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Патент USA US3023972

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March 6, 1962
R. A. STAFFORD
3,023,962
SERIAL-PARALLEL ARITHMETIC UNITS WITHOUT CASCADED CARRIES
Filed May 23, 1957
7 Sheets-Sheet 2‘
r
BYE/0000 5.4/5/1/02/6'
37702%55
March 6, 1962
R. A. STAFFORD
3,023,962
SERiAL-PARALLEL ARITHMETIC umws WITHOUT CASCADED CARRIES
Filed May‘ 23, 1957
7 Sheets-Sheet 3
l‘_905;é'_1
'05,
F4.IG.
‘
IN VEN TOR.
@Oé? % 5277:5020
“Yew/00p 5. {swag/ck
ITTOE/VEVS
March 6, 1962
R. A. STAFFORD
3,023,962
SERIAL-PARALLEL ARITHMETIC UNITS WITHOUT CASCADED CARRIES
Filed May 25', 1957
7 Sheets-Sheet 4
Mamh 6, 1962
R. A. STAFFORD
3,023,952
SERIAL-“PARALLEL ARITHMETIC UNITS WITHOUT CASCADED CARRIES
F‘iied vMay 23, 195-’?
7 Sheets-Sheet 5
6'4
£726. 6a.
I
I
£55021
4c
$7176- . éé.
IN VEN TOR.
£065? ,4. 5721475020
March 6, 1962
R. A.- STAFFORD
3,023,962
SERIAL-PARALLEL ARITHMETIC UNITS WITHOUT CASCADED CARRIES.
Filed May 25, 1957
7 Sheets-Sheet 1
6EN
Q
Unitettt
3,023,962
‘
1
Patented Mar. 6, 19$2
2
number as Well as the number of numbers to be ac
cumulated. This may be expressed as Knm where‘ K is
3,023,962
SERIAL-PARALLEL ARITHMETIC UNITS
WIIHOUT CASCADED CARRIES
a constant, n is the digits per number and m is the'num
Roger A. Stafford, Champaign, 111., assignor, by mesne 5 ber of numbers. It will be shown, however, that ac
cording to the present invention the same accumulation
assignments, to‘Thonipson Rarno Wooldridge‘ Inc,
Cleveland’ 05150, a corporation of Ohio
Filed May 23, 1957’ Ser. No_ 661,158
process is performed only as a function of the number
of numbers to be accumulated, or, as a function of m.
14'Clairns. (Cl. 235-_175)
Thus Raytheon circuit mentioned above requires ?ve
pulse times for n carries in a register of about 20 digits
This invention relates to serial-parallel arithmetic 10 (101:5). If n were double this value (i.e., 40), how
units, and more particularly to an arithmetic unit re-
ever, the factor Kn would be at least doubled so that
quiring relatively few component parts for high-speed
each carry cascading would require 10‘ pulse periods.
operation.
The limitation of the prior art cascading type of con
W'nile the present invention may have general applicatrol adder is even more pronounced when the operations
tion in various types of computer systems, it is particu- 15 of multiplication or division are considered. Speci?cally,
larly useful as employed in various accumulation procin either of these cases, m becomes equal to n. Thus
esses, such as addition and subtraction series, cumulative
the time required to multiply or’to divide becomes pro
multiplication, or division performed as a series of addiportional to n2 because these operations are performed
tions, subtractions or “do nothing” operations. The
as many times as there are digits in a number to be
method and circuit techniques of the invention described 20 operated upon.
below make it possible to accumulate separate partial
The present invention, on the other hand, is adapted
result and carry digits in successive digit time intervals
to perform multiplication or division during an n pulse
without allowing any time for propagating carries in
period, for an increase of speed in the order of n to 1.
order to formacomplete result.
At this point note should be taken of the fact that
The maintenance of partial result and partial carry 25 the accumulated process of the present invention for
registers to maintain a number simply as two factors
addition and subtraction is not complete in terms of a
is relatively easy when it is not necessary to change the
conventional answer, since the technique of the present
sign
of the ciperation. Tihat is, addition or inultiplicatron constanty usrnv a pus sign or constanty using a
inviention
regluires that ifserial combincalttion of the result
an carry igits be pe orrned in or er to generate a
minus sign presents 5no probleni. However, when these 30 complete result.
signs are changed intermittently during a series of computations, it is necessary to perform one of two functions
in order to insure a correct partial carry and result due
However, the time saved during the accumulation
process, in accordance with the technique of the inven
tion, results in an overall saving in time for a complete
to the fact that with prior art it was possible to obtain
operation whenever the number of accumulations times
different signs for different digits in the partial carry reg- 35 the usual 5 pulse time delay for carries exceeds the sum
ister. This'means that the signs of separate digits had
of the number of accumulations and the additional n
to be‘ accounted for separately, or that each time an operation was performed, the carry had to be propagated or
digit time interval to formulate a complete result, in ac
cordance with the present invention.
cascaded. That is, a complete answer would, of necesMoreover, in serial-parallel systems, an n-digit time
srty, have to be obtained each time an operation was 4" interval is required to read operands in and to read
pierforréreg.
In this tease, 1trme would t‘liaushhave to ‘be
ai owe
or propagarng t e carries an t e operation
sut an
would be slow. In other words, prior art devices require
in or read-out period of computation and, accordingly,
oplerandis carry
out sol igits
that the
interval drequired1 to combine r;
may occur uringt e normal rea -
an excessive amount of equipment to carry the signs of
in such systems only the accumulation time need be con
the individual digits or, alternatively, they require that 45 sidered.
time be given to cascading carries to obtain a complete
The importance of the present invention will become
result for each operatron,_thereby lengthening the time
even further evident when it is established below that
recrlir‘ltrlred
to performt ao fsetrirlese rnvenion
of operatronsb
_ ' the
e rmprovemen
in o vratrng
theers tlechnique
accunguflatign
avin an allows
sign ortléie
orm,
an
urt erto1:atinpult
mu ti nlum
ica
necessity of cascading carry signals may best be under- 50 tion and gsimrilltaiaieous accumulation with a prelilious
stood by considering a typical prior art approach. Refresult is permissible. Consequently, a complex series of
erence for this purpose is ‘made to pages 297 through
muted multiplications, additions and subtractions may
:91 and FIGS. l3_—24 of ‘H1gh-$p66(_1 COIIIPUQHE DF-
be performed without any interruption in removing any
vices by Engineering Research Associates, publlsbed H1 55 partial result from the accumulator register until the
1950 bY_MCGraW-H1ll Book Co.
_
_
' end of the entire series of operations, at which time the
Refellflng to Page 297,_1I1 Ramcular, 1t W111 13,6 floted
tha't a Parallel adder W1llh_S1multaI1e0uS cr'fln'y 1? de-
complete result may be read out by combining the re
sult and carry signals previously formulated. Thus,
scribed where, . . . all carries are accomplrshed simulin terms of an accumulation of signed numbers, the
taneously . . .’. In reading further on ‘the same page, It
technique of the invention is adapted to solve a series:
will be observed that “actually, it rise times are required 60
to make 12 carries, and in the Raytheon circuit about
(iNi)i(iNp2)-_'- . . . (:Nn)
?ve pulse times have to be allowed on this account.”
_
Thus, while the circuit described in the above-mentioned
book, representing a typical parallel adder circuit of the
,
,
,
, _
the total lime requlred for accumulatlon bemg " dlglt
_ mules’ 2,15 d‘scussed aboYe'
prior art, is quite complex, yet it cannot function as a 60
_
‘
‘t W111 be: oliserved ‘tithe above SW65 ‘hat 5321c}? mum‘
true parallel adder in terms of a computer digit time interval, since ?ve pulse periods must be allowed in order
ber {nay be either Poslu"e 01' negatlve and that 1} may
be elthel‘ addfifi 01' subtracted‘ Hereafter, Ih‘e $1811 of
to allow the Cascading of the can-y Signals from one end
each number will be assumed to be included in the opera
of the accumulator register to the other.
tion sign so that the operation i-I- (‘+N) or -—(—N) is
Thus in accumulating a series of numbers in the typi- 70 considered to be a plus (+) operation, Whereas the opera—
cal parallel system just described, the time required is
a function of the length of the number of digits in a
tion +(—N) or‘-—'(+N) is considered to be a minus
(—) operation.
3,023,962
3
In a similar manner, the present invention may be
employed to accumulate a series of products, each ac
cumulation operation being performed simultaneously
with the multiplication operation. Thus, the series:
4
It can be shown that the present operation sign logi
cally determines the inherent sign of the new carry signal.
One manner of observing this is to note that where the
input number digit and the previous result and carry
digits are considered to have a weight of one, the new
carry digit must have a weight of two. Consequently, it
the previous result and carry digits have opposite signs, a
may be solved in 2.n.m digit times, where m is the num
carry having a weight of two can only occur with the
aid of the input number, bearing the operation sign. Thus,
ber of multiplier digits and n represents the number of
terms in the series. Here again the operation sign which 10 where the previous result digit is a plus and the previous
carry digit is a minus, a minus operation sign may result
will be referred to hereinafter is considered to be the
in a negative carry-over of two, or a positive operation
combination of all the signs for the corresponding terms.
sign may result in a positive carry‘over of two. The fact,
ThuS1'+(+A1)><*(+B1); +(—A1)><(—B1);
however, that the carry must bear the sign of the opera
15 tion does not mean that the digit in a particular place
must have the same sign. For example, the result digit
may be positive, the carry digit negative, and the opera
tion sign negative, but the total results in the particular
binary place is positive if a one occurs in the result place
20 and zeros in the carry and input digit places.
all have a minus operation sign.
