Weighted-delay column adder and method of organizing same

An adder array for adding two or more input addends, whose bit lengths are not necessarily matched, and a method of configuring the adder array are disclosed. The addends are organized according to bit weight, and bits of equal weight are added in adder columns. Carry-outs are introduced into subsequent, higher weight adder columns according to delay. Thereby, the delay associated with the addition of the addends is minimized. Method and apparatus is disclosed.

TECHNICAL FIELD OF THE INVENTION 
The present invention relates to digital adders, such as may be employed in 
the sum-of-products portion of multipliers, multiplier-adders, such as may 
be embodied in a "Wallace Tree" structure, and in "n" addend adders. 
BACKGROUND OF THE INVENTION 
The simplest form of multiplier is the "AND" gate. The output of such a 
gate may be viewed as the one-bit product of two one-bit inputs. However, 
the utility of using a single AND gate for multiplying is limited to 
multiplicands of only one binary bit, producing a one bit product. Binary 
numbers (multiplicands) of any bit length may be multiplied by a one-bit 
binary number (multiplier) by using a number of "AND" gates equal to the 
number of bits in the multiplicand. Assuming a one-bit multiplier and an 
`n`-bit multiplicand, one input of each of the `n` gates is assigned to 
one bit of the multiplicand while the other input of all of the gates is 
connected to the multiplier. If the outputs of this array of "AND" gates 
is taken as an `n`-bit binary number, it represents the product of the two 
numbers. A simple 1-bit by 8 -bit instance of this type of multiplier is 
shown in FIG. 1. 
If two binary numbers, each having a length of `n` bits, are to be 
multiplied together, the aforementioned scheme may be replicated for each 
bit of the multiplier. Each of the resulting bits represents a partial 
product. These partial products must be summed to arrive at the resulting 
fully resolved product of the multiplier and multiplicand. In the process 
of summings, it must be recognized that each one-bit partial product has a 
binary weight (i.e., significance) associated with it, i.e., the bit value 
will be one or zero, but it represents a binary value which is determined 
by the product of the binary weights of the two bits whose product it 
represents. 
The complexities encountered in the design of parallel adders suitable for 
the sum-of-products application in parallel multipliers led to the 
development of the "Wallace Tree". Simply stated, the Wallace Tree is a 
logical connection of full adders which can simply be replicated (in an 
array) to perform the sum of products calculation. An example of a Wallace 
Tree is the Texas Instrument type SN54LS275 4-bit-by-4-bit binary 
multiplier with 3-state outputs, 7-bit slice Wallace Tree with 3-state 
outputs. 
As is well known, the full adder is a logical device that has three one-bit 
inputs, a one-bit sum output and a one-bit carry out. It is also well 
established that the full adder interposes a delay, hereinafter referred 
to as a one "unit" (of time) delay, between its inputs and outputs. Hence, 
in any interconnection of full adders, such as in a typical Wallace Tree, 
the various columns being added experience different delays due to their 
different logic path lengths. For instance, in a typical Wallace Tree, 
such as the aformentioned SN54LS275, the least significant digit (column) 
passes straight through, experiencing a delay of only one unit, while 
intermediate significant digits experience a greater delay of three units. 
Evidently, as the Wallace Tree is replicated, forming an array to handle 
larger and larger multiplicands and multipliers, greater and greater 
delays are interposed, resulting in slower response times for larger 
arrays. The effective system delay for any such adder, regardless of 
configuration, is the delay incurred in the longest path from input to 
output, since the result may not be considered valid until its last 
component bit has settled to a stable state. 
A significant contributor to the delay in current parallel adder 
configurations is the "carry" logic. Typically, all of the bits of a given 
weight are summed to arrive at intermediate multi-bit sums. For example, 
if seven input bits are to be summed, an array of adders is typically 
configured which will produce a 3 bit binary result whose value may range 
between 0 and 7 (000 and 111 binary). The least significant bit of this 
sum has the same binary weight as the component (input) bits, but the 
other output bits are carried over to neighboring (more significant) 
columns according to their binary weight, where they must be summed with 
other component bits having the same weight. The bit having the same 
weight as its component bits may yet have to be added to other component 
bits at the same weight or combined with other partial multi-bit sums. 
This may, in turn, produce further multi-bit sums which must further be 
added to other component bits. As a result of this process of adding 
intermediate sums together to arrive at other intermediate sums, the 
longest path in the circuit can become quite long. Further, the carry bits 
between columns are typically inserted in the adder chain at a point which 
is logically convenient, but which may not necessarily be the most 
efficient in terms of delay. 
While the Wallace Tree offers a straightforward, systematic, standardized 
approach to multiplication, and multiplier design, it is very gate 
intensive and it is usually not optimized for performance. 
Non-standardized (hand) design of multipliers, adders and multiplier-adders 
is very time consuming and error prone. Various compilers which have been 
employed for this task are limited in their flexibility and optimization 
approaches. 
DISCLOSURE OF THE INVENTION 
It is therefore an object of the invention to provide a technique for 
improving the performance of the "Wallace Tree" section of multipliers, 
multiplier-adders and "n" addend adders. 
It is a further object of the invention to provide an adder capable of 
dealing with dissimilar column heights, wherein each multiplicand or 
multiplier doesn't have to have the same bit width or weight. 
It is a further object of the invention to provide a standardized method of 
designing multiplier-adders. 
It is a further object of the invention to provide a method of designing 
different sizes and features of multipliers, adders, etc. 
It is a further object of the invention to provide an adder allowing for 
unlimited height on any given column. 
It is a further object of the invention to provide an adder that can add 
numbers with different bit width and weight(s). 
It is a further object of the invention to provide an adder that approaches 
having a minimum delay associated with the final result. 
It is a further object of the invention to provide an adder that has a 
highly gate-efficient design. 
It is a further object of the invention to provide a fast, gate-efficient 
"Wallace Tree" with "n" partial products generated from one or more 
multiplicand and multiplier pairs. 
