Abstract:
Present software-defined radios (SDR) employ front end circuits that contain multiple receivers and transmitters for each band of interest, which is inflexible, expensive and power inefficient. A programmable front end circuit is implemented on a CMOS device and is configurable to transmit and receive signals in a wide band of frequencies, thereby providing an adaptable transmitter and receiver operable with current and future wireless networking technologies.

Description:
RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 61/360,579, filed on Jul. 1, 2010. The entire teachings of the above application are incorporated herein by reference. 
    
    
     BACKGROUND 
     A typical mobile phone handset includes a CMOS front end configured for operating with 3G or 4G transmit and receive frequencies. It contains four receivers, each covering a band allocated for cellular service. The two transmitters cover the corresponding bands. Beyond telecommunications transceivers, the phone typically contains separate Bluetooth, WiFi and GPS receivers, which add significantly to cost and consume a substantial amount of power. To address this problem, recently released front-end integrated circuits (ICs) include integrated GPS and WiFi transceivers. However, even with integrated receivers, front end ICs still require large, expensive and power intensive A/D converters and DSPs. Among the deficiencies of this architecture, it is not adaptive to new frequency allocations. New front end ICs must be developed to incorporate hardware changes to receiver and transmitter structures as services and frequency allocations evolve. Moreover, devices are unable to operate across different standards/geographies without redundant hardware, and adding incremental receiver and transmitter structures increases power consumption and cost. 
     As high data rate services become ubiquitous, power consumption and cost will increase greatly, as the digital components required for such services are prohibitively expensive and draw down power quickly. In a market calling for efficiency and low cost, the current mobile handset architecture is pushing the technology in the opposite direction. 
     SUMMARY 
     Embodiments of the invention provide a wideband programmable software defined radio (SDR) front end circuit. In an example embodiment, the front end circuit includes a frequency synthesizer to provide a clock signal having a variable frequency, a transmit path and a receive path. The transmit path includes a first anti-aliasing filter for receiving an analog signal, an upconverter to upconvert an output of the anti-aliasing filter according to the clock signal, and a first programmable bandpass filter to filter an output of the upconverter. The receive path includes a second programmable bandpass filter to filter a received signal, a downconverter to downconvert an output of the anti-aliasing filter according to the clock signal, and a second programmable anti-aliasing filter to filter an output of the downconverter. 
     In further embodiments, the front end circuit may further comprise a programmable transversal filter to provide echo cancellation of the transmit signal from the received signal. A combiner circuit may be configured to combine the received signal with an output of the transversal filter. A balanced hybrid circuit may be configured at both the transmit path and receive path, the hybrid circuit providing electrical symmetry between the transmit path and the receive path and limit transfer of energy from the transmit path to the receive path. Further, a roofing filter may be configured in the receive path to limit a frequency of the received signal to a selected bandwidth. 
     In still further embodiments, the various components of the SDR front end may include at least one programmable biquad circuit or a state variable filter. The biquad circuit may include at least one attenuator, integrator and summer. 
     The biquad attenuator may include a plurality of attenuator blocks, where each block comprises a first switch connected between a signal rail and an output node, a second switch connected between an offset rail and the output node, and a resistive element connected in series between the output node and the first and second switches. 
     The biquad integrator may include first and second p-channel transistors including respective sources coupled in parallel to a first voltage supply terminal and respective drains configured to provide complementary output signals. The integrator may further include first and second variable resistors including respective first terminals coupled to the drains of the first and second p-channel transistors, respectively, and respective second terminals coupled to gates of the second and first p-channel transistors, respectively. Lastly, the integrator may include first and second re-channel transistors including respective drains coupled to the second terminals of the first and second variable resistors, respectively, respective gates configured to receive complementary input signals, and respective sources in electrical communication with a second voltage supply terminal. 
     The biquad summer may include a plurality of N switches connected in parallel between an output node and a ground rail, a resistive element connected in series between a source rail and the output node, each of the switches being controlled by a respective voltage input. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing will be apparent from the following more particular description of example embodiments of the invention, as illustrated in the accompanying drawings and appended slides in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments of the present invention. 
         FIG. 1  is a block diagram of a prior art CMOS front end IC in a typical mobile phone handset. 
         FIG. 2  is a block diagram of an example CMOS front end IC according to one embodiment. 
         FIG. 3  shows first- and second-order canonical forms of a state variable filter. 
         FIG. 4  shows implementation of an arbitrary transfer function  FIG. 5  is a plot of magnitude response of a realistic integrator. 
         FIG. 6  illustrates pole shifting due to the non-ideal integrator response. 
         FIG. 7  shows pole plots for the baseband and RF signals. 
         FIG. 8  is a plot of frequency response of a high-Q bandpass filter. 
         FIG. 9  is a block diagram of a biquad circuit. 
         FIG. 10  is a block diagram of a plurality of cascaded biquad circuits. 
         FIG. 11  is a circuit diagram of an integrator circuit. 
         FIGS. 12A-B  are plots illustrating magnitude and phase response of an integrator circuit. 
         FIG. 13  is a diagram illustrating operation of an attenuator circuit. 
         FIG. 14  is a circuit diagram of a 12-bit attenuator. 
         FIGS. 15A-B  are plots illustrating magnitude and phase response of an attenuator circuit. 
         FIG. 16  is a circuit diagram of a summer circuit. 
         FIGS. 17A-B  are plots illustrating magnitude and phase response of a summer circuit. 
     
