Patent Publication Number: US-2020288242-A1

Title: Speaker excursion protection

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This continuation application claims priority to U.S. patent application Ser. No. 16/402,509, filed May 3, 2019, which application claims priority to India Provisional Patent Application No. 201841046810, filed Dec. 11, 2018, both of which are incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     To enhance bass response, some loudspeaker systems may include a passive radiator or a bass reflex port, in addition to a speaker. A passive radiator is similar to a speaker, but lacks a voice coil and magnet. In a passive radiator, the mass of the cone, the radiation inertia, and the stiffness of the suspension system form a resonant system which interacts with the speaker by coupling through the enclosed air of the speaker housing. A bass reflex port is an open tube that couples the interior of a speaker housing to air outside the speaker housing. 
     SUMMARY 
     Circuitry to protect a speaker system that includes a passive radiator or a bass reflex port from damage caused by excessive excursion is disclosed herein. In one example, an audio circuit includes a speaker housing, an amplifier, a voltage sensor, a current sensor and an excursion control circuit. The speaker housing includes a speaker, and a passive radiator or a bass reflex port. The amplifier is coupled to the speaker. The voltage sensor is coupled to the speaker. The current sensor is coupled to the speaker. The excursion control circuit is coupled to the amplifier, the voltage sensor, and the current sensor. The excursion control circuit includes back electro-magnetic force (EMF) measurement, a back-EMF model, and excursion protection. The back-EMF measurement is to measure back-EMF of the speaker based on voltage measurements received from the voltage sensor and current measurements received from the current sensor. The back-EMF model is updated based on measurements of the back-EMF and is converted to an excursion model. The excursion protection is to limit amplitude of audio signal provided to the amplifier based on the excursion model of the speaker and amplitude of an audio input signal. 
     In another example, an audio circuit includes an amplifier, a voltage sensor, a current sensor, and an excursion control circuit. The voltage sensor is coupled to an output of the amplifier. The current sensor is coupled to the output of the amplifier. The excursion control circuit is coupled to the amplifier, the voltage sensor, and the current sensor. The excursion control circuit includes back-EMF measurement, a back-EMF model, and excursion protection. The back-EMF measurement is to measure back electro-magnetic force of a speaker based on voltage measurements received from the voltage sensor and current measurements received from the current sensor. The back-EMF model is updated based on measurements of the back-EMF and is converted to an excursion model. The excursion protection is to limit amplitude of audio signal provided to the amplifier based on the excursion model of the speaker and amplitude of an audio input signal. 
     In a further example, an audio circuit includes an amplifier, a voltage sensor, a current sensor, and an excursion control circuit. The voltage sensor is coupled to an output of the amplifier. The current sensor is coupled to the output of the amplifier. The excursion control circuit includes a back-EMF measurement circuit, a second order back-EMF filter; a fourth order back-EMF filter, a port detection circuit, a back-EMF model to excursion model conversion circuit, and an excursion protection circuit. The back-EMF measurement circuit is coupled to an output of the voltage sensor and an output of the current sensor. The second order back-EMF filter is coupled to the back-EMF measurement circuit. The fourth order back-EMF filter is coupled to the second order back-EMF filter. The port detection circuit is coupled to the output of the voltage sensor, the output of the current sensor, and an input of the second order filter. The back-EMF model to excursion model conversion circuit is coupled to an output of the second order back-EMF filter and an output of the fourth order back-EMF filter. The excursion protection circuit coupled to an output of the back-EMF model to excursion model conversion circuit, an audio input, and an input of the amplifier. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a detailed description of various examples, reference will now be made to the accompanying drawings in which: 
         FIG. 1  shows a block diagram for an example audio circuit including a speaker system having a passive radiator; 
         FIG. 2  show a block diagram for an example audio circuit including a speaker system having a bass reflex port; 
         FIG. 3  shows an electrical equivalent circuit for a speaker system that includes a passive radiator or a bass reflex port; 
         FIG. 4  shows an example block diagram for an audio circuit that includes an excursion control circuit in accordance with the present disclosure; 
         FIG. 5  shows an example block diagram for a port detection circuit in accordance with the present disclosure; and 
         FIG. 6  shows an example of operation of an excursion control circuit in accordance with the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Certain terms have been used throughout this description and claims to refer to particular system components. As one skilled in the art will appreciate, different parties may refer to a component by different names. This document does not intend to distinguish between components that differ in name but not function. In this disclosure and claims, the terms “including” and “comprising” are used in an open-ended fashion, and thus should be interpreted to mean “including, but not limited to . . . .” Also, the term “couple” or “couples” is intended to mean either an indirect or direct wired or wireless connection. Thus, if a first device couples to a second device, that connection may be through a direct connection or through an indirect connection via other devices and connections. The recitation “based on” is intended to mean “based at least in part on.” Therefore, if X is based on Y, X may be a function of Y and any number of other factors. 
     To improve bass response and sound pressure level, speaker systems that include passive radiators or bass reflex ports are sometime driven hard at low frequencies, which may damage the speaker. Various excursion protection techniques have been applied to reduce such damage. Automatic gain control (AGC) has been applied to limit the peak and root-mean-squared (RMS) power applied to the speaker. However, AGC based systems are overly conservative and significantly reduce the low frequency output by using high pass filters. High pass filters are used because a speaker&#39;s rated power is related to the thermal capacity of the speaker, so driving the speaker even at rated power at low frequencies will result in over excursion and speaker damage. Model based excursion control is also used to reduce speaker damage, and provides good bass and sound pressure level response at low frequencies. However, model based systems suffer from reliability issues. For example, model based systems cannot adapt to variations across speaker lot/temperature/aging, or changes due to opening or closing of the bass reflex port. 
     The excursion control circuits of the present disclosure include an excursion model that is updated based on real-time measurements of speaker current and voltage. Speaker excursion is estimated by filtering audio input using the adaptive excursion model. A limiter circuit compresses the audio signal if the estimated excursion exceeds a safety threshold. The limiting applies a look-ahead control technique to eliminate over excursion of the speaker. The excursion control circuits disclosed herein provide reliable excursion protection while enabling sound pressure levels and bass response that is substantially higher than other techniques. 
       FIG. 1  shows a block diagram for an example audio circuit  100  including a speaker system  102  having a passive radiator  108 . The audio circuit  100  includes the speaker system  102  and a speaker driver circuit  110 . The speaker system  102  includes a speaker  106  and the passive radiator  108 . The speaker  106  includes a magnet and voice coil, while the passive radiator  108  lacks a magnet and voice coil. The speaker system  102  is coupled to the speaker driver circuit  110 . The speaker driver circuit  110  receives an audio input signal  114  and generates an output signal  116  to actuate the speaker  106 . The speaker driver circuit  110  includes an excursion control circuit  112  as disclosed herein to limit excursion of the speaker  106  and prevent damage thereto. 
       FIG. 2  show a block diagram for an example audio circuit  200  including a speaker system  202  having a bass reflex port  208 . The audio circuit  200  includes the speaker system  202  and a speaker driver circuit  210 . The speaker system  202  includes a speaker  206  and the bass reflex port  208 . The speaker system  202  is coupled to the speaker driver circuit  210 . The speaker driver circuit  210  receives an audio input signal  214  and generates an output signal  216  to actuate the speaker  206 . The speaker driver circuit  210  includes an excursion control circuit  212  as disclosed herein to limit excursion of the speaker  206  and prevent damage thereto. 
       FIG. 3  shows an electrical equivalent circuit  300  for a speaker system that includes a passive radiator or a bass reflex port (e.g., the speaker system  102  or the speaker system  202 ). In the electrical equivalent circuit  300 : 
     V(t) is input voltage;
 
I(t) is loop current;
 
X(t) is excursion;
 
Bl is force factor;
 
R E  is DC resistance of the voice coil;
 
R MS  is mechanical dampening of the speaker;
 
C MS  is mechanical compliance of the speaker;
 
M MS  is mechanical mass of the speaker;
 
S D  is radiating area of the speaker;
 
C AB  is housing compliance;
 
R AP  is mechanical dampening of the passive radiator;
 
C AP  is mechanical compliance of the passive radiator;
 
M AP  is mechanical mass of the passive radiator or the bass reflex port;
 
R AL  is the mechanical loss of the port.
 
