Abstract:
An analog equalization filter is disclosed which permits higher speed and linearity than existing designs, allowing for filtering operation to the hundreds of gigahertz range. Possible applications include fixed and adaptive equalization filtering and radio frequency filtering. The filter can be entirely implemented on an integrated circuit chip. The filter is based on transmission line based delay elements and transconductance amplifiers.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority from U.S. Provisional Application entitled “Continuous-Time Multi-Gigahertz Filter Using Transmission Line Delay Elements” Application No. 60/460,679 filed on Apr. 4, 2003. 
    
    
     BACKGROUND 
     1. Technical Field of Invention 
     This present invention relates to analog implementations of finite impulse response (FIR) filters and infinite impulse response (IIR) filters. The filter application examples include but are not limited to equalization of 10 Gb/s and 40 Gb/s fiber optic transmission systems, multi-gigahertz serial chip-to-chip communications, multi-gigahertz serial backplane communications, high-speed network communications, disk drive channel equalization, and filtering for radio frequency (RF) systems. Additionally, the FIR and IIR filter coefficients can be programmed so as to form an adaptive equalizer. 
     2. Background of the Invention and Discussion of Prior Art 
     A FIR or IIR filter is a basic building block of digital filtering systems. Typically for low frequency filtering (less than 100 MHz), as in modems, FIR or IIR filtering can be done with a DSP processor or ASIC using CMOS digital logic. At higher speeds, between 10 MHz and 1 GHz, a digital filtering solution will dissipate very high power and requires large die area, and a CMOS analog solution becomes attractive in order to reduce power and area. However, at speeds of 1 GHz and above, both digital and analog solutions become problematic even in higher speed technologies such as SiGe, InP, and GaAs. 
     A block diagram of a typical FIR filter structure  100  is shown in  FIG. 1 . The FIR filter consists of delay line elements,  101 ,  102 , and  103 , coefficients A, B, C, and D, coefficient multipliers,  104 ,  105 ,  106 , and  107 , and a summing circuit,  8 . The input, X, is successively passed through the fixed delay elements,  101 ,  102 , and  103 . The signals at the successful nodes of the delay elements  101 ,  102 , and  103  are then multiplied by coefficients A, B, C, and D, respectively, before being summed to produce the output, Y. Prior art analog implementations of FIR filters have been published in the literature using switched-capacitor filtering to implement delay elements, multipliers, and summers. Unfortunately, switched-capacitor circuits are limited in speed by the gain-bandwidth product of the operation amplifiers, which is limited by the speed of the technology. Higher gain gives better linearity performance, yet requires the speed to be reduced. 
     Hence, there is also a severe speed versus linearity tradeoff in traditional designs. Other methods for implementing these types of filters exist, but all share high frequency performance limitations. 
     OBJECTS AND ADVANTAGES OF THE INVENTION 
     Accordingly, it is a primary object of the present invention to provide an analog FIR and IIR filter implementation that improves upon the speed of current implementations into the hundreds of gigahertz range with superior linearity to the implementations of the prior art. 
     SUMMARY OF THE INVENTION 
     The present invention achieves the above objectives and advantages by taking advantage of the properties of transmission lines. A transmission line provides a fixed impedance value over a wide range of frequencies with excellent linearity. This easily includes ranges up to and exceeding 100 GHz. In addition, the propagation delay of signals propagating down a transmission line can be precisely controlled only by the dimensions of the transmission line and the physical properties of the medium in which the transmission line is built. The inventor notes that these properties of the transmission line make it an ideal element for the delay stages of the FIR filter and are not subject to the linearity/bandwidth limitations of active delay stages. 
     In general, implementations feature an analog filter circuit. The analog filter include a transmission line delay element and a transconductance element as a building block for fiber optic transmission systems, multi-gigahertz serial chip-to-chip communications, multi-gigahertz serial backplane communications, high-speed network communications, disk drive channel equalization, and filtering for radio frequency (RF) systems. In general, the transmission line delay element has a delay time and time-delays an input signal. The transmission delay line is terminated with a matched or mismatched impedance element according to the filter response requirement. In general, the transconductance element multiplies the time-delayed input signal by a tunable filter coefficient then converts the multiplied time-delayed input signal to a current. The analog filtering is performed by the multiplication function of the transconductance elements. A number of multiplied time-delayed input signals by different delay times of the transmission delay lines are converted to currents by the transconductance elements. The outputs of the transconductance elements couple together to form a current summing node to sum the multiplied or amplified different time-delayed currents into a summed current. In general, the summed current is converted back to a voltage signal by a transimpedance element to produce the filtered output voltage signal. 
     In general, the analog filter is configured to perform filtering functions, such as the IIR filtering. In other implementation, the analog filter is configured to perform FIR filtering functions. 
     These and other implementations can optionally include one or more of the following features. The input can also enter a no-delay transconductance element to be filtered and converted to a no-delay current before summed together with currents produced from different time-delayed input signals. A forward transmission delay line is configured to have at least one forward transmission line delay element for time-delaying the output signal by corresponding at least one forward delay time for using in FIR filtering. A feedback transmission delay line is configured to have at least one feedback transmission line delay element for time-delaying the output signal by corresponding at least one feedback delay time for using in IIR filtering. Each of the forward and feedback transmission delay lines is terminated with a matched or mismatched impedance element according to the filter response requirement. The forward and feedback transmission line delay elements can generate time-delayed input and output signals. The time-delayed input signal and the output signal and the no-delay input signal can be each multiplied by a corresponding transconductance element with a corresponding filter coefficient. The multiplied no-delay input signal, the multiplied time-delayed forward input signal and the multiplied feedback output signal can be converted to corresponding currents by the corresponding transconductance element. The at least one forward current, the at least one feedback current and the no-delay current can be summed into a summed current at the current summing node. A transimpedance amplifier couples to the current summing node to convert the summed node and generate an output voltage signal. The input signal and the output signal can be single ended or differential. In general, the filter coefficient is equal to the transconductance of the transconductance element and is tuned to respond to a system filter requirement specification. These and other implementations can optionally include one or more of the following features. A transconductance element can include a differential transistor pair. A differential gate terminal of the differential transistor pair can couple to the input signal and a differential source terminal of the differential transistor pair can couple to a ground or a negative power supply via a differential current source. A variable resistor can couple between the differential source terminal to tune the transconductance, therefore the filter coefficients of the transconductance element. The variable resistor can be a serial resistor network with a fixed and at least one switched resistor or a parallel resistor network with a fixed and at least one switched parallel resistor. The transimpedance element can be a pair load resistors coupled to the differential drain terminal of the transconductance element. The pair can be replaced by a transimpedance amplifier with feedback impedances to tune the transimpedance amplifier. 
