Abstract:
The present disclosure is directed to an FM demodulator having an extended threshold breakdown point. The FM demodulator uses an arcsin demodulator in combination with a frequency compressive loop to produce a demodulated output signal. The FM demodulator includes three filters that use a coefficient α to determine how the filters behave. The FM demodulator extends the threshold breakdown point of the signal-to-noise ratio of the FM signal beyond traditional levels, allowing the FM demodulator to work at long distances from the broadcasting antenna.

Description:
BACKGROUND 
     1. Technical Field 
     The present disclosure is directed to FM demodulation and, in particular, FM demodulation having an extended threshold using an arcsin demodulator. 
     2. Description of the Related Art 
     Frequency modulation (FM) is a common method of encoding information. FM signals can generally be represented by the equation
 
 S ( t )= A   c  cos(ω c   t +φ( t ))
 
     where A c  is the amplitude, ω c  is the carrier frequency in radians, and φ(t) is the information signal. The information signal can be written as 
     
       
         
           
             
               φ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               2 
               ⁢ 
               π 
               ⁢ 
               
                   
               
               ⁢ 
               
                 k 
                 f 
               
               ⁢ 
               
                 
                   ∫ 
                   0 
                   t 
                 
                 ⁢ 
                 
                   
                     m 
                     ⁡ 
                     
                       ( 
                       τ 
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     τ 
                   
                 
               
             
           
         
       
     
     where m(t) is the modulating signal, and k f  is constant and equal to the peak frequency deviation f d  when m(t)=1. The modulation index, β, can be written as 
     
       
         
           
             β 
             = 
             
               
                 
                   Peak 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   RF 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   frequency 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   deviation 
                 
                 
                   Maximum 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Modulating 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   baseband 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   frequency 
                 
               
               = 
               
                 
                   
                     f 
                     d 
                   
                   
                     f 
                     max 
                   
                 
                 . 
               
             
           
         
       
     
     A common problem with FM demodulation, especially in mobile FM radios, such as those in a vehicle, is that the signal strength can vary significantly as the car moves, causing harsh sounds. In particular, when the FM receiver is too far from the broadcasting antenna, the signal strength can be drastically reduced, making it difficult or impossible for traditional FM demodulators to function. Another problem occurs when the FM signal is reflected off surfaces. When this occurs, the FM receiver will receive two signals simultaneously, one directly from the broadcasting antenna, and another that has been reflected off a nearby surface, such as a building. The reflected signal may be out of phase with the direct signal because of the additional distance traveled, resulting in destructive interference. This destructive interference reduces the strength of the signal that is received at the FM demodulator. One issue associated with these circumstances in particular is the FM threshold effect, which can occur when the amplitude of the noise is comparable to or higher than the amplitude of the FM signal itself. When this occurs, demodulation of the signal rapidly breaks down. 
     The threshold effect can be seen in  FIG. 1 , where the signal-to-noise ratio (S/N or SNR) is generally linear for higher levels of carrier-to-noise ratio (p), but at a certain threshold point, the signal-to-noise ratio has a dramatic downward turn. To the left of this threshold point, the FM demodulator rapidly deteriorates. 
     The output signal-to-noise ratio of an FM system above the threshold region is given by
 
SNR out =3β 2 (β+1)ρ.
 
     However, when the threshold breakdown region is included, the output SNR is 
     
       
         
           
             
               SNR 
               out 
             
             = 
             
               
                 
                   3 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       β 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         β 
                         + 
                         1 
                       
                       ) 
                     
                   
                   ⁢ 
                   ρ 
                 
                 
                   1 
                   + 
                   
                     
                       24 
                       π 
                     
                     ⁢ 
                     
                       β 
                       ⁡ 
                       
                         ( 
                         
                           β 
                           + 
                           1 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ρⅇ 
                       
                         ( 
                         
                           - 
                           ρ 
                         
                         ) 
                       
                     
                   
                 
               
               . 
             
