PATENT DOCUMENT

Abstract:
A method for stereo expansion includes a step to remove the effects of actual relative speaker to listener positioning and head shadow and a step to introduce an artificial effect based on a desired virtual relative speaker to listener positioning using the inter-aural delay and the head-shadow models for the virtual speakers at desired angles relative to the listener thereby creating the impression of a widened and centered sound stage and an immersive listening experience. Known methods drown out vocals and add mid-range coloration thereby defeating equalization. The present method includes the integration of a novel binaural listening model and speaker-room equalization techniques to provide widening while not defeating equalization.

Full Description:
The present application claims the priority of U.S. Provisional Patent Application Ser. No. 60/928,206 filed 7 May, 2007, which application is incorporated in its entirety herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to stereo signal processing and in particular to processing a stereo signal to create the impression of a wide sound stage and/or of immersion. 
     Conventional stereo reproduction, for example television, two-channel speakers such as iPod® speakers, etc., create an impression of a narrow spatial image. The narrow imaging is primarily due to loudspeaker proximity relative to each other and unmatched speaker-room frequency responses. The goal of any multichannel system is to give the listener an immersive or a “listener-is-there” impression. Unfortunately, narrow stereo imaging precludes such an experience. 
     The spatial resolution (i.e., localization ability) of human hearing is at least one degree. It is desirable to manipulate stereo signals to enlarge the stereo sound field and imagery by combining concepts from physical acoustics (for example, room acoustics of the space the listener is located in), signal processing (for example, digital filtering), and auditory perception (for example, spatial localization cues). Stereo expansion will allow listeners to perceive audio signals arriving from a wider speaker separation with high-fidelity through the use of a unique binaural listening model and speaker-room equalization technique. 
     Known stereo signal combining approach (for example, L+α(L−R) and R+α(R−L)) have attempted to expand the acoustic field. Unfortunately, these often result in vocals “drowned out” &amp; midrange coloration. Also, benefits from speaker-room equalization cannot be incorporated because the stereo signal combining is independent of room equalization. Other methods include Head-Related-Transfer-Functions (HRTFs) premised on the localization ability of the human pinna (the visible portion of the ear extending from the side of the head which colors sound based on the arrival angle). However, human pinna vary among listeners and an expansion approach, involving use of specific direction HRTF, is not robust, and equalization is again defeated. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention addresses the above and other needs by providing a method for stereo expansion which includes a step to remove the effects of actual relative speaker to listener positioning and head shadow and a step to introduce an artificial effect based on a desired virtual relative speaker to listener positioning using the inter-aural delay and the head-shadow models for the virtual speakers at desired angles relative to the listener thereby creating the impression of a widened and centered sound stage and an immersive listening experience. Known methods drown out vocals and add mid-range coloration thereby defeating equalization. The present method includes the integration of a novel binaural listening model and speaker-room equalization techniques to provide widening while not defeating equalization. 
     In accordance with one aspect of the invention, there is provided a method including determining speaker angles alpha and beta relative to a listener position wherein said speaker angles are computed using actual stereo speaker spacing and actual listener position, determining actual inter-aural delays between the speakers and the listeners ears, determining the headshadow responses associated with each ear relative to each of the speakers given the speaker angles equalizing the headshadow responses between the speakers and the listener ears, determining virtual speaker angles alpha′ and beta′ relative to listener position, determining virtual inter-aural delays between the speakers and the listeners ears for virtual speaker angles alpha′ and beta′, determining virtual headshadow responses associated with each ear relative to each of the virtual speakers given the virtual speaker angles, determining stereo expansion filters from the headshadow responses and the virtual headshadow responses, converting lattice form filters to shuffler form filters, variable octave complex smoothing the shuffler filters, and converting smoothed shuffler filters to smoothed lattice filters for performing spatialization and preserving the audio quality. 
     In accordance with another aspect of the invention, there is provided a method including (a) determining actual speaker angles alpha and beta relative to listener position centered on the actual speakers wherein said speaker angles are computed using actual stereo speaker spacing and listener position, (b) determining actual inter-aural delays between the speakers and the listener ears, (c) determining the actual headshadow responses associated with each ear relative to each of the speakers given the speaker angles, (d) determining an actual speaker to listener 2×2 matrix transfer function H using the actual inter-aural delays and the actual headshadow responses, (f) determining virtual speaker angles alpha′ and beta′ relative to listener position wherein said virtual speaker angles are computed using a virtual stereo speaker spacing and listener position, (g) determining virtual inter-aural delays between the virtual speakers and the listeners ears for virtual speaker angles alpha′ and beta′ relative to listener position, (h) determining virtual headshadow responses associated with each ear relative to each of the virtual speakers given the virtual speaker angles and, (i) determining a virtual speaker to listener 2×2 matrix transfer function H desired  representing the transfer functions between the virtual speakers and the listener ears, (j) selecting on-diagonal elements of H −1  H desired  as a pair of ipsilateral filters and selecting off-diagonal elements of H −1  H desired  as a pair of contralateral filters, (k) transforming the two pairs of ipsilateral filters and contralateral filters to a single pair of filters RES(1,1) and RES(2,2) to transform a lattice form to a shuffler form, (l) variable octave complex smoothing the pair of filters RES(1,1) and RES(2,2) to obtain smoothed filters sRES(111) and sRES(2,2) to preserve audio quality and spatial widening, and (m) transforming the pair of filters sRES(1,1) and sRES(2,2) back into lattice form for performing spatialization and preserving the audio quality. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
       The above and other aspects, features and advantages of the present invention will be more apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein: 
         FIG. 1  shows an actual relative speaker to listener positioning and head shadow geometry. 
         FIG. 2  shows head shadowing as a function of incidence angle. 
         FIG. 3  shows a head shadow model. 
         FIG. 4  shows a desired relative speaker to listener positioning for creating the impression of a widened and centered sound stage and an immersive listening experience according to the present invention. 
         FIG. 5  is a wide synthesis stereo filter according to the present invention. 
         FIG. 6  is a spatial equalization filter including widening and a phantom center channel shown in a lattice structure according to the present invention. 
         FIG. 7  shows a visualization of relative speaker to listener positioning for creating the impression of a widened and arcing according to the present invention. 
         FIG. 8  shows a shuffler filter representation of the present invention. 
         FIG. 9A  shows unsmoothed filter coefficients for RES(1,1) according to the present invention. 
         FIG. 9B  shows unsmoothed filter coefficients for RES(2,2) according to the present invention. 
         FIG. 10A  shows smoothed filter coefficients for sRES(1,1) according to the present invention. 
         FIG. 10B  shows smoothed filter coefficients for sRES(2,2) according to the present invention. 
         FIG. 11  describes a method according to the present invention. 
     
