Abstract:
Pairs of second-order filters with feedback and cross coupling may be used to implement pairs of complex poles. The cross coupling may be frequency-dependent cross coupling or frequency-independent cross coupling. Frequency independent cross coupling may include coupling an internal node of a biquad filter. The pairs of second-order filters can be used together to form a complex filter. The complex filter can be used to readily provide higher order poles. The resulting complex filter can achieve higher order poles while offering reduced circuit complexity.

Description:
[0001]     This application claims the benefit of U.S. Provisional Application Serial No. 60/529,027, filed Dec. 12, 2003, the entire content of which is incorporated herein by reference. 
     
    
     TECHNICAL FIELD  
       [0002]     The invention relates to complex filters, and more particularly, to complex filters with higher order poles.  
       BACKGROUND  
       [0003]     Complex filters may be useful in a number of applications, such as RF devices. For example, a receiver may use one or more complex filters to reduce noise or filter out adjacent channels. In addition, complex filters may be used to accept a complex signal and separate the real part of the signal from the imaginary part of the signal.  
         [0004]     Circuits that act as complex filters may generate complex poles. A complex pole may be useful in circuits that filter quadrature signals, for example, to provide an asymmetric response about DC. Circuits with complex poles may generate quadrature signals from a single signal and perform amplitude/phase filtering of the quadrature signals. The quadrature signals may be generated by quadrature downconversion or in preparation for quadrature upconversion.  
         [0005]     Several techniques exist for generating a single complex pole. For example, cross coupling may be used between pairs of real poles. Another technique involves converting a pair of ladder-derived real filters into a frequency-shifted complex filter by using cross coupling between the real filters. In both techniques described, frequency-independent cross coupling and single real poles are used.  
       SUMMARY  
       [0006]     In general, the invention is directed to a technique for creating a complex electrical filter, which has an asymmetric response about DC. The complex filter may be especially useful in a wireless communication system. The technique involves the use of pairs of second-order filters, such as biquadratic (biquad) filters, with feedback and cross coupling. The cross coupling may be frequency-dependent cross coupling or frequency-independent cross coupling. Frequency-independent cross coupling may involve coupling to an internal node of the biquad filter. The complex filter can be used to readily provide higher order poles.  
         [0007]     A complex filter in accordance with the invention may provide one or more advantages. For example, the invention allows biquad-derived real filters to be made into complex filters in a straightforward manner. The resulting complex filter can achieve higher order poles while offering reduced circuit complexity. In addition, the shape of a filtered signal can be maintained over a range of operating conditions because pairs of poles in the complex filter move together with changes in operating conditions. Accordingly, the filter may offer reduced sensitivity. In some embodiments, a second-order complex filter may be realized without a differentiator, providing reductions in the chip area consumed by the filter. Additionally, elimination of the differentiator may result in reduced noise.  
         [0008]     In one embodiment, the invention provides a complex filter comprising an input port to receive a complex input signal, a first output port that produces a real output component, a second output port that produces an imaginary output component, a pair of second order filters, wherein each of the second order filters receives a sum of at least a portion of the input signal, and an amplified portion of one of the first and second output components.  
         [0009]     In another embodiment, the invention provides a method comprising receiving a complex input signal, generating a real output component of the complex input signal, generating an imaginary output component of the complex input signal, passing a sum of at least a portion of the complex input signal and an amplified portion of one of the first and second output components through each of a pair of second order filters to produce the complex input signal and the complex output signal.  
         [0010]     In a further embodiment, the invention provides a wireless receiver comprising an antenna to receive a wireless input signal, an amplifier to amplify the wireless input signal, and a complex filter having an input port to receive the wireless input signal, a first output port that produces a real output component of the wireless signal, a second output port that produces an imaginary output component of the wireless signal, and a pair of second order filters, wherein each of the second order filters receives a sum of at least a portion of the input signal, and an amplified portion of one of the first and second output components.  
         [0011]     In another embodiment, the invention provides a complex filter comprising a first biquad filter, a second biquad filter, a first feedback loop between an output and input of the first biquad filter, a second feedback loop between an output and input of the second biquad filter, a cross-coupling between the first biquad filter and the second biquad filter.  
         [0012]     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
     