    
    
     DETAILED DESCRIPTION 
     A description of example embodiments of the invention follows. The teachings of all patents, published applications and references cited herein are incorporated by reference in their entirety. 
     A software-defined radio (SDR) system is a radio communication system that implements in software components that have been typically implemented in hardware, such as filters, amplifiers, mixers and modulators. The software components are typically implemented on embedded computing devices or a personal computer. 
       FIG. 1  illustrates a typical CMOS radio-frequency (RF) front end of a mobile phone handset configured for operating with 3G or 4G transmit and receive frequencies. It contains four receivers, each covering a band (1900 MHz, 1800 MHz, 900 MHz, 850 MHz) allocated for cellular service. The two transmitters cover the corresponding bands. Beyond telecommunications transceivers, the phone typically contains separate Bluetooth, WiFi and GPS receivers, which add significantly to cost and consume a substantial amount of power. 
     SDR technology has matured considerably in the back end (consisting of digital circuits and the software), but the RF front end has not kept pace. This is because of the fragmented frequency allocation for specific given service. For example, the frequency allocation for 4G is shown in Table 1. The table includes only a few operating bands for illustration. 
     
       
         
               
             
               
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Selected frequency allocations for LTE (Long Term Evolution) 
               
             
          
           
               
                   
                 Uplink (MHz) 
                 Downlink (MHz) 
                   
               
               
                   
                 BS 1  receive 
                 BS transmit 
                   
               
               
                 Operating 
                 UE 2  transmit 
                 UE receive 
                 Duplex 
               
             
          
           
               
                 Band 
                 F UL _low 
                 F UL _high 
                 F DL _low 
                 F DL _high 
                 Mode 
               
               
                   
               
             
          
           
               
                 1 
                 1,920 
                 1,980 
                 2,110 
                 2,170 
                 FDD 
               
               
                 2 
                 1,850 
                 1,910 
                 1,930 
                 1,990 
                 FDD 
               
               
                 3 
                 1,710 
                 1,785 
                 1,805 
                 1,880 
                 FDD 
               
               
                 4 
                 1,710 
                 1,755 
                 2,110 
                 2,155 
                 FDD 
               
               
                 5 
                 824 
                 849 
                 869 
                 894 
                 FDD 
               
               
                 6 
                 830 
                 840 
                 875 
                 885 
                 FDD 
               
               
                 7 
                 2,500 
                 2,570 
                 2,620 
                 2,690 
                 FDD 
               
               
                 8 
                 880 
                 915 
                 925 
                 960 
                 FDD 
               
               
                 9 
                 1,749.9 
                 1,784.9 
                 1,844.9 
                 1,879.9 
                 FDD 
               
               
                 10 
                 1,710 
                 1,770 
                 2,110 
                 2,170 
                 FDD 
               
               
                 11 
                 1,428 
                 1,447.9 
                 1,475.9 
                 1,495.9 
                 FDD 
               
               
                 12 
                 698 
                 716 
                 728 
                 746 
                 FDD 
               
               
                 17 
                 704 
                 716 
                 734 
                 746 
                 FDD 
               
               
                 40 
                 2,300 
                 2,400 
                 2,300 
                 2,400 
                 TDD 
               
               
                   
               
               
                   1 Base Station 
               
               
                   2 User Equipment - handset is an example of UE 
               
             
          
         
       
