Patent Publication Number: US-2006020655-A1

Title: Library of low-cost low-power and high-performance multipliers

Description:
PRIORITY  
      The present application claims priority to a provisional patent application entitled “A LIBRARY OF LOW-COST LOW-POWER AND HIGH-PERFORMANCE MULTIPLIERS,” filed on Jun. 29, 2004, and assigned Ser. No. 60/583,948, the contents of which are hereby incorporated by reference. 
    
    
     STATEMENT OF GOVERNMENT INTEREST  
      The present invention was funded, at least in part, by NSF Grant CCR 0073469, Computer Systems Architecture, July 2000 to May 2003. The government has certain rights in the present invention. 
    
    
     BACKGROUND OF THE INVENTION  
      1. Field of the Invention  
      The present invention relates generally to low power high-performance digital circuits and in particular, to highly complexity-effective multiplier triple expansion schemes enabling the construction of a large library of NxN multipliers with input size N ranging from 3 to 99 bits.  
      2. Description of the Related Art  
      Conventional multiplier schemes, including the state-of-the-art approaches (see, R. Montoye et al., “A Double Precision Floating Point Multiplier,” Proc. of 2003 IEEE ISSCC, February, 2003, and N. Itoh et al., “A 600 MHz, 54×54-bit Multiplier With Rectangular styled Wallace Tree”, IEEE JSSCs, Vol. 35, No. 2, February 2001), which produce high-speed, low-power circuits, are usually not feasible for use in the construction of a large library of multipliers. This is because expansive custom design and mask work are required because of the large amount of irregular circuits involved to construct these circuits. Consequently, existing Application Specific Integrated Circuit (ASIC) flexible design-tool libraries lack sufficient capabilities for building a large library of multipliers.  
      Moreover, conventional large multiplier circuits are typically constructed based on the schemes of generation of a single or a few large irregular bit matrices, followed by several stages of reduction of the bits into two numbers using binary-logic. However, these circuits are ineffective in dealing with the irregularity. Accordingly, in order to achieve high-performance level, these multiplier circuits usually require an increased amount of circuit complexity. This increase in circuit complexity not only adds to the multiplier circuit&#39;s design and testing time, but also increases design, optimization and manufacturing costs.  
      Thus, there is a need for borrow parallel counter circuits and highly complexity-effective multiplier triple expansion schemes which can enable the construction of a large library of NxN multipliers with input size N ranging from 3 to 99 bits with minimal cost, effort and complexity.  
     SUMMARY OF THE INVENTION  
      It is, therefore, an object of the present invention to provide borrow parallel counter circuits and highly complexity-effective multiplier triple expansion schemes which enable the construction of a large library of NxN multipliers with input size N ranging from 3 to 99 bits with minimal cost, effort and complexity.  
      It is a further object of the present invention to provide low-cost, compact low-power high-performance multipliers, particularly for a library of different sizes of multipliers including small (e.g., 3 to 11 bits), medium (e.g., 12 to 33 bits), and large (e.g., 34 to 99 bits) multipliers, corresponding unique schemes and circuits.  
      It is a further object of the present invention to provide a library which can be used as a flexible design tool for Designing Application Specific Integrated Circuits (ASIC&#39;s).  
      The novel borrow parallel counter circuits and highly complexity-effective multiplier triple expansion schemes proposed by the present invention enable the construction of a large library of NxN multipliers with an input size N which is preferably between 3 and 99 bits, with low cost and complexity.  
      High Performance Multiplier Circuits and Triple Expansion Schemes are described in R. Lin and R. B. Alonzo, “A Library Of Low-Cost High-Performance Multipliers Using Borrow Parallel Counters And Double-Triple Expansion Schemes,” Proc. Of Workshop On Unique Chips And Systems” (UCAS-1), March, 2005, Austin, Tex., pp. 74-83. R. Lin and R. B. Alonzo, “An Extra-Regular, Compact, Low-Power Multiplier Design Using Triple-Expansion Schemes And Borrow Parallel Counter Circuits,” Proc. of workshop on complexity-effective design (WCED, ISCA), June 2003, the contents of which are incorporated herein by reference.  
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      The foregoing and other objects, aspects, and advantages of the present invention will be better understood from the following detailed description of preferred embodiments of the invention with reference to the accompanying drawings, in which:  
       FIG. 1A  is block diagram illustrating an extra-compact, low-power, high-speed, CMOS circuits  5 _ 1  borrow parallel counter (hereinafter a  5 _ 1  counter), serving as building blocks for parallel arithmetic designs;  
       FIG. 1B  is a detailed block diagram illustrating circuitry which can be substituted in the  5 _ 1  counter of  FIG. 1  to create a  5 _ 1 _ 1  borrow parallel counter (hereinafter a  5 _ 1 _ 1  counter);  
       FIGS. 1C and 1D  are detailed block diagrams illustrating the  5 _ 1  and  5 _ 1 _ 1  borrow parallel counters of  FIGS. 1A and 1B ;  
       FIG. 2A  is a block diagram illustrating a first base multiplier included in a small multiplier sub-library;  
       FIG. 2B  is a block diagram illustrating a second base multiplier included in the small multiplier library;  
       FIGS. 2C-2E  are diagrams illustrating a  6 _ 0 , non-full counter, a  6 _ 1 , full counter, and a  7 _ 0 , full counter, respectively;  
       FIGS. 3A-3C  are diagrams illustrating multiplier triple expansion schemes;  
       FIG. 4  is a diagram illustrating a Level-1 multiplier triple expansion scheme;  
       FIG. 5  is a diagram illustrating a Level-2 multiplier triple expansion scheme;  
       FIG. 6  is a diagram illustrating 2:2 and 3:2 binary counters and their corresponding symbols;  
       FIG. 7  is a diagram illustrating a 6-b high-speed and compact ripple-carry adder SA 6 ;  
       FIGS. 8A-8C  is are diagrams illustrating a modification of a 3m−b (m=6) multiplier into a (3m+1)−b multiplier and a (3m−1)−b multiplier, respectively;  
