Abstract:
A method and apparatus for enhancing cutoff detection of a loudspeaker. The method comprising retrieving a loudspeaker model cutoff and model error, generating a probability distribution of the cutoff frequency based on the retrieved models, and utilizing the generated probability distribution to enhance the detection of the cutoff of the loudspeaker.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    Embodiments of the present invention generally relate to a method and apparatus for loudspeaker cutoff detection. 
         [0003]    2. Background of the Invention 
         [0004]    For applications such as room equalization, loudspeaker equalization and bass management, it is sometimes necessary to measure the frequency response of the loudspeakers. If the low-frequency cutoff of the loudspeakers can be determined, this information can be used to effectively apply bass management, i.e. remove poorly reproduced frequencies and route them to a better loudspeaker such as a subwoofer. However the measured spectrum of a loudspeaker usually contains irregularities caused by reflections and noise, making cutoff detection difficult. 
         [0005]    Therefore, there is a need for an improved loudspeaker cutoff detection method and apparatus. 
       SUMMARY OF THE INVENTION 
       [0006]    Embodiments of the present invention relate to a method and apparatus for enhancing cutoff detection of a loudspeaker. The method comprising retrieving a loudspeaker model cutoff and model error, generating a probability distribution of the cutoff frequency based on the retrieved models, and utilizing the generated probability distribution to enhance the detection of the cutoff of the loudspeaker. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]    So that the manner in which the above recited features of the present invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments. In this application, a computer readable processor is any medium accessible by a computer for saving, writing, archiving, executing and/or accessing data. Furthermore, the method described herein may be coupled to a processing unit, wherein said processing unit is capable of performing the method. 
           [0008]      FIG. 1  is an embodiment of a bass management filter design; 
           [0009]      FIG. 2  is an embodiment of a front left loudspeaker measurement; 
           [0010]      FIG. 3  is an embodiment of an effect of M as a shape parameter for critical u=0.6436, maxiflat u=1.5538 and huge u=1000.0 cases; 
           [0011]      FIG. 4  is an embodiment of a normalized probability of cutoff frequency ω c  with u from 0.5 to 500.0 and ω c  from 1.0 Hz to 748.5 Hz (thatched region is unrealizable); 
           [0012]      FIG. 5  is an embodiment of an audio based method; 
           [0013]      FIG. 6  is a flow diagram depicting an embodiment of a method for enhancing a loudspeaker cutoff detection; and 
           [0014]      FIG. 7  is a flow diagram depicting an embodiment of a normalization method to make a probability distribution. 
       
    
    
     DETAILED DESCRIPTION 
       [0015]    Bass management refers to routing the low frequency part of the signal to the most effective transducer, typically a subwoofer. Thus, the upper cutoff frequency of the subwoofer and lower cutoff frequencies of the other loudspeakers are usually known. If a subwoofer is not available, a technique, such as bass-boost (creates the sensation of more bass) may be applied. Such technique may be utilized when the loudspeaker cutoff is known to be too high. For these and other applications, it is useful to be able to estimate the lower cutoff frequency of regular loudspeakers.  FIG. 1  shows how a loudspeaker measurement is taken with a microphone, analyzed for cutoff frequency, which is then used to design or choose appropriate bass management filters. 
         [0016]    The measurement may be the same as loudspeaker equalization. Loudspeaker equalization refers to filters applied to a signal which are designed to compensate for the loudspeaker response. Generally, a known test signal is applied to the loudspeaker. The output is picked up by a microphone with a known frequency response. The unknown system, such as, amplifier, loud-speaker, environment, may be tested by applying a known test signal and recording the output. The frequency response may be derived using standard techniques. This measured frequency response, used primarily to design equalization filters, in principle may be used for several addition purposes including distance detection, polarity detection and cutoff detection. However, the spectrum of the measured system is typically not smooth, as shown in  FIG. 2 .  FIG. 2  is an embodiment of a measurement for the front left speaker of a 5 speaker plus woofer system. The irregularity in the spectrum makes accurate cutoff estimation difficult. 
         [0017]    The basic approach of this method is to generate a probability distribution of the cutoff frequency based on a model of loudspeaker cutoff and a Gaussian model of error. The error is the difference between the model and measurement. The error is caused by several factors, such as, background noise, measurement error, and room and speaker reflections. Such error may effect choosing the wrong model function. 
         [0018]    The background noise and measurement error are likely to be approximately Gaussian. However, assuming the loudspeaker model is accurate, the largest source of error is usually the room and speaker reflections, which are generally non-Gaussian. Using a Gaussian error model may lead to relatively straight forward mathematical formulations. 
         [0019]    After the loudspeaker model and error model are set, a probability distribution for the cutoff frequency remains, which may also require utilizing cutoff frequency as one of the parameters, applying Bayes&#39; Theorem and eliminating the other “nuisance” parameters. Finally, this distribution can be analyzed and action taken based on the result. 
         [0020]    A closed-box loudspeaker system model is 
         [0000]    
       
         
           
             
               
                 
                   
                     G 
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                       ( 
                       s 
                       ) 
                     
                   
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                         s 
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                         T 
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                         2 
                       
                     
                     
                       
                         
                           s 
                           2 
                         
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                           T 
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                       + 
                       
                         
                           sT 
                           C 
                         
                         / 
                         
                           Q 
                           TC 
                         
                       
                       + 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Q TC  is the total Q of the system at f C , with f C  being the resonance frequency of closed-box system, and T C  is the time constant 1/2πf C . The frequency response of this model is 
         [0000]    
       
         
           
             
               
                 
                   
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                       ( 
                       
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                         ω 
                       
                       ) 
                     
                   
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                         - 
                         
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                           2 
                         
                       
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                   ( 
                   2 
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         [0000]    and the magnitude response is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         G 
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                           ( 
                           
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                           ( 
                           jω 
                           ) 
                         
                       
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                             ( 
                             
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                         _ 
                       
                     
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                     = 
                     
                       
                         
                           
                             ω 
                             4 
                           
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                             T 
                             C 
                             4 
                           
                         
                         
                           
                             
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                                 ′ 
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                                  
                                 1 
                               
                             
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                               T 
                               C 
                               4 
                             
                           
                           - 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               ω 
                               2 
                             
                              
                             
                               T 
                               C 
                               2 
                             
                           
                           + 
                           
                             
                               
                                 ω 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     C 
                                   
                                   / 
                                   
                                     Q 
                                     TC 
                                   
                                 
                                 ) 
                               
                             
                             2 
                           
                           + 
                           1 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Now the cutoff frequency ω c  is defined as the point at which 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                         G 
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                           ) 
                         