    
    
     Corresponding reference characters indicate corresponding components throughout the several views of the drawings. 
     DETAILED DESCRIPTION OF THE INVENTION 
     The following description is of the best mode presently contemplated for carrying out the invention. This description is not to be taken in a limiting sense, but is made merely for the purpose of describing one or more preferred embodiments of the invention. The scope of the invention should be determined with reference to the claims. 
     Left and right speakers (or transduces)  10 L and  10 R and a listener  12  are shown in  FIG. 1 . The speakers  10 L and  10 R receive left and right channel signals X L  and X R  and have a speaker spacing d T . Speaker response measurements may be obtained at a listener position  12   a  centered on the listener head  12  through two channels  h   L,C  and  h   R,C . Signals Y L  and Y R  at listener ear positions  11 L and  11 R are determined based on direct sound based binaural response modeling because localization is governed primarily through direct sound. The distances d L,C  and d R,C  from left speaker  10 L and from the right speaker  10 R respectively to a microphone centered at the listener position  12   a , may be obtained from existing technique (for example, a sample in the first peak in the responses  h   L,C  and  h   R,C ) or setting the distances to nominal values. Speaker angles α and β (where a 90 degree speaker angle is directly in front of the listener) may be computed as: 
     
       
         
           
             α 
             = 
             
               
                 cos 
                 
                   - 
                   1 
                 
               
               ( 
               
                 
                   
                     d 
                     
                       L 
                       , 
                       C 
                     
                     2 
                   
                   + 
                   
                     d 
                     T 
                     2 
                   
                   + 
                   
                     d 
                     
                       R 
                       , 
                       C 
                     
                     2 
                   
                 
                 
                   2 
                   ⁢ 
                   
                     d 
                     
                       L 
                       , 
                       C 
                     
                   
                   ⁢ 
                   
                     d 
                     T 
                   
                 
               
               ) 
             
           
         
       
       
         
           
             β 
             = 
             
               
                 cos 
                 
                   - 
                   1 
                 
               
               ( 
               
                 
                   
                     d 
                     
                       R 
                       , 
                       C 
                     
                     2 
                   
                   + 
                   
                     d 
                     T 
                     2 
                   
                   + 
                   
                     d 
                     
                       L 
                       , 
                       C 
                     
                     2 
                   
                 
                 
                   2 
                   ⁢ 
                   
                     d 
                     
                       R 
                       , 
                       C 
                     
                   
                   ⁢ 
                   
                     d 
                     T 
                   
                 
               
               ) 
             
           
         
       
     
     The signals Y L  and Y R  at each ear position  11 L and  11 R may be represented in terms of the propagation delays and the effects of head shadowing (diffraction or attenuation effects) relative to the responses h L,C =δ L,C  and h R,C =δ R,C  (acoustic direct path propagation responses) at the listener position  12   a  from left and right speakers  10 L and  10 R respectively. 
     The listener  12  is assumed to have a head radius a of approximately nine centimeters, an ear offset γ of approximately ten degrees, and the system to have a sampling frequency of f s . Four headshadowed responses result: 
     1) A headshadowed response H α+γ   L,L (z) results from an observation point being the left ear position  11 L for signals arriving from the left channel (i.e., the angle of the incident wave relative to the left ear position  11 L is α+γ); 
     2) A headshadowed response H π−β+γ   R,L (z) results from an observation point being the left ear position  11 L for signals arriving from the right channel (i.e., the angle of the incident wave relative to the left ear position  11 L is π−β+γ); 
     3) A headshadowed response H π−α+γ   L,R (z) results from an observation point being the right ear position  11 R for signals arriving from the left channel (i.e., the angle of the incident wave relative to the right ear position  11 R is π−α+γ); and 
     4) A headshadowed response H β+γ   R,R (z) results from an observation point being the right ear position  11 R for signals arriving from the right channel (i.e., the angle of the incident wave relative to the right ear position  11 R is β+γ). 
     The signals at each ear position  11 L and  11 R may then be calculated as a function of the headshadowed response as: 
                 Y   L     ⁡     (   z   )       =         z     ⌊       ψ   ⁢           ⁢   L     ,   L     ⌋       ⁢       H     L   ,   C       ⁡     (   z   )       ⁢            H     α   +   γ       L   ,   L       ⁡     (   z   )            ⁢       X   L     ⁡     (   z   )         +       z     ⌊       ψ   ⁢           ⁢   R     ,   L     ⌋       ⁢       H     R   ,   C       ⁡     (   z   )       ⁢            H     π   -   β   +   γ       L   ,   R       ⁡     (   z   )            ⁢       X   R     ⁡     (   z   )                           Y   R     ⁡     (   z   )       =         z     ⌊       ψ   ⁢           ⁢   L     ,   R     ⌋       ⁢       H     L   ,   C       ⁡     (   z   )       ⁢            H     π   -   α   +   γ       L   ,   R       ⁡     (   z   )            ⁢       X   L     ⁡     (   z   )         +       z     ⌊       ψ   ⁢           ⁢   R     ,   R     ⌋       ⁢       H     R   ,   C       ⁡     (   z   )       ⁢            H     β   +   γ       R   ,   R       ⁡     (   z   )            ⁢       X   R     ⁡     (   z   )                         H     L   ,   C       =       H     R   ,   C       =   1           
where:
 
               ψ     L   ,   L       =     {                 a   ⁢           ⁢     cos   ⁡     (     α   +   γ     )       ⁢     f   s       c           0   &lt;   α   ≤       π   2     -   γ                     -   a     ⁢           ⁢     cos   ⁡     (     α   -     π   2     +   γ     )       ⁢     f   s       c               π   2     -   γ     &lt;   α   ≤     π   2             ⁢     
     ⁢     ψ     R   ,   R         =     {                 a   ⁢           ⁢     cos   ⁡     (     β   +   γ     )       ⁢     f   s       c           0   &lt;   β   ≤       π   2     -   γ                     -   a     ⁢           ⁢     cos   ⁡     (     β   -     π   2     +   γ     )       ⁢     f   s       c               π   2     -   γ     &lt;   β   ≤     π   2             ⁢     
     ⁢     ψ     R   ,   L         =     {                   -   a     ⁢           ⁢     cos   ⁡     (       π   2     -   β   +   γ     )       ⁢     f   s       c           0   &lt;   β   ≤       π   2     -   γ                     -   a     ⁢           ⁢     cos   ⁡     (       π   2     -   β   +   γ     )       ⁢     f   s       c               π   2     -   γ     &lt;   β   ≤     π   2             ⁢     
     ⁢   and     ,     
     ⁢       ψ     L   ,   R       =     {               -   a     ⁢           ⁢     cos   ⁡     (       π   2     -   α   +   γ     )       ⁢     f   s       c           0   &lt;   α   ≤       π   2     -   γ                     -   a     ⁢           ⁢     cos   ⁡     (       π   2     -   α   +   γ     )       ⁢     f   s       c               π   2     -   γ     &lt;   α   ≤     π   2                                   
where ψ X,Y  is the actual inter-aural delay between speaker X and ear Y, a is head radius, fs is sample frequency, and c is sound speed. H L,C  and H R,C  are speaker to center of head transfer function matrices and are assumed to be unity here.
 
     The headshadowed models used are range independent. Accuracy may potentially be improved by multiplying by a distance or (room-dependent factor such as D/R) with H θ (ω) as shown in  FIG. 2 . 
     The headshadowed model H θ (ω) may be approximated by a single pole filter Ĥ θ (ω) shown in  FIG. 3  for θ=0 degree (curve  14 ), θ=45 degree (curve  16 ), θ=90 degree (curve  18 ), θ=120 degree (curve  28 ), and θ=150 degree (curve  22 ), applied for f&gt;1.5 kHz: 
     
       
         
           
             
               
                 
                   H 
                   ^ 
                 
                 θ 
               
               ⁡ 
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 1 
                 + 
                 
                   
                     j 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       τ 
                       θ 
                     
                     ⁢ 
                     ω 
                   
                   
                     2 
                     ⁢ 
                     
                       ω 
                       0 
                     
                   
                 
               
               
                 1 
                 + 
                 
                   
                     j 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                   
                   
                     2 
                     ⁢ 
                     
                       ω 
                       0 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               τ 
               θ 
             