       FIGS. 9A and 9B  are diagrams illustrating a partial product matrix of an mxm multiplier (where m=4);  
       FIGS. 10A and 10B  are diagrams illustrating Carry-look-ahead binary counters 3:2L and 3:2NL, and their corresponding symbols;  
       FIGS. 11A-11C  are diagrams illustrating the circuitry of a  6 SA 8  Carry-look-ahead-adder; the structural symbol which indicates a 4-b ripple adder followed by a 2-b carry-look-ahead node and then followed by a 2-b ripple adder; and the abstract symbol which means the small 8-b adder has a critical path including 6 transmission gates (or pass transistors), respectively;  
       FIG. 12  is a diagram illustrating a Carry-look-ahead-adder  6 SA 9 ;  
       FIGS. 13A-13C  are diagrams illustrating a Carry-look-ahead-adder&#39;s  6 SA 10  circuit; the structural symbol which indicates a 3-b ripple adder followed by a 2-b carry-look-ahead node and then followed by a 3-b carry-look-ahead node then a 2-b ripple adder; and the abstract symbol which means the small 10-b adder has a critical path including 6 transmission gates, respectively;  
       FIG. 14  is a diagram illustrating a Carry-look-ahead-adder  7 SA 12 ;  
       FIG. 15  is a diagram illustrating a Carry-look-ahead-adder  8 SA 15 ;  
       FIG. 16  is a diagram illustrating a Carry-look-ahead-adder  8 SA 17 ;  
       FIG. 17  is a diagram of small adders with 1-level Carry-look-ahead nodes: (a)  4 SA 6  (for 6×6-6); (b)  5 SA 8  (for 7×7-8); (c)  6 SA 10  (for 8×8-10) (d)  6 SA 10  (e)  6 SA 11  (f)  7 SA 13  (for 9×9-12) (g)  7 SA 14  (h)  8 SA 15  (for 10×10-b-15) (i)  8 S 16 ( j )  8 SA 16  (k)  8 SA 17  (for 11×11-b-17);  
       FIG. 18  is a diagram illustrating a medium-size 24-b adder for the final addition of an 18×18 multiplier with 2-level look-ahead nodes;  
       FIG. 19  is a diagram illustrating a medium-size 54-b adder for the final addition of a 33×33 multiplier with a 3-level look-ahead nodes, in which the carry-look-ahead structure is shown in horizontal (right to left for LSB to MSB), which is the same as that shown in vertical form as shown in  FIGS. 11   b ,  17 , and  17   e  (for  6 SA 11 );  
       FIG. 20  is a diagram illustrating a large-size 89-b adder for the final addition of a 54×54 multiplier with 3-level look-ahead nodes;  
       FIG. 21  is a diagram illustrating a multiplier redistributing a few (e.g., 10 as shown) partial product bits for (3m+1)×(3m+1) multipliers (where m=5);  
       FIG. 22  is a diagram illustrating a multiplier redistributing and zeroing several (e.g., 6) partial product bits for (3m−1)×(3m−1) multipliers (where m=4);  
       FIG. 23  is a diagram illustrating an input distribution and circuit structure of level-1 carry-save-adder (CSA) of an 18×18 multiplier;  
       FIG. 24  is a diagram illustrating an input distribution and circuit structure of a level-1 carry-save adder (CSA) of a 19×19 multiplier which is modified from the 18×18 multiplier shown in  FIG. 23 ;  
       FIG. 25  is a diagram illustrating an input distribution and circuit structure of level-1 CSA of 17×17 multiplier modified from  FIG. 23 ;  
       FIG. 26  is a diagram illustrating three types of segmented small adders: type-8, type-9, type-10;  
       FIG. 27  is a diagram illustrating an organization of nine 18×18-b virtual multipliers;  
       FIG. 28  is a diagram illustrating outputs from nine 18×18 virtual multipliers to a level-2 CSA counter array of a 54-b multiplier, where level-2 contains an array of borrow parallel counters which is similar to a level-1 CSA but larger;  
       FIG. 29  is a diagram illustrating five types of segmented small adders: type-6, type-7, type-8, type-9, type-10;  
       FIG. 30  is a diagram illustrating an organization of nine 21×21-b virtual multipliers;  
       FIG. 31  is a diagram illustrating outputs generated from nine 21×21 virtual multipliers (i.e., from segmented small adders);  
       FIG. 32  is a diagram illustrating outputs from nine 21×21 virtual multipliers to a level-2 CSA counter array of the 63-b multiplier;  
       FIG. 33  is a diagram illustrating three types of segmented small adders: type-8, type-9, type-10;  
       FIG. 34  is a diagram illustrating an organization of nine 24×24-b virtual multipliers;  
       FIG. 35  is a diagram illustrating outputs generated from nine 24×24 virtual multipliers (i.e., from segmented small adders);  
       FIG. 36  is a diagram illustrating outputs from nine 24×24 virtual multipliers to a level-2 CSA counter array of a 72-b multiplier inputs to CSA of Level-2;  
       FIG. 37  is a diagram illustrating three types of segmented small adders: type-9, type-10, type-11;  
       FIG. 38  is a diagram illustrating an organization of nine 33×33-b virtual multipliers;  
       FIG. 39  is a diagram illustrating outputs from the nine 33×33 virtual multipliers to a level-2 CSA counter array of a 99-b multiplier inputs to CSA of Level-2;  
       FIG. 40  is a diagram illustrating a  5 _ 1 ′ borrow parallel counter ( 5 _ 1  with an extra hidden constant input  1 );  
       FIG. 41  is a diagram illustrating 4×4-b twos complement multipliers, in which a circle followed by an arrow indicates a hidden bit (see  FIG. 9 );  
       FIG. 42  is a diagram illustrating 5×5-b twos complement multipliers, in which a circle followed by an arrow indicates a hidden bit (see  FIG. 9 );  
       FIG. 43  is a diagram illustrating a 6×6-b twos complement multipliers, in which only one  5 _ 1  borrow counter in column 6 is replaced by a  5 _ 1 ′ counter in this modification;  
       FIG. 44  is a diagram illustrating 7×7-b twos complement multipliers, in which only one  6 _ 0  borrow counter in column 7 is replaced by a  6 _ 0 ′ counter in this modification;  
       FIG. 45  is a diagram illustrating 8×8-b twos complement multipliers, in which only one  6 _ 0  borrow counter in column  8  is replaced by a  6 _ 0 ′ counter in this modification;  
       FIG. 46  is a diagram illustrating 9×9-b twos complement multipliers, in which only one  6 _ 0  borrow counter in column  9  is replaced by a  6 _ 0 ′ counter in this modification;  
       FIG. 47  is a diagram illustrating 10×10-b twos complement multipliers, in which only one  6 _ 0  borrow counter in column  10  is replaced by a  6 _ 0 ′ counter in this modification; and  
       FIG. 48  is a diagram illustrating 10×10-b twos complement multipliers, in which only one  7 _ 0  borrow counter in column  11  is replaced by a  7 _ 0 ′ counter in this modification. 
    
