Abstract:
A fully-integrated continuous-time active complex bandpass IF filter that may contain transmission zeros yielding much sharper roll-off than that of an all-pole filter is implemented using transconductors and capacitors only. Each of the filter second-Order sections realizes a pair of complex poles and a may realize a double imaginary axis zero. Since the transconductors are electronically tunable the positions of filter zeros and poles are adjustable using an automatic tuning system. In each filter section the value of different transconductors are modified to separately change the pole frequency, its Q-factor and the zero frequency. Each pole and zero are separately tuned, which achieves a higher level of tuning accuracy than in case where all poles and zeros were adjusted simultaneously.

Description:
TECHNICAL FIELD OF THE INVENTION 
   The present invention relates to fully-integrated continuous-time active complex band-pass filters and their synthesis method using transconductance amplifiers and capacitors that allows direct synthesis of their transfer function poles and transmission zeros. 
   BACKGROUND OF THE INVENTION 
   Complex bandpass (BP) active filters are widely used in integrated receivers. These filters serve primarily as intermediate-frequency (IF) channel select filters with an additional function of providing image rejection. Their ability to reject unwanted image frequencies of the preceding mixer results directly from their non-symmetrical transfer characteristic. Depending on its input signal conditioning, a complex BP filter transmits for positive frequencies and rejects all negative frequencies, or vice-versa, the filter transmits for negative frequencies and blocks all positive frequencies. 
   The achievable image-rejection ratio (IRR) depends on the matching of on-chip components used in the complex filter. These components include resistors, capacitors and transconductors. Also, non-ideal gain of operational amplifiers (opamps) if used for filter synthesis results in IRR degradation. The IRR performance also depends on the choice of synthesis method. Certain methods are more sensitive to component variation than others. Practically achievable IRR of a complex filter is 30–35 dB. If extreme caution is taken to achieve an excellent component matching, or if an IRR is enhanced by a special automatic tuning scheme IRR better than 55 dB is achievable. 
   Complex BP filters can be realized using two distinct synthesis methods: the active ladder simulation and the direct synthesis. Similarly to the classical passive LC ladder synthesis method, in the active ladder simulation the pole frequency and its quality (Q) factor are defined by all filter elements. Contrary, in the direct synthesis method the pole frequency and its Q factor are defined by elements of one particular filter section. Due to its lower sensitivity to the component value variation, the active ladder simulation method is superior to the direct synthesis method. However, the latter results in a simple circuit that is usually easier to integrate. 
   The complex bandpass filters can also be categorized according to the chosen active synthesis method. Two different active synthesis techniques have been used: the active-RC technique described in U.S. Pat. No. 4,914,408 and the gyrator method described in U.S. Pat. No. 6,346,850. In the active-RC method the transfer function is realized using active-RC integrators built with input series resistors and feedback capacitors around opamps. The gyrator method uses voltage-controlled current sources and capacitors to realize integrators. The advantage of the gyrator method over the active-RC method stems from the gyrator method&#39;s ability to adjust the filter pole frequency through adjusting the transconductance of voltage controlled current sources, which is not easily achievable in a R and C arrangement of the active RC filters. 
   Due to their prime application as channel select filters complex BP filters must demonstrate sharp roll-off outside their pass-bands. In wireless receiver system design, BP filter attenuation determines such critical parameters as co-channel and adjacent channel rejection. Steep roll-off is not easily achievable with all-pole transfer functions. Depending on their order all-pole transfer functions may be quite steep, but as illustrated in  FIG. 7 , their roll-off is never as steep as that of filters that contain transmission zeros in their transfer function. For all these reasons, in an integrated receiver design there is a strong need for complex BP filters with transmission zeros. 
   DESCRIPTION OF THE PRIOR ART 
   The arrangement for a complex all-pole (no transmission zeros) bandpass active ladder simulation scheme using active-RC integrators such as one described in U.S. Pat. No. 4,914,408 is illustrated in  FIG. 1   a , and is identified by the numeral  10 . It consists of two identical banks of active-RC integrators  12  connected by the feedback resistors  14 . Without these resistors each of the banks performs a lowpass function. The lowpass filter integrator banks  12  are connected in so called a leapfrog structure as shown in  FIG. 1   b , which is a common technique to simulate lossless ladders using active integrators. Two types of integrators are used. Lossy integrators  16  simulate the ladder first element with the source resistor and the ladder last element with the load resistor. Lossless integrators  18  simulate the reminding ladder elements. Since presented ladder is of 3 rd -Order, only one lossless integrator  18  is necessary. Similar to the classical passive LC ladder synthesis method, in the active ladder simulation a single pole frequency and its quality (Q) factor are defined by all filter elements. Such structures are characterized by low sensitivities of their pole frequencies and Q-factors to the component value variation. However, since on-chip RC time-constants can vary as much as 30–40% the accuracy of cut-off frequency of a lowpass built with active-RC integrators if no tuning is applied is similarly low. When the feedback resistors  14  in  FIG. 1   a  are connected, the complex signals shift the lowpass transfer functions by a frequency inversely proportional to the value of feedback resistor. This frequency shift causes the lowpass in  FIG. 2   a  that is symmetrical around zero frequency to transform into the non-symmetrical (for positive or negative frequencies only) complex bandpass as shown in  FIG. 2   b . Again, if no tuning is available the accuracy of the frequency shift is as low as the on-chip RC time constant. 