           
         
       
     
     The carrier-to-noise ratio, ρ, is given by 
     
       
         
           
             ρ 
             = 
             
               
                 A 
                 c 
                 2 
               
               
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   N 
                   0 
                 
                 ⁢ 
                 
                   B 
                   
                     I 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     F 
                   
                 
               
             
           
         
       
     
     where B IF  is the bandwidth of the IF filter in the receiver and N 0 /2 is the two-sided power spectral density of the white noise. 
     As shown in  FIG. 2 , when the FM threshold effect occurs, the phase angle, θ(t), of the signal can abruptly increase or decrease by 2 TT  radians in a short period of time. This causes an impulse in the signal having an area of 2 TT , and results in a “click” noise that can be heard by the user. The click noise generally indicates that the FM threshold has been reached, and the noise is greater than the signal. 
     There is a need for an FM demodulator that will extend the threshold beyond the current levels. 
     BRIEF SUMMARY 
     One embodiment of the present disclosure is directed to an FM demodulator having an extended threshold breakdown point. The FM demodulator uses an arcsin demodulator in combination with a feedback loop having an error circuit to produce an output signal that has a linear relationship between the signal-to-noise ratio and the carrier-to-noise ratio for a wider range of carrier-to-noise ratios. The FM demodulator also includes three filters that filter signals based on a coefficient α. According to one embodiment, the coefficient α can be adjusted automatically based on signal strength, frequency deviation and distortion levels of the FM signal. 
     According to a further embodiment, a first mixer receives the FM signal and a signal from the error circuit, and produces an error difference signal to the first filter. The first filter then produces a signal to the arcsin demodulator, which demodulates the signal. The arcsin demodulator provides a signal to the second filter, which provides a signal to the error circuit and to a third filter. The third filter produces an output signal of the FM demodulator. 
     A benefit of an FM demodulator in accordance with the present disclosure is an extended threshold breakdown point, allowing the FM demodulator to demodulate FM signals having a low signal strength. With the principles as taught herein, an FM demodulator is able to demodulate an FM signal that would not be possible with traditional FM demodulation methods, such as when the FM receiver is a long distance from the broadcasting antenna, or when destructive interference from reflected signals are present. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         FIG. 1  shows a signal-to-noise ratio of an FM signal having a threshold breakdown effect according to the prior art. 
         FIG. 2  shows a phase of an FM signal having abrupt 2 TT  jumps as a result of the threshold breakdown effect according to the prior art. 
         FIG. 3  shows a signal-to-noise ratio of an FM signal having an extended threshold breakdown according to one embodiment of the present disclosure. 
         FIG. 4A  shows an FM demodulator having a frequency compressive loop according to one embodiment of the present disclosure. 
         FIG. 4B  shows an arcsin FM demodulator according to one embodiment of the present disclosure. 
         FIG. 5  shows an arcsin threshold extension demodulator according to one embodiment of the present disclosure. 
         FIG. 6  shows a simulated performance of the arcsin threshold extension demodulator of  FIG. 5  at a modulating tone frequency of 1 kHz compared to the prior art according to one embodiment of the present disclosure. 
         FIG. 7  shows a simulated performance of the arcsin threshold extension demodulator of  FIG. 5  at a modulating tone frequency of 10 kHz compared to the prior art according to one embodiment of the present disclosure. 
         FIG. 8  shows a simulated performance of the arcsin threshold extension demodulator of  FIG. 5  with a multipath channel compared to the prior art according to one embodiment of the present disclosure. 
         FIG. 9  shows a simulated output signal of the arcsin threshold extension demodulator of  FIG. 5  with a multipath channel compared to the prior art according to one embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description, certain specific details are set forth in order to provide a thorough understanding of the various embodiments of the disclosure. However, one skilled in the art will understand that the disclosure may be practiced without these specific details. In some instances, well-known structures associated with FM demodulation have not been described in detail to avoid obscuring the descriptions of the embodiments of the present disclosure. 
     Unless the context requires otherwise, throughout the specification and claims that follow, the word “comprise” and variations thereon, such as “comprises” and “comprising,” are to be construed in an open, inclusive sense, that is, as “including, but not limited to.” 
     Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment. Thus, the appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. 
     In the drawings, identical reference numbers identify similar features or elements. The size and relative positions of features in the drawings are not necessarily drawn to scale. 
       FIG. 3  shows the signal-to-noise ratio of an FM signal having an extended breakdown threshold  42 . In the graph of  FIG. 3 , the carrier-to-noise ratio ρ is on the X-axis, and the signal-to-noise ratio S/N is on the Y-axis. The linear portion of the S/N ratio extends past the traditional threshold point  40  as the carrier-to-noise ratio ρ decreases. According to one embodiment, the extended threshold  42  may be linear for up to 10 dB longer than the threshold  40  as the carrier-to-noise ratio level decreases. 
       FIG. 4A  is an FM demodulator having a frequency compressive feedback (FMFB) according to one embodiment of the present disclosure. The frequency compressive feedback technique will only be briefly explained here. An FM signal s(t) is received at a first input of a first mixer  20 , which also receives at a second input an error reference signal ε 0 (t). The mixer  20  outputs an error difference signal ε d (t), which is a received at a first filter  22 , generally a band pass filter. The band pass filter  22  outputs a filtered signal x(t), which is received at an FM demodulator  24 . The FM demodulator demodulates the filtered signal x(t) based on the equation 
             β   =         Peak   ⁢           ⁢   RF   ⁢           ⁢   frequency   ⁢           ⁢   deviation       Maximum   ⁢           ⁢   Modulating   ⁢           ⁢   baseband   ⁢           ⁢   frequency       =         f   d       f   max       .             
and outputs a demodulated signal x d (t). The demodulated signal x d (t) is received at a loop compensation filter  26 , which filters the demodulated signal and outputs a detected error signal ε v (t). The detected error signal ε v (t) is fed into an error circuit  28 . The error circuit  28  contains a loop filter  30  that receives the detected error signal, and an FM modulator  32  coupled to the loop filter  30  and configured to provide the error reference signal ε 0 (t) back to the mixer  20 . The loop filter  30  and FM modulator  32  thus create a feedback loop for the FM demodulation system. The detected error signal ε v (t) is also fed into an output compensation filter  34 , which filters the detected error signal, and outputs an output signal.
 