In accordance with the invention, the result digits are
To the present time, in both cascading carries and
produced in a logical network which operates inde
carrying or storing the sign of each carry digit, it has been
pendently of the particular sign combinations of the
assumed that at all times the partial result of any system
input signals; and the carry digits are produced in accord
of this type must always be positive. This mental block
ance with two basic sets of logic, the ?rst set being
has perhaps been a barrier to advances in this art because
utilized when the signs of the previous carry and the
of its historical basis. However, once the arbitrary selec
present operation are the same, and the second set being
tion of a positive sign for the partial carry was made in
utilized when the signs of the previous carry and the
the prior art, it became impossible to perform successive
present operation are different. The two carry sets may
and —(+Al) X (—B1) are all considered to have a plus
(\+)
operation sign
whereas:
--\(+Al)><(+B1);
accumulations without cascading carries unless all num 30 also be considered to exist for the cases where the present
bers were of the same sign. According to the prior art
and previous operation signs are the same and where the
technique, it is impossible to perform subtraction after
present and previous operation signs are different, respec
a positive partial result and carry digits have been accumu
tively.
lated, unless the sign is carried along side each correspond
ing digit, since carries of both signs may remain in the
accumulator register after computation.
According to the basic concept of the present inven
tion, however, no arbitrary selection is made in either
the sign of the partial result or partial carry signals. In
stead it is revealed according to the present invention that
the manner of determining the sign of the carry and
result signals must be established as an algebraic rule de
termined by the operation sign.
According to the invention, it has been established that
The basic method of the invention comprises the
following steps:
(1) Producing a ?rst control signal, which may be
referred to as signal K hereafter, whenever the previ
ous carry sign and present operation sign are the same,
or stated in other words, whenever the previous and
present operation signs are the ‘same;
(2) Generating a ?rst type of carry signal in response
to the ?rst control signal;
(3) Producing a second control signal, which may
be referred to as signal K’ hereafter, whenever the previ
ous carry sign and the present operation sign are differ
cut, or stated in other Words, whenever the previous and
the partial carry signals which are accumulated will al
ways have the same sign it their sign is made to be the
same as the operation sign of the previous operation.
present operation signs are different;
That is, if addition is performed the carry signals are all
(4) Producing a second type of carry signal in re
made positive, and if subtraction is performed the carry
sponse to the second control signal; and
_
signals are all made negative.
(5) Generating all carry signals with a sign implied
50
The next step of the invention was to recognize that
therein corresponding to the present operation sign, and
the partial result in carry digits accumulated must be
generating result signals of opposite sign to respective
generated to have values such that the combination there
carry signals; the carry and result signals produced con
of, considering the digit values thereof to have opposite
signs would result in a complete answer in each digital
place having the desired value and sign.
According to this basic concept then, the partial result
signals impliedly have negative signs during operations
stituting together a complete binary result considering the
binary value of each digit and the implied sign thereof.
In its basic structural form, the invention contemplates
a unit for combining input and previous result and carry
signals in a binary digital place, to form carry-over and
new result signals. The basic unit is duplicated for system
of addition, that is, Where the operation sign is positive.
Thus, by discarding the rigid rule of the prior art where 60 usage so that a plurality thereof are provided, one for
by the partial result digits are always positive and by
each digital place of an input number to be accumulated.
generating these digits to be of opposite implied sign to
Each unit includes a result network and a carry network
the previous operation sign and partial carry signal sign,
including first and second carry circuits operative in re
it has become possible, according to the invention, to
accumulate numbers without cascaded carries by either
addition or subtraction, without the necessity of indicating
the sign for each individual digit, since this sign is known
sponse to ?rst and second levels of a control signal. The
control signal is generated as a function of the present
to be uniform throughout each of the carry and result
a logical ampli?er. In general terms, the control signal is
registers.
Once it was established, according to the present inven
tion, that the signs of all pairs of corresponding partial
result and carry signals should be opposite, it was then
determined that sign possibilities where the partial result
and carry signals are of the same sign need not be con
sidered since they cannot occur.
and previous operation signs and may either be stored in
a flip-?op or other bistable element or obtained through
de?ned to have an “on" or binary one state whenever the
present and next condition of the operation signal are go
ing to be the same. Thus, if the operation signal is now
on and is not going to be turned o?, the control signal is
on; and if the operation signal is now off, and is not
going to be turned on, the control signal is on.
Accordingly, it is an object of the present invention
75
3,023,962
6
to provide a method for accumulating binary result and
division system incorporating, simpli?edv logic different
carry signals in respective binary digital places whereby
from thatshown in HG. 2; and
FIG.v 8 illustrates asuitable form of arithmetic. unit for
the division system of FIG. 7.
Referring now to FIG. 1, it will be noted that input
the signs of the signals, are known as a function of the
previous operation sign.
Another object of the invention is to provide a method
for accumulating result and carry signals. whereby suc
signals Ai, Ri and Ci are applied to a, result network and
cessive input numbers having varying signs may be ac
to a carry network producingv output signals R0 and C0,
cumulated by addition or subtraction without the necessity
respectively. The, carry network is controlled‘ by device
of cascading carries.
K producing complementary control signals K and K’
A further object of the invention is to provide a method 10 and produces either of two types of carries depending
for the high-speed accumulation of input numbers ob
upon the state of these control signals. Control device
viating the consuming requirement of cascaded carries.
K receives input signals as a function of the previous op
eration sign, stored in a device Os, and of the present
Still another object is to provide a simple arithmetic
operation sign. Signals corresponding to the present op
unit which may be employed in a serial-parallel com
puting system wherein binary result and carry digits may 15 eration sign are generated in a network gOs which receives
be accumulated as function of input signals in a respective
certain input information depending uponthe particular
binary place, without any information being required in
operation which is performed.
In the discussion which follows, several logical de?ni
tions will be developed for the various networks shown
in FIG. 1. Before considering the manner in which the
applicable logic may be derived, it is helpful to analyze
a few simple examples illustrating the manner in which
various accumulation operations are performed in accord
the digital place as to carry signals which may result from
carries from other digital places.
Yet a further object is to provide an arithmetic unit
for generating result and carry signals of opposite im
plied signs, the carry signal being assumed to carry the
sign of the operation, the arrangement thereby permitting
the accumulating of input numbers of varying signs
ance with the basic approach of the present invention.
25 As a ?rst example, the binary number +011 (+3) and
through either addition or subtraction.
the binary number —101 (-5) will becombined to form
Still a further object is to provide an arithmetic unit
accumulated result and carry signals. First it must be
which may be employed in a serial~parallel multiplier
assumed that if the number \—~l0l is a previous result,
system wherein successive products may be simultaneously
the previous carries, although zeros, must have been
combined with previous results regardless of the sign or
30 positive in accordance with the basic approach of the
form thereof.
present invention.- It will be noted, however, that it is
An additional object is to provide an arithmetic unit
also possible to assume that the number —101 is a series
which may be employed in a serial-parallel division where
of previous carries, in which case it must be assumed
in successive partial remainders may be formed by addi—
that the result digits were positive and all zeros.
tion, subtraction, or “do nothing” Without the necessity of
Four illustrative examples are shown below. The ?rst
carrying parallel sign information as to the result and
example shows the situation where the number —101 is
carry signals in each place.
assumed to constitute a previous negative result and con
Yet another object is to provide a high-speed parallel
sequently the initial carries must be assumed to be posi
accumulator network which may be operated to accumu
tive even though they are all zero. The second example
late each input number during one digit time interval, no 40 shows the situation where the number -——1‘Oil is assumed
additional time being required for cascading carries.
to constitute initial carries and consequently the initial
A specific object is to provide a high-speed serial-parallel
result is assumed to be positive. The third example illus
multiplier system wherein an n~digit multiplier may be
trates the case where there are both initial result and
multiplied times a multiplicand in 2.n digit time intervals,
carry digits, the operation sign being assumed to be posi
a complete result being obtained in series starting with the 45 tive; and the fourth example illustrates a similar situation
least signi?cant digit thereof, the system permitting simul
where the operation sign is assumed to be negative.
taneous accumulation to a previous result by either addi
EXAMPLE 1
tion or subtraction.
The novel features which are believed to be charac
teristic of the invention, both as to its organization and 50
method of operation, together with further objects and
advantages thereof, will be better understood from the fol~
lowing description considered in connection with the ac
companying drawings. it is to be expressly understood,
however, that the drawings are for the purpose of iilustra- '
tion and description only, and are not intended as a
de?nition of the limits of the invention.
FIG. 1 is a block diagram of an arithmetic unit em
bodying the present invention;
FlG. la is a chart illustrating the sign considerations 60
involved in practicing the method of the invention;
FIG. 2 illustrates in general schematic form the logic
Ai
EXAMPLE 2
4 3 2 l
0 1 1
—
l 0 1
El
Ci
+
R0
——
Al
+
0 0 0
R1
Ct
+
—
1
R0
—
1
0
Co + 0 l 0
Al
Bi
+
—
C1
+
0 l 1
R0
Co
—
l
+ 0 1
1 0
0
1
0
1
l
1
0
Co + 0 l 0
EXAMPLE 3
4 3 2 1
0 0 0
l 0 1
4 3 2 1
0 1 1
0 0 0
EXAMPLE 4
A1
Bi
—
+
4 3' 2 1
0 O 0
O 1 1
+
—-
1 1 0
0. 0
C1
R0
Co
1 0, 1
1
It will be noted in the ?rst three cases that the negative
which may be employed in one embodiment of the in
result signals shown as R0, signifying the output digits
vention;
of the result, represent the binary number —110 or \—-.-6
and that the carry digits Co represent the binary number
+100 or +4. Thus, the complete result is --6 ‘+4 or
—2. In the fourth case, however, the result is v+6, rather
than —6, and the carry signals provide —8, so that the
FIG. 3 is a partial block diagram of a parallel accumu
lator system embodying arithmetic units of the invention;
FIG. 4 illustrates a speci?c form of unit which may be
employed in the system of FIG. 3;
FIGS. 5a and 5b illustrate alternate forms of a serial 70
parallel multiplier system embodying the invention;
FIGS. 6a and 6b illustrate alternate forms of arithmetic
units which may be employed in the embodiments of
FlGS. 5a and 5 b, respectively.