     In general, the analog filter circuit is implemented according to a filter response requirement in selecting the number of delay elements, delay times of the delay elements, filter coefficients, gain of the amplifiers and layout considerations. An analog filter circuit can implement a variety of FIR filters including low pass filters, high pass filters, band pass filters, notch filters, differentiators, or comb filters. The forward FIR filter coefficients can be easily obtained by taking the Fourier Transform of the system filter response. An alaog filter circuit can implement a variety of IIR filters including low pass filters, high pass filters, band pass filters, notch filters, or integrators. The analog filter described can be implemented in high speed, i.e. 1 GHz to over 100 GHz, for equalization in fiber optic transmission systems, multi-gigahertz serial chip-to-chip communications, multi-gigahertz serial backplane communications, high-speed network communications, disk drive channel equalization, and filtering for radio frequency (RF) systems. The analog filter can be implemented on-chip, off-chip, on printed circuit board or on backplane. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is a block diagram of a FIR filter considered as prior art. 
         FIG. 2  is a block diagram of an analog FIR filter. 
         FIG. 3  is a block diagram of an example analog IIR filter. 
         FIG. 4   a  is a block diagram of an example 1-tap analog IIR filter. 
         FIG. 4   b  is a block diagram of an example analog IIR filter. 
         FIG. 5  is a block diagram of an example differential analog IIR filter. 
         FIG. 6   a  is a schematic of an example transconductance and transimpedance circuit. 
         FIG. 6   b  is a schematic of an example single ended transconductance and transimpedance circuit. 
         FIG. 7  is a schematic of an example transconductance and transimpedance circuit. 
         FIG. 8  is a schematic of an example transconductance and transimpedance circuit. 
         FIG. 9  is a schematic of an example analog filter. 
         FIG. 10  is a schematic of an example analog filter. 
         FIG. 11  is a schematic of an example analog filter. 
         FIG. 12  is a floor plan of an example transmission line delay element layout. 
         FIG. 13  is a schematic of an example analog equalizer circuit. 
         FIG. 14  is a schematic of an example backplane system. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 2  shows an example block diagram of an N-tap analog FIR filter  200  wherein N is a positive integer. Taking advantage of the above described transmission line properties, a structure comprised of a transmission line delay element and a transconductance element such as the structure  250  can be used as the building block of an analog filter. For example, an N-tap filter can comprise of a plurality of N delay element and transconductance element structure  250 , a transimpedance element  222 , a termination impedance  221  and a summing node  235  wherein outputs of the N delay and transconductance structure can couple together to form the summing node  235  at an input of the transimpedance element  222 . The analog FIR filter  200  can comprise a plurality of N transmission line delay element and transconductance element structure  250 , a termination impedance element  221 , a transconductance element  210 , a transimpedance element  222 , an input terminal  230  and an output terminal  240  wherein the transmission line delay element and transconductance element structure  250  further comprises a plurality of N delay elements  201 - 20 N, and a plurality of N analog transconductance elements  210 - 21 N. 
     The transmission line delay line elements  201 - 20 N can use a coaxial line, a two-wire line, or a plate line including a stripline, and/or a microstrip line or a combination of the above lines. The characteristic impedance Z o  of the transmission line can be a function of the geometric dimensions and the medium of the transmission line such as the intrinsic impedance η, wherein η=(μ/∈) 1/2  is a function of the permeability μ and the permittivity ∈ of the transmission line medium. For example, the lossless unit characteristic impedance of a coaxial line can be Z o =η*ln(ra/rb)/2π wherein ln is the natural logarithm and ra and rb are the radius of the center and outer conductors of the coaxial line, respectively; and the unit characteristic impedance of a stripline can be Z o =(30π/∈ r ) 1/2 *d/(W e +0.441d) wherein ∈ r  is the stripline permeittivity, d is the height and W e  is an effective width of the center conductor. 
     An analog input signal signal X(tn) at time to can enter the analog FIR filter  200  at the input terminal  230  to produce a filtered analog output signal Y(tn) at the output terminal The analog FIR filter can perform the FIR filtering function to generate the filtered analog output Y(tn) represented by a general FIR filter equation 
                 Y   ⁡     (   tn   )       =       ∑     i   =   0       N   -   1       ⁢       a   ⁡     (     n   ,   1     )       *     X   ⁡     (     tn   -   i     )             ,         
as a function of weighted input and delayed input signals wherein each a(n,i) is a filter coefficient and n is a time index for a time period of Tn. The time period Tn can be an infinitesimal period, for example, on the order of pico seconds for a continuous time analog FIR filter. Y(tn) can be delayed by the propagation time through the analog FIR filter  200 . In particular, the input voltage signal, X(tn), enters at the input terminal  230  of the transmission line that can be configured with a plurality of N delay elements  201 - 20 N that can provide a plurality of N corresponding fixed propagation delay time td 1 -td N . The transmission delay line can be terminated with the termination impedance  221  at the last delay element  21 N. In some implementations, the input signal X(tn) can be a discrete analog signal. In other implementations, the input signal X(tn) can be a continuous time analog signal.
 
     The outputs of the transmission line delay elements  201 - 20 N can be coupled to the inputs of the transconductance elements  211 - 21 N, respectively. The analog transconductance elements,  210 - 21 N can have corresponding transconductances Gm n0 -Gm nN  of positive values that can be used as the absolute values of the corresponding filter multiplier coefficients a(n, 0 ) to a(n,N) to perform multiplication functions, i.e. FIR filtering, of the analog FIR filter  200 . In particular, abs(a(n,i))=Gm n1  for i=0, 1 to N can be tuned for different time periods. In addition, each of the transconductance elements  210 - 21 N can convert a multiplied input voltage signal to a current. The input signal X(tn) can also be coupled to an input of the transconductance element  210 . The outputs of the transconductance elements  210 - 21 N can be coupled together to form the summing node  235  at an input of the transimpedance element  222  to sum a plurality of N+1 multiplied output currents into a summed current. The transimpedance element  222  can be configured to convert the summed current to an output voltage signal Y(tn). In some implementations, the transconductance elements  210 - 21 Nn can be configured as transconductance amplifiers or multistage voltage amplifiers. In some implementations, matching impedances can be coupled to inputs and/or outputs of the transconductance elements  210 - 21 N. In some implementations, the transimpedance element  222  can be configured to be transconductance amplifiers, multistage voltage amplifiers, resistors, impedances, or a combination of resistors, impedances and voltage amplifiers. In some implementations, the transimpedance element  222  implemented as a transimpedance amplifier can amplify the summed current with a gain in addition to convert the current input to a voltage output Y(tn). Further the transimpedance element  222  configured as the transimpedance amplifier of an operation amplifier and an impedance feedback loop can reduce signal noise. In some implementations, the transimpedance element  222  can use a load resistor to convert a current input to a FIR filtered output voltage signal Y(tn). The transmission line can be terminated at the end with impedance element,  221 . In some implementations, the FIR filtered output voltage signal Y(tn) can be an equalized signal. 