    
     BRIEF DESCRIPTION OF DRAWINGS  
       [0013]      FIG. 1  is a block diagram illustrating an exemplary communication system that includes a filter for filtering complex signals.  
         [0014]      FIG. 2  is a block diagram illustrating a basic complex filter accepting a single input signal.  
         [0015]      FIG. 3  is a block diagram illustrating a basic complex filter accepting two input signals.  
         [0016]      FIG. 4  is a diagram illustrating an implementation of a complex filter circuit with two second-order filters.  
         [0017]      FIG. 5  is a diagram illustrating an exemplary second-order filter for use as part of a complex filter circuit.  
         [0018]      FIG. 6  is a diagram illustrating an alternative second-order filter with an auxiliary input.  
         [0019]      FIG. 7  is a schematic diagram illustrating the exemplary second-order filter of  FIG. 6 , with no auxiliary input.  
         [0020]      FIG. 8  is a signal flow diagram illustrating an implementation of a complex filter circuit that includes two second-order filters.  
         [0021]      FIG. 9  is a schematic diagram illustrating an alternative second-order filter, which has an auxiliary input.  
         [0022]      FIG. 10  is a schematic diagram illustrating an alternative second-order filter that can be part of a complex filter circuit.  
         [0023]      FIG. 11  is a graph illustrating a frequency response of an exemplary second-order complex filter. 
     
    
     DETAILED DESCRIPTION  
       [0024]      FIG. 1  is a block diagram illustrating an exemplary system  10  that includes a filter  18  for filtering complex signals. Complex filters are used in a variety of telecommunications applications, such as wireless network access points, communication chips, receivers, and transmitters. System  10 , which may be part of a receiver system, includes an antenna  12 , a low noise amplifier (LNA)  14 , a Radio Frequency-to-Intermediate Frequency (RF-to-IF) mixer  16 , a filter  18 , and an analog-to-digital converter  19 .  
         [0025]     System  10  receives an RF signal via antenna  12 . Antenna  12  passes the signal to the LNA  14 , which amplifies the signal. Mixer  16  processes the amplified RF signal by down-converting the signal from a high RF frequency, such as 5.2 GHz, to an intermediate frequency, such as 10 MHz. In one embodiment, mixer  16  comprises a down mixer and a quadrature mixer, which are cascaded in two stages. In another embodiment, mixer  16  may use complex mixing to separate an imaginary image from the signal.  
         [0026]     In one example, mixer  16  may be configured to process signals transmitted within a wireless network conforming to the IEEE 802.11a, 802.11b, or 802.11g standards. Mixer  16  generates baseband signals for in-phase and quadrature phase components of the RF signal. Mixer  16  passes the amplified signal to filter  18 , which filters out adjacent channels, alternate adjacent channels, and noise. For example, filter  18  may filter out negative frequencies, thereby removing negative frequency images from the signal. In one embodiment, filter  18  further includes a block that limits the dynamic range of system  10 .  
         [0027]     Filter  18  may be configured to relay the filtered signal to another component, such as an analog-to-digital (A/D) converter  19 . A/D converter  19  converts the analog signal to a digital signal for additional processing, e.g., with a demodulation block. The digital signal may be further amplified and processed based on the needs of the system  10 .  
         [0028]     In some applications, system  10  may include pairs of second-order filters with feedback and cross coupling for implementing pairs of complex poles. The cross coupling may be frequency-dependent cross coupling or frequency-independent cross coupling. The second-order filters may simultaneously perform both lowpass and highpass filtering of an input signal to selectively pass signals in particular frequency ranges.  
         [0029]      FIG. 2  is a block diagram illustrating a basic complex filter circuit  20  accepting a single input signal, for purposes of example. Complex filters may accept a complex input signal and separate the real part of the signal from the imaginary part of the signal. As shown in  FIG. 2 , complex filter  22  separates an input signal into an ‘I’ output component  24  and a ‘Q’ output component  26 . The ‘I’ output component  24  is the real component of the input signal and the ‘Q’ output component  26  is the imaginary component of the input signal. In one embodiment, the ‘I’ component  24  leads the ‘Q’ component  26  by approximately 90 degrees. In practice, filter circuit  20  may be modified in a variety of manners. For example, filter  22  may accept more than one input.  
         [0030]      FIG. 3  is a block diagram illustrating a basic complex filter circuit  30  accepting two input signals. Rather than inputting a single input signal into filter  32 , both ‘I’ and ‘Q’ components of an input signal are separately applied to filter  32 . Two or more filters  22  ( FIG. 2 ) may be used together to make a higher order complex filter circuit  32 . For example, two complex filters may be used to make a second order complex filter, such as a biquadratic (biquad) filter. Second order filters, which include Tow-Thomas biquad filters, may be used as described below for implementing pairs of complex poles.  
         [0031]      FIG. 4  is a circuit diagram illustrating implementation of a complex filter circuit  40  with two second-order filters. In particular, the two second-order filters may be all-pole biquad filters. The complex filter  40  may include two channels, an ‘I’ channel and a ‘Q’ channel. The ‘I’ channel corresponds to a real portion of an input signal, while the ‘Q’ channel corresponds to the imaginary portion of an input signal. As shown in  FIG. 4 , complex filter circuit includes biquad filters  42 A,  42 B, differentiators (d/dt)  46 A,  46 B, and amplifiers Afb  43 A, Afb  43 B, Ac  44 A, -Ac  44 B, Hc  45 A, -Hc  45 B.  
         [0032]     An all-pole biquad filter has the following transfer function:  
                 y   x     =       w   0   2         s   2     +     s   ⁢       w   0     Q       +     w   0   2                 =     1         s   2       w   0   2       +     s     w   0   2       +   1             =         ⁢     1     1   +     j   ⁢     w       w   0     ⁢   Q         -       w   2       w   0   2               
 
