Abstract:
An N-bit delta-sigma modulator reduces circuit errors associated with an internal N-bit digital-to-analog converter by replacing the N-bit digital-to-analog converter with a digital feedback circuit comprising a ternary digital-to-analog converter and an (N−1)-bit digital-to-analog converter. The internal N-bit digital-to-analog converter is typically used to generate a global feedback signal that is provided to a first summation circuit. To provide an equivalent function but with better noise transfer characteristics, the digital feedback circuit generates two feedback signals. The ternary digital-to-analog converter is part of a first feedback path that generates a first feedback signal for the first summation circuit, and the (N−1)-bit digital-to-analog converter is part of a second feedback path that generates a second feedback signal for the first summation circuit or a subsequent summation circuit.

Description:
The present application claims priority benefits under 35 U.S.C. §119(e) from U.S. Provisional Application No. 60/863,134, filed on Oct. 27, 2006, entitled “Multi-Bit Delta-Sigma Modulator and Method Thereof,” the entirety of which is incorporated herein by reference. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to delta-sigma modulators and in particular to multi-bit delta-sigma modulators that employ spectral shaping of circuit errors in internal digital-to-analog converters. 
   2. Description of the Related Art 
   Over-sampling delta-sigma modulators are widely used in prior art to achieve high-resolution analog-to-digital conversion despite using a coarse quantizer.  FIG. 1  depicts a functional block diagram of a typical second order delta-sigma modulator  100  comprising: a first summation circuit  110  for subtracting a first feedback signal (or global feedback signal) f 1 (n) from a modulator input signal x(n); a first integrator  120  having a transfer function of approximately 0.5z −1 /(1-z −1 ) for integrating an output of the first summation circuit  110 ; a second summation circuit  130  for subtracting a second feedback signal f 2 (n) from an output of the first integrator  120 ; a second integrator  140  having a transfer function of approximately 2z −1 /(1-z −1 ) for integrating an output of the second summation circuit  130 ; an N-bit quantizer (e.g., analog-to-digital converter or ADC)  150  for digitizing an output of the second integrator  140  into an N-bit modulator output signal y(n); a first N-bit digital-to-analog converter (or DAC)  170  for converting y(n) into the first feedback signal f 1 (n); and a second N-bit DAC  160  for converting y(n) into the second feedback signal f 2 (n). 
   Throughout this disclosure, “(n)” is used to denote a timing index of states or signals of a discrete-time system. When a sampling rate of a modulator input signal x(n) is much higher than a bandwidth of the information of interest, a delta-sigma modulator can spectrally shape its quantization error and greatly suppress its power in the frequency band of interest. Therefore, the delta-sigma modulator can have a high in-band signal-to-quantization-noise-ratio (SQNR) despite using a coarse quantization (i.e., using a small N for the quantizer  150 , the second DAC  160  and the first DAC  170  in  FIG. 1 ). The number of integrators used in a delta-sigma modulator determines an order of the modulator. In general, a higher order modulator allows more aggressive spectral shaping, and thus a better in-band SQNR, but has more instability. 
   In the early days of the history of delta-sigma modulators, 1-bit data conversion (i.e., N=1 for the quantizer  150 , the second DAC  160  and the first DAC  170  in  FIG. 1 ) was widely used. This is because 1-bit data conversion is inherently linear and relatively simple as far as circuit design is concerned. In high-order modulators, however, using single-bit data conversion has many drawbacks (e.g., small usable input range, conditional stability and spurious tonal behavior). The drawbacks can all be greatly alleviated by using multi-bit data conversion (i.e., using N&gt;1). For an ideal N-bit DAC, there are 2 N  output levels that are uniformly spaced. In practice, however, these 2 N  output levels cannot be perfectly uniformly spaced due to mismatches among circuit elements under a limited tolerance in manufacturing. The non-uniformity (or alternatively described as non-linearity) of a DAC used in a delta-sigma modulator results in a source of circuit errors, which unlike the quantization error, cannot be effectively suppressed by the delta-sigma modulator. 
   While there have been many works in the prior art that address the problem of circuit errors in a multi-bit DAC used in a delta-sigma modulator, these works generally involve extensive usage of complex digital algorithms. As a result, they are not highly amenable to very high-speed applications. What is needed is a simple digital algorithm to effectively suppress the circuit errors caused by the non-linearity of a multi-bit DAC in a delta-sigma modulator. 
