Abstract:
A simplified digital implementation of a fourth order Linkwitz-Riley crossover network is provided using approximations and transformations of the classical form. The approximation is particularly beneficial when the crossover frequency is low relative to the digital sampling rate, such as when an audio stream is split between bass and treble at about 30-300 Hz and the sampling frequency is about 100 times the cutoff frequency or higher. Rather than merely cascading two sets of second order filters, such as Butterworth filters, a fourth order transfer function is more directly implemented. Conventional transfer functions are simplified through approximations resulting in the elimination of all except one parameter, c, which is a linear function of the cutoff frequency. Additionally, multipliers are moved in line with the integrator elements. A modulator may be inserted in the processing path at the output of each integrator element if limited fixed precision of the operators is desired while maintaining high performance. A crossover of the inventive design requires a fewer number of state variables, multipliers and adders.

Description:
TECHNICAL FIELD  
       [0001]     The present Invention pertains generally to crossover networks and, in particular, to a simplified implementation of a digital fourth order Linkwitz-Riley network with a low cutoff frequency.  
       BACKGROUND ART  
       [0002]     High quality audio speakers are typically designed to operate best over a limited range of frequencies. Consequently, crossover filters or networks have long been used in audio systems to separate the band of audio frequencies into two or more sub-bands, with each sub-band used to drive a separate speaker. Desired characteristics of crossover network include relatively flat response, rapid roll off at the cutoff frequency or frequencies, minimum phase response and a minimum number of components.  
         [0003]     The following notation will be used herein with respect to the various equations which are presented: 
        f c  is the cutoff frequency (Hz);     f s  is the sampling rate (Hz);     ω c  is the angular cutoff frequency;     ω c =2 Pi*f c  is the angular cutoff frequency in continuous time;     ω c =2 Pi*f c /f s  is the angular cutoff frequency in discrete time;     s is the s transform parameter;     z is the z transform parameter;     z i =1/z: this is implemented as a delay element in discrete time systems;     z t =z i /(1−z i ) : this is implemented as an integrator in discrete time systems.        
 
         [0013]     As is known, a two-way crossover network comprises a low pass filter and a high pass filter (a network for more than two speakers would include one or more intermediate band pass filters). Numerous types of crossover networks have been developed, each with its own transfer function and resulting characteristics. Butterworth, Tchebychev and Bessel filters are among the most widely used. In addition, crossover networks may be implemented in different “orders”. A first order network is relatively simple, has in-phase outputs and has a roll off of 6 dB/octave. Because there is significant output beyond the crossover frequency, the speaker drivers must be able to handle the corresponding energy.  
         [0014]     A second order network, such as the popular Butterworth, is more complex but, as illustrated in the frequency response plots of  FIGS. 1A and 1B , the low pass and high pass elements have sharper roll offs of 12 dB/octave (reducing the demand on the drivers); however, the outputs of the low pass and high pass filters are 180° out of phase, as illustrated in the phase response plots of  FIGS. 1C and 1D . In the FIGs, the cutoff frequency f c  is 10 Hz. The transfer functions of the low pass and high pass continuous time second order Butterworth filters are:  
             BW2_LP   =         ω   c   4           (     s   +         (     -   1     )       1   /   4       ⁢     ω   c         )     ⁢     (     s   -         (     -   1     )       3   /   4       ⁢     ω   c         )                 (   1   )               BW2_HP   =       s   2         (     s   +         (     -   1     )       1   /   4       ⁢     ω   c         )     ⁢     (     s   -         (     -   1     )       3   /   4       ⁢     ω   c         )                 (   2   )             
 
 As illustrated in the plots of  FIGS. 2A and 2B , the low pass filter has 2 complex conjugate poles, while the high pass filter has the same poles as the low pass filter plus two zeroes at 0. A second order Linkwitz-Riley network may be designed from combining second order low pass and high pass Butterworth filters and has the following combined transfer function:  
               LR   ⁢           ⁢   2     =         s   2         (     s   +         (     -   1     )       1   /   4       ⁢     ω   c         )     ⁢     (     s   -         (     -   1     )       3   /   4       ⁢     ω   c         )         +         ω   c   4           (     s   +         (     -   1     )       1   /   4       ⁢     ω   c         )     ⁢     (     s   -         (     -   1     )       3   /   4       ⁢     ω   c         )                   (   3   )             
 
  FIGS. 3A and 3B  are plots of the amplitude and phase responses, respectively, of a second order Linkwitz-Riley network (showing plots of the low pass component, the high pass component and their sum), again having a cutoff frequency of 10 Hz. 
 