     
     As shown in  FIG. 1 , the prior art front end employs multiple receivers and multiple transmitters to cover the bands in the above table. This translates to high power consumption, cost and size. 
     Digital devices such as field programmable gate array (FPGA) and analog-to-digital converters (ADC) cannot operate over large bandwidths or data rates because of the Nyquist criteria. Their power consumption increases with the sampling rate. This translates to expensive thermal management solutions to lower the junction temperature of the device and improve its reliability. For instance, a decrease of 10° C. in operating temperature of the device doubles its component life. High power consumption also translates to implementation cost which is typically between $0.50 and $1.00 per Watt. Large number of gates in the FGPA contributes to propagation delay. 
     Embodiments of this invention address two problems faced by the next generation of handsets operating from 400 MHz to 6 GHz. Handsets operate across multiple bands and offer wideband services. Therefore, the prior art CMOS IC contains multiple receivers and transmitters for each band of interest. Such architecture is expensive and inflexible to accommodate future frequency allocations. In addition, processing high data rate at the Nyquist rate consumes considerable power. This translates to high cost, poor device reliability and short battery life. 
     Comparable prior art analog signal processors are either narrow band or adopt stochastic implementation, and fail to deliver a wideband and deterministic analog signal process. Such processors cannot provide an adaptive filter from 400 MHz to 6 GHz, and a prior art switched capacitor filter does not offer wideband coverage and suffers from problems caused by the presence of the periodic clocking signal. 
       FIG. 2  is a top-level block diagram of a wideband programmable SDR front end in one embodiment of the present invention. The front end may be implemented on a CMOS and SiGe device for low transmit power levels. For high transmit power levels, some components may be located off-chip. Embodiments of the SDR front end can be configured for use in a software defined radio, a spectrum analyzer, an early warning radar system, or in any other application where wideband filtering and signal processing is required. Other embodiments can be used in handsets for cellular telephone use. Operation of the front end is described below. 
     In the transmit path, an anti-aliasing filter  1  receives an input signal from a digital-to-analog converter (DAC) or I/Q DACSs (not shown). The output from the anti-aliasing filter drives an upconverter mixer  3  where the local oscillator frequency is provided by a frequency synthesizer  2 . A programmable band pass filter  4  rejects unwanted sideband and harmonic content from the output of the upconverter  3 . The passband characteristics of this filter can be changed under software control such that it offers low loss insertion loss to the transmit frequency, which is determined by the synthesizer  2 . 
     The output of the bandpass filter  4  drives an amplifier driver  5 , which typically operates in the linear region and, therefore, does not contribute to the spectral regrowth or non linearization of the amplifier driver. The output from the amplifier driver  5  drives the power amplifier  7 , which may be implemented as a component external to the front end. 
     The balanced hybrid circuit  9  is an external component configured to limit the bandwidth of the signal. It offers a low loss path from the output of the power amplifier  7  to the antenna  8  and high insertion loss (or high isolation) to the path from the power amplifier to the combiner  11 . The balanced hybrid  9  offers a low insertion path to the received signal from the antenna to the combiner. The balanced hybrid  9  may not provide acceptable isolation of the transmit power leaking into the low noise amplifier  11 . An isolation of about 70 dB is required, whereas the balanced hybrid may provide an isolation of about 20 to 30 dB across the operating frequency range. 
     Accordingly, additional isolation is provided by utilizing the coupler  6 , transversal filter  10  and combiner  11 . The coupler  6  couples the transmitted signal to the transversal filter  10 . The transversal filter  10  also receives input from the output of the roofing filter  12 . This received signal also contains the transmit signal leaking into the receiver. The transversal filter  12  correlates the two inputs, and adapts the coefficients of the transversal filters to make the correlation disappear. It creates an estimation of the transmit signal leak into the receive signal, which the combiner  11  then combines with the hybrid  9  output to cancel the leak. The transversal filter  10  functions in continuous time, and the cancellation of the transmit power into the receiver also occurs in continuous time. 
     In the receive path, a low noise amplifier  13  is a wide band device that operates across the operating range of the SDR front end. Therefore, it is susceptible to jamming from broad band noise of incoming signals at the antenna  8 . The purpose of the roofing filter  12  is to prevent this from occurring. The roofing filter  12  has a fixed bandwidth of 40 MHz to 80 MHz, but this bandwidth (or the center frequency of the roofing filter) can be moved to any location in the operating range of the SDR front end. Therefore, at any given time, the low noise only noise amplifier only ‘sees’ a selected signal spectrum, the spectrum being selected according to the received signal frequency. 
     A programmable bandpass filter  14  further reduces the bandwidth of the incoming signal to the channel bandwidth of the desired receive signal. A downconverter mixer  15  downconverts the incoming RF signal from the low noise amplifier  13  to base band. The local oscillator frequency is provided by the synthesizer  2 . The programmable anti-aliasing filter  16  is a low pass filter, the filter transfer characteristics of which can be changed under software control based on data rate and presence of interferer in the base band. The filter  16  is provided to maximize the sensitivity of the analog-to-digital converter. 
     The architecture and operation of the components of the SDR front end of  FIG. 2  are described in further detail below, with reference to  FIGS. 3-17 . Additional description of these components may be found in International Application PCT/US11/24542, the entirety of which is incorporated herein by reference. 
     Analytical Framework: The State Variable Filter 
     Wideband Signal Processing (WiSP) is analog signal processing technology; that is, it implements programmable and executable analog computing over bandwidths from 50 MHz to 20 GHz or more. The underlying basis of WiSP is state variable theory, which, when combined with CMOS deep sub-micron technology, makes it possible to extend low-frequency signal processing techniques to micron and millimeter wavelengths. WiSP may be realized in complementary-metal-oxide-semiconductor (CMOS), silicon germanium (SiGe) technology, and silicon-on-insulator (SOI) technology. 
     WiSP is highly accurate because parameters of the state variable machines can be set to 10 bits of accuracy. WiSP is also frequency agile, as changing state variable parameters, such as gain, makes it possible to span the whole frequency band. For example, a state variable machine that is centered about a frequency of 1 GHz may be shifted to a frequency of 10 GHz just by changing the gain parameters. WiSP technology is suitable for both linear time invariant signal processing and time variant signal processing. State variable systems can be used in single input/output mode and in multiple input/multiple output (MIMO) mode—for example, in mimicking a MIMO wireless antenna system. 
       FIG. 3  is a block diagram of first- and second-order canonical forms of a state variable filter (SVF). The SVA structure may be implemented in wideband signal processing. In particular, it may be configured to operate as one or more of the components in the front end circuit of  FIG. 2 , including the anti-aliasing filters, the bandpass filters, and the transversal filter. Such configuration is described below. When configuring the filter, it is an objective to implement an impulse response y(t) or equivalently a transfer function of the form 
                       T     m   ⁢           ⁢   n       ⁡     (   s   )       =             B   m     ⁢     s   m       +       B     m   -   1       ⁢     s     m   -   1         +   …   +     B   0           s   n     +       A     n   -   1       ⁢     s     n   -   1         +   …   +     A   0         ⁢           ⁢     (     m   &lt;   n     )               (   1   )               
that approximates Y(s), the Laplace transform of y(t) or the desired transfer function. Taking a partial fraction expansion of (1) followed by an inverse Laplace transform allows any temporal function to be approximated by a linear combination of complex sinusoids:
 
