Sound reproduction systems

Abstract

A sound reproduction system includes an electro-acoustic transducer and a transducer driver for driving the electro-acoustic transducer. The transducer drive includes a filter which is configured to reproduce at a listener's location an approximation to the local sound field that would be present at the listener's ears in recording space, taking into account the characteristics and intended position of the electro-acoustic transducer relative to the listener's ears. The electro-acoustic transducer includes a first sound emitter which provides an intermediate sound emission channel, and second and third sound emitters providing respective left and right sound emission channels. The first sound emitter is located intermediate of second and third sound emitters. Higher frequencies from at least one of the second and third sound emitters are transmitted closer to the first sound emitter while lower frequencies are transmitted away from the first sound emitter.

Claims

1. A sound reproduction system comprising: an electro-acoustic transducer; and a transducer drive for driving the electro-acoustic transducer in response to an input sound recording, the transducer drive comprising a filter which is configured to reproduce at a listener location an approximation to the local sound field that would be present at a listener's ears in recording space, taking into account characteristics and an intended position of the electro-acoustic transducer relative to the ears of the listener, the electro-acoustic transducer comprising a central sound emitter which provides a central sound emission channel, a left sound emitter which provides a left sound emission channel and a right sound emitter which provides a right sound emission channel, the central sound emitter being located intermediate of left sound emitter and the right sound emitter, the central sound emitter, the left sound emitter and the right sound emitter each arranged to emit a range of frequencies, and both of the left sound emitter and the right sound emitter being such that different frequencies of the range are emitted from different respective azimuthal positions in a frequency distributed arrangement wherein predominantly higher frequencies of the range are transmitted closer to the central sound emitter and predominantly lower frequencies of the range are transmitted away from the central sound emitter, and the central sound emitter arranged to emit said range of frequencies emitted by one or both of the left sound emission channel and the right sound emission channel from substantially a single azimuthal location, as opposed to the frequency distributed arrangement of both of said left sound emission channel and said right sound emission channel.

2. The sound reproduction system as claimed in claim 1 in which at least one of the left sound emitter and the right sound emitter is positioned over a respective azimuthal span or region, and portions of at least one of (i) the left sound emitter and (ii) the right sound emitter having different azimuth directions emit predominantly different frequencies of sound, or predominantly different ranges of frequencies of sound.

3. The sound reproduction system as claimed in claim 1 in which at least one of the left sound emitter and the right sound emitter comprises a plurality of different positioned sound emitter devices, and in use each sound emitter device emitting a respective predominant frequency or a predominant range of frequencies of sound.

4. The sound reproduction system as claimed in claim 1 in which the central sound emitter is provided substantially central of the left sound emitter and the right sound emitter.

5. The sound reproduction system as claimed in claim 1 in which the central sound emitter is located rearwardly of the left sound emitter and the right sound emitter.

6. The sound reproduction system as claimed in claim 1 in which the central sound emitter provides a substantially non-variable frequency output with respect to the spatial extent of the central sound emitter, wherein the frequency substantially does not vary with azimuthal position and a range of frequencies is configured to be emitted therefrom.

7. The sound reproduction system as claimed in claim 1 in which one of the left sound emitter and the right sound emitter provides a substantially non-variable frequency output with respect to the spatial extent of the left sound emitter and the right sound emitter, wherein the frequency substantially does not vary with azimuthal position and a range of frequencies is configured to be emitted therefrom.

8. The sound reproduction system as claimed in claim 1 in which the head related transfer functions of a listener are taken into account.

9. The sound reproduction system as claimed in claim 1 in which the operational transducer frequency/azimuth range is determined by an equation of the form $f=\frac{{\mathrm{nc}}_0}{4r\sin ({}_L)}$ $\mathrm{or}$ $f=\frac{{\mathrm{nc}}_0}{4r\sin ({}_R)}$ where the transducer azimuth angle .sub.L, .sub.R, are the angles subtended at the listener by the central sound emitter, the left sound emitter and the right sound emitter, respectively, where
0<n<4, f: is the frequency, c.sub.0: is the speed of sound, and r: is the equivalent distance between the ears.