     This FMFB system reduces the FM demodulator noise bandwidth by reducing the modulation index by the feedback factor, resulting in an extended threshold. A mathematical analysis of the FMFB system described above shows the reduced modulation index. For simplification, we let:
 
 s ( t )= A   c  cos ω c   t +φ( t ))
 
     where 
     
       
         
           
             
               φ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               2 
               ⁢ 
               π 
               ⁢ 
               
                   
               
               ⁢ 
               
                 k 
                 f 
               
               ⁢ 
               
                 
                   ∫ 
                   0 
                   t 
                 
                 ⁢ 
                 
                   
                     m 
                     ⁡ 
                     
                       ( 
                       τ 
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     τ 
                   
                 
               
             
           
         
       
     
     and we let
 
ε 0 ( t )= A   v  cos(ω c   t+ 0( t ))
 
     where 
     
       
         
           
             
               θ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 K 
                 v 
               
               ⁢ 
               
                 
                   ∫ 
                   0 
                   t 
                 
                 ⁢ 
                 
                   
                     
                       ɛ 
                       v 
                     
                     ⁡ 
                     
                       ( 
                       τ 
                       ) 
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     
                       ( 
                       τ 
                       ) 
                     
                   
                 
               
             
           
         
       
     
     and where K v  is the gain of the FM modulator. This results in 
     
       
         
           
             
               
                 ɛ 
                 d 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   - 
                   
                     1 
                     2 
                   
                 
                 ⁢ 
                 
                   A 
                   c 
                 
                 ⁢ 
                 
                   A 
                   v 
                 
                 ⁢ 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       
                         φ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       - 
                       