   A different arrangement for a complex all-pole (no transmission zeros) bandpass active ladder simulation scheme built with gyrators is described in U.S. Pat. No. 6,346,850. As illustrated in  FIG. 3   a  a gyrator identified by numeral  30  consists of two voltage-controlled current sources  31 . The gyrator method uses gyrators and capacitors to realize integrators. The complex bandpass filter is illustrated in  FIG. 3   b  and identified by numeral  32 . Similarly to the previous method, it consists of two banks of integrators  34  cross-connected by gyrators  36 . Since filter integrators are built with gyrators that consist of voltage-controlled current sources their transconductance can be adjusted to compensate for the process variation of on-chip capacitors and resistors. Also, since this is also an active ladder simulation method a single pole frequency and its quality (Q) factor are defined by all filter elements, which results in low sensitivities of these parameters to the component value variation. 
   Yet another arrangement for a complex all-pole (no transmission zeros) bandpass active ladder simulation scheme described in U.S. Pat. No. 6,441,682 is illustrated in  FIG. 4  and identified by numeral  40 . The scheme uses active-RC integrators similar to those described in U.S. Pat. No. 4,914,408 to build lowpass filters  42 , but instead of using fixed resistors to shift the lowpass function the method uses voltage-controlled current sources  44 . The advantage of voltage-controlled current sources over the fixed resistors is that voltage controlled current sources can be adjusted to compensate for unavoidable process variation of on-chip capacitors and resistors. However, since the remaining lowpass filter circuitry consists of active-RC integrators the accuracy of the lowpass cutoff frequency, with respect to the process variation, is similar to that of U.S. Pat. No. 4,914,408. Nevertheless, due to using the adjustable voltage-controlled current sources, the accuracy of the complex bandpass center frequency is expected to be better, similar to that of U.S. Pat. No. 6,346,850. 
   A different arrangement for a complex all-pole (no transmission zeros) bandpass active direct synthesis scheme has been described in a section of  CMOS Wireless Transceivers  by J. Crols and M. Steyaert, Kluwer Academic Publishers, pp. 193–203, 1997. The scheme is illustrated in  FIG. 5   a  and identified by numeral  50 . It is a feedback scheme that forms real pole using “−1” operators  51  and then translates the real pole into a pair of complex poles using cross-coupled operators “±ω c /ω o ”  52 . The proposed realization is illustrated in  FIG. 5   b  and identified by numeral  54 . Similarly to U.S. Pat. No. 4,914,408 it uses active-RC integrators, but instead of building the whole lowpass filters transfer function in a ladder structure, each pair of complex poles is realized directly. By using feedback resistors R Q    55  in cross-coupled configuration the pole positions are shifted by a frequency vector. This results in a complex bandpass transfer function. However, contrary to all previously described methods the individual pole frequency and its Q-factor values are defined by the component values of one particular filter section only. Therefore, the expected sensitivities of the pole parameters to the component value variation are higher than that of the active ladder simulation method. Also, since the pole frequencies are defined by the on-chip RC time constant, without the tuning, the accuracy of the center frequency and the bandwidth of the resulting complex bandpass is expected to be similar to that of U.S. Pat. No. 4,914,408. However, in the presented circuit the pole tunability is achieved by switching on and off binary weighted capacitors  56 . 
   SUMMARY OF THE INVENTION 
   The present invention is used to implement a fully-integrated continuous-time active complex IF bandpass (BP) filter denoted by numeral  67  in a wireless receiver identified by numeral  60  in  FIG. 6 . The receiver consists of the following components: the low-noise amplifier  62 , the real RF BP filter  64 , the complex mixers  66 , the complex IF BP filter  67  and the variable-gain amplifier  68 . The present invention allows for incorporation of transmission zeros into the complex BP filter transfer function. As illustrated in  FIG. 7 , that yields the complex BP filter  71  with much sharper roll-off than that of an all-pole filter  72 . The filter with transmission zeros  71  achieve the same attenuation of 45 dB, as illustrated by numeral  73 , at 1.3 times its cutoff frequency, illustrated by the numeral  74 , whereas the all-pole filter requires 1.5 times of its cutoff frequency, illustrated by the numeral  75 , to achieve the same attenuation. The filter transfer function is constructed using transconductors and capacitors only. Each of the first-Order filter sections realizes one complex pole and a single zero at jω axis. Therefore, two consecutive filter sections realize a pair of complex poles and a double imaginary axis zero. Since the transconductors are electronically tunable, the positions of filter zeros and poles are adjustable using an automatic tuning system. The value of section transconductors and/or capacitors are modified to separately change the pole frequency, its Q-factor and the zero frequency. Each pole and zero are separately tuned, which achieves a much higher level of tuning accuracy than in case where all poles and zeros were adjusted simultaneously. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For a more complete understanding of the present invention and for further advantages thereof, reference is now made to the following Description of the Preferred Embodiments taken in conjunction with the accompanying Drawings in which: 
       FIGS. 1   a ,  1   b ,  2   a ,  2   b ,  3   a ,  3   b ,  4 ,  5   a , and  5   b  are block diagrams of prior art methods to realize active polyphase filters; 
       FIG. 6  is the block diagram of a wireless receiver equipped with the present active complex IF BP filter with transmission zeros; 
       FIG. 7  illustrates the filter attenuation for a complex IF BP filter with transmission zeros and a complex all-pole IF BP filter; 
       FIG. 8  illustrates the circuit arrangement for a first-Order section of the present active complex all-pole BP filter; 
       FIG. 9  illustrates the circuit arrangement for a second-Order section of the present active complex all-pole BP filter; 
       FIG. 10  illustrates the circuit arrangement for a first-Order section of the present active complex BP filter with transmission zeros; 
       FIG. 11  illustrates the circuit arrangement for a second-Order section of the present active complex BP filter with transmission zeros; 
       FIG. 12  illustrates the circuit arrangement for the present active complex BP filter with transmission zeros for an even-Order lowpass prototype as a cascade of second-Order sections; and 
       FIG. 13  illustrates the circuit arrangement for the present active complex BP filter with transmission zeros for an odd-Order lowpass prototype as a cascade of first- and second-Order sections; 
   

   DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Referring to  FIG. 8 , a first-Order lowpass section of the present active complex all-pole bandpass filter is illustrated, and is generally identified by the numeral  80 . The input signals enter the section inputs  81  and  82 . The input signals are in quadrature such that if the phase of the input  81  is 0 degrees the phase of the input  82  is lagging the input  81  by 90 degrees. The input  81  is labeled I and the input  82  is labeled Q. The outputs of the filter section  80  are also in quadrature. The output  83  with phase 0 degrees is labeled I and the output  84  that is lagging output  83  by 90 degrees is labeled Q. 
   The lowpass section  80  realizes one complex pole and consists of six transconductors and two capacitors. The input transconductors g m0  ( 85   a  and  85   b ) set up the section gain. The 1/g m1  resistors of transconductors ( 86   a  and  86   b ) and capacitors C ( 88   a  and  88   b ) form a real pole at the frequency g m1 /C that is shifted by transconductors g mA    87   a  and  87   b  in a cross-coupled configuration by a frequency vector proportional to g mA . Capacitors C ( 88   a  and  88   b ) can be realized as grounded, differential, or a combination of both types. 
   The transfer function of the lowpass sections  80  from the input  81  to the output  83  is the same as that from the input  82  to the output  84  and can be expressed as: 
                     H   1     ⁡     (     j   ⁢           ⁢   ω     )       =         g   m01         g   m1     +     j   ⁢           ⁢     ω   1     ⁢   C     -     j   ⁢           ⁢     g     m   ⁢           ⁢   A             .             (   1   )               
The equation for H 1  gives the insight into how the actual complex bandpass filter transfer function is constructed. The lowpass prototype pole can be either a real or a complex pole. If a real pole is to be converted the initial g mA  is set to 0. Then, the g mA /C is the actual frequency shift, or the center frequency for the complex bandpass. If a complex pole is being converted the initial g mA  is set to the value of the imaginary part of that pole. Then, the same g m -value is added as in the case of the real pole. The conjugate pole can be realized by switching the polarity of the initial g mA -value to yield the transfer function:
 