                       
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                   = 
                   
                     
                       1 
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using this constraint and solving for T C /Q TC  we have 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ω 
                       c 
                       4 
                     
                      
                     
                       T 
                       C 
                       4 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           
                             ω 
                             c 
                             4 
                           
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                             T 
                             C 
                             4 
                           
                         
                         - 
                         
                           2 
                            
                           
                               
                           
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                             c 
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                               c 
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                               ( 
                               
                                 
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                                   C 
                                 
                                 / 
                                 
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                                   TC 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                         + 
                         1 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       
                         ω 
                         c 
                         4 
                       
                        
                       
                         T 
                         C 
                         4 
                       
                     
                     - 
                     
                       
                         ω 
                         c 
                         4 
                       
                        
                       
                         T 
                         C 
                         4 
                       
                     
                     + 
                     
                       2 
                        
                       
                           
                       
                        
                       
                         ω 
                         c 
                         2 
                       
                        
                       
                         T 
                         C 
                         2 
                       
                     
                     - 
                     1 
                   
                   = 
                   
                     
                       
                         ω 
                         c 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             T 
                             C 
                           
                           / 
                           
                             Q 
                             TC 
                           
                         
                         ) 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       T 
                       C 
                     
                     / 
                     
                       Q 
                       TC 
                     
                   
                   = 
                   
                     
                       
                         
                           ω 
                           c 
                           2 
                         
                          
                         
                           T 
                           C 
                           4 
                         
                       
                       + 
                       
                         2 
                          
                         
                           T 
                           C 
                           2 
                         
                       
                       - 
                       
                         1 
                         
                           ω 
                           c 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substituting (7) into (3) gives 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         G 
                          
                         
                           ( 
                           
                             j 
                              
                             
                                 
                             
                              
                             ω 
                           
                           ) 
                         
                       
                        
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           ω 
                           4 
                         
                          
                         
                           T 
                           C 
                           4 
                         
                       
                       
                         
                           
                             ω 
                             4 
                           
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                             T 
                             C 
                             4 
                           
                         
                         - 
                         
                           2 
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                             ω 
                             2 
                           
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                             T 
                             C 
                             2 
                           
                         
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ω 
                                   c 
                                   2 
                                 
                                  
                                 
                                   T 
                                   C 
                                   4 
                                 
                               
                               + 
                               
                                 2 
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                                   C 
                                   2 
                                 
                               
                               - 
                               
                                 1 
                                 
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                                   c 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         1 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                     = 
                     
                       
                         
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                           C 
                           4 
                         
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                           4 
                         
                       
                       
                         1 
                         - 
                         
                           
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                             2 
                           
                           
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                             c 
                             2 
                           
                         
                         + 
                         
                           
                             T 
                             C 
                             4 
                           
                            
                           
                             ω 
                             c 
                             2 
                           
                            
                           
                             ω 
                             2 
                           
                         
                         + 
                         
                           
                             T 
                             C 
                             4 
                           
                            
                           
                             ω 
                             4 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    eliminating the Q TC  parameter and introducing the cutoff ω c  as a new parameter. 
         [0021]    Equation (8) takes two parameters, T C  and ω c , and one variable ω which represents a frequency. Since the data is taken at discrete bin frequencies, we will usually index this variable with k as ω k  to mean the frequency at the k th  bin which can be interpreted in Hz depending on sampling rate and FFT size. To remain neutral during calculation frequency is measured in bins, i.e. ω k =k. However in this paper the sampling rate is always 48 kHz and the FFT size is always 32,768 giving a conversion factor of ≈1.46475 Hz per bin. It is also convenient to write the cutoff frequency ω c  on the same scale so that ω c =ω k  when c=k. However c need not be restricted to be an integer. 
         [0022]    The T C  parameter determines the shape of the model frequency response once ω c  is fixed. However the effect of T C  depends on ω c . For instance a given value of T C  may make the frequency response peaky for some an and very flat for other ω c . This is due to the fact that scaling ω c  and ω by the same amount α in (8) gives 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         α 
                         4 
                       
                        
                       
                         T 
                         C 
                         4 
                       
                        
                       
                         ω 
                         4 
                       
                     
                     
                       1 
                       - 
                       
                         
                           ω 
                           2 
                         
                         
                           ω 
                           c 
                           2 
                         
                       
                       + 
                       
                         
                           α 
                           4 
                         
                          
                         
                           T 
                           C 
                           4 
                         
                          
                         
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                           c 
                           4 
                         
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                           ω 
                           2 
                         
                       
                       + 
                       
                         
                           α 
                           4 
                         
                          
                         
                           T 
                           C 
                           4 
                         
                          
                         
                           ω 
                           4 
                         
                       
                     
                   
                   ≠ 
                   
                     
                       
                         
                           T 
                           C 
                           4 
                         
                          
                         
                           ω 
                           4 
                         
                       
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                           
                             ω 
                             c 
                             2 
                           
                         
                         + 
                         
                           
                             T 
                             C 
                             4 
                           
                            
                           
                             ω 
                             c 
                             2 
                           
                            
                           
                             ω 
                             2 
                           
                         
                         + 
                         
                           
                             T 
                             C 
                             4 
                           
                            
                           
                             ω 
                             4 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    However making the substitution u−ω c T C  in (8) gives 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         G 
                          
                         
                           ( 
                           
                             j 
                              
                             
                                 
                             
                              
                             ω 
                           
                           ) 
                         
                       
                        
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           
                             u 
                             ′1 
                           
                           
                             ω 
                             c 
                             ′1 
                           
                         
                          
                         
                           ω 
                           4 
                         
                       
                       
                         1 
                         - 
                         
                           
                             ω 
                             2 
                           
                           
                             ω 
                             c 
                             2 
                           
                         
                         + 
                         
                           
                             
                               u 
                               4 
                             
                             
                               ω 
                               c 
                               2 
                             
                           
                            
                           
                             ω 
                             2 
                           
                         
                         + 
                         
                           
                             
                               u 
                               4 
                             
                             
                               ω 
                               c 
                               4 
                             
                           
                            
                           
                             ω 
                             4 
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                     = 
                     
                       
                         
                           u 
                           4 
                         
                          
                         
                           ω 
                           4 
                         
                       
                       
                         
                           ω 
                           c 
                           4 
                         
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             ω 
                             c 
                             2 
                           
                         
                         + 
                         
                           
                             u 
                             4 
                           
                            
                           
                             ω 
                             2 
                           
                            
                           