         [0015]     A fourth order Butterworth network is still more complex and has an even sharper roll off of 24 dB/octave. This network is generally not economically feasible to implement as a passive network.  
         [0016]     A fourth order Linkwitz-Riley network, which is typically designed from two series-connected second order Butterworth filters, retains the sharp roll off advantage of fourth order filters and has the added advantages of having a substantially flat frequency response and having outputs which are 6 dB down at the crossover frequency (instead of only 3 dB for other filters) and in-phase. A fourth order Linkwitz-Riley network may be designed by cascading two second order Linkwitz-Riley networks as follows:  
               LR   ⁢           ⁢   4     =       LR4_HP   +   LR4_LP     =         s   4           (     s   +         (     -   1     )       1   /   4       ⁢     ω   c         )     2     ⁢       (     s   -         (     -   1     )       3   /   4       ⁢     ω   c         )     2         +       ω   c   4           (     s   +         (     -   1     )       1   /   4       ⁢     ω   c         )     2     ⁢       (     s   -         (     -   1     )       3   /   4       ⁢     ω   c         )     2                     (   4   )             
 
  FIGS. 4A and 4B  are amplitude and phase response plots of a continuous time, fourth order Linkwitz-Riley network (showing plots of the low pass component, the high pass component and their sum). 
 
