Abstract:
An apparatus is disclosed for implementing a complex filter of the type represented by a transfer function having a complex pole. The apparatus is capable of creating real and imaginary parts, Y r  and Y i , of a complex output signal in response to receiving real and imaginary parts, X r  and X i , of a complex input signal. The apparatus comprises a plurality of variable resistors that may be tuned to adjust various operating parameters of the complex filter.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     The present invention is related to that disclosed in U.S. patent application Ser. No. 09/778,540 filed on Feb. 7, 2001 by Brian C. Martin entitled “Resistor Tuning Network and Method for Microelectronic RC-Based Filters”. U.S. patent application Ser. No. 09/778,540 is to be assigned to the assignee of the present invention. The disclosures within U.S. patent application Ser. No. 09/778,540 are hereby incorporated by reference for all purposes as if fully set forth herein. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates in general to electronic filter technology and in particular to complex filters implemented in integrated circuits. 
     BACKGROUND OF THE INVENTION 
     Complex filters are useful in certain applications in wireless communications. Complex filters offer selective suppression of positive or negative frequency components of a complex or real signal. This feature of complex filters contrasts with the operation of real filters in that real filters have a transfer function that is symmetric around the direct current (DC) position. The ability of complex filters to suppress positive or negative frequency components enables the suppression of image frequencies of a signal. The suppression of image frequencies is a very important consideration in the design and operation of wireless transceivers. 
     A review of complex signals and complex filters will be useful to understand the present invention. A real signal may have both a positive frequency component and a negative frequency component. For example, a cosine signal cos (ωt) equals (e jω +e −jω )/2 and a sine signal sin (ωt) equals (e jω +e −jω )/2j. The letter j represents the square root of minus one. That is, j 2 =−1. The letter j therefore represents one imaginary unit. 
     A complex signal is a signal that is composed of two real signals in which one of the real signals is multiplied by j. A complex signal therefore has the form: 
     
       
           x ( t )= x   r ( t )+ j x   i ( t )  (1) 
       
     
     where x r (t) is a real signal that represents the real component of the complex signal x(t) and where x i (t) is a real signal that represents the imaginary component of the complex signal x(t). 
     A complex signal x(t) may be amplified by multiplication by a complex constant A+j B. For example, let y(t) be the result of multiplying the complex signal x(t) by the complex constant A+j B. Then 
     
       
           y ( t )=( A+j B )  x ( t )  (2) 
       
     
     where 
     
       
           y ( t )= y   r ( t )+ j y   i ( t ).  (3) 
       
     
     The expression Y r (t) represents a real signal that is the real component of the complex signal y(t) and the expression y i (t) represents a real signal that is the imaginary component of the complex signal y(t). 
     Substituting Equation (1) into Equation (2) and multiplying and equating the real and imaginary parts of the result with y(t) gives: 
     
       
           Y   r ( t )= A x   r ( t )− B x   i ( t )  (4) 
       
     
     and 
     
       
           y   i ( t )= B x   r ( t ) +A x   i ( t ).  (5) 
       
     
     Similarly, a complex signal x(t) may be multiplied by another complex signal z(t) where 
     
       
           z ( t )= Z   r ( t ) +j z   i ( t ).  (6) 
       
     
     The multiplication of x(t) by z(t) is represented by: 
     
       
           y ( t )= z ( t )· x ( t )  (7) 
       
     
     The result of multiplying x(t) by z(t) may be obtained by substituting Equation (6) into Equation (7) and multiplying and equating the real and imaginary parts of the result with y(t). The result is: 
     
       
           y   r ( t )= z   r ( t ) x   r ( t )− z   i ( t ) x   i ( t )  (8) 
       
     
     and 
     
       
           y   i ( t )= Z   r ( t ) x   r ( t )+ z   i ( t ) x   i ( t ).  (9) 
       