   
     
       
         
           
             
               
                 
                   
                     H 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       g 
                       m01 
                     
                     
                       
                         g 
                         m1 
                       
                       + 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ωC 
                       
                       - 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                       
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   Referring to  FIG. 9 , a second-Order biquad section of the present active complex all-pole bandpass filter is illustrated, and is generally identified by the numeral  90 . The biquad section  90  consists of two similar lowpass sections labeled  95  and  96 , each of which realize one conjugate complex pole. The transfer function of the biquad section  90  is a cascade of the two lowpass section transfer functions and can be expressed as: 
   
     
       
         
           
             
               
                 
                   
                     
                       H 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                       
                       ) 
                     
                   
                   * 
                   
                     
                       H 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         g 
                         m01 
                       
                       ⁢ 
                       
                         g 
                         m02 
                       
                     
                     
                       
                         
                           ( 
                           
                             
                               g 
                               m1 
                             
                             + 
                             
                               jω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               C 
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           j 
                           2 
                         
                         ⁢ 
                         
                           g 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         g 
                         m01 
                       
                       ⁢ 
                       
                         g 
                         m02 
                       
                     
                     
                       
                         g 
                         m1 
                         2 
                       
                       + 
                       
                         2 
                         ⁢ 
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           m1 
                         
                         ⁢ 
                         C 
                       
                       + 
                       
                         
                           
                             ( 
                             jω 
                             ) 
                           
                           2 
                         
                         ⁢ 
                         
                           C 
                           2 
                         
                       
                       + 
                       
                         g 
                         
                           m 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           A 
                         
                         2 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
     
   
   The transfer functions of the filter section  90  from the input  91  to the output  93  and from the input  92  to the output  94  are identical and can be expressed in s-domain, with s=jω as: 
                   H   ⁡     (   s   )       =             g   m01     ⁢     g   m02         C   2               g   m1   2     +     g     m   ⁢           ⁢   A     2         C   2       +     s   ⁢       2   ⁢     g   m1       C       +     s   2         .             (   4   )               
H(s) represents a pair of conjugate complex poles with their pole frequency expressed as:
 
                   ω   0     =         g   m1     C     ⁢       1   +       (       g     m   ⁢           ⁢   A         g   m1       )     2                   (   5   )               
and their Q-factor expressed as:
 