    
     DETAILED DESCRIPTION OF THE PRESENT INVENTION  
      The novel borrow parallel counter circuits and highly complexity-effective multiplier triple expansion schemes according to the present invention enable the construction of a large library of NxN multipliers with input size N ranging from 3 to 99 bits with minimal cost and effort.  
      The present invention provides for low-cost, compact, low-power high-performance multipliers, particularly for a library of different sizes of multipliers including small (e.g., 3 to 11 bits), medium (e.g., 12 to 33 bits), and large (e.g., 34 to 99 bits) multipliers, and unique schemes and circuits for these multipliers.  
      A description of the multiplier design, the borrow parallel multiplier library, and the library components will be given below.  
      The present invention provides a scheme to produce complexity-effective, high-speed, low-power, NxN-b multipliers, where N preferably is an positive integer between 3 and 99. Moreover, the present invention enables large multipliers to be generated from smaller multipliers using a unified expansion scheme. Typically, the size of a resulting multiplier is almost tripled in two or fewer steps. A sub-library including nine extra-regularly structured base multipliers (e.g., 3-b to 11-b multipliers) is designed and optimized, which significantly simplifies the library construction. For example, with 6-b base multipliers, an 18-b multiplier is constructed in a first step, and the resulting 18-b multiplier is then used to construct a 54-b, Institute of Electrical and Electronics Engineers (IEEE) standard floating point multiplier in a second step. In a similar fashion, with 7-b and 8-b base multipliers, 21-b and 22-b multipliers are constructed in a first step, and the 21-b or the 22-b multipliers can then be used to construct a 64-b multiplier.  
      The present invention employs both building block circuits (building blocks) and construction schemes, which optimize decompositions and minimize global complexity. The building blocks include a small library of nine base multipliers, each using complementary metal oxide semiconductors (CMOS), large parallel counters including “4-bit 1-hot” logic processing (where 4-bit 1-hot logic processing refers to 4 parallel data paths having only one input (IN) logic high) and borrow-bits, i.e., bits weighted 2 (see R. Lin and R. B. Alonzo, “A Library of Low-Cost High-Performance Multipliers Using Borrow Parallel Counters and Double-Triple Expansion Schemes,” in Proc. of Workshop on Unique Chips and Systems (UCAS-1), March, 2005, pp 74-83, which is incorporated herein by reference). As used herein, unless context indicates otherwise, the term “bit-weight position” refers to a column of a partial product matrix, in which each bit is in the same binary position with respect to the final product. A higher bit-weight position refers to a column in a binary position with higher significance, e.g., in the 2 4th  place, as compared to the 2 3rd  place, and a lower bit-weight position refers to a column in a binary position with lower significance.  
      According to the present invention, the building block circuits are capable of rearranging and balancing input bits in each processing column, and turning irregular multiplication units (e.g., multipliers) into substantially regular single array structured small multipliers, thus greatly reducing the local complexity allocated to each block during the decomposition. This construction scheme optimizes the decomposition, resulting in a natural rectangular-shaped and simply wired structure, thereby effectively minimizing the global complexity.  
      According to the present invention, the overall multiplier construction is a highly regular, modular, one-level or two-level (recursive) process. The multiplier construction trisect-decomposes an input bit matrix and re-positions the partitioned blocks to achieve an optimal design/layout and to improve the self-testability.  
      A block diagram illustrating a  5 _ 1 _ 1  borrow parallel counter ( 5 _ 1  counter) according to the present invention is shown in  FIG. 1A . The  5 _ 1  counter  102  is a parallel counter which can serve as building block for parallel arithmetic designs. The  5 _ 1  counter  102  has a regular distribution of cells and includes a “4-bit-1-hot” logic feature with a logic high and a “borrow bit” of weight 2 (i.e., B-B 2 ). The  5 _ 1  counter  102  includes 5 inputs (A 1 -A 5 ), two outputs (U and L), and three pairs of in-stage input/output bits, X, Y, Z (with contiguous counters close to each other), where the weighted sum of all outputs equals the weighted sum of all inputs. This is more clearly illustrated with reference to Equation 1 below which corresponds to the  5 _ 1  counter. In Equations 1 and 2 below, the variables on the left side of the equation are inputs and in-stage inputs and the variables on the right side of the equation are outputs and in-stage outputs. All variables in all equations are binary variables, and all operations are arithmetic operations except that OR, XOR, AND and prime sign′ (for complement) are logic operations. 
 