 where y is the output, x is the input, w o  is the cut-off frequency of the biquad filter, w is the frequency, and Q is the ‘Q’ factor of the biquad filter. However, a complex all-pole biquad filter has the transfer function:  
               y   x     =     1     1   +     j   ⁢       (     w   -     w   1       )         w   0     ⁢   Q         -         (     w   -     w   1       )     2       w   0   2                   =     1     1   +     j   ⁢     w       w   0     ⁢   Q         -     j   ⁢       w   1         w   0     ⁢   Q         -       w   2       w   0   2       +     2   ⁢         w   1     ⁢   w       w   0   2         -       w   1   2       w   0   2                     
 
 The value w I  is the frequency shift. 
 
         [0033]     Feedback and cross-coupling are used in order for complex filter circuit  40  to produce the complex all-pole biquad filter transfer function. The gains from cross-coupling include a frequency independent term, and a term proportional to s, which is shown by (d/dt) in  FIG. 4 . As discussed in more detail below, the frequency dependent term may instead be implemented by using a frequency independent cross gain into an auxiliary input of a Tow-Thomas biquad.  
         [0034]     A feedback circuit of the ‘I’ channel may include an amplifier Afb  43 A, which connects the output of biquad filter  42 A to the input of biquad filter  42 A. The output of Afb  43 A is added to the output of amplifier Ac  44 A and amplifier Hc  45 A. The sum of Afb  43 A, Ac  44 A, Hc  45 A and an ‘I’ component of the input signal is inputted into biquad filter  42 A. Maintaining consistency with the complex all-pole biquad filter transfer function, the values of amplifiers Afb, Ac, and Hc are as follows:  
         Afb   =       w   1   2       w   0   2         ,       
 
 which is a frequency-independent feedback term with no j;  
         Ac   =       w   1         w   0     ⁢   Q         ,       
 
 which is a frequency-independent, feedback term from the ‘Q’ channel with no j; and  
         Hc   =       2   ⁢     w   1         w   0   2         ,       
 
 which is a frequency-dependent feedback term from the ‘Q’ channel with j. 
 
         [0035]     The value of the output y of biquad filter  42 A may be expressed with respect to the input x′ of the biquad filter.  
               y   =       x   ′     ⁢     1     1   +     j   ⁢     w       w   0     ⁢   Q         -       w   2       w   0   2               ,           ⁢   where                   x   ′     =       x   +     x   ″       =       y   ⁡     (     1   +     jw       w   0     ⁢   Q       -       jw   1         w   0     ⁢   Q       -       w   2       w   0   2       +     2   ⁢         w   1     ⁢   w       w   0   2         -       w   1   2       w   0   2         )       +     x   ″           ,                 =     y   ⁢     (     1   +     jw       w   0     ⁢   Q       -       w   2       w   0   2         )         ,   and             
 
           x   ″     =     y   ⁡     (         w   1   2       w   0   2       +       jw   1         w   0     ⁢   Q       -     2   ⁢         w   1     ⁢   w       w   0   2           )         ,       
 
 where x n  is the sum of Afb, Ac, and Hc. 
 