10. The sound reproduction system as claimed in claim 9 where 0<n<3.9.

11. A sound reproduction system as claimed in claim 9 where 0<n<3.7.

12. The sound reproduction system as claimed in claim 9 where 0.1<n<3.9.

13. The sound reproduction system as claimed in claim 9 where 0.3<n<3.7.

14. The sound reproduction system as claimed in claim 1 in which the transducer drive comprises cross-over filters for distributing signals of the appropriate frequency range to the appropriate sets of sound emitters, the cross-over filters responding to the outputs of an inverse filter of said filter.

15. The sound reproduction system as claimed in claim 1 in which the transducer drive comprises cross-over filters for distributing signals of the appropriate frequency range to the appropriate sets of sound emitters, with an inverse filter of said filter being responsive to the outputs of the cross-over filters.

16. The sound reproduction system as claimed in claim 1, in which the filter may be configured to be a minimum norm solution of the inverse problem.

17. The sound reproduction system as claimed in claim 1, in which the filter is configured to be a pseudoinverse filter.

18. The sound reproduction system as claimed in claim 1, in which the filter is configured to comprise adaptive filters.

19. The sound reproduction system as claimed in claim 1 comprising sub-woofers for responding to very low audio frequencies.

20. The sound reproduction system as claimed in claim 1, in which the number of sound emitter devices for the central sound emitter, the left sound emitter, and the right sound emitter comprise a different number of sound emitter devices to each other.

21. The sound reproduction system as claimed in claim 1, in which the central sound emitter comprises a single sound emitter device without any cross-over filters.

22. The sound reproduction system as claimed in claim 1 comprising a conventional loudspeaker for reproducing sound in a conventional method.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Various embodiments of the invention will now be described, by way of example only, together with a more detailed presentation of prior art arrangements with reference to the accompanying drawings, which show:

(2) FIG. 1Block diagram for binaural reproduction over loudspeaker with system inversion,

(3) FIG. 2The geometry of a 2-source 2-receiver system under investigation,

(4) FIG. 3The definition of azimuth span,

(5) FIG. 4Norm and singular values of the inverse filter matrix H as a function of n. a) Logarithmic scale. b) Linear scale,

(6) FIG. 5Dynamic range loss due to system inversion,

(7) FIG. 6Condition number (C) as a function of n,

(8) FIG. 7Sound radiation by the control transducer pairs with reference to the receiver directions (0 dB and dB).

(9) FIG. 8Principle of the OSD system,

(10) FIG. 9Relationship between source span and frequency for different odd integer number n,

(11) FIG. 10Norm and singular values of the inverse filter matrix H of OSD as a function of frequency,

(12) FIG. 11Sound radiation by the OSD transducer pairs with reference to the receiver directions (0 dB and dB).

(13) FIG. 12Singular values of the inverse filter matrix H as a function of n. Optimal point for 2 channel OSD and 3 channel OSD

(14) FIG. 13Principle of the 3 channel OSD system,

(15) FIG. 14Relationship between source span and frequency for different integer number of n=2, 6, 10, . . . .

(16) FIG. 15Block diagram for binaural reproduction over 3 loudspeakers with system inversion,

(17) FIG. 16The geometry of a 3-source 2-receiver system under investigation,

(18) FIG. 17Norm and singular values of the inverse filter matrix H of the 3 channel case as a function of n. a) Logarithmic scale. b) Linear scale,

(19) FIG. 18Norm and singular values of the inverse filter matrix H of the 3 channel case as a function of n when the sensitivity of the centre channel transducer is increased by a factor of 3 dB. a) Logarithmic scale. b) Linear scale,

(20) FIG. 19Norm and singular values of the inverse filter matrix H of the 3 channel OSD as a function of frequency