     
     Complex signals may be filtered by real filters or by complex filters. A real filter has a real impulse response h r (t). The transfer function H r (jω) is a rational polynomial function of jω. The transfer function H r (jω) can be real only if H r (jω)=H r *(−jω). 
     A complex filter has a complex impulse response 
     
       
           h ( t )= h   r ( t )+ j h   i ( t )  (10) 
       
     
     and a complex transfer function 
     
       
           H ( jω )= H   r ( jω )+ j H   i (   107   ).  (11) 
       
     
     The response of a linear time invariant system to an arbitrary input x(t) can be expressed as the convolution of x(t) and the impulse response h(t) of the system. That is, 
     
       
           y ( t )= h ( t )◯ x ( t )  (12) 
       
     
     where the symbol ◯ represents the convolution operation. Applying the well known time convolution theorem of the Fourier transform to Equation (12) gives: 
     
       
           Y ( jω )= H ( jω )· x ( jω )  (13) 
       
     
     where Y(jω) is a complex output signal and X(jω) is a complex input signal. H(jω) is a rational complex polynomial that is a function of jω. 
     Because the input signal X(jω) is complex, X(jω) is composed of a real part and an imaginary part. 
     
       
           X ( jω )= X   r ( jω )+ j X   i ( jω ).  (14) 
       
     
     where X r (jω) represents the real part of X(jω) and where X i (jω) represents the imaginary part of X(jω). Similarly, because the output signal Y(jω) is also complex, Y(jω) is also composed of a real part and an imaginary part. 
     
       
           Y ( jω )= Y   r ( jω )+ j Y   i ( jω ).  (15) 
       
     
     where Y r (jω) represents the real part of Y(jω) and where Y i (jω) represents the imaginary part of Y(jω). 
     Substituting Equations (11), (14) and (15) into Equation (13) and multiplying and equating the real and imaginary parts of the result to the real and imaginary parts of Y(jω) gives: 
     
       
         Y r ( jω )= H   r ( jω )− X   r ( jω )− H   i ( jω ) X   i ( jω )  (14a) 
       
     
     
       
           Y   i ( jω )= H   r ( jω ) X   r ( jω )+ H   i ( jω ) X   i ( jω )  (15a) 
       
     
     In the time domain Equations (14a) and (15a) give: 
     
       
         y r ( t )= h   r ( t )◯ x   r ( t )− h   i ( t )◯ x   i ( t )  (16) 
       
     
     
       
           y   i ( t )= h   r ( t )◯ x   r ( t )+ h   i ( t )◯ x   i ( t )  (17) 
       
     
     where the symbol ◯ represents the convolution operation. 
     The equation of a transfer function having a complex pole has the form:                H        (     j                 ω     )       =     A     s   +     (     p   ±     j                 q       )                 (   18   )                                
     The letter A represents a constant. The letter s represents the quantity jω. The letter p represents the real part of the complex pole and the letter q represents the imaginary part of the complex pole. Substitution of Equation (18) into Equation (13) gives:                Y        (     j                 ω     )       =       A     s   +     (     p   ±     j                 q       )         ·     X        (     j                 ω     )                 (   19   )                                
     One of the main applications for complex filters is the selective suppression of positive or negative frequency components of a complex or real signal. This may be accomplished by using a bandpass filter that is obtained from the linear frequency transformation of a lowpass filter. A complex lowpass filter that is centered on the direct current (DC) value (i.e., jω=0) of the jω axis of a H(jω)/jω plane may be linearly transformed to create a complex bandpass filter that is centered on another value, jω c , of the jω axis. 
     Using the linear transformation 
     
       
           s=jω−jω   c   (20) 
       
