Abstract:
An image rejection quadratic filter is tunable to filter image frequencies over a wide band. In one embodiment, the image rejection filter includes in-phase and quadrature phase mixers. The image rejection has a fractional transfer function. The image rejection filter has two sub-circuits, wherein the first sub-circuit produces the imaginary component of the transfer function, and the second sub-circuit produces the real component of the transfer function. The first sub-circuit receives the Q signal, and the second sub-circuit receives the I signal. In one embodiment, to create the fractional transfer function, a multi-feedback looped integrator is used.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
   This application claims the benefit of U.S. Provisional Patent Application No. 60/384,283, filed May 29, 2002, entitled “Image Rejection Quadratic Filter.” 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention is directed toward the field of filters, and more particularly toward image rejection notch filters. 
   2. Art Background 
   Typically, receivers employ filters to condition both input signals and internally generated reference signals. For example, bandpass, notch, and low pass are types of filters employed in receivers. The frequency response of a filter refers to the characteristics of the filter that condition the signal input to the filter. For example, a bandpass filter may attenuate an input signal across a pre-determined band of frequencies above and below a center frequency of the filter. Filters are designed to exhibit frequency responses based on one or more circuit parameters. 
   Some receivers are designed to process input signals with a range of input carrier frequencies (e.g., wide band receivers). For example, television receivers must be capable of processing input television signals with carrier frequencies ranging from 55 MHz to 880 MHz. One circuit parameter used to define the frequency response of a filter is the carrier frequency of an input signal. 
     FIG. 1  illustrates one embodiment for an image rejection mixer, including a resistive-capacitive (“RC”) filter output. As shown in  FIG. 1 , a signal is input (e.g., RF input) to the in-phase (“I”) mixer  110  and the quadrature phase (“Q”) mixer  120 . Also input to the (“I”) mixer  110  and quadrature phase (“Q”) mixer  120 , at the local oscillator (“LO”) port, is a local oscillator signal. 
   As shown in  FIG. 1 , the I and Q signals are input to the resistor  140  and capacitor  130 , respectively. The transfer function of the RC filter may be expressed as: 
   
     
       
         
           A 
           = 
           
             
               
                 1 
                 + 
                 
                   
                     ( 
                     
                       - 
                       j 
                     
                     ) 
                   
                   × 
                   S 
                 
               
               
                 1 
                 + 
                 S 
               
             
             = 
             
               
                 ( 
                 
                   1 
                   + 
                   Z 
                 
                 ) 
               
               
                 ( 
                 
                   1 
                   + 
                   S 
                 
                 ) 
               
             
           
         
       
     
     
       
         
           where 
           , 
           
             
 
           
           ⁢ 
           
             S 
             = 
             
               j 
               ⁢ 
               
                   
               
               ⁢ 
               wCr 
             
           
         
       
     
     
       
         
           S 
           = 
           
             
               j 
               × 
               Z 
             
             = 
             
               j 
               × 
               
                 W 
                 / 
                 Wo 
               
             
           
         
       
     
   
     FIG. 2  illustrates a frequency response for the prior art image rejection mixer of  FIG. 1 . The frequency response of  FIG. 2  is normalized. The image frequency, at −1, is attenuated and the desired frequency range is part of the passband of the filter. The transfer function, for the normalized response, may be expressed as: 
   
     
       
         
           
             ( 
             
               1 
               + 
               X 
             
             ) 
           
           
             
               1 
               + 
               
                 X 
                 2 
               
             
           
         
       
     
   
   Conventionally an image rejection mixer applies a single notch in the baseband after demodulation. When the desired signal has a relatively wide bandwidth, the image rejecting single notch can remove only one image frequency point. In other words, the single notch cannot remove a desired band. For this reason, the conventional image rejection mixer requires multiple stages of down conversion. 
   SUMMARY OF THE INVENTION 
   An image rejection quadratic filter is tunable to filter image frequencies over a wide band. In one embodiment, the image rejection filter includes in-phase and quadrature phase mixers. The image rejection has a fractional transfer function. The transfer function of the image rejection filter has the following characteristics:
         a non-recursive numerator and a recursive denominator;   the numerator and the denominator are polynomial equations;   the numerator includes only real terms and the denominator includes only imaginary terms, or the numerator includes only imaginary terms and the denominator includes only real terms;   the numerator and the denominator have the same sign, so that the numerator can be factorized into factors having more zeros;   the transfer function is a second order polynomial or higher; and   the coefficients of the polynomial of the numerator are chosen so that the numerator is factorizable at a real number.       