     The transmission line with delay elements  201 - 21 N can provide linearity for signals of frequencies from around 1 GHz to above 100 GHz. Therefore, the analog FIR filter  200  can extend the bandwidth and improves the linearity performance when compared to the prior art FIR filter shown in  FIG. 1 . The filter coefficients a(n, 0 )-a(n,N) for the time period to can be controlled by modifying the transconductances, Gm n0 −Gm nN , of the analog transconductance amplifiers. The values of the transconductances Gm n0 -Gm nN  can be fixed, programmable, or adaptively controlled. In some implementations, the transmission line delay elements can have switches to change the number of the transmission delay elements. In some implementations, the termination impedance,  221 , can be matched to the transmission line characteristic impedance in order to eliminate reflections. In other implementations, the termination impedance  221 , can be purposely mismatched to induce a reflection to alter the response of the filter. In some implementations, the termination impedance  221  and/or the matching impedances to the inputs and/or outputs of the transconductance elements  210 - 21 N can further comprise resistors, capacitors, inductors or a combination of resistor, capacitor and/or inductor networks. In some implementations, the termination impedance  221  and/or the matching impedances to the inputs and/or the outputs of the transconductance elements  210 - 21 N can be configured to be fixed, programmable or adaptive impedances. 
     The analog FIR filter  200  can be single ended or differential. Since the transconductance coefficients Gm&#39;s can have positive values only, a full differential or a pseudo-differential analog filter can be employed for a general filter requirement with both positive and negative filter coefficients. For a full differential analog FIR filter  200  with differential input X(tn), the transconductance elements  210 - 21 N and the transimpedance elements  201 - 20 N are differential and two transmission delay lines are used. However, a pseudo-differential analog FIR filter can have only a single transmission line for a single ended input signal X(tn) and the differential transimpedance elements  201 - 20 N can convert single ended signals into differential signals. In addition, a differential or pseudo-differential analog FIR filter can have better noise immunity compared to a single ended analog FIR filter  200 . 
     The topology of the analog FIR filter  200  can be selected according to a system filter requirement specification. The analog FIR filter  200  can be applied in high speed backplanes, disk drives, optical systems, video systems, and wireline and wireless communication systems. In some implementations, the analog FIR filter  200  can be used after the number of filter taps, the delay time of each delay element, and the filter coefficients are determined according to a FIR filter system requirement. In some implementations, the transmission line of delay elements and the transconductance elements can be transposed to perform the same FIR filtering function. However, additional transimpedance elements may be needed in order to convert currents back to voltages before entering the transmission line delay lines in a transposed structure of the analog FIR filter  200 . In other implementations, the order, the number of the transmission line delay elements and the transconductance elements can be designed differently from the analog FIR filter  200  or its transposed form but can still perform the same FIR filter functions. 
       FIG. 3  is an example block diagram of an analog N+N′ tap infinite impulse response (IIR) filter  300 , wherein N and N′ can be different positive integers or N can be equal to N′. The same building blocks, i.e. the delay element and transconductance element structure  250 , used in the analog FIR filter  200  of  FIG. 2  can be used to create the analog IIR filter  300  for both a forward propagation filtering section  335  and a feedback propagation, i.e. a feedback loop, filtering section  345 . The IIR filter  300  uses a feedback loop in its filter structure and can perform filter functions that are represented by a general IIR filter equation with an input signal X(tn) and an output signal Y(tn), 
                 Y   ⁡     (   tn   )       =         ∑     i   =   0     N     ⁢       a   ⁡     (     n   ,   i     )       *     X   ⁡     (     tn   -   i     )           +       ∑     j   =   1       N   ′       ⁢       b   ⁡     (     n   ,   j     )       *     Y   ⁡     (     tn   -   j     )               ,         
wherein X(tn) is an analog input signal, Y(tn) is an analog filter output signal, X(tn−i)&#39;s and Y(tn−j)&#39;s are delayed input and output signals, a(n,i)&#39;s and b(n,j)&#39;s are filter coefficients, and n can denote a time index for a time period of Tn. Therefore, a FIR filter such as the analog FIR filter  200  shown in  FIG. 2  can be a special case of the IIR filter  300  when all the filter coefficients b(n,j)=0 for all the j&#39;s. The analog IIR filter  300  can comprise a feedback propagation transmission line of a plurality of N′ delay elements  351 - 35 N′ that can provide a plurality of corresponding delay times td′ 1 −td′ N′ , a forward transmission line of a plurity of delay elements  301 - 30 N that can provide a plurality of corresponding delay times td 1 −td N , a plurality of N+1 transconductance elements  310 - 31 N of a plurality of N+1 corresponding transconductances Gm n0 -Gm nN  and a plurality of N′ transconductor elements  361 - 36 N′ of a plurality of N′ corresponding transconductances Gm′ n1 -Gm′ nN , a forward transmission line termination impedance  321 , a feedback transmission line impedance  322 , a transimpedance element  323 , an input terminal  330  and an output terminal  340 , wherein the transconductance Gm n0 -Gm nN  and Gm′ n1 -Gm′ nN′  are positive values and can be used as the absolute values of corresdponding filter coefficients a(n, 0 )a(n,N) and b(n, 1 )−b(n,N).
 
     The filter operation of the analog IIR filter  300  can be described similarly to that for the FIR filter  200  shown in  FIG. 2 . In particular, the input voltage signal X(tn) can enter the IIR filter  300  and the transconductance element  310  at the input terminal  330 , then propagate through the forward transmission line delay elements  311 - 31 N, and the filtered output voltage signal Y(tn) can be feedback through the feedback transmission line delay elements  351 - 35 N′. The input signal X(tn) and the outputs of the forward and feedback transmission line delay elements  311 - 31 N and  351 - 35 N′ can be coupled to corresponding inputs of the transconductance elements  310 - 31 N and  361 - 36 N′ before being multiplied by the corresponding filter coefficients, i.e., the transconductances Gm n0 -Gm nN  and G n m′ 1 −Gm′ nN′  to perform the IIR filtering and being converted to currents at the outputs of the transconductance elements  310 - 31 N and  361 - 36 N′. The outputs of the transconductance elements  310 - 31 N and  361 - 36 N′ are coupled together and to the input of the tramsimpedance  323  to sum the output currents at the input of the transimpedance  323  which then converts the resulting summed current at its input into the output voltage signal Y(tn) at the filter output terminal  340 . In some implementations, the IIR filtered output voltage signal Y(tn) can be an equalized signal. 