(21) FIG. 20Variable frequency/position transducer,

(22) FIG. 21Discretised variable frequency/position transducer,

(23) FIG. 22An example of frequency/azimuth region and discretisation,

(24) FIG. 23Condition number (C) of the 3 channel case as a function of n,

(25) FIG. 24Condition number (C) of the 3 channel case as a function of n, when the sensitivity of the centre channel transducer is increased by a factor of 3 dB, and

(26) FIGS. 25 to 33 show schematic representations of various sound reproduction systems embodying the three channel OSD arrangement.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

(27) The principle of binaural reproduction over loudspeaker is described below and is illustrated in FIG. 1. The objective of the system is to feed to each ear of the listener independently the binaural signals that contain auditory spatial information as well as the signals associated with sources in a virtual sound environment. However, when loudspeakers are used for this purpose, each loudspeaker feeds its signal to both ears. There is a matrix of acoustic paths between the loudspeakers and the listener's ears, and this can be expressed as a matrix of transfer functions (plant matrix). Independent control of two signals (such as the binaural sound signals) at two receivers (such as the ears of a listener) can be achieved with two electro-acoustic transducers (such as loudspeakers), by filtering the input signals to the transducers with the inverse of the transfer function matrix of the plant. This process is also referred to as system inversion or cross-talk cancellation. The signals and transfer functions involved are defined as follows. Two monopole transducers produce source strengths (volume accelerations) defined by the elements of the complex vector v=[.sub.1(j).sub.2(j)].sup.T. The resulting acoustic pressure signals are given by the elements of the vector w=[w.sub.1(j)w.sub.2(j)].sup.T. This is given by
w=Cv(1)
where C is the plant matrix (a matrix of transfer functions between sources and receivers). The two signals to be synthesised at the receivers are defined by the elements of the complex vector d=[d.sub.1(j)d.sub.2(j)].sup.T. In the case of audio applications, these signals are usually the signals that would produce a desired virtual auditory sensation when fed to the two ears independently. They can be obtained, for example, by recording sound source signals u with a recording head (eg a dummy head) or by filtering the signals u by matrix of synthesised binaural filters A.

(28) Therefore, a filter matrix H which contains inverse filters is introduced (the inverse filter matrix) so that v=Hd where

(29) $\begin{array}{cc}H=\left[\begin{array}{cc}{H}_{11}\left(j\right)& H_{12}\left(j\right)\\ H_{21}\left(j\right)& H_{22}\left(j\right)\end{array}\right]& \left(2\right)\end{array}$
and thus
w=CHd(3)

(30) The inverse filter matrix H can be designed so that the vector w is a good approximation to the vector d with a certain delay [14][15]. When the independent control at two receivers is perfect, CH becomes the identity matrix I. The inverse filter matrix H can also be designed to be a pseudoinverse of the plant matrix C. The filter matrix H can also consist of adaptive filters.

(31) However, the system inversion involved gives rise to a number of problems such as, for example, loss of dynamic range and sensitivity to errors. A simple case involving the control of two monopole receivers with two monopole transducers (sources) under free field conditions is first considered here. The fundamental problems with regard to system inversion can be illustrated in this simple case. The geometry is illustrated in FIG. 2. Note that is a difference of azimuth (azimuth span), not the actual span (FIG. 3).

(32) In the free field case, the plant transfer function matrix can be modelled as

(33) $\begin{array}{cc}C=\frac{{}_0}{4}\left[\begin{array}{cc}{e}^{-jk{l}_1}/l_1& e^{-j{\mathrm{kl}}_2}/l_2\\ e^{-j{\mathrm{kl}}_2}/l_2& e^{-j{\mathrm{kl}}_1}/l_1\end{array}\right]& \left(4\right)\end{array}$
where an e.sup.jt time dependence is assumed with k=/c.sub.0, and where .sub.0 and c.sub.0 are the density and sound speed.

