Fast Multiplier

This invention relates generally to improved means and methods for performing arithmetic operations in a data processing system, and more particularly to an improved high speed binary multiplier provided on an integrated circuit chip.

In designing a high speed binary multiplier on an integrated circuit chip, two important considerations are operating speed and required chips area. Most multiplier designs attempt to make some trade-off between the two, particularly where available chip area is limited, as is usually the case.

The primary object of the present invention is to provide improved means and methods for increasing multiplier operating speed while providing a minimum required chip area.

In a particular preferred embodiment of the invention, a Wallace-type binary tree multiplier is provided in which the partial products of a multiplicand A and a multiplier B are produced and then reduced by successive addition using a plurality of adder levels comprised of full and half adders. This reduction continues until a final set of inputs is produced having no more than two inputs remaining to be added in any column. This final set is then added using two side-by-side final adders to produce the final product. In this preferred embodiment, the particular inputs to be added by the full and half adders at each level are performed in accordance with prescribed rules to provide for fastest overall operating speed and minimum required chip area. In addition, the side-by-side final adders are chosen of prescribed type and length to take advantage of the earlier arrival times of least significant bits as compared to bits of higher significance.

The specific nature of the invention as well as other objects, features, advantages and uses thereof will become evident from the following detailed description of a particular preferred embodiment taken in conjunction with the accompanying drawings.

FIG. 1 illustrates the well known pencil-and-paper method of multiplication applied to binary multiplication.

FIG. 2 illustrates the method of FIG. 1 applied to the multiplication of an 8-bit multiplicand by an 8-bit multiplier.

FIG. 3 (including FIGS. 3A-3F) is a preferred binary multiplier circuit in accordance with the invention.

FIG. 4 is a block diagram illustrating a full adder FA which may be employed in the preferred binary multiplier circuit of FIG. 2.

FIG. 5 is a block diagram illustrating a half adder HA which may be employed in the preferred binary multiplier circuit of FIG. 3.

FIGS. 6, 7, 8, 9 and 10 schematically illustrate how the levels of FIG. 3 provide for progressively reducing the partial products in FIG. 2 to produce two final rows of inputs for application to a final adder stage.

FIG. 11 illustrates a preferred embodiment of the serial adder 20 in FIG. 3.

Like numerals and characters refer to like elements throughout the figures of the drawings.

Initially, reference is directed to FIG. 1 which illustrates the well known paper-and-pencil method applied to the multiplication of a 4-bit the binary multiplicand A=1100 (decimal 12) by the 4-bit binary multiplier B=1101 (decimal 13) to produce the 8-bit product P=10011100 (decimal 156).

FIG. 2 illustrates the application of the well known paper-and-pencil method of FIG. 1 applied to the multiplication of an 8-bit multiplicand A=a.sub.7 a.sub.6 a.sub.5 a.sub.4 a.sub.3 a.sub.1 a.sub.0 by an 8bit multiplier B=b.sub.7 b.sub.6 b.sub.5 b.sub.4 b.sub.3 b.sub.1 b.sub.0 to produce a 16bit product P=p.sub.15 p.sub.14 p.sub.13 p.sub.12 p.sub.11 p.sub.10 p.sub.9 p.sub.8 p.sub.7 p.sub.6 p.sub.5 p.sub.4 p.sub.3 p.sub.2 p.sub.1 p.sub.0 obtained by adding the "ab" partial products in each column.

FIG. 3 is a preferred embodiment of a binary multiplier circuit which illustrates how the present invention may preferably be implemented for performing the multiplication exemplified in FIG. 2.

As shown in FIG. 3, the 8-bit multiplicand A=a.sub.7 a.sub.6 a.sub.5 a.sub.4 a.sub.3 a.sub.2 a.sub.1 a.sub.0 and the 8-bit multiplier B=b.sub.7 b.sub.6 b.sub.5 b.sub.4 b.sub.3 b.sub.2 b.sub.1 b.sub.0 are applied to an initial multiplier 10 for producing the sixty-four partial products a.sub.0 b.sub.0 a.sub.0 b.sub.1 -a.sub.7 b.sub.7 shown in FIG. 2. Note that each partial product is in a respective column wherein each column corresponds to a respective product bit. These partial products are applied to a plurality of adder levels L1 to L4 comprised of full adders FA and half adders HA for successively reducing the number of column inputs applied to each level until level L4 is reached which produces a final set of column inputs wherein no more than two inputs remain to be added in any column. These remaining are then applied to side-by-side final adders 20 and 21 to produce the resulting product P=p.sub.15 p.sub.14 p.sub.13 p.sub.12 p.sub.11 p.sub.10 p.sub.9 p.sub.8 p.sub.7 p.sub.6 p.sub.5 p.sub.4 p.sub.3 p.sub.2 p.sub.1 p.sub.0.