     For a speaker system that includes a passive radiator, R AL →∞. For a speaker system that includes a bass reflex port, R AP →0 and C AP →∞. The parameters of a speaker vary across lots, with temperature, and with age, while the passive radiator and bass reflex ports parameters do not change. 
     The back electro-motive force (EMF) transfer function for a speaker is a fourth order system that is expressed as: 
     
       
         
           
             
               
                 
                   
                     BEMF 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       V 
                        
                       
                         ( 
                         t 
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                     - 
                     
                       
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                           ( 
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                        
                       
                         R 
                         E 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       BEMF 
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     
                       I 
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       H 
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             B 
                             3 
                           
                            
                           
                             s 
                             3 
                           
                         
                         + 
                         
                           
                             B 
                             2 
                           
                            
                           
                             s 
                             2 
                           
                         
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                             B 
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                           s 
                         
                       
                       
                         
                           s 
                           4 
                         
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                             s 
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                             A 
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                           s 
                         
                         + 
                         
                           A 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     For a speaker system that includes a passive radiator: 
     
       
         
           
             
               
                 B 
                 3 
               
               = 
               
                 
                   B 
                    
                   
                     l 
                     2 
                   
                 
                 
                   M 
                   
                     M 
                      
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             , 
             
               
                 B 
                 2 
               
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                   B 
                    
                   
                     l 
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                    
                   
                     R 
                     
                       A 
                        
                       P 
                     
                   
                 
                 
                   
                     M 
                     
                       M 
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                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                 
               
             
             , 
             
               
                 B 
                 1 
               
               = 
               
                 
                   B 
                    
                   
                     
                       l 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           C 
                           AP 
                         
                         + 
                         
                           C 
                           
                             A 
                              
                             B 
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     C 
                     AP 
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 A 
                 3 
               
               = 
               
                 
                   
                     R 
                     
                       A 
                        
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                     M 
                     
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                        
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                     R 
                     
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                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 A 
                 2 
               
               = 
               
                 
                   
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                     + 
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
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                         A 
                          
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                         A 
                          
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                      
                     
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                         M 
                          
                         S 
                       
                     
                   
                   
                     
                       M 
                       AP 
                     
                      
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
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                         M 
                          
                         S 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         1 
                         
                           C 
                           
                             M 
                              
                             S 
                           
                         
                       
                       + 
                       
                         
                           s 
                           d 
                           2 
                         
                         
                           C 
                           AB 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 A 
                 1 
               
               = 
               
                 
                   
                     
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                       + 
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                     
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   × 
                   
                     
                       R 
                       
                         M 
                          
                         S 
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     
                       R 
                       
                         A 
                          
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                         A 
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                   × 
                   
                     1 
                     
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                         M 
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                         S 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         1 
                         
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                             S 
                           
                         
                       
                       + 
                       
                         
                           s 
                           d 
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                     ) 
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 A 
                 0 
               
               = 
               
                 
                   
                     C 
                     
                       A 
                        
                       P 
                     
                   
                   + 
                   
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                        
                       B 
                     
                   
                   + 
                   
                     
                       s 
                       d 
                       2 
                     
                      
                     
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                         M 
                          
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                     M 
                     
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                        
                       S 
                     
                   
                    
                   
                     C 
                     MS 
                   
                    
                   
                     M 
                     
                       A 
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                       P 
                     
                   
                    
                   
                     C 
                     
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                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
     
     For a speaker system that includes a bass reflex port: 
     
       
         
           
             
               
                 B 
                 3 
               
               = 
               
                 
                   B 
                    
                   
                     l 
                     2 
                   
                 
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
               
             
             , 
             
               
                 B 
                 2 
               
               = 
               
                 
                   B 
                    
                   
                     l 
                     2 
                   
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
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                        
                       B 
                     
                   
                    
                   
                     R 
                     
                       A 
                        
                       L 
                     
                   
                 
               
             
             , 
             
               
                 B 
                 1 
               
               = 
               
                 
                   B 
                    
                   
                     l 
                     2 
                   
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 A 
                 3 
               
               = 
               
                 
                   1 
                   
                     
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                         A 
                          
                         B 
                       
                     
                      
                     
                       R 
                       
                         A 
                          
                         L 
                       
                     
                   
                 
                 + 
                 
                   
                     R 
                     
                       M 
                        
                       S 
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 A 
                 2 
               
               = 
               
                 
                   1 
                   
                     
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                         A 
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                         A 
                          
                         B 
                       
                     
                   
                 
                 + 
                 
                   
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                        
                       
                         R 
                         
                           A 
                            
                           L 
                         
                       
                     
                   
                   × 
                   
                     
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                         M 
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                         S 
                       
                     
                     
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                          
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                 + 
                 
                   
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                          
                         S 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         1 
                         
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                              
                             S 
                           
                         
                       
                       + 
                       
                         
                           s 
                           d 
                           2 
                         
                         
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                             A 
                              
                             B 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 A 
                 1 
               
               = 
               
                 
                   
                     R 
                     MS 
                   
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                      
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       
                         C 
                         
                           A 
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                        
                       
                         R 
                         
                           A 
                            
                           L 
                         
                       
                     
                   
                   × 
                   
                     1 
                     
                       
                         M 
                         MS 
                       
                        
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 A 
                 0 
               
               = 
               
                 1 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     C 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
     
     The speaker excursion transfer function is a fourth order system that is expressed as: 
     
       
         
           
             
               
                 X 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               
                 V 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
             
             = 
             
               
                 
                   H 
                   _ 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       
                         B 
                         2 
                       
                       _ 
                     
                      
                     
                       s 
                       2 
                     
                   
                   + 
                   
                     
                       
                         B 
                         1 
                       
                       _ 
                     
                      
                     s 
                   
                   + 
                   
                     
                       B 
                       0 
                     
                     _ 
                   
                 
                 
                   
                     s 
                     4 
                   
                   + 
                   
                     
                       
                         A 
                         3 
                       
                       _ 
                     
                      
                     
                       s 
                       3 
                     
                   
                   + 
                   
                     
                       
                         A 
                         2 
                       
                       _ 
                     
                      
                     
                       s 
                       2 
                     
                   
                   + 
                   
                     
                       
                         A 
                         1 
                       
                       _ 
                     
                      
                     
                       s 
                       1 
                     
                   
                   + 
                   
                     
                       A 
                       0 
                     
                     _ 
                   
                 
               
             
           
         
       
     
     For a speaker system that includes a passive radiator: 
     
       
         
           
             
                 
             
              
             
               
                 
                   
                     B 
                     2 
                   
                   _ 
                 
                 = 
                 
                   
                     B 
                      
                     l 
                   
                   
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                      
                     
                       R 
                       E 
                     
                   
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   
                     B 
                     1 
                   
                   _ 
                 
                 = 
                 
                   
                     B 
                      
                     l 
                      
                     
                       R 
                       
                         A 
                          
                         P 
                       
                     
                   
                   
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                      
                     
                       R 
                       E 
                     
                      
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                   
                 
               
               , 
               
                 
                   
                     B 
                     0 
                   
                   _ 
                 
                 = 
                 
                   
                     B 
                      
                     
                       l 
                        
                       
                         ( 
                         
                           
                             C 
                             
                               A 
                                
                               P 
                             
                           
                           + 
                           
                             C 
                             
                               A 
                                
                               B 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                      
                     
                       R 
                       E 
                     
                      
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     A 
                     3 
                   
                   _ 
                 
                 = 
                 
                   
                     
                       R 
                       
                         A 
                          
                         P 
                       
                     
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                   
                   + 
                   
                     
                       
                         R 
                         
                           M 
                            
                           S 
                         
                       
                       + 
                       
                         
                           B 
                            
                           
                             l 
                             2 
                           
                         
                         
                           R 
                           E 
                         
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
               
               , 
               
                 
 
               
                
               
                 
                   
                     A 
                     2 
                   
                   _ 
                 
                 = 
                 
                   
                     
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                       + 
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                     
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         R 
                         
                           A 
                            
                           P 
                         
                       
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                     
                     × 
                     
                       
                         
                           R 
                           
                             M 
                              
                             S 
                           
                         
                         + 
                         
                           
                             B 
                              
                             
                               l 
                               2 
                             
                           
                           
                             R 
                             E 
                           
                         
                       
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                     
                   
                   + 
                   
                     
                       1 
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           1 
                           
                             C 
                             
                               M 
                                
                               S 
                             
                           
                         
                         + 
                         
                           
                             s 
                             d 
                             2 
                           
                           
                             C 
                             
                               A 
                                
                               B 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   A 
                   1 
                 
                 _ 
               
               = 
               
                 
                   