(34) Now consider the case

(35) $\begin{array}{cc}d=\frac{{}_0{e}^{-j{\mathrm{kl}}_1}}{4l_1}\left[\begin{array}{c}{D}_1\left(j\right)\\ D_2\left(j\right)\end{array}\right]& \left(5\right)\end{array}$
i.e., the desired signals are the acoustic pressure signals which would have been produced by the closer sound source and whose values are either D.sub.1(j) or D.sub.2(j) without disturbance due to the other source (cross-talk). This way the effect of system inversion can be separated from the effects of spherical attenuation due to propagation in space as well as ensuring a causal solution. The elements of H can be obtained from the exact inverse of C, and the magnitude of the elements of H (|H.sub.mn(j)|) show the necessary amplification of the desired signals produced by each inverse filter in H. The maximum amplification of the source strengths can be found from the 2-norm of H (denoted as H) which is the largest of the singular values of H, where these singular values are denoted by .sub.o and .sub.i [13]. Thus
H=max(.sub.o,.sub.i)(6)
.sub.o corresponds to the amplification factor of the out-of-phase component of the desired signals and .sub.i corresponds to the amplification factor of the in-phase component of the desired signals. Plots of .sub.o, .sub.i, and H with respect to frequency are illustrated in FIG. 4. As seen in FIG. 4, H changes periodically and has peaks where k and satisfy the following relationship with even values of the integer number n.

(36) $\begin{array}{cc}kr\sin =\frac{n}{2}& \left(7\right)\end{array}$

(37) The singular value .sub.o has peaks at n=0, 4, 8, . . . where the system has difficulty in reproducing the out-of-phase component of the desired signals and .sub.i has peaks at n=2, 6, 10, . . . where the system has difficulty in reproducing the in-phase component. Around these frequencies, sound signals from control sources interfere destructively with each other, leaving little response left at the ears of the listener. In other words, the signals cancel each other. Therefore, the solution for the inverse, i.e., the amplification required to produce the desired sound pressure at each receiver, becomes substantially large.

FUNDAMENTAL PROBLEMS OF PRIOR ART SYSTEMS BEFORE OPTIMAL SOURCE DISTRIBUTION

3.1 Loss of Dynamic Range

(38) In practice, since the maximum source output is given by H.sub.max, this must be within the range of the system in order to avoid clipping of the signals. The required amplification results directly in the loss of dynamic range illustrated in output levels and dynamic range of the systems are the same. Where H is large, each transducer is emitting very large sound most of which is cancelled by the sound from the other transducers. As a result, the levels of synthesised binaural signals at the listener's ears are significantly smaller than that those without cancellation. The given dynamic range is distributed into the system inversion and the remaining dynamic range that is to be used by the binaural auditory space synthesis, and also most importantly, by the sound source signal itself. Thus the signal to noise ratio of the signals w becomes low. Since the transducers are working much harder than they would normally to produce usual sound levels at the ears, non-linear distortion becomes more significant and is often audible. For the same reason, fatigue of the transducers is more severe. Conventional driver units are not designed to be used in this manner and they can be easily destroyed by fatigue.

(39) Eq. (1) implies that the system inversion (which determines v and leads to the design of the filter matrix H) is very sensitive to small errors in the assumed plant C (which is often measured and thus small errors are inevitable) where the condition number of C, (C), is large. In addition, the reproduced signals w are less robust to small changes in the real plant matrix C, where (C) is large.

(40) The condition number of C is shown in FIG. 6. As seen in FIG. 6, (C) has peaks where Eq. (7) is satisfied with an even value of the integer number n. The frequencies which give peaks of (C) are consistent with those which give the peaks of H.

(41) The calculated inverse filter matrix H is likely to contain large errors due to small errors in the assumed plant matrix C and results in large errors in the reproduced signal w at the receiver. This is because such errors are magnified by the inverse filters but remain not being cancelled in the plant. Even if H does not contain any errors, the reproduction of the signals at the receiver is too sensitive to the small errors within the real plant matrix C to be useful.

(42) Such errors include individual differences of HRTFs, [16]-[18] and misalignment of the head and loudspeakers [19], approximation of filters and regularisation, where a small error is deliberately introduced to improve the condition of matrix to design practical filters [20]. These errors may seem small but it is far too large in practice where (C) is large.