     
       
         
           
             
               
                 
                   
                     
                       T 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             R 
                             i 
                           
                           
                             s 
                             - 
                             
                               p 
                               i 
                             
                           
                         
                       
                       ⇔ 
                       
                         
                           y 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                       ⁢ 
                       
                         
                           R 
                           i 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               p 
                               i 
                             
                             ⁢ 
                             t 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where p i  and R i  are the ith pole and its corresponding residue. This approximation can be made to an arbitrary degree of accuracy by adding additional terms in the summation. 
     The real pole/residue pairs in (2) are realized using the first-order canonical form structure shown in  FIG. 3(   a ). The complex pole/residue pairs occur as complex conjugates and are combined as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             T 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             s 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               R 
                               i 
                             
                             
                               s 
                               - 
                               
                                 p 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               R 
                               i 
                               * 
                             
                             
                               s 
                               - 
                               
                                 p 
                                 i 
                                 * 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   Re 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       R 
                                       i 
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 s 
                               
                               - 
                               
                                 2 
                                 ⁢ 
                                 
                                   Re 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         p 
                                         i 
                                       
                                       ⁢ 
                                       
                                         R 
                                         i 
                                         * 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 s 
                                 2 
                               
                               - 
                               
                                 2 
                                 ⁢ 
                                 
                                   Re 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       p 
                                       i 
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 s 
                               
                               + 
                               
                                 
                                    
                                   
                                     p 
                                     i 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           = 
                           
                             
                               
                                 
                                   b 
                                   1 
                                 
                                 ⁢ 
                                 s 
                               
                               + 
                               
                                 b 
                                 0 
                               
                             
                             
                               
                                 s 
                                 2 
                               
                               + 
                               
                                 
                                   a 
                                   1 
                                 
                                 ⁢ 
                                 s 
                               
                               + 
                               
                                 a 
                                 0 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where all the coefficients (b&#39;s and a&#39;s) are real. These conjugate pairs can thus be realized using the second-order observer canonical form structure shown in  FIG. 3(   b ). Note that an extra b 2  block, corresponding to a term b 2 s 2  in the numerator, is added in  FIG. 3(   b ) to account for cases like a bandstop notch filter of the form: 
                       T   NF     ⁡     (   s   )       =         s   2     +     ω   r   2           s   2     +       ω   r     ⁢     s   /   Q       +     ω   r   2                 (   4   )               
Such first- and second-order structures can then be combined and their outputs summed to realize T mn (s) as shown in  FIG. 4 .
 