Those skilled in the art will recognize that the preferred multiplier circuit illustrated in FIG. 3 is of the general Wallace type described in "A Suggestion for a Fast Multiplier", C. S. Wallace, Vol. 13, No. 14, IEEE Transactions on Electronic Computers (Feb. 1964), pp. 14-17. The present invention enhances the basic Wallace approach in a manner not previously known or taught in the art in order to achieve a significantly better combination of operating speed and required chip area. The manner in which these enhancements are incorporated in the preferred embodiment of FIG. 3 will next be described.

First to be considered is the manner in which the full adders FA and half adders HA in the preferred multiplier circuit of FIG. 3 are chosen for reducing the sums output from each level. The basic rules used in FIG. 3 are as follows. For each column at each level, a full adder FA is connected to three-input groups until less than three inputs remain in the column at that level. If only two inputs remain in a column, or if the column initially has only two inputs, then an adder (preferably a half adder HA) must be used for adding these two inputs if both of the following apply: (1) the adjacent less significant column will produce a carry into this column, and (2) this two-input addition is needed to achieve a 3-to-2 reduction (rounded-down to the nearest integral ratio) of the inputs in this column. Any inputs not added at a level are passed on to the next level.

Before describing how the above rule is applied in the preferred multiplier circuit of FIG. 3, attention is directed to FIGS. 4 and 5 which respectively illustrate conventional implementations for the full adders FA and half adders HA in FIG. 3. In these figures, each gate 6 performs an AND function, gate 7 performs an OR function and each gate 8 performs an exclusive OR function. Note that the full adder FA is slower than the half adder HA since it requires two gate levels compared to one for the half adder HA. Also note that the full adder FA requires significantly more chip area than does the half adder HA.

Reference is next directed to FIGS. 6-9 which schematically illustrate how the above described rules are applied at each level of the multiplier circuit of FIG. 3. FIG. 6 illustrates the inputs to level L1, which are the "ab" partial products shogun in FIG. 2.

FIG. 6 also indicates which "ab" inputs to level L1 are added in each column by use of enclosing loops. If the loop encloses three inputs, a full adder FA (FIG. 4) is used, and if the loop encloses two inputs, a half adder HA (FIG. 5) is used. Each loop includes a designation of the resulting sum and carry of level L1 which are applied as inputs to the next level L2. For example, adding a.sub.2 b.sub.0, a.sub.1 b.sub.1 and a.sub.0 b.sub.2 in column 2 results in a sum i.sub.21 and a carry i.sub.21c, the subscript "c" being used to identify a carry. Note that sum i.sub.21 produces an input in the same column 2 in the next level L2, while carry i.sub.21c produces an input in column 3 of level L2.

FIGS. 7, 8, 9 and 10 illustrate the inputs to levels L2, L3 and L4, respectively, and are arranged similarly to FIG. 6 with loops being provided for a like purpose. Note that the inputs to level L2 shown in FIG. 7 use "i" designations, the inputs to level L3 shown in FIG. 8 use "j" designations, the inputs to level L4 shown in FIG. 9 use "k" designations, and the inputs to the final adders 20, 21 use "1" designations. The first subscript (or first two subscripts for columns greater than 9) of these i,j,k designations identify the column from which the sum or carry was derived. The second subscript is merely used to distinguish sums and carrys derived from the same column.

The manner in which the above rule is applied at each level in FIG. 3 will next be considered in detail with reference to FIGS. 6-10.

It will be seen in FIGS. 3 and 6 that each three-input group in each column of level L1 is added by a full adder FA to produce a sum and a carry. For example, in column 2 of level L1, a.sub.2 b.sub.0, a.sub.1 b.sub.1 and a.sub.0 b.sub.2 are added by a full adder FA to produce a sum i.sub.21 and a carry i.sub.21c. The sum "i.sub.21 " as applied to level L2 in FIG. 7 in the same corresponding column 2, while the carry "i.sub.21c " is applied to the next column 3; in level L2, since that is where it is added. With regard to the remaining two-input groups in the columns of level L1, the above rules do not require any of these to be added in level L1, as will now be explained.

It will be remembered that the above rules state that a two-input group in a column at a level are added if both of the following apply: (1) the adjacent less significant column will produce a carry into this column, and (2) addition is needed to achieve a 3-to-2 reduction (rounded-down to the nearest integer ratio) of the inputs in this column.

Only columns 1, 4, 7, 10, 13 in level L1 have a two-input group for which an add decision has to be made. The two inputs in column 1 meet neither requirement (1) or (2) of the rule. The remaining columns 4, 7, 10 and 13 also are not required to be added. Although they meet requirement (1), they do not meet requirement (2). This will be understood by noting that the maximum number of inputs applied to level L1 is eight (in column 7). Thus, in order to achieve a 3-to-2 reduction, the maxima number of inputs applied to level L2 in any column must be six or less (since an 8-to-6 reduction meets the 3-to-2 reduction requirement). The fact that addition of none of the two-input groups in level L1 are required to meet the rules will become evident by noting that all of the columns in level L2 in FIG. 7 have six or less inputs even though none of the two-input groups are added in level L1.