(43) On the contrary, (C) is small around the frequencies where n is an odd integer number in Eq. (7). Around these frequencies, a practical and close to ideal inverse filter matrix H is easily obtained and the accurate reproduction of intended sound signal is possible.

(44) FIG. 7 shows an example (n2) of far field sound radiation by the control transducers with reference to the receiver directions. The horizontal axis is the inter-source axis and the receivers (ears) are close to the direction of the vertical axis. At frequencies where Eq. (7) is not satisfied with an odd value of the integer number n, as in this example, the sound radiation in directions other than receiver directions can be significantly larger (typically +30 dB40 dB) than those at the receiver directions (0 dB and dB). When the environment is not anechoic, as is normally the case, this obviously results in severe reflection. Reflections from surrounding objects (e.g., furniture, walls, floors, and ceilings) affect the control performance. Although the perceptual aspects of sound localization such as the precedence effect suggest that the performance of this kind of system will be retained to some extent [21], reflected sound with a much higher level than the controlled sound arriving directly at the listener's ears destroys the correct perception.

(45) In addition, the sound radiated in directions other than that of receiver has a peaky frequency response due to the response of inverse filter matrix H and normally results in severe coloration. This contributes to coloured reverberation and makes listening in any other location other than one optimal location impractical.

(46) Equation (7) can be rewritten in terms of the source azimuth span as

(47) $\begin{array}{cc}=2=2\arcsin \left(\frac{n}{2kr}\right)& \left(8\right)\end{array}$

(48) As seen from the analysis above, frequencies with the source span where n is an odd integer number in Eq. (8) give the best control performance as well as robustness.

(49) The Optimal Source Distribution (OSD) introduced the idea of a pair of conceptual monopole transducers whose span varies continuously as a function of frequency (FIG. 8) in order to satisfy the requirement for n to be an odd integer number in Eq. (8) (FIG. 9) at all frequencies (except at very low frequencies) [15]. This relationship is where .sub.i and .sub.o are balanced and the source span becomes smaller as frequency becomes higher. With this concept, the frequency response of the inverse filter becomes flat for all frequencies as shown in FIG. 10. Therefore, there is no dynamic range loss compared to the case without system inversion. This means the system has good signal to noise ratio and the advantage of reduced distortion or fatigue of transducers. The inverse filters have a flat frequency response so there is no coloration at any location in the listening room even outside the intended listening position. When the listener is far away from the intended listening position, the spatial information perceived may not be ideal. However, the spectrum of the sound signals is not changed by the inverse filters. Therefore, the listener can still enjoy the natural production of sound together with some remaining spatial aspects, especially the aspects for which the spectral information is important. As shown in FIG. 11, the sound radiation by the transducer pair in all directions is always smaller than those in the receiver directions, which is also smaller than the sound radiation by a single monopole transducer producing the same sound level at the ears. In contrast to FIG. 7, the system does not radiate excessive sound all around so it is also robust to reflections in a reverberant environment, and these small reflections do not have any coloration other than that caused by the reflecting materials. Note also that (C)=1 which is constant over all frequencies and which is the smallest possible value [13]. The error in calculating the inverse filter is small and the system has very good control over the reproduced signals. The system is also very robust to the changes in plant matrix.

(50) As discussed above, the two-channel OSD essentially uses the frequency span region where the two singular values, representing the in-phase and out-of-phase components of the binaural reproduction process, are balanced in order to overcome the fundamental problems of conventional binaural reproduction over loudspeakers. However, a system which aims to improve this further is proposed in what follows. For convenience, we refer to it as the three channel OSD system in contrast to the earlier OSD that will henceforth be referred as the two channel OSD.

