PATENT DOCUMENT

Abstract:
An acoustic apparatus including circuitry to correct for acoustic cross-coupling of acoustic drivers mounted in a common acoustic enclosure. A plurality of acoustic drivers are mounted in the acoustic enclosure so that motion of each of the acoustic drivers causes motion in each of the other acoustic drivers. A canceller cancels the motion of each of the acoustic drivers caused by motion of each of the other acoustic drivers. A cancellation adjuster cancels the motion of each of the acoustic drivers that may result from the operation of the canceller.

Full Description:
CLAIM OF PRIORITY 
       [0001]    This application is a continuation-in-part of, and claims priority to, U.S. patent application Ser. No. 11/499,014 filed Aug. 4, 2006 and published Feb. 7, 2008 as published Pat. App. US-2008-0031472-A1 and also claims priority to U.S. Provisional Patent App. 61/174,726, filed May 1, 2009. 
     
    
     BACKGROUND 
       [0002]    This specification describes a loudspeaker system in which two or more acoustic drivers share a common enclosure. 
       SUMMARY 
       [0003]    In one aspect, an apparatus includes an acoustic enclosure, a plurality of acoustic drivers mounted in the acoustic enclosure so that motion of each of the acoustic drivers causes motion in each of the other acoustic drivers, a canceller, to cancel the motion of each of the acoustic drivers caused by motion of each of the other acoustic drivers, and a cancellation adjuster, to cancel the motion of each of the acoustic drivers that may result from the operation of the canceller. The cancellation adjuster may adjust for undesirable phase and frequency response effects that result from the operation of the canceller. The 
         [0000]    cancellation adjuster may apply the transfer function matrix 
         [0000]    
       
         
           
             [ 
             
                 
               
                 
                   
                     
                       
                         H 
                         11 
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         H 
                         
                           1 
                            
                           
                               
                           
                            
                           n 
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         H 
                         
                           1 
                            
                           
                               
                           
                            
                           n 
                         
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         H 
                         nn 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    where each of the matrix elements H xy  represents a transfer function from an audio signal V x  applied to the input of acoustic driver x to motion represented by velocity S y  of acoustic driver y. The acoustic drivers may be a components of a directional array. The acoustic drivers may be components of a two-way speaker. 
         [0004]    In another aspect, a method of operating a loudspeaker having at least two acoustic drivers in a common enclosure, includes determining the effect of the motion of a first acoustic driver on the motion of a second acoustic driver; developing a first correction audio signal to correct for the effect of the motion of the first acoustic driver on the motion of the second acoustic driver; determining the effect on the motion of the first acoustic driver of the transducing of the correction audio signal by the second acoustic driver; and developing a second correction audio signal to correct for the effect on the motion of the first acoustic driver of the transducing of the first correction audio signal by the second acoustic driver. The correction audio signal may correct the frequency response and the phase effects on the motion of the first acoustic driver of the transducing of the correction audio signal by the second acoustic driver. The second correction audio signal may be 
         [0000]    
       
         
           
             
               1 
               
                 det 
                  
                 
                     
                 
                  
                 H 
               
             
             , 
           
         
       
     
         [0000]    where H is the transfer function matrix 
         [0000]    
       
         
           
             [ 
             
                 
               
                 
                   
                     
                       
                         H 
                         11 
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         H 
                         
                           1 
                            
                           
                               
                           
                            
                           n 
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         H 
                         
                           1 
                            
                           
                               
                           
                            
                           n 
                         
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         H 
                         nn 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    where the matrix elements H xy  represent the transfer function from an audio signal V x  applied to the input of acoustic driver x to motion represented by velocity S y  of acoustic driver y. The method may further include determining matrix elements H xy  by causing acoustic driver y to transduce an audio signal, and measuring the effect on acoustic driver x of the transducing by acoustic driver y by a laser vibrometer. The method of claim  8 , wherein the motion of acoustic driver is represented by a displacement. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
         [0005]      FIGS. 1A-1D  are block diagrams of an audio system; 
           [0006]      FIG. 2  is a block diagram of an audio system having cross-coupling canceller and a cancellation adjuster; 
           [0007]      FIG. 3  is a block diagram of an audio system showing elements of the canceller; 
           [0008]      FIG. 4  is a block diagram of an audio system showing elements of the canceller and the cancellation adjuster; 
           [0009]      FIG. 5  is a block diagram of an audio system having three transducer; 
           [0010]      FIG. 6  is a block diagram of an alternate configuration of an audio system having a cross-coupling canceller; 
           [0011]      FIG. 7  is s plot of cone velocity vs. frequency; and 
           [0012]      FIG. 8  is a plot of phase vs. frequency. 
       