   
     
       
         
           
             
               
                 Q 
                 = 
                 
                   
                     1 
                     2 
                   
                   ⁢ 
                   
                     
                       
                         1 
                         + 
                         
                           
                             ( 
                             
                               
                                 g 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 g 
                                 m1 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   Referring to  FIG. 10 , a first-Order lowpass section of the present active complex bandpass filter with transmission zeros is illustrated, and is generally identified by the numeral  100 . The input signals enter the filter section inputs  101  and  102 . The input signals are in quadrature such that if the phase of the input  101  is 0 degrees the phase of the input  102  is lagging the input  101  by 90 degrees. The input  101  is labeled I and the input  102  is labeled Q. The outputs of the filter section  100  are also in quadrature. The output  103  with phase 0 degrees is labeled I and the output  104  that is lagging output  103  by 90 degrees is labeled Q. 
   The lowpass section  100  realizes one complex pole and one imaginary axis zero and consists of two unity-gain voltage buffers, six transconductors and four capacitors. The input voltage buffers “ 1 ” ( 105   a  and  105   b ) drive capacitors C 1  ( 106   a  and  106   b ) with their low output impedance. The 1/g m1  resistors made of transconductors  107   a  and  107   b  form with capacitors C 1  ( 106   a  and  106   b ) and C ( 108   a  and  108   b ) a real pole at g m1 /(C 1 +C) that is shifted by transconductors g mA  ( 109   a  and  109   b ) in a cross-coupled configuration by a frequency vector proportional to g mA . Capacitors C ( 108   a  and  108   b ) can be realized as grounded, differential, or a combination of both types. 
   The transfer function of the lowpass sections  100  from the input  101  to the output  103  is the same as that from the input  102  to the output  104  and is expressed as: 
                     H   3     ⁡     (     j   ⁢           ⁢   ω     )       =           j   ⁢           ⁢     ωC   1       +     j   ⁢           ⁢     g   mB             g   m1     +     j   ⁢           ⁢     ω   ⁡     (       C   1     +   C     )         +     j   ⁢           ⁢     g     m   ⁢           ⁢   A             .             (   7   )               
It contains a purely imaginary axis zero at the frequency g mB /C 1  created by feeding forward the input signal across the input capacitors C 1  ( 106   a  and  106   b ). The equation for H 3  gives the insight into how the actual complex bandpass filter transfer function is constructed. The lowpass prototype pole can be either a real or a complex pole. If a real pole is to be converted the initial g mA  is set to 0. Then the g mA /C is the actual frequency shift, or the center frequency for the complex bandpass. If a complex pole is being converted the initial g mA  is set to the value of the imaginary part of that pole. Then the same g m -value is added as in the case of the real pole. The conjugate pole and zero can be realized by switching the polarity of the initial g mA - and g mB -values to yield the transfer function:
 
   
     
       
         
           
             
               
                 
                   
                     H 
                     4 
                   
                   ⁡ 
                   
                     ( 
                     jω 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ωC 
                           1 
                         
                       
                       - 
                       
                         jg 
                         mB 
                       
                     
                     
                       
                         g 
                         m1 
                       
                       + 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ω 
                           ⁡ 
                           
                             ( 
                             
                               
                                 C 
                                 1 
                               
                               + 
                               C 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                       
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
         
       
     
   
   Referring to  FIG. 11 , a second-Order biquad section of the present active complex bandpass filter with transmission zeros is illustrated, and is generally identified by the numeral  110 . The biquad section  110  consists of two similar lowpass sections labeled  115  and  116 , each of which realizes one conjugate complex pole and one imaginary axis zero. The transfer function of the biquad section  110  is a cascade of the two lowpass section transfer functions and can be expressed as: 
   
     
       
         
           
             
               
                 
                   
                     
                       H 
                       3 
                     
                     ⁡ 
                     
                       ( 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                       
                       ) 
                     