                         θ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   A 
                   c 
                 
                 ⁢ 
                 
                   A 
                   v 
                 
                 ⁢ 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       
                         2 
                         ⁢ 
                         
                           
                             ω 
                             c 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       + 
                       
                         φ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       + 
                       
                         θ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     and 
     
       
         
           
             
               x 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 - 
                 
                   1 
                   2 
                 
               
               ⁢ 
               
                 A 
                 c 
               
               ⁢ 
               
                 A 
                 v 
               
               ⁢ 
               
                 sin 
                 ⁡ 
                 
                   ( 
                   
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     - 
                     
                       θ 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     and 
     
       
         
           
             
               
                 ɛ 
                 v 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   
                     2 
                     ⁢ 
                     π 
                   
                 
                 ⁢ 
                 
                   K 
                   D 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   
                     ∂ 
                     t 
                   
                 
               
               - 
               
                 
                   1 
                   
                     2 
                     ⁢ 
                     π 
                   
                 
                 ⁢ 
                 
                   K 
                   D 
                 
                 ⁢ 
                 
                   ∂ 
                   
                     ∂ 
                     t 
                   
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       
                         K 
                         v 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           t 
                         
                         ⁢ 
                         
                           
                             
                               ɛ 
                               v 
                             
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             τ 
                           
                         
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
     Solving for ε v  leads to 
     
       
         
           
             
               
                 ɛ 
                 v 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     1 
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                   ⁢ 
                   
                     K 
                     D 
                   
                 
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         K 
                         D 
                       
                       ⁢ 
                       
                         K 
                         v 
                       
                     
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   
                     ∂ 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   
                     ∂ 
                     t 
                   
                 
                 . 
               
             
           
         
       
     
     Substituting ε v (t) into x(t) leads to 
     
       
         
           
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       A 
                       c 
                     
                     ⁢ 
                     
                       A 
                       v 
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           
                             φ 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               K 
                               v 
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 t 
                               
                               ⁢ 
                               
                                 
                                   
                                     ɛ 
                                     v 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   t 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       A 
                       c 
                     
                     ⁢ 
                     
                       A 
                       v 
                     
                     ⁢ 
                     
                       sin 
                       [ 
                       
                         
                           1 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   1 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                 
                                 ⁢ 
                                 