         [0017]     For digital audio signals, the above-described crossover networks may have digital counterparts. A particular digital implementation of a fourth order Linkwitz-Riley filter is designed as a cascade of second order filters with programmable coefficients. In order for the cutoff frequency to be selectable over a reasonable range, such as 30-300 Hz, with low distortion, the filter coefficients must have very high accuracy.  FIG. 5  illustrates one stage of such a filter  100 . Two such stages would be combined for the low pass section of the crossover network and two more such stages would be combined for the high pass section. The full design requires 16 state variables, 20 multipliers with 20 coefficients and 16 adders. Moreover, in some applications in which the audio stream is highly over-sampled, a first decimation stage may be required, adding to the complexity.  
         [0018]     Consequently, a need remains for a high quality crossover network having an easily implemented design with a minimum number of operations, coefficients and state variables and which does not require a decimation stage.  
       SUMMARY OF THE INVENTION  
       [0019]     The present invention provides a simplified digital implementation of a fourth order Linkwitz-Riley crossover network. The implementation is particularly beneficial when the crossover frequency is low relative to the digital sampling rate, such as when an audio stream is split between bass and treble at about 30-300 Hz and the sampling frequency is about 100 times the cutoff frequency or higher. Rather than merely cascading two sets of second order filters, such as Butterworth filters, a fourth order transfer function is more directly implemented, resulting in a fourth order Direct Form II type of filter in which delay elements have been replaced by integrators having a gain of c. Conventional transfer functions are simplified through approximation resulting in the elimination of all except the one parameter, c, which is a linear function of the cutoff frequency. Additionally, multipliers are moved in line with the integrators. A modulator may be inserted in the processing path at the output of each integrator if limited fixed precision of the operators is desired while maintaining high performance. A crossover of the inventive design requires a fewer number of state variables, multipliers and adders.  
         [0020]     In one implementation, the crossover network of the present invention includes four integrators coupled in series. The crossover frequency of the network is dependent upon a constant c input to each integrator. The outputs of the integrators are summed with the input and the output of the summer provides both the input to the first integrator and a fourth order high pass output. The output of the last integrator provides a fourth order low pass output. A multiplier may be inserted between the output of each integrator and the summer whereby each integrator output is multiplied by a predetermined value before being summed.  
         [0021]     The outputs of the first and second integrators may also be summed with the fourth order high pass output to provide a second order high pass output while the outputs of the second, third and fourth integrators may be summed with the fourth order low pass output to provide a second order low pass output. In this configuration, an additional set of multipliers may be inserted between the output of each integrator and the second set of summers, also to multiply each integrator output by a predetermined value. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0022]      FIGS. 1A and 1B  are amplitude and phase response plots, respectively, of a continuous time, second order low pass Butterworth filter;  
         [0023]      FIGS. 1C and 1D  are amplitude and phase response plots, respectively, of a continuous time, second order high pass Butterworth filter;  
         [0024]      FIGS. 2A and 2B  are plots of the poles and zeros of continuous time, second order low and high pass Butterworth filters, respectively;  
         [0025]      FIGS. 3A and 3B  are amplitude and phase response plots, respectively, of a continuous time, second order Linkwitz-Riley filter;  
         [0026]      FIGS. 4A and 4B  are amplitude and phase response plots, respectively, of a continuous time, fourth order Linkwitz-Riley filter;  
         [0027]      FIG. 5  illustrates one stage of a continuous time, fourth order Linkwitz-Riley filter;  
         [0028]      FIGS. 6A and 6B  are plots of the poles and zeros of discrete time, second order low and high pass Butterworth filters, respectively;  
         [0029]      FIGS. 7A-7D  are amplitude and phase response plots of discrete time, second order low and high pass Butterworth filters;  
         [0030]      FIG. 8  is a block diagram of the Direct Form II structure of a discrete time, second order Butterworth filter;  
         [0031]      FIGS. 9A-9D  are amplitude and phase response plots of discrete time, second order low and high pass Butterworth filters following two approximations;  
         [0032]      FIG. 10  is a block diagram of a mapping of Direct Form II structure of a discrete time, second order Butterworth filter in which each delay element has been replaced by a multiplier in line with an integrator;  
         [0033]      FIG. 11  is a block diagram of discrete time Linkwitz-Riley filter with both second order and fourth order high and low pass outputs;  
         [0034]      FIG. 12  is a block diagram of a modification to the network stage of  FIG. 10  to accommodate a modulator; and  
         [0035]      FIG. 13  is a block diagram of a second order delta-sigma modulator which may be incorporated into the filter of the present invention. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0036]     The present invention is particularly beneficial in digital applications in which the cutoff frequency f c  is substantially lower than the sampling rate f s  (such as less than or equal to 1% of the sampling rate), thereby reducing the risk of unexpected artifacts in the filter response. Consequently, the present invention may be beneficially implemented in various types of digital circuits. One application is to use the present invention to generate offset tracking loops. Another, which will be described herein, is in digital audio in which a low cutoff frequency, such as between 30 Hz and 300 Hz, is well below the sampling rate.  
         [0037]     Discrete time Butterworth filters are obtained by mapping the S-plane into the Z-plane through a bilinear transformation. The transfer functions of second order Butterworth filters are:  
             BW2_LP   =       -     (     2   ⁢       (     1   +     z   i       )     2     ⁢       Cos   ⁡     [       ω   c     2     ]       2     ⁢         Tan   ⁡     [       ω   c     2     ]       4         )         (         -   2     ⁢     (     1   +     z   i   2       )       +     4   ⁢     z   i     ⁢     Cos   ⁡     [     ω   c     ]         +       2     ⁢     (       -   1     +     z   i   2       )     ⁢     Sin   ⁡     [     ω   c     ]           )               (   5   )               BW2_HP   =       -     (     2   ⁢       (       -   1     +     z   i       )     2     ⁢       Cos   ⁡     [       ω   c     2     ]       2       )         (         -   2     ⁢     (     1   +     z   i   2       )       +     4   ⁢     z   i     ⁢     Cos   ⁡     [     ω   c     ]         +       2     ⁢     (       -   1     +     z   i   2       )     ⁢     Sin   ⁡     [     ω   c     ]           )               (   6   )             
 