     
     will result in a bandpass filter that has the form of the lowpass filter but is centered around the frequency ω c . This form of bandpass filter has only the frequency shifted lowpass filter characteristics for positive frequencies. The transfer function of this form of bandpass filter suppresses negative frequency components. 
     Substituting Equation (20) into Equation (19) leads to the following design equations for the real and imaginary parts of the output signal Y(jω). The argument jω in the expressions Y(jω) and X(jω) in Equations (21) and (22) will be omitted for clarity.                Y   r     =         X   r     -         ω   C     A          Y   i       -         p   A          Y   r       ±       q   A          Y   i             j                   ω   A                 (   21   )                 Y   i     =         X   i     -         ω   C     A          Y   r       -         p   A          Y   i       ∓       q   A          Y   r             j                   ω   A                 (   22   )                                
     It would be desirable to provide circuitry on an integrated circuit that is capable of implementing a complex filter of the type represented by a transfer function having a complex pole. In particular, it would be desirable to provide an apparatus for providing the real part Y r (jω) and imaginary part Y i (jω) of an output signal Y(jω) that results from multiplying an input signal X(jω) by a transfer function H(jω) that has a complex pole. 
     SUMMARY OF THE INVENTION 
     The apparatus of the invention comprises circuitry that is capable of implementing a complex filter of the type that is represented by a transfer function having a complex pole. The apparatus creates real and imaginary parts, Y r (jω) and Y i (jω), of a complex output signal from real and imaginary parts, X r (jω) and X i (jω), of a complex input signal by implementing a transfer function that has a complex pole. The apparatus generally comprises an operational amplifier circuit and an input circuit comprising a plurality of input resistors. The input resistors in the input circuit may be variable and may be tuned to adjust various operating parameters of the complex filter. 
     It is an object of the present invention to provide an apparatus for implementing a complex filter of the type that is represented by a transfer function having a complex pole. 
     It is also an object of the present invention to provide an apparatus for providing the real part Y r (jω) and imaginary part Y i (jω) of an output signal Y(jω) that results from multiplying an input signal X(jω) by a transfer function H(jω) that contains a complex pole. 
     It is another object of the present invention to provide an apparatus for implementing a complex filter where the apparatus comprises a plurality of variable resistors that may be tuned to adjust operating parameters of the complex filter. 
     It is a further object of the present invention to provide an apparatus for implementing a complex filter where the apparatus comprises a plurality of variable resistors in which each resistor may be independently tuned to adjust operating parameters of the complex filter. 
     It is yet another object of the present invention to provide an apparatus for implementing a complex filter where the apparatus comprises a plurality of variable resistor pairs in which each resistor pair may be independently tuned to adjust operating parameters of the complex filter. 
     Other objects and advantages of the invention will become apparent as the description proceeds. 
     The foregoing has outlined rather broadly the features and technical advantages of the present invention so that those skilled in the art may better understand the Detailed Description of the Invention that follows. Additional features and advantages of the invention will be described hereinafter that form the subject matter of the claims of the invention. Those skilled in the art should appreciate that they may readily use the conception and the specific embodiment disclosed as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. Those skilled in the art should also realize that such equivalent constructions do not depart from the spirit and scope of the invention in its broadest form. 
