Abstract:
A crosstalk canceler wherein different frequency bands are canceled at different locations so as to allow greater listener movement about the “sweet spot” while maintaining effective crosstalk cancellation. A spectrally smooth canceler equalization is used, reducing artifacts for listeners away from the sweet spot and further enlarging the sweet spot. Finally, the canceler equalization is adapted to either the anticipated or the actual crosscoherence among the input channels, producing a natural equalization regardless of the input.

Description:
BACKGROUND OF THE INVENTION 
     This invention pertains to audio signal processing, and specifically to a system and method for crosstalk cancellation. 
     There are a number of settings in which separate audio signals are prepared for the left and right ears of a listener. Such signals are referred to as binaural signals, and are distinct from stereo signals in that the left and right binaural channels are intended to be heard only by the respective left and right ears of the listener. 
     Binaural signals are typically used to convey spatial information about the sounds presented. It turns out that a sense of sound source location is created by subtle features imposed on the signals arriving at the left and right ears of the listener [5, 6, 7]. By separately processing left-ear and right-ear signals, as illustrated in FIG. 1, a sound source can be made to appear at any desired location in a listener&#39;s perceptual space. 
     Such synthetic spatial audio—commonly referred to as 3D audio—has application to video games, teleconferencing, and virtual environments, wherein each sound may be processed so as to appear to originate from its generating object. Another 3D audio application is placing “virtual” speakers about a listener, for instance in a standard home theater surround sound configuration as shown in FIG.  2 . Here, each of five surround signals  30 ,  40 ,  50 ,  60 ,  70  is processed according to its location  34 ,  44 ,  54 ,  64 ,  74  to form left-ear and right-ear. signals  32 ,  42 ,  52 ,  62 ,  72  and  33 ,  43 ,  53 ,  63 ,  73 , which are summed to form the left-ear and right-ear channels  35  and  36  of a binaural signal. Presenting the binaural signal to a listener over headphones gives the impression of a five-speaker surround system, though only the two binaural channels are used. 
     In all of these applications, headphones or similar transducers are often used to ensure that the left and right binaural-channels are delivered, respectively, to the left and right ears of the listener [5, pp. 217-220]. If the binaural signal were played through stereo speakers configured as shown in FIG. 4, each listener ear would hear both binaural channels. This mixing of the left and right binaural channels, called crosstalk, can significantly degrade the spatial cues in the binaural signal, diminishing the listening experience. 
     There are, however, situations such as in the case of an arcade game where the use of headphones or earphones is impractical, and it is desired to use stereo speakers to present binaural material. In [1], Atal and Schroeder presented a system called a crosstalk canceler for processing a binaural signal to develop a pair of speaker signals that would deliver the original binaural signal to a properly positioned listener. 
     The system relies on differences among the transfer functions between the two speakers and the two ears. The basic idea is to cancel the crosstalk appearing in the right ear from the left speaker by sending a negative filtered version of the left speaker signal out the right speaker. The filtering is such that the crosstalk from the left speaker and the canceling signal from the right speaker arrive at the right ear simultaneously as negative replicas of each other, and sum to zero. Left ear crosstalk from the right speaker is similarly eliminated. 
     The crosstalk canceler proposed in [1] can be very effective, but has several drawbacks which limit its usefulness. First, so that the cancellation signal exactly cancels the crosstalk signal, the listener must be carefully positioned at the so-called sweet spot. In addition, the transition between effective cancellation in the sweet spot and no cancellation out of the sweet spot is very abrupt, making it difficult for listeners to find the sweet spot. Consider a 5 kHz signal having a wavelength of about two inches. The listener only need move his head an inch closer to one speaker than the other to turn the perfect cancellation between the crosstalk and canceling signals into perfect reinforcement between the two. 
     In addition to restricting listener movement, the canceler [1] is sensitive to the shape of the listener&#39;s head and ears. To get effective cancellation, particularly at high frequencies, the canceling signal filter should be tailored to the listener. 
     The second drawback has to do with the timbre or equalization of the canceled signal as compared to that of the original binaural signal. Listeners in the sweet spot sometimes sense that the canceler output is lacking in low-frequency energy compared to the original binaural signal. Listeners away from the sweet spot complain of phase artifacts and a position sensitive equalization. (Note that the apparent equalization away from the sweet spot is important in some applications. For example, consider a television equipped with stereo speakers and virtual surround sound processing as shown in FIG.  3 . While the crosstalk canceler can deliver the virtual surround binaural signal to listener  80  in the sweet spot, the crosstalk canceler should not compromise the listening experience of those away from the sweet spot.) 
     To address the restrictions on listener movement, Cooper and Bauck in [2] proposed a crosstalk canceler which cancels only the low frequencies; the high-frequency portion of the binaural input is sent to the output unchanged. Many audio signals have their energy concentrated below a few kilohertz, so that canceling only those frequencies should not significantly diminish the cancellation effect. Because the wavelengths for the canceled portion of the binaural signal are relatively large, the listener has greater freedom of movement before perceiving a change in cancellation effectiveness. Essentially, the canceler trades a less effective cancellation in the sweet spot for a broader sweet spot. 
     In [3, 4] Cooper and Bauck present a canceler equalization based on the observation that each canceler has a set of so-called “null canceler” frequencies at which the canceling signal filter is orthogonal to—that is, ±90° out of phase from-the direct signal filter. The proposed equalization inverts the sum of the power in the direct and canceling filters at the null canceler frequencies. This equalization is an improvement over the one implied in [1] in that listeners away from the sweet spot hear few artifacts, and those in the sweet spot experience less of a timber change. However; for certain kinds of source material, a timbre change is still noticeable for listeners in and out of the sweet spot. 
     SUMMARY OF THE INVENTION 
     An embodiment of the present invention provides a crosstalk canceler allowing greater listener movement while maintaining effective cancellation, and having an equalization which leaves the input binaural signal uncolored. An embodiment of the present invention provides a canceler that is insensitive to listener head and ear acoustic properties. An embodiment of the present invention broadens the transition between effective cancellation in the sweet spot and no cancellation outside the sweet spot to help listeners find the sweet spot. An embodiment of the present invention develops a canceler that is relatively free of artifacts away from the sweet spot. An embodiment of the present invention adapts the equalization to the input signal so as to minimize timbre changes imposed by the canceler. 