     Similar to the FIR filter  200  in  FIG. 2 , the transconductance elements  310 - 31 N and  361 - 36 N′ can be implemented as transconductance amplifiers or multistage voltage amplifiers, and the multiplication filter coefficients can be controlled by modifying the corresponding transconductances, Gm n0 -Gm nN  and Gm′ n1 -Gm′ nN′  of the analog transconductance amplifiers  310 - 31 N and  361 - 36 N′. In some implementations, matching impedances can be coupled to inputs of the transconductance elements  210 - 21 N. Similarly, the transimpedance element  323  can be implemented with a load resistor, a transimpedance amplifier, multistage voltage amplifiers, resistors, impedances, or a combination of resistors, impedances and voltage amplifiers. In some implementations, the transimpedance element  323  implemented as a transimpedance amplifier and can amplify the input current signal with a gain in addition to convert a current input to a voltage output. Further the transimpedance element  323  configured as the transimpedance amplifier with an impedance feedback loop can reduce signal noise. In some implementations, the transimpedance element  323  can use a load resistor to convert a current input to a voltage output. In some implementations, the transimpedance element  323  can be configured to be fixed, programmable or adaptive. 
     The values of the transconductances Gm and Gm′ can be fixed, programmable, or adaptively controlled. As in the FIR filter  200 , the forward and feedback transmission lines can be terminated with the termination impedance elements,  321  and  322 . The termination impedance,  321  and  322 , can be matched to characteristic impedances of the transmission line, respectively in order to eliminate reflections, or can be purposely mismatched to induce a reflection to alter the response of the filter. In some implementations, the termination impedances  321  and  322  and/or the matching impedances to the inputs and/or of the transconductance elements  310 - 31 N and  361 - 36 N′can further comprise resistors, capacitors, inductors or a combination of resistor, capacitor and/or inductor networks. In some implementations, the termination impedance  321  and  322  and/or the matching impedances to the inputs and/or outputs of the transconductance elements  310 - 31 N and  361 - 36 N′can be configured to be fixed, programmable or adaptive impedances. 
     Similar to the analog FIR filter  200  shown  FIG. 2 , the analog IIR filter  300  can be single ended, pseudo-differential or differential. A single ended analog IIR filter  300  can have only positive or negative filter coefficients and a differential or pseudo-differential analog filter  300  can be employed for a general filter with both positive and negative filter coefficients. 
     In some implementations, the analog IIR filter  300  can be used after the number of filter taps, the delay time of each delay element, and the filter coefficients are determined according to an IIR filter system requirement. In some implementations, the transmission line of delay elements and the transconductance elements can be transposed to perform the same IIR filtering function. However, additional transimpedance elements may be needed in order to convert currents back to voltages before entering the transmission line delay lines in a transposed structure of the analog IIR filter  300 . In other implementations, the order, the number of the transmission line delay elements and the transconductance elements can be designed differently from the analog IIR filter  300  or its transposed form but can still perform the same IIR filter functions. In addition, a pseudo-differential or differential analog IIR filter  300  can have better noise immunity compared to a single ended analog IIR filter. The topology of the analog IIR filter  300  can be selected according to a system filter requirement. 
     When all the filter coefficients Gm′ 1 −Gm′ N′  are zero, the feedback loop section  345  can be removed in the circuit and the analog IIR filter  300  is reduced to the analog FIR filter  200  shown in  FIG. 2 . Therefore, the analog FIR filter  200  can be considered as a special case of the analog IIR filter  300 . 
       FIGS. 4   a  and  4   b  are schematics of an example 1-tap analog IIR filters  400   a  and  400   b  configured to perform IIR filtering functions according to the filter equation (2) Y(tn)=a(n, 1 )*X(tn−1)+b(n, 1 )*Y(tn−1), wherein Y(tn) can be an output signal at a time −tn, X(tn−1) and Y(tn−1) can be delayed input and output signals, and a(n,  1 ) and b(n, 1 ) can be filter coefficients. The IIR filter  400   a  comprises a single transmission line delay element  403 , two transinductance elements  401   a  and  405   a , a transimpedance  402   a  and a termination impedance  404   a , an input terminal  410   a  and an output terminal  420   a . The transconductance elements  401   a  can multiply an input voltage signal Xa(tn) at the input terminal  410   a  by a first filter coefficient Gm 1 , i.e. the transconductance of the transconductance element  401   a , and convert the weighted input signal to a current. The output voltage signal Ya(tn) can be multiplied by a second filter coefficient Gm′  1 , i.e. the transconductance of the transconductance element  405   a  and converted to a current. The IIR filtering function can be performed by the transconductance elements  401   a  and  405   a . The transimpedance element  402   a  can convert the resulting summed current of the output currents from the transcnductance elements  401   a  and  405   a  with a delay time of td. The filtered output signal at the output of the transimpedance element  402   a  can enter the transmission line delay element  403   a  which can be terminated by the termination impedance  404   a . The output signal Y(tn) at the output terminal  420   a  of the transmission line delay element  403   a  can be the output signal Y(tn) of the analog IIR filter  400   a.    
       FIG. 4   b  is another example schematic of an analog IIR filter  400   b  which can also perform the same IIR filter equation (2) Y(tn)=a(n, 1 )*X(tn−1)+b(n, 1 )*Y(tn−1). The analog IIR filter  400   b  can be constructed by using the structure of the analog filter  300  shown in  FIG. 3  for a 2-tap analog IIR filter, i.e. 1-tap for the forward transmission line delay element  402   a  and 1-tap for the feedback transmission line delay element  406   b  wherein the two transmission line delay elements are identical. The forward and feedback transmission line delay elements  402   b  and  406   b  can be terminated by the termination impedance elements  404   b  and  407   b , respectively. An input signal X(tn) can propagate through the forward transmission line delay element  402   b  and be delayed by the delay 370 time td. The transconductance element  401   b  then multiply the delayed output signal of the delay element  402   b  by a first filter coefficient Gm 1 , i.e. the transconductance of the transconductance element  401   b  and convert the filtered signal to a first current signal. The output signal Y(tn) of the analog IIR filter  400   b  can propagate through the feedback transmission line delay element  406   b  with the delay time td. The transconductance element  405   b  can multiply the delayed output signal from the feedback transmission line delay element  406   b  with a second filter coefficient Gm′  1 , i.e. the transconductance of the transconductance element  405   b  and convert the filtered output signal to a second current signal to be combined with the first current signal at an input of the transimpedance element  402   b  before converted back to the output voltage signal Y(tn). In some implementations, the IIR filtered output voltage signal Y(tn) can be an equalized signal. 