 A   1 + A   2 + A   3 + A   4 +2 A   5 +2 Xi+ 4( Yi+ 2 Yi′Zi )= Xo+ 2 Yo+ 4( Yo′Zo+L )+8 U; where Zo=Xi   Equation (1) 
 
      The circuitry contained in insert  106  can be replaced by the circuitry shown in  FIG. 1B  to form a  5 _ 1 _ 1  counter which will be described below.  
      A detailed block diagram illustrating circuitry which can be substituted in the  5 _ 1  counter of  FIG. 1A  to create a  5 _ 1 _ 1  counter is shown in  FIG. 1B . These counters are also known as borrow parallel counters. The  5 _ 1 _ 1  counter  110  is formed by replacing the circuitry in the insert  106  of the  5 _ 1  counter  102  ( FIG. 1A ) with circuitry contained in insert  110 . The  5 _ 1 _ 1  counter includes 5 inputs A 1 -A 5 , (with a difference being that bits A 4 -A 5  are used as borrow bits), two outputs (U and L), and three pairs of in-stage input/output bits, X, Y, Z (with contiguous counters close to each other), where the weighted sum of all outputs equals the weighted sum of all inputs. This is more clearly illustrated with reference to Equation 2 below. 
 
 A   1 + A   2 + A   3 +2 A   4 +2 A   5 +2 Xi+ 4( Yi+ 2 Yi′Zi )= Xo+ 2 Yo+ 4( Yo′Zo+L )+8 U; where Zo=Xi   Equation (2) 
 