                     
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                       + 
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                     
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   × 
                   
                     
                       
                         R 
                         
                           M 
                            
                           S 
                         
                       
                       + 
                       
                         
                           B 
                            
                           
                             l 
                             2 
                           
                         
                         
                           R 
                           E 
                         
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     
                       R 
                       
                         A 
                          
                         P 
                       
                     
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                   
                   × 
                   
                     1 
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         1 
                         
                           C 
                           
                             M 
                              
                             S 
                           
                         
                       
                       + 
                       
                         
                           s 
                           d 
                           2 
                         
                         
                           C 
                           
                             A 
                              
                             B 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             , 
             
               
 
             
              
             
                 
             
              
             
               
                 
                   A 
                   0 
                 
                 _ 
               
               = 
               
                 
                   
                     C 
                     
                       A 
                        
                       P 
                     
                   
                   + 
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                   + 
                   
                     
                       s 
                       d 
                       2 
                     
                      
                     
                       C 
                       
                         M 
                          
                         s 
                       
                     
                   
                 
                 
                   
                     M 
                     MS 
                   
                    
                   
                     C 
                     MS 
                   
                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
     
     For a speaker system that includes a bass reflex port: 
     
       
         
           
             
               
                 
                   B 
                   2 
                 
                 _ 
               
               = 
               
                 
                   B 
                    
                   l 
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     R 
                     E 
                   
                 
               
             
             , 
             
               
                 
                   B 
                   1 
                 
                 _ 
               
               = 
               
                 
                   B 
                    
                   l 
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     R 
                     E 
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                    
                   
                     R 
                     
                       A 
                        
                       L 
                     
                   
                 
               
             
             , 
             
               
                 
                   B 
                   0 
                 
                 _ 
               
               = 
               
                 
                   B 
                    
                   l 
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     R 
                     E 
                   
                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   A 
                   3 
                 
                 _ 
               
               = 
               
                 
                   1 
                   
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                      
                     
                       R 
                       
                         A 
                          
                         L 
                       
                     
                   
                 
                 + 
                 
                   
                     
                       R 
                       
                         M 
                          
                         S 
                       
                     
                     + 
                     
                       
                         B 
                          
                         
                           l 
                           2 
                         
                       
                       
                         R 
                         E 
                       
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   A 
                   2 
                 
                 _ 
               
               = 
               
                 
                   1 
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                        
                       
                         R 
                         
                           A 
                            
                           L 
                         
                       
                     
                   
                   × 
                   
                     
                       
                         R 
                         
                           M 
                            
                           S 
                         
                       
                       + 
                       
                         
                           B 
                            
                           
                             l 
                             2 
                           
                         
                         
                           R 
                           E 
                         
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         1 
                         
                           C 
                           
                             M 
                              
                             S 
                           
                         
                       
                       + 
                       
                         
                           s 
                           d 
                           2 
                         
                         
                           C 
                           
                             A 
                              
                             B 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   A 
                   1 
                 
                 _ 
               
               = 
               
                 
                   
                     1 
                     
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   × 
                   
                     
                       
                         R 
                         
                           M 
                            
                           S 
                         
                       
                       + 
                       
                         
                           B 
                            
                           
                             l 
                             2 
                           
                         
                         
                           R 
                           E 
                         
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                        
                       
                         R 
                         
                           A 
                            
                           L 
                         
                       
                     
                   
                   × 
                   
                     1 
                     
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                        
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   A 
                   0 
                 
                 _ 
               
               = 
               
                 1 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     C 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
     
       FIG. 4  shows an example block diagram for an audio circuit  400  that includes an excursion control circuit  402  in accordance with the present disclosure. The excursion control circuit  402  is an implementation of the excursion control circuit  112  or the excursion control circuit  212 . The audio circuit  400  includes an amplifier  404  coupled to the excursion control circuit  402 , a speaker system  406  that is coupled to the amplifier  404 , a current sensor  408 , and a voltage sensor  410 . The amplifier  404  is a class-D amplifier in some implementations of the audio circuit  400 . The speaker system  406  is an implementation of the speaker system  102  or the speaker system  202 , and includes a passive radiator or a bass reflex port. The current sensor  408  and the voltage sensor  410  are coupled to the amplifier  404 , the speaker system  406  and the excursion control circuit  402 . The current sensor  408  and the voltage sensor  410  respectively measure current and voltage in the speaker system  406 . The excursion control circuit  402  applies the current and voltage measurements to update tracking of a back-EMF filter used to estimate excursion of a speaker in the speaker system  406 . 
     The excursion control circuit  402  includes back-EMF measurement  412 , a back-EMF model  414 , excursion protection  416 , port detection  418 , and back-EMF model to excursion model conversion  420 . The back-EMF measurement  412  is coupled to the current sensor  408  and the voltage sensor  410 , and determines values of Back-EMF of the speaker system  406  per equation (1). The DC resistance R can be temperature dependent, and can vary over time in some examples of the speaker system  406 . In some implementations of the audio circuit  400 , a pilot tone generator  436  injects a very low frequency signal into the amplifier  404  via the summation circuit  438  so that R E  can be estimated. By measuring the low-frequency component of V(t) and I(t) denoted respectively as V lf (t) and I lf (t), the DC resistance R E  can be estimated by 
     
       
         
           
             
               
                 R 
                 E 
               
               = 
               
                 
                   
                     V 
                     lf 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 
                   
                     I 
                     lf 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
               
             
             , 
           
         
       
     
     Since the pilot tone is at very low frequency (close to DC), the measurement of R E  is very accurate and very close to the DC resistance value. The measured back-EMF values are filtered around a resonant frequency of the speaker in some implementations. 
     In the back-EMF model  414 , the fourth order back-EMF filter H(s) is split into a second order filter (H 1 (s))  422  and a fourth order filter (H 2 (s))  424 . 
     
       
         
           
             
               
                 BEMF 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               
                 I 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
             
             = 
             
               
                 H 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               = 
               
                 
                   
                     H 
                     1 
                   
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
                 × 
                 
                   
                     H 
                     2 
                   
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
               
             
           
         
       
     
     The second order filter  422  includes only driver elements (i.e., speaker elements), which are variable. 
     
       
         
           
             
               
                 
                   H 
                   1 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               = 
               
                 
                   
                     B 
                     1 
                     ′ 
                   
                    
                   s 
                 
                 
                   
                     s 
                     2 
                   
                   + 
                   
                     
                       A 
                       1 
                       ′ 
                     
                      
                     s 
                   
                   + 
                   
                     A 
                     0 
                     ′ 
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 B 
                 1 
                 ′ 
               
               = 
               
                 
                   B 
                    
                   
                     l 
                     2 
                   
                 
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
               
             
             , 
             
               
                 A 
                 1 
                 ′ 
               
               = 
               
                 
                   R 
                   
                     M 
                      
                     S 
                   
                 
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
               
             
             , 
             
               
                 A 
                 0 
                 ′ 
               
               = 
               
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         S 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The fourth order filter  424  includes both the driver elements and the passive radiator or bass reflex port elements, which are fixed. The second order filter  422  is coupled to the back-EMF measurement  412 , and is continuously adapted to track measured speaker back-EMF provided by the back-EMF measurement  412  by minimizing the error between the estimated back-EMF generated by the second order filter  422  and the measured speaker back-EMF provided by the back-EMF measurement  412 . Some implementations of the second order filter  422  employ least-squares minimization. The fourth order filter  424  is coupled to the second order filter  422 , and updated based on output of the second order filter  422 . H 2 (s) has a weak effect on H(s) near resonance of the speaker, and filtering of the measured back-EMF about speaker resonance in the back-EMF measurement  412  makes error minimization based on H 1 (s) effective and accurate. 
     Back-EMF error minimization in the second order filter  422  is expressed as: 
       BEMF( s )= H   1 ( s ) H   2 ( s ) I ( s ) 
       BEMF est ( t )= H   1 ( t )⊕ H   2 ( t )⊕ I ( t )
 
       BEMF act ( t )= V ( t )− I ( t ) R   E  
 
       Err( t )=BEMF act ( t )−BEMF err ( t )
 
     For a speaker system  406  that includes a passive radiator, the fourth order filter  424  is described as: 
     
       
         
           
             
               
                 H 
                 2 
               
                
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 
                   s 
                   4 
                 