According to the invention, the Weighted-Delay Column Adder adds (sums) 
input bits on a column-by-column basis, starting from the column having 
the input bits with the least binary-weight (significance). Based on the 
number of input bits in any particular column, an adder chain (column 
array) is configured for each column to terminate the carry-outs as soon 
as possible so that at most two one-bit column results can be added by a 
fast (parallel) adder to obtain a final result. Propagation delays (D) of 
intermediate sums and carry-outs throughout the adder chains are accounted 
for. 
With the weighted-delay column adder of the present invention, the height 
of the various columns (i.e. the number of inputs in any particular 
column) may be dissimilar, and there is no limit to the height of a 
column. By allowing unlimited height on any given column, special "n" 
addend adders can be configured for use in indexing, or in adders that can 
add numbers with different bit width(s) or weight(s). In addition to its 
utility in association with multiplier-adders, or for obtaining the sum of 
(partial) products, such as may result from multiplications such as 
A*B+C*D +E*F+. . . with different bit width and weight(s), the techniques 
disclosed herein result in minimum delay in the final result of binary 
addition and highly gate-efficient designs. 
The Weighted-Delay Column Adder is configured by analyzing columns of 
component (input) bits, such as may be result from the partial products of 
a multiplication, one column at a time, starting from the column 
containing input bits having the least weight or weights (least 
significant column of bits). Each of the input bits arriving at the 
Weighted-Delay Column Adder is considered to have a "zero delay", prior to 
its introduction to the Adder. In the case of multiplying binary numbers 
the input bits would be one-bit partial products of the multiplication. 
One skilled in the art to which this invention most nearly pertains will 
recognize that the input bits (of the addends) may not arrive at the 
weighted-delay column adder at the same time, and their arrival times 
should be accounted for. Hence, the assumption of a "zero" delay for the 
input bits is intended to be for illustrative purposes only. 
Carry-outs from column-to-column (adder chain to adder chain) are 
classified (weighted) according to delay, for instance by taking the 
number of adders in the longest path from the zero-delay inputs to the 
carry-out under consideration. 
The Weighted-Delay Column Adder is designed to add "n" addends of varying 
widths as fast as possible by taking into account each column of bits and 
the delay of each bit as it propagates through the column, from top to 
bottom. Terminating the bits with the longest delay first, the least 
significant column is structured as a chain (vertical array) of adders, 
and its carry-outs are provided to the next significant bit column. The 
next significant bit column is similarly structured, and so on. By knowing 
the number of bits in a column, an adder chain can be constructed to 
producing a valid bit value for that column. 
In the resulting, exemplary hardware configuration of the weighted-delay 
column adder, the construction of the first column (e.g. 2.sup.0) is very 
straightforward, since there are no carries into the 2.sup.0 column, and 
all of the input bits are considered to be zero-delay inputs. For example, 
if the 2.sup.0 column contains seven component input bits, then a "chain" 
of three full-adders can be constructed to arrive at the correct end 
result for the 2.sup.0 column. Three of the 2.sup.0 weighted component 
bits will be placed at the inputs of the first adder, three at the second 
and one at the third. The intermediate sum outputs of the first and second 
adders will be connected to the remaining two inputs of the third adder. 
Each of the three adders will have a carry-out to the 2.sup.1 stage. 
Since the final result (out of the third adder) can experience delays from 
up to two adders, (each adder is considered as imposing a one "unit" 
propagation delay), the final result will have a delay of two units. The 
carry-out bits from the first, second and third adders will have unit 
delays of one, one, and two units, respectively, and will be combined in 
the next more significant adder chain with input bits, intermediate sums 
and/or a partial result thereof having the most nearly matched propagation 
delays. In other words, carry-outs from the previous column are combined 
in the present column (adder chain) at points therein having the most 
nearly matched propagation delay associated therewith. 
One skilled in the art to which this invention most nearly pertains will 
recognize that the actual delay imposed between the inputs and outputs of 
an adder is dependent upon logic states and conditions, as well as upon 
device configuration. Hence, the assumption of each adder imposing a 
one-unit delay is for illustrative purposes only. 
In the following and subsequent (next higher) columns, carry-out bits from 
the previous column are introduced into the present column at a point 
where their delay is closely matched to the delay of the input bits, 
intermediate sums and/or column results occurring in the present column. 
Carry-outs are always terminated in the next more significant column, and 
carry-outs with the longest delays are terminated in the last (bottom 
most) segments of the column, and in some cases are terminated at the end 
of the more significant column, so that their delay has a minimal 
contribution to the overall column delay. Each column, or adder chain is 
constructed from full and/or half adders. 
A significant difference between the technique of the present invention and 
previous approaches is that the end of the adder chain for each column 
produces either one or two one-bit column results. In the case of two 
column results, each result is termed a "partial" result. A final 
ripple-carry adder stage or other similar binary adder, such as a parallel 
adder, may be used to combine the closely delay-matched column results 
into a final result. Because carry-outs from each adder chain are 
terminated in the next more significant adder chain, delays are minimized 
and are not redundantly cascaded. In some cases, the carry-out from the 
previous column having the longest delay associated therewith is 
introduced at the end of the next adder chain and is treated as a partial 
result of the next column. 
Other objects, features and advantages of the invention will become 
apparent in light of the following description thereof.

DETAILED DESCRIPTION OF THE INVENTION 
The present invention is particularly useful for adding partial products, 
such as may be generated by a Wallace Tree Array, but may also be used for 
the implementation of functions which would ordinarily require multiple 
Wallace Trees. 