In level L2 in FIG. 7, only columns 1, 6, 8, 10 and 12 require decisions to be made with regard to adding a two-input group. Column 1 need not be added since it does not meet either requirement (1) or (2) of the rules. The two-input groups in columns 1, 6, 10 and 12 need not be added. This will be evident from FIG. 8 which shows that, even though these two-input groups are not added, none the effected columns in level L3 (FIG. 8) will have inputs which exceed the maximum of 4 dictated by the required 3-to-2 reduction (6-to-4) for level L2. However, the rule requires adding of the two-input group a.sub.7 b.sub.1, i.sub.81 in column 8 of level L2 (FIG. 7), since seven inputs would otherwise result in column 8 of level L3 (FIG. 8), exceeding the maximum of 4. By adding a.sub.7 b.sub.1, i.sub.81, the column 8 inputs to level L3 are held to the maximum of 4.

In level L3, decisions with respect to adding two-input groups need only be made for columns 1, 5, 11 and 14. Columns 1 and 14 need not be added since they do not meet either requirement (1) or (2) of the rules. The two-input groups in columns 5, 11 and 14 also need not be added. Although they meet requirement (1), they do not meet requirement (2). This will be understood by noting that the required 3-to-2 reduction ratio (rounded-down to the nearest integer) is met for level L3 by providing a 4-to-3 reduction (the nearest rounded-down integer ratio). This reduction is met for these columns since, even though these two input groups are not added, none of the effected columns in level L4 (FIG. 9) have inputs which exceed the maximum of 4 dictated by the required 4-to-3 reduction for level L3.

In level L4, adding decisions need to be made only for columns 1, 6, 7, 13 and 14. Columns 1, 13 and 14 need not be added, since neither requirement (1) nor (2) of the rule is met for these columns. However, the two-input groups in columns 6 and 7 need to be added in order to meet the 3-to-2 reduction requirement for level L4, otherwise three inputs would be applied to the effected columns of final adders 20 and 21, rather than the no more than two inputs per column applied to these adders 21 and 22 shown in FIG. 10.

It has thus been explained how additions of inputs are performed at each level of the preferred multiplier circuit in accordance with the prescribed rules. Although the preferred multiplier circuit of FIG. 3 performs only those two-input additions required by the rules, it is within the scope of the present invention to allow addition of a remaining two-input group in a column even though not required by the rules. For example, this may be done for routing convenience. Such instances should be limited to prevent exceeding a desired predetermined minimum chip area.

The remaining parts of the preferred multiplier circuit of FIG. 3 to be considered are the final adders 21 and 22, which receive the two rows of inputs shown in FIG. 10. Note that the least significant product bit P.sub.0 is obtained directly from the partial product a.sub.0 b.sub.0 in column "o" and need not be included in the final addition.

Adder 21, receives the inputs from column 1-5 (FIG. 10) and is chosen to be simply a conventional serial adder (without lookahead), such as illustrated in FIG. 11. Adder 22 is chosen to be a conventional carry lookahead adder, which receives the output carry C.sub.4 from serial adder 21 and also the inputs from columns 6-15 (FIG. 10). This choice of length for the serial adder 21 in the preferred multiplier circuit of FIG. 3 is approximately 1/3 of the total number of columns required to be added in this final adder-stage, which has been found to be a preferred length for a Wallace-type multiplier. This choice of the two adders 21 and 22 takes advantage of signal flow through levels L1-L4 of the multiplier circuit of FIG. 3 which causes more significant inputs in FIG. 10 to arrive progressively later than less significant inputs. The number of stages provided for the serial adder 20 (FIG. 11) is chosen so that its output carry C.sub.5 (FIG. 3) arrives no later than the arrival of the column 6 inputs applied to the carry lookahead adder 21. It is most advantageous that the number of stages for the serial adder 20 be chosen so that its output carry c.sub.5 arrives at approximately the same time as the arrival of the column 6 inputs. Note that the serial adder 20 shown in FIG. 11 advantageously employs half adders, thereby permitting faster generation of the output carry C.sub.5.

The above choice of the adders 20 and 21 in FIG. 2, as described above, provides a further significant enhancement in multiplier performance. The serial adder 20, because of its simplicity, requires relatively less chip area; yet, its slower addition does not detract from overall adder speed, since its length is advantageously chosen so that its output carry C.sub.5 is produced at approximately the same time as the arrival of the column 6 inputs applied to the carry lookahead adder 22. Also, because the use of the serial adder 20 permits a smaller carry lookahead adder 21 to be used, it will be faster and require less chip area. As is well known, the propagation delay of a conventional carry lookahead adder increases progressively as the number of bit positions it is required to handle increases.

It is to be understood that the present invention is not limited to the preferred embodiment of the invention described herein, since many variations in construction arrangement, use and operation are possible within the scope of the invention. For example, the invention can readily be adapted for use when the original multiplier and multiplicand are represented in two's complement binary encoded format. Also, the invention may be adapted for use with other types of multipliers, such as a Booth multiplier.

Accordingly, the present invention is to be considered as encompassing all modifications, variations and adaptations coming within the scope of the invention as defined by the appended claims.