     To provide greater listener freedom of movement, the basic idea is to cancel different frequency bands at different locations, rather than to cancel all frequency bands at the same location as is currently practiced. In this way, changes in listener position do not eliminate cancellation, but shift the part of the signal canceled. In addition, this widening of the sweet spot creates a smooth transition between regions of effective cancellation and no cancellation. 
     The expectation in canceling different frequency bands at different locations is that while the set of listener positions where some cancellation occurs is broader, the cancellation is everywhere less effective than at the sweet spot of a traditional canceler. That the sweet spot of the new canceler is larger than that of traditional cancelers was verified in listening tests using virtual surround sound, speaker spreader, and one-channel signals as the binaural input. Surprisingly, the inventive canceler was perceived to have nearly as effective cancellation in the sweet spot as the traditional canceler. 
     In analyzing the signal arriving at a listener&#39;s ears from a traditional canceler, it was discovered that unless the listener is precisely positioned, the signal arrives with a timbre change compared to the original binaural signal, irrespective of the cancellation effectiveness. A similar timbre change appears when the acoustic characteristics of the listener&#39;s head and ears are not those used in designing the crosstalk canceler, regardless of listener position. 
     The inventive canceler has an equalization which takes into account the signal arriving at the ears of a variety of listeners positioned in a range of locations. The inventive equalization is the one minimizing the timbre change over an expected range of listener positions and listener acoustic characteristics. Whereas the power spectrum of the traditional crosstalk canceler equalization has a number of peaks and valleys, that of the inventive equalization is by comparison smooth. 
     The timbre of output from cancelers using the inventive equalization, in fact, is less sensitive to listener position or acoustic properties than is that from the traditional canceler [1]. In addition, the inventive equalization has the unexpected benefit of reducing artifacts for listeners outside the sweet spot. 
     Finally, it was noted that binaural signals having a large monophonic component seemed to require an equalization with more bass emphasis than did binaural signals with a small monophonic component. Based on this observation, a canceler equalization was developed which depends on the percentage of monophonic signal energy in the input binaural signal. In this way, the canceler equalization may be adapted to the binaural input. 
     One embodiment of the invention is a crosstalk canceler providing greater listener freedom of movement comprising an input audio signal, two output channels, and a network of filters designed to eliminate crosstalk at the ear of a listener at different listener positions for different frequency bands of the input audio signal. 
     Another embodiment of the invention is a crosstalk canceler equalization which is less sensitive to listener acoustic characteristics and listener position, said equalization being a spectrally smooth version of an input equalization, the details of which may be optionally determined by anticipated ranges of listener acoustic characteristics and listener positions. 
     An additional embodiment of the invention is a crosstalk canceler having an equalization designed to leave unchanged at the output the power spectrum of a Gaussian binaural input with a specified crosscoherence. Another aspect of this embodiment is a canceler in which the crosscoherence of the input binaural signal is sensed and used to adapt the characteristics of the canceler. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows a synthetic spatial audio display. 
     FIG. 2 shows a binaural, virtual surround sound system. 
     FIG. 3 shows a stereo speaker virtual surround sound system. 
     FIG. 4 shows the crosstalk geometry. 
     FIG. 5 shows a crosstalk canceler. 
     FIG. 6 shows a lattice crosstalk canceler. 
     FIG. 7 shows a shuffler crosstalk canceler. 
     FIG. 8 shows a butterfly crosstalk canceler. 
     FIGS. 9 a  and  9   b  show a crosstalk remover example. 
     FIG. 10 shows an incomplete crosstalk cancellation example. 
     FIG. 11 shows a crosstalk equalization example. 
     FIG. 12 shows a crosstalk equalization error example. 
     FIG. 13 shows an inventive sweet spot position example. 
     FIG. 14 shows example transfer function ratio magnitudes. 
     FIG. 15 shows example transfer function ratio phase delays. 
     FIGS. 16 a  and  16   b  shoe an inventive mixing filter example. 
     FIG. 17 shows sweet spot crosstalk energy. 
     FIGS. 18 a  and  18   b  show an inventive mixing filter example. 
     FIG. 19 shows example sweet spot crosstalk energy. 
     FIGS. 20 a  and  20   b  show a example inventive residual energy minimizing equalization. 
     FIG. 21 shows inventive smoothed and interpolated equalizations systems. 
     FIG. 22 shows a smoothed equalization example. 
     FIG. 23 shows an interpolated equalization example. 
     FIG. 24 shows inventive reduced feedback equalization systems. 
     FIG. 26 shows example inventive equalizations. 
     FIG. 27 shows a system for adapting crosstalk canceler equalization to signal characteristics. 
     FIGS. 28 a  and  28   b  show a system and an example inventive equalization approximation. 
     FIG. 29 shows a system for mixing filter evaluation. 
     FIG. 30 shows a system for optimizing sweet spot trajectory. 
     FIG. 31 shows a system for mixing filter optimization. 
     FIG. 32 shows a system for computing transfer function means. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     For clarity, the invention will be described with respect to the symmetric two-speaker, one-listener crosstalk scenario of FIG.  4 . Modifications needed to apply the invention to asymmetric crosstalk geometries, to multiple listeners, or to more than two speakers will be readily apparent to those skilled in the art. In the following, references to listener position or ear position refer also to listener orientation as well as other geometric factors including speaker position and orientation. In addition, in the following equivalent time-domain and frequency-domain quantities and operations are used interchangeably; any technique discussed or description given in one domain is meant to apply in the other. Finally, the functions “mean” and “average” are to be understood in their general sense, for instance being weighted or unweighted arithmetic, geometric, or trimmed means and the like. 
     Crosstalk Cancellation 
     To better appreciate aspects of the present invention, the traditional crosstalk canceler will be described in detail. Referring to FIG. 4, consider two speakers  100  and  102  symmetrically placed about listener  110  at an angle θ  112  with respect to listener axis  111 . Signals applied to the speakers will arrive at the listener&#39;s ears transformed according to near-ear and far-ear transfer functions ν(ω)  104  and φ(ω)  105  embodying, among other effects, the speaker radiation, speaker-listener propagation effects, and acoustic characteristics of the listener. Denoting by s l (t) and s r (t) the left and right speaker signals  101  and  103 , the signals l l (t)  106  and l r (t)  109  appearing at the listener&#39;s left and right ears  107  and  108  are given by 
     
