Abstract:
Direct incorporation of transmission zeros into a continuous-time active complex BP filter transfer function yields a filter having much sharper roll-off than that of an all-pole filter. The ladder filter is constructed using transconductors and capacitors only. The filter center frequency, its bandwidth and positions of transmission zeros can be electronically varied using tunable transconductors. The positions of zeros are changed by modifying cross-coupled differential transconductors connected between capacitors in parallel with the series inductors. Since all transconductors used in the filter are electronically tunable an automatic tuning system conveniently adjusts the filter center-frequency and its Q-factor.

Description:
TECHNICAL FIELD OF THE INVENTION 
   The present invention relates to fully-integrated continuous-time active complex band-pass filters with transmission zeros and their synthesis method using transconductance amplifiers and capacitors that allows conversion of a passive ladder prototype of a real lowpass with transmission zeros into an active complex bandpass filter with transmission zeros. 
   BACKGROUND OF THE INVENTION 
   Continuous-time active complex bandpass (BP) filters are widely used in the integrated receivers. They serve primarily as intermediate-frequency (IF) channel select filters with 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, it 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 a complex filter. They include resistors, capacitors and transconductors. Also, non-ideal gain of 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 the 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 a special automatic compensation scheme is applied the IRR can reach 55–60 dB. 
   Active complex bandpass 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 each pole frequency and its quality (Q) factor are defined by all filter components. Contrary, for the direct synthesis method the pole frequency and its Q factor are defined only by the components 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 and it is the preferred method for realization of complex active bandpass filters. 
   Active complex bandpass filters can also be categorized according to the active synthesis method chosen for their implementation. Two different synthesis techniques have been used so far: the active-RC technique used in U.S. Pat. No. 4,914,408 and the gyrator method used in U.S. Pat. No. 6,346,850. It should be noted that both techniques were widely known prior to their application in the above-mentioned Patents. In active-RC method the transfer function is realized using active-RC integrators built with input series resistors and feedback capacitors around operational amplifiers (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 its ability to adjust the filter pole frequencies by adjusting the transconductance with a control voltage or current, which is not easily achievable in a fixed 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 their 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. 16 , their roll-off is never as steep as that of filters that contain transmission zeros. For all these reasons, in an integrated receiver design there is a strong need for active complex BP filters with transmission zeros. Ladders offer a wide range of transfer functions: both all-pole prototypes and filters with transmission zeros. 
   DESCRIPTION OF THE PRIOR ART 
   The arrangement for a complex all-pole (no transmission zeros) bandpass (BP) active ladder scheme using active-RC integrators such as one described in U.S. Pat. No. 4,914,408 is illustrated in  FIG. 1 , 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 (LP) function. 
   The LP filter integrator banks— 12  are connected in so called a leapfrog structure as shown in  FIG. 2  and identified by the numeral  20 , which is a common technique to simulate lossless ladders using active integrators. Two types of integrators are used. Lossy integrators— 22  simulate the ladder first element—the source resistor, and the ladder last element—the load resistor. Lossless integrators— 24  simulate the reminding ladder elements. Since presented ladder is of third-Order, only one lossless integrator— 24  is necessary. Similarly 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 component value variation. However, since on-chip RC time-constants can vary as much as 30–40% the accuracy of the cut-off frequency of a LP built with active-RC integrators is similarly low if no tuning is applied. 
   When the feedback resistors— 14  in  FIG. 1  are connected, the complex signals shift the LP transfer functions by a frequency ω c  inversely proportional to the value of feedback resistors. This frequency shift by a vector jω c  causes the LP in  FIG. 3  that is symmetrical around zero frequency to transform into the non-symmetrical (transmits for positive frequencies only) complex BP as shown in  FIG. 4 . Again, if no tuning is applied the accuracy of the frequency shift is as low as that of the on-chip RC time constant. 
   A different arrangement for a complex all-pole (no transmission zeros) BP active ladder simulation scheme built with gyrators is described in U.S. Pat. No. 6,346,850. As illustrated in  FIG. 5  a gyrator identified by a numeral  50  consists of two voltage-controlled current sources— 52 . The gyrator method uses gyrators and capacitors to realize integrators. The complex BP filter is illustrated in  FIG. 6  and identified by numeral  60 . Similarly to the previous method, it consists of two banks of integrators— 62  cross-connected by gyrators— 64 . 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 component value variation. 
   Yet another arrangement for a complex all-pole (no transmission zeros) BP active ladder simulation scheme described in U.S. Pat. No. 6,441,682 is illustrated in  FIG. 7  and identified by numeral  70 . The scheme uses active-RC integrators similar to those described in U.S. Pat. No. 4,914,408 to build LP filters— 72 , but instead of using fixed resistors to shift the LP function the method uses voltage-controlled current sources— 74 . 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 LP filter circuitry consists of active-RC integrators the accuracy of the LP 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 BP center frequency is expected to be better, similar to that of U.S. Pat. No. 6,346,850. 