                                   K 
                                   D 
                                 
                                 ⁢ 
                                 
                                   K 
                                   v 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           φ 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     This results in the modulation index of the original signal φ(t) being reduced by the factor 1/(1+1/(2 TT )K D K v ), providing an extension of the threshold. 
       FIG. 4B  shows an arcsin demodulator  50  according to one embodiment of the present disclosure. The Arcsin demodulator  50  includes an inverse circuit  52 , which receives the incoming signal, and produces an inverse of the absolute value of the signal. An in-phase signal I is received at a first input of a mixer  54 , and the inverse of the absolute value of the signal produced by the inverse circuit  52  is received at the second input of the mixer  54 , which outputs a mixed in-phase signal. Another mixer  56  receives a quadrature phase signal Q at a first input, and the inverse of the absolute value of the signal produced by the inverse circuit  52  at a second input, and outputs a mixed quadrature phase signal. The mixer  54  outputs the mixed in-phase signal to an inverse circuit  58 , which produces an inverse z-transform of the mixed in-phase signal. The mixer  56  outputs the mixed quadrature phase signal to another inverse circuit  60 , which produces an inverse z-transform of the mixed quadrature phase signal. A mixer  62  receives at a first input the inverse z-transform of the mixed in-phase signal, and at a second input the mixed quadrature phase signal. Another mixer  64  receives at a first input the inverse z-transform of the mixed quadrature phase signal, and at a second input the mixed in-phase signal. A summing circuit  66  receives the output of the mixer  62  at a first input, and the output of the mixer  64  at a second input, and is configured to output a summed signal that is the output signal of the mixer  64  subtracted from the output signal of the mixer  62 . The summed signal from the summing circuit  66  is provided to a divider circuit  68 , which divides the summed signal by 2 TT . The divided signal from the divider circuit  68  is then provided to an arcsin circuit  70 , which produces a demodulated signal. 
       FIG. 5  shows an arcsin threshold extension FM demodulator  80  that incorporates the arcsin demodulator  50 . The arcsin threshold extension FM demodulator  80  uses a frequency compressive feedback loop, also referred to as FMFB, in conjunction with the arcsin demodulator  50  to extend the threshold breakdown point of an FM signal. 
     In the arcsin threshold extension FM demodulator  80 , an FM signal s(t) is received at a first input of a mixer circuit  82 . The mixer circuit  82  also receives at a second input an error reference signal ε 0 (t), and outputs an error difference signal ε d (t), which is a received at a filter  84 . The filter  84  is generally a low pass filter that filters out frequencies based on a coefficient α, but can be any suitable filter such as a band pass filter. According to one embodiment, the filter  84  filters frequencies according to the equation 
                 (     1   -   α     )       1   -     α   ⁢           ⁢     z     -   1             ,         
where α is based in part on a signal strength indicator, frequency deviation, and distortion. According to some embodiments, the coefficient α will be pre-set when the arcsin threshold extension FM demodulator  80  is manufactured. However, in other embodiments the coefficient α is adjustable on the fly, changing according to the presently detected signal strength. In some embodiments, the coefficient α will automatically change based on the detected signal strength, frequency deviation or distortion, and in other embodiments the coefficient will be user adjustable. The coefficient α is generally between 0.1 and 0.9, and preferably between 0.2 and 0.85. If the signal strength is weak, the coefficient α will be higher, preferably between 0.5 and 0.9. For a medium or strong signal strength, the coefficient α will be a lower value, preferably between 0.1 and 0.5.
 
     The filter  84  outputs a filtered signal x(t), which is received at the arcsin demodulator  50 . The arcsin demodulator  50  demodulates the filtered signal x(t), and outputs a demodulated signal x d (t). In some embodiments, an absolute value circuit  86  also receives the filtered signal x(t) when the signal is below a certain threshold V thr . The absolute value circuit  86  produces an absolute value of the filtered signal x(t). A mixer  88  receives at a first input the demodulated signal x d (t) and at a second input the absolute value of the filtered signal x(t). The mixer  88  outputs a signal to a loop compensation filter  90 , which filters out frequencies from the received signal based on the coefficient α. According to one embodiment, the loop compensation filter  90  filters frequencies based on the equation 
     
       
         
           
             
               
                 1 
                 - 
                 
                   α 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     z 
                     
                       - 
                       1 
                     
                   
                 
               
               
                 1 
                 - 
                 
                   z 
                   
                     - 
                     1 
                   
                 
               
             
             . 
           
         
       
     
     The filter  90  outputs a detected error signal ε v (t). According to some embodiments, a normalizing circuit  92  is coupled to the output of the filter  90 , and normalizes the detected error signal ε v (t). 
     The detected error signal ε v (t) is fed into a feedback loop  94 , which is coupled to the mixer circuit  82 . The feedback loop  94  contains an error circuit  96 , which receives the detected error signal ε v (t), and produces the error reference signal ε 0 (t) to the second input of the mixer circuit  82 . According to one embodiment, the error circuit  96  has a delay circuit  98  configured to delay the detected error signal ε v (t), an integrator circuit  100  coupled to the delay circuit  98  and configured to integrate the signal, and an FM modulator  102 , configured to modulate the signal. The FM modulator  102  is coupled to the second input of the mixer circuit  82 , and provides the error reference signal ε 0 (t) to the mixer  82 . The delay circuit  98  is generally a z-transform delay in the form of z −1 , but may be any suitable delay. The integrator circuit  100  is also generally performed by a z-transform, in the form of 1/(1−z −1 ), but may be any other suitable integration technique. 
     The detected error signal ε v (t) is also received at a filter  104 . The filter  104  is generally an output compensation filter configured to filter frequencies based on the coefficient α. Preferably, the filter  104  is configured to filter based on the equation 
                 1   -     α   ⁢           ⁢     z     -   1             1   -   α       .         
The filter  104  produces an output signal, which is then received by stereo equipment or the like.
 