                 + 
                 
                   
                     B 
                     3 
                     ″ 
                   
                    
                   
                     s 
                     3 
                   
                 
                 + 
                 
                   
                     B 
                     2 
                     ″ 
                   
                    
                   
                     s 
                     2 
                   
                 
                 + 
                 
                   
                     B 
                     1 
                     ″ 
                   
                    
                   s 
                 
                 + 
                 
                   B 
                   0 
                   ″ 
                 
               
               
                 
                   s 
                   4 
                 
                 + 
                 
                   
                     B 
                     3 
                     ″ 
                   
                    
                   
                     s 
                     3 
                   
                 
                 + 
                 
                   
                     B 
                     2 
                     ″ 
                   
                    
                   
                     s 
                     2 
                   
                 
                 + 
                 
                   
                     B 
                     1 
                     ″ 
                   
                    
                   s 
                 
                 + 
                 
                   A 
                   0 
                   ″ 
                 
               
             
           
         
       
       
         
           
             
               
                 B 
                 3 
                 ″ 
               
               = 
               
                 
                   
                     R 
                     
                       A 
                        
                       P 
                     
                   
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                 
                 + 
                 
                   
                     R 
                     
                       M 
                        
                       S 
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 B 
                 2 
                 ″ 
               
               = 
               
                 
                   
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                     + 
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 + 
                 
                   
                     
                       R 
                       
                         A 
                          
                         P 
                       
                     
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                   
                   × 
                   
                     
                       R 
                       
                         M 
                          
                         S 
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         1 
                         
                           C 
                           
                             M 
                              
                             S 
                           
                         
                       
                       + 
                       
                         
                           s 
                           d 
                           2 
                         
                         
                           C 
                           
                             A 
                              
                             B 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               B 
               1 
               ″ 
             
             = 
             
               
                 
                   
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                     + 
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   
                     R 
                     
                       M 
                        
                       S 
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
               + 
               
                 
                   
                     R 
                     
                       A 
                        
                       P 
                     
                   
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                 
                 × 
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         s 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               B 
               0 
               ″ 
             
             = 
             
               
                 
                   
                     C 
                     
                       A 
                        
                       P 
                     
                   
                   + 
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
                 
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
               × 
               
                 1 
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
               
                
               
                 ( 
                 
                   
                     1 
                     
                       C 
                       
                         M 
                          
                         S 
                       
                     
                   
                   + 
                   
                     
                       s 
                       d 
                       2 
                     
                     
                       C 
                       AB 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               A 
               0 
               ″ 
             
             = 
             
               
                 
                   C 
                   
                     A 
                      
                     P 
                   
                 
                 + 
                 
                   C 
                   
                     A 
                      
                     B 
                   
                 
                 + 
                 
                   
                     s 
                     d 
                     2 
                   
                    
                   
                     C 
                     
                       M 
                        
                       S 
                     
                   
                 
               
               
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
                  
                 
                   C 
                   
                     M 
                      
                     S 
                   
                 
                  
                 
                   M 
                   
                     A 
                      
                     P 
                   
                 
                  
                 
                   C 
                   
                     A 
                      
                     P 
                   
                 
                  
                 
                   C 
                   
                     A 
                      
                     B 
                   
                 
               
             
           
         
       
     
     For a speaker system  406  that includes a bass reflex port, the fourth order filter  424  is described as: 
     
       
         
           
             
               
                 H 
                 2 
               
                
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 
                   s 
                   4 
                 
                 + 
                 
                   
                     B 
                     3 
                     ″ 
                   
                    
                   
                     s 
                     3 
                   
                 
                 + 
                 
                   
                     B 
                     2 
                     ″ 
                   
                    
                   
                     s 
                     2 
                   
                 
                 + 
                 
                   
                     B 
                     1 
                     ″ 
                   
                    
                   s 
                 
                 + 
                 
                   B 
                   0 
                   ″ 
                 
               
               
                 
                   s 
                   4 
                 
                 + 
                 
                   
                     B 
                     3 
                     ″ 
                   
                    
                   
                     s 
                     3 
                   
                 
                 + 
                 
                   
                     B 
                     2 
                     ″ 
                   
                    
                   
                     s 
                     2 
                   
                 
                 + 
                 
                   
                     A 
                     1 
                     ″ 
                   
                    
                   s 
                 
                 + 
                 
                   A 
                   0 
                   ″ 
                 
               
             
           
         
       
       
         
           
             
               
                 B 
                 3 
                 ″ 
               
               = 
               
                 
                   1 
                   
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                      
                     
                       R 
                       
                         A 
                          
                         L 
                       
                     
                   
                 
                 + 
                 
                   
                     R 
                     
                       M 
                        
                       S 
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 B 
                 2 
                 ″ 
               
               = 
               
                 
                   1 
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       
                         R 
                         
                           A 
                            
                           L 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   × 
                   
                     
                       R 
                       
                         M 
                          
                         S 
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         1 
                         
                           C 
                           
                             M 
                              
                             S 
                           
                         
                       
                       + 
                       
                         
                           s 
                           d 
                           2 
                         
                         
                           C 
                           
                             A 
                              
                             B 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               B 
               1 
               ″ 
             
             = 
             
               
                 
                   1 
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   
                     R 
                     
                       M 
                        
                       S 
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
               + 
               
                 
                   1 
                   
                     
                       R 
                       
                         A 
                          
                         L 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         s 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               B 
               0 
               ″ 
             
             = 
             
               
                 1 
                 
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
               × 
               
                 1 
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
               
                
               
                 ( 
                 
                   
                     1 
                     
                       C 
                       
                         M 
                          
                         S 
                       
                     
                   
                   + 
                   
                     
                       s 
                       d 
                       2 
                     
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               A 
               1 
               ″ 
             
             = 
             
               
                 
                   1 
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   
                     R 
                     
                       M 
                        
                       S 
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
               + 
               
                 
                   1 
                   
                     
                       R 
                       
                         A 
                          
                         L 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   1 
                   
                     
                       M 
                       
                         M 
                          
                         s 
                       
                     
                      
                     
                       C 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               A 
               0 
               ″ 
             
             = 
             
               1 
               
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
                  
                 
                   C 
                   
                     M 
                      
                     S 
                   
                 
                  
                 
                   M 
                   
                     A 
                      
                     P 
                   
                 
                  
                 
                   C 
                   
                     A 
                      
                     B 
                   
                 
               
             
           
         
       
     
     The excursion protection  416  is coupled to the back-EMF model to excursion model conversion  420 , an audio input  434 , and the amplifier  404 . The excursion model generated by the back-EMF model to excursion model conversion  420  is provided to the excursion protection  416 . The speaker excursion model is also decomposed in second order and fourth order filters. 
     
       
         
           
             
               
                 X 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               
                 V 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
             
             = 
             
               
                 
                   H 
                   _ 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       H 
                       1 
                     
                     _ 
                   
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
                 × 
                 
                   
                     
                       H 
                       2 
                     
                     _ 
                   
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     H 
                     1 
                   
                   _ 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       B 
                       0 
                     
                     _ 
                   
                   ′ 
                 
                 
                   
                     s 
                     2 
                   
                   + 
                   
                     
                       
                         
                           A 
                           1 
                         
                         _ 
                       
                       ′ 
                     
                      
                     s 
                   
                   + 
                   
                     
                       
                         A 
                         0 
                       
                       _ 
                     
                     ′ 
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     B 
                     0 
                   
                   _ 
                 
                 ′ 
               
               = 
               
                 
                   B 
                    
                   l 
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     R 
                     E 
                   
                 
               
             
             , 
             
               
                 
                   
                     A 
                     1 
                   
                   _ 
                 
                 ′ 
               
               = 
               
                 
                   
                     R 
                     
                       M 
                        
                       S 
                     
                   
                   + 
                   
                     
                       B 
                        
                       
                         l 
                         2 
                       
                     
                     
                       R 
                       E 
                     
                   
                 
                 
                   M 
                   
                     M 
                      
                     S 
                   
                 
               
             
             , 
             
               
                 
                   
                     A 
                     0 
                   
                   _ 
                 
                 ′ 
               
               = 
               
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         s 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The fourth order excursion filter for a speaker with a passive radiator is expressed as: 
     
       
         
           
             
                 
             
              
             
               
                 
                   
                     H 
                     2 
                   
                   _ 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               = 
               
                 
                   
                     s 
                     4 
                   
                   + 
                   
                     
                       
                         
                           B 
                           3 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       3 
                     
                   
                   + 
                   
                     
                       
                         
                           B 
                           2 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       2 
                     
                   
                   + 
                   
                     
                       