   Different arrangement for a complex all-pole (no transmission zeros) BP filter using the active direct synthesis scheme has been described in a paper by J. Crols and M. Steyaert, “An Analog Integrated Polyphase Filter for High Performance Low-IF Receiver,”  Proc. IEEE VLSI Circuit Symposium , Kyoto, pp. 87–88, 1995 illustrated in  FIG. 8  and identified by numeral  80 . It is a negative feedback scheme that forms real pole using “−1” operators— 82  and then translates the real pole into a pair of complex poles using cross-coupled operators “±ω c /ω o ”— 84 . The proposed realization is illustrated in  FIG. 9  and identified by numeral  90 . Similarly to U.S. Pat. No. 4,914,408 it uses active-RC integrators, but instead of building the whole LP filters transfer function in a ladder structure, each pair of complex poles is realized directly. By using feedback resistors R Q — 92  in cross-coupled configuration the pole positions are shifted by a frequency vector. This results in a complex BP 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 BP is expected to be similarly low as that of U.S. Pat. No. 4,914,408. However, in the presented practical circuit switching on and off binary weighted capacitors— 94 , provides with the pole tunability. 
   Practical realizations for active complex all-pole (no transmission zeros) BP filters using both the direct synthesis and active ladder simulation methods have been presented in numerous conference and journal papers. Most of the presented complex BP filters use g m —C technique, some a combination of g m —C and passive RC elements, and some active-RC integrators. Since most of the presented applications for complex active BP filters do not require very steep transfer functions only few attempts to implement transmission zeros have been made up to date. 
   A practical realization of an active complex all-pole (no transmission zeros) ladder BP filter and the method of converting an all-pole LP ladder prototype into an active all-pole complex BP filter have been presented in a paper by P. Andreani, S. Mattisson, B. Essink; A CMOS g m —C Polyphase Filter with High Image Band Rejection,  Proc. IEEE  26 th European Solid - State Circuits Conf , pp. 244–247, September 2000. In particular, the conversion of a passive LP shunt-C series-L shunt-C all-pole third-Order ladder prototype illustrated in  FIG. 10  and identified by numeral  100  into an active all-pole third-Order LP ladder structure illustrated in  FIG. 11  and identified by numeral  110 , and then into an active all-pole sixth-Order complex BP filter illustrated in  FIG. 12  and identified by numeral  120  has been demonstrated. For illustration purpose L 2  realizations  FIGS. 11 and 12  are encircled with a dotted line. 
   As illustrated in  FIG. 13  and identified by numeral  130 , in order to obtain a steep transfer function required by some system specifications transmission zeros are introduced by cascading an all-pole LP prototype— 132  with a notch prototype— 134 , and then transforming the two into an active all-pole complex BP and active complex notch pair. This approach results in a sub-optimal solution requiring more circuitry, yielding overall less robust design and worse quality of the implemented transmission zeros that suffer from low Q&#39;s (losses) and parasitic capacitance— 136  distorting the desired transfer function. 
   As illustrated in  FIG. 14  and identified by numeral  140 , a simpler and more robust approach would be to incorporate a transmission zero directly into the original LP ladder transfer function in a form of a series capacitor C 2  in parallel with the inductor L 2  and then to convert it into an active complex BP filter. Note that using this technique for a fifth-Order LP prototype by adding a capacitor C 4 — 142  in parallel with L 4  a second zero can be created, which further improves the steepness of transfer function. Introducing zeros directly into the original LP ladder results also in much better quality of implemented zeros and the elimination of additional loss and parasitic capacitance distorting the desired transfer function. The method of converting a LP ladder prototype with transmission zeros into an active g m -C complex BP filter with transmission zeros will be presented in the present disclosure. 
   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  156  in a low-IF wireless receiver such as one identified by numeral  150  in  FIG. 15 . Contrary, to other methods the present invention allows for direct incorporation of transmission zeros into a complex BP filter transfer function. It is performed by converting a passive real lowpass (LP) ladder prototype with transmission zeros into an active g m -C complex BP filter with transmission zeros. As illustrated in  FIG. 16  for a LP prototype, that yields a filter with transmission zeros— 161  having much sharper roll-off than that of an all-pole filter— 162 . The present filter is constructed using transconductors and capacitors only. The filter center frequency, its bandwidth and positions of transmission zeros can be individually tuned using tunable transconductors. Modifying the additional transconductors to those of an active LP prototype changes its center frequency. To change the filter bandwidth only the transconductors of an active LP prototype need to be adjusted. The positions of zeros can be changed separately by modifying the cross-coupled differential transconductors between capacitors in parallel with the series inductors. Since all transconductors used in the present filter are electronically tunable an automatic tuning system can conveniently adjust the filter center-frequency and its Q-factor for process variation. 