                             ω 
                             c 
                             2 
                           
                         
                         + 
                         
                           
                             u 
                             4 
                           
                            
                           
                             ω 
                             4 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    so that after scaling ω and ω c  by α it becomes the case that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         α 
                         4 
                       
                        
                       
                         u 
                         4 
                       
                        
                       
                         ω 
                         4 
                       
                     
                     
                       
                         
                           
                             
                               
                                 α 
                                 4 
                               
                                
                               
                                 ω 
                                 c 
                                 4 
                               
                             
                             - 
                             
                               
                                 α 
                                 4 
                               
                                
                               
                                 ω 
                                 2 
                               
                                
                               
                                 ω 
                                 c 
                                 2 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 α 
                                 4 
                               
                                
                               
                                 u 
                                 4 
                               
                                
                               
                                 ω 
                                 2 
                               
                                
                               
                                 ω 
                                 c 
                                 2 
                               
                             
                             + 
                             
                               
                                 α 
                                 4 
                               
                                
                               
                                 u 
                                 4 
                               
                                
                               
                                 ω 
                                 4 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         u 
                         4 
                       
                        
                       
                         ω 
                         4 
                       
                     
                     
                       
                         ω 
                         c 
                         4 
                       
                       - 
                       
                         
                           ω 
                           2 
                         
                          
                         
                           ω 
                           c 
                           2 
                         
                       
                       + 
                       
                         
                           u 
                           4 
                         
                          
                         
                           ω 
                           2 
                         
                          
                         
                           ω 
                           c 
                           2 
                         
                       
                       + 
                       
                         
                           u 
                           4 
                         
                          
                         
                           ω 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the shape is persevered. 
         [0023]    Note that u should be constrained to physically realizable values derived from the constraint T C /Q TC ≧0 from (7). We also have from (7) that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ω 
                           c 
                           2 
                         
                          
                         
                           T 
                           C 
                           4 
                         
                       
                       + 
                       
                         2 
                          
                         
                           T 
                           C 
                           2 
                         
                       
                       - 
                       
                         1 
                         
                           ω 
                           c 
                           2 
                         
                       
                     
                     ≥ 
                     0 
                   
                    
                   
                     
 
                   
                    
                   or 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ω 
                         c 
                         4 
                       
                        
                       
                         T 
                         C 
                         4 
                       
                     
                     + 
                     
                       2 
                        
                       
                           
                       
                        
                       
                         ω 
                         c 
                         2 
                       
                        
                       
                         T 
                         C 
                         2 
                       
                     
                   
                   ≥ 
                   1 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    so completing the square we have 
         [0000]        w   c   4   T   C   4 +2ω c   2   T   C   2 +1≧1   (14)
 
         [0000]      and 
         [0000]      ( w   c   2   T   C   2 +1) 2 ≧2.   (15)
 
         [0000]      Thus 
         [0000]      ω c   2   T   C   2 +1≧√{square root over (2)}  (16)
 
         [0000]      and finally 
         [0000]        u=ω   c   T   C   ≧√{square root over (√{square root over (2)}−1)}≈0.643594252905582742.    (17)
 
         [0000]    Thus, u cannot be less than the critical value ≈0.6436. 
         [0024]    Another important value of u is that which makes the frequency response maximally flat. A flat response is often a goal in loudspeaker design, so the value of u that achieves this will likely be a good value for a loudspeaker model. 
         [0025]    The maxiflat value of u can be found by plugging the denominator of (1) into the quadratic formula and making the discriminant 0 as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     0 
                     = 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     C 
                                   
                                   
                                     Q 
                                     TC 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               4 
                                
                               
                                 T 
                                 C 
                                 2 
                               
                             
                           
                         
                         ⇒ 
                         
                           
                             T 
                             C 
                             2 
                           
                           
                             Q 
                             TC 
                             2 
                           
                         
                       
                       = 
                       
                         4 
                          
                         
                           T 
                           C 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     and  since 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       4 
                        
                       
                         T 
                         C 
                         2 
                       
                     
                     = 
                     
                       
                         
                           T 
                           C 
                           2 
                         
                         
                           Q 
                           TC 
                           2 
                         
                       
                       = 
                       
                         
                           
                             ω 
                             c 
                             2 
                           
                            
                           
                             T 
                             C 
                             4 
                           
                         
                         + 
                         
                           2 
                            
                           
                             T 
                             C 
                             2 
                           
                         
                         - 
                         
                           1 
                           
                             ω 
                             c 
                             2 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     we  have 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           ω 
                           c 
                           2 
                         
                          
                         
                           T 
                           C 
                           4 
                         
                       
                       - 
                       
                         2 
                          
                         
                           T 
                           C 
                           2 
                         
                       
                       - 
                       
                         1 
                         
                           ω 
                           c 
                           2 
                         
                       
                     
                     = 
                     0 
                   
                    
                   
                     
 
                   
                    
                   
                     so  that 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     T 
                     C 
                     2 
                   
                   = 
                   
                     
                       
                         1 
                         + 
                         
                           2 
                         
                       
                       
                         ω 
                         c 
                         2 
                       
                     
                     . 
                     
                       
 
                     
                      
                     
                       and 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   u 
                   = 
                   
                     
                       
                         T 
                         C 
                       
                        
                       
                         ω 
                         c 
                       
                     
                     = 
                     
                       
                         
                           1 
                           + 
                           
                             2 
                           
                         
                       
                       ≈ 
                       
                         1.5537739740300374 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0026]    Equation (10) may need to be scaled by an amplitude A in order to best fit the data. This is important since generally the amplitude of the data is unknown. Thus we can define our basic model to be 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Am 
                       k 
                     
                      
                     
                       ( 
                       
                         
                           ω 
                           c 
                         
                         , 
                         u 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       A 
                        
                       
                         
                            
                           
                             G 
                              
                             
                               ( 
                               
                                 jω 
                                 k 
                               
                               ) 
                             
                           
                            
                         
                         2 
                       
                     
                     = 
                     
                       A 
                        
                       
                           
                       
                        
                       
                         
                           
                             
                               u 
                               4 
                             
                              
                             
                               ω 
                               k 
                               4 
                             
                           
                           
                             
                               ω 
                               c 
                               4 
                             
                             - 
                             
                               
                                 ω 
                                 k 
                                 2 
                               
                                
                               
                                 ω 
                                 c 
                                 2 
                               
                             
                             + 
                             
                               
                                 u 
                                 4 
                               
                                
                               
                                 ω 
                                 k 
                                 2 
                               
                                
                               
