Company Restrited 2000/002 Tehnial Note: 2000/002 Title: Author(s): Position Independent Stereo Sound Reprodution Part I: State-of-the-art on PI-stereo Josep A. Ródenas and Ronald M. Aarts Part of projet: Sound Reprodution Improvement (CRB 0854) Customer: Consumer Eletronis Keywords: Abstrat: Sweet spot widening, PI-stereo, loudspeakers arrays, analog and digital filtering tehniques, time/intensity trading experiments, frequeny independent diretivity patterns, sound soures. In this report the orretion to ompensate for the degradation of the stereophoni illusion due to off-enter listening is investigated. The main idea here is that the diretivity pattern of a loudspeaker has a well defined shape suh that a good stereo reprodution in ahieved in a large listening area. This appliation is known as Position Independent (PI) Stereo. Analog and digital filtering tehniques are designed and applied to individual drivers of linear loudspeaker arrays in order to obtain a frequeny independent radiation pattern of a speifi shape. This shape is adapted to the time/intensity trading mehanism of the human ear (psyhoaousti experiments) within a wide listening area. The drivers were fed by analog or digital filters (with oeffiients whih were alulated by means of a numerial optimisation proedure) in suh a way that both loudspeakers together radiated sound in a broad range of listening positions in aordane with the time-intensity trading results. Conlusions: Koninklijke Philips Eletronis N.V The outome of this work showed that optimal diretivity patterns for loudspeaker arrays in stereophoni appliations an be very useful for sweet spot widening. Listening experiments demonstrated the orret entral sound loalisation for a number of lateral listening positions. The stereo sound sensation, in partiular the plaement of entral voies, was listening position independent within a large area. This PI-stereo feature an be best applied to any systems where a good stereo sound reprodution in a large listening area is required, suh as: TV-sets, Hi-Fi s, multimedia, iii

4 2000/002 Company Restrited home theatre and portable audio. A future report, presenting the results of several listening experiments with the PI-stereo system for different stereo base setups, will ompare the robustness of the analog and digital PI-stereo implementations presented in this report. iv Koninklijke Philips Eletronis N.V. 2000

5 Company Restrited 2000/002 Contents 1 Introdution Optimal stereo pereption in a wider area PI-stereo setup and basis Loudspeaker array design for PI-stereo Optimal diretivity patterns Calulation of the target funtion Aoustial theory on sound soures Simple soure Simple linear array Filtering tehiques for driving the loudspeaker arrays Optimisation problem: Digital FIR filters Equivalent analog filtering for PI-stereo Lowpass filtering Allpass filtering Theoretial study for analog phase shifting Digital IIR implemetation from the analog system Digital FIR implementation from the measured transfer funtions Experiments and measured diretivity patterns Comparisons between the measured and the theoretial diretivity patterns 27 7 Conlusions and future work 29 Referenes 30 Distribution Koninklijke Philips Eletronis N.V v

6 2000/002 Company Restrited vi Koninklijke Philips Eletronis N.V. 2000

7 Company Restrited 2000/002 1 Introdution The basis of stereophony is the ability to reate phantom images. What do we mean by phantom image? It is known that the brain loates a mono signal originated from a single soure by omparing the differenes in the arrival time and intensity of that signal at eah ear. What if that same mono signal is played through two loudspeakers on either side of the listener? (see Figure 1). If this happens, then the sound appears to ome from midway between the two loudspeakers. Thus, the brain is fooled by assuming that the sound omes from diretly in front sine the time differenes of the signal arriving at eah ear are the same. This is alled a phantom image [1]. Generally it is onsidered as a serious artefat of the present stereo system that the listener is aware of the stereophoni illusion only in a limited region. Optimum stereo pereption only happens if the listener is plaed exatly in the median plane between the two loudspeakers. If the head is moved away from the median plane then the stereo effet deteriorates. This off-entre listening problem beomes even more serious when the distane between the loudspeakers is not large in omparison with the deviations from the entre position as our in multimedia PC monitor and TV appliations; the later has normally a wider stereo base but some ases a smaller stereo base is desired. Thus, if the head is moved laterally, the sound rapidly seems to ome from the nearest loudspeaker only. This is mainly beause of two additional effets: the intensity of the nearest loudspeaker at the listener s head is highest, and its wavefront arrives earlier (law of the first wavefront or preedene effet) [1, 2] (Figure 1). In general, it an be stated that orret loalisation within a wide listening area is benefiial for all appliations where good sound is required: audio, video and ar stereo, but that the pratial realisation of this goal is impossible with a normal stereo setup. LEFT Sound Image RIGHT Move Distorted Sound Image Sweet-Spot Figure 1: Normal stereo setup and optimal listening in the sweet spot. Koninklijke Philips Eletronis N.V