   
     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: 
       FIG. 1  is a block diagram of a prior art all-pole complex bandpass active ladder; 
       FIG. 2  is a schematic diagram of a prior art all-pole complex bandpass active ladder using active-RC integrators in leapfrog configuration; 
       FIG. 3  is an illustration of a prior art real lowpass transfer function; 
       FIG. 4  is an illustration of a prior art real lowpass transfer function shifted by a vector jω c  that becomes a complex bandpass transfer function; 
       FIG. 5  is a block diagram of a prior art gyrator; 
       FIG. 6  is a block diagram of a prior art active all-pole complex bandpass ladder filter built with gyrators and capacitors; 
       FIG. 7  is a block diagram of a prior art active all-pole complex bandpass ladder filter using active-RC integrators and voltage-controlled current sources; 
       FIG. 8  is a block diagram of a prior art direct complex pole synthesis method; 
       FIG. 9  is a circuit diagram of a prior art the direct complex pole synthesis method using active-RC integrators; 
       FIG. 10  is a schematic diagram for a third-Order passive LC ladder all-pole lowpass prototype of a prior art ladder synthesis method; 
       FIG. 11  is a schematic diagram for a third-Order active g m -C ladder all-pole lowpass prototype of a prior art ladder synthesis method; 
       FIG. 12  is a schematic diagram for a sixth-Order active g m -C ladder all-pole complex bandpass filter of a prior art ladder synthesis method; 
       FIG. 13  is a schematic diagram for a fifth-Order passive LC ladder all-pole lowpass prototype combined with a separate notch of a prior art ladder synthesis method; 
       FIG. 14  is a schematic diagram illustrating a method of improving the prior art method by allowing a transmission zero in the ladder lowpass prototype; 
       FIG. 15  is the block diagram of a wireless low-IF receiver equipped with the present active complex IF BP filter with transmission zeros; 
       FIG. 16  illustrates the comparison between the attenuation of two LP prototype filters: one with transmission zeros and other—all-pole prototype; 
       FIG. 17  is a schematic diagram of a passive third-Order lowpass LC ladder prototype with a transmission zero; 
       FIG. 18  is a schematic diagram of an active third-Order lowpass g m -C ladder with a transmission zero; 
       FIG. 19  is a schematic diagram of an active sixth-Order complex bandpass g m -C ladder with two transmission zeros; 
       FIG. 20  is a schematic diagram of a grounded C with a single-ended shifting transconductor illustrating the idea of complex shifting of a grounded C; 
       FIG. 21  is a schematic diagram of a floating C with a differential shifting transconductor illustrating the idea of complex shifting of a floating C; 
       FIG. 22  is a schematic diagram of differential shifting transconductors for transmission-zero-generating differential series C&#39;s; 
       FIG. 23  is an illustration of a real lowpass transfer function with a transmission zero; 
       FIG. 24  is an illustration of a lowpass transfer function with a transmission zero shifted by a vector jω c  that becomes a complex bandpass transfer function with two transmission zeros; 
       FIG. 25  is a schematic diagram of an active third-Order lowpass g m -C ladder with a transmission zero built with a C-LC section; 
       FIG. 26  is a schematic diagram of an active (2N+1) th-Order lowpass g m -C ladder with up to N transmission zeros built with N separate C-LC, or C-L sections and a shunt C section; 
       FIG. 27  is a schematic diagram of an active sixth-Order complex bandpass g m -C ladder with two transmission zeros built with a double C-LC section including shifting transconductors; and 
       FIG. 28  is a schematic diagram of an active 2*(2N+1) th-Order complex bandpass g m -C ladder with up to 2N transmission zeros built with N separate double C-LC, or C-L sections including shifting transconductors and a double shunt C section with shift. 
   

   DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   The present invention is used to implement a fully-integrated continuous-time active complex IF bandpass (BP) filter denoted by numeral  156  in a low-IF wireless receiver identified by numeral  150  in  FIG. 15 . It consists of a low-noise amplifier (LNA)— 151 , a tunable RF BP filter— 152 , a RF polyphase filter— 153 , a double-complex mixer— 154 , a pair of summers— 155 , a complex IF BP filter— 156 , and a pair of variable gain amplifiers (VGA&#39;s)— 157 . The main function of the complex IF BP filter is IF channel selection and providing the receiver with a substantial image rejection—the task performed jointly with the double-complex mixer and the pair of summers. 
   Contrary to other methods the present invention allows to directly incorporate transmission zeros into a complex BP filter transfer function. As illustrated in  FIG. 16  for a lowpass (LP) prototype, that yields the filter— 161  with much sharper roll-off than that of an all-pole filter— 162 . The filter with transmission zeros— 161  achieve the attenuation of 45 dB, illustrated by numeral  163 , at 1.3 times of its cutoff frequency, illustrated by the numeral  164 , whereas the all-pole filter requires 1.5 times of its cutoff frequency, illustrated by the numeral  165 , to achieve the same attenuation. It should be noted that although the both filters are of seventh-Order they use different approximations. The all-pole filter— 162  has much larger passband ripple indicating that its pole Q&#39;s are higher that that of the filter— 161  with transmission zeros. Since high Q&#39;s should be avoided for practical filter implementation—the filter  161  is preferred over the filter— 162 . 
   Referring to the  FIG. 17 , a third-Order LP LC-ladder prototype with a transmission zero of the present active complex BP filter with two transmission zeros is illustrated, and is generally identified by the numeral  170 . Since it is a lossless LC ladder all its internal components with exception to terminations R S  and R L  are reactances. The transmission zero is generated by the resonance between L 2  and C 2 . Since in an active implementation all inductors are realized with g m -C gyrators, a selected LP prototype uses the fewest number of inductors. 
   Assuming the termination resistors R S =R L =R 1  and C 3 =C 1  the transfer function of the LP prototype  170  can be expressed as 
   
     
       
         
           
             
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   Referring to the  FIG. 18 , an active realization of third-Order LP LC-ladder prototype with a transmission zero is illustrated, and is generally identified by the numeral  180 . The input transconductor— 182  serves as a voltage-to-current converter. The termination resistors R S  and R L  are realized as 1/g m  resistors. The floating inductor L 2 — 184  is realized with two pairs of gyrators and a grounded capacitor C L . For illustration purpose L 2  realization is encircled with a dotted line. 
   With R S =1/g m1 , R L =1/g m3 , C 3 =C 1 , L 2 =C L /g m2   2 , and g m0  of the V-I converter the transfer function of the LP prototype— 180  can be expressed as 
               H   2     ⁡     (   s   )       =     A   ⁢       (       s   2     +       g   m2   2         C   L     ⁢     C   2           )         (     s   +       g   m1       C   1         )     ⁢     (       s   2     +     s   ⁢       g   m1       2   ⁢     (       C   1     +     2   ⁢     C   2         )           +       2   ⁢     g   m2   2           C   L     ⁡     (       C   1     +     2   ⁢     C   2         )           )                 
where
 