'
FIGS. 7a and 7b are schematic diagrams illustrating a
answer in this case is —8 +6 or —2 again.
It will be noted that a- digit is provided for the result
R0 in the same place as the corresponding input digits
Ri, Ci and Ai and that the carry digit for the same
accumulation is shifted to the left to constitute a weight
of two. Thus, reviewing column 1 of the ?rst example,
it is noted that —l-Ai, —Ri, —i-Ci, have the values l‘, l
3,023,962
and 0, respectively, and that the result of this combina
When the multiplier digit is 0, the A1 digits are not
shown and only the result and carry digits are combined.
The complete result of this cumulative multiplication
'and +Ai cancel. Consequently, R0 is 0 and C0, in the
is noted to be the number 1 . . . 1100, which is the
next place, is 0.
Considering column 2 of Example 1 it will be noted Ul complement of the binary number 0 . . . 0100. A
complemented number results in view of the fact that the
that a 1 appears in input digit +Ai and US are in the
result digits are negative, but that the true answer is
digits Ri and Ci. Consequently, the answer for the par
positive (+4 or +100). This situation causes no diffi
ticular place is +1. According to the basic technique of
culty in view of the fact that by de?nition the result of
the invention, then, a carry must be developed which has
any process in accordance with the present invention
the sign of the Operation—which is the sign of Ai(+) in
must bear a sign opposite to the previous operation sign.
this case. Furthermore, it must be noted that the carry
It, therefore, it had been desired to avoid generating a
weight is twice that of the result weight since it is carried
complementary result, the signs of the numbers could be
over to the next place. This means that Co in the next
changed so that the original input numbers would have
place is 1, with a weight of 2, and R0 in column 2, is
also a 1. The total result considered on the basis of the 15 been Ai, +Ri, and —Ci. This procedure is unimportant,
however, in view of the fact that the form of the cumula
weight in the second column is eifectively a +2 for signal
tion provides a zero output signal, since the ones in -—Ri
C0 and a ---1 1for signal R0. The difference between these
two signals is the desired answer of +1. In a similar
manner an answer of —1 is required in the third place
tive result need not be considered until the answer must
Ci=+0, and Ai=+0. This is accomplished by insert
out serially.
be read out, and at this time note need only be taken of
the sign of the previous operation, and a simple process
(third column) as the combination of the digits Ri=-1, 20 remains to recomplement the ?nal answer, as it is shifted
R0 and a 0 in the fourth column place for carry-over
Referring now again to FIG. 1, it should be clear that
the unit shown therein operates upon input and previous
It will now be shown that the answer of --2 need not
units of the type shown in FIG. 1 are required, one for
ing a --1 in the third column place for output number
result and carry digits existing in the same place or
signal C0. The total weight referenced to column 3 is
25 column. Thus, in performing Example 1 above, three
then: 2.0—1=--1.
be modi?ed during a subsequent operation and may be
accumulated into the next operation without any dii?oulty.
To illustrate this point, let us now multiply +010 (+2)
each column of the problem. In this utilization of
units, the result signals produced by each unit are
back to the input circuit thereof after storage, and
represented by the number A1, times the number +011 30 carry signals produced by the unit are shifted to the
the
fed
the
left
(+3) considered to be the number Bl. It will be shown
in order to constitute a carry-over to the next place. The
precise manner of utilization of the basic units of the in
vention, as well as several other examples which will
previous result [Ri=—6(—0110) and Ci=+4(+0100)
further clarify the features of the invention will be con
‘for a total of 1-2] left in the result and carry registers
thus the ?nal answer is —2+2><3=+4. This is shown 35 sidered below with reference to FIGS. 3, 5a, and 5b.
that this product will be simultaneously accumulated to the
The basic logical operation of the unit shown in FIG.
in Example 5 below.
1 will be developed in terms of logical algebra which may
be derived from a “truth” table as will be discussed below;
reference being made ?rst to FIG. 1a where the various
EXAMPLE 5
A1
4
+ 0
3
0
2
1
1
0
R1
—
0
1
l
0
0 Product
Place
Bl (Multiplier)
__" 1'—"_
possible sign combinations are introduced.
As indicated in FIG. 1a, only four sign possibilities
are present in view of the fact that the invention requires
the sign of the previous result and carry signals Ri and Ci
to be opposite. It will also be noted in FIG. la that
whenever the operation sign is positive, the new carry
signal must also be positive and consequently the new
result signal is negative; whereas whenever the operation
sign signal is negative, the new carry signal becomes nega
tive and the new result signal is positive.
It will be noted that the sign possibilities shown in FIG.
_'_"_
01
+ 0
1
0
0
]
A1
+ 0
0
l
0
i
R2
——
0
0
0
C2
Al
+ U
+ 0
0
0
0
1
0<——L
0
R3
0
l
1
0
2
0
l
3
0
—-
0
O
l
C3
+ 0
0
l
0
R4
—
0
0
l
C4
+ 0
0
1
0
R5
—-
0
0
1
1
C5
+
1
0
i
E
l
N
4
0
1a are broken into two groups, namely, those where the
Answer-(11 . . . 1l00)=+4
In this example, it is assumed that the result digits are
shifted to the right as they are formed, and that the
carry digits are accumulated in the same place as the in
put digits. Thus, the digits A1, R1, and C1 (‘0, 0, 0)
sign of the previous carry and the operation sign are
the same, and those where the sign of the previous carry
and the operation sign are different. It will be noted
* that where the previous carry and input signals have the
same sign, a control signal K=1 is shown, whereas where
appearing in column 1 are combined to form the new
the previous carry and input signals have different signs,
digits R2 and C2 (0, 0) in columns 0 and 1, respectively.
It will be noted, then, that the product digits appear serial
the control signal K=0 is shown.
The logical derivation of the signals for the unit shown
in FIG. 1 may then be derived from the “truth” table,
ly after successive accumulations in column 0 for the suc
cessive results.
Each addition ‘of the multiplicand to the previous result
as represented by the series of digits R and C is per
such as is shown in Table 1 below.
Table 1
formed in the same manner as in the examples considered
above. Thus, in response to the ?rst digit of B1 (multi
plier) the multiplicand A1 (0010) is added to series
R1—-(0110) and C1+(‘0l00).
K =1
Signs
(1)
This produces a new
gas +
result series R2~—~(0000) and (32+ (0000). The answer,
Cj +
1
Total
Si
(2) Value
I;
Digit Values
5
Total
(1)
(2)
—
+ R0
-
Value
_.
-
A3
a;
oi
R0
00
0
0
0
0
0
0
l
1
0
0
O
1
1
0
1
1
1
0
+
C”
it will be noted is advanced to the right so that the R2
digit appearing in column 0 constitutes the least signi? 70
cant digit of the product. This process is repeated for
the second multiplier digit to combine A1, 112-, and
(I)
(2)
(3)
+
—
—
+
(4)
0
+
—
—
+
1
1
0
0
l
1
0
0
C2—, to produce the new result and carries series:
(5)
+
—
l
l
0
0
1
1
+
—-
R3—(0010) and CS-i-(OOIO). The second least signi?
cant product digit of 0 is also produced atthis time. 75
(6)
+
—
-2
1
0 -
1
0
1
+
—
2
(7)
(8)
-
0
l
1
1
1
l
0
1
0
1
t)
1
—
0
1
+
0
+
1
3,023,962
Table 1--Cont1nued
K=0
Signs
Total
3) (4) Value
"gs i
1
EXAMPLE 1A
Signs
(3)
(4)
—
+ R0
Digit Values
_
c1 -
__~
+
Aj
Rj
Cj
R0 00
+
3 2 1
Total
Value
A1
._
0°
+
1
R1
-
0
G1
+
0
Input
R0
—
1
Output
Co +
(1)
(2)
(a)
—
+
+
-
+
-
+
+
—
-
(4)
(5)
(6)
(7)
(s)
0
o
0
0
0
0
1
1
0
0
0
1
1
0
1
1
0
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
o
2
1
1
1
1
1
0
1
0
1
1
1
SignCombination (1)
Total
5
Ro+Oo=-l+2=+l
1
0
—
+
+
-
+
-
+
+
—
-
1
1
EXAMPLE 2A
0
1
1O
4 3 2
0
Al
+
0
2
1
R1 +
0
Ci
—
1
Ro
—
It will be noted that Table l is divided into two sec
Co +
1
Input
Sign Combination (8)
Total
Output
Ro-l-Co=-~1+0=—1
(J
tions designated by K=l and K=0, respectively. In the 15
In Example 10:, sign combination (1) is present‘ where
section where K=1, two sign combinations designated as
signals Ai, Ri, and Ci have implied signs of +— and
(l) and (2) are noted. In sign combination (1), the
+, respectively. Since, according to the invention,‘ the
signals gOs, Rj, Cj, R0, and Co, have the following
new carry, C0, must bear the sign of the operation which
signs: +-+——+. In sign combination (2), these signals
is performed upon signal Ai, it is positive. The inven
have the signs: —+-—+—. It will be noted that K=l
tion also speci?es that the new result signal R0 must
occurs when the signs of Cj and gOs are the same, that
bear an implied sign opposite to that of the carry and
is they are either both plus or both minus.
therefore it is negative. Thus, the total output value is
In the section where K=0, two sign combinations (3)
and (4) are shown as: ++—-——+ and ——++t—-, tor _ +1, and is accomplished by generating a relative weight
signals gOs, Rj, Cj, R0, and Co. In this situation the 2° signal of —1 ‘for signal R0 and a relative weight signal
of ‘+2 for signal Co. It may be observed, that the terms
signs of OJ’ and gOs are noted to be opposite. In all
partial result and carry signals, as employed herein, per
cases, whether K=1 or K=O, the invention speci?es that
tain to the binary digits or equivalent signals, which must
signals Rj and Cj must be of opposite signs, and that
be combined to generate a total value. Thus, R0=l
signals R0 and C0 must be of opposite signs. This sign,
of course, is not actually generated but is implied in the C20 0 in binary place 2 of Example 1:: and C0=l, in binary
place 3 of the same example, are partial result and carry
state of the signal having a value of l or 0.
digits.