                         
                           B 
                           1 
                         
                         _ 
                       
                       ″ 
                     
                      
                     s 
                   
                   + 
                   
                     
                       
                         B 
                         0 
                       
                       _ 
                     
                     ″ 
                   
                 
                 
                   
                     s 
                     4 
                   
                   + 
                   
                     
                       
                         
                           B 
                           3 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       3 
                     
                   
                   + 
                   
                     
                       
                         
                           B 
                           2 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       2 
                     
                   
                   + 
                   
                     
                       
                         
                           B 
                           1 
                         
                         _ 
                       
                       ″ 
                     
                      
                     s 
                   
                   + 
                   
                     
                       
                         A 
                         0 
                       
                       _ 
                     
                     ″ 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     
                       B 
                       3 
                     
                     _ 
                   
                   ″ 
                 
                 = 
                 
                   
                     
                       R 
                       
                         A 
                          
                         P 
                       
                     
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                   
                   + 
                   
                     
                       
                         R 
                         
                           M 
                            
                           S 
                         
                       
                       + 
                       
                         
                           B 
                            
                           
                             l 
                             2 
                           
                         
                         
                           R 
                           E 
                         
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
               
               , 
               
                 
 
               
                
               
                 
                   
                     
                       B 
                       2 
                     
                     _ 
                   
                   ″ 
                 
                 = 
                 
                   
                     
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                       + 
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                     
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         R 
                         
                           A 
                            
                           P 
                         
                       
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                     
                     × 
                     
                       
                         
                           R 
                           
                             M 
                              
                             S 
                           
                         
                         + 
                         
                           
                             B 
                              
                             
                               l 
                               2 
                             
                           
                           
                             R 
                             E 
                           
                         
                       
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                     
                   
                   + 
                   
                     
                       1 
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           1 
                           
                             C 
                             
                               M 
                                
                               S 
                             
                           
                         
                         + 
                         
                           
                             s 
                             d 
                             2 
                           
                           
                             C 
                             
                               A 
                                
                               B 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   B 
                   1 
                 
                 _ 
               
               ″ 
             
             = 
             
               
                 
                   
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                     + 
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   
                     
                       R 
                       
                         M 
                          
                         S 
                       
                     
                     + 
                     
                       
                         B 
                          
                         
                           l 
                           2 
                         
                       
                       
                         R 
                         E 
                       
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
               + 
               
                 
                   
                     R 
                     
                       A 
                        
                       P 
                     
                   
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                 
                 × 
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         s 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     B 
                     0 
                   
                   _ 
                 
                 ″ 
               
               = 
               
                 
                   
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                     + 
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         s 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     A 
                     0 
                   
                   _ 
                 
                 ″ 
               
               = 
               
                 
                   
                     C 
                     
                       A 
                        
                       P 
                     
                   
                   + 
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                   + 
                   
                     
                       s 
                       d 
                       2 
                     
                      
                     
                       C 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     C 
                     
                       M 
                        
                       S 
                     
                   
                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
     
     The fourth order excursion filter for a speaker and bass reflex port is expressed as: 
     
       
         
           
             
                 
             
              
             
               
                 
                   
                     H 
                     2 
                   
                   _ 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               = 
               
                 
                   
                     s 
                     4 
                   
                   + 
                   
                     
                       
                         
                           B 
                           3 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       3 
                     
                   
                   + 
                   
                     
                       
                         
                           B 
                           2 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       2 
                     
                   
                   + 
                   
                     
                       
                         
                           B 
                           1 
                         
                         _ 
                       
                       ″ 
                     
                      
                     s 
                   
                   + 
                   
                     
                       
                         B 
                         0 
                       
                       _ 
                     
                     ″ 
                   
                 
                 
                   
                     s 
                     4 
                   
                   + 
                   
                     
                       
                         
                           B 
                           3 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       3 
                     
                   
                   + 
                   
                     
                       
                         
                           B 
                           2 
                         
                         _ 
                       
                       ″ 
                     
                      
                     
                       s 
                       2 
                     
                   
                   + 
                   
                     
                       
                         
                           A 
                           1 
                         
                         _ 
                       
                       ″ 
                     
                      
                     s 
                   
                   + 
                   
                     
                       
                         A 
                         0 
                       
                       _ 
                     
                     ″ 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     
                       B 
                       3 
                     
                     _ 
                   
                   ″ 
                 
                 = 
                 
                   
                     1 
                     
                       
                         C 
                         AB 
                       
                        
                       
                         R 
                         AL 
                       
                     
                   
                   + 
                   
                     
                       
                         R 
                         
                           M 
                            
                           S 
                         
                       
                       + 
                       
                         
                           B 
                            
                           
                             l 
                             2 
                           
                         
                         
                           R 
                           E 
                         
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
               
               , 
               
                 
 
               
                
               
                 
                   
                     
                       B 
                       2 
                     
                     _ 
                   
                   ″ 
                 
                 = 
                 
                   
                     1 
                     
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   + 
                   
                     
                       1 
                       
                         
                           R 
                           AL 
                         
                          
                         
                           C 
                           AB 
                         
                       
                     
                     × 
                     
                       
                         
                           R 
                           
                             M 
                              
                             S 
                           
                         
                         + 
                         
                           
                             B 
                              
                             
                               l 
                               2 
                             
                           
                           
                             R 
                             E 
                           
                         
                       
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                     
                   
                   + 
                   
                     
                       1 
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           1 
                           
                             C 
                             
                               M 
                                
                               S 
                             
                           
                         
                         + 
                         
                           
                             s 
                             d 
                             2 
                           
                           
                             C 
                             
                               A 
                                
                               B 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   B 
                   1 
                 
                 _ 
               
               ″ 
             
             = 
             
               
                 
                   1 
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   
                     
                       R 
                       
                         M 
                          
                         S 
                       
                     
                     + 
                     
                       
                         B 
                          
                         
                           l 
                           2 
                         
                       
                       
                         R 
                         E 
                       
                     
                   
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
               
               + 
               
                 
                   1 
                   
                     
                       R 
                       AL 
                     
                      
                     
                       C 
                       AB 
                     
                   
                 
                 × 
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         s 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     B 
                     0 
                   
                   _ 
                 
                 ″ 
               
               = 
               
                 
                   1 
                   
                     
                       M 
                       
                         A 
                          
                         P 
                       
                     
                      
                     
                       C 
                       
                         A 
                          
                         B 
                       
                     
                   
                 
                 × 
                 
                   1 
                   
                     M 
                     
                       M 
                        
                       S 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         C 
                         
                           M 
                            
                           S 
                         
                       
                     
                     + 
                     
                       
                         s 
                         d 
                         2 
                       
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     A 
                     1 
                   
                   _ 
                 
                 ″ 
               
               = 
               
                 
                   
                     1 
                     
                       
                         M 
                         
                           A 
                            
                           P 
                         
                       
                        
                       
                         C 
                         
                           A 
                            
                           B 
                         
                       
                     
                   
                   × 
                   
                     
                       
                         R 
                         
                           M 
                            
                           S 
                         
                       
                       + 
                       
                         
                           B 
                            
                           
                             l 
                             2 
                           
                         
                         
                           R 
                           E 
                         
                       
                     
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                   
                 
                 + 
                 
                   
                     1 
                     
                       
                         R 
                         AL 
                       
                        
                       
                         C 
                         AB 
                       
                     
                   
                   × 
                   
                     1 
                     
                       
                         M 
                         
                           M 
                            
                           S 
                         
                       
                        
                       
                         C 
                         MS 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     A 
                     0 
                   
                   _ 
                 
                 ″ 
               
               = 
               
                 1 
                 
                   
                     M 
                     MS 
                   
                    
                   
                     C 
                     MS 
                   
                    
                   
                     M 
                     
                       A 
                        
                       P 
                     
                   
                    
                   
                     C 
                     
                       A 
                        
                       B 
                     
                   
                 
               
             
           
         
       
     
     In the speaker system  406 , H 1 (s) and H 2 (s) are in the back-EMF domain 
     
       
         
           
             
               ( 
               
                 
                   BEMF 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
                 
                   I 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
               
               ) 
             
             , 
           
         
       
     
     and  H 1   (s) and  H 2   (s) are in the excursion domain 
     
       
         
           
             
               ( 
               
                 
                   X 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
                 
                   V 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
               
               ) 
             
             . 
           