   The image rejection filter has two sub-circuits, wherein the first sub-circuit produces the imaginary component of the transfer function, and the second sub-circuit produces the real component of the transfer function. The first sub-circuit receives the Q signal, and the second sub-circuit receives the I signal. In one embodiment, to create the fractional transfer function, a multi-feedback looped integrator is used. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  illustrates one embodiment for an image rejection mixer, including a resistive-capacitive (“RC”) filter output. 
       FIG. 2  illustrates a frequency response for the prior art image rejection mixer of  FIG. 1 . 
       FIG. 3  is a schematic diagram illustrating one embodiment for a second order image rejection mixer. 
       FIG. 4  illustrates one embodiment for a second order image rejection filter that has a split pole. 
       FIG. 5  illustrates another embodiment for a second order image rejection filter that has the same two poles. 
       FIG. 6  is a schematic diagram illustrating one embodiment for a third order image rejection mixer. 
       FIG. 7  illustrates one embodiment for a frequency response of a third order image rejection filter. 
       FIG. 8  is a schematic diagram illustrating one embodiment for a fourth order image rejection mixer. 
       FIG. 9  illustrates one embodiment for a frequency response of a fourth order image rejection filter. 
       FIG. 10  is a schematic diagram illustrating another embodiment for a fourth order image rejection mixer. 
       FIG. 11  illustrates one embodiment for a frequency response of the fourth order image rejection filter of  FIG. 10 . 
   

   DETAILED DESCRIPTION 
   The disclosure of U.S. Provisional Patent Application 60/384,283, filed May 29, 2002, entitled “Image Rejection Quadratic Filter” is hereby expressly incorporated herein by reference. 
   An “image signal” is a product of a mixer. The image signal results from mixing an RF signal with a local oscillator signal. For example, an RF input signal with a fundamental frequency of 880 MHz is mixed with a local oscillator having a frequency of 660 MHz to produce a first harmonic at 220 MHz (RF (880 Mhz)−LO (660 Mhz)=220 Mhz). In turn, this first harmonic, centered around 220 MHz, mixes with the local oscillator frequency of 660 MHz to produce the image at 440 MHz. The image frequencies require suppression for proper operation of the circuit. 
   In one embodiment, the image signal is suppressed using an image rejection quadratic filter. The image rejection quadratic filter is tuned based on the input channel of the RF signal. In one embodiment, the image rejection quadratic filter is tuned to filter the RF signal among a range of frequencies between 110 Mhz and 440 Mhz (i.e., the band of frequencies for the image frequency). The image rejection quadratic filter attenuates the RF signal at the image frequency. One embodiment for tuning a filter, including an RC filter, is described in U.S. patent application Ser. No. 10/448,605 entitled “Methods and Apparatus for Tuning Using Successive Approximation”, inventor Lance M. Wong, filed currently herewith, and is expressly incorporated herein by reference. 
     FIG. 3  is a schematic diagram illustrating one embodiment for a second order image rejection mixer. As shown in  FIG. 3 , a signal is input (e.g., RF input) to the in-phase (“I”) mixer  320  and the quadrature phase (“Q”) mixer  310 . Also input to the (“I”) mixer  320  and quadrature phase (“Q”) mixer  310 , at the local oscillator (“LO”) port, is a local oscillator signal. The local oscillator signal, generated from a emitter, such as a voltage-controlled oscillator, is phase shifted to generate the in-phase and the quadrature phase components of the local oscillator signal. The in-phase and the quadrature phase LO components are input to the in-phase (“I”) mixer  320  and the quadrature phase (“Q”) mixer  310 , respectively. The in-phase (“I”) mixer  320  and the quadrature phase (“Q”) mixer  310  generate mixed I and Q signals. 
   For this embodiment, the second order image rejection mixer comprises a plurality of transconductance amplifiers. The transconductance amplifiers consist of transistors (e.g., bipolar transistors) capacitors and resistors. As shown in  FIG. 3 , the I mixed signal, output from I mixer  320 , is input to the base of transistor  380 . The emitter of transistor  380  is coupled to resistor  376  of transistor  380 . A first transconductance amplifier also includes a second transistor,  390 . The emitter of transistor  390  is coupled to resistor  372 , and the collector is coupled to capacitor  340 . Capacitor  340  has a capacitance of “C1”, and capacitor  340  is coupled to receive the output of Q mixer  310 . 
   The capacitor  340  also couples the output of Q mixer  310  to the input to a second transconductance amplifier (i.e., input to the base of transistor  370 ). The second transconductance amplifier generates the imaginary component of the transfer function. The second transconductance amplifier contains transistors  370  and  365 , and resistors  368  and  374 . The output of the I mixer  320  is inverted in inventor  330  for subsequent input to capacitor  350 . Capacitor  350  has a value set to “C2.” Transistor  360  receives, at its base, the output of capacitor  350 . The circuit  300  is biased with current sources as shown in  FIG. 3 . The output of the image rejection mixer  300  is labeled as “A” on  FIG. 3 . 
   The second order image rejection filter  300  of  FIG. 3  has a transfer function as follows: 
                 A   =       1   +     j   ×   S   ⁢           ⁢   1   ×   S   ⁢           ⁢   2         1   +     S   ⁢           ⁢   1     +     S   ⁢           ⁢   1   ×   S   ⁢           ⁢   2                     =         (     1   +   Za     )     ×     (     1   +   Zb     )           (     1   +   Sa     )     ×     (     1   +   Sb     )                           where   ,     
     ⁢       S   ⁢           ⁢   1     =     j   ⁢           ⁢   w   ⁢           ⁢   C   ⁢           ⁢   1   ⁢   R                     S   ⁢           ⁢   2     =     j   ⁢           ⁢   w   ⁢           ⁢   C   ⁢           ⁢   2   ⁢   R           
The numerator of the transfer function is real number, whereas the denominator of the transfer function is a complex number. Also:
 1 +S 1 +S 1 ×S 2=(1 +Sa )×(1 +Sb )   Sa=j×Za=j×W/Wa      Sb=j×Zb=j×W/Wb    
   The capacitor value C1 corresponds to capacitor  340  ( FIG. 3 ), and capacitor value C2 corresponds to capacitor  340 . The resistor value, R, is the value for resistors  376 ,  372 ,  374 , and  368 . 
     FIG. 4  illustrates one embodiment for a second order image rejection filter that has a split pole. A normalized transfer function for this embodiment is as follows: 
   