     The architecture shown in  FIG. 4  can be realized by implementing various filters, for example, bandpass elliptic filters for channel selection (in MATLAB and/or Electronic Workbench) to validate the analysis. 
     Cure of Non-Idealities: Pre-Compensation 
     Ideally the integrator block in the previous section has a frequency response of 1/s (i.e., “linear” magnitude response when drawn in the log-log scale). However the magnitude response of a realistic integrator is usually not linear for all frequencies, rather it presents characteristics as shown in  FIG. 5  (two-pole model). Instead of 1/s, the transfer function of such an integrator has the following form 
                         T   ^     int     ⁡     (   s   )       =       1     s   +     ω   a         ·       ω   b       s   +     ω   b                   (   5   )               
where ω a  and ω b  are poles usually caused by the intrinsic resistances and capacitances of MOS FETs. Correspondingly, the ith term of equation (2) becomes
 
                               T   ^     i     ⁡     (   s   )       =         R   i     ·     1     s   +     ω   a         ·       ω   b       s   +     ω   b             1   -       p   i     ·     1     s   +     ω   a         ·       ω   b       s   +     ω   b                           =         R   i     ⁢     ω   b           s   2     +       (       ω   a     +     ω   b       )     ⁢   s     +       (       ω   a     -     p   i       )     ⁢     ω   b                         (   6   )               
which in general leads to two poles as shown in  FIG. 5 :
 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       ^ 
                     
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   , 
                   
                     
                       
                         p 
                         ^ 
                       
                       
                         i 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     = 
                     
                       - 
                       
                         
                           
                             
                               ω 
                               a 
                             
                             + 
                             
                               ω 
                               b 
                             
                           
                           2 
                         
                         [ 
                         
                           1 
                           ∓ 
                           
                             
                               1 
                               - 
                               
                                 
                                   4 
                                   ⁢ 
                                   
                                     
                                       ω 
                                       b 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           ω 
                                           a 
                                         
                                         - 
                                         
                                           p 
                                           i 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         a 
                                       
                                       + 
                                       
                                         ω 
                                         b 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     In  FIG. 6 , the black crosses (x) represent the desired poles p i  and the red crosses (x) represent the actual poles ({circumflex over (p)} i1  and {circumflex over (p)} i2 ) that can be achieved when a non-ideal integrator is used. Notice that {circumflex over (p)} i1  and {circumflex over (p)} i2  are symmetric around the line Re[s]=−ω a +ω b )/2. 
     With the two poles defined in equation (7), a partial expansion of equation (6) can be readily obtained as 
                         T   ^     i     ⁡     (   s   )       =         R   ^     i     ⁡     (       1     s   -       p   ^       i   ⁢           ⁢   1           -     1     s   -       p   ^       i   ⁢           ⁢   2             )               (   8   )               
the new residue is given by
 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       ^ 
                     
                     i 
                   
                   = 
                   
                     
                       
                         
                           R 
                           i 
                         
                         ⁢ 
                         
                           ω 
                           b 
                         
                       
                       
                         
                           
                             p 
                             ^ 
                           
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         - 
                         
                           
                             p 
                             ^ 
                           
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           R 
                           i 
                         
                         ⁢ 
                         
                           ω 
                           b 
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   ω 
                                   a 
                                 
                                 + 
                                 
                                   ω 
                                   b 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             4 
                             ⁢ 
                             
                               
                                 ω 
                                 b 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ω 
                                     a 
                                   
                                   - 
                                   
                                     p 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     When ω b  is much larger than ω a  and |p i |, {circumflex over (p)} i1  is usually close to the original pole p i  whereas {circumflex over (p)} i2  is far away and negligible. In order to correct for the deviation due to such a non-ideality, we can preset the pole (denoted by p i,pre ) such that {circumflex over (p)} i1  becomes exactly the desired pole p i . That is 
                     -           ω   a     +     ω   b       2     [     1   -       1   -       4   ⁢       ω   b     ⁡     (       ω   a     -     p     i   ,   pre         )             (       ω   a     +     ω   b       )     2             ]       =     p   i             (   10   )               
which leads to:
 
                     p     i   ,   pre       =       ω   a     +           ω   a     +     ω   b         ω   b       ⁢     p   i       +       1     ω   b       ⁢     p   i   2                 (   11   )               
With such a preset pole, clearly {circumflex over (p)} i1 =p i . Correspondingly we have {circumflex over (p)} i2 =−(ω a +ω b )−p i . So the actual implementation of the transfer function becomes (substituting into equation (8)):
 
     
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         ^ 
                       
                       i 
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           R 
                           i 
                         