      Detailed block diagrams illustrating the  5 _ 1  and  5 _ 1 _ 1  borrow parallel counters of  FIGS. 1A and 1B  are shown in  FIGS. 1C and 1D .  
      Three other borrow parallel counter variants are termed  6 _ 0 ,  6 _ 1  and  7 _ 0  (not shown), and can be synthesized by the  5 _ 1  or  5 _ 1 _ 1  circuits shown in  FIGS. 1A and 2B , with the addition of one or two 3:2 counters (which is a type of x:2 counter). The  5 _ 1 ,  5 _ 1 _ 1 ,  6 _ 0 ,  6 _ 1  and  7 _ 0  counters each have a similar layout height which is approximately equal to a height of a 3:2 counter, but each counter differs in layout width. Moreover, the  5 _ 1 ,  5 _ 1 _ 1 ,  6 _ 0 ,  6 _ 1  and  7 _ 0  counters have speed differences which are not greater than the delay of a single 3:2 counter. The  6 _ 0 ,  6 _ 1  and  7 _ 0  counters are illustrated in  FIGS. 2C-2E , respectively.  
      Having the borrow bits each weighted 2 or more makes it possible to form small virtual (i.e., two numbers in output) multipliers (i.e., base multipliers), ranging from 3 to 11 bits each, in a structure having a single array of counters (e.g., see  FIG. 2 ), with many desirable properties. These properties include having a perfectly rectangular shape (or substantially rectangular shape), substantially equal height, substantially equal delay, low power consumption, high speed, extra compact dimensions, and a simple CMOS construction.  
      When used as building blocks for the design and construction of larger multipliers (e.g., large multipliers with up to 99 bits), the base “virtual multipliers” turn irregular small multiplication units (e.g., the virtual and non-virtual multipliers having small and large sizes) into regular blocks of circuits, thus greatly reducing the local complexity of the large multipliers. The term “virtual multiplier” as used herein refers to a multiplier without the results of the final stage partial product reduction being added. The term “virtual product” as used herein refers to the results of the final stage partial product reduction of the virtual multiplier.  
      By adding a ripple-carry adder or a simple carry-look-ahead adder to each base virtual multiplier, the base multiplier sub-library is formed. The base multiplier sub-library will be described in further detail below with reference to  FIGS. 2A-2B  below.  
      A block diagram illustrating a first base multiplier included in a small-multiplier sub-library is shown in  FIG. 2A . The first base multiplier  200 A (also known as a 6×6-b partial product generation unit) includes a plurality of parallel base virtual multipliers  212 - 217 , a 3:2 counter  222 , and an XOR (exclusive or) gate  224 . The base virtual multipliers  212 - 217  correspond to major columns  2  through  7 , respectively, where the columns refer to corresponding columns of the partial product matrix of the 6×6 base multiplier. In the following example, the matrix has 11 columns  0  to  10 , with columns  0 ,  1  and  8 ,  9 ,  10  degraded, and as such are not counted as major columns. The XOR gate  224  (which corresponds to column  9 ) inputs 2 bits as shown and outputs a result to the base virtual multiplier  217 . A 3:2 counter  222  is coupled to the base virtual multiplier  215 . The 3:2 counter sums input bits a, b, and c and outputs a two bit result s and c so that a+b+c=2c+s. The base virtual multipliers  213 ,  214 , and  216  are  5 _ 1  multipliers and the base virtual multipliers  215  and  217  are  5 _ 1 _ 1  multipliers.  
      Additionally, the base virtual multiplier  212  can be either a  5 _ 1  or a  5 _ 1 _ 1  multiplier. Each of the base virtual multipliers  212 - 217  receives a given number of input bits as shown in  FIG. 2A .  
      Borrow bits of weight  2  are denoted by B-B 2 , borrow bits of weight  4  (for Yi) are denoted by B-B 4  and borrow bits of weight  8  (for Zi) are denoted by B-B 8  and outputs a result. Each of the base virtual multipliers  212 - 217  operates as described above with reference to  FIGS. 1A and 1B , and therefore, for the sake of clarity, no further description will be given. Borrow bits B-Bs, shown in offset, rearrange and balance inputs to each column so that only one of nearly identical base virtual counters  212 - 217  is needed in each column  0 - 9 . The outputs of base virtual multipliers  212 - 217  are input into a 6-bit ripple-carry adder  220  which outputs bits P 5  to P 13 , of a partial product P 0-13 , which is the output of the first base multiplier  200 A. The simple structures eliminate almost all irregularity inherent in such arithmetic units, providing a perfect base for larger multiplier designs.  
      A block diagram illustrating a second base multiplier included in a small-multiplier sub-library is shown in  FIG. 2B . The second base multiplier  200 B (also known as a 7×7-b partial product generation unit) is similar to the first base multiplier, with a difference being the substitution of an 8-bit carry-look ahead adder instead of a 6 bit ripple-carry adder which is used in the first base multiplier  200 . The second base multiplier  200 B includes a plurality of parallel base virtual multipliers  212 B- 219 B, a 3:2 counter  222 B, and an XOR (exclusive or) gate  224 B.  
      The base virtual multipliers  212 B- 219 B correspond to columns  2  through  9  (of the partial product matrix of the 6×6-b multiplier), respectively. The XOR gate  224 B (which corresponds to column  9 ) inputs 2 bits as shown and outputs a result to the base virtual multiplier  217 B. A 3:2 counter  222 B is coupled to the base virtual multiplier  215 B. The base virtual multipliers  212 B is a  5 _* multiplier,  213 B and  214 B are  5 _ 1  multipliers, the base multipliers  215 B and  219 B are  5 _ 1 _ 1  multipliers, the base multipliers  216 B and  217 B are  6 _ 1  multipliers, and the base multiplier  218 B is a  6 _ 1  multiplier. Each of the base virtual multipliers  212 B- 219 B receives a given number of input bits as shown in  FIG. 2B , and outputs a result. Each of the base virtual multipliers  212 B- 219 B operates as described above with reference to  FIGS. 1A and 1B , and therefore, for the sake of clarity, no further description will be given. Borrow bits B-Bs, shown in offset, rearrange and balance inputs to each column so that only one of the nearly identical base virtual counters  212 B- 219 B is needed in each column  0 - 9 . The outputs of base virtual multipliers  212 B- 219 B are input into a 8-bit ripple-carry adder  220 B, which outputs bits P 5  to P 13  of a partial product P 0-13 , which is the output of the first base multiplier  200 A.  
      For a more detailed description of base multipliers, see U.S. Patent Publication No. 2004/0172439 A1, entitled “Unified Multiplier Triple-Expansion Scheme And Extra Regular Compact Low-Power Implementations With Borrow Parallel Counter Circuits,” to R. Lin (the &#39;439 Publication), the contents of which are incorporated by reference.  
      The other base multipliers belonging to the base multiplier library are similar to the first and second base multipliers described above and therefore, for the sake of clarity, are not shown.  
      According to the present invention, a triple expansion scheme optimizes the multiplier decomposition, resulting in naturally rectangular shapes and simple circuit wiring, thus effectively minimizing global complexity of the design of multipliers. The Simulations indicate that significant reductions can be achieved on overall design cost, power, and VLSI (very large scale integrated circuit) area, which is at least 25% smaller, and is much simpler than conventional multipliers. A comparison of multipliers according to the present invention with conventional multipliers is shown in Table 1 below.  
                                   TABLE 1                           area - scaled *   operation       process *   self       multiplier   relative value   frequency - tech   power   complexity   testable                                                            6-bit   borrow parallel       1 GHz-0.18 μm, 1.8 V           yes           binary   1836 μm 2  - 1   1 GHz-0.18 μm, 1.8 V   0.83 μW   high   yes       54-bit   triple expanded           NA *           rectangular styled   0.98 mm 2  - 2   0.6 GHz-0.18 μm, 1.8 V   NA   high   no           Wallace tree [7]           limited switch   0.15 mm 2  - 1   2 GHz-0.13 μm, 1.2 V   522 mW   high   no           dynamic logic [8]                  
 