    
    
     DETAILED DESCRIPTION 
       [0013]    Though the elements of several views of the drawing are shown and described as discrete elements in a block diagram and may be referred to as “circuitry”, unless otherwise indicated, the elements may be implemented as one of, or a combination of, analog circuitry, digital circuitry, or one or more microprocessors executing software instructions. The software instructions may include digital signal processing (DSP) instructions. Unless otherwise indicated, signal lines may be implemented as discrete analog or digital signal lines, as a single discrete digital signal line with appropriate signal processing to process separate streams of audio signals, or as elements of a wireless communication system. Unless otherwise indicated, audio signals may be encoded in either digital or analog form. For convenience, “radiating sound waves corresponding to channel x” will be expressed as “radiating channel x.” 
         [0014]    Referring to  FIG. 1A , there is shown a block diagram of an acoustic system. Audio signal source  10 A is coupled to acoustic driver  12 A that is mounted in enclosure  14 A. Audio signal source 10B is coupled to acoustic driver  12 B that is mounted in enclosure  14 B. Acoustic enclosure  14 A is acoustically and mechanically isolated from acoustic enclosure  14 B. Driving acoustic driver  12 A by an audio signal represented by voltage V 1  results in desired motion S 1  which results in the radiation of acoustic energy. The motion can be expressed as a velocity or a displacement; for convenience, the following explanation will express motion as a velocity. Driving acoustic driver  12 B by an audio signal represented by voltage V 2  results in desired motion S 2 . 
         [0015]    In the audio system of  FIG. 1B , audio signal source  10 A is coupled to acoustic driver  12 A. Audio signal source  10 B is coupled to acoustic driver  12 B. Acoustic drivers  12 A and  12 B are mounted in enclosure  14 , which has the same volume as enclosures  14 A and  14 B. Driving acoustic driver  12 A by an audio signal represented by voltage V 1  results in motion S 1 ′ which may not be equal to desired motion S 1  because of acoustic cross-coupling, either through the air volume in the shared enclosure or mechanical coupling through the shared enclosure, or both. Similarly, driving acoustic driver  12 B by an audio signal represented by voltage V 2  results in motion S 2 ′ which may not be equal to desired motion S 2 . 
         [0016]    The effect of cross-coupling can be seen in  FIG. 1C , in which applying an acoustic signal represented by voltage V 1  to acoustic driver  12 A and applying no signal (indicated by the dashed line between audio signal source  10 B and acoustic driver  12 B) to acoustic driver  12 B results in cross-coupling induced motion S cc  of acoustic driver  12 B. In  FIG. 1D , applying an acoustic signal represented by voltage V 2  to acoustic driver  12 B and applying no signal (indicated by the dashed line between audio signal source  10 A and acoustic driver  12 A) to acoustic driver  12 A results in cross-coupling induced motion S cc  of acoustic driver  12 A. For the purpose of the explanations following, transfer function H 11  is the transfer function from voltage V 1 . to velocity S 1 , transfer function H 12  is the transfer function from voltage V 2  to velocity S 1 , transfer function H 21  is the transfer function from voltage V 1 . to velocity S 2 , and transfer function H 22  is the transfer function from voltage V 2  to velocity S 2 . In the explanations that follow, an acoustic driver with an audio signal applied (such as acoustic driver  12 A of  FIG. 1C  and acoustic driver  12 B of  FIG. 1D ) will be referred to as a “primary acoustic driver”; an acoustic driver without a signal applied (for example acoustic driver  12 B of  FIG. 1C  and acoustic driver  12 A of  FIG. 1D ) that moves responsive to an audio signal being applied to a primary acoustic driver will be referred to as a “secondary acoustic driver”. 
         [0017]      FIG. 2  includes the elements of  FIG. 1B , and in addition includes a canceller  16 , cancellation adjuster  15 , and conventional signal processor  17 . The canceller  16  modifies the input audio signals U 1  and U 1  to cancel transfer function H 12  and transfer function H 21  (as indicated by the dashed lines) to provide modified signals V 1  and V 2  which result in the desired motion S 1  and S 2  of acoustic drivers  12 A and  12 B, respectively. The cancellation adjuster  15  adjusts the signal to cancel undesirable effects that may result from the operation of the canceller, such as effects on the phase or on the frequency response. The conventional signal processor  17  includes processing that is not related to cross-coupling cancellation, for example equalization for room effects; equalization for undesired effects on frequency response of the acoustic drivers, amplifiers, or other system components; time delays; array processing such as phase reversal or polarity inversions; and the like. Canceller  16 , cancellation adjuster  15 , and conventional signal processor  17  can be in any order. For clarity, conventional signal processor  17  will not be shown in subsequent figures. 
         [0018]    Actual implementations of acoustic system of  FIG. 2  is most conveniently performed by a digital signal processor. 
         [0019]      FIG. 3  shows the canceller  16  in more detail; cancellation adjuster  15  is not shown in this view and will be discussed below. Canceller  16  includes canceling transfer function C 11  coupling signal U 1  and summer  18 A, canceling transfer function C 21  coupling signal U 1  and summer  18 B, canceling transfer function C 22  coupling signal U 2  and summer  18 B, canceling transfer function C 12  coupling signal U 2  and summer  18 A. Summer  18 A is coupled to acoustic driver  12 A and summer  18 B is coupled to acoustic driver  12 B. 
         [0020]    Canceling transfer functions C 11 , C 21 , C 22 , and C 12  can be derived as follows. The relationships of  FIGS. 1C and 1D  can be expressed mathematically as 
         [0000]    
       
      
       H 
       11 
       ·V 
       1 
       +H 
       12 
       ·V 
       2 
       =S 
       1  
      
     
         [0000]    
       
      
       H 
       21 
       ·V 
       1 
       +H 
       22 
       ·V 
       2 
       =S 
       2  
      
     
         [0021]    The notation can be simplified by transforming this set of linear equations into matrix form. The transfer function matrix H contains all transmission paths in the system: 
         [0000]    
       
         
           
             H 
             = 
             
               [ 
               
                 
                   