8 2000/002 Company Restrited 1.1 Optimal stereo pereption in a wider area There an be many situations where it is desired the diretivity of a loudspeaker to have a well defined form suh that a good stereo sound reprodution is ahieved in a large listening area [3, 4, 5, 6]. The idea of ahieving an inrease in the listening area (sweet spot widening) in a stereo setup, also known as Position Independent (PI) stereo, has been briefly introdued and studied at the begining of the 90 s [7, 8, 9]. In [7], the old idea of time/intensity trading, realised by an appropriate radiation pattern for the loudspeakers for a two hannel stereo arrangement, was firstly presented. The main idea was the following: if the listener moves to the left, the sound intensity from the right loudspeaker inreases, while that of the left loudspeaker dereases in suh a way that the virtual sound image remains in the middle. This an also be seen as a sort of automati balane ontrol depending on the position of the listener [10]. In this study, Goossens firstly determined the required diretivity pattern, whih was presribed by psyhoaousti laws and experiments onduted in the anehoi room for audio and video stereo setups; and seondly, he designed a PI-stereo system in a broad frequeny range (200 Hz to 12 khz) by means of two linear loudspeaker arrays with 4 drivers and filtering for eah audio hannel. In [8] the analog PI-stereo implementation was widely presented. The influene of the analog filtering to obtain an optimal diretivity pattern for the loudspeakers was desribed. Several measurements in the anehoi room for the diretivity patterns of the loudspeakers were undertaken. Results demonstrated some small disrepanies from the theoretial models at higher frequenies. However, listening tests showed that the PI-stereo effet was quite overdone, that is the virtual sound image did not remain exatly in the enter, for some positions. In [9] some more listening experiments were desribed with the aim to investigate and learly understand the influene of the off-entre listening for nomal stereo setups and for small stereo base setups as for HDTV and multimedia appliations. New time/intensity trading experiments reported a different range of trading ratios, whih were muh lose to those reported in the open literature (see Figure 5), ompared to those of Goossens, whih bring us to think that those experiments, and therefore that data, were not aurate enough. The purpose of this report is to summarise the urrent state-of-the-art on PI-stereo sound reprodution and to larify some unlear parts in the previous works. Our goal is to desribe and design new robust digital implementations for the PI-stereo system so as to ahieve a better performane of the system in a larger listening area. Therefore, some digital and analog filtering tehniques whih are applied to the individual drivers of the loudspeaker arrays to obtain a frequeny independent diretivity pattern of a speifi shape will be desribed. Let us first introdue the desription of the PI-stereo system and how to ahieve an optimal target funtion for the diretivity patterns of the loudspeakers. 2 PI-stereo setup and basis PI-stereo is basially two loudspeaker arrays, eah fitted into a one abinet, with a frequeny independent diretivity pattern of a speial shape whih has been designed suh 2 Koninklijke Philips Eletronis N.V. 2000

9 Company Restrited 2000/002 Figure 2: Optimal listening area for PI-stereo reprodution appliations. that a good stereo reprodution is ahieved in a large listening area (see Figure 2). The system works as follows: a listener at position A reeives a stronger signal from the box R than from box L, so that this intensity differene ompensates the time of arrival differene from box L and R, whih is know as time/intensity trading [2, 9]. That is the sound pressure should not be onstant for all listening positions. The desired shape an be obtained by psyhoaousti experiments. Another important onsideration to keep in mind for the PI-stereo system is that when listening to PI-stereo, the loudspeaker boxes are fae on, not a 30 degrees pointing inwards to the listener as is normal stereo. A standard listening setup for PI-stereo for audio and video sets is shown in Figure Loudspeaker array design for PI-stereo The design of loudspeaker array abinets using several drivers is not new in the literature. It has been proved that an adjustable diretivity pattern for a loudspeaker an be realised by using an array of drivers positioned at a small distane from eah other [6, 11]. In [7] a very lever loudspeaker array design for PI-stereo sound reprodution was desribed. To simplify the pratial realisation of the system, a pair of loudspeaker array abinets were equipped with a pair of drivers for eah frequeny range (high and mid). All speakers were mounted in a m W H D abinet with internal volume of 12 liters (Figure 3(a)). It was also deided to split the audible frequeny band into 3 ranges. The mid frequeny range loudspeakers must have a reasonable flat Sound Pressure Level between 200 Hz and 2 khz, and a uniform horizontal omnidiretional diretivity pattern in this frequeny Koninklijke Philips Eletronis N.V

10 2000/002 Company Restrited H B H A H A M B M A M A M B H B Left LS Central Woofer Right LS (a) (b) Figure 3: Desription of the loudspeaker abinets for PI-stereo. (a) A piture of the physial loudspeaker array design. (b) The system omprises two abinets with two loudspeaker arrays (mid and high ranges) with two drivers (A and B). The woofer only reprodues the low frequenies of the audio signal. range. The same applies for the high frequeny range loudspeakers whih due to physial inter-driver distane for the tweeters, the frequeny band goes from 2 khz to 12 khz. There is not need for frequenies below 200 Hz beause sound loalisation is not required there (low frequenies an be reprodued in the PI-stereo system by a entral woofer) (see Figure 3(b)). These drivers will be fed by analog or digital filters in suh a way that both loudspeakers together radiated sound in a broad range of listening positions in aordane with the time/intensity trading experiments. Let us next introdue a method whih alulates an optimal target funtion for the diretivity patterns of the loudspeakers based on time/intensity trading experiments. 3 Optimal diretivity patterns Differenes between the ear input signals are neessary in order for auditory events to appear laterally, away from the median plane. Interaural time differenes (ITD) and interaural level differenes (ILD) are the main parameters, whih affet the way that we pereive the diretion of sound and hene for the evoation of a phantom soure. A frequent used tehnique for the measurement of the equivalene between ILD and ITD are the time/intensity trading experiments, that is, asking a subjet whih time differene is equivalent to a ertain level differene [1, 2]. As we have pointed out previously, in order to alulate the optimal diretivity pattern for the loudspeakers, two listening tests in the anehoi room were onduted [7, 9]. During 4 Koninklijke Philips Eletronis N.V. 2000

11 Company Restrited 2000/ m Left TV-SET Right 2.5 m Virtual Center Left TV-SET Right 1.5m Virtual Center Sweet-Spot Line 13L 12L 11C 12R 13R 0.75m 2.19m 23L 22L 21C 22R 23R Sweet-Spot Line 33L 32L 31C 32R 0.5m 33R (a) (b) Figure 4: PI-stereo setups for the two experiments: (a) Goossens, (b) Aarts. these tests, the differenes in intensity levels between right R and left L loudspeakers for different listening positions were measured (see Figure 4). From these experiments, an optimal diretivity pattern for the sound soures of the loudspeakers an be determined, whih yields to the appropriate level differenes, neessary for orret stereo sound reprodution for all listening positions within a given area. We introdue next a lever approah to the problem of finding the optimal digital filter oeffiients to drive the arrays of loudspeakers so as to obtain the required diretivity pattern. This approah has been previously introdued by Goossens [7] and it onsits of finding first a theoretial target fution for the diretivity pattern and then to fit the optimal FIR filters to this diretivity pattern. 3.1 Calulation of the target funtion Goossens [7] defined an ad ho mathematial expression for a target funtion (L t ) for the sound pressure levels for the diretivity patterns of the loudspeakers whih an approximate any data from time/intensity trading experiments (see Figure 5). This mathematial expression whih depends on two linear parameters (A and B) and one nonlinear parameter (C) was defined as follows (note that if C is fixed, then the system beomes a linear system): L t 20 log[a B sin C ] (1) In order to optimise and fit this target funtion to any kind of time/intensity trading data, we an use the ombination of the well known nonlinear least squares method and the linear optimisation problem, whih solve and give the values for A, B and C [12, 13]. Koninklijke Philips Eletronis N.V