                         ⁢ 
                         
                           ω 
                           b 
                         
                       
                       
                         
                           2 
                           ⁢ 
                           
                             p 
                             i 
                           
                         
                         + 
                         
                           ω 
                           a 
                         
                         + 
                         
                           ω 
                           b 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           1 
                           
                             s 
                             - 
                             
                               p 
                               i 
                             
                           
                         
                         - 
                         
                           1 
                           
                             s 
                             + 
                             
                               ( 
                               
                                 
                                   ω 
                                   a 
                                 
                                 + 
                                 
                                   ω 
                                   b 
                                 
                               
                               ) 
                             
                             + 
                             
                               p 
                               i 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In order to obtain the desired transfer function T i (s), the residue R i  has to be also preset. It can be easily seen that replacing R i  by R i,pre =R i (2p i +ω a +ω b )/ω b  makes the first term right, as presented 
                         T   ^     i     ⁡     (   s   )       =             R   i       s   -     p   i         -       R   i       s   +     (       ω   a     +     ω   b       )     +     p   i           ⇔         y   ^     i     ⁡     (   t   )         =         R   i     ⁢     ⅇ       p   i     ⁢   t         -       R   i     ⁢     ⅇ       -     (       ω   a     +     ω   b     +     p   i       )       ⁢   t                     (   13   )               
When ω b  is much greater than ω a  and |p i |, the second term on the right-hand side is negligible and the desired transfer function is obtained.
 
     Above derivations are only valid for a simple two-pole model. In a real CMOS design, it is not unusual for multiple poles to exist. Furthermore, both left- and right-plane zeros may exist. 
     CMOS Implementation 
     The SDR front-end architecture in  FIG. 2  allows for the creation of a CMOS chip containing a large number of first- and second-order blocks, whose coefficients can be programmed to realize any transfer function to a specifiable degree of accuracy. The first-order block is actually a special case of the second-order block (when b 2 =b 0 =a 0 =0). Because it is extremely rare for multiple real poles to appear in a realistic filter design (in nearly every case, zero or one first-order block is required), only the second-order blocks (which is referred to as the biquad herein) may be necessary. 
       FIG. 8  is a plot illustrating calculated pre-layout simulation results, including frequency response (magnitude and phase), of a high-Q bandpass filter. A UWB RFID transceiver may operates in the frequency range from 3 GHz to 10 GHz. Only single-biquad filters are built for the purpose of identifying and notching out in-band interferers and they are optimized to have high quality values for use in the UWB band only. Example filters carry this research further, solving challenging design issues that arise when implementing systems that require multiple biquads. 
     An integrator, which is the core component of the biquad circuit, follows the feedforward-regulated cascode operational transconductance amplifier structure. The attenuator (for a and b coefficients) is based on the standard R-2R ladder network that tends to lose accuracy and bandwidth when large source impedance presents. 
     Components of the SDR front-end of  FIG. 2  include programmable processors consisting of multiple second-order differential equation engines, each employing an architecture including one or more state variable filters or biquad circuits. Such architecture and related circuitry may be found in U.S. Pub. No. 2011/0051782, the entirety of which is incorporated by reference. This architecture can implement any desired impulse response or transfer function to a specifiable degree of accuracy by invoking more or fewer engine blocks. Each engine is an analog block containing programmable components, whose parameters are set, controlled, and optimized through algorithms running on a low data rate wideband DSP in the control path to 12 bit accuracy. These engines are implemented as biquad circuits, which are themselves composed of three smaller circuit types: the Integrator, Attenuator, and Summer. 
       FIG. 9  is a block diagram of a biquad circuit. As with the state variable filter of  FIG. 3 , the biquad circuit may be configured to operate as one or more of the components in the front end circuit of  FIG. 2 , including the anti-aliasing filters, the bandpass filters, and the transversal filter. Such configuration is described below. A mapping exists between the coefficients of the biquad (a&#39;s and b&#39;s) and the coefficients of the desired transfer function, while the gain (G) scales the transfer function in frequency. Therefore, adjusting coefficients and the gain parameter alters the filter shape, bandwidth, and center frequency. 
     The fundamental components of the biquad are the integrator, attenuator, and summer. A broadband self-tuned integrator using feedforward-regulated topology has been constructed. This integrator has a high bandwidth, high linearity and low intermodulation distortion which make it suitable especially for applications at microwave frequencies. The characteristics of the biquad output y(t) can be changed by altering the transfer function, T(s), of the biquad. This is achieved by changing the values of the attenuators, which are comprised of a 0 , a 1 , b o , b 1  and b 2 . The center frequency of the transfer function is swept by changing the gain G of the integrators. The values of the attenuators and the gain of the integrators may be digitally controlled by a Serial Peripheral Interface (SPI), which has 12-bit accuracy. 
       FIG. 10  is a block diagram of a plurality of biquad circuit in a cascaded configuration, demonstrating how the biquad acts as a fundamental building block whose series constructions implement circuits of arbitrary function and order. This architecture allows for a programmable and dynamic implementation of any specified filter and transfer function. 
       FIG. 11  is a circuit diagram of an integrator circuit illustrates an integrator circuit that may be implemented in the biquad circuits of  FIGS. 9 and 10 . The integrator is a broadband self-tuned integrator using feedforward-regulated topology. The integrator has a high bandwidth, high linearity and low intermodulation distortion which make it suitable especially for applications at microwave frequencies. 
     Small signal analysis shows that the transfer function of this circuit is of the form: 
               T   ⁡     (   s   )       =         Ω   0     ·     (       s   /     z   1       -   1     )     ·     (       s   /     z   2       -   1     )           s   ·     (       s   /   p     +   1     )       +     a   0               
where
 