      In Table 1, “area—scaled relative value” refers to a scaled-for-technology based on Montoye&#39;s teachings; “operation frequency-tech” refers to the operational frequencies; “power” refers to power consumption of the multiplier; “process complexity” refers to the complexity of the multiplier and takes into account the amount of custom design-layout necessary, the difficulty of implementing the technology and the cost to both design and implement; and “self testable” refers to the stability of the multiplier.  
      The triple expansion method optimizes only one column of a plurality of CSA block columns in a multiplier processing a plurality of bit inputs. The method provides a first level of application of a triple expansion scheme PxP, where P is (3 m+z1), m is an integer multiplier, and z1 is {0, 1, −1}; and when required expanding the first level of application according to a ExE, where E is (3P+z2) and z2 is {0, 1, −1}.  
      Efficient small multipliers of any magnitude may be considered as bases for the triple expansion to yield large multipliers. In an exemplary embodiment, the present invention has adopted two types of 6×6 and 7×7 multipliers shown in  FIGS. 2A and 2B , respectively. The multipliers  200 A and  200 B of  FIGS. 2A and 2B  respectively are borrow parallel small multipliers, which use a single array of borrow parallel counters. The multiplier circuits will be described in detail below. Both multipliers receive two 6-bit input numbers, J and K generate a small partial product bit matrix, and then reduce it into two numbers P (p10-p0) and Q (q10-q5), so that J*K=P+Q* 2 ** 5 . The (4,2)−(3,2) based 6×6 multiplier  150  of  FIG. 4A  uses slightly fewer transistors, while the borrow parallel 6×6 multiplier  152  of  FIG. 4B  has a more compact layout and mainly performs logic with 4b-1-hot signals that feature lower switching activity and use fewer hot lines.  
      Diagrams illustrating multiplier triple expansion schemes are shown in  FIGS. 3A-3C . An MxM multiplier  300 A is constructed using 9 smaller multipliers M 1 -M 9  (e.g., 6×6-b multipliers) and large carry-save adder  304 A. The multiplier&#39;s  300 A inputs  302 A include words J and K each having a given width (e.g., 6 bits). Using a trisect decomposition approach, the inputs J and K are trisected into input group-bits or six-bit segments, partitioned and distributed to the multipliers M 1 -M 9 . The multipliers M 1 -M 9  then form partial product matrices (e.g., 6×6-b matrices) and 9 products (e.g., 12-b products) which are then input into the large carry-save adder  304 A which computes a final product.  
      Multiplier  300 B in  FIG. 3B  is a 18-18-b multiplier and has two 18-b inputs J and K and includes 9 6×6 multipliers M 1 B-M 9 B (whose connections are shown) which output their results to a Level-1 small carry-save adder  304 B.  
      Multiplier  300 C is a 54×54-b multiplier which is similar to the multipliers  300 A and  300 B shown in  FIGS. 3A and 3B  with the following differences. J and K are each 54-b inputs, multipliers M 1 C-M 9 C are each 18×18-b, and a Level-2 small carry save adder  304 C is used to add the outputs of multipliers M 1 C-M 9 C.  
      A diagram illustrating a Level-1 multiplier triple expansion scheme is shown in  FIG. 4 . An 18×1 8-b virtual multiplier  400  includes nine 6×6-b multipliers  402 , an array of counters including  5 _ 1   s    404  in the middle and 3:2s in each end  410  and a segmented simple adder  408 . Note that by replacing the segmented simple adder with a carry-look-ahead adder, an 18×18 multiplier is obtained. To construct an NxN multiplier for some N(&lt;34), one or two of the dotted areas  406  may be used for adder layout when necessary.  
      A diagram illustrating a Level-2 multiplier triple expansion scheme is shown in  FIG. 5 . A 54×54-b multiplier  500  includes nine 18×18-b multipliers  502  plus an array of counters including  5 _ 1   s  and  6 _ 1   s    504  in the middle and 3:2s  510  in the ends, plus a carry look-ahead fast adder  508 . Note that dotted areas  506  may be used for adder layout.  
      A diagram illustrating 2:2 and 3:2 binary counters and their corresponding symbols is shown in  FIG. 6 .  
      A diagram illustrating a 6-b high-speed and compact ripple-carry adder SA 6  is shown in  FIG. 7 . The adder inputs (which are the outputs of bit a matrix reduction network or a CSA array, i.e., generated from the borrow parallel counters) and outputs bits S 0 -S 6 .  
      Diagrams illustrating a modification of a 3m-b (where m=6) multiplier into a (3 m+1)-b multiplier and a (3 m− 1 )-b multiplier are shown in  FIGS. 8A-8C , respectively.  
      A diagram illustrating a partial product matrix of an mxm multiplier (where m=4) is shown in  FIGS. 9A-9B . The original partial product matrix  900 A is shown in  FIG. 9A , and a modified matrix  900 B is shown in  FIG. 9B . The modified matrix  900 B is a modified for 2&#39;s complement form inputs, and each solid circle represents the complement of an initially generated bit and a hidden-bit  1  is added on column m=4 (There are 7 columns from  0  to  6 ). For a more information see, C. R. Baugh and B. A. Wooley, “A Two&#39;s Complement Parallel Array Multiplication Algorithm,” IEEE Tran. on Computers, Vol. C-22, pp. 1045-1047, 1973.  
      The Multiplier Library  
      The multiplier library includes the following components:  
      (1) NxN Multipliers  
      Base Multipliers (3-b to 11-b Multipliers)  
      Each base multiplier includes :(a) an array of borrow parallel counters (including one or more optional 3:2 counters) which serves as a virtual base multiplier; and 
          (b) a ripple-carry or a single-level carry-look-ahead adder, which produces the final product (see  FIGS. 2A and 2B ). 
 