     Before undertaking the Detailed Description of the Invention, it may be advantageous to set forth definitions of certain words and phrases used throughout this patent document: The terms “include” and “comprise” and derivatives thereof, mean inclusion without limitation, the term “or” is inclusive, meaning “and/or”; the phrases “associated with” and “associated therewith,” as well as derivatives thereof, may mean to include, be included within, interconnect with, contain, be contained within, connect to or with, couple to or with, be communicable with, cooperate with, interleave, juxtapose, be proximate to, to bound to or with, have, have a property of, or the like; and the term “controller,” “processor,” or “apparatus” means any device, system or part thereof that controls at least one operation. Such a device may be implemented in hardware, firmware or software, or some combination of at least two of the same. It should be noted that the functionality associated with any particular controller may be centralized or distributed, whether locally or remotely. Definitions for certain words and phrases are provided throughout this patent document. Those of ordinary skill should understand that in many instances (if not in most instances), such definitions apply to prior, as well as future uses of such defined words and phrases. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taking in conjunction with the accompanying drawings, wherein like numbers designate like objects, and in which: 
     FIG. 1 schematically illustrates an advantageous embodiment of a first circuit for representing a first portion of a complex filter with a complex pole; 
     FIG. 2 schematically illustrates an advantageous embodiment of a second circuit for representing a second portion of a complex filter with a complex pole; 
     FIG. 3 schematically illustrates an alternate advantageous embodiment of the present invention comprising a differential form of the circuit shown in FIG. 1; and 
     FIG. 4 schematically illustrates an alternate advantageous embodiment of the present invention comprising a differential form of the circuit shown in FIG.  2 . 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 1 schematically illustrates a first circuit  100  for representing a first portion of a complex filter with a complex pole. FIG. 2 schematically illustrates a second circuit  200  for representing a second portion of a complex filter with a complex pole. As will be more fully described, first circuit  100  produces the value of the real part Y r of the output of a complex filter described in Equation (21) and second circuit  200  produces the value of the imaginary part Y i of the output of a complex filter described in Equation (22). 
     Circuit  100  comprises four resistors coupled in parallel. The four resistors in circuit  100  include resistor  140  having a value of R1 ohms, resistor  150  having a value of R2 ohms, resistor  160  having a value of R3 ohms, and resistor  170  having a value of R4 ohms. Resistors  140 ,  150 ,  160  and  170  are variable resistors and may be tuned by a resistor tuning device. One example of a resistor tuning device for tuning integrated circuit resistors is set forth and described in U.S. patent application Ser. No. 09/778,540 filed by Brian C. Martin on Feb. 7, 2001 entitled “Resistor Tuning Network and Method for Microelectronic RC-Based Filters.” The disclosures within U.S. patent application Ser. No. 09/778,540 are hereby incorporated by reference for all purposes as if fully set forth herein. 
     An output signal from each of the four variable resistors  140 ,  150 ,  160  and  170  is combined and fed into the inverting input of op amp  120 . The output of op amp  120  is coupled to the inverting output of op amp  120  through capacitor  130  having a value of C farads. Capacitor  130  may be a variable capacitor that may be tuned by a capacitor tuning device. 
     As shown in FIG. 1, the input signal X r  is connected to resistor  140 . The output signal −Y r  is obtained from the output of op amp  120  and is fed back to resistor  150 . As will be described more fully below, op amp  220  in FIG. 2 produces output signal −Y i . The output signal −Y i  is obtained from the output of op amp  220  in FIG.  2  and fed back to resistor  170  in FIG.  1 . Lastly, the value of ±Y i  (obtained from the value of output signal −Y i ) is provided to resistor  160 . It is noted that both polarities of the output signals Y r  and Y i  are necessary to correctly implement circuit  100  and circuit  200 . 
     The equation for the output −Y r  of circuit  100  is:                -     Y   r       =         X   r     sR1C     -         Y   r     sR2C     ±       Y   i     sR3C       -       Y   i     sR4C               (   23   )                                
     By substituting jω for s in Equation (23) it may be demonstrated that circuit  100  implements Equation (21) where 
     