Consider the case of A times B, plus C times D, where A, B, C and D are 
each 4-bit binary numbers. A, B, C and D are readily expressed as follows: 
EQU A=a.sub.3 2.sup.3 +a.sub.2 2.sup.2 +a.sub.1 2.sup.1+ a.sub.0 2.sup.0 
EQU B=b.sub.3 2.sup.3 +b.sub.2 2.sup.2 +b.sub.1 2.sup.1+ b.sub.0 2.sup.0 
EQU C=c.sub.3 2.sup.3 +c.sub.2 2.sup.2 +c.sub.1 2.sup.1+ c.sub.0 2.sup.0 
EQU D=d.sub.3 2.sup.3 +d.sub.2 2.sup.2 +d.sub.1 2.sup.1+ d.sub.0 2.sup.0 
The first partial product of A times B may be expanded as follows, arranged 
as binary bits a.sub.n and b.sub.n in columns of 2.sup.n : 
______________________________________ 
2.sup.6 2.sup.5 
2.sup.4 2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
b.sub.0 a.sub.3 
b.sub.0 a.sub.2 
b.sub.0 a.sub.1 
b.sub.0 a.sub.0 
b.sub.1 a.sub.3 
b.sub.1 a.sub.2 
b.sub.1 a.sub.1 
b.sub.1 a.sub.0 
b.sub.2 a.sub.3 
b.sub.2 a.sub.2 
b.sub.2 a.sub.1 
b.sub.2 a.sub.0 
b.sub.3 a.sub.3 
b.sub.3 a.sub.2 
b.sub.3 a.sub.1 
b.sub.3 a.sub.0 
______________________________________ 
Similarly, the second partial product of C times D may be expanded as 
follows: 
______________________________________ 
2.sup.6 2.sup.5 
2.sup.4 2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
d.sub.0 c.sub.3 
d.sub.0 c.sub.2 
d.sub.0 c.sub.1 
d.sub.0 c.sub.0 
d.sub.1 c.sub.3 
d.sub.1 c.sub.2 
d.sub.1 c.sub.1 
d.sub.1 c.sub.0 
d.sub.2 c.sub.3 
d.sub.2 c.sub.2 
d.sub.2 c.sub.1 
d.sub.2 c.sub.0 
d.sub.3 c.sub.3 
d.sub.3 c.sub.2 
d.sub.3 c.sub.1 
d.sub.3 c.sub.0 
______________________________________ 
Looking at the totality of the partial products A times B and C times D, 
column-by-column, results in: 
______________________________________ 
2.sup.6 2.sup.5 
2.sup.4 2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
b.sub.0 a.sub.3 
b.sub.0 a.sub.2 
b.sub.0 a.sub.1 
b.sub.0 a.sub.0 
b.sub.1 a.sub.3 
b.sub.1 a.sub.2 
b.sub.1 a.sub.1 
b.sub.1 a.sub.0 
b.sub.2 a.sub.3 
b.sub.2 a.sub.2 
b.sub.2 a.sub.1 
b.sub.2 a.sub.0 
b.sub.3 a.sub.3 
b.sub.3 a.sub.2 
b.sub.3 a.sub.1 
b.sub.3 a.sub.0 
d.sub.0 c.sub.3 
d.sub.0 c.sub.2 
d.sub.0 c.sub.1 
d.sub.0 c.sub.0 
d.sub.1 c.sub.3 
d.sub.1 c.sub.2 
d.sub.1 c.sub.1 
d.sub.1 c.sub.0 
d.sub.2 c.sub.3 
d.sub.2 c.sub.2 
d.sub.2 c.sub.1 
d.sub.2 c.sub.0 
d.sub.3 c.sub.3 
d.sub.3 c.sub.2 
d.sub.3 c.sub.1 
d.sub.3 c.sub.0 
______________________________________ 
Evidently, the columns have different "heights". This is more graphically 
represented by "collapsing" the columns, as follows: 
______________________________________ 
2.sup.6 2.sup.5 
2.sup.4 2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
b.sub.0 a.sub.3 
b.sub.1 a.sub.2 
b.sub.1 a.sub.3 
b.sub.2 a.sub.1 
b.sub.0 a.sub.2 
b.sub.2 a.sub.2 
b.sub.3 a.sub.0 
b.sub.1 a.sub.1 
b.sub.2 a.sub.3 
b.sub.3 a.sub.1 
d.sub.0 c.sub.3 
b.sub.2 a.sub.0 
b.sub.0 a.sub.1 
b.sub.3 a.sub.2 
d.sub.1 c.sub.3 
d.sub.1 c.sub.2 
d.sub.0 c.sub.2 
b.sub.1 a.sub.0 
b.sub.3 a.sub.3 
d.sub.2 c.sub.3 
d.sub.2 c.sub.2 
d.sub.2 c.sub.1 
d.sub.1 c.sub.1 
d.sub.0 c.sub.1 
b.sub.0 a.sub.0 
d.sub.3 c.sub.3 
d.sub.3 c.sub.2 
d.sub.3 c.sub.1 
d.sub.3 c.sub.0 
d.sub.2 c.sub.0 
d.sub.1 c.sub.0 
d.sub.0 c.sub.0 
______________________________________ 
Since each of the bits a.sub.n, b.sub.n, c.sub.n or d.sub.n is a single 
bit, each of the partial products b.sub.n a.sub.n and d.sub.n c.sub.n is a 
single bit, and may be obtained with an "AND" gate, as discussed 
hereinbefore. 
All of the partial products are assumed to arrive as input bits at the 
weighted-delay column adder at the same time, with "zero" delay as 
discussed hereinbefore, and the delay through each individual adder in the 
various adder chains is assumed to be one-unit (of time). 
The columns (2.sup.n) are each added with the weighted-delay technique of 
the present invention. 
First, the columns of input bits are "decomposed", one-by-one, according to 
the delays of the bits in each of the columns. An adder chain is 
configured for each column to arrive at either a one- or two-component 
column result, where both components, if present, will have a binary 
weight which is equal to the weight (binary significance) of that column. 
If a column has a two-component result, one of the components will be the 
longest-delay carry-out from the previous column. 
Initially, for example, all of the columns are assumed to be full of 
zero-delay input bits, as shown in the following where each input bit is 
represented by a "0": 
______________________________________ 
2.sup.6 2.sup.5 
2.sup.4 2.sup.3 
2.sup.2 2.sup.1 
2.sup.0 
______________________________________ 
0 
0 
0 0 0 
0 0 0 
0 0 0 0 0 
0 0 0 0 0 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
______________________________________ 
The 2.sup.0 column is processed first. Since there are only two input bits 
in this columns, they may be added with a half-adder. This would produce a 
one-unit delay column result (represented by the "1" beneath the 2.sup.0 
column, below) and a one-unit delay carry-out (represented by the "1" at 
the top of the 2.sup.1 column, below) which is placed in the 2.sup.1 
column, as follows: 
______________________________________ 
2.sup.6 
2.sup.5 
2.sup.4 
2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
0 
0 
0 0 0 
0 0 0 1 &lt;=one-delay carry-out from 2.sup.0 column 
0 0 0 0 0 
0 0 0 0 0 
0 0 0 0 0 0 
0 0 0 0 0 0 
1 &lt;=one-delay column result 
______________________________________ 
Next, the 2.sup.1 column is processed. Three of the four zero-delay input 
bits (represented by "0", above) are added in a full adder, producing a 
one-delay intermediate sum (not shown) and a one-delay carry-out (shown 
below as "1") which is placed in the 2.sup.2 column. 