           A   =         2   ⁢     g   m0           g   m1     +     g   m3         ⁢         g   m1     ⁢     C   2           C   1     ⁡     (       C   1     +     2   ⁢     C   2         )                 
the center frequency ω o  and quality factor Q of complex poles are given by
 
                   ω   o     =       g   m2     ⁢       2       C   L     ⁡     (       C   1     +     2   ⁢     C   2         )                         Q   =         2   ⁢     g   m2         g   m1       ⁢         2   ⁢     (       C   1     +     2   ⁢     C   2         )         C   L                       
and the zero frequency ω z  is given by
 
             ω   z     =       g   m2           C   L     ⁢     C   2                 
With g m0 =2g m  and g m1 =g m2 =g m3 =g m  the transfer function becomes
 
               H   3     ⁡     (   s   )       =           2   ⁢     g   m     ⁢     C   2           C   1     ⁡     (       C   1     +     2   ⁢     C   2         )         ⁢     (       s   2     +       g   m   2         C   L     ⁢     C   2           )           (     s   +       g   m       C   1         )     ⁢     (       s   2     +     s   ⁢       g   m       2   ⁢     (       C   1     +     2   ⁢     C   2         )           +       2   ⁢     g   m   2           C   L     ⁡     (       C   1     +     2   ⁢     C   2         )           )               
Note that with s→0 H 3 (s)→1, which is expected for a LP transfer function.
 