12 2000/002 Company Restrited Figure 5: Time/intensity trading results of the experiments arried out in [9]. The system we have is then the following: 1 sin C 1 1 sin C 2 1 sin C n A B L t 1 L t 2 L t n (2) Thus, in matrix form this redues to: T x L t (3) where T is the [N 2] matrix of the target funtion to optimise, x is the [1 2] vetor of the linear parameters to find and L t is the [1 N] vetor of the experimental data to fit. This problem an be solved first finding out the nonlinear parameter C and then the linear parameters A and B. This results in the best estimation for the system whih gives the minimum error in the fit to T by min x T x R Lt 2 (4) 2 We then obtain the next target funtions whih approximate the orret diretivity patterns for the louspeakers for the Goossens time/intensity trading data: 6 L t 20 log[ sin 3 ] (5) Koninklijke Philips Eletronis N.V. 2000

13 Company Restrited 2000/002 and for the Aarts time/intensity trading data: L t 20 log[ sin 3 ] (6) Figure 6 shows the polar plots for the optimised diretivity patterns for the alulated time/intensity trading data. Figure 6: Theoretial diretivity patterns for the optimal target funtions obtained from the given listening experiments onduted by Goossens and Aarts. 4 Aoustial theory on sound soures Let us now briefly review the basis on wavefronts and sound soures when designing a number of linear loudspeaker arrays. It is well known that the harater of the indiret sound is strongly ontrolled by the diretivity pattern of the loudspeaker. In general terms, this required diretivity pattern is not the same for the diret and indiret sound and different optimisation proedures have to be applied. As previously said, an ajustable diretivity pattern for a loudspeaker an be realised by using an array of drivers, positioned at a small distane from eah other. In this setion, we introdue the foundations of the aoustial theory required to understand the design of linear loudspeaker arrays as it is desribed in our appliation. 4.1 Simple soure Let us define the expression for the aousti pressure of a simple soure: p r t j "! 0 4# r Q se j $&% t' kr( (7) Koninklijke Philips Eletronis N.V

14 /002 Company Restrited where the quantity Q s, whih is known as the omplex soure strength of the spherial soure, is defined as the produt of its surfae area and veloity amplitude, i.e., Q s 4# r 2 U 0. Where U 0 is the veloity amplitude, r is the radius distane, is the angular frequeny ( 2# f ), k is the wavenumber (k *) ),! 0 is the density of the air, and at time t. 4.2 Simple linear array We onsider here a linear array of N simple soures with adjanent elements spaed a distane d apart as shown in Figure 7. If all soures have the same soure strength and radiate waves with the same phase, then the i th soure generates a pressure wave of the form A r i e j $+% t' kr i(, where r i is the distane from this soure to r, and the oeffiient A is given by A j%-, 0 4. Q s. The resultant pressure at the field point is the summation: p r t N i/ 1 A r i e j $&% t' kr i( (8) This is known as the priniple of superposition whih shows that the pressure at a field point is the sum of the pressures produed by the individual soure. To (r, 0 ) r 1 r 2 r 3 r 4 r 5 r 6 r N r 2 d sin d L 243 N 5 16 d N Figure 7: Geometry used for a linear array with N simple soures. If we restrit attention to the far field (speified by the ondition r 7 L where L N 1 d is the length of the array), we an approximate by assuming all r i are parallel. Then r i r 1 8 i 1 :9 r, where 9 r d sin. The distane to the enter of the array an be expressed as r 1 r 1 2 L) d :9 r. In the far field, r i in the denominador an be replaed with r and then the equation takes the form: 8 N p r t A r e' j 2 1 $ L; d( k< r j e $&% t' kr( e j $ i' 1( k< r (9) i/ 1 Koninklijke Philips Eletronis N.V. 2000

15 ? Company Restrited 2000/002 Using trigonometri identities, we get sin p r t A r e j $+% t' kr( N 2 k< r sin 1 2 k< r (10) The pressure on the axis ( = 0) is p r 0 t N A r e j $+% t' kr( (11) and has the maximum possible pressure amplitude: P ax r N A r (12) Identifiation of the diretional fator: H 1 sin N 2 kd sin N sin 1 2 kd sin (13) allows us to write the amplitude of the pressure in the familiar form: P r P ax r H (14) Notie that the denominator of H may vanish if 1 2 kd > sin > m#, but the numerator vanishes also, and the pressure amplitude beomes P ax r. Thus, we an have more than one major lobe. The angles of these our for: > sin > m d m 0 1 2@ : : A d? (15) This result an be restated as >B9 r > m?, whih reveals that the radiated pressure is maximised at those angles for whih the distanes from the field point to the adjanent array elements differ by integral numbers of wavelenghts. 5 Filtering tehiques for driving the loudspeaker arrays In this setion we onsider the design of filtering tehniques (digital and analog) to obtain the alulated optimal target funtion in a wide listening area. Firstly, we fous on the optimisation problem of estimating the required oeffiients for the FIR filters whih ahieve the given target funtion. Seondly, we show how the obtained digital filters an easily be impletemented in an analog way by using a ombination of a lowpass filter with an allpass filter. Finally, we introdue some digital IIR and FIR filter equivalents for the designed analog filters. Koninklijke Philips Eletronis N.V