         
       
     
     The back-EMF model  414  (including 2 nd  order filter H 1 (s) and 4 th  order filter H 2 (s)) is converted to excursion model (including 2 nd  order filter  H 1   (s) and 4 th  order filter  H 2   (s)) by the back-EMF model to excursion model conversion  420 . Although these equations are represented by the continuous time variable “s”, the actual implementation uses the discrete time variable “z”. The back-EMF model to excursion model conversion  420  is coupled to the second order filter  422  and the fourth order filter  424 . Conversion from back-EMF domain to excursion domain includes sample rate conversion where H 1 (s) and  H 2   (s) are computed at a lower sample rate 
     
       
         
           
             
               ( 
               
                 
                   e 
                   . 
                   g 
                   . 
                 
                 , 
                 
                   
                     f 
                     s 
                   
                   N 
                 
               
               ) 
             
             , 
           
         
       
     
     and  H 1   (s) and  H 2   (s) are applied at the audio sampling rate f s  such as for example 48 KHz. The estimation of back-EMF parameters H 1 (s) and H 2 (s) at a lower sampling rate significantly reduces the processing load of the illustrative system  400 . In some implementations, N=8 and the lower sampling rate equals 6 KHz. The back-EMF to excursion mapping is expressed as: In the Back-EMF domain: 
     
       
         
           
             
               
                 BEMF 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
               
                 I 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
             
             = 
             
               
                 H 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       H 
                       1 
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                    
                   
                     
                       H 
                       2 
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         H 
                         1 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                      
                     
                        
                       
                         s 
                         = 
                         
                           2 
                            
                           fs 
                            
                           
                             
                               1 
                               - 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               1 
                               + 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                      
                     
                       
                         × 
                         
                           
                             H 
                             2 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                        
                       
                          
                         
                           s 
                           = 
                           
                             2 
                              
                             fs 
                              
                             
                               
                                 1 
                                 - 
                                 
                                   z 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   z 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       
                           
                       
                        
                       
                         
                           H 
                           1 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         b 
                         0 
                         ′ 
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             z 
                             
                               - 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     
                       1 
                       + 
                       
                         
                           a 
                           1 
                           ′ 
                         
                          
                         
                           z 
                           
                             - 
                             1 
                           
                         
                       
                       + 
                       
                         
                           a 
                           2 
                           ′ 
                         
                          
                         
                           z 
                           
                             - 
                             2 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     For a speaker and passive radiator: 
     
       
         
           
             
               
                 H 
                 2 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         a 
                         1 
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         s 
                         1 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         s 
                         2 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               
                 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 a 
                                 1 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 a 
                                 2 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 s 
                                 1 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 s 
                                 2 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                     
                   
                 
                 
                   
                     
                       
                         b 
                         0 
                         ′ 
                       
                       × 
                       
                         
                           S 
                           d 
                           2 
                         
                         
                           2 
                            
                           
                             fsBl 
                             2 
                           
                            
                           
                             M 
                             ap 
                           
                            
                           
                             C 
                             
                               a 
                                
                               b 
                             
                             2 
                           
                            
                           
                             K 
                             S 
                           
                         
                       
                        
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               z 
                               
                                 - 
                                 1 
                               
                             
                           
                           ) 
                         
                         4 
                       
                     
                   
                 
               
             
           
         
       
     
     For a speaker and bass reflex port: 
     
       
         
           
             
               
                 H 
                 2 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         a 
                         1 
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         s 
                         1 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         s 
                         2 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               
                 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 a 
                                 1 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 a 
                                 2 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 s 
                                 1 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 s 
                                 2 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           b 
                           0 
                           ′ 
                         
                         × 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             S 
                             d 
                             2 
                           
                           
                             B 
                              
                             
                               l 
                               2 
                             
                              
                             
                               R 
                               al 
                             
                              
                             
                               C 
                               ab 
                               2 
                             
                              
                             
                               K 
                               s 
                             
                           
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               z 
                               
                                 - 
                                 1 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                           3 
                         
                       
                       - 
                       
                         
                           b 
                           0 
                           ′ 
                         
                         × 
                         
                           
                             S 
                             d 
                             2 
                           
                           
                             2 
                              
                             
                               fsBl 
                               2 
                             
                              
                             
                               M 
                               ap 
                             
                              
                             
                               C 
                               ab 
                               2 
                             
                              
                             
                               K 
                               s 
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                           4 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     In the excursion domain: 
     
       
         
           
             
               
                 X 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
               
                 V 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
             
             = 
             
               
                 
                   H 
                   _ 
                 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       
                         H 
                         1 
                       
                       _ 
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                    
                   
                     
                       
                         H 
                         2 
                       
                       _ 
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         
                           H 
                           1 
                         
                         _ 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                      
                     
                        
                       
                         s 
                         = 
                         
                           2 
                            
                           fs 
                            
                           
                             
                               1 
                               - 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               1 
                               + 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                      
                     
                       
                         × 
                         
                           
                             
                               H 
                               2 
                             
                             _ 
                           
                            
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                        
                       
                          
                         
                           s 
                           = 
                           
                             2 
                              
                             fs 
                              
                             
                               
                                 1 
                                 - 
                                 
                                   z 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   z 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       
                         
                           
                             H 
                             1 
                           
                           _ 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         
                           f 
                           s 
                         
                          
                         
                           BlR 
                           E 
                         
                       
                     
                     × 
                     
                       
                         
                           
                             
                               
                                 b 
                                 0 
                               
                               _ 
                             
                             ′ 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         2 
                       
                       
                         1 
                         + 
                         
                           
                             
                               
                                 a 
                                 1 
                               
                               _ 
                             
                             ′ 
                           
                            
                           
                             z 
                             
                               - 
                               1 
                             
                           
                         
                         + 
                         
                           
                             
                               
                                 a 
                                 2 
                               
                               _ 
                             
                             ′ 
                           
                            
                           
                             z 
                             
                               - 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     where “s” is the continuous time variable and “z” is the discrete time variable. The {b′ 0 ,a′ 1 ,a′ 2 } parameters are continuously adapted by minimizing the error between the estimated back-EMF generated by the second order filter  422  and the measured speaker back-EMF provided by the back-EMF measurement  412  (as described herein).
 
For a speaker and passive radiator:
 
     
       
         
           
             
               
                 
                   H 
                   2 
                 
                 _ 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         
                           
                             a 
                             1 
                           
                           _ 
                         
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             a 
                             2 
                           
                           _ 
                         
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         s 
                         1 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         s 
                         2 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               
                 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   
                                     a 
                                     1 
                                   
                                   _ 
                                 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     a 
                                     2 
                                   
                                   _ 
                                 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 s 
                                 1 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 s 
                                 2 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             b 
                             0 
                           
                           _ 
                         
                         ′ 
                       
                       × 
                       
                         
                           S 
                           d 
                           2 
                         
                         
                           2 
                            
                           
                             fsBl 
                             2 
                           
                            
                           
                             M 
                             ap 
                           
                            
                           
                             C 
                             ab 
                             2 
                           
                            
                           
                             K 
                             S 
                           
                         
                       
                        
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               z 
                               
                                 - 
                                 1 
                               
                             
                           
                           ) 
                         
                         4 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               K 
               s 
             
             = 
             
               
                 4 
                  
                 
                   f 
                   s 
                   2 
                 
               
               + 
               
                 2 
                  
                 
                   f 
                   s 
                 
                  
                 
                   
                     R 
                     ap 
                   
                   
                     M 
                     ap 
                   
                 
               
               + 
               
                 
                   
                     C 
                     ap 
                   
                   + 
                   
                     C 
                     ab 
                   
                 
                 
                   
                     M 
                     ap 
                   
                    
                   
                     C 
                     ap 
                   
                    
                   
                     C 
                     ab 
                   
                 
               
             
           
         
       
       
         
           
             
               s 
               1 
             
             = 
             
               
                 
                   
                     - 
                     8 
                   
                    
                   
                     f 
                     s 
                     2 
                   
                 
                 + 
                 
                   2 
                    
                   
                     
                       
                         C 
                         ap 
                       
                       + 
                       
                         C 
                         ab 
                       
                     
                     
                       
                         M 
                         ap 
                       
                        
                       
                         C 
                         ap 
                       
                        
                       
                         C 
                         ab 
                       
                     
                   
                 
               
               
                 
                   4 
                    
                   
                     f 
                     s 
                     2 
                   
                 
                 + 
                 
                   2 
                    
                   
                     f 
                     s 
                   
                    
                   
