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2.6.1 Datapath Elements
Figure 2.21
shows some typical datapath symbols for an adder (people rarely use the
IEEE standards in ASIC datapath libraries). I use heavy lines (they are
1.5 point wide) with
a stroke to denote a data bus (that flows in the horizontal direction in
a datapath), and regular lines (0.5 point)
to denote the control signals (that flow vertically in a datapath). At the
risk of adding confusion where there is none, this stroke to indicate a
data bus has nothing to do with mixed-logic conventions. For a bus, A[31:0]
denotes a 32-bit bus with A[31] as the leftmost or most-significant bit
or MSB , and A[0] as the least-significant bit or LSB
. Sometimes we shall use A[MSB] or A[LSB] to refer to these bits. Notice
that if we have an n -bit bus and
LSB = 0,
then MSB =
n – 1.
Also, for example, A[4] is the fifth bit on the bus (from the LSB). We use
a ' S ' or 'ADD' inside the symbol to denote
an adder instead of '+', so we can attach '' or '+/' to the inputs for a
subtracter or adder/subtracter.
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FIGURE 2.21 Symbols
for a datapath adder. (a) A data bus is shown by a heavy line (1.5 point)
and a bus symbol. If the bus is n
-bits wide then MSB =
n – 1.
(b) An alternative symbol for an adder. (c) Control signals are
shown as lightweight (0.5 point)
lines. |
Some schematic datapath symbols
include only data signals and omit the control signalsbut we must not forget
them. In Figure 2.21, for example, we may need to explicitly tie CIN[0]
to VSS and use COUT[MSB] and COUT[MSB 1]
to detect overflow. Why might we need both of these control signals? Table 2.11
shows the process of simple arithmetic for the different binary number representations,
including unsigned, signed magnitude, ones' complement, and two's complement.
TABLE 2.11 Binary
arithmetic. |
Operation |
Binary Number Representation |
Unsigned |
Signed
magnitude |
Ones'
complement |
Two's
complement |
|
no change |
if positive
then MSB = 0
else MSB = 1 |
if negative then
flip bits |
if negative then
{flip bits; add 1} |
3 = |
0011 |
0011 |
0011 |
0011 |
3 = |
NA |
1011 |
1100 |
1101 |
zero = |
0000 |
0000 or 1000 |
1111 or 0000 |
0000 |
max. positive = |
1111 = 15 |
0111 = 7 |
0111 = 7 |
0111 = 7 |
max. negative = |
0000 = 0 |
1111 = 7 |
1000 = 7 |
1000 = 8 |
addition =
S =
A + B
= addend + augend
SG(A) = sign
of A |
S = A + B |
if SG(A) = SG(B)
then S = A + B
else { if
B < A
then S = A B
else S = B A} |
S =
A + B + COUT[MSB]
COUT is carry out |
S = A + B |
addition result:
OV = overflow,
OR = out
of range |
OR = COUT[MSB]
COUT is carry out |
if SG(A) = SG(B)
then OV = COUT[MSB]
else OV = 0
(impossible) |
OV =
XOR(COUT[MSB],
COUT[MSB1]) |
OV =
XOR(COUT[MSB], COUT[MSB 1]) |
SG(S) = sign
of S
S =
A + B |
NA |
if SG(A) = SG(B)
then SG(S) = SG(A)
else { if
B < A
then SG(S) = SG(A)
else SG(S) = SG(B)} |
NA |
NA |
subtraction =
D =
A B
= minuend
subtrahend |
D = A B |
SG(B) = NOT(SG(B));
D = A + B |
Z = B
(negate);
D = A + Z |
Z = B
(negate);
D = A + Z |
subtraction result :
OV = overflow,
OR = out
of range |
OR = BOUT[MSB]
BOUT is borrow out |
as in addition |
as in addition |
as in addition |
negation :
Z = A
(negate) |
NA |
Z = A;
SG(Z) = NOT(SG(A)) |
Z = NOT(A) |
Z = NOT(A) + 1 |
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