         [0038]     The low pass filter has two complex conjugate poles plus two zeroes at −1, while the high pass filter has the same poles as the low pass filter plus two zeroes at 1 (see  FIGS. 6A and 6B ). The response at the cutoff frequency is maintained by pre-warping the frequency axis (see  FIGS. 7A-7D ).  
         [0039]     The discrete time Butterworth filters can be implemented in a digital system by using delay elements for the variable z i =1/z. For instance a Direct Form II structure would use the coefficients of the following rational functions:  
             BW2_LP   =             (         2   ⁢           ⁢       Cos   ⁡     [       ω   c     2     ]       2     ⁢         Tan   ⁡     [       ω   c     2     ]       4           2   +       2     ⁢     Sin   ⁡     [     ω   c     ]             +       4   ⁢           ⁢     z   i     ⁢       Cos   ⁡     [       ω   c     2     ]       2     ⁢         Tan   ⁡     [       ω   c     2     ]       4           2   +       2     ⁢     Sin   ⁡     [     ω   c     ]             +                     2   ⁢     z   i   2     ⁢       Cos   ⁡     [       ω   c     2     ]       2     ⁢         Tan   ⁡     [       ω   c     2     ]       4           2   +       2     ⁢     Sin   ⁡     [     ω   c     ]             )             (     1   -       4   ⁢           ⁢     z   i     ⁢     Cos   ⁡     [     ω   c     ]           2   +         2   ⁢               ⁢     Sin   ⁡     [     ω   c     ]             +         z   i   2     ⁡     (     2   -       2     ⁢     Sin   ⁡     [     ω   c     ]           )         2   +       2     ⁢     Sin   ⁡     [     ω   c     ]               )               (   7   )               BW2_HP   =       (         2   ⁢           ⁢       Cos   ⁡     [       ω   c     2     ]       2         2   +       2     ⁢     Sin   ⁡     [     ω   c     ]             -       4   ⁢           ⁢     z   i     ⁢       Cos   ⁡     [       ω   c     2     ]       2         2   +       2     ⁢     Sin   ⁡     [     ω   c     ]             +       2   ⁢     z   i   2     ⁢       Cos   ⁡     [       ω   c     2     ]       2         2   +       2     ⁢     Sin   ⁡     [     ω   c     ]               )       1   -     (         4   ⁢           ⁢     z   i     ⁢     Cos   ⁡     [     ω   c     ]           2   +         2   ⁢               ⁢     Sin   ⁡     [     ω   c     ]             +         z   i   2     ⁡     (     2   -       2     ⁢     Sin   ⁡     [     ω   c     ]           )         2   +       2     ⁢     Sin   ⁡     [     ω   c     ]               )                 (   8   )             
 
         [0040]     The Direct Form II filter has the structure shown in  FIG. 8  in which b 0 , b 1 , b 2  are the coefficients of the numerator polynomial while a 1 , a 2  are the coefficients of the denominator polynomial. These coefficients are fairly complicated. However, if the cutoff frequency of the filters is much less than the sampling rate (i.e. f c &lt;&lt;f s &lt;=&gt;ω c &lt;&lt;2 Pi), two approximations may be made which greatly simplify the coefficients.  
         [0041]     First, the two zeroes at −1 of the low pass filter have little effect on this filter response, because the attenuation around the Nyquist frequency is very high. Thus, the low pass filter may be safely approximated by removing two zeroes at −1 and adjusting for the gain coming from these zeroes:  
             BW2_LP   =       (     8   ⁢     z   i   2     ⁢       Cos   ⁡     [       ω   c     2     ]       2     ⁢         Tan   ⁡     [       ω   c     2     ]       4         )             (     (     2   +       2     ⁢     Sin   ⁡     [     ω   c     ]           )                   (     1   -       4   ⁢           ⁢     z   i     ⁢     Cos   ⁡     [     ω   c     ]           (     2   +       2     ⁢     Sin   ⁡     [     ω   c     ]           )       +         z   i   2     ⁡     (     2   -       2     ⁢     Sin   ⁡     [     ω   c     ]           )         (     2   +       2     ⁢     Sin   ⁡     [     ω   c     ]           )         )     )                     (   9   )             
 