   SUMMARY OF THE INVENTION 
   The present invention solves these and other problems by providing a multi-bit delta-sigma modulator that uses a digital circuit comprising a ternary DAC to generate a feedback signal, thereby reducing non-linearity errors typically associated with a multi-bit feedback DAC. In one embodiment, the multi-bit delta-sigma modulator comprises a first summation circuit, a first integrator circuit, a second summation circuit, a filter circuit, a quantizer and a feedback circuit. The first summation circuit generates a first intermediate signal based on a summation of a first set of signals that includes a modulator input signal and a first feedback signal. The first integrator circuit integrates the first intermediate signal to generate a first integrated signal. The second summation circuit generates a second intermediate signal based on a weighted sum of a second set of signals that includes the first integrated signal and a second feedback signal. The filter circuit includes a second integrator circuit and processes the second intermediate signal to generate a filtered signal. The quantizer digitizes the filtered signal to generate an N-bit (or multi-bit) modulator output signal (e.g., N is an integer greater than one). 
   The feedback circuit receives the N-bit modulator output signal to generate the first feedback signal using a ternary DAC in a first feedback path. The feedback circuit further comprises an (N−1)-bit DAC in a second feedback path. In one embodiment, the second feedback path generates an additional feedback signal that is provided as part of the first set of signals to the first summation circuit. In another embodiment, the second feedback path generates the second feedback signal that is provided to the second summation circuit. 
   In one embodiment, the feedback circuit further comprises an (N−1)-bit accumulator that receives (N−1)-bit least significant bits of the N-bit modulator output signal. The accumulator comprises an (N−1)-bit adder and a delay element to generate an (N−1)-bit cumulative sum signal and a 1-bit carry signal. A 1-bit adder receives the 1-bit carry signal and a most significant bit of the N-bit modulator output signal to generate a 2-bit ternary signal that is provided as an input to the ternary DAC in the first feedback path to generate the first feedback signal. In one embodiment, an output of the (N−1)-bit adder is provided as an input to the (N−1)-bit DAC and an output of the (N−1)-bit DAC is provided to a differentiator to generate the additional feedback signal for the first summation circuit. In another embodiment, a delayed version of the output of the (N−1)-bit adder is provided to the (N−1)-bit DAC and an output of the (N−1)-bit DAC is multiplied by a scaling factor to generate the second feedback signal for the second summation circuit. 
   In one embodiment, the second set of signals further includes a third feedback signal. For example, an N-bit DAC can be configured to receive the N-bit modulator output signal and to generate the third feedback signal that is provided to the second summation circuit. Additional feedback paths coupled between the modulator output and one of the summation circuits or the filter circuit are possible. 
   In one embodiment, the filter circuit further includes a third summation circuit that is configured to generate a third intermediate signal based on a weighted sum of a third set of signals including an output of the second integrator circuit and an input of the second integrator circuit. Alternately, the third set of signals can include the output of the second integrator circuit and an output of the second summation circuit. Other feed forward compensation paths that bypass one or more of the summation circuits are also possible. 
   In one embodiment, the filter circuit includes at least one additional summation circuit and at least one additional integrator circuit to increase an order of the multi-bit delta-sigma modulator. The integrators in the multi-bit delta-sigma modulator can be continuous-time integrators or discrete-time integrators. For example, a discrete-time integrator can be a differential switch-capacitor integrator implemented using a pair of input capacitors, a differential operational amplifier, a pair of feedback capacitors, and a plurality of switches controlled by a plurality of clock signals. In one embodiment, the first summation circuit is combined with the first integrator circuit in a single differential switch-capacitor circuit with a first pair of capacitors coupled to the modulator input signal, a second pair of capacitors coupled to an output of the ternary DAC, a pair of feedback capacitors, a differential operational amplifier and a plurality of switches controlled by a plurality of clock signals. Similarly, the second summation circuit can be combined with the second integrator circuit in a single switch-capacitor circuit with the weighted sum of the second set of signals implemented by setting a capacitor ratio in the single switch-capacitor circuit. 