Table 1 represents all possible binary combinations of
It will be understood that the digit values which are
the input signals Aj, Rf, and C]'. In this sense it may be
mentioned herein are related to a particular binary place.
referred to as a “truth” table. This table will be con
sidered hereafter as a basic reference for determining 35 Thus, the values of —1 and +2 for output signals R0
and C0 are referenced to digital position 2, in Example
the value of binary output digits according to the inven
1a. The carry bears the weight of two since it is to be
tion, in the examples which follow. The table has been
carried over to the next higher digital place 3, whereas
developed by combining the binary input digits accord
the result bears the weight of 1 in column 2.
ing to their weight and sign to develop a total weight for
Another example as to how binary digits are combined
these digits, the total weight being then represented as
according to the invention is shown in Example 2a above.
the same value by the respective output digits R0 and C0.
In this case, sign combination (3) is illustrated where
Consider, for example, the section of Table 1 where
Ai, Ri, and Ci have signs: ++ and ——, respectively.
K=l and where sign combinations in group (1) are em
The carry output signal, C0, then bears the sign of At
ployed. In row (1) the total value is 0 since all of the
input signal Aj, Rj, and Cj are 0, and thus output digits 45 which is +, and the result signal bears the opposite sign
which is —-. Reference again to the section of Table 1
R0 and C0 are 0. In this case the signs are not desig
where K=0 indicates that the combination of 0 0‘ 1 for
nated.
the input signals Ai, Ri, and Ci, has the total value of
In row (2) only input signal Cj has a value of 1, and
—l. This is developed in the output signals through
the sign for column (1) is + so that the total value of
the input is +1. If sign group (2) were considered the
total value in row (1) is —l. The signs of output
signals R0 and C0, for the condition K=1, will be noted
to be the same as those for signals Rj and Cj, consider
signal Ro=—1 and signal C0=0, referenced to binary
place or column 3.
In the examples which are given below, following the
derivation of the basic units of the invention, reference
will be made again to Table 1 to show how certain out
ing the respective sign group.
The value of output digits R0 and C0 may be deter 55 put signals are generated. This can be done quite easily,
however, without the use of Table 1 by giving the new
mined by observing the sign of the input signals which
carry the sign of the operation, and then determining the
correspond to the total value. For example, in row (2),
value which the output signals must assume in order to
the signs + -— appears for the total value of binary '1, for
accomplish the proper total value. Thus if the total in
sign combinations (1) and (2), respectively. These signs
put value is -1, and the new carry is to be negative with
correspond to the sign of Cj which is also the sign of
output signal Co. This means that in order to obtain a
a positive result digit, both the new carry and the new
total value of 1 for the particular combination of input
result digits must be binary 1 to accomplish the desired
signals it is necessary to generate both a carry signal
Co, which has a weight of 2, and a result signal R0,
which has a weight, of the opposite sign, of 1. In other
total value. In a similar manner, if the total value is +1,
and the new carry is to be positive, both result and carry
digits must again be binary 1.
words, for sign combination (1), output signals R0 and
C0 in row (2) represent: -—l+2; and for sign combina
tion (2) these signals represent +1—2, the total value
in the ?rst case being +1 and in the second case, for
It may be desirable in some cases to expand the logical
expression for the carry signal to a two-level logical form
tively.
and input signals have opposite signs, may be derived by
as follows:
sign combination (2), the total value is —1.
70
In terms of actual computation, the summation of in
put signals and the production of output signals may
appear as illustrated in the Examples 1a and 2a, below.
The logical de?nition of the carry output signal for the
These correspond to Examples 1 and 2 above, respec
case where control signal K=O, and the previous carry
3,023,9ea
11
observing that the carry is binary 1 whenever the Previ
ous result signal Ri is a binary 1, and either the previous
12
it to 1, this latter condition being indicated by the ex
pression (10s)’=1.
In one form of mechanization, then, the signals K
and K’ may be derived through a complementary ampli<
?er driven by a logical network mechanized in accord
ance with the above equation. This type of mechaniza
Co=Ri.(Ci’-l-Ai)-l-Ci'.Ai
tion is employed for multiplication and division, as shown
below. It may be desirable, however, in other situ
or in two-level logic as follows:
ations, to employ a ?ip-?op to store signals K and to
C0=Ri.Ci'+Ri.Ai+Ci’.Ai
10 generate the complemented control signals K and K’.
A complete carry signal covering all possible input sign
in this situation, provision must be made for resetting
carry is a 0 or the previous input signal is a 1; and signal
Co is also 1 if the previous carry Ci is a 0 and the input
signal is a binary 1. This may be expressed as follows:
m
combinations may now be expressed as the logical com
the flip-?op to O in response to the condition:
bination of the two carry signals derived above, separated
by the control signals K and K’, where the signal K is
assumed to be “on” for the condition K: 1, and signal
The O-setting condition, it will be noted, indicates that
K’ is assumed to be “on” for the condition K=0. This
may be written as follows:
Co=K.(Ri’.Ci+Ri'.Ai+Ci.Ai)
+K'.(Ri.Ci'+Ri.Ai+Ci'.Ai)
Table 1 may be used to derive a different set of logic
for the carry C0. Speci?cally, it is to be noted that when
A1‘ is l and Ri and Ci are alike, i.e., both are 0 or both
are 1, C0 is 1. This takes care of Co for either K=1
or K=0 in the second and last rows of digits shown in
Table 1. For K=l, the l’s for C0 in the third and fourth
rows correspond ‘to the conditions for Ri=0 and Ci=1.
Similarly, for the 1’s in the C0 column for the condition
K=O in the ?fth and sixth rows, the conditions that
Ri=1 and Ci=0 are true. Thus, C0 may alternatively
be represented in a different set of logic as follows:
At this point note must be taken of the fact that the
algebraic expressions above do not necessarily constitute
particular circuit mechanization de?nitions since the
0K=Os.(00s)+0s’.(l0s)
K=0 whenever the previous and present operation signs
differ.
The general logical expressons derived above are shown
in a general mechanized form in FIG. 2 as another mode
of expression rather than as a particular example of
20 utilization. In FiG. 2 each “and” function is generated
through an “and” circuit shown as a semi-circular draw
ing representation with a dot in the center. Thus, the
logical “and” terms Ai’.Ri'.Ci', Ai.Ri'.Ci’, Ai'.Ri.Ci',
Ai’.Ri'.Ci are provided by “and” circuits 21, 22, 23 and
24 producing the corresponding output signals. The
signals of “and” circuits 21-24 are then combined through
an “or” circuit 25, all “or” circuits being shown as semi
circular drawing representation with a plus (+) in the
center thereof.
It will be noted in FIG. 2 that result signal R0 is
produced independently of the condition of the control
signal K, but that two different carry signals are pro
duced, being combined under the control of the signals
K and K’ to form a complete carry Co.
The arrangement of FIG. 2 shows the utilization of
‘be employed must each be considered separately. The
a storage ?ip-?op producing signals K and K’ so that
both l-setting and O-setting logic must be employed, as
particular de?nition of the invention, but rather to de?ne
a generic class of logical structure. Thus, the expressions
for the parallel accumulation of positive or negative
numbers by either addition or subtraction is shown,
various cases where the basic unit of the invention may
discussed above. The particular logic for entering the
reason for this will become apparent as various different
utilizations of the arithmetic unit are introduced below. 40 operation sign into storage device Os is not shown since
this logic varies with each utilization of the invention.
Accordingly, it will be understood that the logical algebra
Reference is now made to FIG. 3 wherein a system
introduced above is in no way intended to constitute a
for the result and carry are intended to be generic to a
multitude of other forms which may be derived there
from by well known algebraic techniques.
Since the sign of the carry input signal was determined
by the previous operation sign Os, it may be stated that
the control signal K assumes a binary 1, or “on” state
whenever the previous and present operation sign sig
nals are the same.
in a similar manner, it may be
stated that the control signal K becomes binary O or
“off” whenever the previous and present operation sig
nals are different. it will be understood, however, that
the de?nition of control signal K in terms of the pre
vious carry and present operation sign is equally ap
plica-blev
If it is assumed, then, that each operation sign signal
is to be entered into a storage device, such as a flip-?op,
provision being also made for combining the result and
carry signals in a ?nal serial operation to form a com
plete result. As indicated in FlG. 3, the system in
cludes an input number register consisting of a plurality
of storage elements Al . . . Ai . . . An for receiving
the number, and a storage element As for receiving the
sign of the input number. Control signals Su and Ad
representing operations of sn'btractions and addition, re
spectively, are also received by the system and applied
to gating circuit 505' controlling operation sign storage
device Os.
including
carry
The system also includes a result register
storage
register
devices
including
Cl . _. . Ci . _ . C12.
R1
. . . Ri . . . Rn
carriage
storage
and
a
devices
The input, result, and carry digits
are applied to respective full adder-subtracters FASI . . .