         [0042]     Second, the coefficients of the Direct Form II implementation are only a function of the angular cutoff frequency ω c . If f c &lt;&lt;f s , ω c  is close to 0 and the coefficients may be approximated by a second order Taylor series of ω c  around 0:  
             BW2_LP   =     {           poly   ,     z   i     ,               {           0   ,     {         b   0     =   0     ,       b   1     =   0     ,       b   2     =     (       ω   c   2     +       0   ⁡     [     ω   c     ]       3       )         }     ,               {               ⁢         a   0     =   1     ,       a   1     =     (       -   2     +       2     ⁢     ω   c       +       0   ⁡     [     ω   c     ]       3       )       ,                     ⁢       a   2     =     (     1   -       2     ⁢     ω   c       +     ω   c   2     +       0   ⁡     [     ω   c     ]       3       )               }           }           }             (   10   )               BW2_HP   =     {           poly   ,     z     i   ,                   {           0   ,     {               ⁢         b   0     =     (     1   -       ω   c       2       +       ω   c   2     4     +       0   ⁡     [     ω   c     ]       3       )       ,                     ⁢         b   1     =     (       -   2     +       2     ⁢     ω   c       -       ω   c   2     2     +       0   ⁡     [     ω   c     ]       3       )       ,                     ⁢       b   2     =     (     1   -       ω   c       2       +       ω   c   2     4     +       0   ⁡     [     ω   c     ]       3       )               }     ,               {               ⁢         a   0     =     (   1   )       ,       a   1     =     (       -   2     +       2     ⁢     ω   c       +       0   ⁡     [     ω   c     ]       3       )       ,                     ⁢       a   2     =     (     1   -       2     ⁢     ω   c       +     ω   c   2     +       0   ⁡     [     ω   c     ]       3       )               }           }           }             (   11   )             
 
         [0043]     The numerator coefficients of the high pass filter can be further approximated to a 0 th  order.  
             BW2_HP   =     {               b   0     =     (     1   +       0   ⁡     [     ω   c     ]       1       )       ,                   b   1     =     (       -   2     +       0   ⁡     [     ω   c     ]       1       )       ,                 b   2     =     (     1   +       0   ⁡     [     ω   c     ]       1       )             }             (   12   )             
 
         [0044]     The following simplified transfer functions for the low pass and high pass filters may now be derived:  
             BW2_LP   =         ω   c   2     ⁢     z   i   2         1   +       (       -   2     +       2     ⁢     ω   c         )     ⁢     z   i       +       (     1   -       2     ⁢     ω   c       +     ω   c   2       )     ⁢     z   i   2                   (   13   )               BW2_HP   =       1   -     2   ⁢     z   i       +     z   i   2         1   +       (       -   2     +       2     ⁢     ω   c         )     ⁢     z   i       +       (     1   -       2     ⁢     ω   c       +     ω   c   2       )     ⁢     z   i   2                   (   14   )             
 
         [0045]     These approximations create little distortion for f s &gt;=100 f c . For processing a direct digital stream (DSD) (in which an audio stream is encoded using a very high sampling rate, such as 64 to 128 times the baseband rate) in an audio application, the sample rate is typically f s =128*48 KHz. The cutoff frequency of the filter to implement this value would be around 100 Hz. In this case, f s =61,440 f c  and the approximations work well. As an example, the low pass and high pass filter transfer functions are plotted in  FIGS. 9A-9D  for f s =100 f c .  
         [0046]     The previous transfer functions reflect the coefficients of a Direct Form II implementation. When f c &lt;&lt;f s , some variable changes may be used to simplify the transfer functions still further which results in small modifications to the filters structure. It may then be observed that the poles of the transfer functions are close to 1. A first variable change, 1/z t =z −1 &lt;=&gt;z t =z i /(1−z i ), preserves the structure and characteristics of the filters but translates the poles by −1 in the 1/z t  plane. In other words, in this plane, the poles are translated close to 0. The dynamic range required to implement the coefficients is dramatically reduced.  
             BW2_LP   =         ω   c   2     ⁢     z   t   2         1   +       2     ⁢     ω   c     ⁢     z   t       +       ω   c   2     ⁢     z   t   2                   (   15   )               BW2_HP   =     1     1   +       2     ⁢     ω   c     ⁢     z   t       +       ω   c   2     ⁢     z   t   2                   (   16   )             
 