     
       
         
           
             
               ( 
               
                 1 
                 + 
                 X 
               
               ) 
             
             ⁢ 
             
               ( 
               
                 1 
                 + 
                 
                   0.25 
                   ⁢ 
                   X 
                 
               
               ) 
             
           
           
             
               1 
               + 
               
                 
                   X 
                   2 
                 
                 ⁢ 
                 
                   
                     1 
                     + 
                     
                       
                         ( 
                         
                           0.25 
                           ⁢ 
                           X 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
   
   For this embodiment, the capacitor value, C1 , which corresponds to capacitor  340  ( FIG. 3 ), has a value of 1.25 Co, and capacitor value C2 , which corresponds to capacitor  340 , has a value of 0.2 Co. The value, Co, corresponds to the value of “C” to normalize the image at “−1.” 
     FIG. 5  illustrates another embodiment for a second order image rejection filter that has the same two poles. As shown in  FIG. 5 , a desired band of frequencies (e.g., television channel) is centered around “+1”, and the image from the I,Q mixers is centered around “−1.” For this embodiment, the normalized transfer function for this embodiment is as follows: 
   
     
       
         
           
             
               ( 
               
                 1 
                 + 
                 X 
               
               ) 
             
             2 
           
           
             1 
             + 
             
               X 
               2 
             
           
         
       
     