The one-delay carry-out from the 2.sup.0 column, the remaining zero-delay 
input and the intermediate sum are then added in a second full adder to 
arrive a two-delay column result (shown as "2", below the 2.sup.1 column) 
and a two-delay carry-out (shown as "2", below in the 2.sup.2 column) 
which is placed in the 2.sup.2 column, as follows: 
__________________________________________________________________________ 
2.sup.6 
2.sup.5 
2.sup.4 
2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
__________________________________________________________________________ 
0 2 &lt;=two-unit delay carry out from 2.sup.1 column 
0 1 &lt;=one-unit delay carry out from 2.sup.1 column 
0 0 0 
0 0 0 
0 0 0 0 
0 0 0 0 
0 0 0 0 0 
0 0 0 0 0 
2 &lt;=two-unit delay column result for 2.sup.1 
__________________________________________________________________________ 
column 
Hence, it can already be seen that the carry-outs from a "previous" column 
are inserted for addition into the "next", or present column, and their 
delays are matched as nearly as possible with the delays of intermediate 
sums in the present column. For instance, in the 2.sup.1 column, the 
one-unit delay carry-out from the 2.sup.0 column is added (e.g. in a full 
adder) with the one-unit delay intermediate sum in the 2.sup.1 column, and 
with the zero-unit delay input ebit in the 2.sup.1 column which has a 
lesser delay. 
Next, the 2.sup.2 column is processed. Two full-adders may be employed to 
add the six zero-delay input bits ("0" , above), arriving at two one-delay 
intermediate sums (not shown) and two one-unit delay carry outs (each 
represented by "1", below) which are placed (as carry-ins) in the 2.sup.3 
column. The one-delay carry-out from the 2.sup.2 column and the two 
one-delay intermediate sums may be added with a full adder, producing a 
two-delay column result (shown as "2", below) and a two-delay carry-out 
(shown as "2", below) which is placed in the 2.sup.3 column. The two-delay 
carry-out from the 2.sup.1 column is treated as a "partial" column result 
along with the result (also considered to be a "partial" column result) 
from the last adder in this chain. These two (partial) column results can 
be combined into a final result using a final stage ripple-carry adder 
(shown in FIG. 21, and described hereinafter). 
______________________________________ 
2.sup.6 
2.sup.5 
2.sup.4 
2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
2 &lt;=two-unit delay from previous column 
1 &lt;=one-unit delay from previous column 
1 &lt;=one-unit delay from previous column 
0 
0 
0 0 
0 0 
0 0 0 
0 0 0 
0 0 0 0 
0 0 0 0 
2 &lt;=two-delay result out of 2.sup.2 column 
2 &lt;=two-delay carry-out from 2.sup.1 
______________________________________ 
column 
Again, it is evident that the carry-outs from the previous (2.sup.1) column 
are combined in or at the end (bottom) of the present (2.sup.2) column 
adder chain at points where their delays are closely matched to the delays 
of the intermediate sums or partial results thereof. In the example given 
above, the two-delay carry out from the 2.sup.1 column is treated as a 
partial column result of the 2.sup.2 column. It is further demonstrated 
that having two partial results at the end of the adder chain is 
acceptable, and these two partial results will be combined (added) as 
described hereinafter with respect to FIG. 2I. 
Next, the 2.sup.3 column is processed. Six of the eight zero-delay input 
bits ("0", above) are summed using full adders, producing two one-delay 
carry-outs (shown below) which are placed in the 2.sup.4 column, and two 
one-delay intermediate sums. The two remaining zero-delay input bits and 
one of the one-delay carry-outs from the previous 2.sup.2 column are 
summed, producing a two-delay intermediate sum (not shown) and a two-delay 
carry-out (shown below). The two one-delay intermediate sums and the 
remaining one-delay carry-out from the previous column are added in a full 
adder producing a two-delay partial column result (shown below) and a 
two-delay carry-out (shown below) which is placed in the 2.sup.4 column. 
The two-delay intermediate sum and the two-delay carry-out from the 
previous column are added in a half adder to produce a three-unit delay 
partial result (shown below) and a three-unit delay carry-out (shown 
below) which is placed in the next higher column. 
______________________________________ 
2.sup.6 
2.sup.5 
2.sup.4 2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
3 &lt;=three-delay carry-out from 2.sup.3 column 
2 &lt;=two-delay carry-out from 2.sup.3 column 
2 &lt;=two-delay carry-out from 2.sup.3 column 
1 &lt;=one-delay carry-out from 2.sup.3 column 
1 &lt;=one-delay carry-out from 2.sup.3 column 
0 
0 
0 0 
0 0 
0 0 0 
0 0 0 
3 &lt;=three-delay partial result 
2 &lt;=two-delay partial result 
______________________________________ 
It will be seen that the three-unit delay carry-in to the 2.sup.4 column, 
shown above at the top of the column, will be treated as a partial result 
in the 2.sup.4 column, and summed with a second partial result of the 
adder chain of the 2.sup.4 column. 
Next, the 2.sup.4 column is processed. The six zero-delay input bits (shown 
above) are added in a pair of full adders, producing two one-delay 
intermediate sums (not shown), and two one-delay carry-outs (shown below) 
which are placed in the 2.sup.5 column. The two intermediate sums and one 
of the one-delay carries from the 2.sup.3 column are added in a full adder 
producing a two-delay carry-out (shown below) which is placed in the 
2.sup.5 column and a two-delay intermediate sums (not shown). The 
two-delay intermediate sum, the remaining one-delay carry-in from the 
2.sup.3 column, and one of the two-delay carry-ins from the 2.sup.3 column 
are added in a full adder, producing a three-delay carry-out (shown below) 
which is placed in the 2.sup.5 column and a three delay intermediate sum. 