       
           R 1=1/( A C )  (24) 
       
     
     
       
           R 2=1/( p C )  (25) 
       
     
     
       
           R 3=1/( q C )  (26) 
       
     
     and 
     
       
           R 4=1/(ω c   C ).  (27) 
       
     
     Turning now to FIG. 2, one sees that circuit  200  also comprises four resistors coupled in parallel. The four resistors in circuit  200  include resistor  240  having a value of R1 ohms, resistor  250  having a value of R2 ohms, resistor  260  having a value of R3 ohms, and resistor  270  having a value of R4 ohms. Resistors  240 ,  250 ,  260  and  270  are variable resistors and may be tuned by a resistor tuning device. 
     An output signal from each of the four variable resistors  240 ,  250 ,  260  and  270  is combined and fed into the inverting input of op amp  220 . The output of op amp  220  is coupled to the inverting input of op amp  220  through capacitor  230  having a value of C farads. Capacitor  230  may be a variable capacitor that may be tuned by a capacitor tuning device. 
     As shown in FIG. 2, the input signal X i  is connected to resistor  240 . The output signal −Y i  is obtained from the output of op amp  220  and is fed back to resistor  250 . As previously described, op amp  120  in FIG. 1 produces output signal −Y r . The output signal −Y r  is obtained from the output of op amp  120  in FIG.  1  and fed back to resistor  270  in FIG.  2 . Lastly, the value of ∓Y r  (obtained from the value of output signal −Y r ) is provided to resistor  260 . 
     The equation for the output −Y i  of circuit  200  is:                -     Y   i       =         X   i     sR1C     -         Y   i     sR2C     ∓       Y   r     sR3C       +       Y   r     sR4C               (   28   )                                
     By substituting jω for s in Equation (28) it may be demonstrated that circuit  200  implements Equation (22) where 
     