 (2) Mid-Size Virtual Multipliers and Multipliers (12-b to 33-b Multipliers) 
       

      Each mid-size virtual multiplier includes: 
          (a) nine base multipliers of either the same type or no more than two different types (e.g., having  5 _ 1  multipliers or a  5 _ 1  and a  5 _ 1 _ 1  multipliers, etc.);     (b) an array of borrow parallel counters (including one or more 3:2 counters located in two end positions) which serves as a one-stage carry-save addition operator reducing no more than 5 input bits in each column into an output of two bits; and,     (c) a segmented ripple-carry or a single-level carry-look-ahead adder, i.e., an array of smaller adders, which produces the final product plus a few extra bits. Two short ripple-carry adders over lapped at one bit, which is an extra bit in designated columns so that no two extra bits will be produced in the same column when they reach to the next stage (e.g., see  FIG. 4 ). This can be controlled by a simple location-related scheme. Each mid-size multiplier is the same as a mid-size virtual multiplier, except that its final adder is not segmented but is a one- or two-level carry-look-ahead final adder, which produces the final product. 
 
 (3) Large-Size Multipliers (34-b to 99-b Multipliers) 
       

      Each large-size multiplier includes: 
          (a) nine midsize virtual multipliers of the same type or no more than two types;     (b) an array of borrow parallel counters (including one or more optional 3:2 counters in two end positions) which serves as a one-stage carry-save addition operator reducing no more than 6 input bits in each column into an output of two bits; and     (c) a three-level fast carry-look ahead final adder which produces the final product (e.g., see  FIG. 5 ). 
 
 (4) The Binary Counters and Adders 
       

      The present invention modifies the 2:2-3:2 counters which are disclosed in U.S. Patent Publication No. 2001/0,056,455, entitled “A Family Of High Performance Multipliers And Matrix Multipliers,” to R. Lin, which is incorporated herein by reference, to build the above multipliers with ripple carry adders (i.e., for triple expansion cases as opposed to double expansion cases.) (see  FIG. 6 ). The binary counters and the constructed adders (see  FIG. 7 ) include the following features: 
          (a) simple and compact, with a good layout that can well match a  5 _ 1  counter layout;     (b) high speed on carry propagation;     (c) low power. A simulation has shown that each small adder or segmented adder used in the above library components has a delay comparable to a single  5 _ 1  counter delay (about 650 ps with a 0.18 mm, 1.8 V technology).        