     Both the analog IIR filters  400   a  and  400   b  perform the same IIR filtering but the IIR filter  400   b  requires one more transmission line delay element and one more termination element compared with that of the analog IIR filter  400   a . In other implementations, a transposed structure of the analog filter  400   b  can be used. 
     If the feedback loops of the analog IIR filters  400   a  and  400   b  are eliminated, the analog filters  400   a  and  400   b  can be FIR filters and filter  400   b  can represent a transposed filter  400   a . Similarly, analog FIR filters such as FIR filter  200  can be implemented in different filter structures. 
     Similarly, the analog IIR filters  400   a  and  400   b  can be single ended, pseudo-differential or differential. A single ended analog IIR filter  400   a  or  400   b  can perform a filter function of only positive or negative filter coefficients. A pseudo-differential or differential analog IIR filter  400   a  or  400   b  can perform a filter function with both positive and negative filter coefficients. Similarly, the transconductance elements  401   a ,  405   a ,  401   b  and/or  405   b  can be fixed, programmable and adaptive. In some implementations, matching impedances can be coupled to inputs and/or outputs of the transconductance elements  401   a ,  405   a ,  401   b  and/or  405   b . In some implementations, the transimpedance element  402   a , and/or the termination impedance  404   a  can be configured to be fixed, programmable or adaptive. In some implementations, the matching impedances coupled to the inputs and/or the outputs of the transconductance elements  401   a ,  405   a ,  401   b  and/or  405   b  can be configured to be fixed, programmable or adaptive impedances. 
     For a general IIR or FIR filter, differential transconductance and transimpedance elements can be used to perform filtering with both positive and negative filter coefficients.  FIG. 5  is a schematic of an example differential analog IIR filter  500 . The differential analog IIR  500  can be a 5-tap IIR filter with 3 taps from the forward transmission line configured by forward delay element pairs  501 - 503 , and  2  taps from the feedback transmission line comprising forward delay element pairs  501 - 505 , wherein each pair of the delay elements  501 - 505  comprises two identical delay elements for a differential signal. The analog IIR filter  500  is similar to the analog filter  300  shown in  FIG. 3  except that the input signal X(tn), the output signal Y(tn), transconductance elements  510 - 515  and the transimpedance element  523  are differential to perform filtering according to the filter equation (3) from a system requirement specification 
               Y   ⁡     (   tn   )       =         ∑     i   =   0     3     ⁢       a   ⁡     (     n   ,   i     )       *     X   ⁡     (     tn   -   i     )           +       ∑     j   =   1     2     ⁢       b   ⁡     (     n   ,   j     )       *       Y   ⁡     (     tn   -   j     )       .                 
The use of differential elements can allow the implementation of filter equations with both positive and negative filter coefficients as well as for better noise immunity. By coupling differential outputs from the delay elements  510 ,  511 ,  513  and  515  and cross coupling, or swapping, differential outputs from the delay elements  502  and  504 , to differential inputs of transconductance amplifiers  512  and  514 , i.e. a positive output of a delay element couples to a negative input of a transconductance amplifier and a positive output of a delay element couples to a negative input of a transconductance amplifier, the analog IIR filter  500  can perform the above filter function with positive multiplication coefficients a(n, 0 ), a(n, 1 ), a(n, 3 ) and b(n, 2 ), and negative multiplication coefficients a(n, 2 ) and b(n, 1 ). In some implementations, the feedback filter coefficients b(n,  1 ) and b(n, 2 ) can be zero to provide an analog FIR filtering function. Therefore, the differential analog IIR filter  500  can be a differential analog FIR filter by removing a feedback loop  525 , i.e. the feed forward section  520  can be a differential analog FIR filter, i.e. an analog FIR filter is a special case of an analog IIR filter.
 
     In some implementations, each of the transconductance elements  510 - 515  can be configured to be fixed, programmable or adaptive. In some implementations, the transimpedance element  523  and/or the termination impedance  521  and  522  can be configured to be fixed, programmable or adaptive. In some implementations, matching impedances can be coupled to inputs and/or outputs of the transconductance elements  510 - 515 . In some implementations, matching impedances coupled to the inputs and/or outputs of the transconductance elements  510 - 515  can be configured to be resistors, capacitors, inductors and/or combination networks of at least one type of the resistors, capacitors, and inductors and to be fixed, programmable or adaptive impedances. In some implementations, the input signal X(tn) can be single ended but can be converted to a differential signal by the differential transconductance elements  510 - 515  and the differential transimpedance element  523  for a general filter implementation. In this case, each of the transmission line delay elements  501 - 505  can include only a single delay element. 
       FIGS. 6   a  and  6   b  show example schematics of transconductance and transimpedance circuits  600   a  and  600   b .  FIG. 6   a  shows a transconductance and transimpedance circuit  600   a  for single ended input voltage signals V in1 −V inm . The transconductance and transimpedance circuit  600   a  can be used for the analog FIR or IIR filters as described above. The transconductance and transimpedance circuit  600   a  comprises a differential input terminal  630   a , a differential output terminal  640   a , a transconductance circuit  610   a  configured to include a plurality of M differential transconductance circuit segments  603   a  and a transimpedance circuit  620   a , wherein M is an integer and the M differential circuit segments  603   a  can have corresponding transconductance Gm 1 −Gm M . Each of the transconductance Gm of a transconductance circuit segment  603   a  can be used as an absolute value of a filter coefficient and equal to 1/[(gm*Rs/2+1)*2] wherein Rs is a variable resistor and gm is the transconductance of the transistor  601   a  and equal to k p *(Wc/Lc)*(Vgs−Vt) with k p  a fabrication process dependent constant, We and Lc channel dimensions, Vgs a gate to source voltage and Vt a threshold voltage of the transistor  601 . Specifically, an absolute value of each filter coefficient, i.e. abs(c(k))=Gm k , can be used to multiply the single ended input voltage signal V ink  wherein k=0 to N and each V ink  can be an output signal from a transmission line delay element, for example, delay element  20   k  shown in FIG., delay element  30   k  or  35   k  shown in  FIG. 3 . V bias1 −V biasM  are corresponding differential bias voltages input to the differential transconductance circuit  610   a.    