       
           R 1=1/( A C )  (29) 
       
     
     
       
           R 2=1/( p C )  (30) 
       
     
     
       
           R 3=1/( q C )  (31) 
       
     
     and 
     
       
           R 4=1/(ω c   C ).  (32) 
       
     
     The value R1 is the same for both resistor  140  and resistor  240 . The value R2 is the same for both resistor  150  and resistor  250 . Similarly, the value R3 is the same for both resistor  160  and resistor  260 . Lastly, the value R4 is the same for both resistor  170  and resistor  270 . 
     Two resistors with the same value form a resistor pair. That is, resistor  140  and resistor  240  form a first resistor pair where each resistor has a value of R1 ohms. Resistor  150  and resistor  250  form a second resistor pair where each resistor has a value of R2 ohms. Resistor  160  and resistor  260  form a third resistor pair where each resistor has a value of R3 ohms. Lastly, resistor  170  and resistor  270  form a fourth resistor pair where each resistor has a value of R4 ohms. 
     By keeping the resistors distinct (as opposed to combining resistor  160  and resistor  170 , for example) each resistor pair controls a different aspect of the complex pole independently of the other resistor pairs. Specifically, the value of R1 ohms sets the gain of the filter. The value of R2 ohms sets the value of the real part of the complex pole. The value of R3 ohms sets the value of the imaginary part of the complex pole. The value of R4 ohms controls the low pass to band pass translation of the complex pole. That is, the value of R4 ohms sets the center frequency. 
     The advantageous embodiment of the present invention shown in FIG.  1  and in FIG. 2 employs resistors and capacitors that are variable in value (i.e., tunable). The use of tunable resistors and capacitors provides several advantages. First, integrated continuous time filters generally suffer in performance due to variations in the resistor values and variations in the capacitor values that occur during the manufacturing process. Compensation for such variations can be accomplished by tuning the resistors and capacitors. This type of tuning is usually performed by varying the capacitor values because there are fewer elements to tune. 
     Second, if the resistors are made independently tunable, then the filter can be tuned (1) to compensate for op amp imperfections by tuning the R2 and R3 values, or (2) to adjust the gain by tuning the R1 value, or (3) to readjust the center of the filter by tuning the R4 value. 
     Tuning the R1 value allows the filter to be incorporated into an automatic gain control (AGC) loop for use in radios and similar equipment. Tuning the R4 value to change the center frequency allows a radio, for example, to maximize performance by centering the filter more exactly on a desired signal. 
     As previously described, one advantageous embodiment of the invention comprises independently tunable resistors. Another alternate advantageous embodiment of the present invention may comprise independently tunable resistor pairs. Yet another alternate advantageous embodiment of the present invention may comprise resistors in which only one resistor pair is independently tunable. Still another alternate advantageous embodiment of the present invention may comprise resistors connected in an R-2R ladder. Another alternate advantageous embodiment of the present invention may comprise resistors that are digitally tunable. 
     In general complex poles are used in conjugate pairs. Conjugate pairs of complex poles may be formed by cascading two complex pole circuits. For example, in circuit  100  of FIG.  1  and in circuit  200  of FIG. 2 it is possible to use the positive polarity of Y i  at the input of resistor  160  and the negative polarity of Y r  at the input of resistor  260  to implement one of the conjugates. The other conjugate can be created by using the negative polarity of Y i  at the input of resistor  160  and the positive polarity of Y r  at the input of resistor  260 . In general, most filter functions can be implemented by cascading several stages of complex poles. 
     FIG. 3 schematically illustrates an alternate advantageous embodiment of the present invention comprising a differential form  300  of circuit  100  shown in FIG.  1 . Circuit  300  produces both a positive and negative value of the real part Y r  of the output of a complex filter described in Equation (21) 
     FIG. 4 schematically illustrates an alternate advantageous embodiment of the present invention comprising a differential form  400  of circuit  200  shown in FIG.  2 . Circuit  400  produces both a positive and negative value of the imaginary part Y i  of the output of a complex filter described in Equation (22). 
     Circuit  300  of FIG. 3 comprises two sets of four resistors coupled in parallel. The first set of four resistors in circuit  300  includes resistor  345  having a value of R1 ohms, resistor  350  having a value of R2 ohms, resistor  355  having a value of R3 ohms, and resistor  360  having a value of R4 ohms. Resistors  345 ,  350 ,  355  and  360  are variable resistors and may be tuned by a resistor tuning device. 
     An output signal from each of the four variable resistors  340 ,  350 ,  355  and  360  is combined and fed into the inverting input of op amp  320 . The inverting output of op amp  320  is coupled to the inverting input of op amp  320  through capacitor  330  having a value of C farads. Capacitor  330  may be a variable capacitor that may be tuned by a capacitor tuning device. 
     As shown in FIG. 3, the input signal X r  is connected to resistor  345 . The output signal −Y r  is obtained from the inverting output of op amp  320  and is fed back to resistor  350 . As will be described more fully below, op amp  420  in FIG. 4 produces output signal −Y i . The output signal −Y i  is obtained from the output of op amp  420  in FIG.  4  and fed back to resistor  360  in FIG.  3 . Lastly, the value of ±Y i  (obtained from the value of output signal −Y i ) is provided to resistor  355 . It is noted that both polarities of the output signals Y r  and Y i  are necessary to correctly implement circuit  300  and circuit  400 . 