                     
                       H 
                       11 
                     
                   
                   
                     
                       H 
                       12 
                     
                   
                 
                 
                   
                     
                       H 
                       21 
                     
                   
                   
                     
                       H 
                       22 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0022]    The input voltages are grouped into a vector v and the velocity or displacement into a vector S. In matrix notation, the system is described as 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         H 
                         11 
                       
                     
                     
                       
                         H 
                         12 
                       
                     
                   
                   
                     
                       
                         H 
                         21 
                       
                     
                     
                       
                         H 
                         22 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 ( 
                 
                   
                     
                       
                         V 
                         1 
                       
                     
                   
                   
                     
                       
                         V 
                         2 
                       
                     
                   
                 
                 ) 
               
             
             = 
             
               ( 
               
                 
                   
                     
                       S 
                       1 
                     
                   
                 
                 
                   
                     
                       S 
                       2 
                     
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    Or simply 
         [0000]    
       
      
       H·{right arrow over (V)}={right arrow over (S)} 
      
     
         [0023]    The relation between the input voltage and output voltage of the canceller is described by the linear equations: 
         [0000]    
       
      
       C 
       11 
       ·U 
       1 
       +C 
       12 
       ·U 
       2 
       =V 
       1  
      
     
         [0000]    
       
      
       C 
       21 
       ·U 
       1 
       +C 
       22 
       ·U 
       2 
       =V 
       2  
      
     
         [0024]    Or in matrix notation 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         C 
                         11 
                       
                     
                     
                       
                         C 
                         12 
                       
                     
                   
                   
                     
                       
                         C 
                         21 
                       
                     
                     
                       
                         C 
                         22 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 ( 
                 
                   
                     
                       
                         U 
                         1 
                       
                     
                   
                   
                     
                       
                         U 
                         2 
                       
                     
                   
                 
                 ) 
               
             
             = 
             
               ( 
               
                 
                   
                     
                       V 
                       1 
                     
                   
                 
                 
                   
                     
                       V 
                       2 
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               C 
               · 
               
                 U 
                 → 
               
             
             = 
             
               V 
               → 
             
           
         
       
     
         [0025]    The velocities of the acoustic drivers can now be expressed as a function of the input voltages to the canceller. 
         [0000]    
       
         
           
             
               H 
               · 
               C 
               · 
               
                 U 
                 → 
               
             
             = 
             
               
                 
                   S 
                   → 
                 
                  
                 
                   
 
                 
                 ⇔ 
                 
                   
                     [ 
                     
                       
                         
                           
                             H 
                             11 
                           
                         
                         
                           
                             H 
                             12 
                           
                         
                       
                       
                         
                           
                             H 
                             21 
                           
                         
                         
                           
                             H 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   
                     [ 
                     
                       
                         
                           
                             C 
                             11 
                           
                         
                         
                           
                             C 
                             12 
                           
                         
                       
                       
                         
                           
                             C 
                             21 
                           
                         
                         
                           
                             C 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   
                     ( 
                     
                       
                         
                           
                             U 
                             1 
                           
                         
                       
                       
                         
                           
                             U 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         S 
                         1 
                       
                     
                   
                   
                     
                       
                         S 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0026]    The overall system transfer function is described by the product of H and C. We can simplify this equation by defining a matrix T, which describes the entire system transfer function. 
         [0000]    
       
      
       H·C=T  
      
     
         [0027]    With this, the equation of the input-output relationship of the system can be simplified to: 
         [0000]    
       
         
           
             
               T 
               · 
               
                 U 
                 → 
               
             
             = 
             
               
                 
                   S 
                   → 
                 
                  
                 
                   
 
                 
                 ⇔ 
                 
                   
                     [ 
                     
                       
                         
                           
                             T 
                             11 
                           
                         
                         
                           
                             T 
                             12 
                           
                         
                       
                       
                         
                           
                             T 
                             21 
                           
                         
                         
                           
                             T 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   
                     ( 
                     
                       
                         
                           
                             U 
                             1 
                           
                         
                       
                       
                         
                           
                             U 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         S 
                         1 
                       
                     
                   
                   
                     
                       
                         S 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0028]    T also includes operations of conventional signal processor  17  and cancellation adjuster  15 . 
         [0029]    Assuming that the desired system transfer function T and the matrix H are known, the equation above can be solved for the canceller matrix C: 
         [0000]    
       
      
       C=H 
       −1 
       ·T  
      
       
         
           
             where H −1  is the matrix inverse of H: 
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     H 
                     
                       - 
                       1 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               H 
                               11 
                             
                           
                           
                             
                               H 
                               12 
                             
                           
                         
                         
                           
                             
                               H 
                               21 
                             
                           
                           
                             
                               H 
                               22 
                             
                           
                         
                       
                       ] 
                     
                     
                       - 
                       1 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         
                           
                             H 
                             11 
                           
                           · 
                           
                             H 
                             22 
                           
                         
                         - 
                         
                           
                             H 
                             12 
                           
                           · 
                           
                             H 
                             21 
                           
                         
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               H 
                               22 
                             
                           
                           
                             
                               H 
                               12 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 H 
                                 21 
                               
                             
                           
                           
                             
                               H 
                               11 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               H 
                               22 
                             
                           
                           
                             
                               - 
                               
                                 H 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 H 
                                 21 
                               
                             
                           
                           
                             