                                 ω 
                                 c 
                                 2 
                               
                             
                             + 
                             
                               
                                 u 
                                 4 
                               
                                
                               
                                 ω 
                                 k 
                                 4 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substituting (22) into (23) gives 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Am 
                       k 
                     
                      
                     
                       ( 
                       
                         ω 
                         c 
                       
                       ) 
                     
                   
                   = 
                   
                     A 
                      
                     
                         
                     
                      
                     
                       
                         
                           
                             ( 
                             
                               3 
                               + 
                               
                                 2 
                                  
                                 
                                   2 
                                 
                               
                             
                             ) 
                           
                            
                           
                             ω 
                             k 
                             4 
                           
                         
                         
                           
                             ω 
                             c 
                             4 
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                 + 
                                 
                                   2 
                                    
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               ω 
                               k 
                               2 
                             
                              
                             
                               ω 
                               c 
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 3 
                                 + 
                                 
                                   2 
                                    
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               ω 
                               k 
                               4 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    as a maxiflat loudspeaker model depending only on parameters of amplitude A and cutoff frequency ω c . 
         [0027]      FIG. 3  illustrates the effect of u on the frequency response for the basic model (23) with ω c  set at 100 Hz and A=1. Shown are frequency responses with the critical value of (17) where u=0.6436 and maxiflat value of (22) where u=1.5538 and a “huge” value of u=1000.0. In the critical value case, the resonance peak heads toward ∝ while the maxiflat case is close to the huge case, but below the cutoff frequency drops off more rapidly. 
         [0028]    By error we mean the difference between the model and the measured value. For this error, a gaussian model is assumed. Letting D represent our data, which is the squared magnitude of a measured loudspeaker spectrum X, letting d k  represent the data at frequency bin index k, i.e. d k =|X[k]| 2 , letting m k , A, u and ω c  represent the model and parameters used in (23) and letting I represent our models for loudspeakers and error, the likelihood for a particular set of parameters can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         
                           D 
                            
                           A 
                         
                         , 
                         u 
                         , 
                         
                           ω 
                           c 
                         
                         , 
                         I 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       k 
                     
                      
                     
                         
                     
                      
                     
                       
                         1 
                         
                           
                             σ 
                             k 
                           
                            
                           
                             
                               2 
                                
                               π 
                             
                           
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                     σ 
                                     k 
                                     2 
                                   
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     d 
                                     k 
                                   
                                   - 
                                   
                                     
                                       Am 
                                       k 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ω 
                                           c 
                                         
                                         , 
                                         u 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where σ k  is the standard deviation of the noise at index k. Here the “noise” is really the error at each frequency bin which can be frequency dependent. These σ k  can be treated as a set of additional parameters, but for now we will assume these are known since doing so doesn&#39;t affect the rest of the derivations. The σ k  can be thought of as a weighting on the frequency, a smaller σ k  value indicates more certainty about the d k  value and thus the error at that frequency counts more. As a frequency weighting, these σ k  can also be modified to force the algorithm to weigh some frequencies more than others. Conversely, if there is no reason to emphasize the contribution at any frequency, all σ k  can be set to the same value. 
         [0029]    Equation (25) is called the likelihood of the parameters, since the data is fixed and the parameters can vary. It can be interpreted as saying that the probability of the data given the loudspeaker model, gaussian noise model, and a set of model parameters is just the product of the independent probability densities that gaussian noise makes up the difference between the model with those parameters and the data. The parameter values which maximize the probablity of the data are those that minimize the sum of the squared differences with the data, and are known as the least squares solution. 
         [0030]    Bayes&#39; Theorem follows directly from the definition of conditional probability as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                            
                           
                             ( 
                             AB 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             P 
                              
                             
                               ( 
                               
                                 A 
                                  
                                 B 
                               
                               ) 
                             
                           
                            
                           
                             P 
                              
                             
                               ( 
                               B 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             P 
                              
                             
                               ( 
                               
                                 B 
                                  
                                 A 
                               
                               ) 
                             
                              
                             
                               P 
                                
                               
                                 ( 
                                 A 
                                 ) 
                               
                             
                           
                           ⇒ 
                           
                             P 
                              
                             
                               ( 
                               
                                 B 
                                  
                                 A 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               P 
                                
                               
                                 ( 
                                 
                                   A 
                                    
                                   B 
                                 
                                 ) 
                               
                             
                              
                             
                               P 
                                
                               
                                 ( 
                                 B 
                                 ) 
                               
                             
                           
                           
                             P 
                              
                             
                               ( 
                               Λ 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where A and B can be basically any statements for which conditional probability makes sense. Applying (26) to (25) gives 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         A 
                         , 
                         u 
                         , 
                         
                           
                             ω 
                             c 
                           
                            
                           D 
                         
                         , 
                         I 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           P 
                            
                           
                             ( 
                             
                               
                                 D 
                                  
                                 A 
                               
                               , 
                               u 
                               , 
                               
                                 ω 
                                 c 
                               
                               , 
                               I 
                             
                             ) 
                           
                         
                          
                         
                           P 
                            
                           
                             ( 
                             
                               A 
                               , 
                               u 
                               , 
                               
                                 
                                   ω 
                                   c 
                                 
                                  
                                 I 
                               
                             
                             ) 
                           
                         
                       
                       
                         P 
                          
                         
                           ( 
                           
                             D 
                              
                             I 
                           
                           ) 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0031]    Thus, in addition to the likelihood P(D|A, u, ω c , I) given by (25), we need a prior probability P(A, u, ω c |I) and a normalizing term P(D|I) in order to get our posterior probability P(A, u, ω c |D, I). However another step is then to eliminate the “nuisance” parameters A and u to give the posterior probability of the cutoff frequency P(w c |D, I). The elimination of A as a “nuisance” parameter can be achieved by exact marginalization. 
         [0032]    Let {θ} be a set of parameters, A be a scale (amplitude) parameter, d k  be the k th  data value and m k  be the model value at index k with parameters {θ}. Then using the gaussian error model we have 
         [0000]    
       
         
           
             
               P 
                
               
                 ( 
                 
                   
                     D 
                      
                     
                       { 
                       θ 
                       } 
                     
                   
                   , 
                   A 
                   , 
                   I 
                 
                 ) 
               
             
             = 
             
               
                 ∏ 
                 k 
               
                
               
                   
               
                
               
                 
                   1 
                   
                     
                       σ 
                       k 
                     
                      
                     
                       
                         2 
                          
                         π 
                       
                     
                   
                 
                  