16 C C 2000/002 Company Restrited 5.1 Optimisation problem: Digital FIR filters It is well known that a FIR filter has a transfer funtion H z of the form: N' 1 H z h k z ' k (16) k/ 0 FIR filters are always stable mainly for two reasons: (i) they don t have any poles outside the unit irle; (ii) the sum of the absolute values of the samples of the impulse response is finite. FIR filters an have an exatly linear phase harateristi whih gives a onstant group delay D g. Table 1: Properties of FIR filters in omparison to IIR filters. Some of the properties of FIR filters in omparison to IIR filters are shown in Table 1. As previously mentioned, Goossens [7] introdued an optimisation method, based on the least squares method, to alulate the required FIR filter oeffiients that optimise a given target funtion. Let us now revise this optimisation problem of estimating the required FIR filter oeffiients whih ahieve the target funtions desribed in Setion 3.1. We onsider the general ase of having a linear array of N equal and equidistant omnidiretional sound soures separated by a distane d. From Equation (7) and given that eah sound soure is driven by a FIR filter h ne m of M oeffiients with N number of drivers, the total sound pressure level P ke l for a frequeny k and an angle :F is given by: P G k :F N M h ne me j% k n' NH 1 d 2 sinikjg' ml 1 fs (17) n/ 1 m/ 1 where f s is the sampling frequeny, and is the angular frequeny whih varies from 1 -- k and is the observation angle whih varies from 1 --NM. The last expression an also be writen as: P O k :F N M h ne m os k n n/ 1 m/ 1 N 1 j sin k n 2 N 1 2 d sin :F d sin :F m 1 f s m 1 f s (18) Sine we want to obtain a sound pressure level equivalent to that given by the theoretial target funtions in Equations (5) and (6), we an define the required sound pressure level L F ke for the target funtion as: 10 L FP ke LF t e ' j% ML k 2 fs 1 LF t M 1 os k j sin k 2 f s M 1 2 f s (19) Koninklijke Philips Eletronis N.V. 2000

17 F F F F Company Restrited 2000/002 where the phase term orresponds to a onstant group delay of T M 1 N) 2 and LF t are the sound pressure levels at a angles M given by the target funtions desribed in Setion 3.1. We an now formulate the least squares optimisation problem, so as to find the h ne m FIR oeffiients for the different array drivers, as follows: T H L (20) where the vetor H of dimensions [N M 1] ontains the filter oeffiients and the matrix T of dimensions [2K L N M] is defined by Equation (18) as: T$ k' 1( LQ T$ KQ k' 1( LQ ER$ n' 1( MQ m os k n ER$ n' 1( MQ m sin k n N 1 2 N 1 2 d sin SF d sin :F m 1 f s (21) m 1 f s (22) and the vetor L of dimensions [2K L L $ k' 1( LQ L $ KQ k' 1( LQ E 1 E 1 whih give the following matrix form system: 1] by Equation (19) as: F F L t M 1 os k 2 f s (23) L t M 1 sin k 2 f s (24) T 11 T 12 : S T 1E N M T 21 T 22 : S T 2E N M T 2K LE 1 T 2K LE 2 : S T 2K LE N M h 11 h 1M h 21 h N M L 11 L 1L L 21 L 2K L (25) The resulted FIR oeffiients by the optimisation method are real for frequenies 0 and f s ) 2: f k 0 : a n0 a n1 a n2 a n3 -- a nm (26) f k f s ) 2 : a n0 a n1 a n2 a n3 --UT a nm (27) We all these real values a n and Va n. Hene from Equation (17) it follows that the diretivity patterns are symmetri around W 0 for those frequenies: f k 0 : P FX ke f k f s ) 2 : P FX ke N a n Y[Z 1 irle (28) n/ 1 N n/ 1 Va n e j. f s n' NH 1 2 Y[Z d sinikj > P F ke :F > > P F ke N :F > (29) Koninklijke Philips Eletronis N.V

18 _ d 2000/002 Company Restrited This an ause problems if the frequeny independent target funtion is strongly asymmetri. The most pratial solution to overome this diffiulty in the optimisation problem is turning the whole array by an angle 0, whih symply orresponds to a rotation of the target funtion by an angle of 0, that is replaing all s by 0. This method results on the alulation of the filter oeffiients h ne m (n 2 and m 20) for the two sound soures or drivers (we will refer them from now on as being A and B). We performed some simulations in MatLab and they onverged to the optimum hoie of filter oeffiients for eah of the two frequeny bands (mid and high). Figure 8 shows the impulse responses of the resulted FIR filters for the target funtions from Equations (5) and (6). As observed in the figure, the impulses responses of the A and B filters are ross-symmetri, thus the magnitudes of the transfer funtions are the same, only the phase differs. This an be seen in Figure 9 whih shows the frequeny responses of the filters. This observation will be a key point, later on, so as to onsider an analog equivalene to the A and B filter responses. Figure 10 illustrates the phase differene response between the two A and B filters for both drivers in the mid and high frequeny ranges. Using the Goossens target funtion for the mid range filters, the phase differene \^]`_ A a B ranges from 159b (200 Hz) to 87 3b (2 khz), and for the high range filters it goes from 142 5b (2 khz) to 85 5b (12 khz). For the Aarts target funtion, the following values are obtained: 172 5b to 134b (mid range) and 166b to 126 5b (high range). The design of the individual arrays of drivers for the loudspeaker abinets to ahieve the sweet spot widening feature has been previously introdued in Setion 2.1. As we previously said, we deided to split the audible frequeny band in 3 frequeny ranges (see Figure 11). 1. Lower than frequeny f ] L 200 Hz: the left and right hannels signals are added (beomes a mono signal). No optimal diretivity pattern is required sine low frequenies are not loalisable. 2. Between frequenies f ] L 200 Hz and f ] H hannels are proessed separately. 3. Greater than frequeny f H ] 2 khz (mid range): Left and right 2 khz (high range): Idem as in the mid range. Thus, the pair of loudspeaker abinets (left and right) were equipped with a pair of drivers, whih were separated with a given distane to ahieve the two frequeny ranges (high and mid), so as to ahieve the desired diretivity pattern. The low frequenies were reprodued by means of a woofer (see Figure 3(b)). The drivers were finally fed by the alulated digital FIR filters in suh a way that both loudspeakers together radiate sound in a broad range of listening positions in aordane with the time/intensity trading experiments. Conerning the digital rossover filters, it an be pointed out that the asade of two idential Butterword filters [14] meet the required tolerable impedane to the power amplifier driving it in order to be frequeny independent. Thus, the resulting network behaves as an all-pass filter. For example, the asading of two seond-order Butterworth filters to obtain a fourth-order, 24-dB/otave slope rossover network (see Figure 12). 12 Koninklijke Philips Eletronis N.V. 2000