     The differential transconductance circuit  610   a  further comprises a plurality of M differential transistors  601   a , a plurality of M differential current sources  602   a , and a plurality of M variable resistors  604   a  of resistances Rs 1 −Rs M , coupled between a plurality of M corresponding source terminals  611   a + and  611   a − of the M differential transistors, that can be used to tune the transconductances Gm&#39;s of the transconductance circuit segments  603   a . Herein a plurality of M differential gate terminals  630   a  of corresponding M differential transistors  601   a  can couple to a plurality of M corresponding input signals V in0 −V inm  and a plurality of corresponding M bias voltages V bias1 —V biasM  according to a sign of a corresponding filter coefficient of the general FIR or IIR filter equation. Specifically, an input signal V in  can be coupled to a positive input terminal  630   a + and a corresponding bias voltage V bias  to a negative input terminal  630   a − for the input signal V in  to be multiplied by a positive filter coefficient, and the input signal V in  can cross couple to the negative terminal  630   a - and the bias voltage V bias  to the positive terminal  630   a + for the input signal V in  to be multiplied by a negative filter coefficient. A plurality of M differential source terminals of the corresponding M differnential transistors  601   a  further couple to a negative power supply or a ground via a corresponding plurality of M differential current sources  602   a . The input signals V in1 −V inm  and the bias voltages V bias1 −V biasM  at the differential input terminals  630   a  can be multiplied by corresponding filter coefficients Gm 1 −Gm M , and converted to currents at the drain terminals of the transistors  601   a  which are connected together at nodes  607   a  as a summer to sum the output filtered currents. 
     The transimpedance circuit  620   a  comprises load resistors  606   a  of resistance R 1 , coupled between the node  605   a  and the power supply Vdd, can convert the resulting summed differential current to a differential output voltage V out+  and V out− , at a differential output terminal  640   a , wherein the positive output voltage V out+  can be at the drain of the transistors  601   a − and the negative output voltage V out−  can be at the drain of the transistors  601   a +. In other implementations, one of the differential output Vout+ and Vout− at the differential output terminal  640  can be used for a single ended output if a single ended output is desired. 
     In some implementations, the transconductance and transimpedance circuit  600   a  can be fully single ended for analog IIR or FIR filters whose filter functions can require only all positive or all negative filter coefficients. However, a full single ended FIR or FIR analog filter can have noise compared to a pseudo-differential or a full differential analog filter. The toplogy of the transconductance and transimpedance circuit  600   a  can be selected according to a system filter requirement specification. 
     The variable resistor networks  604   a  can tune the absolute values of filter coefficients, i.e. the transconductances, Gm 1 −Gm M  of the transconductance circuit  610   a  according to a system filter requirement. Each of the variable resistor Rs f −Rs M  can be a resistor network configured as, for example, a serial resistor network  611   a  or a parallel resistor network  612  with a fixed resistor of resistance Rs f  and a plurality of l switchable resistors of resistances Rs v1 −Rs vl  controlled by switches S 1  to S l  wherein l can be a positive integer. In some implementations, the resistor network  604   a  can be employed to increase a net resistance from the fixed resistor value Rs f  by switching in resistances Rs v1 −Rs vl . In other implementations, the resistor network  604   a  can decrease its net resistance Rs from the fixed resistor value Rs f  by switching in parallel resistors Rs v1 −Rs vl . For example, each Rs v  can be Rs f /2 i  for i=1−l and the total resistance Rs of the serial resistor network  604   a  can be between Rs f  and Rs f *2*(1−1/2 l-1 ). Similarly, the total resistance of the parallel resistor network  604   a  can be between Rs f  and Rs f /(2 l-1 −1). The examples are exemplary and some implementations can use resistor and inductor, resistor and capacitors or a combination of resistor, inductor and capacitor networks to tune the transconductances Gm 1 −Gm M  for a FIR or IIR filter response requirement. Therefore, the transconductance circuit  610  can be fixed, programmable or adaptive. In some implementations, additional switches can be employed to vary the load resistors  606   a  of the transimpedance circuit  620   a  for a programmable or an adaptive transimpedance circuit. In some implementations, matching impedances can be coupled to inputs and/or outptus of the transconductance circuit segments  603   a . In some implementations, matching impedances coupled to the inputs and/or the outputs of the transconductance circuit segments  603   a  can be fixed, programmable or adaptive impedances and can comprise resistors, capacitors, inductors and/or combination networks of resistors, capacitors, inductors and/or combination networks of at least two types of resistors, capacitors, and inductors. 
       FIG. 6   b  show a schematic of an example single ended transconductance and transimpedance circuit  600   b . The transconductance and transimpedance circuit  600   b  is similar to a branch of the differential transconductance and transimpedance circuit  600   a  except that the variable resistors Rs 1 −Rs M  are coupled between corresponding source terminals of transistors  610   b  and a negative power supply or a ground. The filter operation of the transconductance and transimpedance circuit  600   b  is similar to that of the transconductance and transimpedance circuit  600   a  except the values of the transconductances Gm 1 −Gm N  of the transconductance transconductance circuit segments  603   b . Each of the transconductances Gm 1 −Gm N  of a transconductance circuit element can be Gm=1/(gm*Rs/2+1) wherein gm is the transistor transconductance and Rs is a variable resistor network as described above. However, an output voltage signal −V out  at an output terminal  640   b  of the transconductance and transimpedance circuit  600   b  can be 180 degree out of phase to an input signal V in  at an input terminal  630   b . Since the transconductances Gm 1 −Gm N  can be only be positive, and a 180° phase shifter or an inverter can be needed to obtain the output signal V out  for a filter of positive filter coefficients. For an analog IIR filter of negative filter coefficients, the output signal −V out  at the output terminal  640   b  can be used. 
     Similarly, the transconductance circuit  610   b  can be fixed, programmable or adaptive. In some implementations, additional switches can be employed to vary the load resistors  606   b  of the transimpedance circuit  620   b  for a programmable or adaptive transimpedance circuit. In some implementations, matching impedances can be coupled to inputs and/or outputs of the transconductance circuit segments  603   b . In some implementations, matching impedances coupled to the inputs and the outputs of the transconductance circuit segments can be configured to comprise resistors, capacitors, inductors and/or combination networks of resistors, capacitors, inductors and/or combination networks of at least two types of resistors, capacitors, and inductors; and configured to be fixed, programmable or adaptive impedances. 