             = 
             
               
                 ( 
                 
                   1 
                   + 
                   
                     
                       τ 
                       min 
                     
                     2 
                   
                 
                 ) 
               
               + 
               
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         τ 
                         min 
                       
                       2 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   cos 
                   ( 
                   
                     
                       θ 
                       
                         θ 
                         min 
                       
                     
                     ⁢ 
                     180 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               τ 
               min 
             
             = 
             0.1 
           
         
       
       
         
           
             
               θ 
               min 
             
             = 
             150 
           
         
       
     
     The signals Y L  and Y R  at each ear may then be represented in matrix form as: 
               [           Y   L               Y   R           ]     =     H   ⁡     [           X   L               X   R           ]             
where the actual speaker to listener matrix transfer function H, including both inter-aural delays and headshadow responses, is:
 
             H   =     [             z       ψ   ⁢           ⁢   L     ,   L       ⁢         H   ^       α   +   γ       L   ,   L       ⁡     (   z   )                 z       ψ   ⁢           ⁢   R     ,   L       ⁢         H   ^       π   -   β   +   γ       R   ,   L       ⁡     (   z   )                     z       ψ   ⁢           ⁢   L     ,   R       ⁢         H   ^       π   -   α   +   γ       L   ,   R       ⁡     (   z   )                 z       ψ   ⁢           ⁢   R     ,   R       ⁢         H   ^       β   +   γ       R   ,   R       ⁡     (   z   )               ]           
where the headshadow models Ĥ θ (ω) may be minimum phase.
 
     Additionally, an equalization filter matrix G(z) may be designed to counteract the effects of “regular” stereo perception using a joint minimum-phase approach disclosed in “An Alternative Design for Multichannel and Multiple Listener Room Equalization” S. Bharitkar, Proc. 2004 38 th  IEEE Asilomar Conference on Signal, Systems, and Computers, Pacific Grove, Calif., November 2004 to minimize artifacts: 
               [           Y   L               Y   R           ]     =     HG   ⁡     [           X   L               X   R           ]             
and when G(z) is formed as H −1 (z):
 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       Y 
                       L 
                     
                   
                 
                 
                   
                     
                       Y 
                       R 
                     
                   
                 
               
               ] 
             
             = 
             
               [ 
               
                 
                   
                     
                       X 
                       L 
                     
                   
                 
                 
                   
                     
                       X 
                       R 
                     
                   
                 
               
               ] 
             
           
         
       
     
     A wide stereo synthesis visualization  24  according to the present invention is shown in  FIG. 4 . A left synthesized (or virtual) speaker  10 L′ is shown displaced a distance p 1  to the left of the speaker  10 L, and a right synthesized (or virtual) speaker  10 R′ is shown displaced a distance p 2  to the right of the speaker  10 L. Given p 1  and/or P 2 , the distances d L,C ′ and d R,C ′ from the synthesized speakers to the microphone position are computed as:
 
 d   L,C ′=√{square root over (( p   1   +d   L,C  cos α) 2 +( d   L,C  sin α) 2 )}{square root over (( p   1   +d   L,C  cos α) 2 +( d   L,C  sin α) 2 )}
 
 d   R,C ′=√{square root over (( p   2   +d   R,C  cos β) 2 +( d   L,C  sin α) 2 )}{square root over (( p   2   +d   R,C  cos β) 2 +( d   L,C  sin α) 2 )}
 
     Virtual speaker angles α′ and β′ are computed: 
     
       
         
           
             
               tan 
               ⁢ 
               
                   
               
               ⁢ 
               
                 α 
                 ′ 
               
             
             = 
             
               
                 
                   d 
                   
                     L 
                     , 
                     C 
                   
                 
                 ⁢ 
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 α 
               
               
                 
                   p 
                   1 
                 
                 + 
                 
                   
                     d 
                     
                       L 
                       , 
                       C 
                     
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   α 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               tan 
               ⁢ 
               
                   
               
               ⁢ 
               
                 β 
                 ′ 
               
             
             = 
             
               
                 
                   d 
                   
                     L 
                     , 
                     C 
                   
                 
                 ⁢ 
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 α 
               
               
                 
                   p 
                   2 
                 
                 + 
                 
                   
                     d 
                     
                       R 
                       , 
                       C 
                     
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   β 
                 
               
             
           
         
       
     
     It is generally (but not necessarily) desired that the listener  12  perceives themself to be centered on the speakers  10 L′ and  10 R′. In order to achieve the centered perception, the virtual speaker angles α′ and β′ should be perceived as being approximately equal, which is equivalent to:
 
 p   1   +d   L,C  cos α= p   2   +d   R,C  cos β
 
     The desired left and right signals Y L ′ and Y R ′ at the listener ear positions  11 L and  11 R in matrix representation are: 
               [           Y   L               Y   R           ]     =       H   desired     ⁡     [           X   L               X   R           ]             
where a speaker to listener matrix transfer function H desired  is determined from the virtual inter-aural delays Δ X,Y  and the virtual headshadow responses:
 
     
       
         
           
             
               H 
               desired 
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         z 
                         
                           Δ 
                           
                             L 
                             , 
                             L 
                           
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               H 
                               ^ 
                             
                             
                               
                                 α 
                                 ′ 
                               
                               + 
                               γ 
                             
                             
                               L 
                               , 
                               L 
                             
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                       
                     
                   
                   
                     
                       
                         z 
                         
                           Δ 
                           
                             R 
                             , 
                             L 
                           
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               H 
                               ^ 
                             
                             
                               π 
                               - 
                               
                                 β 
                                 ′ 
                               
                               + 
                               γ 
                             
                             
                               R 
                               , 
                               L 
                             
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                       
                     
                   
                 
                 
                   
                     
                       
                         z 
                         
                           Δ 
                           
                             L 
                             , 
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                       ⁢ 
                       
                          
                         
                           
                             
                               H 
                               ^ 
                             
                             
                               π 
                               - 
                               
                                 α 
                                 ′ 
                               
                               + 
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                               L 
                               , 
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                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                       
                     
                   
                   
                     
                       
                         z 
                         
                           Δ 
                           
                             R 
                             , 
                             R 
                           
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               H 
                               ^ 
                             
                             
                               
                                 β 
                                 ′ 
                               
                               + 
                               γ 
                             
                             
                               R 
                               , 
                               R 
                             
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     Virtual inter-aural delays Δ L,L , Δ R,R , Δ L,R , and Δ R,L  based in the positions of the virtual speakers  10 L′ and  10 R′ and incorporated in left and right channels  h   L,C  and  h   R,C , are: 
               Δ     L   ,   L       =     ⌊         (       -     d     L   ,     C   ′           +     δ     L   ,   L         )     ⁢     f   s       c     ⌋                   Δ     R   ,   R       =     ⌊         (       -     d     R   ,     C   ′           +     δ     R   ,   R         )     ⁢     f   s       c     ⌋                 where   ,     
     ⁢       δ     L   .   L       =     {               a   ⁢           ⁢     cos   ⁡     (       α   ′     +   γ     )               0   &lt;     α   ′     ≤       π   2     -   γ                   -   a     ⁢           ⁢     cos   ⁡     (       α   ′     -     π   2     +   γ     )                   π   2     -   γ     &lt;     α   ′     ≤     π   2             ⁢     
     ⁢     δ     R   ,   R         =     {                 a   ⁢           ⁢     cos   ⁡     (       β   ′     +   γ     )               0   &lt;     β   ′     ≤       π   2     -   γ                   -   a     ⁢           ⁢     cos   ⁡     (       β   ′     -     π   2     +   γ     )                   π   2     -   γ     &lt;     β   ′     ≤     π   2             ⁢     
     ⁢   and   ⁢     
     ⁢     Δ     R   ,   L         =         ⌊         (       -     d     R   ,     C   ′           +     δ     R   ,   L         )     ⁢     f   s       c     ⌋     ⁢     
     ⁢     Δ     L   ,   R         =       ⌊         (       -     d     L   ,     C   ′           +     δ     L   ,   R         )     ⁢     f   s       c     ⌋     ⁢     
     ⁢   where         ,     
     ⁢       δ   RL     =     {               -     a   ⁡     (       π   2     -     β   ′     +   γ     )               0   &lt;     β   ′     ≤       π   2     -   γ                 -     a   ⁡     (       π   2     -     β   ′     +   γ     )                   π   2     -   γ     &lt;     β   ′     ≤     π   2             ⁢     
     ⁢     δ     L   ,   R         =     {           -     a   ⁡     (       π   2     -     α   ′     +   γ     )               0   &lt;     α   ′     ≤       π   2     -   γ                 -     a   ⁡     (       π   2     -     α   ′     +   γ     )                   π   2     -   γ     &lt;     α   ′     ≤     π   2                                     
and where the virtual inter-aural delays Δ X,Y  are in units of samples.
 
     A wide synthesis stereo filter  25  according to the present invention and corresponding to the visualization of  FIG. 4  is shown in  FIG. 5 . The filters  26 ,  28 ,  30 , and  32  represent the elements of H desired  and serve to create the desired wide stereo perception. The equalization filter G(z)  38  receives the summed outputs of the filters  26  and  30 , and  38  and  32 , summed at  34  and  36  respectively and serves to reduce or eliminate the effects of regular stereo perception. 
     Surround synthesis may be obtained by substituting -γ for γ to obtained: 
     
       
         
           
             
               Δ 
               
                 L 
                 , 
                 L 
               
             
             = 
             
               ⌊ 
               
                 
                   
                     ( 
                     
                       
                         - 
                         
                           d 
                           
                             L 
                             , 
                             
                               C 
                               ′ 
                             
                           
                         
                       
                       + 
                       
                         δ 
                         
                           L 
                           , 
                           L 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     f 
                     s 
                   
                 
                 c 
               
               ⌋ 
             
           
         
       
       
         
           
             
               Δ 
               
                 R 
                 , 
                 R 
               
             
             = 
             
               ⌊ 
               
                 
                   
                     ( 
                     
                       
                         - 
                         
                           d 
                           
                             R 
                             , 
                             
                               C 
                               ′ 
                             
                           
                         
                       
                       + 
                       
                         δ 
                         
                           R 
                           , 
                           R 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     f 
                     s 
                   
                 
                 c 
               
               ⌋ 
             
           
         
       
       
         
           
             where 
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   
                     
                       δ 
                       
                         L 
                         . 
                         L 
                       
                     
                     = 
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               α 
                               ′ 
                             
                             - 
                             γ 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     0 
                     &lt; 
                     
                       α 
                       ′ 
                     
                     ≤ 
                     
                       π 
                       2 
                     
                   
                 
               
               
                 
                   
                     
                       δ 
                       
                         R 
                         , 
                         R 
                       
                     
                     = 
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               β 
                               ′ 
                             
                             - 
                             γ 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       0 
                       &lt; 
                       
                         β 
                         ′ 
                       
                       ≤ 
                       
                         π 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               Δ 
               
                 R 
                 , 
                 L 
               
             
             = 
             
               ⌊ 
               
                 
                   
                     ( 
                     
                       
                         - 
                         
                           d 
                           
                             R 
                             , 
                             
                               C 
                               ′ 
                             
                           
                         
                       
                       + 
                       
                         δ 
                         
                           R 
                           , 
                           L 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     f 
                     s 
                   
                 
                 c 
               
               ⌋ 
             
           
         
       
       
         
           
             
               Δ 
               
                 L 
                 , 
                 R 
               
             
             = 
             
               ⌊ 
               
                 
                   
                     ( 
                     
                       
                         - 
                         
                           d 
                           
                             L 
                             , 
                             
                               C 
                               ′ 
                             
                           
                         
                       
                       + 
                       
                         δ 
                         
                           L 
                           , 
                           R 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     f 
                     s 
                   
                 
                 c 
               
               ⌋ 
             
           
         
       
       
         
           
             where 
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   
                     
                       
                         
                           
                             δ 
                             RL 
                           
                           = 
                           
                             - 
                             
                               a 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     π 
                                     2 
                                   
                                   - 
                                   
                                     β 
                                     ′ 
                                   
                                   - 
                                   γ 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           0 
                           &lt; 
                           
                             β 
                             ′ 
                           
                           ≤ 
                           
                             π 
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                               
                           
                           ⁢ 
                           
                             
                               δ 
                               
                                 L 
                                 , 
                                 R 
                               
                             
                             = 
                             
                               - 
                               
                                 a 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       π 
                                       2 
                                     
                                     - 
                                     
                                       α 
                                       ′ 
                                     
                                     - 
                                     γ 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           0 
                           &lt; 
                           
                             α 
                             ′ 
                           
                           ≤ 
                           
                             π 
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
 
                     
                   
                 
               
             
           
         
       
     
     A phantom center channel filter  39  according to the present invention providing widening along with generating a phantom center is shown in a lattice structure in  FIG. 6 . A pair of ipsilateral filters  42  and  48  and a pair of contralateral filters  44  and  46  may be determined from the 2×2 matrix G*H desired , where G includes H −1 . G and H desired  are computed as described above. In the general case, the pair of ipsilateral filters  42  and  48  are the diagonal terms of G*H desired , and the contralateral filters  44  and  46  are the off-diagonal terms of G*H desired . In special cases where the listener  12  is centered on the speakers  10 L and  10 R, the two diagonal terms are equal and the two off diagonal terms are equal so that the ipsilateral filters  42  and  48  may be obtained from the first row and first column of the frequency response matrix G*H desired  and the contralateral filters  44  and  46  may be obtained from the first row and second column of the frequency response matrix G*H desired . The matrix G*H desired  is computed at various frequency values and the inverse Fourier transform is taken to obtain the ipsilateral filters  42  and  48  and the contralateral filters  44  and  46  in the time domain. 
     The matrix G*H desired  is a 2×2 matrix for each frequency point. If there are 512 frequency points we obtain 512 matrices of 2×2 size. In the listener centered case, only the element in the first row and first column from each of the 512 2×2 matrices is taken to form a frequency response vector for the ipsilateral filters  42  and  48 . The frequency response vector is inverse Fourier transformed to obtain the ipsilateral time domain filters  42  and  48 . The process is repeated to obtain the contralateral filters  44  and  46  but selecting the element in the first row and second column. A second equalization filter G′  40 ,  50  provides the phantom center. The phantom center channel filter  39  may process either the inputs to a room equalizer or process the outputs of the room equalizer. 
     The method of the present invention may further be expanded to provide a perception of arcing. An arced stereo synthesis visualization  55  according to the present invention is shown in  FIG. 7 . A desired relative speaker to listener positioning for creating the impression of a widened and arcing according to the present invention is provided by a second left synthesized (or virtual) speaker  10 L″ shown displaced a distance p 1  to the left and δp 1  ahead of the speaker  10 L, and a second right synthesized (or virtual) speaker  10 R″ shown displaced a distance p 2  to the right and δp 2  ahead of the speaker  10 L. The following equations result: 
             Λ   =       tan     -   1       (       δ     p   ⁢           ⁢   1         p   1       )                   z   2     =       p   1   2     +     δ     p   1                     Ω   =     π   -   Λ   -   α                   d     LW   ,   C     2     =       d     L   ,   C     2     +     z   2     -     2   ⁢     zd     L   ,   C       ⁢   cos   ⁢           ⁢   Ω                   Δ   =       cos     -   1       (         z   2     +     d     LW   ,   C     2     -     d     L   ,   C     2         2   ⁢     zd     LW   ,   C           )                   α   ′     =     Δ   -   Λ           
where these terms may be substituted into the above equations for computing the inter-aural delays Δ X,Y  obtain widening and arcing according to the present invention.
 
     The methods of the present invention may further be expanded to include where: 
     the binaural modeled equalization matrix G(z) is lower order modeled with existing techniques; 
     simple delays and shadowing filters (one poll) are implemented; 
     the stereo-expansion system compensates for speaker room effects simultaneously; 
     multi-position and robustness is obtained with least-squares based binaural equalization filter matrix G(z), spatial derivatives/difference constraints etc. 
     speech—music discrimination for center channel synthesis with PC=−d T /2 and/or integrating with X L +X R  approach; 
     potential to pre-integrated with PrevEQ by using head diffraction model engaged beyond 1.5 kHz (that is, intensity differences) with speaker only response; 
     using all pass filters with group delays T 1   f&lt;1.5 kHz =c 1  and T 2   f&gt;1.5 kHz =c 2  for Δ L,R  (Δ R,L ); 
     torso modeling; and 
     distance or room-based function multiplying head-diffraction model. 
     The lattice form can be transformed to the shuffler form (as in Bauck et al, “Prospects of Transaural Recording,” Journal of Audio Eng. Soc., vol. 37 (1/2), January/February 1989). For example, assuming a 2×2 matrix X having elements S and A: 
             X   =     [         S       A           A       S         ]           
where S is the ipsilateral transfer function and A is the contralateral function The inverse Y of X is:
 
             Y   =       X     -   1       =       1       S   2     -     A   2         ⁡     [         S         -   A               -   A         S         ]               
and Y can be factored using eigenvalue/eigenvector decomposition as:
 
     
       
         
           
             Y 
             = 
             
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                   
                   ] 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 S 
                                 + 
                                 A 
                               
                               ) 
                             
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 S 
                                 - 
                                 A 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Note, in this form there are only two filters (i.e., 1/(2(S+A)) and 1/(2(S−A)) located diagonally instead of four filters. The closer these are to a value unity, the net transfer function Y since Y=[1 0;0 1] becomes relatively lossless at all frequencies which implies no distortion or artifacts. In this case the output as Y=[2 0;0 2] which implies YL=2*XL and YR=2*XR (i.e., the left channel is transmitted to the output simply gain changed by a factor of 2 and the right channel is transmitted to the output gain changed by a factor of 2). 
     Incorporating this concept into the present system, the inverse G=H^ (−1)  may be multiplied with H desired  and factored into shuffler form as:
 
RES =G*H desired  =H^ (−1) *H desired  =Y*H desired  
 
with H desired  being represented as H desired  =[L M;M L] where L and M are the desired ipsilateral and contralateral transfer functions (i.e., including the inter-aural delays and headshadow responses). Thus the resulting filters in lattice form can be expressed as:
 
     
       
         
           
             
               
                 
                   RES 
                   = 
                     
                   ⁢ 
                   
                     ( 
                     
                       1 
                       / 
                       
                         
                           
                             ( 
                             
                               
                                 S 
                                 ^ 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               - 
                               
                                 A 
                                 ^ 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             ) 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 S 
                                 - 
                                 A 
                               
                               ; 
                               
                                 
                                   - 
                                   A 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 S 
                               
                             
                             ] 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               M 
                             
                             ; 
                             
                               M 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               L 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     ( 
                     
                       1 
                       / 
                       
                         
                           ( 
                           
                             
                               S 
                               ^ 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             - 
                             
                               A 
                               ^ 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           ) 
                         
                         [ 
                         
                           
                             SL 
                             - 
                             
                               AM 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               SM 
                             
                             - 
                             AL 
                           
                           ; 
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     SM 
                     - 
                     
                       AL 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       SL 
                     
                     - 
                     AM 
                   
                   ] 
                 
               
             
           
         
       
     
     The above may be factored using eigen decomposition into: 
     
       
         
           
             
               
                 
                   RES 
                   = 
                     
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           RES 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               , 
                               1 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ; 
                       
                         0 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           RES 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               , 
                               2 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           1 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         ; 
                         
                           1 
                           - 
                           1 
                         
                       
                       ] 
                     
                     [ 
                     
                       
                         
                           
                             ( 
                             
                               L 
                               + 
                               M 
                             
                             ) 
                           
                           / 
                           2 
                         
                         * 
                         
                           ( 
                           
                             S 
                             + 
                             A 
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       ; 
                     