 In Equations 15 and 16, z t =z i /(1−z i ) is the transfer function of an integrator. The low pass and high pass transfer functions obtained through this variable change may thus be implemented by using a Direct Form II filter structure, replacing each delay element (z i ) by an integrator (z t ) and using the coefficients from the functions above. 
 
         [0047]     With the previous structure, it is still necessary to calculate two coefficients to implement the filters: √2 ω c  and ω c   2 . In addition, the two coefficients have very different dynamic ranges and implementing the square root and squaring operations forces the coefficients to use high precision to avoid the distortion introduced by the coefficients&#39; quantization. Both problems may be solved by making a second set of variable changes: c=√2ω c , z t c=cz t . The filter structure is somewhat changed by adding a multiplier (c, being a linear function of the angular cutoff frequency) inline with the integrators. The following transfer functions result:  
             BW2_LP   =         z   t     ⁢     c   2         2   ⁢     (     1   +       z   t     ⁢   c     +         z   t     ⁢     c   2       2       )                 (   17   )               BW2_HP   =     1     1   +       z   t     ⁢   c     +         z   t     ⁢     c   2       2                 (   18   )             
 
         [0048]     The result is a set of Direct Form II based filters in which each delay element z −1  has been replaced by a multiplier inline with an integrator element z t c. The filter stage  1000  of  FIG. 10  illustrates this replacement. In the original form, a delay register  1002  receives the input, multiplied by a coefficient c in a multiplier  1004 , and its fed back output through the adder  1006 . Making the variable changes has the advantage of requiring only the single coefficient c which is a linear function of the cutoff frequency, instead of the original complicated set of coefficients. Thus, the cutoff frequency is easier to tune and coefficient quantization and dynamic range issues associated with multiple coefficients are avoided. Moreover, because ω c  is assumed to be small, c is correspondingly small and little precision is required for the multipliers (fewer bits required). Beginning with a Direct Form II topology, performing the z t  mapping and moving the multipliers in line with the integrator paths, the dynamic range may be kept relatively constant across the state variables and a c coefficient squaring operation may be avoided.  
         [0049]     Assuming f c &lt;&lt;f s , a discrete time second order Linkwitz-Riley network may be implemented with the previously approximated discrete time Butterworth transfer functions:  
                 LR2_HP   +   LR2_LP     =       1     1   +       z   t     ⁢   c     +         z   t     ⁢     c   2       2         +         z   t     ⁢     c   2         2   ⁢     (     1   +       z   t     ⁢   c     +         z   t     ⁢     c   2       2       )             ⁢     
     ⁢   or           (   19   )                 LR2_LP   LR2_HP     =         z   t     ⁢     c   2       2             (   20   )             
 
 The high pass filter output has a single unity forward path. The low pass filter may easily be obtained from the high pass filter after going through two z t c elements and dividing by two. Also, the numerator and denominator coefficients are all simple powers of two and may be implemented at minimal cost in hardware. 
 
