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Multiplication and division are taught in elementary school, yet
they are still being worked on for computer applications.
The earliest computers just provided add and subtract with
conditional Branch, leaving the programmer to write multiply
and divide subroutines.
Early computers used bit-serial methods that required about
N squared clock times for multiplying or dividing N-bit numbers.
With a parallel adder, the time for multiply was reduced to
N/2 clock times (Booth algorithm) and division N clock times.
Todays computers use parallel, combinational, circuits for
multiply and divide. These circuits still take too long for
signals to propagate in one clock time. The combinational
circuits are "pipelined" so that a multiply or divide can be
completed every clock time.
Consider multiplying unsigned numbers 1010 * 1100 (10 times 12)
Using a hand method would produce:
1010
* 1100
---------
0000 <- think of the multiplier bit being "anded" with
0000 the multiplicand. A 1-bit "and" in digital logic
1010 is like a 1-bit "multiply".
1010
---------
01111000 4-bits times 4-bits produces an 8-bit product
When adding by hand, we can add the middle columns four bits and
produce a sum bit and possibly a carry. In hardware the number
of input bits is fixed. From the previous lecture, we could use
four 4-bit adders with additional "and" gates to do the multiply.
A better design incorporates the "and" gate to do a 1-bit multiply
inside the previous lectures full adder. With this single building
block, that is easy to replicate many times, we get the following
parallel multiplier design.
The 4-bit by 4-bit multiply to produce an 8-bit unsigned product is
The component madd circuit is
The VHDL source code is pmul4.vhdl
The VHDL test driver is pmul4_test.vhdl
The VHDL output is pmul4_test.out
The Cadence run file is pmul4_test.run
The partial Makefile is Makefile.pmul4_test
Notice that the only component used to build the multiplier
is "madd" and some uses of "madd" have constants as inputs.
It is technology dependent whether the same circuit is used
or specialized, minimized, circuits are substituted.
Division is performed by using subtraction. A sample unsigned binary
division of an 8-bit dividend by a 4-bit divisor that produces
a 4-bit quotient and 4-bit remainder is:
1010 <- quotient
/---------
1100/ 01111011 <- dividend
-1100
-----
0110
-0000
------
1101
-1100
------
0011
-0000
-----
0011 <- remainder
With a parallel adder and a double length register, serial division
can be performed. Conventional division requires a trial subtraction
and possibly a restore of the partial remainder. A non restoring
serial division requires N clock times for a N-bit divisor.
The schematic for a parallel 8-bit dividend divided by 4-bit divisor
to produce an 4-bit quotient and 4-bit remainder is:
Notice that the building block is similar to the 'madd' component
in the parallel multiplier. The 'cas' component is the same full
adder with an additional xor gate.
The VHDL test driver is divcas4_test.vhdl
The VHDL output is divcas4_test.out
The Cadence run file is divcas4_test.run
The partial Makefile is Makefile.divcas4_test
Divide can create on overflow condition. This is typically handled by
separate logic in order to keep the main circuit neat. There is a
one bit preshift of the dividend in the manual, serial and parallel
division. Thus, no dividend bit number seven appears on the parallel
schematic.
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