                     
                       R 
                       ap 
                     
                     
                       M 
                       ap 
                     
                   
                 
                 + 
                 
                   
                     
                       C 
                       ap 
                     
                     + 
                     
                       C 
                       ab 
                     
                   
                   
                     
                       M 
                       ap 
                     
                      
                     
                       C 
                       ap 
                     
                      
                     
                       C 
                       ab 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               s 
               2 
             
             = 
             
               
                 
                   4 
                    
                   
                     f 
                     s 
                     2 
                   
                 
                 - 
                 
                   2 
                    
                   
                     f 
                     s 
                   
                    
                   
                     
                       R 
                       ap 
                     
                     
                       M 
                       ap 
                     
                   
                 
                 + 
                 
                   
                     
                       C 
                       ap 
                     
                     + 
                     
                       C 
                       ab 
                     
                   
                   
                     
                       M 
                       ap 
                     
                      
                     
                       C 
                       ap 
                     
                      
                     
                       C 
                       ab 
                     
                   
                 
               
               
                 
                   4 
                    
                   
                     f 
                     s 
                     2 
                   
                 
                 + 
                 
                   2 
                    
                   
                     f 
                     s 
                   
                    
                   
                     
                       R 
                       ap 
                     
                     
                       M 
                       ap 
                     
                   
                 
                 + 
                 
                   
                     
                       C 
                       ap 
                     
                     + 
                     
                       C 
                       ab 
                     
                   
                   
                     
                       M 
                       ap 
                     
                      
                     
                       C 
                       ap 
                     
                      
                     
                       C 
                       ab 
                     
                   
                 
               
             
           
         
       
     
     For a speaker and bass reflex port: 
     
       
         
           
             
               
                 
                   H 
                   2 
                 
                 _ 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         
                           
                             a 
                             1 
                           
                           _ 
                         
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             a 
                             2 
                           
                           _ 
                         
                         ′ 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         s 
                         1 
                       
                        
                       
                         z 
                         
                           - 
                           1 
                         
                       
                     
                     + 
                     
                       
                         s 
                         2 
                       
                        
                       
                         z 
                         
                           - 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               
                 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   
                                     a 
                                     1 
                                   
                                   _ 
                                 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     a 
                                     2 
                                   
                                   _ 
                                 
                                 ′ 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 s 
                                 1 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 s 
                                 2 
                               
                                
                               
                                 z 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           
                             
                               b 
                               0 
                             
                             _ 
                           
                           ′ 
                         
                         × 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             S 
                             d 
                             2 
                           
                           
                             B 
                              
                             
                               l 
                               2 
                             
                              
                             
                               R 
                               al 
                             
                              
                             
                               C 
                               
                                 a 
                                  
                                 b 
                               
                               2 
                             
                              
                             
                               K 
                               S 
                             
                           
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               z 
                               
                                 - 
                                 1 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                           3 
                         
                       
                       - 
                       
                         
                           
                             
                               b 
                               0 
                             
                             _ 
                           
                           ′ 
                         
                         × 
                         
                           
                             S 
                             d 
                             2 
                           
                           
                             2 
                              
                             
                               fsBl 
                               2 
                             
                              
                             
                               M 
                               ap 
                             
                              
                             
                               C 
                               
                                 a 
                                  
                                 b 
                               
                               2 
                             
                              
                             
                               K 
                               S 
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 z 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                           4 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 K 
                 s 
               
               = 
               
                 
                   4 
                    
                   
                     f 
                     s 
                     2 
                   
                 
                 + 
                 
                   2 
                    
                   
                     f 
                     s 
                   
                    
                   
                     1 
                     
                       
                         C 
                         
                           a 
                            
                           b 
                         
                       
                        
                       
                         R 
                         al 
                       
                     
                   
                 
                 + 
                 
                   1 
                   
                     
                       M 
                       
                         a 
                          
                         p 
                       
                     
                      
                     
                       C 
                       
                         a 
                          
                         b 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 s 
                 1 
               
               = 
               
                 
                   
                     
                       - 
                       8 
                     
                      
                     
                       f 
                       s 
                       2 
                     
                   
                   + 
                   
                     2 
                     
                       
                         M 
                         ap 
                       
                        
                       
                         C 
                         ab 
                       
                     
                   
                 
                 
                   
                     4 
                      
                     
                       f 
                       s 
                       2 
                     
                   
                   + 
                   
                     2 
                      
                     
                       f 
                       s 
                     
                      
                     
                       1 
                       
                         
                           R 
                           al 
                         
                          
                         
                           C 
                           ab 
                         
                       
                     
                   
                   + 
                   
                     1 
                     
                       
                         M 
                         ap 
                       
                        
                       
                         C 
                         ab 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 s 
                 2 
               
               = 
               
                 
                   
                     4 
                      
                     
                       f 
                       s 
                       2 
                     
                   
                   - 
                   
                     2 
                      
                     
                       f 
                       s 
                     
                      
                     
                       1 
                       
                         
                           C 
                           ab 
                         
                          
                         
                           R 
                           al 
                         
                       
                     
                   
                   + 
                   
                     1 
                     
                       
                         M 
                         ap 
                       
                        
                       
                         C 
                         ab 
                       
                     
                   
                 
                 
                   
                     4 
                      
                     
                       f 
                       s 
                       2 
                     
                   
                   + 
                   
                     2 
                      
                     
                       f 
                       s 
                     
                      
                     
                       1 
                       
                         
                           C 
                           ab 
                         
                          
                         
                           R 
                           al 
                         
                       
                     
                   
                   + 
                   
                     1 
                     
                       
                         M 
                         ap 
                       
                        
                       
                         C 
                         ab 
                       
                     
                   
                 
               
             
           
         
       
     
     In the back-EMF to excursion mapping, the parameters { b 0   ′, a 1   ′, a 2   ′} of  H 1   (z) and  H 2   (z) are derived from the parameters {b 0 ′,a 1 ′, a 2 ′} of H 1 (z) and H 2 (z). 
     
       
         
           
             
               
                 
                   b 
                   0 
                 
                 _ 
               
               ′ 
             
             = 
             
               
                 4 
                  
                 
                   b 
                   1 
                   ′ 
                 
                  
                 
                   KK 
                   ′ 
                 
               
               
                 
                   
                     ( 
                     
                       K 
                       + 
                       
                         K 
                         ′ 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                       - 
                       
                         K 
                         ′2 
                       
                     
                     ) 
                   
                    
                   
                     a 
                     1 
                     ′ 
                   
                 
                 + 
                 
                   
                     
                       ( 
                       
                         K 
                         - 
                         
                           K 
                           ′ 
                         
                       
                       ) 
                     
                     2 
                   
                    
                   
                     a 
                     2 
                     ′ 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         a 
                         1 
                         ′ 
                       
                       + 
                       
                         a 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                    
                   γ 
                    
                   
                       
                   
                    
                   
                     K 
                     ′ 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   a 
                   1 
                 
                 _ 
               
               ′ 
             
             = 
             
               
                 2 
                  
                 
                   ( 
                   
                     
                       ( 
                       
                         
                           K 
                           2 
                         
                         - 
                         
                           K 
                           ′2 
                         
                       
                       ) 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             K 
                             2 
                           
                           + 
                           
                             K 
                             ′2 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         1 
                         ′ 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             K 
                             2 
                           
                           - 
                           
                             K 
                             ′2 
                           
                         
                         ) 
                       
                        
                       
                         a 
                         2 
                         ′ 
                       
                     
                   
                   ) 
                 
               
               
                 
                   
                     ( 
                     
                       K 
                       + 
                       
                         K 
                         &#39; 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                       - 
                       
                         K 
                         ′2 
                       
                     
                     ) 
                   
                    
                   
                     a 
                     1 
                     ′ 
                   
                 
                 + 
                 
                   
                     
                       ( 
                       
                         K 
                         - 
                         
                           K 
                           ′ 
                         
                       
                       ) 
                     
                     2 
                   
                    
                   
                     a 
                     2 
                     ′ 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         a 
                         1 
                         ′ 
                       
                       + 
                       
                         a 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                    
                   γ 
                    
                   
                       
                   
                    
                   
                     K 
                     ′ 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   a 
                   2 
                 
                 _ 
               
               ′ 
             
             = 
             
               
                 
                   
                     ( 
                     
                       K 
                       - 
                       
                         K 
                         ′ 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                       - 
                       
                         K 
                         ′2 
                       
                     
                     ) 
                   