       
           l   l ( t )=ν( t )* s   l ( t )+φ( t )* s   r ( t ),  (1) 
       
     
     
       
           l   r ( t )=φ( t )* s   l ( t )+ν( t )* s   r ( t ),  (2) 
       
     
     where * represents convolution, and ν(t) and φ(t) are the near-ear and far-ear impulse responses, that is, the inverse Fourier transforms of the near-ear and far-ear transfer functions ν(ω) and φ(ω). Expressed in the frequency domain, the listener ear sound pressure signals are 
     
       
           l (ω)= C (ω) s (ω),  (3) 
       
     
     where l(ω) and s(ω) are columns containing the listener ear signal and speaker signal Fourier transforms,                  l                   (   ω   )       =     [             l   l                     (   ω   )                   l   r                     (   ω   )             ]       ,                  s                   (   ω   )       =     [             s   l                     (   ω   )                   s   r                     (   ω   )             ]       ,           (   4   )                                
     and C(ω), the crosstalk matrix, contains the speaker-listener transfer functions,                C                   (   ω   )       =       [           υ                   (   ω   )             φ                   (   ω   )                 φ                   (   ω   )             υ                   (   ω   )             ]     .             (   5   )                                
     It is clear that unless the far-ear transfer function φ(ω) is zero, a binaural signal applied directly to the speakers will exhibit crosstalk. However, as discussed above, crosstalk may be removed by processing the binaural signal so as to anticipate the changes imposed in propagating from the speakers to the listener. 
     Consider the processing shown in FIG.  5 . Binaural channels b l (ω)  120  and b r (ω)  121  are processed by canceler filter network  122  to produce crosstalk canceled speaker signals s l (ω)  123  and s r (ω)  124 , which, in turn arrive at the ears of the listener transformed by the near-ear and far-ear transfer functions comprising the crosstalk matrix C(ω). The listener ear signals l(ω) are easily related to the binaural signal b(ω), 
     
       
           l (ω)= C (ω) s (ω)= C (ω) X (ω) b (ω),  (6) 
       
     
     where b(ω) is the column of binaural channel signal transforms,                  b                   (   ω   )       =     [             b   l                     (   ω   )                   b   r                     (   ω   )             ]       ,           (   7   )                                
     and where the matrix transfer function X(ω) is referred to as the canceler matrix. Note that if the inverse of the crosstalk C(ω) is realizable, setting the canceler to the crosstalk inverse, 
     
       
           X (ω)= C   −1 (ω),  (8) 
       
     
     will produce left and right listener ear signals l l (ω)  129  and l r (ω)  130  equal to the respective input left and right binaural channels b l (ω)  120  and b(ω)  121 . 
     The canceler inverse may be expressed in terms of the near-ear and far-ear transfer functions,                  X                   (   ω   )       =         C     -   1                       (   ω   )       =       [           υ                   (   ω   )               -   φ                     (   ω   )                   -   φ                     (   ω   )             υ                   (   ω   )             ]           υ   2                     (   ω   )       -       φ   2                     (   ω   )               ,           (   9   )                                
     and implemented in the lattice architecture of FIG.  6 . Here, binaural inputs  140  and  141  are applied to filters  142 ,  143 ,  144 , and  145 , each implementing the transfer function contained in the corresponding element of the canceler matrix (9). The filter outputs are combined to form canceled speaker outputs  152  and  153 . 
     Note that for the crosstalk inverse to exist, the near-ear and far-ear transfer functions cannot be identical at any frequency. If this were the case, any canceling signal arriving at one ear would cancel the original signal in the other ear. Also, note that for X(ω) to be realizable, the quantity ν 2 (ω)−φ 2 (ω) needs to be minimum phase. If this is not the case, then its minimum phase equivalent may be used to form its inverse in (9), and the signals appearing in the ear of the listener will be the binaural channel signals shifted in phase by the allpass component of ν 2 (ω) φ 2 (ω). 
     The canceler may also be formed by noting that the crosstalk matrix can be decomposed in terms of the sum and difference of the near-ear and far-ear transfer functions,                  C                   (   ω   )       =         1   2                [         1       1           1         -   1           ]     ·     [             υ                   (   ω   )       +     φ                   (   ω   )             0           0           υ                   (   ω   )       -     φ                   (   ω   )               ]     ·                [         1       1           1         -   1           ]         ,           (   10   )                                
     where the diagonalizing matrix              F   =     [         1       1           1         -   1           ]             (   11   )                                
     is referred to as the shuffler matrix. Noting that the shuffler matrix F is twice its own inverse, the crosstalk canceler X(ω) can be written as                  X                   (   ω   )       =         C     -   1                       (   ω   )       =         1   2                [         1       1           1         -   1           ]     ·     [           1       υ                   (   ω   )       +     φ                   (   ω   )               0           0         1       υ                   (   ω   )       -     φ                   (   ω   )                 ]     ·                [         1       1           1         -   1           ]           ,           (   12   )                                
     leading to the shuffler canceler architecture shown in FIG.  7 . In this canceler implementation, the sum and difference of binaural input, channels  160  and  161  are filtered by shuffler sum filter  164  and shuffler difference filter  165 , respectively, the outputs of which are summed and differenced to form the canceled speaker outputs  170  and  171 . The advantage of this architecture is that only two filters are needed, rather than the four required by the lattice canceler shown in FIG.  6 . 
     The crosstalk inverse may also be decomposed as follows,                    C     -   1                       (   ω   )       =       [         1           -   ρ                     (   ω   )                   -   ρ                     (   ω   )           1         ]     ·     1     υ                   (   ω   )         ·     1     1   -       ρ   2                     (   ω   )               ,           (   13   )                                
     where ρ(ω) is the ratio of the far-ear transfer function to the near-ear transfer function, 
     