         [0036]     For ease of explanation, techniques for calculating only the values of ‘I’ channel components are described. ‘Q’ channel components are calculated using similar techniques. In particular, the same principles used for the ‘I’ channel may be applied to the ‘Q’ channel, with jX as the input and jY as the output. Accordingly,  FIGS. 5-7 ,  9  and  10  generally illustrate one-half of a complex filter for ease of illustration. The other half may be formed by another version of the illustrated filter to form the complex filter.  
         [0037]     The complex poles of complex filter circuit  40  are at:  
           -       w   0       2   ⁢   Q         ±       w   0     ⁢       1   -     1     4   ⁢     Q   2             *   j       +       w   1     ⁢   j         
 
 The pair of complex poles corresponding to complex filter circuit  40  may move together as operating conditions change. In other words, the shape of a filtered signal may be maintained because pairs of real and imaginary poles move together, tracking one another. 
 
         [0038]      FIG. 5  is a diagram illustrating an exemplary second-order filter  50  that can be part of a complex filter. In particular, filter  50  may be a Tow-Thomas biquad filter, which can be used in the complex filter circuit  40  described above. Filter  50  includes inverting integrators  52 ,  54 , and an inverter  57 . In addition, filter  50  includes gains  51 ,  53 ,  55 , and  56 , and summations  58 A and  58 B. Taken together, these components form an exemplary implementation of the unity-gain all-pole biquad transfer function. The transfer function of filter  50  is:  
           V   OUT       V   IN       =       w   0   2         s   2     +       w   0     Q     +     w   0   2               
         [0039]      FIG. 6  is a diagram illustrating an alternative second-order filter  60  with an auxiliary input. The filter  60  may be a Tow-Thomas biquad filter, which may be used in the complex filter circuit  40  described above. Filter  60  is a modified version of filter  50 . In particular, filter  60  includes an auxiliary input that is added to filter  50 . Filter  60  includes inverting integrators  62 ,  64 , and an inverter  67 . In addition, filter  60  includes gains  61 ,  63 ,  65 ,  66 , and  68 , and summations  69 A,  69 B, and  69 C. Together, these components form an exemplary implementation of the unity-gain all-pole biquad transfer function from the main input V IN  to the output V OUT , as well as an implementation of a differentiated version of the unity-gain all-pole biquad transfer function from the auxiliary input V AUX  to the output V OUT . The transfer function of filter  60  from auxiliary input V AUX  to output V OUT  is:  
           V   OUT       V   AUX       =         sw   0   2         s   2     +         w   0     Q     ⁢   s     +     w   0   2         =     s   ⁢       V   OUT       V   IN                 
 As seen in the transfer function of filter  60 , bringing a signal though the auxiliary input is equivalent to bringing s*the signal into the primary input (i.e., s multiplied by the signal into the primary input). This is equivalent to bringing the signal through a differentiator into the primary input. 
 
         [0040]      FIG. 7  is a schematic diagram illustrating the exemplary second-order filter  70  (shown conceptually as filter  50  in  FIG. 5 ), which has no auxiliary input. Filter  70  may be a Tow-Thomas filter and fully-differential to allow inversion by cross-coupling. As shown in  FIG. 7 , the second-order filter  70  may include operational amplifiers  71 A,  71 B, resistor  76 , resistor  77 , resistor  78 , resistor  79 , resistor  80 , and capacitor  81 . More particularly, operational amplifier  71 A comprises input ports  72 A,  73 A, output port  74 A, and a capacitor  75 A connecting output port  74 A to input port  72 A. Likewise, operational amplifier  72 B comprises input ports  72 B,  73 B, output port  74 B, a capacitor  75 B connecting output port  74 B to input port  72 B, and resistor  78  also connecting output port  74 B to input port  72 B. Resistor  77  connects the output  74 A of operational amplifier  71 A to the input  72 B of operational amplifier  71 B. In addition, resistor  76  feeds the output  74 B of operational amplifier  71 B to the input  72 A of operational amplifier  71 A.  
         [0041]     The ‘I’ component of the input signal passes through resistor  79  on the way to input  72 A of operational amplifier  71 A. Likewise, the ‘Q’ component of the output signal passes through resistor  80  in parallel with capacitor  81  on the way to input  72 A of operational amplifier  71 A. Resistor  80  and capacitor  81  may together form a differentiator. Some exemplary relationships of circuit elements in filter  70  are as follows:  
               R   79     =       ⁢     1       w   o     ⁢     C               75   ⁢   A                           R   76     =       ⁢       w   o         (       -     w   1   2       +     w   o   2       )     ⁢     C     75   ⁢   A                         R   77     =       ⁢       -   1         w   o     ⁢     C     75   ⁢   B                         R   78     =       ⁢     Q       w   o     ⁢     C     78   ⁢   B                         R   80     =       ⁢     Q       w   1     ⁢     C     75   ⁢   A                         C   81     =       ⁢     2   ⁢     C     75   ⁢   A       ⁢       w   1       w   0                   
 