                               H 
                               11 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0000]    det H is the determinant of matrix H: 
         [0000]      det  H=H   11   ·H   22   −H   12   ·H   21    
         [0000]    Written out in matrix notation: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             C 
                             11 
                           
                         
                         
                           
                             C 
                             12 
                           
                         
                       
                       
                         
                           
                             C 
                             21 
                           
                         
                         
                           
                             C 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               H 
                               22 
                             
                           
                           
                             
                               - 
                               
                                 H 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 H 
                                 21 
                               
                             
                           
                           
                             
                               H 
                               11 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               T 
                               11 
                             
                           
                           
                             
                               T 
                               12 
                             
                           
                         
                         
                           
                             
                               T 
                               21 
                             
                           
                           
                             
                               T 
                               22 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         T 
                                         11 
                                       
                                       · 
                                       
                                         H 
                                         22 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       21 
                                     
                                      
                                     
                                       H 
                                       12 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         T 
                                         12 
                                       
                                       · 
                                       
                                         H 
                                         22 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       22 
                                     
                                     · 
                                     
                                       H 
                                       12 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           T 
                                           11 
                                         
                                       
                                       · 
                                       
                                         H 
                                         21 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       21 
                                     
                                     · 
                                     
                                       H 
                                       11 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           T 
                                           12 
                                         
                                       
                                       · 
                                       
                                         H 
                                         21 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       22 
                                     
                                     · 
                                     
                                       H 
                                       11 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0000]    Thus, the coefficients of C are 
         [0000]    
       
         
           
             
               C 
               11 
             
             = 
             
               
                 
                   
                     
                       
                         T 
                         11 
                       
                       · 
                       
                         H 
                         22 
                       
                     
                     - 
                     
                       
                         T 
                         21 
                       
                       · 
                       
                         H 
                         12 
                       
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   12 
                 
               
               = 
               
                 
                   
                     
                       T 
                       12 
                     
                     · 
                     
                       H 
                       22 
                     
                   
                   - 
                   
                     
                       T 
                       22 
                     
                     · 
                     
                       H 
                       12 
                     
                   
                 
                 
                   det 
                    
                   
                       
                   
                    
                   H 
                 
               
             
           
         
       
       
         
           
             
               C 
               21 
             
             = 
             
               
                 
                   
                     
                       
                         - 
                         
                           T 
                           11 
                         
                       
                       · 
                       
                         H 
                         21 
                       
                     
                     + 
                     
                       
                         T 
                         21 
                       
                       · 
                       
                         H 
                         11 
                       
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   22 
                 
               
               = 
               
                 
                   
                     
                       - 
                       
                         T 
                         12 
                       
                     
                     · 
                     
                       H 
                       21 
                     
                   
                   + 
                   
                     
                       T 
                       22 
                     
                     · 
                     
                       H 
                       11 
                     
                   
                 
                 
                   det 
                    
                   
                       
                   
                    
                   H 
                 
               
             
           
         
       
     
         [0000]    The denominators in these fractions are the same. 
         [0031]    The concept described above with canceller matrix and target function can be universally applied to enclosures with more than two acoustic drivers. For a system with n acoustic drivers the transfer function from the electrical inputs to the velocities of the cones would be described by an nxn matrix. The elements on the main diagonal describe the actively induced cone motion. All other elements describe the acoustic cross-coupling between all cones. The equalization matrix will also be an nxn matrix. 
         [0032]    It should be noted that this method can be applied to systems with different acoustic drivers, for example a loudspeaker system with a mid-range acoustic driver and a bass acoustic driver sharing the same acoustic volume. This will result in an asymmetric transfer function matrix but can be solved using the same methods. 
         [0033]    The elements in the target function matrix can describe arbitrary responses, such as general equalizer functions. This also allows to control the relative amplitude and phase of all transducers (e.g. for acoustic arrays). 
         [0034]    C can be calculated in either frequency or time domain. When the coefficients of the target matrix have been determined and the voltage to velocity or displacement transfer functions H xx  have been measured, the coefficients of C are derived from those functions as described above. 
         [0035]    Solving in the time domain always yields stable and causal filters. For this, the corresponding impulse responses for the matrix elements are determined. In this case, inverses of the impulse responses are determined by least-mean-squares (LMS) approximation. Information on LMS approximations can be found in Proakis and Manolakis,  Digital Signal Processing: Principles, Algorithms and Applications  Prentice Hall; 3rd edition (Oct. 5, 1995), ISBN-10: 0133737624, ISBN-13: 978-0133737622. The impulse responses can also be determined by other types of recursive filters. 
         [0036]    The general solution for a 2×2 target matrix (a system with two acoustic drivers) is: 
         [0000]    
       
         
           
             
               H 
               · 
               C 
               · 
               
                 U 
                 → 
               
             
             = 
             
               
                 
                   S 
                   → 
                 
                  
                 
                   
 
                 
                 ⇔ 
                 
                   
                     [ 
                     
                       
                         
                           
                             H 
                             11 
                           
                         
                         
                           
                             H 
                             12 
                           
                         
                       
                       
                         
                           
                             H 
                             21 
                           
                         
                         
                           
                             H 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   
                     [ 
                     
                       
                         
                           
                             C 
                             11 
                           
                         
                         
                           
                             C 
                             12 
                           
                         
                       
                       
                         
                           
                             C 
                             21 
                           
                         
                         