       
         ρ(ω)=φ(ω)/ν(ω).  (14) 
       
     
     The corresponding canceler may be implemented in two stages using the butterfly architecture shown in FIG.  8 . The first stage  192  is referred to as the crosstalk remover or mixing stage, and adds to each binaural channel a filtered version of the other binaural channel; its transfer function is given by                  R                   (   ω   )       =     [         1           -   r                     (   ω   )                   -   r                     (   ω   )           1         ]       ,           (   15   )                                
     where r(ω) is referred to as the mixing filter. The second stage  193 , which may be applied either before or after the first stage, equalizes the output, and is called the canceler equalization; its transfer function is 
     
       
           Q (ω)= q (ω) I,   (16) 
       
     
     where I is the identity matrix, and q(ω) is the equalization filter. By setting the mixing filter. to the transfer function ratio 
     
       
           r (ω)=ρ(ω),  (17) 
       
     
     and the equalization filter to the product 
     
       
           q (ω)=1/[ν(ω)(1−ρ 2 (ω))],  (18) 
       
     
     the butterfly architecture of FIG. 8 will implement the canceler inverse. 
     To understand the function of the mixing stage R(ω), consider the example shown in FIG. 9 a . Binaural signal channels  200  and  201  are applied to mixing stage  202 , which produces speaker signals  207  and  208  in response. These signals propagate to the listener, appearing as listener ear signals  215  and  216 . For purposes of illustration, the near-ear transfer function here is one (ω)=1, and the far-ear transfer function is a scaled pure delay φ(ω)=ρe −jωr . In this example, the mixing filter r(ω) is set to the transfer function ratio ρ(ω)=φ(ω)/ν(ω)=ρe −jωr . 
     Referring to FIG. 9 b , pulse  230  applied to the left binaural channel appears directly at the left speaker as pulse  232 . It also appears delayed and scaled according to −ρ(ω) at the right speaker as pulse  235 . The listener left ear will hear pulse  232  directly from the left speaker via near-ear transfer function  211  ν(ω)=1. The left ear will also hear pulse  235 , delayed and scaled according to far-ear transfer function  213  φ(ω)=ρe −jωr . The listener right ear will hear pulse  232  from the left speaker via far-ear transfer function  212 , and pulse  235  directly via near-ear transfer function  214 . 
     Note that pulses  241  and  242  arriving at the right ear cancel. Pulse  241  arriving from the left speaker via far-ear transfer function  213  is delayed and scaled by the same amount as pulse  235  by mixing filter  203  and near-ear transfer function  214 . Therefore, signals applied to left binaural input  200  do not appear at the listener&#39;s right ear. Similarly, right binaural channel signals will be canceled at the listener&#39;s left ear. More generally, when the mixing filter r(ω) is set to the ratio of the near-ear and far-ear transfer functions, binaural signals processed according to the mixing stage R(ω) (15) will appear at the listener&#39;s ears without crosstalk. 
     Note that listener ear signals  215  and  216  are not the original binaural signal channels  200  and  201 ; each ear contains an echo of its respective binaural channel  239  and  243  as a residual effect of canceling crosstalk. The purpose of the equalization is now clear: In addition to inverting the near-ear transfer function (referred to as “naturalization” in [3, 4]), the equalizer must eliminate the echo. As shown in FIG. 11, the echo at the listener ear may be removed by adding a series of echoes to the binaural signal. If the echoes are properly spaced in time and filtered, then the chain binaural signal echoes arriving from the far speaker will exactly cancel all but the first of the binaural signal instances arriving directly from the near speaker. 
     Inventive Crosstalk Removal 
     The canceler sensitivity to listener position and listener acoustic characteristics discussed above is seen to result from discrepancies between the mixing filter r(ω) and the transfer function ratio ρ(ω). As illustrated in FIG. 10, the crosstalk signal is the crosstalk binaural channel (i.e., the left binaural channel at the right ear or the right binaural channel at the left ear) filtered by φ(ω)−r(ω)ν(ω). As the listener moves, the transfer functions φ(ω) and ν(ω) change, and, unless those changes are anticipated by the mixing filter r(ω), the canceling signal radiated from the near-ear speaker will not cancel crosstalk from the far-ear speaker. 
     To give the listener some freedom of movement while maintaining effective (though not complete) crosstalk cancellation, Cooper and Bauck set the mixing filter to a low-pass filtered version of the transfer function ratio, r(ω)=ρ(ω)h(ω), h(ω) being a low-pass filter with a cutoff frequency above 600 Hz and below 10 kHz. In doing so, crosstalk is canceled only below the cutoff frequency. However, since low frequencies have relatively long wavelengths, ρ(ω) is somewhat insensitive to listener position at low frequencies. As a result, the listener is afforded a degree of freedom of movement without noticeably changing canceler effectiveness. 
     The present invention gives the listener freedom of movement by canceling different frequency bands at different listener positions. For instance, low frequencies might be canceled at a speaker separation angle of θ=10°, and high frequencies at an angle of θ=30°. Doing so provides a measure of cancellation over a range of anticipated listener positions; listener position changes do not eliminate cancellation, but simply shift the part of the signal canceled. An additional benefit of distributing the cancellation location is that a smooth transition between regions of effective cancellation and no cancellation is created. 
     Changing the cancellation geometry as a function of frequency may be accomplished by setting the mixing filter to the transfer function ratio evaluated at a frequency-dependent geometry as shown in FIG.  29 , 
     