   In one embodiment, the ternary DAC is implemented in a differential circuit topology using two 3-to-1 multiplexers. For example, the 2-bit ternary signal provided to the input of the ternary DAC is coupled to select lines of both 3-to-1 multiplexers. Three reference voltages are provided in a different order to input terminals for each of the 3-to-1 multiplexers. The first 3-to-1 multiplexer generates a positive end of a differential signal and the second 3-to-1 multiplexer generates a negative end of the differential signal. 
   In one embodiment, a method for suppressing circuit errors associated with an N-bit feedback DAC in a multi-bit delta-sigma modulator comprises replacing the N-bit feedback DAC with an (N−1)-bit DAC and a ternary DAC. The method comprises performing a first weighted summation on a first set of signals including a modulator input signal and a first feedback signal to generate a first intermediate signal. The first intermediate signal is then integrated to generate a first integrated signal. The method further comprises performing a second weighted summation on a second set of signals including the first integrated signal and a second feedback signal to generate a second intermediate signal. The second intermediate signal is filtered (e.g., integrated) to generate a filtered signal. The filtered signal is digitized to generate an N-bit modulator output signal with N being an integer greater than one. A cumulative summation on (N−1) least significant bits of the N-bit modulator output signal is performed to generate an (N−1)-bit sum signal and a 1-bit carry signal. The 1-bit carry signal is added to a most significant bit of the N-bit modulator output signal to generate a 2-bit ternary signal. The ternary DAC converts the 2-bit ternary signal into the first feedback signal. 
   In one embodiment, the (N−1)-bit sum signal is converted into the second feedback signal using a feedback path comprising the (N−1)-bit DAC. For example, an (N−1)-bit adder is used to add the (N−1) least significant bits of the N-bit modulator output signal with the (N−1)-bit sum signal. An output of the (N−1)-bit adder is delayed to generate the (N−1)-bit sum signal that is provided to an input of the (N−1)-bit DAC. An output of the (N−1)-bit DAC may be scaled to generate the second feedback signal. In an alternate embodiment, the output of the (N−1)-bit adder is provided to the (N−1)-bit DAC and the output of the (N−1)-bit DAC is differentiated to generate an additional feedback signal that is combined with the first set of signals to generate the first intermediate signal. Other feedback signals can be generated by performing a digital-to-analog conversion of the N-bit modulator output signal. The multi-bit delta-sigma modulator can also use feed forward compensation to process the modulator input signal. 
   For purposes of summarizing the invention, certain aspects, advantages, and novel features of the invention have been described herein. It is to be understood that not necessarily all such advantages may be achieved in accordance with any particular embodiment of the invention. Thus, the invention may be embodied or carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other advantages as may be taught or suggested herein. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A general architecture that implements the various features of the invention will now be described with reference to the drawings. The drawings and the associated descriptions are provided to illustrate embodiments of the invention and not to limit the scope of the invention. Throughout the drawings, reference numbers are re-used to indicate correspondence between referenced elements. 
       FIG. 1  illustrates a functional block diagram of a prior art second order multi-bit delta-sigma modulator. 
       FIG. 2A  illustrates a functional block diagram of a first embodiment of a multi-bit delta-sigma modulator comprising a ternary DAC in a feedback path. 
       FIG. 2B  illustrates a functional block diagram of a second embodiment of a multi-bit delta-sigma modulator comprising a ternary DAC in a feedback path. 
       FIG. 2C  illustrates a functional block diagram of a third embodiment of a multi-bit delta-sigma modulator comprising a ternary DAC in a feedback path. 
       FIG. 3  illustrates one embodiment of a voltage-mode ternary DAC. 
       FIG. 4  illustrates one embodiment of a discrete-time integrator. 
       FIG. 5  illustrates one embodiment of a differential switch-capacitor circuit that combines the functionalities of a summation circuit and an integrator circuit. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   The present invention relates to a method and an apparatus for reducing circuit errors associated with a feedback DAC in a multi-bit delta-sigma modulator. While the specifications describe several example embodiments of the invention, it should be understood that the invention can be implemented in many ways and is not limited to the particular examples described below or to the particular manner in which any features of such examples are implemented. 