FASJ . . . FASn which are mechanized the same as the
and that this device includes l-setting and (l-setting cir 60 unit shown in FIG. 2 in accordance with the ?rst basic
logic introduced above and are controlled by signals K
cuits, referred to hereafter as 10s and 00s, respectively,
and K’ produced by device K. At the termination of
then signal K may be de?ned as follows:
each accumulation operation, which consists of the suc
cessive combination of a plurality of input numbers A
to the previous partial result and carry signals, the result
and carry signals are shifted right under the control of
This expression may be interpreted as stating that signal
a shift control device X, and are applied to a full sub
K is 1 if a previous operation sign 0s=1 is stored in a
tracter unit FS which produces a series of output sig~
?ip-?op or other storage device, and if this storage de
nals corresponding to the accumulated total result. Full
vice is now to remain in the “on” or 1 state, since no
signal is to be applied to its O-Setting input circuit [as 70 subtracter FS produces a series of carry signals which
are stored in storage device Cs being applied thereto
indicated by the condition (0Os)'=l)l. The other
through a gating circuit gCs. It will be shown that the
condition for signal 1 being equal to K occurs when
storage logic for full subtracter FS may be the same
the previous operation sign Os is stored in a ?ip-?op or
as that part of the logic included in full adder-subtracters
other storage device is a 0, as indicated by Os’=l, and
FAS during the control condition where signal K=l.
uorsignpal is to be applied to the storage device to set
3,028,982
13
14
The particular details specifying the structure for shift
control circuit X and various means for. synchronizing
the operation of this device with the control signal in
it is important to note that Where complementary num
bers may be allowed, the most signi?cant place must be
reserved for a binary digit having a- sign opposite to
puts are not shown and will not be discussed since, such
arrangements are now well known in the art. The novel
the remainder of the number. This means that arithmetic
computation must not be performed in the most- signi?~
cant place. To be further particular, it will be noted
that in Example 7, where no complementary numbers
are employed, the S-Weight place includes all zeros at
features of the invention reside entirely in the manner
in which partial result and carry signals are accumulated.
Various types of prior art circuits are available for add
ing two binary series through a shifting operation in
the start, and that no addition or subtraction is performed
order to develop a single output binary series or present 10 which requires an arithmetic total result having a digit in
ing the sum or difference of the two binary numbers.
this place. Thus, in Example ‘6, any digit in the S-Weight
The operation of the system shown in FIG. 3 may
place results from the use of complementary numbers,
best be illustrated by considering a few speci?c examples,
and is not confused with the arithmetic computation.
as shown below. In Examples 6 and 7 the numbers
Referring to Example 6, in particular, the addition of
—5, +3, and +4 are accumulated, all numbers being 15
the signals A1 and R1 in the S-Weight place results in
represented in complementary form to illustrate a differ.
ent type of operation, so that —5 is expressed as the
an answer of —1, which is expressed as a binary 1 in
the 8-weight place for signal ——R2. Then, When signals
binary number +1011, +3 is expressed as the binary
A2 and R2 in the S-Weight place are added, it must be
number —ll0l, and +4 is the binary number --1100.
A simple way of interpreting complemented binary num 20 recalled that the most signi?cant A2 digit is positive,
the lower place digits being negative, whereas all of the
bers is to consider the most signi?cant place as having
R2 digits are negative. This results then in a zero result
a sign opposite to the other places. Thus, binary weights
are given to the successive columns as 8, 4, 2 and 1,
digit R3 in the same place. However, although for ex
so that the number +1011 may be considered to be
planatory purposes only, it is helpful to note the sign of
—l000 +011 or —8+3=—5. In a similar manner, 25 the most signi?cant digit, in any case the logic for com
the binary number --1101 may be expressed as +1000
bining all digits is always the same.
-—101 or +8—5=+3.
The partial accumulation may be noted after each
Example 7, shown below, performs the same operation,
operation. For example, after —5 is accumulated, the
signals R2 and C2 will represent the numbers ~11 and
absolute value plus sign. In this case, the operation sign 30 +6, respectively, corresponding to the entry of a ~—5.
is a function of the sign of the input number Ai as well as
After the accumulation of —5 and +3, it will be noted
the accumulation sign. Thus, it will be noted that when
that signals R3 are all zero and signals C3 represent a
operation sign Os appears as minus, it may have been
—2. Finally, the last result is R3 equalto +14 and C4
either a plus number which is subtracted, or a minus
number which is added; whereas if the operation sign is 35 equal to —12 so that the total result, subtracting C4
namely, —5+3+4=2, but all numbers therein are in
from R4 is a +2.
In Example 7 the ?rst accumulation results in the num
ber R2 equal to +5 and C2 equal to —10 corresponding
9, shown below, are similar to Ex
to the entry of —5; the second accumulation results in
in that the same numbers are
signs are reversed. This example 40 the number R3 equal to —12 and C3 equal to +10 for
a total result of —2; and the ?nal accumulation results
in R4 equal to —2 and C4 equal to +4 for a total result
plus, it may indicate either that a minus number is
subtracted or that a plus number is added.
Examples 8 and
amples
6 and 7
employed but the
is +5—4—3=-2.
EXAMPLE 6
K
Os
+
A1
0
8 4 2 l
1 0 1 1
K
—5
+R1 0 0 0 0
—C1
-
A2
0
—R2
Os
—
A1
1
1 1 0 1
0
1
1
—C1
+3
.
+
0
+02 0 l. 1 0
—
A3 1 1 0 0
+R3 0 0 0 0
—O3 0 0 1 0
1
+4
+
1
K
Os
-—
+
—
(J
1 0 1
1
1 1 0 1
0
K
~—C2
l 0 l 0
1
0
0
—3
—
—4
—
1
8 4 2 1
Al
0
1 0 l
+3.1 0 O 0 0
-—C1 0 0 0 0
0
+C30010
noted that the signals A1, R1 and C1 in column 1 are
1 1 0
section designated by control signal K=0, it will be noted
that where the same combination of binary values oc
curs for signal Ai, Ci and Ri, the output signals R0‘ and
Co are both binary 1. This, then, corresponds to'the
60 digits of R2 and C2 in columns 1 and 2, respectively, of
Example 6. In terms of the result logic, the digit R2
1
occurs upon satisfaction of the condition Ri'.Ci’.Ai=1.
The carry output signals is caused by satisfying the logical
A2
A3
0 0 1
0
1
0
0
+R3 l l 0 0
+R4 0 0 1 0
-—C4 0 1 0 0
—-2
1, 0, and 0, respectively, and that control signal K is a
0. Referring again to Table 1, and in particular to the
—-R2.0101
+G210>10
—O31010
t0} 1 0 0 0
1 1 l 0
Os
+
0
—R 0 0 0 0
R4
1
0 0 1 0
+5
+R21011
—C20l10
A3
1 0
EXAMPLE 9
+Rl 0 0 0 0
—C1 0 0 0 0
A2
O
0
+2
8 4 2 1
A1
result signals.
It is helpful in analyzing the logical operation of the
invention to consider various logical steps in Examples
6 through 9. in Example 6, for example, it will be
A2 0 0 1 1
+R2
v-R4 0 0 1 0
+04 0 1 0 0 55
EXAMPLE 8
1
taken of the fact that an answer of —2 results where
the negative carry signals are greater than the positive
0 0 0 0
A3 0 1 0 0
-—R3 1 1 0 0
+03 1 0 1 0
+R4 1 1 1 0
~04 1 1 0 0
0 0 1 0
8 4 2 l
1 0 1
0
+R1 0 0 0 0
0 0 0 0
1
of +2.
It is believed that Examples 8 and 9 may be readily
followed from the above discussion particular note being
EXAMPLE 7
1 1
1 0
While, for the most part, Examples 6 through 9 are
similar to Examples 1 through 4, indicated above, a few
important distinctions should be noted. In the ?rst place,
condition Ci’.Ai=1.
65
Another typical example of the logical operation of
the invention is found in the transformation of the input
signals A2, R2 and C2 in column 1 of Example 6 which
are 1, 1 and 0, respectively, into the output signals R3
and C3 in columns 1 and 2, the output signals being 0
and 1, respectively. It will be noted that K is equal to 0
again and reference to the corresponding portion of Table
1 indicates that the correct result has been formed. In
terms of the logic, the result signal is 0 since none of the
75 conditions of the-result logic are satis?ed, and the carry
3,023,962
15
,
over signal is 1 since the condition Ri.Ai=l has been
satis?ed.
A similar logical operation may be made where the
control signal K=l as, for example, where signals A3,
R3 and C3 have the values 1, 0, and 0 in column 4 of
Example 6, the result thereof being R4=l and C4=l.
EXAMPLE 6A
EXAMPLE 7A
e-—-—t
+R0
—-Co
4—-—-t
l l 1 0
1 1 0 0
—Os000>0
+Rt
This corresponds to Table 1 and illustrates the case where
the result condition Ri'.Ci’.Ai=l is satis?ed and the
carry condition Ri’.Ai=1 has been satis?ed.
0 0 l. 0
=+2
-R0
+00
0 0 1 0
0 1 0 0
+03
11000
+Rt
1 1 1 ‘l 0
EXAMPLE 8A
4___...t
The logical progression of control signal K should also 10
be noted. It will be observed that whenever the operation
<--——-t
+R00l10
+Ro
U010
—Gsl0000
—Cs
11000
—O0
1 0 0 0
==-—2
EXAMPLE 9A
-—O0
0 l U 0
+Rt 11 1 l 0
=~2
+Rt l 1 1 l 0
=—2
sign signal Os changes from a plus to a minus or from a
minus to a plus, the following control signal is a zero;
It will be noted that in Examples 7a, 8a and 9a, the re
whereas whenever signal Os is a plus and remains a plus
sult appears in complementary form so that it must be
or is a minus and remains a minus, the following con 15 subtracted from zero in order to obtain an absoluted
trol signal K is a 1. It will also be observed that signal K
number. The sign of the complete result is therefore
may be de?ned by noting the comparison between the
opposite to that indicated as the sign of the series Rt.
present signal Os and the carry input signal sign. If these
In Example 6a the result of +2 is generated as the abso
luted binary value series, whereas in Example 711 a com
plementary series is generated as indicated by a binary l
in the most signi?cant or leftmost digit position. The full
subtracter just described may be employed where numbers
are stored as absolute value plus sign in a memory device,
signs are different,_K is equal to 0, if they are the same,
K is equal to 1. It will be understood, therefore, that
either de?nition of signal K is suitable and further that
two basic types of circuit mechanization are possible.