                 
                   
                     exp 
                      
                     
                       ( 
                       
                         
                           - 
                           
                             1 
                             
                               2 
                                
                               
                                 σ 
                                 k 
                                 2 
                               
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 d 
                                 k 
                               
                               - 
                               
                                 Am 
                                 k 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0033]    Note that parameter A appears as a scale term outside of the model itself, which only takes parameters {θ}. Then we have 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         
                           D 
                            
                           
                             { 
                             θ 
                             } 
                           
                         
                         , 
                         A 
                         , 
                         I 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       ( 
                       
                         
                           ∏ 
                           k 
                         
                          
                         
                             
                         
                          
                         
                           1 
                           
                             
                               σ 
                               k 
                             
                              
                             
                               
                                 2 
                                  
                                 π 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       exp 
                       ( 
                       
                         - 
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                             
                               1 
                               
                                 2 
                                  
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     d 
                                     k 
                                   
                                   - 
                                   
                                     Am 
                                     k 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     ( 
                     
                       
                         ∏ 
                         k 
                       
                        
                       
                           
                       
                        
                       
                         1 
                         
                           
                             σ 
                             k 
                           
                            
                           
                             
                               2 
                                
                               π 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     exp 
                     ( 
                     
                       
                         ∑ 
                         k 
                       
                        
                       
                         
                           1 
                           
                             2 
                              
                             
                               σ 
                               k 
                               2 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 d 
                                 k 
                                 2 
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               Λ 
                                
                               
                                   
                               
                                
                               
                                 d 
                                 k 
                               
                                
                               
                                 m 
                                 k 
                               
                             
                              
                             
                               
                                 Λ 
                                 2 
                               
                                
                               
                                 m 
                                 k 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ( 
                       
                         
                           ∏ 
                           k 
                         
                          
                         
                             
                         
                          
                         
                           1 
                           
                             
                               σ 
                               k 
                             
                              
                             
                               
                                 2 
                                  
                                 π 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       exp 
                       ( 
                       
                         - 
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                             
                               1 
                               
                                 2 
                                  
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                             
                              
                             
                               d 
                               k 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                     
                    
                   
                     exp 
                     ( 
                     
                       - 
                       
                         
                           ∑ 
                           k 
                         
                          
                         
                           
                             1 
                             
                               2 
                                
                               
                                 σ 
                                 k 
                                 2 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   Ad 
                                   k 
                                 
                                  
                                 
                                   m 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   A 
                                   2 
                                 
                                  
                                 
                                   m 
                                   k 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ( 
                       
                         
                           ∏ 
                           k 
                         
                          
                         
                             
                         
                          
                         
                           1 
                           
                             
                               σ 
                               k 
                             
                              
                             
                               
                                 2 
                                  
                                 π 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       exp 
                       ( 
                       
                         - 
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                             
                               1 
                               
                                 2 
                                  
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                             
                              
                             
                               d 
                               k 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       exp 
                       ( 
                       
                         
                           - 
                           
                             
                               A 
                               2 
                             
                             ( 
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   m 
                                   k 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           A 
                           ( 
                           
                             
                               ∑ 
                               k 
                             
                              
                             
                               
                                 1 
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                                
                               
                                 d 
                                 k 
                               
                                
                               
                                 m 
                                 k 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0034]    From Bayes&#39; Theorem we have 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         
                           { 
                           θ 
                           } 
                         
                         , 
                         
                           A 
                            
                           D 
                         
                         , 
                         I 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         P 
                          
                         
                           ( 
                           
                             
                               { 
                               θ 
                               } 
                             
                             , 
                             
                               A 
                                
                               I 
                             
                           
                           ) 
                         
                       
                        
                       
                         P 
                          
                         
                           ( 
                           
                             
                               D 
                                
                               
                                 { 
                                 θ 
                                 } 
                               
                             
                             , 
                             A 
                             , 
                             I 
                           
                           ) 
                         
                       
                     
                     
                       P 
                        
                       
                         ( 
                         
                           D 
                            
                           I 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         P 
                          
                         
                           ( 
                           
                             
                               { 
                               θ 
                               } 
                             
                              
                             I 
                           
                           ) 
                         
                       
                        
                       
                         P 
                          
                         
                           ( 
                           
                             A 
                              
                             I 
                           
                           ) 
                         
                       
                        
                       
                         P 
                          
                         
                           ( 
                           
                             
                               D 
                                
                               
                                 { 
                                 θ 
                                 } 
                               
                             
                             , 
                             A 
                             , 
                             I 
                           
                           ) 
                         
                       
                     
                     
                       ∫ 
                       
                         ∫ 
                         
                           
                             P 
                              
                             
                               ( 
                               
                                 D 
                                 , 
                                 
                                   { 
                                   θ 
                                   } 
                                 
                                 , 
                                 
                                   A 
                                    
                                   I 
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               { 
                               θ 
                               } 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             A 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         P 
                          
                         
                           ( 
                           
                             
                               { 
                               θ 
                               } 
                             
                              
                             I 
                           
                           ) 
                         
                       
                        
                       
                         P 
                          
                         
                           ( 
                           
                             
                               D 
                                
                               
                                 { 
                                 θ 
                                 } 
                               
                             
                             , 
                             A 
                             , 
                             I 
                           
                           ) 
                         
                       
                        
                       
                         P 
                          
                         
                           ( 
                           
                             A 
                              
                             I 
                           
                           ) 
                         
                       
                     
                     
                       ∫ 
                       
                         
                            
                           
                             { 
                             θ 
                             } 
                           
                         
                          
                         
                           ∫ 
                           
                             
                                
                               
                                 AP 
                                  
                                 
                                   ( 
                                   
                                     
                                       { 
                                       θ 
                                       } 
                                     
                                      
                                     I 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               P 
                                
                               
                                 ( 
                                 
                                   
                                     D 
                                      
                                     
                                       { 
                                       θ 
                                       } 
                                     
                                   
                                   , 
                                   A 
                                   , 
                                   I 
                                 
                                 ) 
                               
                             
                              
                             
                               P 
                                
                               
                                 ( 
                                 AI 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    with the integration ranges and prior probabilities appropriately chosen for the parameters. We would like to marginalize A. 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         
                           
                             { 
                             θ 
                             } 
                           
                            
                           D 
                         
                         , 
                         I 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     ∫ 
                     
                        
                       
                         AP 
                          
                         
                           ( 
                           
                             
                               { 
                               θ 
                               } 
                             
                             , 
                             
                               A 
                                
                               D 
                             
                             , 
                             I 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         P 
                          