         [0050]     To obtain a fourth order Linkwitz-Riley network, the previous structure may be used as a first stage to obtain the second order Linkwitz-Riley low pass and high pass filters. Two more comparable structures are then cascaded with the low pass and high pass outputs, respectively, of the first structure to obtain the fourth order low and high pass outputs. However, a simpler structure may be obtained by directly realizing a fourth order Linkwitz-Riley high pass filter from the fourth order polynomial equations, then recreating the other outputs by adding forward paths tapped from the already existing z t c elements outputs. The number of operations and the number of state variables required are both reduced. The feedback path of the structure forces the filters outputs to have four poles. In addition to providing a fourth order network, second order Linkwitz-Riley outputs may also be provided by canceling two of the poles with forward paths.  
         [0051]     The following equations illustrate the filter coefficients from the feedback paths used to implement the fourth order Linkwitz-Riley high pass filter output and illustrate how the other outputs are obtained by adding ztc delayed versions through forward paths.  
               LR4_HP   +   LR4_LP     =     (       1     1   +     2   ⁢     z   t     ⁢   c     +     2   ⁢     z   t     ⁢     c   2       +       z   t     ⁢     c   3       +         z   t     ⁢     c   4       4         +         z   t     ⁢     c   4         4   ⁢     (     1   +     2   ⁢     z   t     ⁢   c     +     2   ⁢     z   t     ⁢     c   2       +       z   t     ⁢     c   3       +         z   t     ⁢     c   4       4       )           )             (   21   )             
 
 where the coefficients of the feedback paths are 1, 2, 2, 1, ¼.  
               LR2_HP   LR4_HP     =     1   +       z   t     ⁢   c     +         z   t     ⁢     c   2       2               (   22   )             
 
 where the coefficients of the forward path for LR2_HP are 1, 1, ½, 0, 0.  
               LR2_LP   LR4_HP     =           z   t     ⁢     c   2       2     +         z   t     ⁢     c   3       2     +         z   t     ⁢     c   4       4               (   23   )             
 
 where the coefficients of the forward path for LR2_LP are 0, 0, ½, ½, ¼.  
               LR4_LP   LR4_HP     =         z   t     ⁢     c   4       4             (   24   )             
 
 where the coefficients of the forward path for LR4_LP are 0, 0, 0, 0, ¼. 
 
         [0052]      FIG. 11  is a block diagram of an embodiment of a network or filter  1100  of the present invention in which the transfer functions of Equations 21-24 are implemented and in which the replacement illustrated in  FIG. 10  has been made. The network  1100  includes a first summer  1102  coupled to receive a digital signal IN; the output of the first summer  1102  comprises the fourth order high pass filter output HP 4 . As used herein, the term “coupled” may refer to an indirect relationship in which two components may be separated by one or more intermediary components, whereby a signal may pass through and be processed or altered by the intermediary component(s), as well as to a direct electrical connection between two components, whereby a signal passes directly from one to the other. If second order high and low pass outputs are desired, optional second and third summers  1104 ,  1106  may be included in the network  1100 . The second summer  1104  has an input coupled to receive the output HP 4  of the first summer  1102 ; the output of the second summer  1104  comprises the second order high pass filter output HP 2 . The third summer  1106  has inputs described hereinbelow and an output which comprises the second order low pass filter output LP 2 . A first multiplier  1108  has an output coupled to the second input of the first adder  1102 .  
         [0053]     The network  1100  further includes four stages  1110 ,  1120 ,  1130  and  1140 ; a first output of each stage is received by an input (or set of inputs) of the first multiplier  1108 . Second outputs of the first and second stages  1110  and  1120  are received by a second input (or set of inputs) of the second summer  1104  and second outputs of the third and fourth stages  1130  and  1140  are received by a second input (or set of inputs) of the third summer  1106 . The second output of the fourth stage  1140  also comprises the fourth order low pass filter output LP 4 .  
         [0054]     Each stage  1110 ,  1120 ,  1130  and  1140  includes a first integrator element z t c  1112 ,  1122 ,  1132 ,  1142  to receive the output from the previous stage (or, in the case of the first stage  1110 , to receive the HP 4  output from the first summer  1102 ). Each integrator element  1112 ,  1122 ,  1132 ,  1142  is a function of the single coefficient c. Their outputs comprise the transfer functions I 1 , I 2 , I 3 , I 4  and are multiplied by a first set of values in second multipliers  1114 ,  1124 ,  1134 ,  1144 . The outputs of the second multipliers  1114 ,  1124 ,  1134 ,  1144  are input into the first multiplier  1108 . If the second order outputs are desired, the outputs I 1  and I 2  are also multiplied by another set of values in third multipliers  1116  and  1126  whose outputs are input into the second summer  1104  which outputs the second order high pass filter output HP 2 . The outputs I 3 , I 4  are also multiplied by another set of values in third multipliers  1136 ,  1146  whose outputs, along with the output of the second stage multiplier  1126 , are input into the third summer  1106  which outputs the second order low pass filter output LP 2 .  
         [0055]     In the embodiment illustrated, the inputs to the multipliers in  FIG. 11  are multiplied by:  
                                                   Multiplier Number   Value                           1108   −1            1114   2           1116   1           1124   2           1126   ½           1134   1           1136   ½           1144   ¼           1146   ¼                      
 