These two logical de?nitions for signal K must occur in
view of the fact that each new carry signal bears the sign
if an additional word time interval is utilized to recom
of the previous operation sign so that etfectively the com 25 piement numbers in complementary form by subtracting
parison between the present operation sign and the present
from zero. The sign of the answer is the same as the sign
carry sign is really a comparison between the present op
of the result series, unless the most signi?cant digit indi
eration sign and the previous operation sign.
cates a complementary number, in which case the true
It will be understood that while in Examples 6 through 9
sign is opposite to that of the result series.
only four binary digits are shown and only three accumu 30
A serial subtracter is thus provided wherein a single
set of logic is operative for either subtractive case:
lations are illustrated, in fact the invention most likely
will be employed with binary numbers in the order of
C0—R0 or R0——C0. Thus, it will be understood that the
basic unit of the invention is important for serial usage
magnitude of 20 signi?cant digits, or more, and that
as well as for the serial-parallel or parallel usage.
accumulations of ten or more input numbers is certainly
Reference is now made to FIG. 4 where the general
feasible. The only limitation in any accumulation proc
form of logic of the unit for the various places of the
ess is the capacity of the accumulation register.
parallel add-subtract unit of FIG. 3 is shown. It will
It will be recognized that the accumulation process in—
be noted in FIG. 4 that the carry network logic is dif
troduced above is performed without cascading any car
ries from one place to the next, since no carry is formed 40 ferent from that of FIG. 2.
It is in fact mechanized ‘from the preferred and simpli
as a function of any carry other than that in the same
?ed logic:
place.
After completing the accumulation as in Examples 6
through 9, the complete result may be formed by serially
combining the accumulated result digits and carry digits
The result logic, however, is somewhat simpli?ed in view
of the fact that in parallel accumulation, each result signal
through a full subtracter. One manner of accomplishing
this is to employ a subtracter of the type which is utilized
is formed as a function of a previous result signal in the
signals referred to hereafter as C0 to constitute a serial
one for each term R1" to de?ne the l-setting control sig
same place. When logical storage elements, such as ?ip
when control signal K is equal to 1. It will be recalled
?ops, are utilized, this means that the state of the ?ip-?op,
or other storage device", need not be changed if it is al
that in this case the carry input signals and the input digits
have the same sign and the previous result and input 50 ready in the state required for the next logic. In terms
of logic, this means that the normal result logic may be
digits have different signs. This type of full subtracter
simpli?ed by substituting a zero for each term Ri and a
may be employed by considering the accumulated carry
input number, and the accumulated result signals referred 55 nal for a ?ip-?op. Thus, assuming a ?ip-?op Rj which is
to receive the jth result signal, and that the ?ip-flop has 1
to hereafter as R0, as a previous result series. In per
and 0 input circuits de?nable as 1R1‘ and OR)‘, respective
forming a serial subtraction upon Co and] R0 to form the
ly, the input logic may be expressed as follows:
series of digits Rt representing the total result, a carry
series Cs is generated. The signal series Rt and Cs are
de?ned as follows:
60
This expression may also be considered to be a half
sum in terms of the input number and carry. It will also
be noted in FIG. 4 that a particular form for the input
logic of a storage device Os is shown which may be
The carry logic Cs may be entered into a ?ip-?op in ac 65 expressed logically as follows:
cordance with simpli?ed l-setting and O-setting functions
as follows:
70
The operation of this type of subtracter may be illus
trated by considering the manner in which the various
results of Examples 6 through 9 are combined, as shown
in Examples 6a through 9a, respectively:
This logic indicates that the operation sign is to be rep
resented as a binary 1, indicating a minus (—) operation
whenever the operation to be performed is addition, rep
resented by a control signal Ad, and the sign of the input
number is negative, represented by the signal As: 1,
75 Whenever a subtraction is to be performed, indicated by
n
17
3,023,962
18
the control signal Su, and the sign of the input member is
positive, indicated by signal As'=l. The complementary
partial product register. The manner in which the ?rst
digit in the multiplicand is combined with its correspond
ing digits in the other registers will then be determined
by both the multiplier digit value and sign, and also the
accumulation sign for the particular product. As in the
condition for the operation sign should be apparent from
this explanation.
In referring to FIG. 4 it will be noted that each “and”
function of the ‘above expression is mechanized by means
other cases where the unit of the invention is employed,
control signals K and K’ determine which of the two
being represented by a semi-circular enclosure with a dot
types of carries are formulated in the various full adder
(.) therein. Thus signals Su and As’ are applied to an
subtracters. A serial product passes through the least sig
“and” circuit in gate gOs of FIG. 4 and signals Ad and 10 ni?cant register place, referred to as R0, during two suc
As are applied to another “and" circuit. The output sig
cessive n-digit time intervals, starting with the least signi?
nals of these two “and” circuits are combined in an “or”
cant product digit. It will be noted that each product
of an “and” circuit which may be of a conventional type
circuit designated as a semi-circular enclosure with a plus
digit is shifted forward through the corresponding full
sign (+) therein. This then develops the function 10s,
adder-subtracter as it is formed to a corresponding result
and an ampli?er is employed to invert signal 10s to de 15 digit storage stage.
velop its complement 10s’ which is used in other logic.
The embodiment of FIG. 5b operates on a somewhat
Signal 10s is passed through the ampli?er uninverted and
different principle from that shown in FIG. 5a in that the
is supplied to the corresponding input circuit of device Os.
full adder-subtracter signals are only employed when the
The manner in which the various signal entries may
multiplier digit is a binary 1. When the multiplier digit
be accomplished during respective computing intervals
through the means of well-known types of synchronizing
circuits will not be discussed since this technique forms
20 is a zero, the partial result and carry signals are shifted to
the right and applied to a full subtracter FS. Thus, in
the embodiment of FIG. 5b, the signals of full adder-sub
no part of the present invention and can be readily sup
tracters
plied by those skilled in the art.
The result and carry logic shown in FIG. 4 is modi?ed
FASl . . . FASj . . . FASn
are
applied
to
“and” circuits 51a . . . 5ja _. . . Sna, respectively. These
“and” circuits pass the full adder-subtracter signs in re
sponse to signal Sx produced by a device Sx which serially
to introduce the right shifting operation required to apply
signals R0 and C0 to the full subtracter P8. In general
terms and in terms of a right shifting control signal X,
this shift function may be expressed as follows:
receives the multiplier digit signals. The result digit sig
nals are shifted to the right through “and” circuits
30
51b . . . Sjb . . . Snb controlled by signal Sx' produced
by device Sx, whenever the multiplier digit signal applied
thereto is a zero. In this manner, the multiplicand is ac~
cumulated to the previous partial product when the mul
tiplier digit is a binary 1 and the partial result and carry
signals are shifted forward to full subtracter FS when the
The above general terms de?ne an operation such that, 35 multiplier digit is zero.
when signal X equals 1, functions gRj and gCj (further
The shifting of the carry signals is controlled by a flip
de?ned below) are entered into ?ip-flops Rf and Ci, re—
?op CX, which is assumed to be in an “on” state when
spectively. When X’ is equal to l, shifting is performed
ever the multiplier digit is zero and right shifting is to be
since the logic speci?es that each ?ip-?op receives the
performed,
and to assume a zero state whenever the multi
40
signals from the next higher place ?ip-?op. Thus if j is
plier digit is a one and the normal carry function is to be
equal to l ?ip-?op R1 receives output signal R2 of Hip
entered into the associated ?ip-?op. However, unless
?op R2 and flip-flop C2 receives output signal C3 from
other shifting functions are involved, this function may
flip-flop C3. The speci?c connections of FIG. 4 may be
be performed simply by the use of signals Sx and 8x’. _
determined from the discussion which follows, but will
An ‘additional carry storage ?ip-?op Co is introduced
not be reviewed in detail since the manner in which logic
so that full subtracter FS combines input signals R0 and
is interpreted in terms of structure is now fully understood
C0 to produce the complete product, carries Cs being gen
by those skilled in the computing art.
erated ‘by device FS and stored in device Cs.
In other respects, the logic for the unit employed in a
The manner in which multiplication is performed in
parallel accumulator is similar to that introduced above
in the basic logical discussion and therefore will not be 50 accordance with the basic principles of the present inven
tion is shown in Examples 10 through 13.
described further. A complete set of logic for the various
units in FIG. 4 is summarized as follows:
EXAMPLE 10
55
OS
0
4 3 2 1 0 B1
A1 0
—Rl 0
1
+01 0
60
E‘; 0
I
9-1
Reference is now made to FIGS. 5a and Sb showing two
23 c:
alternate systems for performing serial-parallel multiplica
tion in accordance with the present invention. Consider 65
ing FIG. 5a in particular, it will be noted that the multi
plicand is entered into a register including storage devices
Hot-1 HD GHQ
O0 O0
Col-4 COD-l
r-4
H
COhPpH-il‘yOc0CG
Al . . . Aj . . . An, the digits thereof being transferred
through a plurality of gating circuits
gAl....gA]'...gAn
H C
70
to a partial product register including a plurality of
full adder-subtracters mechanized in accordance with the
principles introduced above. In operation, the multipli
cand is added to the previous partial product stored in the 75
H
H
+
dé
o
iii;
Q
HOG: 0H
H0O ON
o 0
H
l
grasses
20
19
to ‘be subtracted as multiplied, the sign Os is minus. The
number Bl shown vertically adjacent to successive sets
of the numbers A, R, and C, is the multiplier and is noted
EXAMPLE 11
OHIO
1
to be the binary number 101 or 5. The previous rem-ain
der, left over from another operation, is noted to comprise
the number R1=—1 and Cl=+6(0ll0) the total of
O
which is +5. In Example 10 the multiplicand is etiec
tively a series of zeros whenever the multiplier digit is
zero in view of the gating operation of “and” circuits
w
H
10 gAl . . . gAj . . . gAn. In Example 10, then, the full
HCOG0
m
HO Hail
OH Idol
1
I-lHQ
adder-subtracters operate continuously in the same man
ner as in addition and subtraction discussed above except
owOHveI-cl:
O
Db-'
m
that the result digits are shifted ahead as formed, and
the carry digits remain in the same place. Thus, it will
15 be noted that the digits Al, R1 and C1 in column 1
(0, 1, O), combine to form the result digit R2 equal to 1
in column 0, and carry digit C2 equal to 1 in column 1.