     FIG. 2A  illustrates a multi-bit delta-sigma modulator  200 A in accordance with one embodiment of the present invention. The multi-bit delta-sigma modulator  200 A comprises a first summation circuit  210 A that generates a first intermediate signal r 1 (n) based on a summation of a first set of signals. For example, the first summation circuit  210 A subtracts a first feedback signal a 1 (n) and a second feedback signal a′ 2 (n) from a modulator input signal x(n) to generate the first intermediate signal r 1 (n). The first intermediate signal r 1 (n) is integrated by a first integrator circuit  220  to generate a first integrated signal I 1 (n). By way of example, the first integrator circuit  220  has a transfer function of about 0.5z −1 /(1-z −1 ). The first integrated signal I 1 (n) is provided to a second summation circuit  230 A that generates a second intermediate signal r 2 (n) based on a weighted sum of a second set of signals. For example, the second summation circuit  230 A subtracts a third feedback signal (or local feedback signal) a 3 (n) from the first integrated signal I 1 (n) to generate the second intermediate signal r 2 (n). 
   The multi-bit delta-sigma modulator  200 A further comprises a filter circuit  240  to process the second intermediate signal r 2 (n) to generate a filtered signal I 2 (n).  FIG. 2A  shows the filter circuit  240  comprising a second integrator to illustrate an example of a second order delta-sigma modulator, but the filter circuit  240  can include additional integrators and summation circuits to increase the order of the modulator. Furthermore, the first integrator circuit  220  and the integrator of the filter circuit  240  are shown as discrete-time integrators, but the present invention is also applicable to continuous-time integrators. By way of example, the filter circuit  240  has a transfer function of about 2z- −1 /(1-z −1 ). The filtered signal (or second integrated signal) I 2 (n) is provided to an N-bit quantizer  250  that digitizes the filtered signal I 2 (n) to generate an N-bit modulator output signal y(n). N is preferably an integer greater than one. 
   The multi-bit delta-sigma modulator  200 A comprises multiple feedback circuits. A first feedback circuit  295 A generates the first feedback signal a 1 (n) and second feedback signal a′ 2 (n) that is provided to the first summation circuit  210 A. A second feedback circuit  260  generates the third feedback signal a 3 (n) that is provided to the second summation circuit  230 A. Additional feedback circuits can be used to generate additional feedback signals for the first summation circuit  210 A, the second summation circuit  230 A or additional summation circuits in the filter circuit  240 . In one embodiment, the second feedback circuit  260  and any additional feedback circuits are N-bit DACs with inputs coupled to the N-bit modulator output signal y(n). 
   In one embodiment, the first feedback circuit  295 A comprises an (N−1)-bit adder  280 , a delay circuit (e.g., a unit-sample delay element)  265 , an (N−1)-bit DAC  270 , a differentiator  275 A, a 1-bit adder  293  and a ternary DAC  290 . The (N−1)-bit adder  280  along with the delay circuit  265  perform a cumulative summation on (N−1) least significant bits of the N-bit modulator output signal (i.e., y L (n)) to generate an (N−1)-bit sum signal s(n) and a 1-bit carry signal c(n). The (N−1)-bit sum signal s(n) is provided to the delay circuit  265  to generate a delayed sum signal s(n−1) that is provided to an input of the (N−1)-bit adder  280 . The (N−1)-bit sum signal s(n) is also provided to the (N−1)-bit DAC  270  to generate an analog output a 2 (n) that is further processed by the differentiator  275 A to be the second feedback signal a′ 2 (n). The 1-bit carry signal c(n) and a most significant bit of the N-bit modulator output signal (i.e., y M (n)) is provided to the 1-bit adder  293  to generate a 2-bit ternary signal d(n). The ternary DAC  290  processes the ternary signal d(n) to generate the first feedback signal a 1 (n). 
   In comparing the multi-bit delta-sigma modulator  200 A in  FIG. 2A  to the multi-bit delta-sigma modulator  100  in  FIG. 1 , one of the changes is that the first feedback circuit  295 A is used to substantially replace the first N-bit DAC  170 . The following discussion illustrates how the first feedback circuit  295 A with the ternary DAC  290  and the (N−1)-bit DAC  270  functions in a similar manner and is mathematically equivalent to the first N-bit DAC  170 . 
   For example, the N-bit modulator output signal y(n) has 2 N  possible values: {0, 1, 2, . . . 2 N−1 }. Assuming for now that all of the DACs in  FIGS. 1 and 2A  are ideal and the weight of a least significant bit (LSB) is approximately Δ, the first feedback signal f 1 (n) in the modulator  100  of  FIG. 1  is defined by the following equation:
 
 f   1 ( n )= y ( n )·Δ  (1)
 
   In the modulator of  FIG. 2A , the second feedback signal a′ 2 (n) is obtained by performing a differentiation operation (1-z −1 ) on the analog output a 2 (n) from the (N−1)-bit DAC  270 . The second feedback signal in  FIG. 2A  is defined by the following equation:
 
 a′   2 ( n )= a   2 ( n )− a   2 ( n− 1)   (2)
 
   The (N−1) least significant bits of the N-bit modulator output signal and the most significant bit of the N-bit modulator output signal can be represented by the following mathematical expressions:
 
 y   L ( n )=mod( y ( n ),2 N−1 )   (3)
 
 y   M ( n )=[ y ( n )− y   L ( n )]/2 N−1    (4)
 
   The (N−1)-bit adder  280  along with the unit-sample delay  265  perform a cumulative summation on the (N−1) least significant bits of the N-bit modulator output signal to generate the (N−1)-bit sum signal s(n) and the 1-bit carry signal c(n). The (N−1)-bit sum signal s(n) and the 1-bit carry signal c(n) can be represented by the following mathematical expressions:
 
 s ( n )=mod( y   L ( n )+ s ( n− 1),2 N−1 )   (5)
 
 c ( n )=[ y   L ( n )+ s ( n− 1)− s ( n )]/2 N−1    (6)
 
   The 2-bit ternary signal d(n) is a sum of c(n) and y M (n) as shown in the following equation:
 
 d ( n )= c ( n )+ y   M ( n )   (7)
 
   The weight of the LSB of the (N−1)-bit sum signal s(n) is also approximately Δ, but the weight of the 1-bit carry signal c(n) is approximately 2 N−1  times higher since it represents the carry from the (N−1)-bit adder  280 . The weight of y M (n) is also 2 N−1  times higher since it is the most significant bit of the N-bit modulator output signal y(n). Thus, the analog output signal a 2 (n) from the (N−1)-bit DAC  270  and the first feedback signal a 1 (n) in  FIG. 2A  can be represented by the following mathematical expressions:
 
 a   2 ( n )= s ( n )·Δ  (8)
 
 a   1 ( n )= d ( n )·(2 N−1 Δ)=[ c ( n )+ y   M ( n )]·(2 N−1 Δ)   (9)
 
   Based on equations (2) and (8), the second feedback signal a′ 2 (n) in  FIG. 2A  can be rewritten as follows:
 
 a′   2 ( n )= a   2 ( n )− a   2 ( n− 1)= s ( n )Δ− s ( n− 1)Δ  (10)
 
   Based on equations (10), (9), (6) and (4), the second feedback signal a′ 2 (n) in  FIG. 2A  can be simplified as follows: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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   From equations (1) and (11), the first feedback signal f 1 (n) in  FIG. 1  is shown to be mathematically equivalent to a summation of the first feedback signal a 1 (n) and the second feedback signal a′ 2 (n) in  FIG. 2A . Thus, the first feedback circuit  295 A in  FIG. 2A  with the ternary DAC  290  and the (N−1)-bit DAC  270  is functionally equivalent to the first N-bit DAC  170  in  FIG. 1  if all of the DACs are ideal.
 
 f   1 ( n )= a′   2 ( n )+ a   1 ( n )   (12)
 
   When DACs are not ideal, the multi-bit delta-sigma modulator  200 A of  FIG. 2A  with the first feedback circuit  295 A performs better than the multi-bit delta-sigma modulator  100  of  FIG. 1 . In the multi-bit delta-sigma modulator  100  of  FIG. 1 , circuit errors from the first N-bit DAC  170  introduce additional noises at the input of the first summation circuit  110 . These additional noises are affected by the same transfer characteristics as the modulator input signal, and thus in-band components of the additional noises cannot be suppressed at the modulator output. 
   The multi-bit delta-sigma modulator  200 A of  FIG. 2A  can suppress DAC circuit errors. For example, circuit errors from the (N−1)-bit DAC  270  introduce additional noises at the input of the differentiator  275 A and the additional noises are subject to high-pass filtering by the differentiator  275 A before being combined with the modulator input signal at the first summation circuit  210 A. Thus, in-band components of the additional noises due to the (N−1)-bit DAC  270  can be suppressed at the modulator output. 
   The output of the ternary DAC  290  is provided to the first summation circuit  210 A. Thus, any circuit errors from the ternary DAC  290  will introduce additional noises at the input of the first summation circuit  210 A. Any additional noises will be affected by the same transfer characteristics as the modulator input signal. Thus, the in-band components of any additional noises from the ternary DAC  290  cannot be suppressed at the modulator output. However, the ternary DAC  290  can be inherently linear with minimal non-linearity related circuit errors when implemented in a fully differential circuit topology. Therefore, the modulator output signal in  FIG. 2A  can be relatively free of noises due to circuit errors in feedback DACs. 
     FIG. 2B  illustrates another embodiment of a multi-bit delta-sigma modulator  200 B that advantageously eliminates the differentiator  275 A. In  FIG. 2A , the differentiator  275 A is followed by the first integrator circuit  220 . A differentiator followed by an integrator effectively does nothing to a signal, aside from introducing a delay and a possible scaling factor. For example, the combined effect of the differentiator  275 A and the first integrator circuit  220  in  FIG. 2A  to the analog output a 2 (n) is 0.5z −1  (i.e., a scaling by a factor of 0.5 together with a unit-sample delay). Thus, the differentiator  275 A can be removed by bypassing the first integrator circuit  220  and introducing a proper scaling and delay to the analog output a 2 (n) of the (N−1)-bit DAC  270 . 
   In  FIG. 2B , a first summation circuit  210 B generates a first intermediate signal r′ 1 (n) based on a summation of a modulator input signal x(n) and a first feedback signal a 1 (n). The first intermediate signal r′ 1 (n) is integrated by a first integrator circuit  220  to generate a first integrated signal I′ 1 (n). The first integrated signal I′ 1 (n) is provided to a second summation circuit  230 B that generates a second intermediate signal r 2 (n) based on a weighted sum of a second set of signals. For example, the second summation circuit  230 B subtracts a second feedback signal 0.5a″ 2 (n−1) and a third feedback signal a 3 (n) from the first integrated signal I′ 1 (n) to generate the second intermediate signal r 2 (n). The second intermediate signal r 2 (n) is processed by the filter circuit  240  to generate the filtered signal I 2 (n). The filter signal (or second integrated signal) I 2 (n) is provided to an N-bit quantizer  250  that digitizes the filter signal I 2 (n) to generate an N-bit modulator output signal y(n). 
   The multi-bit delta-sigma modulator  200 B comprises a first feedback circuit  295 B that generates the first feedback signal a 1 (n) for the first summation circuit  210 B and the second feedback signal 0.5a″ 2 (n−1) for the second summation circuit  230 B. A second feedback circuit (e.g., an N-bit DAC)  260  generates the third feedback signal a 3 (n) that is also provided to the second summation circuit  230 B. 
   In one embodiment, the first feedback circuit  295 B comprises an (N−1)-bit adder  280 , a delay circuit  265 , an (N−1)-bit DAC  270 , a scaling factor  275 B, a 1-bit adder  293  and a ternary DAC  290 . The (N−1)-bit adder  280  along with the delay circuit  265  perform a cumulative summation on (N−1) least significant bits of the N-bit modulator output signal (i.e., y L (n)) to generate an (N−1)-bit sum signal s(n) and a 1-bit carry signal c(n). The (N−1)-bit sum signal s(n) is provided to the delay circuit  265  to generate a delayed sum signal s(n−1) that is provided to an input of the (N−1)-bit adder  280 . The delayed sum signal s(n−1) is also provided to the (N−1)-bit DAC  270  to generate a delayed analog output a″ 2 (n−1) that is scaled by the scaling factor  275 B to be the second feedback signal 0.5a″ 2 (n−1). The 1-bit carry signal c(n) and a most significant bit of the N-bit modulator output signal (i.e., y M (n)) is provided to the 1-bit adder  293  to generate a 2-bit ternary signal d(n). The ternary DAC  290  processes the ternary signal d(n) to generate the first feedback signal a 1 (n). 
   The multi-bit delta-sigma modulator  200 B of  FIG. 2B  functions equivalently as the multi-bit delta-sigma modulator  200 A of  FIG. 2A . The first feedback circuit  295 B of  FIG. 2B  advantageously eliminates the differentiator  275 B used in the first feedback circuit  295 A of  FIG. 2A  by providing the delayed sum signal s(n−1) to the (N−1)-bit DAC  270 , properly scaling the analog output of the (N−1)-bit DAC  270 , and providing the resulting second feedback signal to the second summation circuit  230 B rather than the first summation circuit  210 B. 
     FIG. 2C  illustrates yet another embodiment of a multi-bit delta-sigma modulator  200 C. The multi-bit delta-sigma modulator  200 C of  FIG. 2C  is substantially similar to the multi-bit delta-sigma modulator  200 B of  FIG. 2B  except the second feedback circuit  260  is eliminated and a feed forward path is added. The present invention is applicable to multi-bit delta-sigma modulators that use feed forward compensation, local feedback compensation or a combination of both. 
   By way of example,  FIG. 2C  shows a multi-bit delta-sigma modulator  200 C with a feed forward path that bypasses the filter circuit  240  comprising at least one integrator circuit. The feed forward path comprises a gain block  235  that receives an input from an output of a second summation circuit  230 C (or an input of the filter circuit  240 ) and generates a feed forward signal that is combined with an output of the filter circuit  240  at a third summation circuit  245 . This feed forward path is functionally equivalent to the local feedback path comprising the N-bit DAC  260  in  FIG. 2B . Other feed forward paths that bypass the first integrator circuit  220  or other integrator circuits (not shown) are also possible. 
     FIG. 3  illustrates one embodiment of a ternary DAC  300  implemented in a fully differential circuit topology. The ternary DAC  300  receives a 2-bit ternary signal (or control word DATA) having three possible values (e.g., 0, 1, 2) and outputs a corresponding differential signal comprising a positive end (VOUTP) and a negative end (VOUTN). The ternary DAC  300  uses three reference voltages: VREFP, VCM and VREFN. The positive end of the differential signal is coupled to one of the three reference voltages via a first multiplexer  310  and the negative end of the differential signal is coupled to one of the three reference voltages via a second multiplexer  320 . The multiplexers  310 ,  320  are controlled by the control word DATA. The output of the ternary DAC  300  is the voltage difference between the positive end and the negative end (i.e., the respective outputs of the multiplexers  310 ,  320 ). Table 1 shows the possible outputs of the ternary DAC  300 . The three possible outputs of the ternary DAC  300  are uniformly spaced. Thus, the ternary DAC  300  is inherently linear regardless of the values of reference voltages. 
   
     
       
             
             
             
             
           
         
             
               TABLE 1 
             
             
                 
             
             
               DATA 
               VOUTP 
               VOUTN 
               DAC OUTPUT 
             
             
                 
             
           
           
             
               0 
               VREFN 
               VREFP 
               VREFN-VREFP 
             
             
               1 
               VCM 
               VCM 
               0 
             
             
               2 
               VREFP 
               VREFN 
               VREFP-VREFN 
             
             
                 
             
           
        
       
     
   
     FIG. 4  illustrates one embodiment of a discrete-time integrator implemented as a fully differential switch-capacitor integrator. A pair of input capacitors (Cin 1 , Cin 2 ) is coupled to inputs of a fully differential operational amplifier  400 . A pair of feedback capacitors (Cfb 1 , Cfb 2 ) is coupled to outputs of the fully differential operational amplifier  400 . A plurality of switches are controlled by a plurality of clock signals to implement the discrete-time integrator function. 
     FIG. 5  illustrates one embodiment of an integrator circuit that includes a summation circuit function. The integrator circuit is implemented using a discrete-time integrator topology. A summation circuit that precedes the integrator circuit and is configured to receive a feedback signal from a DAC can be effectively combined with the integrator into a single fully differential switch-capacitor circuit similar to the one shown in  FIG. 5 . For example, the first summation circuit  210 B and the first integrator  220  can be combined in a single fully differential switch-capacitor circuit that has a first pair of input capacitors (e.g., Cin 1 , Cin 2 ) for receiving the modulator input signal and a second pair of input capacitors (e.g., Cin 3 , Cin 4 ) for receiving the first feedback signal generated by a DAC. The second summation circuit  230 B can also be combined with the second integrator in the filter circuit  240  in a single fully differential switch-capacitor circuit with a first pair of input capacitors for receiving the first integrated signal, a second pair of input capacitors for receiving the second feedback signal and a third pair of input capacitors for receiving the third feedback signal. In one embodiment, the scaling factor  275 B can also be included in the switch-capacitor circuit by setting the capacitor ratios between the pairs of input capacitors. 
   While certain embodiments of the inventions have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the inventions. Indeed, the novel methods and systems described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions, and changes in the form of the methods and systems described herein may be made without departing from the spirit of the inventions. The accompanying claims and their equivalents are intended to cover such forms or modifications as would fall within the scope and spirit of the inventions.