       FIG. 7  shows an example schematic of a full differential transconductance and transimpedance circuit  700  for a differential input signal V in . The transconductance and transimpedance circuit  700  is similar to the transcondcutance and transimpedance circuit  600  shown in  FIG. 6  except that the input signals V in1 −V inm  are differential signals coupled to differential input terminals  630  of the differential transcondcutance and transimpedance circuit  700  and the differential output signal V ont  can be coupled to the differential output terminal  740  of the transconductance and transimpedance circuit  700 . Similarly to the description of  FIG. 6 , a differential input signal V in + and V in − can be coupled to the positive input terminal  730 + and negative input terminal  730 −, respectively to be multiplied by a positive filter coefficient. Further, the differential input signal V in + and V in − can be coupled to the negative input terminal  730 − and positive input terminal  730 +, respectively to be multiplied by a negative filter coefficient. A filter coefficient used to multiply a differential input signal in a full differential transconductance circuit segment  703  can be 1/(gm*Rs/2+1) wherein gm is the transconductance of the transistor  701 . Similarly, the transconductance circuit  710  can be fixed, programmable or adaptive. In some implementations, additional switches can be employed to vary the load resistors  706  of the transimpedance circuit  720  for a programmable or an adaptive transimpedance circuit. In some implementations, matching impedances can be coupled to inputs and or outputs of the transconductance circuit segments  603 . In some implementations, matching impedances coupled to the inputs and/or the outputs of the transconductance circuit segments  603  can comprise resistors, capacitors, inductors and/or combination networks of resistors, capacitors, inductors and/or combination networks of at least two types of resistors, capacitors, and inductors; and be configured to be fixed, programmable or adaptive impedances. 
       FIG. 8  shows an example schematic of a differential transconductance and transimpedance circuit  800 . The differential transconductance and transimpedance circuit  800  comprises a transconductance circuit  810  which is similar to the transconductance circuit  710  shown in  FIG. 7 , and a transimpedance circuit  820  which can be a transimpedance operational amplifier to replace the load resistors  706  of resistance R 1  shown in  FIG. 7  to terminate the transconductance circuit  810  and to convert summed differential currents I out  at differential node  805  to voltage output signals V out  at output terminals of the transimpedance operational amplifier  820 , i.e. also output terminals of the transconductance and transimpedance circuit  800 . The transimpedance operational amplifier  820  can include an operational amplifier  806  configured into a feedback system with feedback impedances  807 . The transimpedance amplifier  820  can further be configured to provide additional filtering function to attenuate unwanted signals such as noises. Similarly, the transimpedance operational amplifier  820  can also replace the load resistors  606  shown in  FIG. 6  for a pseudo-differential analor IIR or FIR filters. The transconductance circuit  810  can be fixed, programmable or adaptive. In some implementations, feedback impedance  807  can be configured to be programmable or adaptive for a programmable or an adaptive transimpedance amplifier  820 . In some implementations, matching impedances can be coupled to inputs of the transconductance circuit segments  803 . In some implementations, matching impedances coupled to inputs of the transconductance circuit segments  803  can be configured to be fixed, programmable or adaptive impedances. 
       FIG. 9  is a of an example schematic of an analog FIR filter  900 . The analog FIR filter  900  is similar to the analog FIR filter  200  except the additional filter control circuit  923 . The addition of a control circuit  923  allows the filter to adaptively be changed to compensate for time-varying phenomenon. The control circuit  923  can change the filter response by tuning the filter coefficients, i.e. tuning the transconductances Gm 0 −Gm N  of the transconductance elements  910 - 91 N. For example, the tunable transconductance elements  910 - 91 N can be configured to include tunable resistor networks  611  or  612  as shown in  FIG. 6  for the transconductance and transimpedance circuits  610 ,  710  or  810  shown in  FIG. 6 ,  7  or  8 , respectively. The control circuit  923  can, for example, control the switches S 1 -S l  to switch corresponding resistors Rs v1 −Rs vl  to increase or decrease the total resistance of the resistor network  611  or  612  shown in  FIG. 6 , respectively, to tune the corresponding transconductances Gm 1 −Gm N  therefore the filter coefficients used by the transconductance elements  910 - 91 N according to a system filter response requirement specification. 
     In some implementations, the control circuit  923  can comprise a switch table for controlling the switches S 1 −S l  shown in  FIG. 6 . In some implementations, the control circuit  923  can determine the control of the switches S 1 −S l  dynamically for an adaptive transconductance circuit using information from the transconductance circuit  910 - 91 N. In some implementations, the control circuit  923  can also control the transimpedance elements  922  and/or the termination impedance  921  for fixed, programmable or adaptive transimpedance and/or termination impedance elements, respectively. In some implementations, matching impedances can be coupled to inputs and/or outputs of the transconductance circuit elements  910 - 91 N. In some implementations, matching impedances coupled to the inputs and the outptus of the transconductance elements  910 - 91 N can comprise resistors, capacitors, inductors and/or combination networks of resistors, capacitors, inductors and/or combination networks of at least two types of resistors, capacitors, and inductors; and can be configured to be fixed, programmable or adaptive impedances. 
     In some implementations, the input signal X(tn) and the output signal Y(tn) can be single ended. In some implementations, the input signal X(tn) can be single ended but the output signal Y(tn) can be differential. In other implementations, both the input and output signals are differential. In some implementations, additional switches can be employed to add or remove delay elements and corresponding transcnductance elements for more programmability of the analog filters. 
       FIG. 10  is a schematic of an example 5-tap analog IIR filter  1000  for a single ended input signal X(tn). The analog IIR filter  1000  is similar to the analog IIR filter  300  except the additional control circuit  1024  which is similar to the control circuit  923  shown in  FIG. 9 . Similarly, the filter coefficients for the transconductance elements  1010 - 1015  can be tuned by the control circuit  1024  by switching S 1 −S l  to switch corresponding resistors Rs v1 −Rs vl  in the resistor network  611  or  612  shown in  FIG. 6  to tune transconductors Gm 0 −Gm 5  of corresponding transconductance elements  1010 - 1015 . 
     In some implementations, the control circuit  1024  can comprise a switch table for controlling the switches S 1 −S l  shown in  FIG. 6 . In some implementations, the control circuit  1024  can determine the control of the switches S 1 −S l  dynamically for an adaptive transconductance elements  1010 - 1015 . In some implementations, additional switches can be employed to add or remove delay elements and corresponding transconductance elements for more programmability of the analog filters. In some implementations, the control circuit can also control the transimpedance elements  1023  and/or the termination impedance elements  1021  and  1022  for fixed, programmable or adaptive transimpedance and/or termination impedance elements, respectively. In some implementations, matching impedances can be coupled to inputs and/or outputs of the transconductance elements  1010 - 1015 . In some implementations, matching impedances coupled to the inputs and/or the of the transconductance elements  1010 - 1015  can comprise resistors, capacitors, inductors and/or combination networks of resistors, capacitors, inductors and/or combination networks of at least two types of resistors, capacitors, and inductors; and can be configured to be fixed, programmable or adaptive impedances. 
       FIG. 11  is a schematic of an example full differential 5-tap analog IIR filter  1100  with differential input signals X(tn). The differential analog IIR filter  1100  is similar to the analog IIR filter  500  shown in  FIG. 5  except the additional control circuit  1124  which is similar to the control circuit  1124  shown in  FIG. 1100 . Similarly, the filter coefficients for the differential transconductance elements  1110 - 1115  can be tuned by the control circuit  1124  by switching S 1 −S l  to switch corresponding resistors Rs v1 −Rs vl  in the resistor network  611  or  612  shown in  FIG. 6  to tune transconductors Gm 0 −Gm 5  of corresponding transconductance elements  1110 - 1115 . 