                           
                             C 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   · 
                   
                     ( 
                     
                       
                         
                           
                             U 
                             1 
                           
                         
                       
                       
                         
                           
                             U 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         S 
                         1 
                       
                     
                   
                   
                     
                       
                         S 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             C 
                             11 
                           
                         
                         
                           
                             C 
                             12 
                           
                         
                       
                       
                         
                           
                             C 
                             21 
                           
                         
                         
                           
                             C 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               H 
                               22 
                             
                           
                           
                             
                               - 
                               
                                 H 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 H 
                                 21 
                               
                             
                           
                           
                             
                               H 
                               11 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               T 
                               11 
                             
                           
                           
                             
                               T 
                               12 
                             
                           
                         
                         
                           
                             
                               T 
                               21 
                             
                           
                           
                             
                               T 
                               22 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         T 
                                         11 
                                       
                                       · 
                                       
                                         H 
                                         22 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       21 
                                     
                                      
                                     
                                       H 
                                       12 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         T 
                                         12 
                                       
                                       · 
                                       
                                         H 
                                         22 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       22 
                                     
                                     · 
                                     
                                       H 
                                       12 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           T 
                                           11 
                                         
                                       
                                       · 
                                       
                                         H 
                                         21 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       21 
                                     
                                     · 
                                     
                                       H 
                                       11 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           T 
                                           12 
                                         
                                       
                                       · 
                                       
                                         H 
                                         21 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       T 
                                       22 
                                     
                                     · 
                                     
                                       H 
                                       11 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0000]    This is the same solution as described above. 
         [0037]    Ideally, each acoustic driver&#39;s motion would be dependent on its corresponding input signal only. This would be represented as: 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         H 
                         11 
                       
                     
                     
                       
                         H 
                         12 
                       
                     
                   
                   
                     
                       
                         H 
                         21 
                       
                     
                     
                       
                         H 
                         22 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         C 
                         11 
                       
                     
                     
                       
                         C 
                         12 
                       
                     
                   
                   
                     
                       
                         C 
                         21 
                       
                     
                     
                       
                         C 
                         22 
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       T 
                       11 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       T 
                       22 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    Only the diagonal elements of the target matrix are non-zero here.
 
The solution of this system is
 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             C 
                             11 
                           
                         
                         
                           
                             C 
                             12 
                           
                         
                       
                       
                         
                           
                             C 
                             21 
                           
                         
                         
                           
                             C 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               H 
                               22 
                             
                           
                           
                             
                               - 
                               
                                 H 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 H 
                                 21 
                               
                             
                           
                           
                             
                               H 
                               11 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               T 
                               11 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               T 
                               22 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               
                                 T 
                                 11 
                               
                               · 
                               
                                 H 
                                 22 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   T 
                                   22 
                                 
                               
                               · 
                               
                                 H 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   T 
                                   11 
                                 
                               
                               · 
                               
                                 H 
                                 21 
                               
                             
                           
                           
                             
                               
                                 T 
                                 22 
                               
                               · 
                               
                                 H 
                                 11 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0000]    Thus, the coefficients of C are 
         [0000]    
       
         
           
             
               C 
               11 
             
             = 
             
               
                 
                   
                     
                       T 
                       11 
                     
                     · 
                     
                       H 
                       22 
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   12 
                 
               
               = 
               
                 
                   
                     - 
                     
                       T 
                       22 
                     
                   
                   · 
                   
                     H 
                     12 
                   
                 
                 
                   det 
                    
                   
                       
                   
                    
                   H 
                 
               
             
           
         
       
       
         
           
             
               C 
               21 
             
             = 
             
               
                 
                   
                     
                       - 
                       
                         T 
                         11 
                       
                     
                     · 
                     
                       H 
                       21 
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   22 
                 
               
               = 
               
                 
                   
                     T 
                     22 
                   
                   · 
                   
                     H 
                     11 
                   
                 
                 
                   det 
                    
                   
                       
                   
                    
                   H 
                 
               
             
           
         
       
     
         [0000]    Which can be expressed as: 
         [0000]    
       
         
           
             
               C 
               11 
             
             = 
             
               
                 1 
                 
                   det 
                    
                   
                       
                   
                    
                   H 
                 
               
               · 
               
                 T 
                 11 
               
               · 
               
                 H 
                 22 
               
             
           
         
       
       
         
           
             
               C 
               12 
             
             = 
             
               
                 
                   
                     1 
                     
                       det 
                        
                       
                           
                       
                        
                       H 
                     
                   
                   · 
                   
                     T 
                     22 
                   
                   · 
                   
                     ( 
                     
                       - 
                       
                         H 
                         12 
                       
                     
                     ) 
                   
                 
                  
                 
                   
 
                 
                  
                 
                   C 
                   21 
                 
               
               = 
               
                 
                   1 
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                 · 
                 
                   T 
                   11 
                 
                 · 
                 
                   ( 
                   
                     - 
                     
                       H 
                       21 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               C 
               22 
             
             = 
             
               
                 1 
                 
                   det 
                    
                   
                       
                   
                    
                   H 
                 
               
               · 
               
                 T 
                 22 
               
               · 
               
                 H 
                 11 
               
             
           
         
       