                    
                   
                     a 
                     1 
                     ′ 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       K 
                       - 
                       
                         K 
                         ′ 
                       
                     
                     ) 
                   
                    
                   
                     a 
                     2 
                     ′ 
                   
                 
                 - 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         a 
                         1 
                         ′ 
                       
                       + 
                       
                         a 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                    
                   γ 
                    
                   
                       
                   
                    
                   
                     K 
                     ′ 
                   
                 
               
               
                 
                   
                     ( 
                     
                       K 
                       + 
                       
                         K 
                         &#39; 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         K 
                         2 
                       
                       - 
                       
                         K 
                         ′2 
                       
                     
                     ) 
                   
                    
                   
                     a 
                     1 
                     ′ 
                   
                 
                 + 
                 
                   
                     
                       ( 
                       
                         K 
                         - 
                         
                           K 
                           ′ 
                         
                       
                       ) 
                     
                     2 
                   
                    
                   
                     a 
                     2 
                     ′ 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         a 
                         1 
                         ′ 
                       
                       + 
                       
                         a 
                         2 
                         ′ 
                       
                     
                     ) 
                   
                    
                   γ 
                    
                   
                       
                   
                    
                   
                     K 
                     ′ 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 K 
                 = 
                 
                   
                     2 
                      
                     
                       f 
                       s 
                     
                   
                   N 
                 
               
               , 
               
                 
                   K 
                   ′ 
                 
                 = 
                 
                   2 
                    
                   
                     f 
                     s 
                   
                 
               
               , 
               
                 γ 
                 = 
                 
                   
                     B 
                      
                     
                       l 
                       2 
                     
                   
                   
                     
                       M 
                       
                         M 
                          
                         S 
                       
                     
                      
                     
                       R 
                       E 
                     
                   
                 
               
             
           
         
       
     
     In these equations it is assumed that H 1 (z) and H 2 (z) are computed at lower sampling rate 
     
       
         
           
             
               f 
               s 
             
             N 
           
         
       
     
     but me technique is equally applicable for implementations where H 1 (z), H 2 (z) and  H 1   (z),  H 2   (z) are computed at the same audio sampling rate f s , i.e., N=1. 
     The excursion protection  416  applies the excursion model received from the back-EMF model to excursion model conversion  420  in conjunction with an audio input signal to estimate speaker excursion resulting from the audio input signal. The estimated speaker excursion is expressed as: 
       Xest( t )=   H   1   ( t )⊕   H   2   ( t )⊕In( t )
 
     where In(t) is the audio input signal. 
     If X est (t) exceeds a predetermined excursion limit (Xmax), then the excursion protection  416  limits the output signal amplitude as: 
     
       
         
           
             Out 
              
             
               
                 ( 
                 t 
                 ) 
               
               = 
               
                 
                   In 
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 × 
                 
                   
                     
                       X 
                       max 
                     
                     
                       
                         X 
                         est 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
     If X est (t) does not exceed the predetermined excursion limit, the output signal is equal to the input signal (Out(t)=In(t)). The excursion protection  416  includes a delay buffer  432  coupled to the audio input  434  and the amplifier  404 . The delay buffer  432  delays the audio input signal to allow a determination of excursion based on the audio input signal and attenuation of the audio signal before the audio signal is provided to the amplifier  404 . The delay buffer  432  provides 3-10 milliseconds of delay in some implementations of the excursion control circuit  402 . 
     A speaker system  406  with a bass reflex port is a fourth order system 
     
       
         
           
             
               
                 
                   BEMF 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
                 
                   I 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
               
               = 
               
                 
                   
                     H 
                     1 
                   
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
                  
                 
                   
                     H 
                     2 
                   
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
               
             
             . 
           
         
       
     
     However, if the port is closed (e.g., blocked by an object external to the speaker system  406 ), then the system becomes second order 
     
       
         
           
             
               
                 
                   BEMF 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
                 
                   I 
                    
                   
                     ( 
                     s 
                     ) 
                   
                 
               
               = 
               
                 
                   H 
                   1 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
             
             . 
           
         
       
     
     To accommodate variable port state, the excursion control circuit  402  includes the port detection  418 . The port detection  418  detects whether the port is open or closed. The port detection  418  is coupled to the voltage sensor  410 , the current sensor  408 , and the second order filter  422 . 
       FIG. 5  shows an example block diagram for a port detection circuit  500  in accordance with the present disclosure. The port detection circuit  500  is an implementation of the port detection  418 . In the port detection circuit  500 , BEMF(s) and I(s) are passed through an N-band filter bank  502  near the port frequency, and the magnitude of 
     
       
         
           
             
               BEMF 
                
               
                 ( 
                 s 
                 ) 
               
             
             
               I 
                
               
                 ( 
                 s 
                 ) 
               
             
           
         
       
     
     is computed in each band. For each band, the computed magnitude of 
     
       
         
           
             
               BEMF 
                
               
                 ( 
                 s 
                 ) 
               
             
             
               I 
                
               
                 ( 
                 s 
                 ) 
               
             
           
         
       
     
     is compared to a port closed reference  504  and a port open reference  506 . The comparisons produce a port open metric and a port closed metric. 
     
       
         
           
             PortOpenMetric 
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 N 
               
                
               
                 
                    
                   
                     
                       
                         BEMF 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                       
                         I 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                      
                     
                       
                         
                            
                           i 
                         
                          
                         
                           - 
                           
                             
                               BEMF 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                             
                               I 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                          
                       
                       PortOpen 
                     
                   
                    
                 
                 2 
               
             
           
         
       
       
         
           
             PortClosedMetric 
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 N 
               
                
               
                 
                    
                   
                     
                       
                         BEMF 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                       
                         I 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                      
                     
                       
                         
                            
                           i 
                         
                          
                         
                           - 
                           
                             
                               BEMF 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                             
                               I 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                          
                       
                       PortClosed 
                     
                   
                    
                 
                 2 
               
             
           
         
       
     
     The port detection circuit  500  deems the port open if the PortClosedMetric exceeds the PortOpenMetric, and closed if the PortOpenMetric exceeds the PortClosedMetric. 
     The excursion control circuit  402  applies output of the port detection  418  to select second order or fourth order operation. The excursion control circuit  402  includes a switch  426  that is controlled by output of the port detection  418 . The switch  426  is coupled to the current sensor  408 , the fourth order filter  424 , the port detection  418 , and the second order filter  422 . If the port is closed, the switch  426  routes current measurements from the current sensor  408  to the second order filter  422 . If the port is open, the switch  426  routes the filtered output obtained by passing the current measurements from the current sensor  408  through the fourth order filter  424  to the second order filter  422 . 
     Similarly, the excursion protection  416  includes a switch  428  that is controlled by the port detection  418 . The switch  428  is coupled to the port detection  418 , the back-EMF model to excursion model conversion  420 , and the audio input  434 . If the port is open, the switch  428  routes excursion estimates based on applying fourth order excursion model and second order excursion model to the audio input signal to the limiter  430  for use in determining whether the excursion exceeds the predetermined maximum. If the port is closed, the switch  428  routes excursion estimates based on applying only second order excursion model to the audio input signal to the limiter  430  for use in determining whether the excursion exceeds the predetermined maximum. The limiter  430  controls the gain applied to audio signal received from the delay buffer  432 . The limiter  430  is coupled to the switch  428 , the delay buffer  432 , and the amplifier  404 . 
       FIG. 6  shows an example of operation of the excursion control circuit excursion control circuit  402 . Signal  602  is an audio input signal received by the excursion control circuit  402 . At time  604 , the amplitude of the  602  increases to a level that would cause over-excursion of the speaker in the speaker system  406 . The signal  606  is the output of the excursion control circuit  402  (e.g., the signal provided to the amplifier  404 ). At time  604 , the excursion control circuit  402  detects the potential over-excursion and attenuates (applies a gain attenuation to) the  602  to prevent damage to the speaker. 
     Some implementations of the excursion control circuit  402  omit the port detection  418 , the switch  426 , and/or the switch  428  if the  406  includes a passive radiator rather than a bass reflex port. Some implementations of the  402  include a digital signal processor (DSP) to provide the functions described herein. For example, a DSP executes instruction read from memory to provide the back-EMF measurement  412 , the back-EMF model  414 , the back-EMF model to excursion model conversion  420 , and/or the excursion protection  416 . 
     The above discussion is meant to be illustrative of the principles and various embodiments of the present invention. Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. It is intended that the following claims be interpreted to embrace all such variations and modifications.