   
   For this embodiment, the capacitor value, C1 , has a value of 2 Co, and capacitor value, C2 , has a value of 0.5 Co. 
     FIG. 6  is a schematic diagram illustrating one embodiment for a third order image rejection mixer. As shown in  FIG. 6 , a signal is input to an in-phase (“I”) mixer  620  and quadrature phase (“Q”) mixer  610 . A local oscillator signal is also input to the (“I”) mixer  620  and quadrature phase (“Q”) mixer  610 , at the local oscillator (“LO”) ports. The local oscillator signal is phase shifted to generate the in-phase and the quadrature phase components (e.g., x 1  and xj) of the local oscillator signal. The I mixer  620  and Q mixer  610  generate mixed I and Q signals. 
   For this embodiment, the third order image rejection mixer comprises multiple cascaded transconductance amplifiers. As shown in  FIG. 6 , the I mixed signal, output from I mixer  620 , is input to a first transconductance amplifier. Specifically, the I mixed signal is input to the base of transistor  654 , and the emitter of transistor  654  is coupled to resistor  670 . The emitter of transistor  660  is coupled to resistor  680 , and the collector is coupled to capacitor  652 . The capacitance of capacitor  652  has value of “3 Co.” Capacitor  652  is coupled to receive the output of Q mixer  610 . 
   The capacitor  652  also couples the output of Q mixer  610  to the input of a second transconductance amplifier (i.e., input to the base of transistor  650 ). The second transconductance amplifier provides an imaginary component to the transfer function. The second transconductance amplifier consists of transistors  650  and  644 , resistors  646  and  648 , and capacitor  651 . The output of I mixer  620  is inverted in inventor  640  for subsequent input to capacitor  651 . Capacitor  651  has a value set to “Co.” 
   The third order image rejection mixer also comprises a third transconductance (g m ) amplifier. The third transconductance amplifier also provides an imaginary component to the transfer function. For this embodiment, the third transconductance amplifier consists of transistors  642  and  638 , resistors  634  and  636 , and capacitor  665 . Capacitor  665  has a value set to “Co/3.” The output of Q mixer  610  is inverted in inventor  630  for subsequent input to capacitor  665 . The emitter of transistor  638  is coupled to resistor  634 , and the collector is coupled to capacitor  665 . Transistor  632  receives, at its base, the output of capacitor  665 . The output of the image rejection mixer  600  is coupled to the emitter of transistor  632 . The output of image rejection mixer  600  drives the bases of transistors  638 ,  644  and  660 . The third order image rejection response circuit  600  is biased using current sources as shown in  FIG. 6 . 
   The third order image rejection filter  600  of  FIG. 6  has a transfer function as follows: 
   
     
       
         
           
             
               
                 A 
                 = 
                 
                   
                     1 
                     + 
                     
                       j 
                       × 
                       3 
                       ⁢ 
                       S 
                     
                     - 
                     
                       3 
                       ⁢ 
                       S 
                       × 
                       S 
                     
                     - 
                     
                       j 
                       × 
                       31 
                       × 
                       S 
                       × 
                       
                         S 
                         / 
                         3 
                       
                     
                   
                   
                     1 
                     + 
                     
                       3 
                       ⁢ 
                       S 
                     
                     + 
                     
                       3 
                       ⁢ 
                       S 
                       × 
                       S 
                     
                     + 
                     
                       3 
                       ⁢ 
                       S 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                       × 
                       S 
                       × 
                       
                         S 
                         / 
                         3 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                   
                   
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           where 
           , 
           
             S 
             = 
             
               j 
               ⁢ 
               
                   
               
               ⁢ 
               wCR 
             
           
         
       
     
   
   The capacitor value, C, corresponds to the “C” designations in capacitors  652 ,  651  and  665  ( FIG. 6 ). The resistor value, R, is the value for resistors  670 ,  680 ,  648 ,  636  and  634 . 
     FIG. 7  illustrates one embodiment for a frequency response of a third order image rejection filter. The frequency response graph of  FIG. 7  is normalized. As shown in  FIG. 7 , a desired band of frequencies for the signal is centered around “+1”, and the image from the I,Q mixers is centered around “−1.” For this embodiment, the normalized transfer function for this embodiment is as follows: 
   
     
       
         
           
             
               ( 
               
                 1 
                 + 
                 X 
               
               ) 
             
             3 
           
           
             
               ( 
               
                 1 
                 + 
                 
                   X 
                   2 
                 
               
               ) 
             
             
               3 
               / 
               2 
             
           
         
       
     