                         
                           ( 
                           
                             
                               { 
                               θ 
                               } 
                             
                              
                             I 
                           
                           ) 
                         
                       
                        
                       
                         ∫ 
                         
                           
                              
                             
                               AP 
                                
                               
                                 ( 
                                 
                                   
                                     D 
                                      
                                     
                                       { 
                                       θ 
                                       } 
                                     
                                   
                                   , 
                                   A 
                                   , 
                                   I 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             P 
                              
                             
                               ( 
                               
                                 A 
                                  
                                 I 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       ∫ 
                       
                         
                            
                           
                             { 
                             θ 
                             } 
                           
                         
                          
                         
                           P 
                            
                           
                             ( 
                             
                               
                                 { 
                                 θ 
                                 } 
                               
                                
                               I 
                             
                             ) 
                           
                         
                          
                         
                           ∫ 
                           
                             
                                
                               
                                 AP 
                                  
                                 
                                   ( 
                                   
                                     
                                       D 
                                        
                                       
                                         { 
                                         θ 
                                         } 
                                       
                                     
                                     , 
                                     A 
                                     , 
                                     I 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               P 
                                
                               
                                 ( 
                                 
                                   A 
                                    
                                   I 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ∫ 
               
                 
                    
                   A 
                 
                  
                 
                     
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       
                         
                           - 
                           
                             C 
                             0 
                           
                         
                          
                         
                           A 
                           2 
                         
                       
                       + 
                       
                         
                           C 
                           1 
                         
                          
                         A 
                       
                     
                     ) 
                   
                 
               
             
             = 
             
               
                 
                   
                     π 
                   
                    
                   
                     crf 
                     ( 
                     
                       
                         
                           2 
                            
                           
                               
                           
                            
                           A 
                            
                           
                               
                           
                            
                           
                             C 
                             0 
                           
                         
                         - 
                         
                           C 
                           1 
                         
                       
                       
                         2 
                          
                         
                           
                             C 
                             0 
                           
                         
                       
                     
                     ) 
                   
                 
                 
                   2 
                    
                   
                     
                       C 
                       0 
                     
                   
                    
                   
                     exp 
                      
                     
                       ( 
                       
                         - 
                         
                           
                             C 
                             1 
                             2 
                           
                           
                             4 
                              
                             
                                 
                             
                              
                             
                               C 
                               0 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               + 
               constant 
             
           
         
       
     
         [0035]    So, if we choose a flat prior for P(A|I) and a range of (−∞, ∞) we have 
         [0000]    
       
         
           
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 ∞ 
               
                
               
                   
               
                
               
                 
                    
                   A 
                 
                  
                 
                     
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       
                         
                           - 
                           
                             C 
                             0 
                           
                         
                          
                         
                           A 
                           2 
                         
                       
                       + 
                       
                         
                           C 
                           1 
                         
                          
                         A 
                       
                     
                     ) 
                   
                 
               
             
             = 
             
               
                 
                   π 
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       
                         C 
                         1 
                         2 
                       
                       
                         4 
                          
                         
                             
                         
                          
                         
                           C 
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   C 
                   0 
                 
               
             
           
         
       
       
         
           
             and  with 
           
         
       
       
         
           
             
               C 
               0 
             
             = 
             
               ( 
               
                 
                   ∑ 
                   k 
                 
                  
                 
                     
                 
                  
                 
                   
                     1 
                     
                       2 
                        
                       
                         σ 
                         k 
                         2 
                       
                     
                   
                    
                   
                     m 
                     k 
                     2 
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               C 
               1 
             
             = 
             
               ( 
               
                 
                   ∑ 
                   k 
                 
                  
                 
                     
                 
                  
                 
                   
                     1 
                     
                       σ 
                       k 
                       2 
                     
                   
                    
                   
                     d 
                     k 
                   
                    
                   
                     m 
                     k 
                   
                 
               
               ) 
             
           
         
       
       
         
           
             we  have 
           
         
       
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         
                           AP 
                            
                           
                             ( 
                             
                               
                                 D 
                                  
                                 
                                   { 
                                   θ 
                                   } 
                                 
                               
                               , 
                               A 
                               , 
                               I 
                             
                             ) 
                           
                         
                       
                        
                       
                         P 
                          
                         
                           ( 
                           
                             A 
                              
                             I 
                           
                           ) 
                         
                       
                     
                   
                   = 
                     
                    
                   
                     
                       ( 
                       
                         
                           ∏ 
                           k 
                         
                          
                         
                             
                         
                          
                         
                           1 
                           
                             
                               σ 
                               k 
                             
                              
                             
                               
                                 2 
                                  
                                 π 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       exp 
                       ( 
                       
                         - 
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                               
                           
                            
                           
                             
                               1 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                             
                              
                             
                               d 
                               k 
                               2 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         ∞ 
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           A 
                         
                          
                         
                             
                         
                          
                         
                           exp 
                           ( 
                           
                             
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ( 
                                 
                                   
                                     ∑ 
                                     k 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       1 
                                       
                                         2 
                                          
                                         
                                           σ 
                                           k 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       m 
                                       k 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               A 
                                
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     k 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       1 
                                       
                                         σ 
                                         k 
                                         2 
                                       
                                     
                                      
                                     
                                       d 
                                       k 
                                     
                                      
                                     
                                       m 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                          
                         P 
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             A 
                              
                             I 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ( 
                       
                         
                           ∏ 
                           k 
                         
                          
                         
                             
                         
                          
                         
                           1 
                           
                             
                               σ 
                               k 
                             
                              
                             
                               
                                 2 
                                  
                                 π 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       exp 
                       ( 
                       
                         - 
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                               
                           
                            
                           
                             
                               1 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                             
                              
                             
                               d 
                               k 
                               2 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         
                             
                         
                          
                         
                           π 
                         
                       
                       
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                               
                           
                            
                           
                             
                               1 
                               
                                 2 
                                  
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                             
                              
                             
                               m 
                               k 
                               2 
                             
                           
                         
                       
                     
                      
                     
                       exp 
                       ( 
                       
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                 
                                   1 
                                   
                                     σ 
                                     k 
                                     2 
                                   
                                 
                                  
                                 
                                   d 
                                   k 
                                 
                                  
                                 
                                   m 
                                   k 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             
                               ∑ 
                               k 
                             
                              
                             
                               
                                 1 
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                                
                               
                                 m 
                                 k 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             so  that 
           
         
       
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         
                           
                             { 
                             θ 
                             } 
                           
                            
                           D 
                         
                         , 
                         I 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       
                         P 
                          