The answer appears in complementary form since the
most signi?cant digit is a binary 1 and represents the
+116
- C6
20 number —25.
I
In Example 11 no multiplicand digits are shown at such
times as the multiplier digit is zero. In this case, the re
sult and carry registers are shifted to the right and the
Oo
Answer 1 0 O 1 1 1
25 generates carry signals Cs. It will ‘be noted that the
carry digits in column 0 for C4 must be reset to zero
EXALIPLE 12
+(-3)X(-—7)—-2=+19
K Os
H
o
4 3 2 1 0 B1
I
I
F5
55 us
car-Io r-n d
O0 ODu-0Ipr!“Q Or-IO H140
GO OH
O H 0°C 0 -‘
result series Rt is formed through full subtracter FS which
—25
1
H O
after a multiplier digit of 1 appears, since the carry signal
remaining is left over from a previous shifting operation
which applies only to the case when the multiplier digit
30 is zero. Thus, when a multiplier digit of 1 appears all
carrys are generated through the full adder-subtracters.
In other respects, the operation of the various arithmetic
units is the same for both embodiments shown in FIGS.
5a and Sb.
c: O
35
Examples 12 and 13 illustrate the operation of serial
parallel multipliers according to the present invention
when complementary numbers are employed, and corre
M H
spond to the operation of the embodiments of FIGS. 5a
and 5 b, respectively.
40
one v-uso
o
w a:
c
I
O
o
0 00
w o
o
C1 in column 4 results in an answer of —-1 so that digit
R2 in column 3 is a 1 and digits C2 in column 4 is a 0.
EXAMPLE 13
+(-3)X(-7)—2=+19
Os
1
4
A1 1
-R1 0
As in the case where complementary numbers are added
and subtracted, note must be taken of the opposite sign
of the input digit A1’ in its most signi?cant place. Thus,
the number —3 represented as the binary number 1101
is considered to have a positive operation sign during the
time the least signi?cant multiplier digits are applied, but
45 the most signi?cant digits thereof are negative. Conse
quently, the combination of the input digits A1, R1 and
2 1 0 Rt B1
It will be noted that the operation sign signals change
plus (+) to minus (——) after receipt of the most
signi?cant multiplier digit. The reason for this is that the
multplier (-7) appears in complementary form as 1001,
50' from
1
+01 0
which is —8 +1, so that the result digits formed after the
' most signi?cant binary 1 is received have signs opposite
55 to those formed during the entering of the least signi?cant
1
multiplier digit. Thus, the answer is
-R3
+03 0
+++-—
011101
60 which is equal to +24 —-5 or +19.
Two forms of result and carry logic will be shown
below corresponding to the structure for the embodiments
Hid
I
Om
of FIGS. 5a and Sb. Accordingly, the logical de?nitions
will be prefaced by the symbols (5a) and (5b), respec
65 tively, to indicate the desired correspondence.
The result logic for the embodiments of FIG. 5a is
based upon a gated series of digits Al . . . Aj . . . An
+++———
AnswerU 1 1 10 1 = +25 —5 = +19
Examples 10 and 11 illustrate the solution of the same
so that, considering the multiplier digits as being repre
sented by symbol M, each input digit to a result network
70 may he represented as (M .Aj). Thus, in the embodiment
of FIG. 5a the input logic for flip-flop Rj——1 may be rep
cumulative multiplication problem utilizing the embodi
ments of FIGS. 5a and 5b, respectively. In Example 10
the multiplicand is the number A (6:0110). Since it is 75.
resented as follows:
21
8,023,962
In the result network for the embodiment of FIG. 5b,
the result function is continuously formed and either
shifted into ?ip-?op Rj—-l under the control of signal Sx,
or the full adder-subtracter is effectively bypassed under
the control of signal Sx' in which case signal Rj is shifted
into ?ip-?op Rj—1. The logic for this approach is speci
?ed as follows:
22 '
Signal Rj= is also applied to this network so that the re
sult signal is passed through the network in response to
the signal Sx and the previous result digit Rj is passed
through the network in response to signal Sx’.
In other respects, the general form of the networks
shown in FIGS. 6a and 6b will be readily understood from
the logic introduced above.
Reference is now made to FIGS. 7a and 7b constitut
ing together a partial schematic and block diagram of a
division system incorporating arithmetic units mechanized
In a similar manner the carry logic may appear in 10
in accordance with the present invention. The division
either of two general forms as follows:
system will not be described in any detail beyond that
which is necessary to show the manner in which the basic
arithmetic units are incorporated therein, since the sys
15 tem features thereof are covered in copending application
for “High-Speed Division System” by Roger A. Stafford,
It will be noted, however, that each carry function is
written in terms of the previous carry signal stored in
the corresponding ?ip-flop. Thus, a simpli?cation is pos
sible in view of the nature of the flip-?op in that no sig
nal need be applied to the ?ip-?op to change its state
if it is already in the state desired. For this reason,
Cj is an expression for lCj may be set equal to Zero,
and C1" may be set equal to zero in an expression for
0C1‘ as follows for both cases (5a) and (5b):
?led May 23, 1957, Serial No. 661,157. This copending
application need not ‘be considered at this time in view
of the fact that the following description adequately
points out the manner in which division is performed
utilizing the basic arithmetic units of the present inven
tion.
In the division system, the divisor is entered into stor
age devices Al . . . Aj . . . An, which may be noted
25 to be shown in reversed order with respect to the previous
?gures. The reason for this is that the most signi?
cant divisor digit is assumed to have entered into storage
device A1 and sucessively lower place digits to have
been entered into storage elements to the right thereof
30 with the least signi?cant digit of the dividend being en
tered into device An.
In a similar manner, the partial remainder register is
shown reversed with respect. to the previous drawing
representation in order to indicate that the most signi?
35 cant digit of the partial remainder is entered intov storage
Occasionally it might be possible to set Cj' in an ex
pression for lCj in either of the cases (5a) or (5b)
equal to 1, and 0]‘ in the case of OCj equal to 1 since
it is normally assumed that it will be necessary to set
1C1‘ if the condition Cj’ is true and to set ()Cj if the
condition Cj is true. However, in both of the cases
(5a) and (5b), the assumption of such conditions for
simpli?cation would simply mean that under certain gat
ing conditions, a signal would be applied both to the
device R0, and that successively lower place remainder
digits then are entered into storage devices having suc
cessively higher digit representations. This manner of
utilization of the structure for division makes it possible
to share the full adder-subtractor logic for multiplica
tion. In this manner, the partial remainder, which in
itially corresponds to the dividend, is continuously shift
ed to the left with respect to the divisor so that succes
sive additions and subtractions may be performed to form
IQ‘ and OCj terminals of ?ip-?op Cj simultaneously. For 45 quotient digits, each addition and subtraction being per
formed during a digit time interval without cascading
this reason, it is necessary to use 0]" in 10]‘ and Ci in
carries.
0C1‘.
As indicated in FIGS. 6a and 6b, signals K and K’
In the division system each carry produced by a full
may be generated through an inverting ampli?er, in
adder-subtractor xmust be shifted to the left by two digital
which case the input logic therefor corresponds to the 1 50 places so as to represent a weight of twice the correspond
setting logic for ?ip-?op K. The ampli?er logic may
ing result digit which is shifted to the left by only one
be speci?ed as follows:
digital position. It will be understood, of course, that
other means of accomplishing the partial remainder shift
with respect to the divisor are possible. For example,
The input logic for flip-?op Os is a function 0 finput sig 55 if the divisor is shifted to the right, the partial remain
der signals may be left stationary, and the result digits
nals Ac, As and Ms corresponding, respectively, to the
accumulation sign, multiplicand sign and multiplier sign.
Note must be taken, of course, of the change in the multi
plier sign which occurs upon receipt of a binary l in
are then reentered into the same place and the carry
digits are shifted to the left by one place. This division
operation may be made consistent with multiplication if
the most signi?cant digit position for a complementary 60 the multiplicand for multiplication is then shifted
to the left.
number, as discussed above. The general form for the
Returning again to FIGS. 7a and 7b, it will be noted
that certain signals of the divisor register and the partial
remainder register are applied through a gating circuit
and the others are positive, or if all three signs are nega
65 gN to a storage device N which produces output signals
tive. This will be recognized to be similar to the result
N and N’ indicating, respectively, that the partial re
network logic above and may be represented as follows:
input logic de?ning signal Os speci?es that the sign is
negative, if any one of the three input signs is negative
mainder register is to operate as a series of full adder
subtracters, or as a shifting register. Thus, in this respect,
device N is similar to device Sx introduced above for
In comparing FIGS. 6a and 6b, it will be noted that in 70 the multiplication system. The letter “N” is employed
FIG. 6a the input digit signals are represented as M .A]',
in order to indicate that when N=l, the division step is
whereas the signals Aj are applied directly in FIG. 6b.
eifectively a “do nothing” operation, where the remainder
It will be noted that the result signal produced by the
is shifted and neither addition nor subtraction is per
network of FIG. 6b is passed through an ampli?er and
formed. The purpose of this “do nothing” or 0 step
then applied to a network controlled by signals Sx andSx’. 75 will be understood when the division example is con
0,023,002
23
register when the operation sign is positive.
As in the previous examples, each time the operation
tion where addition or subtraction is to be performed.
As in the other systems employed in the present inven
sign is negative, the new carry which is formed is nega
tion, the operation sign is entered into device Os which
controls device K as before.
24
a minus quotient digit —Q is entered into the quotient .
sidered below. Signal N’ then indicates the other situa
Device K is shown as an
ampli?er illustrating a division system where control sig
nals K and K’ are produced without delay, although it
is shown in the above—mentioned copending application
that a flip-?op storage technique may also be employed.
The de?nition of the sign signal is quite complex and 10
tive and the result signals are positive. Thus, series R2
in Example 14 is positive and series C2 is negative, fol
lowing the negative operation sign.