 In the above expressions, the subscripted number refers to the reference number of the corresponding component illustrated in  FIG. 7 . 
 
         [0042]      FIG. 8  is a diagram illustrating an implementation of another complex filter circuit  85 . As shown in  FIG. 8 , complex filter circuit  85  includes two second-order filters. In particular, the two second-order filters may be modified Tow-Thomas biquad filters. Each biquad filter receives an auxiliary input along with a primary input. An ‘I’ channel corresponds to a real portion of an input signal, while a ‘Q’ channel corresponds to the imaginary portion of an input signal. As shown in  FIG. 4 , complex filter circuit includes biquad filters  86 A,  86 B, and amplifiers Afb  87 A, Afb  87 B, Ac  89 A, -Ac  89 B, Hc  88 A, -Hc  88 B.  
         [0043]     The function of complex filter  85  is substantially the same as complex filter  40  shown in  FIG. 4 . However, there are some features in complex filter  85  that distinguish it from complex filter  40 . For example, complex filter  85  allows a simple gain to be used rather than a differentiator. In addition, the auxiliary inputs into biquad filters  86 A,  86 B cause the gain of complex filter circuit  85  to be frequency independent.  
         [0044]      FIG. 9  is a schematic diagram illustrating an alternative second-order filter  90  (shown conceptually as filter  60  in  FIG. 6 ), which has an auxiliary input. Filter  90  may be a modified Tow-Thomas filter. As shown in  FIG. 9 , the second-order filter  90  may include operational amplifiers  91 A,  91 B, resistor  96 , resistor  97 , resistor  98 , resistor  99 , resistor  100 , and resistor  101 . More particularly, operational amplifier  91 A comprises input ports  92 A,  93 A, output port  94 A, and a capacitor  95 A connecting output port  94 A to input port  92 A. Likewise, operational amplifier  92 B comprises input ports  92 B,  93 B, output port  94 B, a capacitor  95 B connecting output port  94 B to input port  92 B, and resistor  98  also connecting output port  94 B to input port  92 B. Resistor  97  connects the output  94 A of operational amplifier  91 A the input  92 B of operational amplifier  91 B. In addition, resistor  96  feeds the output  94 B of operational amplifier  91 B to the input  92 A of operational amplifier  91 A.  
         [0045]     The ‘I’ component of the input signal passes through resistor  99  on the way to input  92 A of operational amplifier  91 A. Likewise, the ‘Q’ component of the output signal passes through resistor  100  on the way to input  92 A of operational amplifier  91 A. Additionally, the auxiliary input signal passes through resistor  101  on the way to input  92 B of operational amplifier  91 B. Some relationships of circuit elements in complex filter  90  are as follows:  
               R   99     =       ⁢     1       w   o     ⁢     C     95   ⁢   A                         R   96     =       ⁢       w   o         (       -     w   1   2       +     w   o   2       )     ⁢     C     95   ⁢   A                         R   97     =       ⁢       -   1         w   o     ⁢     C     95   ⁢   B                         R   98     =       ⁢     Q       w   o     ⁢     C     95   ⁢   B                         R   100     =       ⁢     Q       w   1     ⁢     C     95   ⁢   A                         R   101     =       ⁢       w   0       2   ⁢     w   1     ⁢     C     95   ⁢   B                     
 
 In the above expressions, the subscripted number refers to the reference number of the corresponding component illustrated in  FIG. 9 . 
 