                         
                           ( 
                           
                             
                               { 
                               θ 
                               } 
                             
                              
                             I 
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             ∏ 
                             k 
                           
                            
                           
                             1 
                             
                               
                                 σ 
                                 k 
                               
                                
                               
                                 
                                   2 
                                    
                                   π 
                                 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         exp 
                         ( 
                         
                           - 
                           
                             
                               ∑ 
                               k 
                             
                              
                             
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                     σ 
                                     k 
                                     2 
                                   
                                 
                               
                                
                               
                                 d 
                                 k 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           
                               
                           
                            
                           
                             π 
                           
                         
                         
                           
                             
                               ∑ 
                               k 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                     σ 
                                     k 
                                     2 
                                   
                                 
                               
                                
                               
                                 m 
                                 k 
                                 2 
                               
                             
                           
                         
                       
                        
                       
                         exp 
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   k 
                                 
                                  
                                 
                                   
                                     1 
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                    
                                   
                                     d 
                                     k 
                                   
                                    
                                   
                                     m 
                                     k 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                 
                                   1 
                                   
                                     σ 
                                     k 
                                     2 
                                   
                                 
                                  
                                 
                                   m 
                                   k 
                                   2 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     
                       ∫ 
                       
                         
                            
                           
                             { 
                             θ 
                             } 
                           
                         
                          
                         
                           P 
                            
                           
                             ( 
                             
                               
                                 { 
                                 θ 
                                 } 
                               
                                
                               I 
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             
                               ∏ 
                               k 
                             
                              
                             
                               1 
                               
                                 
                                   σ 
                                   k 
                                 
                                  
                                 
                                   
                                     2 
                                      
                                     π 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           exp 
                           ( 
                           
                             - 
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   d 
                                   k 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             
                                 
                             
                              
                             
                               π 
                             
                           
                           
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   m 
                                   k 
                                   2 
                                 
                               
                             
                           
                         
                          
                         
                           exp 
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     k 
                                   
                                    
                                   
                                     
                                       1 
                                       
                                         σ 
                                         k 
                                         2 
                                       
                                     
                                      
                                     
                                       d 
                                       k 
                                     
                                      
                                     
                                       m 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   k 
                                 
                                  
                                 
                                   
                                     1 
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                    
                                   
                                     m 
                                     k 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           P 
                            
                           
                             ( 
                             
                               
                                 { 
                                 θ 
                                 } 
                               
                                
                               I 
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             
                               ∏ 
                               k 
                             
                              
                             
                               1 
                               
                                 
                                   σ 
                                   k 
                                 
                                  
                                 
                                   
                                     2 
                                      
                                     π 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           exp 
                           ( 
                           
                             - 
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   d 
                                   k 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             
                                 
                             
                              
                             
                               π 
                             
                           
                           
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   m 
                                   k 
                                   2 
                                 
                               
                             
                           
                         
                          
                         
                           exp 
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     k 
                                   
                                    
                                   
                                     
                                       1 
                                       
                                         σ 
                                         k 
                                         2 
                                       
                                     
                                      
                                     
                                       d 
                                       k 
                                     
                                      
                                     
                                       m 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   k 
                                 
                                  
                                 
                                   
                                     1 
                                     
                                       σ 
                                       k 
                                       2 
                                     
                                   
                                    
                                   
                                     m 
                                     k 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       
                         P 
                          
                         
                           ( 
                           
                             D 
                              
                             I 
                           
                           ) 
                         
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0036]    Thus, the marginalization leaves a new equation for the likelihood of the parameters as follows: 
         [0000]    
       
         
           
             
               P 
                
               
                 ( 
                 
                   
                     D 
                      
                     u 
                   
                   , 
                   
                     ω 
                     c 
                   
                   , 
                   I 
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     ∏ 
                     k 
                   
                    
                   
                       
                   
                    
                   
                     1 
                     
                       
                         σ 
                         k 
                       
                        
                       
                         
                           2 
                            
                           π 
                         
                       
                     
                   
                 
                 ) 
               
                
               
                 exp 
                 ( 
                 
                   - 
                   
                     
                       ∑ 
                       k 
                     
                      
                     
                         
                     
                      
                     
                       
                         1 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             σ 
                             k 
                             2 
                           
                         
                       
                        
                       
                         d 
                         k 
                         2 
                       
                     
                   
                 
                 ) 
               
                
               
                 
                   
                       
                   
                    
                   
                     π 
                   
                 
                 
                   
                     
                       ∑ 
                       k 
                     
                      
                     
                         
                     
                      
                     
                       
                         1 
                         
                           2 
                            
                           
                             σ 
                             k 
                             2 
                           
                         
                       
                        
                       
                         m 
                         k 
                         2 
                       
                     
                   
                 
               
                
               
                 exp 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                             
                               1 
                               
                                 σ 
                                 k 
                                 2 
                               
                             
                              
                             
                               d 
                               k 
                             
                              
                             
                               m 
                               k 
                             
                           
                         
                         ) 
                       
                       2 
                     
                     
                       2 
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           k 
                         
                          
                         
                           
                             1 
                             
                               σ 
                               k 
                               2 
                             
                           
                            
                           
                             m 
                             k 
                             2 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where m k  is short for the model m k (u, ω c ). 
         [0037]    Since there are only two remaining parameters, this can be shown in a 2-dimensional graph. For the spectrum shown in  FIG. 2 , a gray-scale plot of (28) is shown in  FIG. 4  with black indicating the highest likelihood at each u level. The thatched region at the bottom lies below the critical u value from (17) and is physically unrealizable. The isolated horizontal black line above the critical region indicates the maxiflat u value from (22). When the u value is above the maxiflat value, the peak likelihood region for ω c  is very stable, since the shape of the model frequency response changes only slightly as u increases. However when u goes below the maxiflat value the model frequency response quickly becomes peaky, and the most probable region for ω c  becomes less stable, changing value and jumping to a higher region before jumping to the lowest frequency as u nears its critical value. 
         [0038]    Exact marginalization over u looks very difficult and numerically integrating over u also seems computationally expensive. However u doesn&#39;t affect the shape of the speaker rolloff very much beyond some low values which cause a large resonance in the model spectrum, as indicated by  FIG. 3 . Since loudspeaker designers try to avoid such resonances, it is unlikely such values for u will explain the data well. Instead we hold the value constant at a reasonable value such the maxiflat value u=1.5538, and evaluate the probability of ω c  on this basis. 
         [0039]    In this example we have 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                            
                           
                             ( 
                             
                               
                                 
                                   ω 
                                   c 
                                 
                                  
                                 D 
                               
                               , 
                               I 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   ∏ 
                                   k 
                                 