19 d Company Restrited 2000/002 (a) (b) Figure 8: Impulse responses for the digital FIR filters for N ] data. (b) Using Aarts data. 20. (a) Using Goossens Koninklijke Philips Eletronis N.V

20 d 2000/002 Company Restrited (a) Figure 9: Frequeny responses for the digital FIR filters for N ] data. (b) Using Aarts data. (b) 20. (a) Using Goossens (a) Figure 10: Phase differenes for the digital FIR filters for N ] data. (b) Using Aarts data. (b) 20. (a) Using Goossens 14 Koninklijke Philips Eletronis N.V. 2000

21 d Company Restrited 2000/002 H A D/A LS HA f H Highpass H B D/A LS HB Audio Channel M A D/A LS MA f L f H Bandpass M B D/A LS MB Crossover Filters Left + Right Channels f L Lowpass D/A LS W Figure 11: A shemati blok diagram for the proessing of the audio hannels (left and right). The frequeny range splitting is ahieved by means of digital rossover filters. Filters H A and H B for the high frequeny range and filters M A and M B for the mid frequeny range are obtained by the proposed method. These filters drive the 2 different arrays of loudspeakers, L S HA, L S HB and L S MA, L S MA. Low frequenies of the audio hannels (left and right) are reprodued by a woofer. Figure 12: Frequeny response of the optimum 4th-order Butterword rossover filters. Koninklijke Philips Eletronis N.V

22 d 2000/002 Company Restrited 5.2 Equivalent analog filtering for PI-stereo Goossens [7] also introdued a very easy and simple equivalent analog implementation whih realises the adequate (frequeny dependent) phase rotation for PI-stereo. The system onsisted basially on the following: first, the signal passes through a ommon lowpass filter for magnitude orretion, and then the neessary phase orretions are done for eah driver separately by appropriate allpass filters (see Figure 13). The most important onditions for these filters are twofold: (i) orret slope and -3 db point for the LP filter, and (ii) orret phase differene for all frequenies for the two AP filters. Figure 14 shows the analog PI-stereo iruit for the two drivers of loudspeakers previously desribed. Audio Signal LowPass AllPass A AllPass B Magnitude Corretion Phase Corretion Drivers Figure 13: Blok shema of the analog implementation for PI-stereo. Figure 14: Analog PI-stereo iruit for 2 drivers. 16 Koninklijke Philips Eletronis N.V. 2000

23 d s ] k ] ] ] ] Company Restrited 2000/ Lowpass filtering We now derive the transfer funtion for the lowpass filter shown in Figure 14. Therefore, in the Laplae domain, we have the following: I in e sf ] V in e sf a Ug R 1 ] I out e sf ] Ug a V out e sf R 2hih 1j sc 1 with Working out this formula, we finally obtain the next transfer funtion: H e sf ] V out m sn V inm sn Ug Ul 0 (30) ] g R 2 R 1 1 1l R 2 C 1 s (31) The gain of the system is h H e 0f h ] DC a R 2 j R 1 and the phase is _ e Hf The utoff frequeny for the lowpass filter is then f ] b 1j e 2o R 2 C 1 f. tan g 1 e fj f b f. If we take as values for the mid range frequenies: R 1 =3.3 kp, R 2 =10 kp and C 1 =33 nf; and for the high range frequenies: R 1 =12 kp, R 2 =22 kp and C 1 =1.8 nf, we then obtain the following utoff frequeny values for the lowpass setions: f b =482 Hz (mid) and f b =4020 Hz (high). Figure 15 shows the frequeny response of the lowpass setions for the high and mid range frequenies. Figure 15: Frequeny response of the lowpass filters Allpass filtering Now, we derive the transfer funtion for the allpass filters (phase shifters) using equal resitors (R ] R ] 3 R ] 4 R 5 ): and Ug I in e sf ] e a V in sf Ug R ] V in e sfrq V out e sf 2 Ul I out e sf ] V in e sf 1 q s RC 2 k ] a e Ug V out sf R with Ug (32) Ul (33) Koninklijke Philips Eletronis N.V

24 _ ] d ] 2000/002 Company Restrited From these formulae, we an easily obtain as a transfer funtion the following: H e sf ] V out m sn V inm sn ] 1g RC 2 s 1l RC 2 s (34) The gain of the system is h H e 0f h ] DC 1 and the phase is shifted by: e Hf tan g 1 ena fj f b f a tan g 1 e fj f b f a 2 tan g 1 e fj f b f (35) Figure 16: Frequeny responses for the allpass filters. Figure 17: Frequeny responses for the whole analog PI-stereo iruit. These results an also be applied to the other allpass setion whih drives the seond loudspeaker. The differene between the two filters is symply the value of the apaitors C 2 and C 3. For the allpass filters, the utoff frequenies depend on the values of the apaitors for both drivers, and they are given by f ] 1 1j e 2o RC 2 f and f ] 2 1j e 2o RC 3 f. Therefore, in order to have a orret phase differene for all required frequenies for both allpass setions, a hoie for the values of the frequenies f 1 and f 2 must be made. If we take as a values for the mid range frequenies: R=10 kp, C 2 =1 t F and C 3 =3.3 nf; and 18 Koninklijke Philips Eletronis N.V. 2000