The remaining two-delay carry from the 2.sup.3 column and the three-delay 
intermediate sum are added together in a half adder, producing a 
four-delay partial result (shown below) and a four-delay carry-out (shown 
below) which is placed in the 2.sup.5 column. The three-delay carry-in 
from the 2.sup.3 column is taken together with the four-delay partial 
result as a pair, will be combined in a final ripple-carry adder (FIG. 
2I). 
______________________________________ 
2.sup.6 
2.sup.5 
2.sup.4 
2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
4 &lt;=four-delay carry from 2.sup.4 column 
3 &lt;=three-delay carry from 2.sup.4 column 
2 &lt;=two-delay carry from 2.sup.4 column 
1 &lt;=one-delay carry from 2.sup.4 column 
1 &lt;=one-delay carry from 2.sup.4 column 
0 
0 
0 0 
0 0 
4 &lt;=four-delay partial result 
3 &lt;=three-delay carry-in, treated as partial 
______________________________________ 
result 
Again, it will be seen that the four-unit delay carry-out from the 2.sup.4 
column will be placed at the end of the 2.sup.5 column to be combined, as 
one of two partial results with the partial result thereof, leading to a 
final result when added by the final stage adder (FIG. 2I). Further, it is 
evident that the three and four unit delays of the partial results is a 
remarkable achievement for a column adding eleven bits. 
Next, the 2.sup.5 column is processed. Three of the four zero-delay input 
bits (shown above) are added together in a full adder, producing a 
one-delay carry-out (shown below) which is placed in the 2.sup.6 column 
and a one-delay intermediate sum. The remaining zero-delay input bit, the 
one-delay intermediate sum and one of the two one-delay carry-ins from the 
2.sup.4 column are summed in another full adder, producing a two-delay 
carry-out (shown below) which is placed in the 2.sup.6 column and a 
two-delay intermediate sum. The two-delay intermediate sum, the remaining 
one-delay carry-in from the 2.sup.4 column, and the two-delay carry-in 
from the 2.sup.4 column are summed in another full adder, producing a 
three-delay carry-out (shown below) which is placed in the 2.sup.6 column 
and a three-delay intermediate sum. The three-delay intermediate sum and 
the three-delay carry-in from the 2.sup.4 column are added in a 
half-adder, producing a four-delay carry-out (shown below) which is placed 
in the 2.sup.6 column and a four-delay partial result. The four-delay 
carry-in from the previous column and the four-delay partial result of the 
present column are taken together as partial results of the 2.sup.5 column 
which will be combined in a final ripple carry adder stage. 
______________________________________ 
2.sup.6 
2.sup.5 
2.sup.4 
2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
4 &lt;=four-unit delay carry from 2.sup.5 column 
3 &lt;=three-unit delay carry from 2.sup.5 column 
2 &lt;=two-unit delay carry from 2.sup.5 column 
1 &lt;=one-unit delay carry from 2.sup.5 column 
4 &lt;=four-delay partial result 
4 &lt;=four-delay carry-in, treated as partial result 
______________________________________ 
Finally, the 2.sup.6 column is processed. The two zero-delay input bits 
(shown above) and the one-delay carry-in from the 2.sup.5 column are 
summed in a full adder, producing a two-unit delay carry-out (shown below) 
for which a new, 2.sup.7 -weight column is created, and a two-delay 
intermediate sum. The two-delay intermediate sum, and the two and 
three-delay carry-ins from the 2.sup.5 column are summed, producing a 
four-delay carry-out (shown below) which is placed in the new 2.sup.7 
-weight column, and a four-delay partial result. The four-delay partial 
result and the four-delay carry-in from the 2.sup.5 column are taken 
together as partial results for the 2.sup.6 column, and will be combined 
in a final ripple carry adder stage. 
______________________________________ 
2.sup.7 
2.sup.6 2.sup.5 
2.sup.4 
2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
4 &lt;=four-delay carry-out 
2 &lt;=two-delay carry-out 
4 &lt;=four-delay partial result 
4 &lt;=four-delay carry-in 
______________________________________ 
Typically, this process of column decomposition and reduction via an adder 
chain would be repeated until the most significant column (which may be 
composed of nothing but carries-in from the previous, next less 
significant column) has fewer than three inputs (including carry-ins, if 
any). In the example given above, the 2.sup.7 column has only two inputs, 
so the process is complete. The fully processed data consists of 8 columns 
of one or two single-bit, closely delay-matched partial results as 
follows: 
______________________________________ 
2.sup.7 
2.sup.6 2.sup.5 
2.sup.4 2.sup.3 
2.sup.2 
2.sup.1 
2.sup.0 
______________________________________ 
4 4 4 4 3 2 2 1 
2 4 4 3 2 2 
______________________________________ 
At this point, a final result may be obtained with a ripple-carry (fast) 
adder. Starting with the least significant column which has two partial 
results, the fast adder is designed. In this case, both the 2.sup.0 and 
the 2.sup.1 columns have only one result, so these bits may bypass the 
fast adder and be treated as a final result as the 2.sup.0 and 2.sup.1 
weight component bits, respectively, of the final sum of the binary 
numbers being added by the weighted-delay column adder. This leaves six 
columns which must be summed. For each column with two components, except 
for the least significant, a full adder is used, where two inputs are used 
for the two (partial result) components and one input is used for the 
carry-out from the addition of the two partial results of the next less 
significant column. All of the one component columns will use a half 
adder, where one input is for the column result and the other input is for 
the carry-out from the previous stage. Since the least significant column 
has two components, but there is no carry from the previous column, a half 
adder is used where both inputs are for the column component results. 
Recognizing and reacting to the fact that one or more of the columns may 
produce only one result, and designing the final fast-adder stage with 
this fact in mind, represents an optimal construction. However, it is 
within the scope of this invention that all of the columns of the 
weighted-delay column adder may be configured to have two (partial result) 
outputs, and the final fast-adder stage can be designed accordingly, with 
full adders in all of the columns except for the least significant (e.g. 