       FIGS. 6 and 7  show simulated results of the arcsin threshold extension FM demodulator (Arcsin TED)  80  of  FIG. 5  compared to the prior art.  FIG. 6  shows the prior art and arcsin threshold extension FM demodulator  80  both with a modulating tone, f m , of 1 kHz, a frequency deviation, f d , of 75 kHz, and with the coefficient α of the arcsin threshold extension FM demodulator  80  set to 0.75. The solid line shows the signal-to-noise and distortion ratio (SINAD) of the prior art as the carrier-to-noise ratio ρ increases, while the dash-dot line shows the SINAD of the arcsin threshold extension FM demodulator  80 . The dashed line of  FIG. 6  shows the SINAD improvement of the arcsin threshold extension FM demodulator  80  over the prior art. It can be seen from  FIG. 6  that the arcsin threshold extension FM demodulator  80  shows a significant improvement in SINAD over the prior art, especially from ρ values of 0 dB to 5 dB.  FIG. 7  shows the performance of the arcsin threshold extension FM demodulator  80  compared to the prior art with a modulating tone, f m , of 10 kHz and the coefficient α set to 0.5. In  FIG. 7  it can be seen that the arcsin threshold extension FM demodulator  80  again has a significant SINAD improvement over the prior art, especially for ρ values of 5 dB to 10 dB. 
       FIG. 8  shows the performance of the arcsin threshold extension FM demodulator  80  and the prior art for an FM signal having a multipath channel. Because FM signals are frequently broadcast in cities for use by vehicle radios, a common occurrence in FM signals is for a vehicle to receive the signal as two paths. One path is the direct signal from the broadcasting tower, and a second path is a reflection of the signal off a nearby building. The reflected second signal is delayed by a certain amount because of the additional distance traveled to reflect off the nearby building. When these two signals meet at a vehicle&#39;s antenna, they may be constructive or destructive, depending on the delay of the second signal. When the signals are out of phase, they are destructive and the second delayed signal will reduce the amplitude of the signal received at the vehicle&#39;s antenna, often resulting in “null” zones for a vehicle. 
       FIG. 8  shows the SINAD for the arcsin threshold extension FM demodulator  80  and the prior art across varying levels of carrier-to-noise ratio ρ. A multi-path channel is simulated by adding a second simulated signal having a delay of 22 μseconds and an amplitude of 0.8 of the original signal. The simulated multipath performance in  FIG. 8  is with a modulating tone of 1 kHz, a frequency deviation of 75 kHz, and a coefficient α of 0.75.  FIG. 8  shows that the arcsin threshold extension FM demodulator  80  has considerable improvement over the prior art across all levels of carrier-to-noise ratio ρ, and especially above ρ values of 0 dB. 
       FIG. 9  shows the demodulated output signals from the arcsin threshold extension FM demodulator  80  and the prior art under the simulated multipath conditions of  FIG. 8  for a carrier-to-noise ratio ρ of 10 dB. It can be seen from  FIG. 9  that the arcsin threshold extension FM demodulator  80  has a much smoother output than the prior art. The assorted sharp peaks and valleys of the prior art output are reduced or removed in the arcsin threshold extension FM demodulator  80  output. 
     The various embodiments described above can be combined to provide further embodiments. These and other changes can be made to the embodiments in light of the above-detailed description. In general, in the following claims, the terms used should not be construed to limit the claims to the specific embodiments disclosed in the specification and the claims, but should be construed to include all possible embodiments along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by the disclosure.