25 d k _ y Company Restrited 2000/002 for the high range frequenies: R=10 kp, C 2 =820 nf and C 3 = 390 pf, we then obtain the following utoff frequeny values for the allpass setions: f 1 =16 Hz, f 2 =4.82 khz (mid) and f 1 =19.4 Hz, f 2 =40.8 khz (high). Figure 16 illustrates the frequeny responses of the allpass setions for the high and mid range frequenies. Sine the phase differene from the lowpass filter is the same for both drivers A and B, we only need to math the phase differene \ from the allpass filters: \u]v_ ] B a 2 tan g 1 e fj f 1 f"q 2 tan g 1 e fj f 2 f (36) A a The frequeny responses of the whole system (LP+AP setions) an be observed on Figure 17. From this figure it an be seen that for the mid range frequenies, \ ranges from around 165b (200 Hz) to 132b (2 khz). For the high range frequenies, \ ranges from 173b (2 khz) to 136b (12 khz) Theoretial study for analog phase shifting Linear array of two simple soures We next present the theoretial investigation of the nature of the sound field produed when a simple seondary point monopole soure is introdued in order to ontrol the radiation of an existing primary point monopole soure, both soures radiating sound at the same single frequeny [15, 16]. Let us onsider a simple example of an aousti doublet whih onsists of two simple soures of equal strengths, separated by a distane d, vibrating with the same frequeny (Figure 18). The diretivity pattern of this array of sound soures depends on the distane between the two soures and the phase between them. pz r{kxs d sinx x 2 r 1 1 dw 2 dw 2 r 2 r 2 Figure 18: Geometry used for two simple soures separated by a distane d. It is a simple matter to apply the priniple of linear superposition to the sound fields generated by two point simple soures having omplex strength Q s. The omplex pressure produed by the interferene of these two sound fields is given by pe r k } tf ] j~r 0 4o r 1 Q s e j m+ tg kr 1 n q j~r 0 4o r 2 Q s e j m& tg kr 2 n (37) Koninklijke Philips Eletronis N.V

26 k Œ Œ d 2000/002 Company Restrited where r 1 and r 2 are the radial distanes from the soure point to the observation point from the two soures, respetively, and } is the angle defined in Figure 18. If we onsider the arrangement of soures in the figure, and restrit our attention to observation positions in the far field, where the radial distane r is muh greater than both the wavelength of the sound radiated and the distane d separating the two soures, then we an make some useful approximations for the distanes r 1 and r 2. As illustrated in Figure 18, we an write d r 1 r q sin d 2 and r 2 r sin a 2. These approximations an be made for the values of r 1 and r 2 appearing in the omplex exponential terms in Equation (37). These terms govern the relative phase of the pressure flutuations produed by the two soures. The values of r 1 and r 2 appearing in the denominator in Equation (37) affet only the amplitude of the pressure in the far field, and as r ƒ we an make the further approximation r r 1 r 2. Two simple soures with a phase shift One desribed the allpass filtering, we an now apply it to the problem of simple sound soures. We onsider the system of two simple sound soures with a phase shift between them as shown in Figure 19. Looking at the Equation (36) it an be onfirmed that the phase differene \ between the two drivers A and B is \u]v A a B ] 2[a artaneo A~ f q artaneo B~ f ] (38) where A=RC 2 and B=RC 3 are the time onstants for the allpass filters. Thus, the omplex pressure produed by the interferene of these two sounds fields with phase shift is given by pe r k } tf ] j -ˆ 0 4 r 1 Q s e j m& tg kr 1 n q j -ˆ 0 4 r 2 Q s e j m+ tg kr 2 l Š n (39) P b rž r 1 Point Soure H A P A r 2 d r H B B d 2 sin Figure 19: Two simple soures with a phase shift between them. Considering d very small (in our system, the distane d is 0.08 m (mid) and m (high)), we an approximate r 1 and r 2 to r in the denominator, and to r ] 1 r q 20 d sin 2 and Koninklijke Philips Eletronis N.V. 2000

27 d k k ] k k k k k Company Restrited 2000/002 r 2 ] r a d sin 2 in the omplex exponential terms. Thus, Equation (39) beomes: pe r k } Considering 8] beomes: tf ] j~r 0 4o r Q se j m+ tg krg k d sin 2 n q k d sin 2 and p s e r k } pe r k } tf tf p s e r k } j~r 0 4o r Q se j m+ tg krl k d sin 2 l"š n (40) ] j -ˆ 0 4 r Q se j m& tg krn, therefore the last equation tf e j q e j m g l Š n (41) where the omplex exponential terms give the diretivity response of the this array of sound soures so as to orient the main lobe in the diretion of the given phase shift. Now, let us normalise Equation (41) in order to remove the onstants using a simple soure whih is twie the original soure length: p re f e r k } tf ] j~" 0 4o r 2Q se j m& tg krn (42) Therefore, we get the following equation: pm r tn p ref m r tn ] 1 2 e j q e j m g l"š n (43) and the final sound pressure level (S P L) in dbs is given by S P L ] 20 log p } e r tf p k } ref e r tf k In onlusion, analysing Equation (43) for some speifi key values we observe the following results: (44) š The S P L on the main lobe ( } ] 0) for no phase shift ( \ ] 0) œ[ pj p ] re f thus S P L ] 0. š At the side of the loudspeaker ( } ] ožj 2) when there is no phase shift ( \4] the frequeny of sound is suh that d ]vÿ j 2 œ pj p ] re f 0. 1 and 0) and Figure 20 shows simulations of the theoretial polar plots of Equation (43) for different frequenies (unlike the optimal target funtion shown in Setion 3.1, the funtion in Equation (43) is frequeny dependent). The frequeny range onsidered goes from 200 Hz - 10 khz. Note here that all polar plots have been normalised at 30b whih is the enter of the onsidered working region (0b a 60b ) for the loudspeakers for the PI-stereo setup as explained in [8]. Later on, we will ompare these theoretial polar plots with the optimal target funtions presented in Setion 3.1. Koninklijke Philips Eletronis N.V