                   
                 
               
             
             
               
                 
                   
                       
                     ⁢ 
                     
                       0 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             L 
                             - 
                             M 
                           
                           ) 
                         
                         / 
                         2 
                       
                       * 
                       
                         ( 
                         
                           S 
                           - 
                           A 
                         
                         ) 
                       
                     
                     ] 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         1 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       ; 
                       
                         1 
                         - 
                         1 
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     The resulting shuffler filter is shown in  FIG. 8  where the two filters RES(1,1)  62  and RES(2,2)  64 , one in each channel, are transformed from the lattice structure of 
       FIG. 6 . The sum  58  of signals XL and XR is provided to RES(1,1) and the difference  60  of signal XR−XL is provided to RES(2,2)  64 . The signal XL is provided to the phantom gain G′  68  and the signal XR is provided to the phantom gain G′  70 . The difference  72  of the output of G′ 68 plus RES(1,1) 62 minus RES(2,2)  64  is output as YL and the sum  74  of the output of G′  70  plus RES(1,1)  62  plus RES(2,2)  64  is output as YL. 
     Examples of unsmoothed filters RES(1,1) and RES(2,2) are shown before smoothing in  FIGS. 9A and 9B . Smoother filters sRES(1,1) and sRES(2,2) are shown after complex smoothed (joint magnitude and phase) using a variable-octave complex smoother to remove unwanted temporal (magnitude and phase) variations that result in artifacts in the reproduced sound quality in  FIGS. 10A and 10B . In this example, the smoothing is 4 octave wide smoothing to remove unnecessary temporal variations so as to approximate a Kronecker delta function. This feature, in essence, provides a tradeoff between amount of spatialization and audio fidelity. The variable-octave complex smoothing allows high-resolution frequency smoothing in regions of the frequency response of the filter by retaining perceptual features in the frequency response of each of the filters which are dominant for accurate localization, while at the same time performing temporal smoothing to allow each filter to converge to a delta function such that RES matrix is close to [1 0;0 1] at each frequency bin for maintaining audio fidelity. The variable-octave complex-domain smoother is described in “Variable-Active Complex Smoothing for Loudspeaker-room Response Equalization” published in Proceedings of IEEE International Conference Consumer Electronics, Las Vegas Nev., January 2008, authored by S. Bharitkar, C. Kyriaskakis, and T. Holman. 
     For example, a complex-domain ⅓ octave full-band (0 Hz to Fs/2 where Fs=sampling frequency in Hz) smoothing may be performed, or 2-octaves wide full-band smoothing may be performed, or 1/12 th -octave smoothing between 1 kHz and 10 kHz may be performed (as the headshadow functions of  FIG. 2  show variations in this region) and 2-octave complex (joint magnitude and phase) smoothing may be performed in the other region (viz., [0 Hz, 1 kHz)U(10 kHz, Fs/2)). Subsequently, the smoothed filters sRES are transformed back into the lattice form of  FIG. 6  by the following transformation (where sRES(x,x) is the corresponding smoothed filter of the shuffler form RES(x,x)). 
     The resulting filters are: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         
                           [ 
                           
                             
                               1 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             ; 
                             
                               1 
                               - 
                               1 
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 sRES 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                             ; 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 sRES 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             1 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           ; 
                           
                             1 
                             - 
                             1 
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     [ 
                     
                       
                         
                           sRES 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               , 
                               1 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             sRES 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 , 
                                 2 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             sRES 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           sRES 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               , 
                               2 
                             
                             ) 
                           
                         
                       
                       ; 
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       sRES 
                       ⁡ 
                       
                         ( 
                         
                           1 
                           , 
                           1 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         sRES 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             , 
                             2 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         sRES 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             , 
                             1 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       sRES 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           , 
                           2 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     A method for providing a stereo-widened sound in a stereo speaker system is described in  FIG. 11 . The method includes determining speaker angles alpha and beta relative to a listener position wherein said speaker angles are computed using stereo speaker spacing and listener position at step  100 , determining inter-aural delays between the speakers and the listeners ears at step  102 , determining the headshadow responses associated with each ear relative to each of the speakers given the speaker angles at step  104 , equalizing the headshadow responses between the speakers and the listener ears at step  106 , determining virtual speaker angles alpha′ and beta′ relative to listener position at step  108 , determining virtual inter-aural delays between the speakers and the listeners ears for virtual speaker angles alpha′ and beta′ at step  110 , determining virtual headshadow responses associated with each ear relative to each of the virtual speakers given the virtual speaker angles at step  112 , determining stereo expansion filters from the headshadow responses and the virtual headshadow responses at step  114 , converting lattice form filters to shuffler form filters at step  116 , variable octave complex smoothing the shuffler filters at step  118 , and converting smoothed shuffler filters to smoothed lattice filters for performing spatialization and preserving the audio quality. 
     While the invention herein disclosed has been described by means of specific embodiments and applications thereof, numerous modifications and variations could be made thereto by those skilled in the art without departing from the scope of the invention set forth in the claims.