It will be noted that whenever signal N=1, the result
and carry registers are shifted to the left. In these cases
the divisor signals are not shown since they are not com
bined through the full addensubtracters. Whenever
N=O, either addition or subtraction is performed, each
result signal is shifted to the left by one binary place
as it is formed, and each carry signal is shifted to the
left by two binary places as it is formed. In this manner,
Os’ indicate negative and positive operation signs, respec
each partial remainder is effectively shifted to the left
tively.
with respect to the divisor.
Before returning again to a further description of FIGS.
It is assumed, for the purpose of Example 14, that the
7a and 7b, and then FIG. 8, it is helpful to consider a
result and carry digits in columns 12, 13 and 14 are
speci?c division example as follows:
sensed in order to determine the next operation step.
20
Whenever it is noted that the digits in these columns
EXAMPLEM
will not be considered in any detail herein since the
necessary description is found in the above-mentioned
co-pending application, and it is only necessary to note
for present purposes that the two output signals Os and
constitute a negative total result, the operation sign is
2ss1/91=a1
KNOs
_
made positive, whereas whenever these digits indicate a
14131211100s1054s21+Q-Q
A1
101101100000
positive result the operation sign is made negative. When
10
+R1001011000001011
0 25 the digits in columns 12, 13 and 14 represent a zero total
result and consequently do not indicate either positive
1
+B2000 000110010100
-o2
00001100000
0
or negative signs, a “do nothing" step is performed dur
ing which times signal N=1, and 0’s are shifted into
1
+R3000 001100101000
+03
00011000000
0
both the +Q and -—Q registers.
1
+R4000 011001010000
-o4
00110000000
0
00
+115
10110110000
+R5000110010100000
1
~01
-o5
-
00
1
00
000000000
01100000000
A6
—R0000
010011000001
0
+R7001001100000000
-o7
0100110000000
0
0110111000000
-
10
—01
-
+
00
-
00
+
1
1
010000
+R1001
10
00
A1
054321
A2
101001
00
000000
1
010000
+R2001
110010
-o2
0000
01
A3
1
+na111
010000
000100
-03
00
0000
A4
since it need not be known. In a similar manner, the
35 operation sign Os is not shown at such times as the “do
nothing” steps are performed, so that the logic for gating
network gOs may be simpli?ed to permit the genera
tion of any signal desired during such intervals.
In considering the speci?c sequence of Example 14,
series -A1, +R1, and —C1, results in digits +R2 and
105/5=21
987
K is not speci?ed during those times that signal N=1,
40 it will be noted that the ?rst combination of the signal
EXAMPLE15
KNOs
It will be noted that, as in the other operations, signal
K=1 whenever the present and previous operation signs
are the same or whenever the present operation sign is
the same as the previous carry sign. The value of signal
101101100000
+00
30
Q
+1
+1
—1
-—C2 in columns 12, 13 and 14 which are all O’s. This
condition remains for three successive digit time inter
vals during which the “do nothing” steps are performed
45 and O’s are entered into the quotient digit registers. When
the signal series A5, R5 and C5 are present in the re
spective registers, it is noted that the partial remainder is
negative, so that a plus operation sign is attributed to the
series A5. Thus, a minus quotient digit is entered into
50 register —Q and the new carries formed have implied
positive signs resulting in signal series +C6.
1
010000
—R4000
+04
01
101000
0100
+1
A5
1
+R5001
010000
010000
—1
4:5
1000
answer is a plus binary number +100001 and a minus
55 binary number —000010, or +33——2=31 corresponding
to the binary number +011111.
Example 15 is similar to Example 14 in most respects
10
—110001
00000
0
+00
0000
_
01
The complete answer is determined by combining the
plus and minus quotient digits serially. E?ectively, the
except that no “do nothing” steps occur until the last
operation and a digit appears in the most signi?cant place
60 of the remainder register shown as column 9 in this ex
In Example 14 the binary number 1011011, correspond
ing to the decimal number 91, constitutes the divisor and
is entered into the A register, and in particular into stor
ample, noted speci?cally in series +R3. In this case
it will be noted that the associated carry signal C3 is en
tered as a minus sign rather than a number.
The reason
for this is more fully described in the above-mentioned
age devices A6 through A12, the other signals being
entered being binary O’s. The dividend is the binary 65 copending application relating to the division system,
wherein general and speci?c techniques for modifying the
number 101100000101, corresponding to the decimal
logic for the most signi?cant result and carry places are
number 2821, and is initially entered into the partial re
introduced whereby these digits represent the complete
mainder register appearingin storage devices R1 through
value in the place. This technique considerably simpli
R12.
It will be noted that the ?rst operation sign Os is as 70 ?es the decision logic which is necessary to formulate the
operation sign signal Os, the “do nothing” signal N, and
sumed to be minus and that the ?rst quotient digit en
the carry control signal K.
tered is +Q=1. Thereafter, whenever N equals 0, indi
It will be noted that the quotient is shown as a series
cating that either addition or subtraction is to be per
of
plus and minus signals in Example 15 to illustrate an
formed, a plus quotient digit +Q is entered into the
quotient register when the operation sign is negative, and 75 other manner of de?nition. The complete answer, then,
3,023,962
Y
256
25
is +l10l0—00101 or +l0l01, corresponding to the deci
simpli?ed in view of certain assumptions which may be
made with respect to that end of the register.
mal number 21.
While a variety of speci?c forms are possible, the basic
result logic may be expressed, as above, as follows:
Carry Cn——3 may be simpli?ed in view of the fact that
there is no input carry from any previous place, it being
assumed that carry Cn—1=0. This, then, may be ex
pressed as follows:
to
This basic logic is found in each result network except
full adder-subtracters FASl, FASn-l, and FASn which
may be simpli?ed for various reasons pointed out below. 10
Since each full adder-subtracter logic is either shifted to
the left under control of signal N’ or is bypassed under
control of signal N, in which case signal Rj—-1 constitutes
signal Rj, the complete logic going into each of storage
In a similar manner, carry Cn-2 may be simpli?ed and
further reduced in view of the fact that no other carry
devices R1 through Rn—3 may be expressed as follows:
may be shifted into this position during the “do nothing”
operation. This, then, provides the function:
to
The logic for storage device R0 may be simpli?ed in
view of the assumption that the most signi?cant divisor
digit A1 is always a binary 1. This may be accomplished
by always shifting divisor to the left until a binary 1 ap
pears in this position. With this assumption, the logic
It will be understood that while the arithmetic unit
shown in FIG. 8 represents most of the stages of the par
tial remainder register, the modi?cations necessary to
25 cover the simpli?cation cases are also considered to fall
may be simpli?ed as follows:
within the general de?nition of the “jth” stage. Thus, as
pointed out above, the general schematic representation
or" the unit is not intended to constitute a structural de?~
nition thereof but rather a convenient means of show
The input logic for storage device Rn—2 may be sim
pli?ed by noting that the carry is zero in this place.
In a
ing its general form, with the understanding that logical
variations may be made either through algebraic manipu
lation or through simpli?cation which may be made by
similar manner the carry may be assumed to be zero in
the input logic for storage device Rn-l.
may be expressed as follows:
These cases
various assumptions.
It is to be understood that the expression “bi-level con
35 trol signals” as used herein includes all de?nitions of K
which are possible. As will be seen hereinafter, K may
be de?ned in several different ways.
An interesting variation which may occur in the logical
de?nition of control signals K and K’ is shown in FIG.
of division and left in this state, thereby causing zeros to 40 70 as employed in the division system. In this case,
both signals Os and K are generated through networks
be entered into full adder-substracter, or the least signi?
and associated ampli?ers, without delay. It is estab
cant half of a double length dividend may be shifted into
device Rn.
lished, however, in the above-mentioned copending ap—
plication that delay or storage logic may be employed for
The manner in which the logic for storage device R00
division as well. Where signals are generated without
is derived is quite complex and will not be considered
delay, it is still necessary to store each operation sign
here, reference being made to the above-mentioned co
so that both present and previous operation signs may be
pending application. It is suf?cient to note, for present
analyzed in forming signals K and K’. In order to avoid
purposes, that this signal corresponds to the most signi?
cascading logical networks, however, signal K may be
cant digit position in Examples 14 and 15 above and may
de?ned directly in terms of the same logic utilized to
be formulated as a complete digit value, the sign thereof
Storage device R11 is either set to zero at the ?rst step
de?ne the operation sign signal. The general manner of
obtaining this derivation will be shown here by assuming
that the present operation sign is generated through a net
work gOs and that this signal is then shifted into opera
tion sign storage device Os through an ampli?er stage
which also provides suitable signals gOs and 30s’ for
various other points of utilization. With this de?nition,
being represented by carry signal C00. It is shown, how
ever, in the above-mentioned copending application that
many alternative schemes are possible. A suitable logi
cal de?nition for the input circuit driving storage device
R00 is submitted here in order to illustrate a typical form
of such a circuit and to generally familiarize the reader
with the problem involved. This logic appears as follows:
signal K may then be expressed as follows:
60
This function indicates that K, as before, is equal to 1
whenever the present operation sign, now represented by
either of signals gOs or (gOs)', is the same as the previ
As in the case of the development of the result signals,
the carry signals are either entered into associated storage
ous operation sign represented by signals Os and Os’.
The \logic for the present operation sign control signal
devices in response to signal N' or they are shifted to the
gOs may assume any of a multitude of forms, depending
left in response to signal N. The general expression for
upon the particular division scheme desired, one suitable
form which is derived in the above-mentioned copending
this operation may be written as follows:
70 application being expressible as follows:
to
Cj——2=N'.EK.(Rj'.Cj+Cj.Aj+Rj’.Aj)+K’.(Rj.Cj'
i+Rj.Aj+Cj’.Aj) ] +N.Cj—1
The above general carry logic applies to carry storage
stages C0 through Cn—-4, carries 01-3 and oft-2 being 75
A speci?c de?nition for signal K may then be derived
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