   
     FIG. 8  is a schematic diagram illustrating one embodiment for a fourth order image rejection mixer. An input signal is input to an in-phase (“I”) mixer  802  and quadrature phase (“Q”) mixer  804 . A local oscillator signal is phase shifted to generate in-phase component (x 1 ) and quadrature phase component (xj). The local oscillator signal (x 1 ) is input to I mixer  802 , and the local oscillator signal (xj) is input to quadrature phase Q mixer  610  at each respective local oscillator (“LO”) ports. The I mixer  620  and Q mixer  610  generate mixed I and Q signals. 
   For this embodiment, the fourth order image rejection mixer comprises a series of cascaded transconductance amplifiers. As shown in  FIG. 8 , the I mixed signal, output from I mixer  620 , is input to a first transconductance amplifier. The first transconductance amplifier includes transistors  852  and  843 . The I mixed signal is input to the base of transistor  852 , and the emitter of transistor  852  is coupled to resistor  850 . The emitter of transistor  843  is coupled to resistor  842 , and the collector is coupled to capacitor  854 . The capacitance of capacitor  854  has value of “4 Co.” Capacitor  652  is also coupled to receive the output of Q mixer  804 . 
   The capacitor  854  also couples the output of Q mixer  804  to the input of a second transconductance amplifier. The second transconductance amplifier consists of transistors  838  and  832 , resistors  840  and  836 , and capacitor  830 . The output of I mixer  802  is inverted in inventor  828  for subsequent input to capacitor  830 . Capacitor  830  has a value set to “1.5 C0.” 
   The fourth order image rejection mixer also comprises a third transconductance (g m ) amplifier. For this embodiment, the third transconductance amplifier consists of transistors  826  and  820 , resistors  824  and  822 , and capacitor  818 . The output of Q mixer  804  is inverted in inventor  840  for subsequent input to capacitor  818 . The emitter of transistor  826  is coupled to resistor  824 . The emitter of transistor  820  is coupled to resistor  822  and the collector is coupled to capacitor  818 . 
   For this embodiment, the fourth order image rejection mixer further comprises a fourth transconductance (g m ) amplifier. The fourth transconductance amplifier consists of transistors  812  and  815 , resistors  814  and  810 , and capacitor  817 . Transistor  812  receives, at its base, the output of capacitor  818 . The output of I mixer  802  is input to capacitor  817 . The emitter of transistor  812  is coupled to resistor  814 . The emitter of transistor  815  is coupled to resistor  810  and the collector is coupled to capacitor  817 . The output of the image rejection mixer  800  is coupled to the emitter of transistor  808 . The output of image rejection mixer  800  drives the bases of transistors  843 ,  832 ,  820  and  815 . 
   The fourth order image rejection filter  800  of  FIG. 8  has a transfer function as follows: 
   
     
       
         
           
             
               
                 A 
                 = 
                 
                   
                     
                       
                         
                           1 
                           + 
                           
                             j4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             S 
                           
                           - 
                           
                             4 
                             ⁢ 
                             S 
                             × 
                             1.5 
                             ⁢ 
                             S 
                           
                           - 
                           
                             j4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             S 
                             × 
                             1.5 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             S 
                             × 
                             
                               S 
                               / 
                               1.5 
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           4 
                           ⁢ 
                           S 
                           × 
                           1.5 
                           ⁢ 
                           S 
                           × 
                           
                             S 
                             / 
                             1.5 
                           
                           × 
                           
                             S 
                             / 
                             4 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           1 
                           + 
                           
                             4 
                             ⁢ 
                             S 
                           
                           + 
                           
                             4 
                             ⁢ 
                             S 
                             × 
                             1.5 
                             ⁢ 
                             S 
                           
                           + 
                           
                             4 
                             ⁢ 
                             S 
                             × 
                             1.5 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             S 
                             × 
                             
                               S 
                               / 
                               1.5 
                             
                           
                           + 
                           
                             4 
                             ⁢ 
                             S 
                             × 
                           
                         
                       
                     
                     
                       
                         
                           1.5 
                           ⁢ 
                           S 
                           × 
                           
                             S 
                             / 
                             1.5 
                           
                           × 
                           
                             S 
                             / 
                             4 
                           
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                   
                   
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           where 
           , 
           
             S 
             = 
             
               j 
               ⁢ 
               
                   
               
               ⁢ 
               wCR 
             
           
         
       
     