     In some implementations, the control circuit  1124  can comprise a switch table for controlling the switches S 1 −S l  shown in  FIG. 6 . In some implementations, the control circuit  1124  can determine the control of the switches S 1 −S l  shown in  FIG. 6  dynamically. In some implementations, additional switches can be employed to add or remove delay elements and corresponding transcnductance elements for more programmability of the analog filters. In some implementations, matching impedances can be coupled to inputs of the transconductance elements  1110 - 1115 . In some implementations, matching impedances can be coupled to inputs of the transconductance elements  1110 - 1115 . In some implementations, matching impedances coupled to inputs of the transconductance elements  1110 - 1115  can be fixed, programmable or adaptive. In some implementations, the transimpedance element  1123  and/or the termination impedances  1121  and  1122  can be fixed, programmable or adaptive. 
       FIG. 12  shows a floor plan of an example 3-tap analog FIR filter  1200  with a single transmission delay line chip layout using a snaked configuration to reduce chip area. The analog FIR filter  1200  is similar to the analog FIR filter  200  for the tap number N=3. The transmission line delay elements. The transmission line delay elements can be, for example, configured in the snaked structure  1201 - 1203  for a microstrip  1204  which is a planar transmission line structure. The lossless unit characteristic impedance Z o  can be a function of an effective dielectric constant ∈ e  and geometry dimensions of the microstip line  1204 , i.e. Z o =f(W,d)/∈ e   1/2 , wherein W is the width and d is the depth of the microstrip line  1204 . The microstrip  1204  can be particularly useful for implementing an analog IIR or FIR filters in an integrated circuit. In some implementations, a coaxial line, a two-wire line, a plate line such as a stripline, a microstrip or a plate waveguide, bond wire, printed circuit board, can be used to implement the delay elements. A delay time td of a transmission line delay element can be designed using the characteristic impedance and according to a system filter requirement specification. 
       FIG. 13  shows a schematic of an example analog equalizer circuit  1300  employing analog filter techniques described above. The analog equalizer circuit  1300  can be a 1-tap FIR filter and comprises two differential transconductance elements  1310 , a transimpedance element  1320  including a load resistor R  1306 , a transmission delay element  1307 , a termination impedance  1308 , an input terminal  1330  and an output terminal  1340 . The transconductance element  1320  can further comprise a differential transistor pairs  301  and  302 , differential current source pairs  303  and  304 , and variable resistor networks Rs 1 −Rs 3 . The analog equalizer  1300  circuit can be configured to represent a FIR filter equation (4) of V out (tn)=a(n, 1 )*V in (tn)+a(n−1,2)*V in (tn−1) wherein V in (tn) and V out (tn) are single ended input and output signals, respectively, V in (tn−1) is a time-delayed input signal, a(n, 1 ) can have a positive value and a(n, 2 ) can have a negative value. Using the disclosed techniques described above, the input signal V in (tn) can couple to a positive input terminal of  1330  the time-delayed input signal V in (tn−1) cross coupled to a negative input terminal of  1335 . The output signal V out (tn) can perform filtering function described in the filter equation (4) which can represent a differentiator or an analog high pass filter. This analog high pass filter for the filter equation (4), i.e. the analog equalizer  1300 , can be used as an equalizer to equalize a signal degraded as filtered by a low pass filter. 
       FIG. 14  shows a diagram of an example of an equalized high speed backplane system  1400 . The backplane channel  1412  can act like a low pass filter  1401 . A transmit signal  1410  can be buffered by a buffer  1411  then can propagate through a backplane  1412  which can degrade the transmitted signal with a low pass filtering function into a degraded signal  1413 . An analog receive equalizer  1414  can use a high pass filter  1402  such as the analog equalizer circuit  1300  shown in  FIG. 13  to equalize the backplane low pass filter  1401  into a all pass filter  1403  and to restore the degraded signal  1413  to the transmit signal  1410 . 
     In some implementations, the analog equalizer  1414  can be single ended. In some implementations, the analog equalizer  1414  can be differential. In some implementations, the analog equalizer can be fixed. In some implementations, the analog equalizer can be programmable or adaptively controlled. In some implementations, the analog equalizer  1414  can be replaced at the receiving side of the backplane as a receive equalizer. In other implementations, the analog equalizer can be a transmit equalizer. However, the receive equalizer can provide better equalization functions than the transmit equalizer. 
     The analog IIR and FIR filters using techniques of this disclosure can be implemented for low-pass, high-pass, notch, band-pass and/or band-stop filters or equalizers with very good linearity in high speed system for frequencies around 1 GHz to above 100 GHz, for example, applications in high speed backplanes, disk drives, optical systems, video systems, wireline and wireless communication systems. These and other implementations can optionally include one or more of the following features. The system can include any combination of one or more components for receivers, transmitters, and transceivers, in which an analog IIR or FIR filter can be coupled to any of the one or more components or their sub-components. Control circuits may include a digital circuit or a microprocessor. Any of the methods, designs, and techniques described herein can also be implemented in a system, an apparatus, a printed circuit board, a circuit, device, a machine, or in any combination thereof. 
     In some implementations, the positions of switches, resistors, or other components can be exchanged from the disclosed figures with minimal change in circuit functionality. Various topologies for circuit models can also be used, other than what is shown in the figures. The exemplary designs shown are not limited to CMOS process technology, but may also use other process technologies, such as BiCMOS (Bipolar-CMOS) process technology, or Silicon Germanium (SiGe) technology. In some implementations, switches can be implemented as transmission gate switches. The circuits can be single-ended or fully-differential circuits. 
     Those skilled in the art will recognize that certain modifications to the intended patent are intended to be within the scope of this patent. These include additional components added to the inputs of the transconductance amplifiers to improve input matching. Those skilled in the art will also recognize that the invention does not depend on the type of transmission line, or if it is implemented on the chip substrate, the package substrate, or the printed circuit board. Those skilled in the art will also recognize that there are many variations of transconductance and transimpedance amplifiers, and that there can be multiple stages of amplification and conversions between voltage and current that are intended to be within the scope of this patent. Those skilled in the art will recognize that other embodiments that utilize the transmission line as a delay element in their filters, including finite impulse response (FIR) or infinite impulse response (IIR) filters, or feed-forward equalization (FFE) filters or decision feedback equalization (DFE) filters, are intended to be within the scope of this patent. These and other modifications, which are obvious to those skilled in the art, are intended to be included within the scope of the present invention. Other implementations are within the scope of the following claims.