(51) Now we try to make use of the lowest value (6 dB, at points B in FIG. 12) of each of the two singular values, rather than where two singular values are balanced at 3 dB (at points A in FIG. 12). When the azimuth span of two transducers becomes 0, the lowest value of the singular values .sub.i is given. In other words, there is one transducer in the median plane. (FIG. 13). This may be viewed as in effect the addition of a third transducer for the binaural reproduction over loudspeakers. When the third transducer is added around the median plane of the listener, we have found that this can relax the condition for the in-phase component significantly, since this should give in effect the lowest singular value over the entire frequency range. With specific reference to FIG. 13 there is provided a first transducer 10 which provides a central channel, a second transducer 11 which provides a left channel and a third transducer which provides a right channel. As is shown schematically in FIG. 13 each of second and third transducers extends over a particular azimuthal directions and at positions progressively closer to the first transducer 10 predominantly higher frequencies are emitted. So, at distal end portion 11b the lowest frequencies are predominantly emitted whereas at the proximal end portion 11a predominantly the highest frequencies are emitted. From the listener's perspective the first transducer 10 is positioned intermediate of the second transducer 11 and the third transducer 12.

(52) Since the condition for the in-phase component has now been relaxed, we now can use the optimal value (points B in FIG. 12) for the singular value .sub.o, the out-of-phase component, rather than compromised (balanced) point between .sub.o and .sub.i which is the optimal combination of the singular values for two channel OSD. The lowest value of .sub.o is given at n=2, 6, 10, . . . . Therefore, the three channel OSD makes use of one of the points B, and stretches the point over entire the frequency range above it by introducing the idea of conceptual monopole transducers whose position varies continuously as a function of frequency, satisfying the requirement for n=2, 6, 10, . . . in Eq. (8) (FIG. 14) at all frequencies except very low frequencies. This is in contrast to the two channel OSD in which one of points A (where n=1, 3, 5, . . . ) is stretched over the entire frequency range.

(53) In order to see the effect of this additional transducer, we consider the simple case again where monopole transducers are used for binaural reproduction as in section 2.2 but this time with another transducer added on the median plane. The block diagram and geometry are illustrated in FIG. 15 and FIG. 16. Eq. 4 becomes

(54) $\begin{array}{cc}C=\frac{{}_0}{4}\left[\begin{array}{ccc}{e}^{-jk{l}_1}/l_1& e^{-j{\mathrm{kl}}_3}/l_3& e^{-j{\mathrm{kl}}_2}/l_2\\ e^{-jk{l}_2}/l_2& e^{-{\mathrm{jkl}}_3}/l_3& e^{-j{\mathrm{kl}}_1}/l_1\end{array}\right]& \left(10\right)\end{array}$
where an e.sup.jt time dependence is assumed with k=c.sub.0, and where .sub.0 and c.sub.0 are the density and sound speed.

(55) Note that the system is under-determined in that there can be a number of choices of the inverse filter matrix which produces no error [22] [23]. Among them, the minimum norm solution would be the most straightforward choice as well as giving the best performance with regard to the fundamental problems described in Section 3.13.3. Therefore, the following examples use the minimum norm solution.

(56) The 2-norm of H (H) and the two singular values .sub.o and .sub.i with respect to frequency are illustrated in FIG. 17. Compared with FIG. 4, the peaks of the singular value .sub.i at n=2, 6, 10, . . . where the system has difficulty in reproducing the in-phase component, have almost disappeared in FIG. 17. The level difference of about 3 dB between the values of .sub.i and .sub.o at n=2, 6, 10, . . . is due to the fact that two transducers can work on reproducing the out-of-phase component of the binaural signal whereas there is only one transducer available for the in phase component.

(57) Having a third transducer for two point reproduction (i.e the mathematically under-determined case), the balance between the two singular values .sub.o and .sub.i can be changed independently by changing the relative sensitivity of the transducer of the centre channel with respect to those on the left and right. This is an important aspect which the three channel OSD possesses which in contrast the two channel OSD does not. If the sensitivity of the centre channel transducer is increased by the factor of {square root over (2)}, the two singular values .sub.o and .sub.i become equal to each other at n=2, 6, 10, . . . and that is shown in FIG. 18.