     
         [0038]    Common coefficients can be moved out of the canceller system, leaving coefficients that are different from unity only in the cross-paths. Referring to  FIG. 4 , the operations represented by transfer functions  30 A and  32 A, and  30 B, and  32 B comprise the operations performed by cancellation adjuster  15 . In other implementations, elements  30 B and  32 B (the target transfer functions elements T 11 −T nn ), may be applied by the canceller  16 . Performing transfer function elements T 11 −T nn  in either the cancellation adjuster  15  or the canceller  16  means that signal processing not related to cross-coupling, for example, for example equalization for room effects, equalization for undesired effects on frequency response of the acoustic drivers, amplifiers, or other system components, time delays, array processing such as phase reversal or polarity inversions, and the like can be done by the canceller  16  or the cancellation adjuster  15 , which eliminates the need for the conventional signal processor  17  of  FIG. 2 . 
         [0039]    If both acoustic drivers are driven by a single input (for example in a directional array), the elements of the second column in T are zero because the array is only driven by one input: 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         H 
                         11 
                       
                     
                     
                       
                         H 
                         12 
                       
                     
                   
                   
                     
                       
                         H 
                         21 
                       
                     
                     
                       
                         H 
                         22 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         C 
                         11 
                       
                     
                     
                       
                         C 
                         12 
                       
                     
                   
                   
                     
                       
                         C 
                         21 
                       
                     
                     
                       
                         C 
                         22 
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       T 
                       11 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     
                       T 
                       21 
                     
                   
                   
                     0 
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    The solution is 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             C 
                             11 
                           
                         
                         
                           
                             C 
                             12 
                           
                         
                       
                       
                         
                           
                             C 
                             21 
                           
                         
                         
                           
                             C 
                             22 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               H 
                               22 
                             
                           
                           
                             
                               - 
                               
                                 H 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 H 
                                 21 
                               
                             
                           
                           
                             
                               H 
                               11 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               T 
                               11 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               T 
                               21 
                             
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       1 
                       
                         det 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   T 
                                   11 
                                 
                                 · 
                                 
                                   H 
                                   22 
                                 
                               
                               - 
                               
                                 
                                   T 
                                   21 
                                 
                                 · 
                                 
                                   H 
                                   12 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     T 
                                     11 
                                   
                                 
                                 · 
                                 
                                   H 
                                   21 
                                 
                               
                               + 
                               
                                 
                                   T 
                                   21 
                                 
                                 · 
                                 
                                   H 
                                   11 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0000]    The elements of C are 
         [0000]    
       
         
           
             
               C 
               11 
             
             = 
             
               
                 
                   
                     
                       
                         T 
                         11 
                       
                       · 
                       
                         H 
                         22 
                       
                     
                     - 
                     
                       
                         T 
                         21 
                       
                       · 
                       
                         H 
                         12 
                       
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   12 
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
               C 
               21 
             
             = 
             
               
                 
                   
                     
                       
                         - 
                         
                           T 
                           11 
                         
                       
                       · 
                       
                         H 
                         21 
                       
                     
                     + 
                     
                       
                         T 
                         21 
                       
                       · 
                       
                         H 
                         11 
                       
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   22 
                 
               
               = 
               0 
             
           
         
       
     
         [0040]    A special case of this operating mode is stopping the motion of the second cone, as described previously. In this case, T 21  is also 0. The elements of C are 
         [0000]    
       
         
           
             
               C 
               11 
             
             = 
             
               
                 
                   
                     
                       T 
                       11 
                     
                     · 
                     
                       H 
                       22 
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   12 
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
               C 
               21 
             
             = 
             
               
                 
                   
                     
                       - 
                       
                         T 
                         11 
                       
                     
                     · 
                     
                       H 
                       21 
                     
                   
                   
                     det 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   22 
                 
               
               = 
               0 
             
           
         
       
     
         [0000]    In this case, the term 
         [0000]    
       
         
           
             
               T 
               11 
             
             
               det 
                
               
                   
               
                
               H 
             
           
         
       
     
         [0000]    is common to both elements and can be moved out in front of the system, leaving only H 22  and −H 21  as filter terms. 
         [0041]      FIG. 5  shows an implementation with three acoustic drivers,  12 A,  12 B, and  12 C, three input signals,  10 A,  10 B, and  10 C, sharing a common enclosure  14 . This implementation includes the elements of  FIG. 3 , and in addition there are canceling transfer functions C 31 , C 32 , and C 33 , coupling input signals U 1 , U 2 , and U 3 , respectively, with a summer  18 C, canceling transfer function C 13  coupling input signal U 3  with summer  18 A, and canceling transfer function C 12  coupling input signal U 3  with summer  18 B. Summer  18 C is coupled to acoustic driver  12 C. 
         [0042]    Again, the system can be described in matrix notation: 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         H 
                         11 
                       
                     
                     
                       
                         H 
                         12 
                       
                     
                     
                       
                         H 
                         13 
                       
                     
                   
                   
                     
                       
                         H 
                         21 
                       
                     
                     
                       
                         H 
                         22 
                       
                     
                     
                       
                         H 
                         23 
                       
                     
                   
                   
                     
                       
                         H 
                         31 
                       
                     
                     
                       
                         H 
                         32 
                       
                     
                     
                       
                         H 
                         33 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         C 
                         11 
                       
                     
                     
                       
                         C 
                         12 
                       
                     
                     
                       
                         C 
                         13 
                       
                     
                   
                   
                     
                       
                         C 
                         21 
                       
                     
                     