2.sup.0 column). In fact, with this criteria in mind, the configuration of 
the weighted-delay column adder itself becomes less critical, since the 
final fast-adder stage would be more "generic", having been designed to 
deal with two partial results from each column (adder chain). 
HARDWARE EMBODIMENT 
With attention to FIGS. 2A through 2I, the hardware solution for the 
above-described example of A times B plus C times D is shown, 
column-by-column, where A, B, C, and D are all 4-bit unsigned binary 
numbers. It should be understood that the example used herein is for 
illustrative purposes only, and that the scope of the invention extends 
beyond the particular embodiment disclosed herein. 
FIG. 2A is a diagram of the multiplication array (32 "AND" gates organized 
in groups of four to form eight 4-by-1 multipliers 101, 102, 103, 104, 
105, 106, 107 and 108). FIGS. 2B through 2I depict the Weighted-Delay 
Column Adder of the current invention, where FIGS. 2B through 2H depict 
the weighted column adder chains and FIG. 2I depicts the final stage 
ripple-carry adder. 
FIG. 2A shows the hardware 100 comprising the array of one-by-four bit 
multipliers used to arrive at the partial products which will be input to 
the Weighted-Delay Column adder. Each of the multipliers is constructed 
from four "AND" gates in the manner of the 1-bit by 8-bit multiplier shown 
in FIG. 1. 
In FIG. 2A, the component bits of A: 110, 111, 112, and 113 (a.sub.0, 
a.sub.1, a.sub.2, and a.sub.3, respectively) are each multiplied by each 
of the component bits of B: 115, 116, 117, and 118 (b.sub.0, b.sub.1, 
b.sub.2, and b.sub.3, respectively) to arrive at 16 partial products 
(b.sub.0 a.sub.3, b.sub.0 a.sub.2. . . ): 140, 141, 142, 143, 145, 146, 
147, 148, 150, 151, 152, 153, 155, 156, 157, and 158. In like manner, the 
component bits of C: 120, 121, 122, and 123 are each multiplied by each of 
the component bits of D: 125, 126, 127, and 128 to arrive at 16 additional 
intermediate products: 160, 161, 162, 163, 165, 166, 167, 168, 170, 171, 
172, 173, 175, 176, 177, and 178. The intermediate product bits will be 
added in a Weighted-Delay Column Adder, as described with respect to FIGS. 
2B through 2H, the results of which addition will be combined in a final 
adder stage to form the completed result, as described with respect to 
FIG. 2I. 
FIG. 2B shows the adder chain 200 for the least significant column 
(2.sup.0), in this example, for adding the partial products b.sub.0 
a.sub.0 and d.sub.o c.sub.0. The partial product 143 (b.sub.0 a.sub.o) is 
applied to the "A" input of a half-adder 202, and the partial product 163 
(d.sub.o c.sub.o) is applied to the "B" input of the adder 202. The 
one-bit result (Sigma) is output on a line 206, and has a weight 
(significance) of 2.sup.0 and a delay (D) of one-unit. The carry-out 
(C.sub.o) appears on a line 204, and has binary weight (significance) of 
2.sup.1 and a delay of one-unit. The carry out on the line 204 will be 
applied in the addition of the next higher significant column (2.sup.1), 
as described with respect to FIG. 2C. 
One skilled in the art to which the present invention most nearly pertains 
will recognize that the delay values of one-unit are approximations used 
for illustrative purposes only, and that the actual delay between the 
inputs and outputs of half- or full-adders is dependent upon state 
conditions and device-specific speeds. Hence, the delays discussed herein 
should be considered "maximum" delays through the various adders of the 
adder chains. The designer will be able to take these factors into account 
in the implementation of this invention, based on the description 
contained herein. 
FIG. 2C shows the adder chain 300 for adding the partial products in the 
next higher (more) significant column (2.sup.1), in other words, the 
addition of d.sub.1 c.sub.o, d.sub.o c.sub.1, b.sub.1 a.sub.o and b.sub.o 
a.sub.1, and the carry-out 204 from the previous column (described with 
respect to FIG. 2A). The partial product 168 (d.sub.1 c.sub.o) is applied 
to the "A" input of a full adder 302, the partial product 162 (d.sub.o 
c.sub.1) is applied to the "B" input of the adder 302, and the partial 
product 148 (b.sub.1 a.sub.0) is applied to the "C.sub.i " input of the 
adder 302. The 1-bit partial result is output on a line 304, and has a 
weight of 2.sup.1 and a delay value (D) of one-unit, and the carry-out 
(C.sub.o) appears on a line 306 with a binary weight of 2.sup.2 and a 
delay of one-unit. The carry-out on the line 306 will be applied in the 
addition of the next higher significant column (2.sup.2), as described 
hereinafter. 
The intermediate sum (result) on the line 304 from the adder 302 is applied 
to the "A" input of a full adder 308, the partial product on line 142 
(b.sub.o a.sub.1)is applied to the "B" input of the full adder 308 and the 
carry out on the line 204 from the addition of the previous column is 
applied to the "C" input of the full adder 308. The 1-bit result is output 
on a line 310, has a binary weight of 2.sup.1 and has a "maximum" delay of 
two units, since bits propagating through the adder chain may have been 
operated on by two adders 302 and 308 (or 202 and 308 in the case of the 
carry-out 204), each interposing a propagation delay of one-unit. 
Similarly, the 1-bit carry out (C) is provided on a line 312, and has a 
binary weight of 2.sup.2 and a maximum delay of two-units. The carry-out 
on the line 312 will be applied in the addition of the next higher 
significant column (2.sup.2), as described hereinafter. 
It should be noted, throughout the Figures, that the partial products can 
be applied to the inputs (A, B, C) of the adders in any order. It should 
also be noted that the partial products within one column may be 
interchanged in any order for the purpose of addition. In similar fashion, 
within one column, carry-ins from the previous column having the same 
delay value may be interchanged in any order. 