   Referring to the  FIG. 19 , an active realization of a sixth-Order complex BP with two transmission zeros is illustrated, and is generally identified by the numeral  190 . It consists of two identical LP I and Q filters identified by the numerals  192  and  194  respectively and fed with shifted by 90 degrees I and Q inputs. In complex notation if I input is V i  then Q input becomes jV i . Two types of shifting transconductors are used in the filter— 190 . For each of the three pairs of grounded capacitors C 1  and C L  of the two filters— 192  and  194  a pair of single-ended transconductors— 196  with opposite polarities −g ma  and g ma  is applied and for the pair of zero generating floating capacitors C 2  a pair of differential transconductors— 198  with opposite polarities −g mb  and g mb  is used. 
   As illustrated in  FIG. 20  the effect of a transconductor— 196  in  FIG. 19  on a grounded capacitor C 1  is shifting its frequency response by the vector jω c =jg ma /C 1 . The explanation of this property is as follows: without the transconductor g ma  the current through C 1  is I=jωC 1 V. By adding the current of g ma  the total current through C 1  becomes I*=j(ωC 1 −g ma )V. Hence, the admittance of C 1  without g ma  is Y=jωC 1 , and with g ma  is Y*=j(ωC 1 −g ma )=jC 1 (ω−g ma /C 1 )=jC 1 (ω−ω c ), where ω c =g ma /C 1 . If the filter contained several different values of grounded capacitors the g ma &#39;s should be adjusted for each capacitor such that ω c  was always constant. It should be noted that although the shifting transconductors— 196  in  FIG. 19  are presented as single-ended if I and Q signals are differential the whole filter structure becomes differential meaning that the transconductors— 196  need to become also differential. 
   As illustrated in  FIG. 21  the effect of transconductors— 198  in  FIG. 19  on floating capacitor C 2  is shifting its frequency response by the vector jω c =jg mb /C 2 . Again, the explanation of this property is as follows: without the transconductor g mb  the current through C 2  is I=jωC 2 (V 1 −V 2 ). By adding the current of g mb  the total current through C 2  becomes I*=j(ωC 2 −g mb )(V 1 −V 2 ). Hence, the admittance of C 2  without g mb  is Y=jωC 2 , and with g mb  is Y=j(ωC 2 −g mb )=jC 2 (ω−g mb /C 2 )=jC 2 (ω−ω c ), where ω c =g mb /C 2 . If the filter contained several different values of floating capacitors the g mb &#39;s should be adjusted for each capacitor such that ω c  was always constant. It should be noted that although the shifting transconductors— 198  in  FIG. 19  are presented as differential if I and Q signals are also differential the whole filter structure becomes differential meaning that the transconductors— 198  need to become double-differential at the input and output. 
   As illustrated in  FIG. 22 , and generally identified by the numeral  220 , the simplest way to realize the transconductors— 198  in  FIG. 19  is to use two separate differential transconductors −g mb — 222  and g mb — 224  for each pair of differential voltages (V 1 −V 2 ) +  and (V 1 −V 2 ) −  between I-filter— 226  and Q-filter— 228 . These double differential transconductors— 222  and  224  are one of the main features of the present invention that allows building practical active tunable complex bandpass filters with transmission zeros. 
   Referring to the  FIG. 23 , the transfer function of a real LP prototype with transmission zeros in  FIG. 18  is illustrated. The two transmission zeros are symmetrical at ±jω z . As illustrated in  FIG. 24  after applying the shifting as explained in  FIGS. 20 and 21  the transfer function of the present complex BP filter in  FIG. 19  is moved by vector jω c . The two transmission zeros are now positioned at j(ω c ±jω z ). 
   Referring to the  FIG. 25 , an active realization of a third-Order LP prototype filter with a transmission zero is illustrated, and is generally identified by the numeral  250 . It contains a shunt C—series LC section identified by the numeral  252  and marked with a dotted line. By repeating this section a higher order LP filters with transmission zeros can be implemented. Since the filter is of third-Order its last section—C 1  is formed by a shunt C— 254 . 
   Referring to the  FIG. 26 , an active realization of a (2N+1) th-Order LP prototype filter with up to N transmission zeros is illustrated, and is generally identified by the numeral  260 . Without of loosing generality, it may contain N shunt C—series LC, or shunt C—series L sections identified by the numerals  262 . The shunt C—series L sections are simply sections without transmission zero. Since the filter— 260  is of an odd-Order its last section—C 1  is formed by a shunt C— 264 . 
   Referring to the  FIG. 27 , an active realization of a sixth-Order complex BP filter with two transmission zeros is illustrated, and is generally identified by the numeral  270 . It consists of two third-Order LP filters— 250  of  FIG. 25  with I, Q inputs and outputs and shifting circuitry connected between them. The filter— 270  contains two shunt C—series LC sections identified by the numerals  272 , a grounded C shifting circuit— 274  consisting of two pairs of single-ended transconductors: first to shift the actual shunt C and second to shift the grounded C of the active series inductor, and a series C shifting circuit— 276  consisting of a pair of differential transconductors. Sections  272 ,  274  and  276  form together a double C-LC section with shift and are marked with dotted lines. By repeating this section a higher order BP filters with transmission zeros can be implemented. Since the LP prototype— 250  in  FIG. 25  is of a third-Order the filter last section—a double shunt C-section with shift is formed by a pair of shunt C&#39;s— 277  together with a pair of single-ended shifting transconductors— 278 . 
   Referring to the  FIG. 28 , an active realization of a 2*(2N+1) th-Order BP filter with up to 2N transmission zeros is illustrated, and is generally identified by the numeral  280 . Without of loosing generality, it may contain N double shunt C—series LC, or shunt C—series L sections all equipped with shift circuits and identified by the numeral  282 . The double shunt C—series L sections are simply sections without transmission zeros. Each time the series C is omitted the associated shifting circuit— 276  in  FIG. 27  is also dropped. The last filter section is a double shunt C-section with shift identified by the numeral  284 . 
   Other features that are considered as characteristic for the invention are set forth in the appended claims. 
   Although the invention is illustrated and described herein as embodied in a circuit configuration for fully-integrated continuous-time active complex band-pass filters with transmission zeros built with transconductance amplifiers and capacitors, it is, nevertheless, not intended to be limited to the details shown because various modifications and structural changes may be made therein without departing from the spirit of the invention and within the scope and range of equivalents of the claims.