               z   1     =       g     m   ⁢           ⁢   1         C     g   ⁢           ⁢   d   ⁢           ⁢   1                       z   2     =       (       g     m   ⁢           ⁢   3       +     1   /   R       )       C     g   ⁢           ⁢   d   ⁢           ⁢   3                       Ω   0     =         g     m   ⁢           ⁢   1       ·     (       g     m   ⁢           ⁢   3       +     1   /   R       )                   (       C     g   ⁢           ⁢   s   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   1       +     C     d   ⁢           ⁢   s   ⁢           ⁢   1         )     ·     (       g     d   ⁢           ⁢   s   ⁢           ⁢   3       +     1   /   R       )       +                   (       C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     d   ⁢           ⁢   s   ⁢           ⁢   3         )     ·     (       g     d   ⁢           ⁢   s   ⁢           ⁢   1       +     1   /   R       )       +       C     g   ⁢           ⁢   d   ⁢           ⁢   3       ·     (       g     m   ⁢           ⁢   3       +     2   /   R       )                             p   =                 (       C     g   ⁢           ⁢   s   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   1       +     C     d   ⁢           ⁢   s   ⁢           ⁢   1         )     ·     (       g     d   ⁢           ⁢   s   ⁢           ⁢   3       +     1   /   R       )       +                   (       C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     d   ⁢           ⁢   s   ⁢           ⁢   3         )     ·     (       g     d   ⁢           ⁢   s   ⁢           ⁢   1       +     1   /   R       )       +       C     g   ⁢           ⁢   d   ⁢           ⁢   3       ·     (       g     m   ⁢           ⁢   3       +     2   /   R       )                     (       C     g   ⁢           ⁢   s   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   1       +     C     d   ⁢           ⁢   s   ⁢           ⁢   1         )     ·     (       C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     d   ⁢           ⁢   s   ⁢           ⁢   3         )       -     C     g   ⁢           ⁢   d   ⁢           ⁢   3     2                       a   0     =           (       g     d   ⁢           ⁢   s   ⁢           ⁢   1       +     1   /   R       )     ·     (       g     d   ⁢           ⁢   s   ⁢           ⁢   3       +     1   /   R       )       -       (       g     m   ⁢           ⁢   3       +     1   /   R       )     /   R                   (       C     g   ⁢           ⁢   s   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     g   ⁢           ⁢   d   ⁢           ⁢   1       +     C     d   ⁢           ⁢   s   ⁢           ⁢   1         )     ·     (       g     d   ⁢           ⁢   s   ⁢           ⁢   3       +     1   /   R       )       +                   (       C     g   ⁢           ⁢   d   ⁢           ⁢   3       +     C     d   ⁢           ⁢   s   ⁢           ⁢   3         )     ·     (       g     d   ⁢           ⁢   s   ⁢           ⁢   1       +     1   /   R       )       +       C     g   ⁢           ⁢   d   ⁢           ⁢   3       ·     (       g     m   ⁢           ⁢   3       +     2   /   R       )                       
The resistor R is usually chosen to be small compared to 1/g m1 , 1/g m3 , 1/g ds1 , and 1/g ds3 . For deep submicron CMOS technology (for example 130 nm or below), it is usually true that C gs  dominates all the parasitic capacitance. In view of this, we have:
 
               z   1     &gt;       g     m   ⁢           ⁢   1         C     g   ⁢           ⁢   s   ⁢           ⁢   1         ≈     2   ⁢     π   ·     f   T                       z   2     &gt;     1     R   ·     C     g   ⁢           ⁢   d   ⁢           ⁢   3                       p   ≈     1     R   ·     C     g   ⁢           ⁢   d   ⁢           ⁢   3                 
with f T  being the unity gain frequency which is usually large. For small R, z 2  and p are usually very large. The transfer function can hence be approximated as
 