   
   The numerator includes real number terms, and the denominator includes imaginary numbers. The capacitor value, C, corresponds to the “C” designations in capacitors  854 ,  830 ,  818  and  817  ( FIG. 8 ). The resistor value, R, is the value for resistors  850 ,  842 ,  840 ,  836 ,  824 ,  822 ,  814  and  810 . The fourth order image rejection response circuit  800  is biased with current sources as shown in  FIG. 8 . 
     FIG. 9  illustrates one embodiment for a frequency response of a fourth order image rejection filter. The frequency response graph of  FIG. 9  is normalized. As shown in  FIG. 9 , a desired band of frequencies for the signal is centered around “+1”, and the image from the I,Q mixers is centered around “−1.” For this embodiment, the normalized transfer function for this embodiment is as follows: 
   
     
       
         
           
             
               ( 
               
                 1 
                 + 
                 X 
               
               ) 
             
             ^ 
             4 
           
           
             
               ( 
               
                 1 
                 + 
                 
                   X 
                   2 
                 
               
               ) 
             
             2 
           
         
       
     
   
     FIG. 10  is a schematic diagram illustrating another embodiment for a fourth order image rejection mixer. Circuit  900  of  FIG. 10  is the same as circuit  800  of  FIG. 8 , except the sub-circuits in circuit  900  are tuned to generate the expressions 1/Sa, 1/1Sb, 1/Sc, and 1/Sd, wherein:
 Sa=jwCaR Sb=jwCbR Sc=jwCcR Sd=jwCdR 
   The fourth order image rejection filter  900  of  FIG. 10  has a transfer function as follows: 
   
     
       
         
           
             
               
                 A 
                 = 
                 
                   
                     1 
                     + 
                     jSa 
                     - 
                     
                       Sa 
                       × 
                       Sb 
                     
                     - 
                     
                       jSa 
                       × 
                       Sb 
                       × 
                       Sc 
                     
                     + 
                     
                       Sa 
                       × 
                       Sb 
                       × 
                       Sc 
                       × 
                       Sd 
                     
                   
                   
                     1 
                     + 
                     Sa 
                     + 
                     
                       Sa 
                       × 
                       Sb 
                     
                     + 
                     
                       Sa 
                       × 
                       Sb 
                       × 
                       Sc 
                     
                     + 
                     
                       Sa 
                       × 
                       Sb 
                       × 
                       Sc 
                       × 
                       Sc 
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         Z 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         
                           0.25 
                           ⁢ 
                           Z 
                         
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         
                           0.25 
                           ⁢ 
                           Z 
                         
                       
                       ) 
                     
                   
                   
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         S 
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         
                           0.25 
                           ⁢ 
                           S 
                         
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         1 
                         + 
                         
                           0.25 
                           ⁢ 
                           S 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
   
   The numerator includes real number terms, and the denominator includes imaginary numbers. The capacitor values Ca, Cb, Cc and Cd correspond to the “Ca”, “Cb”, “Cc” and “Cd” designations in capacitors  950 ,  956 ,  946  and  936  ( FIG. 10 ). The resistor value, R, is the value for resistors  968 ,  966 ,  960 ,  961 ,  950  and  938 . 
     FIG. 11  illustrates one embodiment for a frequency response of the fourth order image rejection filter of  FIG. 10 . The frequency response graph of  FIG. 11  is normalized. As shown in  FIG. 11 , a desired band of frequencies for the signal is centered around “+1”, and the image from the I,Q mixers is centered around “−1.” For this embodiment, the normalized transfer function for this embodiment is as follows: 
   
     
       
         
           
             
               
                 ( 
                 
                   1 
                   + 
                   X 
                 
                 ) 
               
               2 
             
             ⁢ 
             
               
                 ( 
                 
                   1 
                   + 
                   
                     X 
                     / 
                     4 
                   
                 
                 ) 
               
               2 
             
           
           
             
               
                 ( 
                 
                   1 
                   + 
                   X 
                 
                 ) 
               
               2 
             
             ⁢ 
             
               ( 
               
                 1 
                 + 
                 
                   X 
                   / 
                   
                     4 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
   
   As described herein, the invention achieves wideband image rejection. This allows the IF frequency to be as low as possible. It also deepens the attenuation at the image frequency band. 
   Although the present invention has been described in terms of specific exemplary embodiments, it will be appreciated that various modifications and alterations might be made by those skilled in the art without departing from the spirit and scope of the invention.