28 d 2000/002 Company Restrited Figure 20: Theoretial diretivity pattern plots for different frequenies. 22 Koninklijke Philips Eletronis N.V. 2000

29 d ] Company Restrited 2000/ Digital IIR implemetation from the analog system Let us now onsider a digital implementation for the analog PI-stereo implementation using the well known digital filter design tehnique: bilinear transformation. It is well known that the transfer funtion H e zf of a digital IIR filter in the z-plane has the form: H e zf ] b 0 q b 1 z g 1 qv - - Uq b n z g n 1 q a 1 z g 1 q - - Aq a n z g n (45) whih is equivalent to the standard differene equation: ye kf n b i ue k a i 0 if a n a i ye k a if (46) i 1 As disussed previously, we an basially find two families of digital filters: the finite impulse response, or FIR filters; and the infinite impulse response filters, or IIR filters. The great distintion between IIR filters and FIR filters is that in the IIR filters the poles in the system H e zf are outside the origine of the z-plane. In ertain respets, this gives a diret orrespondene with onventional analog filters, whose design also starts from a desription in terms of poles and zeros. We know that most of the IIR approximation shemes are based on transforming the design of an analog filter by mappping the omplex s-plane into the z-plane. A number of approximation methods are formulated as multi-parameter optimisation problems and use a omputer to solve the resulting equations. As mentioned before, the most widely used method of designing disrete filters, starting from analog filters, is known as the bilinear transformation. In this transformation we map the s-plane on to the z-plane. Here, we use the substitution: s ] 2 1 a z g 1 T 1 q z g (47) 1 and hene: z ] 2 q st 2 a st Therefore, the transfer funtions of the 2nd-order IIR filters that we alulated have the form: H e zf ] b 0 l b 1 z 1l b 2 z 2 1l a 1 z 1l a 2 z 2 (49) (48) N MID A MID B HIGH A HIGH B b b b a a Table 2: Digital IIR filter oeffiients. Koninklijke Philips Eletronis N.V

30 š d 2000/002 Company Restrited Figure 21: Frequeny responses for the IIR filters for the PI-stereo system. In Table 2, we present the IIR filter oeffiients for both drivers A and B for the two frequeny ranges (mid and high). Figure 21 shows the frequeny responses of the IIR filters when using a sampling frequeny of 44.1 khz. It an be seen that the behaviour of the phase response is very similar to that one of the analog implementation, whih was one of the important requirements. 5.4 Digital FIR implementation from the measured transfer funtions In this setion we present the frequeny behaviour for the different drivers of the loudspeaker arrays (M A M B H A and H B) for the analog PI-stereo implementation. It was thought that it ould k be a k nie idea to find some equivalent digital transfer funtions to these measured analog transfer funtions. Therefore, four digital transfer funtions (that is, digital FIR filters with N ] 300 oeffiients) for these drivers were omputed and estimated using MatLab methods. The results are shown in Figure 22, where it an be observed how the measured and estimated frequeny responses losely approximate in the required frequeny bands. 6 Experiments and measured diretivity patterns One of the data haraterising a loudspeaker system is the diretional response whih desribes the relation between frequeny, radiation angle and sound level. The most usual ways to measure and plot diretional harateristis are: 24 Measure and plot frequeny spetra at stated angles (frequeny harateristi). The sound pressure level as funtion of the frequeny with the radiation angle as fixed parameter is plotted on a frequeny form. Koninklijke Philips Eletronis N.V. 2000

31 d š Company Restrited 2000/002 Figure 22: Measured and estimated frequeny responses for the digital FIR implementation. The frequeny response for the 4 drivers (MA, MB, HA and HB) was measured and then some equivalent digital FIR filters were estimated. Measure and plot polar diagrams at stated frequenies (polar diagram). The sound pressure level as funtion of the raditation-angle with the frequeny (band) as fixed parameter is plotted on a polar form. In our setup, we used the first method for onduting our measurements. It is important to point out here that only the analog PI-stereo implementation (thus, the analog filters) was tested and measured. The measured diretivity patterns for the loudspeakers using the digital PI-stereo implementations (thus, the digital filters) will be presented in a future report. With this method, frequeny responses were measured at the stated angles. The frequeny response measurements were done with a pseudo random white noise generator and with the help of a omputer (PC) ontrolled by a digital frequeny analyser. The spetrum was measured fast and the result was stored in a omputer. The omputer also ontrolled the turntable so that the measurement ould be repeated at suessive angles automatially. This setup gave us the possibility to display and plot the results afterwards using one of the different measuring methods previously mentioned. The measurement setup used for the experiments is shown in Figure 23. The loudspeaker to be measured was plaed on a turntable faing a mirophone (1/ free field B&K). Koninklijke Philips Eletronis N.V