                                  
                                 
                                   1 
                                   
                                     
                                       σ 
                                       k 
                                     
                                      
                                     
                                       
                                         2 
                                          
                                         π 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               exp 
                               ( 
                               
                                 - 
                                 
                                   
                                     ∑ 
                                     k 
                                   
                                    
                                   
                                     
                                       1 
                                       
                                         2 
                                          
                                         
                                           σ 
                                           k 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       d 
                                       k 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 
                                     
                                 
                                  
                                 
                                   π 
                                 
                               
                               
                                 
                                   
                                     ∑ 
                                     k 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       1 
                                       
                                         2 
                                          
                                         
                                           σ 
                                           k 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       m 
                                       k 
                                       2 
                                     
                                   
                                 
                               
                             
                              
                             
                               exp 
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ∑ 
                                         k 
                                       
                                        
                                       
                                         
                                           1 
                                           
                                             σ 
                                             k 
                                             2 
                                           
                                         
                                          
                                         
                                           d 
                                           k 
                                         
                                          
                                         
                                           m 
                                           k 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       k 
                                     
                                      
                                     
                                       
                                         1 
                                         
                                           σ 
                                           k 
                                           2 
                                         
                                       
                                        
                                       
                                         m 
                                         k 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               P 
                               ( 
                               
                                 
                                   ω 
                                   c 
                                 
                                  
                                 I 
                               
                               ) 
                             
                           
                           
                             P 
                              
                             
                               ( 
                               
                                 D 
                                  
                                 I 
                               
                               ) 
                             
                           
                         
                       
                        
                     
                     
                       u 
                       = 
                       1.5538 
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0040]    The prior P(ω c |I) can be thought of as a weighting based on our belief about what likely values of ω c  should be. This can be flat over a reasonable range, or have some proprietary shape based on many loudspeaker evaluations. It is also useful to make the prior P(ω c |I) discrete with the same set of frequencies bins used for the data. Thus the prior can state P(ω c |I)=0 if c≠k for all frequency bin indexes k, effectively sampling the continuous probability density at some subset of frequency bins. Since the same prior is built into the normalizing denominator, this is a way to move from a continuous distribution to a discrete one defined only at bin frequencies. 
         [0041]    A high level implementation of this method is shown in  FIG. 5 . The loudspeaker spectrum is assumed available from other processes, but if not, these can be calculated by taking the FFT of a test signal recording. 
         [0042]    For implementation it is useful to take the log of (29) which gives 
         [0000]    
       
         
           
             
               
                 
                   
                     log 
                      
                     
                       ( 
                       
                         P 
                          
                         
                           ( 
                           
                             
                               
                                 ω 
                                 c 
                               
                                
                               D 
                             
                             , 
                             I 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         log 
                         ( 
                         
                           ( 
                           
                             
                               ∏ 
                               k 
                             
                              
                             
                                 
                             
                              
                             
                               1 
                               
                                 
                                   σ 
                                   k 
                                 
                                  
                                 
                                   
                                     2 
                                      
                                     π 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           k 
                         
                          
                         
                             
                         
                          
                         
                           
                             1 
                             
                               2 
                                
                               
                                 σ 
                                 k 
                                 2 
                               
                             
                           
                            
                           
                             d 
                             k 
                             2 
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           log 
                            
                           
                             ( 
                             π 
                             ) 
                           
                         
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           log 
                           ( 
                           
                             
                               ∑ 
                               k 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                     σ 
                                     k 
                                     2 
                                   
                                 
                               
                                
                               
                                 m 
                                 k 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 k 
                               
                                
                               
                                 
                                   1 
                                   
                                     σ 
                                     k 
                                     2 
                                   
                                 
                                  
                                 
                                   d 
                                   k 
                                 
                                  
                                 
                                   m 
                                   k 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             
                               ∑ 
                               k 
                             
                              
                             
                               
                                 1 
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                                
                               
                                 m 
                                 k 
                                 2 
                               
                             
                           
                         
                       
                       + 
                       
                         log 
                          
                         
                           ( 
                           
                             P 
                              
                             
                               ( 
                               
                                 
                                   ω 
                                   c 
                                 
                                  
                                 I 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           log 
                            
                           
                             ( 
                             
                               P 
                                
                               
                                 ( 
                                 
                                   D 
                                    
                                   I 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which can be considered as 
         [0000]    
       
         
           
             
               
                 
                   
                     log 
                      
                     
                       ( 
                       
                         P 
                          
                         
                           ( 
                           
                             
                               
                                 ω 
                                 c 
                               
                                
                               D 
                             
                             , 
                             I 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                        
                       
                         log 
                         ( 
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                               
                           
                            
                           
                             
                               1 
                               
                                 2 
                                  
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                             
                              
                             
                               m 
                               k 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               ∑ 
                               k 
                             
                              
                             
                               
                                 1 
                                 
                                   σ 
                                   k 
                                   2 
                                 
                               
                                
                               
                                 d 
                                 k 
                               
                                
                               
                                 m 
                                 k 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       
                         2 
                          
                         
                             
                         
                          
                         
                           
                             ∑ 
                             k 
                           
                            
                           
                             
                               1 
                               
                                 σ 
                                 k 
                                 2 
                               
                             
                              
                             
                               m 
                               k 
                               2 
                             
                           
                         
                       
                     
                     + 
                     
                       log 
                        
                       
                         ( 
                         
                           P 
                            
                           
                             ( 
                             
                               
                                 ω 
                                 c 
                               
                                
                               I 
                             
                             ) 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       constant 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
         [0043]    Since log(x) is a monotonically increasing function of x and the constant term doesn&#39;t affect the location of the maximum value, one approach is just to find the ω c  which maximizes (31), ignoring the constant term. If a uniform probability is assumed for the prior probability P(ω c |I), then this term can be left out as well. 
         [0044]    A block diagram of this implementation is given in  FIG. 6 . Here u is set to 1.5538 as an example. The data[k] is the squared magnitude of the spectrum, which can be calculated beforehand or directly when needed from the k th  value of the spectrum times its conjugate. The maximum value val determines the best cutoff best_ω c . Optionally the values at each ω c  can be stored in an array for further processing. 
         [0045]    If the results are stored for further processing, it is often desirable to convert the values to a probability distribution summing to 1. A way of doing this is shown in  FIG. 7 , which uses the max value found in  FIG. 6 . 
         [0046]    Although any loudspeaker model function can be used in principle, an implementation of the loudspeaker model is given by (23). 
         [0047]    While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.