                       
                         C 
                         22 
                       
                     
                     
                       
                         C 
                         23 
                       
                     
                   
                   
                     
                       
                         C 
                         31 
                       
                     
                     
                       
                         C 
                         32 
                       
                     
                     
                       
                         C 
                         33 
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       T 
                       11 
                     
                   
                   
                     
                       T 
                       12 
                     
                   
                   
                     
                       T 
                       13 
                     
                   
                 
                 
                   
                     
                       T 
                       21 
                     
                   
                   
                     
                       T 
                       22 
                     
                   
                   
                     
                       T 
                       23 
                     
                   
                 
                 
                   
                     
                       T 
                       31 
                     
                   
                   
                     
                       T 
                       32 
                     
                   
                   
                     
                       T 
                       33 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    The solution is 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       C 
                       11 
                     
                   
                   
                     
                       C 
                       12 
                     
                   
                   
                     
                       C 
                       13 
                     
                   
                 
                 
                   
                     
                       C 
                       21 
                     
                   
                   
                     
                       C 
                       22 
                     
                   
                   
                     
                       C 
                       23 
                     
                   
                 
                 
                   
                     
                       C 
                       31 
                     
                   
                   
                     
                       C 
                       32 
                     
                   
                   
                     
                       C 
                       33 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 1 
                 
                   det 
                    
                   
                       
                   
                    
                   H 
                 
               
               · 
               
                 [ 
                 
                     
                 
                  
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   H 
                                   22 
                                 
                                 · 
                                 
                                   H 
                                   33 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 23 
                               
                               · 
                               
                                 H 
                                 32 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     H 
                                     12 
                                   
                                 
                                 · 
                                 
                                   H 
                                   33 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 13 
                               
                               · 
                               
                                 H 
                                 32 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   H 
                                   12 
                                 
                                 · 
                                 
                                   H 
                                   23 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 13 
                               
                               · 
                               
                                 H 
                                 22 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     H 
                                     21 
                                   
                                 
                                 · 
                                 
                                   H 
                                   33 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 23 
                               
                               · 
                               
                                 H 
                                 31 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                 · 
                                 
                                   H 
                                   33 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 13 
                               
                               · 
                               
                                 H 
                                 31 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     H 
                                     11 
                                   
                                 
                                 · 
                                 
                                   H 
                                   23 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 13 
                               
                               · 
                               
                                 H 
                                 21 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 
                                   H 
                                   21 
                                 
                                 · 
                                 
                                   H 
                                   32 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 22 
                               
                               · 
                               
                                 H 
                                 31 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     H 
                                     11 
                                   
                                 
                                 · 
                                 
                                   H 
                                   32 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 12 
                               
                               · 
                               
                                 H 
                                 31 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   H 
                                   11 
                                 
                                 · 
                                 
                                   H 
                                   22 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 12 
                               
                               · 
                               
                                 H 
                                 21 
                               
                             
                           
                         
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         T 
                         11 
                       
                     
                     
                       
                         T 
                         12 
                       
                     
                     
                       
                         T 
                         13 
                       
                     
                   
                   
                     
                       
                         T 
                         21 
                       
                     
                     
                       
                         T 
                         22 
                       
                     
                     
                       
                         T 
                         23 
                       
                     
                   
                   
                     
                       
                         T 
                         31 
                       
                     
                     
                       
                         T 
                         32 
                       
                     
                     
                       
                         T 
                         33 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
       With 
       [0043]      det  H=H   11   ·H   22   ·H   33   −H   11   ·H   23   ·H   32   −H   21   ·H   12   ·H   33   +H   21   ·H   13   ·H   32   −H   31   ·H   12   ·H   23   −H   31   ·H   13   ·H   22    
         [0000]    The final solutions for the elements of C are lengthy terms that are not shown here. 
         [0044]    The derivation of cancellation transfer functions for implementations with three acoustic drivers sharing the same enclosure can be applied to implementations with more than three acoustic drivers. 
         [0045]    The elements of H are determined using a cone displacement or velocity measurement. Laser vibrometers are particularly useful for this purpose because they require no physical contact with the cone&#39;s surface and do not affect its mobility. The laser vibrometer outputs a voltage that is proportional to the measured velocity or displacement. 
         [0046]    For an enclosure with two acoustic drivers, transfer function H 11  is measured by connecting two power amplifiers (not shown) to the two acoustic drivers and driving acoustic driver  12 A with the measurement signal. Acoustic driver  12 B is connected to its own amplifier that is powered up but which does not get an input signal. The laser vibrometer measures the cone motion of acoustic driver  12 A. Transfer function h 12  is measured by using the same setup and directing the laser at Driver  2 . 
         [0047]    The same technique can be used to measure transfer function H xy  in a system with y acoustic drivers by causing acoustic driver y to transduce an audio signal and measuring the effect on acoustic driver x using the laser vibrometer. 
         [0048]    Transfer function H 22  is measured like transfer function H 11 , only that now the amplifier of acoustic driver  12 A has no input signal and acoustic driver  12 B gets the measurement signal. Transfer function H 21  is then determined by directing the laser vibrometer at acoustic driver  12 A again while exciting acoustic driver  12 B. 
         [0049]    A simpler system for the compensation of cross-talk in an enclosure includes adding a phase inverted transfer function of voltage U 1  to velocity S 2  to the input voltage of Acoustic driver  12 B. This solution is shown in  FIG. 6 . The embodiment of  FIG. 5  is similar to the embodiment of  FIGS. 2 and 3 , but does not have the cancellation adjuster  15 . The conventional signal processor  17  of  FIG. 2  is not shown in  FIG. 5 . 
         [0050]    In the implementation of  FIG. 6 , canceller  16  includes a first filter  116 A, coupling audio signal source  10 A and summer  18 - 2 , and a second filter  116 B coupling audio signal source  10 B and summer  18 - 1 . In the embodiment of  FIG. 2 , the movement S 1  and S 2  of acoustic drivers  12 A and  12 B, respectively, in the absence of filters  116 A and  116 B can be expressed as 
         [0000]        S   1   =U   2   ·H   12   +U   1   ·H   11    (1) 
         [0000]        S   2   =U   1   ·H   21   +U   S   ·H   22    (2) 
         [0000]    now we can define functions based on the transfer functions H 12  , H 21 , H 11  and H 22  as: 
         [0000]    
       