               T   ⁡     (   s   )       =     ≈       Ω   0       s   +     a   0                 
Note that a 0 =0 if R is chosen to be
 
             R   =         g     m   ⁢           ⁢   3       -     g     d   ⁢           ⁢   s   ⁢           ⁢   1       -     g     d   ⁢           ⁢   s   ⁢           ⁢   3             g     d   ⁢           ⁢   s   ⁢           ⁢   1       ·     g     d   ⁢           ⁢   s   ⁢           ⁢   3                 
This further simplifies the transfer function to
 
               T   ⁡     (   s   )       ≈       Ω   0     s           
which is exactly the response of a perfect integrator. Note that the transistors are usually chosen such that g m1 &gt;g m3  hence
 
     
       
         
           
             
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     This implies that the unity gain frequency of the integrator is approximately the same as that of the technology. 
       FIGS. 12A-B  illustrate the frequency response of the integrator as implemented in TSMC&#39;s 65 nm CMOS. The unity frequency is about 60 GHz. The magnitude has 20 dB/dec of roll off from about 10 MHz to 60 GHz while the phase is approximately −90° (within ±10°) from 50 MHz to 10 GHz. 
       FIG. 13  illustrates a “linear in voltage” attenuator with respective input and output signals. Attenuators are devices that reduce a signal in proportion to a given binary number specified by a processor or DSP. They are designed to be either “linear in voltage” or “linear in dB.” A “linear in dB” attenuator is similar to a “linear in voltage” attenuator, except the attenuation is carried out in dBs. Attenuator precision is 12 bits. 
       FIG. 14  shows a block diagram of the attenuator. A summing circuit that consists of a single resistor R and N transistors has been created. The proposed summing block can be used in circuit topologies that require broadband analog signal processing. The number of transistors N is determined by the number of input signals that will be summed. 
     The M bit (in this example M=24) attenuator provides N bit (in this example N=12) accuracy. The maximum resistor (and hence FET switch) ratio is approximately 64. Initially resistors R, 2R, 4R, 8R, 16R, 32R, 64R may be used, and resistors of value 59R, 53R, 47R, 43R, 41R, 37R, 31R may be added, where the numbers 59, 47, 43, 41, 37, 31 are prime to the numbers 2, 4, 8, 16, 32, and 64. The resulting distribution is dithered to get the best possible distribution (maximum range of fill) in the 2N bins using x=5%. Resistors of value R may be added until we have M resistors (here we add 10 resistors of value R). 
       FIGS. 15A-B  illustrate the frequency domain performance of the attenuator for a specific loss pattern. In general, the attenuator exhibits greater than 10 GHz of bandwidth at all other attenuation values. 
       FIG. 16  illustrates a summer circuit that may be implemented in the biquad circuit described above with reference to  FIG. 9 . The summing circuit comprises a single resistor R and N transistors. The summer can be used in circuit topologies that require broadband analog signal processing. The number of transistors N is determined by the number of input signals that will be summed. 
     The summer receives input signals V 1 , V 2 , . . . , V N , and provides output signal V out , all of which contain the DC and the AC terms. The resistor R sets the DC current through the summing network and contributes to the overall gain of the summing block. By using superposition and therefore taking into account one transistor at the time, we are left to analyze a common source (CS) amplifier. By ignoring the DC bias term at the output and focusing only on the AC term, the output for CS amplifier is given by
 
ν o =−             m ν in ( R∥r   o ).  Equation 1
 
In Equation 1, term            m  is the gain of the transistor (i.e., transconductance) and r o  is the output resistance of the transistor. Assuming that r o &gt;&gt;R the overall output of the circuit in  FIG. 1  is given by
 
ν out =−(           m1 ν 1 +           m2 ν 2 + . . . +           mN ν N ) R   Equation 2
 
In Equation 2, the            m  terms that accompany the input signals can be viewed as the summing coefficients. Because the resistor R is fixed, we can adjust the summing coefficients by changing the transistor gain            m . The transistor gain            m  can be expressed in terms of the transistor width W,

     
       
         
           
             
               
                 
                   
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     According to Equation 3, the transistor gain is directly proportional to the transistor width and therefore by varying the transistor width we can adjust the summing coefficients. The bandwidth of the summing circuit is determined by the CS amplifier bandwidth. 
       FIGS. 17A-B  illustrate the frequency response of the summer as implemented in TSMC&#39;s 65 nm CMOS. 
     While this invention has been particularly shown and described with references to example embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention.