Abstract:
A multiplexer is implemented in multiple stages. In an exemplary embodiment, a four-to-one multiplexer is transformed into two four-input logic functions to facilitate a field-programmable-gate-array implementation.

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to digital logic circuits and more specifically to an apparatus and method for implementing a multiplexer. 
     BACKGROUND OF THE INVENTION 
     Multiplexers are used extensively in many applications such as computers and telecommunications. For example, multiplexers comprise the primary portion of the register file in a field-programmable-gate-array (FPGA) implementation of a microprocessor. The problem sometimes arises, however, that the multiplexer function does not map efficiently onto four-input look-up tables, which are the basic logical building block of many FPGAs. Specifically, some FPGA synthesis tools use additional logic available in the majority of FPGAs to fit, for example, a four-to-one multiplexer into two four-input look-up tables. Unfortunately, use of this additional logic makes it unavailable for other uses and complicates the process of placement and routing. It is thus apparent that there is a need in the art for an improved multiplexer implementation. 
     SUMMARY OF THE INVENTION 
     An apparatus and associated method are provided for implementing a multiplexer. The multiplexing function is divided into two stages, the first of which outputs either a selection result or a selection signal. The selection result corresponds to one of a first set of data inputs to the first stage. The second stage selects one of a second set of data inputs to the second stage in response to the selection signal. 
    
    
     Other aspects and advantages of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, illustrating by way of example the principles of the invention. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram and associated truth table of a four-to-one multiplexer that may be implemented using the invention. 
     FIG. 2A is a partial truth table associated with a first stage of a four-to-one multiplexer implementation in accordance with an exemplary embodiment of the invention. 
     FIG. 2B is a partial truth table associated with a second stage of a four-to-one multiplexer implementation in accordance with an exemplary embodiment of the invention. 
     FIG. 3A is a complete truth table associated with a first stage of a four-to-one multiplexer implementation in accordance with an exemplary embodiment of the invention. 
     FIG. 3B is a complete truth table associated with a second stage of a four-to-one multiplexer implementation in accordance with an exemplary embodiment of the invention. 
     FIG. 4 is a circuit diagram corresponding to the truth tables of the exemplary embodiment of the invention shown in FIGS. 2A,  2 B,  3 A, and  3 B. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Although the invention will be described in connection with an exemplary embodiment involving a four-to-one multiplexer, the invention may be extended to any N-to-one multiplexer circuit, where N is a positive-integer power of four. As those skilled in the art will recognize, an N-to-one multiplexer has log 2  (N) selection inputs for selecting which input appears at the output. The general approach is as follows. The N data inputs are divided into two sets of        N   2                          
     data inputs associated with a first stage and a second stage, respectively. In the first stage, a first logic function operates upon the first set of data inputs, along with the log 2  (N) selection inputs, to produce an output that serves a dual purpose. In the second stage, a second logic function operates upon the second set of data inputs, at least one of the log 2  (N) selection inputs, and the output of the first stage to produce the final output, which is the desired one of the N inputs. Under a first set of states of the selection inputs, the output of the first stage is a selection result corresponding to one of the data inputs in the first set of data inputs. In this case, the data input in the first set of data inputs to which the selection result corresponds ultimately appears as the output of the second stage. Under a second set of states of the selection inputs, the output of the first stage is a selection signal. In this case, the second stage selects one of the data inputs in the second set of data inputs in response to the selection signal. Each stage has a total of N inputs, including selection inputs. The advantages of such a two-stage multiplexer implementation will become apparent as an exemplary embodiment of the invention is described. 
     FIG. 1 is a block diagram and associated truth table of a four-to-one multiplexer that may be implemented using the invention. Four-to-one multiplexer  100  operates upon data inputs  105  and selection inputs  10  to produce output ƒ. The truth table in FIG. 1 shows the relationship between all the possible states of selection inputs  110  and the output ƒ. Although FIG. 1 shows one particular convention for logic polarities and order of selection inputs  10 , the invention may be applied to multiplexers defined according to different conventions. A Boolean expression corresponding to the block diagram of FIG. 1 is              f   =         a   0          c   0          c   1       +       a   1            c   0     _          c   1       +       a   2          c   0            c   1     _       +       a   3            c   0     _                         c   1     _     .                 (Equation   1)                                
     As explained previously, Equation 1 may be divided into two stages, each having N=4 total inputs (data and selection). A Boolean expression for the first stage may be written as follows:                f   1     =         c   1          (         a   0          c   0       +       a   1            c   0     _         )       +       c   0                         c   1     _     .                 (Equation   2)                                
     The second stage may then be represented in Boolean form by              f   =         c   1          f   1       +           c   1     _          (         a   2          f   1       +       a   3            f   1     _         )       .               (Equation   3)                                
     FIGS. 2A and 2B show partial truth tables corresponding to Equation 2 and Equation 3, respectively, in accordance with an exemplary embodiment of the invention. Note that for two combinations of selection inputs  110  in FIG. 2A, ƒ 1  is a selection result equal to one of the two first-stage data inputs, α 0  and α 1 . In this case, the applicable value (α 0  or α 1 ) ultimately appears as the output ƒ. For the other two combinations, ƒ 1  is a selection signal. In this case, the second stage ultimately selects α 3 (ƒ 1 =logical “0”) or α 2 (ƒ 1 =logical “1”) as the output ƒ in response to the selection signal. FIG. 2B shows the output ƒas a function of second-stage inputs ƒ 1  and c 1 . 
     FIGS. 3A and 3B are complete truth tables corresponding to Equation 2 and Equation 3, respectively, in accordance with an exemplary embodiment of the invention. In this form, with four total inputs to each stage, the four-to-one multiplexer may be mapped readily onto two four-input FPGA look-up tables. 
     FIG. 4 is a circuit diagram corresponding to Equations 2 and 3 and the associated truth tables shown in FIGS. 2A,  2 B,  3 A, and  3 B in accordance with an exemplary embodiment of the invention. The circuit in FIG. 4 is divided into first stage  405  and second stage  410 . Each stage comprises two two-to-one multiplexers  415  connected as indicated. Note that, in each stage, one input serves the function of both data input and selection input. In the first stage, that input is c 0 ; in the second, ƒ 1 . Selection input c 1  determines whether ƒ 1  or one of α 2  and α 3  appears at the output of the second stage. 
     As mentioned previously, the invention may also be applied to larger multiplexers such as a 16-to-one. In this case, each stage may be implemented using two four-to-one multiplexers. 
     The foregoing description of the present invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed, and other modifications and variations may be possible in light of the above teachings. The exemplary embodiment was chosen and described in order to best explain the principles of the invention and its practical application to thereby enable others skilled in the art to best utilize the invention in various embodiments and various modifications as are suited to the particular use contemplated. It is intended that the appended claims be construed to include other alternative embodiments of the invention except insofar as limited by the prior art.