       
           r (ω)=ρ(w,θ(ω)),  (19) 
       
     
     where θ(ω), called the sweet spot trajectory, specifies the frequency-dependent crosstalk geometry at which the transfer function ratio is evaluated. The mixing filter thus designed can be implemented directly as mixing filter  182  and  183  in mixing stage  192  of the butterfly canceler in FIG.  8 . It can also be used in forming the canceler matrix X(ω), and implemented as a lattice, shuffler, or other canceler. Equivalently, shuffler or lattice cancelers, (12) or (9), or other cancelers, may be designed directly based on a frequency-dependent geometry. 
     Details of the sweet spot trajectory θ(ω) depend on, among other factors, the desired listener and speaker positions, and the binaural source material. In one embodiment, shown in FIG. 13, the sweet spot center is moved further from the speakers with increasing frequency. By changing the sweet spot center location more rapidly with decreasing frequency, this embodiment attempts to maintain a constant, but acceptable, level of crosstalk within the extended sweet spot. In another embodiment, the magnitude and phase of the mixing filter are determined from separate sweet spot center trajectories. 
     In FIG.  14  and FIG. 15, example transfer function ratio magnitudes and phase delays are shown as functions of frequency for listener positions along the listener axis. Mixing filters based on the inventive sweet spot trajectory  280  and prior art constant sweet spot trajectories  281  and  282  are shown in FIG.  13 . Note that the inventive mixing filter takes on the characteristics of the closer prior art filter at low frequencies and those of the farther prior aft filter at high frequencies 
     The total energy in the crosstalk signal at an ear of a listener positioned at θ is given by 
     
       
           E   c (θ)=∫ 0   π |ν(ω, θ) r (ω)−φ(ω, θ)| 2   dω,   (20) 
       
     
     where ν(ω, θ) and φ(ω, θ) are the near-ear and far-ear transfer functions to the ear of the listener at θ. The crosstalk energy is plotted in FIG. 17 for the mixing filters implied by the sweet spot center trajectories of FIG.  13 . Note that the inventive sweet spot  300  is somewhat more extended than that of the prior art canceler  301  (corresponding to constant sweet spot  281 ), and of comparable extent to that of prior art canceler  302  (corresponding to constant sweet spot  282 ). 
     In another embodiment of, the invention, the sweet spot trajectory θ(ω) is designed to maximize the area over which the listener can move while maintaining a minimum level of crosstalk rejection or maximum level of uncanceled crosstalk energy. In another embodiment, θ(ω) is chosen to minimize the maximum crosstalk energy experienced by a listener located in a given region. In optimizing the sweet spot trajectory θ(ω) as shown in FIG. 30, note that it may be useful to weight the crosstalk energy in frequency or position to give more importance to certain spectral bands or listener positions, or to account for the canceler equalization. For instance, the power spectrum of many sounds approximates a 1/ω characteristic away from DC, so that in optimizing the sweet spot trajectory, it is useful to weight the crosstalk energy away from DC by 1/ω. 
     Another approach shown in FIG. 31 is to find the optimal mixing filter directly, rather than using θ(ω) to parameterize the solution. In this embodiment of the invention, the crosstalk energy is written in terms of the mixing filter and the near-ear and far-ear transfer functions at each frequency and crosstalk geometry of interest, 
     
       
           E   c (θ, ω)=γ(ω)·|ν(ω, θ) r (ω)−φ(ω, θ)| 2 ,  (21) 
       
     
     where γ(ω) represents the product of the equalization filter power and the anticipated signal power at frequency w. The mixing filter r(ω) is then taken to be the one optimizing some aspect of the crosstalk energy E c (θ, ω). One choice is to minimize the maximum weighted energy over some set of canceler geometries or listener characteristics,                    r   ^                     (   ω   )       =     Arg        [       min     r                   (   ω   )              {       max     θ   ∈   Θ            {       ∫   0   π          w                   (     θ   ,   ω     )                     E   c                     (     θ   ,   ω     )                        ω         }       }       ]         ,           (   22   )                                
     where ω(θ, ω) is a weighting reflecting the importance of eliminating crosstalk energy at frequency ω and geometry θ, and Θ represents the range of canceler geometries and listener characteristics under consideration. Another choice is to maximize the area over which the weighted crosstalk energy is less than a given level,                    r   ^                     (   ω   )       =     Arg        [       max     r                   (   ω   )              {       ∫     θ   ∈   Θ            1                   (         ∫   0   π          w                   (     θ   ,   ω     )                     E   c                     (     θ   ,   ω     )                        ω         &lt;     υ                   (   θ   )         )                        θ         }       ]         ,           (   23   )                                
     where 1(|·) is an indicator function, taking on a value of 1 if the condition is true and 0 otherwise, and the quantity ν(θ) specifies the maximum acceptable crosstalk energy level as a function of position. Alternatively, the maximum acceptable crosstalk energy level could depend on frequency as well as position,                  r   ^                     (   ω   )       =       Arg        [       max     r                   (   ω   )              {       ∫     θ   ∈   Θ              ∫   0   π          1                   (         E   c                     (     θ   ,   ω     )       &lt;     υ                   (     θ   ,   ω     )         )                        ω             θ           }       ]       .             (   24   )                                
     Still another optimization choice is to find the mixing filter minimizing the total crosstalk energy in a given region,                    r   ^                     (   ω   )       =     Arg        [       min     r                   (   ω   )              {       ∫     θ   ∈   Θ              ∫   0   π          w                   (     θ   ,   ω     )                     E   c                     (     θ   ,   ω     )                        ω             θ           }       ]         ,           (   25   )                                
     where the weighting ω(θ, ω) weights the importance of having effective cancellation at a given frequency and speaker-listener geometry. 
     As an example, FIG. 18 shows the magnitude  450  and phase delay  460  of the prior art mixing filter designed to cancel crosstalk at the ears of a listener positioned on the listener axis twice as far from the line joining the speakers as the distance separating the speakers. Also shown are the magnitude and phase delay of the filter minimizing the total crosstalk energy. (25)  451 ,  461  and minimizing the maximum crosstalk energy (22)  452 ,  462  for listeners on the listener axis between 1.5 and 2.5 times the speaker separation from the speaker axis. Note that magnitude of the optimal mixing filters is similar to that of prior art mixing filters for listener positions closer to the speakers than that used to generate prior art mixing filter magnitude  450 . By contrast, the phase delay of the inventive mixing filters is more like that of prior art mixing filters associated with positions further from the speakers than that used to form prior art mixing filter phase delay  460 . The crosstalk energy associated with the inventive and prior art mixing filters of FIG. 18 is plotted as a function of position in FIG.  19 . The minimizer of the maximum crosstalk energy over the region  452 ,  462  provides the widest sweet spot  472 . The prior art crosstalk has the smallest sweet spot  470  and the most abrupt transition between regions of effective cancellation and little cancellation. 
     Another optimization choice is suggested by the observation that listeners prefer cancelers having a gentle transition between areas of effective cancellation and no cancellation over cancelers with a more abrupt transition. To accommodate this preference, the mixing filter may be optimized so that the slope (derivative with respect to position) of the crosstalk energy in the transition region is minimized. 
     It should be noted that the optimal mixing filter {circumflex over (r)}(ω) (25) may be expressed in closed from,                    r   ^                     (   ω   )       =           μ   φ                     (   ω   )                     μ   v                       (   ω   )     *       +       σ     φ                   v   *                         (   ω   )               μ   v                     (   ω   )                     μ   v                       (   ω   )     *       +       σ     vv   *                       (   ω   )             ,           (   26   )                                
     where * denotes complex conjugation, μ φ (ω) and μ ν (ω) are the near-ear and far-ear transfer function means over position, 
     
       
         μ φ (ω)=∫ω(θ, ω)φ(ω, θ) dθ,   (27) 
       
     
     
       
         μ ν (ω)=∫ω(θ, ω)ν(ω, θ) dθ,   (28) 
       
     
     and σ νν *(ω) and σ φν *(ω) are variances over position, 
     
       
         σ νν .(ω)=∫ω(θ, ω)|ν(ω)−μ ν (ω)| 2   dθ,   (29) 
       
     
     
       
         σ φν *(ω)=∫ω(θ, ω)[φ(ω)−μ φ (ω)][ν(ω)−μ ν (ω)]* dθ.   (30) 
       
     
     Note that the optimal mixing filter has a magnitude and phase approximating that of the mean over position of the transfer function ratio ρ(ω, θ), with the magnitude reduced at frequencies where the transfer function ratio changes rapidly with position. This motivates another embodiment of the invention shown in FIG. 32, wherein the magnitude or phase of the mixing filter is given by the respective means over position of the magnitude or phase of the transfer function ratio filter, possibly reducing the mixing filter magnitude at any selected frequency by an amount dependent on the transfer function ratio position variance (i.e., the sensitivity of the transfer function ratio to changes in listener position) at that frequency. 
     Inventive Equalization 
     Listener freedom of movement is also restricted by the canceler equalization. As illustrated in FIG. 11, the equalization associated with the crosstalk matrix inverse removes the unwanted binaural signal echo by creating two chains of canceling echoes. Unfortunately, as shown in FIG. 12, the resulting listener ear signals are very sensitive to listener position, which determines the relative alignment and strength of the two chains through the near-ear and far-ear transfer functions. 
     Therefore, an embodiment of the invention balances the desire to maintain the original binaural signal equalization with the need to accommodate varying crosstalk geometries and listener characteristics. The inventive canceler equalization achieves this balance by optimizing the equalization over a set of anticipated listener positions and characteristics. This approach differs from that of the prior art, which uses a single crosstalk geometry in designing the canceler equalization. 
     The binaural channel signal appearing at the ear of the listener is filtered by 
     
       
           q (ω)(ν(ω, θ)−φ(ω, θ) r (ω)), 
       
     
     q(ω) being the canceler equalization filter, r(ω) the canceler mixing filter, and ν(ω, θ) and φ(ω, θ) the near-ear and far-ear transfer functions evaluated at the crosstalk geometry and listener characteristics θ. Ideally, the binaural channel would appear at the listener unfiltered; the energy in the difference between the unit transfer function and that imposed on the binaural channel, called the equalization residual is given by 
     
       
           E   q (ω, θ)=| q (ω)(ν(ω, θ)−φ(ω, θ) r (ω))−1| 2 .  (31) 
       
     
     In one embodiment of the invention, the equalization q(ω) is optimized to minimize the equalization residual E q (ω, θ) over a distribution of crosstalk geometries and listener characteristics ρ(θ),                    q   ^                     (   ω   )       =     Arg        [       min     q                   (   ω   )              {       ∫     θ   ∈   Θ              ∫   0   π          ℘                   (   θ   )                     E   q                     (     θ   ,   ω     )                        ω                        θ           }       ]         ,           (   32   )                                
     This solution is available in closed form,                  q   ^                     (   ω   )       =         ∫     ℘                   (   θ   )                     (       v                   (     ω   ,   θ     )       -     φ                   (     ω   ,   θ     )                   r                   (   ω   )         )             θ           ∫     ℘                   (   θ   )                              v                   (     ω   ,   θ     )       -     φ                   (     ω   ,   θ     )                   r                   (   ω   )              2                        θ           .             (   33   )                                
     Denoting by μ ν (ω) and μ φ (ω) the means of the near-ear and far-ear transfer functions with respect to ρ(θ), 
     
       
         μ φ (ω)=∫ρ(θ)φ(ω, θ) dθ,z   (34) 
       
     
     
       
         μ ν (ω)=∫ρ(θ)ν(ω, θ) dθ,z   (35) 
       
     
     and by σ νν *(ω), ρ φφ *(ω), and σ φν *(ω) the variances with respect to ρ(θ) 
     
       
         σ νν *(ω)=∫ρ(θ)|ν(ω)−μ ν (ω)| 2   dθ,   (36) 
       
     
     
       
         σ φφ *(ω)=∫ρ(θ)|φ(ω)−μ φ (ω)| 2   dθ,   (37) 
       
     
     
       
         σ φν *(ω)=∫ρ(θ)[φ(ω)−μ φ (ω)][ν(ω)−μ ν (ω)]* dθ,   (38) 
       
     
     the optimal equalization may be written as                    q   ^                     (   ω   )       =       1       μ   v                     (   ω   )         ·     1     1   -     r                   (   ω   )                     μ   φ                       (   ω   )     /     μ   v                       (   ω   )       +     [           σ     vv   *                       (   ω   )       +              r                   (   ω   )            2                     σ     φφ   *                       (   ω   )       -     2                 ℜ        {     r                   (   ω   )                     σ     φ                   v   *                         (   ω   )       }             μ   v                     (   ω   )                     μ   v                     (   ω   )     *     (     1   -     r                   (   ω   )                     μ   φ                       (   ω   )     /     μ   v                       (   ω   )         )         ]             ,           (   39   )                                
     where R{·} is the real part of its argument. By comparison to the prior art equalization,                  q                   (   ω   )       =       1     v                   (   ω   )         ·     1     1   -     r                   (   ω   )                   φ                     (   ω   )     /   v                     (   ω   )               ,           (   40   )                                
     the optimal equalization (39) generates similar train of echoes, but with a shorter time constant (since the bracketed term is nonnegative), particularly in those parts of the spectrum where the near-ear and far-ear transfer functions are sensitive to position changes. In the frequency domain, the magnitude of the optimal equalization will appear smoothed relative to that of the prior art equalization. Note that the greater the sensitivity to position changes or listener characteristics exhibited by ν(ω) and φ(ω), or the greater the range of expected geometries and listeners ρ(θ), the more smoothed the optimal equalization magnitude compared to the prior art equalization. 
     As an example, FIG. 20 shows the prior art equalization magnitude  340  along with that of two optimal equalizations. Equalization  341  is designed to minimize the expected equalization residual for listeners uniformly distributed on the listener axis between 1.5 and 2.5 times the speaker separation distance from the speaker axis; equalization  342  minimizes the equalization residual for listeners between 1.0 and 2.5 times the speaker separation from the speaker axis. The equalization residual as a function of listener position is also shown in FIG.  20 . The inventive equalization residuals  344 ,  345  achieve their minima over wider ranges of listener position than does the prior art equalization residual  343 . In addition, away from the sweet spot center, the inventive equalization residuals are smaller than the prior art equalization residual. 
     The observation that the optimal equalization magnitude is essentially a smoothed version of the prior art equalization magnitude leads to the inventive equalizations shown in FIG.  21  and FIG.  24 . In the embodiment shown in FIG. 21, the inventive canceler equalization spectrum is a smoothed or interpolated version of the spectrum of an input canceler equalization. Note that the smoothing or interpolation may be applied to the entire spectrum, or may be restricted to all but the naturalization, 1/|ν(ω)| 2 . A smoothed canceler equalization spectrum may be found by applying a running mean (arithmetic, geometric, trimmed or other means may be applied) to a prior art equalization spectrum                       q                   (   ω   )            2     =       1            υ                   (   ω   )            2       ·       1     1   +            r                   (   ω   )                   φ                     (   ω   )     /   υ                     (   ω   )            2     -     2                 ℜ        {     r                   (   ω   )                   φ                     (   ω   )     /   υ                     (   ω   )       }           .               (   41   )                                
     It may be equivalently found as the spectrum associated with the appropriately windowed version of the prior art equalization impulse response. In FIG. 22, example prior art equalization  350  is shown along with inventive smoothed equalizations  351 ,  352 . Smoothed equalizations  351 ,  352  were formed by critical band smoothing of the prior art power spectrum using smoothing bandwidths of 1.0 and 2.0 critical bands, respectively. 
     An interpolated spectrum may be found by interpolating in the prior art equalization power spectrum points where the quantity r(ω)φ(ω)/ν(ω) achieves the same phase. The resulting power spectrum is given by                           q   ^                     (   ω   )            2     =       1            υ                   (   ω   )            2       ·     1     1   +            r                   (   ω   )                   φ                     (   ω   )     /   υ                     (   ω   )            2     -     2                 α             r                   (   ω   )                   φ                     (   ω   )     /   υ                     (   ω   )                      ,           (   42   )                                
     where αε[−1, 1] which determines the points of the prior art equalization interpolated. Several example interpolated equalization magnitudes  361 ,  362  are plotted in FIG. 23 along with the prior art equalization magnitude  360 ; interpolation points  363  are marked. 
     The embodiment of FIG. 24 augments a prior art canceler equalization implementation with an additional filter α(ω) which has the effect of reducing feedback, thereby smoothing the spectrum of the prior art canceler. So as to approximate the optimal equalization, feedback should be preferentially reduced in those frequency bands where the feedback is largest. In one instance, a filtered version of the output is added to the feedback path of the prior art equalization,                    q   ^                     (   ω   )       =       1     υ                   (   ω   )         ·     1     1   -     r                   (   ω   )                   φ                     (   ω   )     /   υ                     (   ω   )       +     α                   (   ω   )               ,           (   43   )                                
     where α(ω) is a filter having a phase generally similar to that of r(ω)φ(ω)/ν(ω); it&#39;s presence selectively reduces decay time. In another instance, feedback is reduced directly,                    q   ^                     (   ω   )       =       1     υ                   (   ω   )         ·     1     1   -     α                   (   ω   )                   r                   (   ω   )                   φ                     (   ω   )     /   υ                     (   ω   )               ,           (   44   )                                
     where α(ω) is a filter (preferably minimum phase) having a magnitude no greater than one; it reduces decay time by limiting the amount of feedback at any given frequency. Note that it is possible to adjust both instances of α(ω) above so that the resulting equalization approximates the optimal equalization (39). 
     Another consideration in crosstalk canceler equalization is the apparent coloring of the binaural signal experienced by. those listeners outside the sweet spot. To minimize equalization artifacts for these listeners, the approach taken here is to equalize the canceler so as to be compatible with—i.e., pass unchanged in equalization—certain classes of input signals. For example, many signals including virtual surround binaural signals have a large fraction of their energy common to both binaural channels. In this case, a crosstalk canceler equalized to pass unchanged monophonic signals would be appropriate. The response of a crosstalk canceler X(ω)=q(ω)R(ω) to a two-channel monophonic signal b(ω)=m(ω)1 is 
     
       
           s (ω)= q (ω)(1 −r (ω)) m (ω)1.  (45) 
       
     
     Setting the equalization to                q                   (   ω   )       =     1     1   -     r                   (   ω   )                   (   46   )                                
     leaves the canceler output equal to the canceler-input for monophonic inputs. 
     Consider a binaural input b(ω) composed of zero-mean Gaussian random processes having identical power spectra P b (ω) and crosscoherence η,                  E        {     b                   (   ω   )                   b                     (   ω   )     T       }       =       P   b                       (   ω   )                [         1       η             η   *         1         ]         ,           (   47   )                                
     where E{·} is the expectation operator and · T  is the Hermetian transpose. (Note that the binaural channel crosscoherence η is the energy in the product of the binaural channel signals normalized by the mean of the individual channel signal energies, so that it takes on values in the range [−1, 1]. The energies, and therefore η, may be evaluated as functions of frequency, or they may represent the total energy over the band.) The total power appearing at the output of a canceler X(ω)=q(ω)R(ω)—the sum of the left and right channel output powers—in response to the Gaussian input b(ω) is 
     
       
           E{s (ω) T   s (ω)}=2 |q (ω)| 2 P b (ω)(1 +|r (ω)| 2 −2 R{ηr (ω)}).  (48) 
       
     
     Accordingly, the inventive equalization has a power given by                         q                   (   ω   )            2     =     1     1   +            r                   (   ω   )            2     -     2                 ℜ        {     η                 r                   (   ω   )       }             ,           (   49   )                                
     so as to leave the total power of a random process with channel crosscoherence η unchanged at the output. It is worth pointing out that if the input binaural signal were a deterministic signal decomposed into sum—that is, monophonic—and difference components, with η measuring the percentage monophonic energy less the percentage difference energy, the equalization (49) leaves the total output power unchanged. 
     Note that if the input were monophonic, the channel crosscoherence η would be one, and the equalization power would be that of the monophonic compatible equalization above,                       q                   (   ω   )            2     =       1     1   +            r                   (   ω   )            2     -     2                 ℜ        {     r                   (   ω   )       }           .             (   50   )                                
     If the input channels were statistically independent, the channel crosscoherence would be zero, and the inventive equalization power would be                       q                   (   ω   )            2     =       1     1   +            r                   (   ω   )            2         .             (   51   )                                
     The inventive equalization magnitude is plotted in FIG. 26 for a range of binaural channel crosscoherence values η. 
     In many cases, the channel crosscoherence will be approximately known a priori. For instance, movie soundtracks presented in binaural virtual surround sound format as shown in FIG. 3 typically have a channel crosscoherence in the range ηε[0.8, 0.9]. In one embodiment, if the channel crosscoherence is not known a priori, the listener may tune the canceler equalization to his liking by adjusting the channel crosscoherence value used to determine the equalization power. In another embodiment, shown in FIG. 27, the binaural channel crosscoherence is sensed (possibly as a function of frequency) and used to adjust the canceler equalization. Alternatively, the percentage of sum and difference energies may be used to set η. 
     Because of the manner in which the equalization power (49) depends on the binaural channel crosscoherence η, it is difficult to adapt the equalization filter to real-time changes in η. However, the embodiment of FIG. 28 shows an equalization filter comprising two filters in a feedback delay network which has a magnitude approximating that of (49). By setting the delay τ to the near-ear-far-ear arrival time difference implied by the mixing filter r(ω), and by designing the filters α(ω) and β(ω) to have magnitudes that. approximate                       α                   (   ω   )            =     γ   -       [       γ   2     -   1     ]       1   2           ,                γ   =       1   +            r                   (   ω   )            2         2                        η                 r                   (   ω   )                          (   52   )                        β                   (   ω   )            =       [       1   +            α                   (   ω   )            2         1   +            r                   (   ω   )            2         ]       1   2         ,           (   53   )                                
     the resulting system  441  will closely approximates the desired equalization filter q(ω)  440 , as shown in the example of FIG.  28 . Note that the approximation remains valid even under rather crude approximations to the magnitude characteristics specified for α(ω) and β(ω) above. For the approximation of FIG. 28, the filters α(ω) and β(ω) were designed by matching the specified magnitudes only at DC, the band edge, and at 3 kHz. 
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