(58) The singular value .sub.i at n=0, 4, 8, . . . is always smaller than that of at n=2, 6, 10, . . . where all three transducers can contribute to the reproduction of in phase component. The 2-norm of H (H) and the two singular values .sub.o and .sub.i of the 3 channel OSD with respect to frequency are illustrated in FIG. 19.

(59) The three channel OSD requires, for the transmission of the left and right channels, monopole type transducers whose position varies substantially continuously as frequency varies, similar to the case with the two channel OSD. This may, for example, be realised by exciting a substantially triangular shaped plate whose width varies along its length. The requirement of such a transducer is that a certain frequency or a certain range of frequencies of vibration is excited most at a particular position having a certain width such that sound of that frequency is radiated mostly from that position (FIG. 20). The centre channel can either be a conventional monopole transducer which emits all the frequency components of the sound from one point. Alternatively the same type of transducer as the left and right channel can also be used to provide the centre channel as well.

(60) From Eq. (7), the range of source direction is given by the frequency range of interest as can be seen from FIG. 14. A smaller value of n gives a smaller source azimuth for the same frequency. Therefore, the smallest source azimuth .sub.h for the same high frequency limit is given by n=2 and this is about 4 to give control of the sound field at two positions separated by the distance between two ears (about 0.13 m for KEMAR dummy head) up to a frequency of 20 kHz.

(61) Eq. (7) can also be rewritten in terms of frequency as

(62) $\begin{array}{cc}f=\frac{{\mathrm{nc}}_0}{4r\sin }& \left(11\right)\end{array}$

(63) The smallest value of n gives the lowest frequency limit for a given source direction. Since sin 1,

(64) $\begin{array}{cc}f\frac{{\mathrm{nc}}_0}{4r}& \left(12\right)\end{array}$
ie, the physically maximum source azimuth of .sub.L=.sub.R=90 gives the low frequency limit, f.sub.1, associated with this principle. A smaller value of n gives a lower low frequency limit so the system given by n=2 is normally the most useful among those with n=2, 6, 10, . . . . The low frequency limit given by n=2 of a system designed for all average human is about f.sub.i=700 Hz, which is higher than that for two channel OSD where it is about 350 Hz. Below the low frequency limit of three channel OSD, the performance gradually approaches that of two channel OSD, becoming identical below the low frequency limit of two channel OSD.

(65) In FIG. 17 and FIG. 18, the slope of the singular values around the ideal frequency/azimuth line are a lot shallower, forming a U shaped valley rather than a V shaped valley in the case of two channels shown in FIG. 4. This means the three channel OSD is much more robust to errors than the two channel OSD.

(66) The fundamental behaviour is the same for the more realistic case where various other factors such as the Head Related Transfer Function come into effect as in the case with the two channel OSD.

(67) The discretisation of the Optimal Source Distribution can also be used for the three channel OSD in a similar way to the two channel case. In practice, whilst a monopole transducer whose position varies continuously as a function of frequency may not be easily available it is possible to realize a practical system based on the underlying principle by discretising the transducer span. With a given span, the frequency region where the amplification is relatively small and plant matrix C is well conditioned is relatively wide around the optimal frequency.

(68) Therefore, by allowing n to have some width, say (0<<2), a certain transducer span can nevertheless be allocated to cover a certain range of frequencies where control performance and robustness of the system is still reasonably good (FIG. 22). Consequently, it is possible to discretise the continuously varying transducer position into a finite number of discrete transducer positions, and at each position there is provided a transducer unit. With reference to FIG. 21 there is shown a possible realisation of discretised arrangement in which transducers 111, 112, 113 and 114 provide a left channel, transducers 120, 121, and 123 provide a right channel and transducers 100 and 101 provide an intermediate channel. Each of the transducers forming the left channel emit a predominant frequency, or a predominant frequency band, in respect of frequencies which increase the closer a particular transducer is to the transducer forming the intermediate channel. The transducers forming the right channel are arranged in similar fashion. As is evident from FIG. 21, implementation of an embodiment of the invention need not necessary require that equal numbers of transducer units are required for each of right and left channels.