32 d 2000/002 Company Restrited Loudspeaker Box Mirophone 1.15m Turntable 2.25 m 1.72m Mesh Plane 2.6 m Figure 23: Setup for the loudspeaker diretivity pattern measurements. Sine the devie to be measured was a loudspeaker, the signal of a pseudo random white noise generator, with a noise pattern repetition time of 2 se., was amplified and fed to the devie. The mirophone signal went to the digital frequeny analyser (HP 3562A) whih was ontrolled by the omputer. The turntable was started and the frequeny analyser started measuring. The omputer also ontrolled the turntable. Spetrum measurements were done at every 5b so that a total of 72 measurements were realised in one revolution of the turntable. After eah measurement, the frequeny response from 25 Hz to 20 khz was stored in the omputer. It is important to relate the pressure on the aousti axis in the far field to the available aousti power. The usual measure of the output of a soure is the sound pressure level SPL whih expresses the axial response of the soure in dbs. For the measured diretivity pattern plots a right response was onsidered, that is the main lobe is on the right-hand side and the minor lobe is on the left-hand side. Ideally the diretivity pattern plot for the left loudspeaker box will be the exat mirror image of the polar plot for the right loudspeaker box. To ahieve this pratially, the only differene between the left and right loudspeaker boxes is that the signals are sent to opposite drivers in the boxes. Another important onsideration here is that when listening to PI-stereo the loudspeaker boxes are fae on, not at 30b pointing inwards to the listener as in normal stereo. To ompensate for this, the frequeny responses of the loudspeaker boxes were taken at 30b lokwise for the left and 30b anti-lokwise for the right box so that the frequeny response in the middle of the working region was onsidered as opposed to the response at the edge. We have to keep in mind here that all polar plots were normalised at -30b (at the entre of the working region for the right loudspeaker). By normalising, we mean that the polar plots are drawn relative to one point, in our ase, the sound level at -30b. This an give misleading results as all the levels in all the frequeny bands at -30b are plotted at the same level, even though they may not atually be at the same level. Normalising the plots has to do with the response of the PI-stereo amplifier one the signal has been equalised over the frequeny range [8]. 26 Koninklijke Philips Eletronis N.V. 2000

33 d Company Restrited 2000/002 Figure 24: Measured sound pressure level at 0 degrees for the PI-stereo setup. In [8], it was widely demonstrated that the effet of the analog PI-stereo system is dependant on two apaitors (C a and C b ) for the two drivers (A and B) in the allpass setions. It was observed that neither raising or lowering the value of C a improves the orrelation between the theoretial and the optimal target polar plots. From the omparisons, it was also mentioned that by hoosing the values for C a and C b, a trade was needed between the level required in the region 0b a 30b with the level required in the region 30b a 60b. It was observed that by hanging the values of the apaitors, from those used at the present time, the polar plots do not beome a better math with the optimal target funtion. Figure 24 shows the mesured (in the anehoi room) axial sound pressure level urve for the right loudspeaker box (Nb 4). It an be observed an almost maximally flat response in the interested frequeny band (from 200 Hz - 12 khz) for the loudspeaker. The measured diretivity radiated pattern plots from 200 Hz up to 10 khz for the right box are shown in Figure 25. It is important to note that the measured diretivity polar plots are very similar in the working region (0b a 60b ) to those measured and displayed in [7, 8]. 6.1 Comparisons between the measured and the theoretial diretivity patterns In this setion we ompare the measured diretivity patterns, obtained by using the analog PI-stereo implementation, with the orresponding theoretial diretivity patterns presented in Setion Two approahes, for omparison, are possible: š Theoretial diretivity patterns onsidering two point soures (Figure 20). š Optimal target funtions as explained in Setion 3.1 (Figure 6). From the omparison between the theoretial diretivity patterns shown in Figure 20 and the measured diretivity patterns shown in Figure 25, it an be seen that the polar plots Koninklijke Philips Eletronis N.V

34 d 2000/002 Company Restrited Figure 25: Measured diretivity pattern plots for the analog PI-stereo amplifier. 28 Koninklijke Philips Eletronis N.V. 2000

35 d Company Restrited 2000/002 are nearly idential in the working region. The measured polar plots for frequeny ranges from 200 Hz up to 2500 Hz orresponds pretty well with the theoretial polar plots, however for high frequeny ranges, the differenes in level for the region between 0b a 30b beome bigger (up to 6 db). In onlusion, it an be said that the theoretial polar plots give a lose representation of the measured polar plots for frequenies between 200 Hz Hz, but not as good representation at higher frequenies (from frequenies 3200 Hz Hz). From the omparison between the optimal target funtion (in here, only for the one given by Goossens in Figure 6) and the measured polar plots, it an be seen that they have responses whih are lose to the target one in the region between 0b a 40b, but as the frequeny inreases, the main lobe beomes more rounded in this region and less like the optimal target funtion. It was also observed that the polar plots for the high frequeny range (2 khz - 10 khz) was seen to be more similar than the polar plots for the mid frequeny range (200 Hz - 2 khz). From the omparison between the optimal target funtion and the theoretial diretivity patterns, it an be seen that they are very similar in the region between 0b a 45b for almost all frequeny ranges, and in the region 45b a 60b the differenes between the polar plots beome bigger rising up to approximately 5 db. It was also observed that the theoretial polar plots were more similar to the optimal target plot than the measured polar plots. 7 Conlusions and future work Two reprodution hannels are the minimal ondition for stereo sound reprodution. The ideal position of the listener is alled the sweet spot. Intensity stereophony provides a stable and well defined image position for a listener at the sweet spot. Imaging with intensity stereophony an be understood by studying the loal wave field near the sweet spot. It has been shown that stereophony annot fulfill several demands: imaging is restrited to an extremely narrow listening area. Outside this area, spatial and temporal distortion of the sound image ours. In this report, we have desribed and developed a new stereo sound reprodution system that offers a natural high quality sound image in a large listening area (sweet spot widening). Digital or analog filtering tehniques have been applied to individual drivers of linear loudspeaker arrays in order to obtain a frequeny independent diretivity pattern having some optimal shape. This optimal shape was adapted to some time/intensity trading experiments for enlarging the sweet spot area. Listening experiments have shown that the PI-stereo system worked as predited by the theoretial models, although at higher frequenies there were some disrepanies due to the simpliity of the point soure model. It was also demonstrated the orret entral sound loalisation for voies and other effets for a number of listening positions. The degradation of the position independent stereo image for lateral positions was aeptable and was observed to be muh more better than normal stereo. The stereo sound sensation, in partiular the plaement of entral voies, was listening position independent within Koninklijke Philips Eletronis N.V