         [0046]     Filter  90  is very similar to filter  70  shown in  FIG. 7 . However, filter  90  does not use a differentiator. Instead, filter  90  adds an auxiliary input signal that passes through resistor  101 . Moreover, filter  90  allows a simple gain to be used in place of the differentiator. Lack of a differentiator in complex filter  90  may reduce the chip area required for the second-order complex filter. Additionally, there may be less noise without the differentiator.  
         [0047]      FIG. 10  is a schematic diagram illustrating an alternative second-order filter  110  that may be part of a complex filter. This complex filter is not specifically a Tow-Thomas filter, but rather represents a general filter. In addition, filter  110  accepts no auxiliary input. Second-order filter  110  is similar to the filter shown in  FIG. 7 , except that feedback resistor  78  has been replaced by capacitor  118 . Feedback resistor  78  connected output port  74 B to input port  72 B, whereas capacitor  118  connects the output  114 B of operational amplifier  111 B to input  112 A of operational amplifier  111 A. The Q factor of the filter is then set by the ratio of capacitors  115 B and  118 . In general capacitors can be made to match well, leading to a precise value for the Q factor. The value of resistor  117  can tune all parameters of the filter as long as all capacitors track each other. As shown in  FIG. 10 , the cross gains include a frequency dependent term.  
         [0048]     As shown in  FIG. 10 , the second-order filter  110  may include operational amplifiers  111 A,  111 B, resistor  116 , resistor  117 , resistor  118 , resistor  119 , resistor  120 , and capacitor  121 . More particularly, operational amplifier  111 A comprises input ports  112 A,  113 A, output port  114 A, and a capacitor  115 A connecting output port  114 A to input port  112 A. Likewise, operational amplifier  112 B comprises input ports  112 B,  113 B, output port  114 B, and a capacitor  115 B connecting output port  114 B to input port  112 B. Resistor  117  connects the output  114 A of operational amplifier  111 A the input  112 B of operational amplifier  111 B. In addition, resistor  116  and capacitor  118  feed the output  114 B of operational amplifier  111 B to the input  112 A of operational amplifier  111 A.  
         [0049]     The ‘I’ component of the input signal passes through resistor  119  on the way to input  112 A of operational amplifier  111 A. Likewise, the ‘Q’ component of the output signal passes through resistor  120  in parallel with capacitor  121  on the way to input  112 A of operational amplifier  111 A. Resistor  120  and capacitor  121  may together form a differentiator. Some relationships of circuit elements in complex filter  110  are as follows:  
               R   119     =       ⁢     1       w   o     ⁢     C     115   ⁢   A                         R   116     =       ⁢       w   0         (       -     w   1   2       +     w   o   2       )     ⁢     C     115   ⁢   A                         R   117     =       ⁢       -   1         w   o     ⁢     C     115   ⁢   B                         R   120     =       ⁢     Q       w   1     ⁢     C     115   ⁢   A                         C   121     =       ⁢     2   ⁢     C     115   ⁢   A       ⁢       w   1       w   0                       C   118     =       ⁢       C     115   ⁢   A       Q               
 
 In the above expressions, the subscripted number refers to the reference number of the corresponding component illustrated in  FIG. 10 . 
 
         [0050]      FIG. 11  is a graph illustrating a frequency response  130  of an exemplary second-order complex filter. The second-order complex filter may behave as a bandpass filter. In particular, the filter may filter out any unwanted frequencies. In one example, the filter allows only signals within a band of real frequencies to pass. The frequency response  130  shows a band of frequencies that are allowed to pass through a complex filter. As shown, the magnitude  131  of the frequency response  130  is greatest between frequency  132  and frequency  133 . In one embodiment, the maximum magnitude  131  of the frequency response may be approximately zero decibels. Depending on the characteristic of the bandpass filter, a number of frequency ranges may pass through the filter. For example, frequency  132  may be approximately zero MHz, and frequency  133  may be approximately 20 MHz.  
         [0051]     The frequency response  130  of the second-order complex filter may be substantially the same regardless of how the filter is implemented. For example, a Tow-Thomas implementation using a differentiator, a Tow-Thomas implementation using the auxiliary input, a high-Q biquad using a differentiator, a high-Q biquad using an auxiliary input, a different type of biquad using a differentiator, or a different type of biquad whose first stage is an integrator, all give the same frequency response. By design, any of the filters described above may conform to the principles of a bandpass filter.  
         [0052]     Various embodiments of the invention have been described. These and other embodiments are within the scope of the following claims.