FIG. 2D shows the adder chain 400 for the next higher significant column 
(2.sup.2). Partial products b.sub.0 a.sub.2, b.sub.1 a.sub.1, and b.sub.2 
a.sub.0, on lines 141, 147, 153, respectively are connected to inputs "A", 
"B", and "C", of adder 402, respectively, producing a one-delay 
intermediate sum 406 and a one-delay carry-out 408. Partial products 
d.sub.0 c.sub.2, d.sub.1 c.sub.1, and d.sub.2 c.sub.0 on lines 161, 167, 
and 173, respectively, are connected to inputs "A", "B", and "C", of adder 
404, respectively, producing a one-delay intermediate sum 410 and a 
one-delay carry-out 412. The two intermediate sums 406 and 410 and the 
one-delay carry 306 from the previous stage (FIG. 2C) are connected to 
inputs "A", "B", and "C" of adder 414, respectively, producing a two-delay 
partial result 416 and a two-delay carry 418. The two-delay carry-out 312 
from the previous stage is treated as a second partial result of this 
adder chain, and the two partial results will be summed at 2.sup.2 weight 
in a final fast adder stage (FIG. 2I). The two one-delay carries 408 and 
412, and the two-delay carry 418 will be passed on to the next significant 
column (2.sup.3), as described with respect to FIG. 2E. 
FIG. 2D is particularly illustrative of the method of bringing in the 
carry-out (312) from the previous column into the present column at the 
end thereof to be combined with a similarly delay-weighted column result 
(416) thereof. 
FIG. 2E shows the adder chain 500 for the next higher significant column 
(2.sup.3). Partial products b.sub.0 a.sub.3, b.sub.1 a.sub.2, and b.sub.2 
a.sub.1, on lines 140, 146, and 152, respectively are connected to the 
"A", "B", and "C" inputs, respectively of full adder 502, producing a 
one-delay intermediate sum on a line 506 and a one-delay carry-out on a 
line 508. Partial products b.sub.3 a.sub.0, d.sub.0 c.sub.3, and d.sub.1 
c.sub.2, on lines 158, 160, and 166, respectively, are connected to the 
"A", "B", and "C" inputs of a full adder 504, producing a one-delay 
intermediate sum 510 and a one-delay carry-out 512. The two one-delay 
intermediate sums 506 and 510, and the one-delay carry-in 412 from the 
previous column are connected to the "A", "B", and "C" inputs of a full 
adder 514, producing a two-delay, 2.sup.3 -weight partial result on a line 
520 and a two-delay carry-out on a line 522. The partial product d.sub.2 
c.sub. 1 on line 172, the partial product d.sub.3 c.sub.0 on line 178 and 
the one-delay carry-in on the line 408 are connected to the "A", "B", and 
"C" inputs of a full adder 513, respectively, producing a two-delay 
intermediate sum 516 and a two-delay carry-out 518. The other one-delay 
carry-out 412 from the previous column on the line 418 and the 
intermediate sum on the line 516 are connected to inputs "A" and "B" of a 
half-adder 524, producing a partial column result 526 with a binary weight 
of 2.sup.3 and a three-delay carry-out 528. All of the carries out of this 
column have a binary weight of 2.sup.4 and are carried over to the next 
most significant column, as described with respect to FIG. 2F. 
FIG. 2E is particularly illustrative of the method of obtaining two partial 
results (520, 526) from the present adder chain. In this case, all of the 
carry-outs from the previous column were brought into the adder chain for 
addition with input bits and intermediate sums arrived at therein at 
points where their delays most nearly matched the delays of the input bits 
and intermediate sums. 
FIG. 2F shows the adder chain 600 for the next more (higher) significant 
column (2.sup.4). Partial products b.sub.1 a.sub.3, b.sub.2 a.sub.2, and 
b.sub.3 a.sub.1, on lines 145, 151, and 157, respectively, are connected 
to inputs "A", "B", and "C" of full adder 602, respectively, producing a 
one-delay intermediate sum 606 and a one-delay carry-out 608. Partial 
products d.sub.1 c.sub.3, d.sub.2 c.sub.2, and d.sub.3 c.sub.1, on lines 
165, 171, and 177, respectively, are connected to inputs "A", "B", and "C" 
of full adder 604, respectively, producing a one-delay intermediate sum 
610 and a one-delay carry-out 612. The two intermediate sums 606 and 610 
and the one-delay carry from the previous column on line 508 are connected 
to inputs "A", "B", and "C", respectively, of full adder 614, producing a 
two-delay intermediate sum 616 and a two-delay carry-out 616. The 
intermediate sum 616, the one-delay carry 512 and the two-delay carry 518 
are connected to inputs "A", "B", and "C", respectively, of full adder 
620, producing a three-delay intermediate sum 622 and a three-delay 
carry-out 624. The intermediate sum 622 and the remaining two-delay carry 
522 from the previous (2.sup.3) column are connected to the "A" and "B" 
inputs of a half adder 626, producing a four-unit delay partial result 628 
and a four-unit delay carry-out 630. The three-delay carry 528 from the 
previous column and the partial result 628 are taken as two partial 
results of this column, and are provided to a final ripple-carry adder 
stage, as described with respect to FIG. 2I. All of the carries out of 
this column: 608, 612, 618, 624, and 630; are passed on the next more 
significant column (2.sup.5), as described with respect to FIG. 2G. 
Again, in FIG. 2F it is well illustrated that the longest (highest 
delay-weighted) carry-out (528) from the previous column is conveniently 
brought in to the present column at the end thereof, and treated as one of 
two partial results of that column. 