      The Modification of 3m-B Multipliers into (3 m+1)-B And (3 m−1)-B Multipliers  
      Each 3m-b multiplier can be modified to yield a (3 m+1)-b or a (3 m−1)-b. Very little modification is needed in layout for each of them.  FIG. 8  illustrates the process briefly.  
      (1) The self-test programs Generic test programs exist. Due to the highly regular and modular structure, a test is partitioned into testing each borrow parallel counter and each 3:2 counter.  
      (2) 2&#39;s Complement NxN Multipliers  
      Each NxN multiplier can be modified easily to obtain a two&#39;s complement multiplier by introducing two borrow counter variants  5 _ 1 ′ and  6 _ 0 ′, which are the same as  5 _ 1  and  6 _ 0  counters except that each contains an extra hidden input 1 (e.g., a logic 1). Simulations show that the features of the modified circuits (e.g., inputs, circuits, layout, etc. other than the extra inputs which are equal to a logic 1) are the same as those of the original circuits. The scheme for this process is based on C. R. Baugh and B. A. Wooley, “A Two&#39;s Complement Parallel Array Multiplication Algorithm”, IEEE Tran. on Computers, Vol. C-22, pp. 1045-1047, 1973, which is incorporated herein by reference, and is as illustrated in  FIGS. 9A and 9B .  
      (3) Pipelined Multipliers  
      Each NxN multiplier can also be modified easily to obtain a pipelined multiplier (more meaningfully for none-base N&gt;11 multipliers). For a mid-size multiplier, four-stage pipelining may be used. Stages  1  and  2  are for the two steps of base multiplier operation, i.e., generating two numbers and then the product; Stages  3  and  4  are for level-1 CSA operation and the final addition. Each stage has about the same delay (less than 1 ns). For a large-size multiplier, six-stage pipelining may be used. Stages  1  to  3  are the same as those for a mid-size multiplier. Stage  4  generates a final product plus a few extra bits for each mid-size multiplier. Stages  5  and  6  are for level-2 CSA operation and the final addition. Each stage has about the same delay (less than 1 ns).  
      Other Detailed Library Components and Drawings  
      (1) Carry-Look-Ahead Adders  
      Modified tiny shift switch binary 2:2 and 3:2 counters (e.g., shown in  FIG. 6 ) can be directly used (with an extra output bit p added) to construct carry-look-ahead adders as shown in FIGS.  10  to  20 .  
      (2)The Modification of 3m-b Multipliers into (3 m+1)-b and (3 m−1)-b Multipliers  
       FIG. 21  illustrates the partial product bit matrix generated by two (3 m+1)-b numbers for m=5. With the indicated re-arrangement (as shown by the 10 arrows), there are nine square partial product matrices. Six of them are 5×5-b, and three of them are 6×6-b. Therefore, the process can be realized using hardware which is similar to that shown in  FIG. 8A  (note: sizes are slightly different). For a more detailed description of this rearrangement, see the &#39;439 Publication.  
       FIG. 22  shows the partial product bit matrix generated by two (3 m−1)-b numbers for m=4. With the indicated re-arrangement (by  6  arrows plus 2 zero bits), there are nine square partial product matrices. Six of them are 4×4-b, and three of them are 5×5-b. Therefore, the process can be realized using hardware which is similar to that shown in  FIG. 8C  (note: sizes are also slightly different).  
      The CSAs modifications for the carry-save reduction are illustrated in FIGS.  23  to  25 .  FIG. 23  shows the 18×18 multiplier carry-save reduction.  FIG. 24  shows the 19×19 barray-save reduction slightly modified from  FIG. 23 .  FIG. 25  shows the 17×17 barray-save reduction slightly modified from  FIG. 23 .  
      (3)The Organization of Balanced Segmented Adders  
      FIGS.  26  to  28  show a 54×54 multiplier;  
      FIGS.  29  to  32  show a 63×63 multiplier;  
      FIGS.  33  to  36  show a 72×72 multiplier; and  
      FIGS.  37  to  39  show a 99×99 multiplier.  
      (4) Borrow parallel counters for 2&#39;s complement multipliers  
       FIG. 40  illustrates a modified  5 _ 1  borrow parallel counter denoted by  5 _ 1 ′, which is the same as a regular  5 _ 1  counter except that its input includes a hidden 1, i.e. it implements 1+A 1 +A 2 +A 3 +A 4 +2A 5 +2Xi+4(Yi+2Yi′Zi)=Xo+2Yo+4(Yo′Zo+L)+8U; (and Zo=Xi). Since a  6 _ 0  is synthesized by a  5 _ 1  counter and a 3:2 counter, the  6 _ 0 ′ and  7 _ 0 ′ counters can be constructed by a  5 _ 1 ′ counter with a 3:2 and a  5 _ 1 ′ counter with two 3:2 counters respectively.  
      Modified small multipliers 4-b to 11-b from NxN-b multipliers for n between 4 to 11 are shown in FIGS.  41  to  48  to 2&#39;s complement NxN multipliers.  
      While the invention has been shown and described with reference to a certain preferred embodiment 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 spirit and scope of the invention as defined by the appended claims.