(69) The difference of the slope around the ideal frequency/span relationship has advantages here again in many ways. For the same given tolerance width of n, the error will be much smaller than that in the two channel OSD. So the same level of discretisation gives a better approximation to the ideal case for the three channel OSD. For the same level of approximation, the discretisation can be coarser hence saving resources. The maximum width of n, which is the maximum allowance for , becomes twice that in the two channel OSD, i.e. 0<<2. In general, the performance of the discretised three channel OSD is much better due to the fact that the valley in FIG. 17 and FIG. 18 is U shaped rather than V shaped.

(70) The condition number for the case shown in FIG. 17 and FIG. 18 is plotted in FIG. 23 and FIG. 24 respectively. The condition number is smaller in FIG. 24 than in FIG. 23 around the ideal frequency/azimuth region. On the other hand, The case shown in FIG. 23 could have a smaller maximum condition number over the operational frequency/azimuth region when is larger than 1. These characteristics may be taken into consideration when the discretised three channel OSD is derived from them.

(71) Reference will now be made to FIGS. 25 to 32 which show various further realisations of sound reproduction systems embodying the three channel OSD arrangement.

(72) Turning initially to FIG. 25, this shows one way to realise the arrangement of FIG. 21, in which each transducer of each channel arrangement 200, 201 and 202 is connected to a respective cross-over filter of a respective cross-over filter arrangement 210, 211 and 212.

(73) FIG. 26 shows a variant embodiment of that shown in FIG. 25 in which the centre channel 200 is provided by a single full range transducer. Furthermore the left channel 202 is provided now with a reduced number of transducers, namely two transducers. It will be appreciated however that each of the left and right channel could include any number of transducers.

(74) FIG. 27 shows a three channel OSD arrangement in which an inverse filter, H.sub.h and H.sub.1 is provided for each band C.sub.h and C.sub.1. In this arrangement one of each of a high frequency transducer and a lower frequency transducer is provided for each of the left channel, the right channel and the central channel.

(75) FIG. 28 is a variant embodiment to that shown in FIG. 27 in which cross-over filtering is effected before inverse filtering is effected.

(76) FIG. 29 shows an arrangement which may be viewed as a combination of the three channel OSD and the known two channel OSD, resulting in the system having unequal numbers of channels for each frequency band.

(77) FIG. 30 is a variant of the arrangement of FIG. 29 in which cross-over filtering is effected before inverse filtering.

(78) FIG. 31 is an arrangement similar to that of FIG. 29 in which three high frequency transducers and two low frequency transducers are provided.

(79) FIG. 32 is a variant embodiment of that shown in FIG. 31 in which cross-over filtering is effected before inverse filtering.

(80) With reference to FIG. 33 there is shown yet a further embodiment in which the centre channel and the right channel transducers each emit the entire frequency range from substantially the same (respective) location. For the left channel however higher frequencies are emitted closer to the central channel transducer and lower frequencies are emitted further away from the central channel transducer. In a variant of this embodiment the transducer arrangement of the right channel is replaced by the transducer arrangement of the left channel of FIG. 33, and the transducer arrangement of the left channel is replaced by the transducer arrangement of the right channel of FIG. 33.

(81) A new binaural reproduction system has been described which overcomes the fundamental problems with system inversion by utilising three-channels of transducers with variable position with respect to frequency.

(82) This system can most easily be realised in practice by discretising the theoretical continuously variable transducer span which results in multi-way sound control system.

(83) The three channel OSD arrangement finds application in numerous ways and in particular in the field of home audio. A particularly advantageous implementation is in the context of the transducers of portable media devices, such as mobile telephones and portable gaming devices, and so enhances the listener's experience of sound emitted thereby. Some portable media devices (such as MP3 players) are capable of being interfaced with a separate speaker arrangement (sometimes known as a docking station). Such speaker arrangements would benefit from being adapted to implement the three channel OSD arrangement.

REFERENCES

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