         
           
             
               G 
               12 
             
             = 
             
               
                 
                   
                     H 
                     12 
                   
                   
                     H 
                     11 
                   
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 
                   G 
                   21 
                 
               
               = 
               
                 
                   H 
                   21 
                 
                 
                   H 
                   22 
                 
               
             
           
         
       
     
         [0000]    and apply G 21  at filter  116 A and G 12  at filter  116 B, resulting in modified movements S′ 1  and S′ 2  as: 
         [0000]    
       
      
       S′ 
       1 
       =S 
       1 
       −U 
       2 
       ·G 
       12 
       ·H 
       11  
      
     
         [0000]        S′   2   =S   2   −U   1   ·G   21   ·H   22 . 
         [0000]    Substituting equations (1) and (2) for S 1  and S 2  respectively gives 
         [0000]    
       
         
           
             
               S 
               1 
               ′ 
             
             = 
             
               
                 
                   U 
                   2 
                 
                 · 
                 
                   H 
                   12 
                 
               
               + 
               
                 
                   U 
                   1 
                 
                 · 
                 
                   H 
                   11 
                 
               
               - 
               
                 
                   U 
                   2 
                 
                 · 
                 
                   
                     H 
                     12 
                   
                   
                     H 
                     11 
                   
                 
                 · 
                 
                   H 
                   11 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               S 
               2 
               ′ 
             
             = 
             
               
                 
                   U 
                   1 
                 
                 · 
                 
                   H 
                   12 
                 
               
               + 
               
                 
                   U 
                   2 
                 
                 · 
                 
                   H 
                   22 
                 
               
               - 
               
                 
                   U 
                   2 
                 
                 · 
                 
                   
                     H 
                     12 
                   
                   
                     H 
                     22 
                   
                 
                 · 
                 
                   
                     H 
                     22 
                   
                   . 
                 
               
             
           
         
       
     
         [0000]    The first and third terms cancel, resulting in 
         [0000]        S′   1   =U   1   ·H   11  and 
         [0000]        S′   2   =U   2   ·H   22 , 
         [0000]    Which means that the cross-coupling effects have been eliminated. 
         [0051]    The system of  FIG. 6  provides close results (typically within 1 dB) in the common case in which the cone motion induced by cross-coupling is small relative to the cone motion induced by the direct signal and/or in the case in which the acoustic drivers are nearly identical, which is often the case of the elements of a directional array. In the case of directional arrays, experiments suggest that the cross-talk terms in the matrix Hare in the order of −10 dB. Usually the signal of the canceling transducer is attenuated by 3 to 10 dB. The system of  FIG. 6  is substantially equivalent to the system disclosed in U.S. patent application Ser. No. 11/499,014. 
         [0052]      FIG. 7  shows measurements illustrating the effect of the canceller. Curve  20  is the cone velocity of a primary acoustic driver. (Curve  20  is substantially identical with the canceller  16  in operation as it is with the canceller  16  not in operation.) Curve  22  shows the cone velocity of a secondary driver without the canceller  16  in operation, essentially showing the cross-coupling effect. Curve  24  shows the cone velocity of the secondary acoustic driver with the canceller  16  in operation. Curve  24  is approximately 10 to 20 dB less than curve  22 , indicating that the canceller reduces the effect of the cross-coupling by 10 to 20 dB. 
         [0053]      FIG. 8  shows the effect on phase of canceller  16 . In the test illustrated in  FIG. 7 , it is assumed that a constant phase difference of 90 degrees is to be maintained across the entire frequency range. The 90 degree phase shift can be created by filtering the signal with a Hilbert transform. Curve  26  shows the phase difference between the cone velocity of a primary driver and the cone velocity of a secondary driver with the canceller  16  not operating and with a Hilbert transform introduced into the secondary path. Below resonance (for this system approximately 190 Hz), the phase difference varies significantly from 90 degrees. Curve  28  shows the phase difference between the cone velocity of a primary driver and the cone velocity of a secondary driver with the canceller  16  operating and with a Hilbert transform introduced into the secondary path. The phase difference varies from 90 degrees by less than 10 degrees over most of the range of operation of the audio system. 
         [0054]    Numerous uses of and departures from the specific apparatus and techniques disclosed herein may be made without departing from the inventive concepts. Consequently, the invention is to be construed as embracing each and every novel feature and novel combination of features disclosed herein and limited only by the spirit and scope of the appended claims.