     As in the case of circuit  100 , the equation for the output −Y r  of circuit  300  is:                -     Y   r       =         X   r     sR1C     -         Y   r     sR2C     ±       Y   i     sR3C       -         Y   i     sR4C     .               (   33   )                                
     Circuit  300  also comprises a second set of four resistors coupled in parallel. The second set of four resistors in circuit  300  includes resistor  365  having a value of R1 ohms, resistor  370  having a value of R2 ohms, resistor  375  having a value of R3 ohms, and resistor  380  having a value of R4 ohms. Resistors  365 ,  370 ,  375  and  380  are variable resistors and may be tuned by a resistor tuning device. 
     An output signal from each of the four variable resistors  365 ,  370 ,  375  and  380  is combined and fed into the non-inverting input of op amp  320 . The non-inverting output of op amp  320  is coupled to the non-inverting input of op amp  320  through capacitor  340  having a value of C farads. Capacitor  340  may be a variable capacitor that may be tuned by a capacitor tuning device. 
     As shown in FIG. 3, the input signal −X r  is connected to resistor  365 . The output signal Y r is obtained from the non-inverting output of op amp  320  and is fed back to resistor  370 . The output signal Y i  is obtained from the non-inverting output of op amp  420  in FIG.  4  and fed back to resistor  380  in FIG.  3 . Lastly, the value of ∓Y i  (obtained from the value of output signal Y i ) is provided to resistor  375 . It is noted that both polarities of the output signals Y r  and Y i  are necessary to correctly implement circuit  300  and circuit  400 . 
     The equation for the output +Y r  of circuit  300  is:                +     Y   r       =       -       X   r     sR1C       +         Y   r     sR2C     ∓       Y   i     sR3C       +         Y   i     sR4C     .               (   34   )                                
     Thus it is seen that circuit  300  of FIG. 3 comprises a differential form of circuit  100  of FIG.  1 . It is noted and understood that the resistors and capacitors in circuit  300  may be tuned in the same manner as that described for tuning the resistors and capacitors in circuit  100  and in circuit  200 . 
     Lastly, circuit  400  of FIG. 4 comprises two sets of four resistors coupled in parallel. The first set of four resistors in circuit  400  includes resistor  445  having a value of R1 ohms, resistor  450  having a value of R2 ohms, resistor  455  having a value of R3 ohms, and resistor  460  having a value of R4 ohms. Resistors  445 ,  450 ,  455  and  460  are variable resistors and may be tuned by a resistor tuning device. 
     An output signal from each of the four variable resisters  445 ,  450 ,  455  and  460  is combined and fed into the inverting input of op amp  420 . The inverting output of op amp  420  is coupled to the inverting input of op amp  420  through capacitor  430  having a value of C farads. Capacitor  430  may be a variable capacitor that may be turned by a capacitor tuning device. 
     As shown in FIG. 4, the input signal X i  is connected to resistor  445 . The output signal −Y i  is obtained from the inverting output of op amp  420  and is fed back to resistor  450 . Op amp  320  in FIG. 3 produces output signal −Y r . The output signal −Y r  is obtained from the output of op amp  320  in FIG.  3  and fed back to resistor  460  in FIG.  4 . Lastly, the value of ∓Y r  (obtained from the value of output signal −Y r ) is provided to resistor  455 . It has previously been noted that both polarities of the output signals Y r  and Y i  are necessary to correctly implement circuit  300  and circuit  400 . 
     As in the case of circuit  200 , the equation for the output −Y i  of circuit  400  is:                -     Y   i       =         X   i     sR1C     -         Y   i     sR2C     ∓       Y   r     sR3C       +         Y   r     sR4C     .               (   35   )                                
     Circuit  400  also comprises a second set of four resistors coupled in parallel. The second set of four resistors in circuit  400  includes resistor  465  having a value of R1 ohms, resistor  470  having a value of R2 ohms, resistor  475  having a value of R3 ohms, and resistor  480  having a value of R4 ohms. Resistors  465 ,  470 ,  475  and  480  are variable resistors and may be tuned by a resistor tuning device. 
     An output signal from each of the four variable resistors  465 , 470 ,  475  and  480  is combined and fed into the non-inverting input of op amp  420 . The non-inverting output of op amp  420  is coupled to the non-inverting input of op amp  420  through capacitor  440  having a value of C farads. Capacitor  440  may be a variable capacitor that may be tuned by a capacitor tuning device. 
     As shown in FIG. 4, the input signal −X i  is connected to resistor  465 . The output signal Y i  is obtained from the non-inverting output of op amp  420  and is fed back to resistor  470 . The output signal Y r  is obtained from the non-inverting output of op amp  320  in FIG.  3  and fed back to resistor  480  in FIG.  4 . Lastly, the value of ±Y r  (obtained from the value of output signal Y r ) is provided to resistor  475 . It has previously been noted that both polarities of the output signals Y r  and Y i  are necessary to correctly implement circuit  300  and circuit  400 . 
     The equation for the output +Y i  of circuit  400  is:                Y   i     =       -       X   i     sR1C       +         Y   i     sR2C     ±       Y   r     sR3C       -         Y   r     sR4C     .               (   36   )                                
     Thus it is seen that circuit  400  of FIG. 4 comprises a differential form of circuit  200  of FIG.  2 . It is noted and understood that the resistors and capacitors in circuit  400  may be tuned in the same manner as that described for tuning the resistors and capacitors in circuit  100  and in circuit  200 . 
     The above examples and description have been provided only for the purpose of illustration, and are not intended to limit the invention in any way. As will be appreciated by the skilled person, the invention can be carried out in a great variety of ways, employing more than one technique from those described above, all without exceeding the scope of the invention.