         [0056]      FIG. 12  illustrates a modification of the network stage  1000  illustrated in  FIG. 10 , in the z −1  form. An optional second delay z −1    1008  may be inserted in line between the summer  1006  and the first delay  1002  to process an interleaved two-channel (stereo) audio-stream. Multipliers  1010  and  1012  at the output of the stage  1000  scale the output for adding to the scaled outputs of the other stages, as illustrated in  FIG. 11 .  
         [0057]     Still referring to the network stage of  FIG. 12 , when high performance is desired but with limited fixed precision of the integrator output multipliers  1010  and  1012 , such as in a high quality audio digital-to-analog converter, a second order delta-sigma modulator  1300  may be inserted at the output I. In  FIG. 11 , a modulator may be inserted at the output of each integrator  1112 ,  1122 ,  1132 ,  1142 . A modulator may be inserted whether each network stage includes a single delay or integrator (in the configuration of  FIG. 11 ) or includes the optional second delay  1008  or integrator (in the configuration of  FIG. 12 ).  FIG. 13  is a block diagram of the modulator  1300  which is stable, easy to implement and has good noise rejection properties. The modulator  1300  preferably has a signal transfer function of 1 and a noise transfer function of (1−z −1 ) 2 . More specifically, the modulator  1300  comprises two stages and includes a first summer  1302  having an input to receive the associated integrator output I ( FIG. 12 ). An output of the first summer  1302  is coupled to an input of a quantizer  1304  having an output coupled to the output of the filter stage (in the illustrated example, the output of the quantizer  1304  would be coupled to the input of the second filter stage  1120  of  FIG. 11  as well as to the output multiplier(s)  1114 ,  1116  of the first stage  1110 ). The output of the quantizer  1304  is coupled to an input of a multiplier  1306  having a gain of −1. The output of the multiplier  1306 , as well as the input IN, are input to a second summer  1308 .  
         [0058]     The output of the second summer  1308  is coupled to an input of a third summer  1310 . A first delay element  1312  is coupled to the output of the third summer  1310 . The output of the first integrator  1312  is fed back to the third summer  1310  and is also coupled to inputs of fourth and fifth summers  1314 ,  1320 . A second delay element  1316  is coupled to the output of the fourth summer  1314 . The output of the second delay element  1316  is fed back to the fourth summer  1314  and is also coupled to an input of the fifth summer  1320 . The output of the fifth summer  1320  is coupled to an input of the first summer  1302 .  
         [0059]     The cutoff frequency is still controllable through the use of the single coefficient c, where c=√2*ω c =√2*2π*(f c /f s ). For example, if f c  is 100 Hz and f s  is 128*48 KHz, c will equal 1.44×10 −4 .  
         [0060]     The objects of the invention have been fully realized through the embodiments disclosed herein. Those skilled in the art will appreciate that the various aspects of the invention may be achieved through different embodiments without departing from the essential function of the invention. The particular embodiments are illustrative and not meant to limit the scope of the invention as set forth in the following claims. Moreover, although described above with respect to an apparatus, the need in the art may also be met by a method of processing signals.