                   
                   * 
                   
                     
                       H 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       jω 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         
                           
                             ( 
                             jω 
                             ) 
                           
                           2 
                         
                         ⁢ 
                         
                           C 
                           1 
                           2 
                         
                       
                       + 
                       
                         g 
                         mB 
                         2 
                       
                     
                     
                       
                         
                           ( 
                           
                             
                               g 
                               m1 
                             
                             + 
                             
                               jω 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     C 
                                     1 
                                   
                                   + 
                                   C 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       - 
                       
                         
                           j 
                           2 
                         
                         ⁢ 
                         
                           g 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                               
                               ) 
                             
                             2 
                           
                           ⁢ 
                           
                             C 
                             1 
                             2 
                           
                         
                         + 
                         
                           g 
                           mB 
                           2 
                         
                       
                       
                         
                           g 
                           m1 
                           2 
                         
                         + 
                         
                           2 
                           ⁢ 
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               g 
                               m1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   C 
                                   1 
                                 
                                 + 
                                 C 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                               
                               ) 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   C 
                                   1 
                                 
                                 + 
                                 C 
                               
                               ) 
                             
                             2 
                           
                         
                         + 
                         
                           g 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                           2 
                         
                       
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 9 
                 ) 
               
             
           
         
       
     
   
   The transfer functions of the filter sections  110  from the input  111  to the output  113  and from the input  112  to the output  114  are identical and can be expressed in s-domain, with s=jω as: 
                   H   ⁡     (   s   )       =             s   2     ⁢     C   1   2       +     g   mB   2           g   m1   2     +     2   ⁢       sg   m1     ⁡     (       C   1     +   C     )         +         s   2     ⁡     (       C   1     +   C     )       2     +     g     m   ⁢           ⁢   A     2         =         C   1   2         (       C   1     +   C     )     2       ⁢           g   mB   2       C   1   2       +     s   2               g   m1   2     +     g     m   ⁢           ⁢   A     2           (       C   1     +   C     )     2       +     s   ⁢       2   ⁢     g   m1         (       C   1     +   C     )         +     s   2                     (   10   )               
H(s) represents pair of conjugate complex poles and a double imaginary axis zero with their pole frequency expressed as:
 
                   ω   0     =         g   m1         C   1     +   C       ⁢       1   +       (       g   mA       g   m1       )     2                   (   11   )               
their Q-factor expressed as:
 
                 Q   =       1   2     ⁢       1   +       (       g     m   ⁢           ⁢   A         g   m1       )     2                   (   12   )               
and the zero frequency expressed as:
 
   
     
       
         
           
             
               
                 
                   ω 
                   z 
                 
                 = 
                 
                   
                     g 
                     mB 
                   
                   
                     C 
                     1 
                   
                 
               
             
             
               
                 ( 
                 13 
                 ) 
               
             
           
         
       
     
   
   Referring to  FIG. 12 , the present active complex BP filter with transmission zeros is illustrated, and is generally identified by the numeral  120 . As illustrated in  FIG. 12 , the present complex BP  120  has an even-Order lowpass (LP) prototype. In such a case, each of its biquad sections  125 ,  126  and  127  realizes a pair of complex poles and may add no zeros, a single, a double, or two different imaginary axis zeros to the overall transfer function. 
   Referring to  FIG. 13 , the present active complex BP filter with transmission zeros is illustrated, and is generally identified by the numeral  130 . As illustrated in  FIG. 13 , the present complex BP  130  has an odd-Order LP prototype. In such a case, the lowpass section  135  realizes a real pole and it can implement no zeros, or a single imaginary axis zero. Each of its biquad sections  136  and  137  realizes a pair of complex poles and may add no zeros, a single, a double, or two different imaginary axis zeros to the overall transfer function. 
   Other alteration and modification of the invention will likewise become apparent to those of ordinary skill in the art upon reading the present disclosure, and it is intended that the scope of the invention disclosed herein be limited only by the broadest interpretation of the appended claims to which the inventor is legally entitled.