FIG. 2G shows the adder chain 700 for the next more significant column 
(2.sup.5). Partial products b.sub.2 a.sub.3, b.sub.3 a.sub.2, and d.sub.2 
c.sub.3 on lines 150, 156, and 170, respectively, are applied to inputs 
"A", "B", and "C" of full adder 702, producing an intermediate sum 704 and 
a carry-out 706, both having a one-unit propagation delay through the 
adder 702. The intermediate sum 704, partial product d.sub.3 c.sub.2, and 
the one-delay carry-in 608 from the previous column, are applied to inputs 
"A", "B", and "C" of a full adder 708, respectively, producing a two-delay 
intermediate sum 710 and a two-delay carry-out 712. The intermediate sum 
710, the other one-delay carry-in 612 and the two-delay carry-in 618 are 
applied to inputs "A", "B", and "C", respectively, of a full adder 714, 
producing a three-delay intermediate sum 716 and a three-delay carry-out 
718. The "maximum" delay through the adder 714 is considered to be 
three-units, since the carry-in 618 and the intermediate sum 710 each may 
have maximum delays of two-units. The intermediate sum 716 and the 
three-delay carry-in 624 from the previous (2.sup.4) column are applied to 
inputs "A" and "B", respectively, of a half-adder 720, producing a 
four-delay partial result 722 and a four-delay carry-out 724. The partial 
result 722 and the four-delay carry in 630 from the previous stage are 
taken together as partial results with 2.sup.5 weight and are applied to a 
final ripple-carry adder stage, as described with respect to FIG. 2I. All 
of the carries out of this stage: 706, 712, 718, and 724; are passed on 
the next more significant column (2.sup.6), as described with respect to 
FIG. 2H. 
FIG. 2H shows the adder chain 800 for the next more significant column 
(2.sup.6) of the weighted-delay column adder. Partial products b.sub.3 
a.sub.3 and d.sub.3 c.sub.3 on lines 155 and 175, respectively, and the 
one-delay carry-in 706 from the previous (2.sup.5) column are applied to 
inputs "A", "B", and "C", respectively, of a full adder 802, producing a 
two-delay intermediate sum 804 and a two-delay carry-out 806. The 
intermediate sum 804 and the two- and three-unit delay carry-ins from the 
previous column on lines 712 and 718, respectively, are applied to inputs 
"A", "B", and "C", respectively, of a full adder 808, producing the 
four-delay partial result 810 and a four-delay carry-out 812. The 
four-delay carry-in 724 from the previous column and the partial result 
810 are taken together as partial results having 2.sup.6 -weight and are 
applied to a final ripple-carry adder stage, as described with respect to 
FIG. 2I. 
The two carry-outs 806, 812 out of this column (adder chain 800) are passed 
on to the next column at 2.sup.7 -weight, which column, having only these 
two one-bit binary inputs, is further passed on at 2.sup.7 -weight 
directly to the final ripple-carry adder stage, as describe with respect 
to FIG. 2I. 
FIG. 2I shows an embodiment 900 for the final ripple-carry adder stage. The 
ripple-carry adder comprises a half adder 902 and five full adders 904, 
906, 908, 910, and 912, arranged in order of the binary weight 
(significance) of partial (column) results to be added. The carry-outs 
from each of these adders, except for the last one (e.g. the adder 912 to 
which the highest binary weight is assigned) are fed directly into the 
carry-in of the next more significant adder. That is, carry-out 914 from 
half adder 902 is connected to the "C" input of the full-adder 904, the 
carry-out 916 from the full-adder 904 is applied to the "C" input of full 
the adder 906, the carry-out 918 from the full-adder 906 is applied to the 
"C" input of the full-adder 908, the carry-out 920 from the full-adder 908 
is applied to input "C" of the full-adder 910, and the carry-out 922 from 
the full-adder 910 is applied to the "C" input of full-adder 912. 
The 2.sup.2 -weight, two-delay partial results from the 2.sup.2 column 
(adder chain 400) on the lines 416 and 312 are applied to the "A" and "B" 
inputs of the half-adder 902, which produces a final result having 2.sup.2 
-weight and three-unit delay on a line 936. The 2.sup.3 -weight 
three-delay partial result on line 526 and the 2.sup.3 -weight two-delay 
partial result on the line 520 are applied to the "A" and "B" inputs of a 
full-adder 904, and the three-delay carry 914 from the less significant 
2.sup.2 adder 902 is applied to the "C" input thereof, thereby producing a 
final result having 2.sup.3 -weight a four-unit delay on a line 934. The 
2.sup.4 -weight four-unit and three-unit delay partial results on lines 
628 and 528, respectively, are applied to inputs "A" and "B", 
respectively, of a full adder 906 while the four-delay carry 916 from the 
less significant adder 904 is applied to the "C" input of the adder 916, 
thereby producing final 2.sup.4 -weight five-unit delay result on a line 
932. The 2.sup.5 -weight four-delay partial results on lines 722 and 630, 
are applied to inputs "A" and "B", respectively, of the full-adder 908, 
while five-delay carry 918 from the 2.sup.4 -weight adder 906 is applied 
to input "C" thereof, providing a final 2.sup.5 -weight six-delay result 
on a line 930. The 2.sup.6 -weight partial results on lines 810 and 724 
are applied to the "A" and "B" inputs of the full-adder 910 while the 
carry 920 from the adder 908 is applied to input "C" thereof, thereby 
providing a final 2.sup.6 -weight seven-delay result on a line 928. The 
2.sup.7 -weight two-unit and four-unit delay partial results on lines 806 
and 812, respectively, are applied to inputs "A" and "B", respectively, of 
full adder 912, and the seven-delay carry 922 from the 2.sup.6 -weight 
adder 910 is applied to input "C" thereof, producing final 8-delay 2.sup.7 
and 2.sup.8 -weight results on lines 926 and 924, respectively. 
The 2.sup.0 -weight one-delay result of the least significant adder chain 
200 is read directly off of the line 206, and is valid as the 2.sup.0 
-weight bit of the final result of the addition of binary numbers being 
added by the weighted-delay column adder. Similarly, the 2.sup.1 -weight 
two-delay result out of the 2.sup.1 adder chain 300 is valid as the 
2.sup.1 -weight bit of the final result. 
In the examples above, the one and zero unit delays are intended for 
illustrative purposes only. Inputs having different delay values would 
have to be accounted for. For instance, input bits with greater than zero 
delay would need to be inserted further down the adder chain. 
As shown above, by matching the delays of input bits and carry-ins to input 
bits, intermediate sums and partial results in each column, the final sum 
of the binary addition is arrived at with minimum delay while retaining a 
gate-efficient design. 
Partial products arriving for addition at the weighted-delay column adder 
of the present invention can be provided from a multiplier employing a 
modified Booth algorithm to reduce the number of input bits. 
Having thus described the invention, many modifications thereto will become 
apparent to those skilled in the art to which it pertains without 
deviating from the spirit and scope of the appended claims.