Description
The purpose of this project is for you to build 16-bit multiplication and 16-bit division hardwares
as we discussed in class in logisim.
Introduction to the Multiplication Hardware
The 16-bit multiplication hardware that we discussed in class is shown below:
M_Ready
Product
32
Multiplier 1
Multiplicant
Mul
16
16
1
Clock
Multiplication
Hardware
You can consider the above circuit as a sub-circuit named multiplication which contains the
following input/output:
• Multiplicand: a 16-bit input
• Multiplier: a 16-bit input
• Mul (1-bit input): This input will be one if the instruction is the multiplication instruction
• Clock (1-bit input)
• Product (32-bit output)
• M Ready (1-bit output): This output will be 1 if the product is ready
Note that we require to have the output M Ready because the multiplication instruction will take
multiple clock cycles to produce a result. Ideally, if a CPU see the instruction mul, it will set the
appropriate Multiplicant and Multiplier. Then, it will set Mul to 1 and wait until the signal
M Ready to turn to 1 before it continues to the next instruction. The circuit inside will be the same
as the multiplication hardware discussed in class as shown below:
Multiplicand
Shift left
Product
Write
Control test
Shift right
Multiplier
32 bits
16 bits
32−bit Adder
32 bits
1
Inside the multiplication hardware, you need three registers, Multiplicand (32-bit), Multiplier
(16-bit), and Product (32-bit). For these registers, you do not have build them from scratch.
Simply use the register component under “Memory”. Similarly, for the 32-bit adder, simply use
the one supplied by the logisim. Note that the above hardware is for multiplying two 16-bit
numbers and produce a 32-bit result. The flowchart of this hardware is shown below:
Start
Multiplier0
1. Test
1a. Add multiplicand to product and
place the result in Product register
2. Shift the Multiplicand register left 1 bit
3. Shift the Multiplier register right 1 bit
Done
Multiplier0 = 1 Multiplier0 = 0
No: < 16 repetitions
Yes: 16 recetitions
16nd rep?
Recall that in the first step, this hardware have to load the top 16-bit of the multiplicand register
with 0s and the bottom 16-bit with Multiplicand, load the product register with 0s, and load the
multiplier register with the Multiplier. After all three registers are loaded with proper values,
then the algorithm can start as follows.
1. product = product + (multiplicand ∗ multiplier0
): In this step, if multiplier0
is 0, we
actually perform product = product + 0. But if multiplier0
is 1, we perform product =
product + multiplicand. This can be done by adding a 32-bit (2-input) multiplexer. This
multiplexer has two inputs, one from the multiplicand and another one is imply a 32-bit constant 0. Simply use the Least Significant Bit (LSB) of the multiplier register (multiplier0
)
to choose which one to go to the output as shown below:
Multiplicand Product
u
x
m
0
multiplier0
Note that before the algorithm starts, you must clear the product register which can be
done in two ways:
2
(a) by writing 0. So, you also need another multiplexer to choose whether you want to write
0 or output from 32-bit adder to the product register as shown below:
Multiplicand
u
x
m
u
x
m
Product 0
multiplier0
0
Clear
Product
(b) use the Clear input pin of the register. Simply set it to 1 and the content will be cleared.
2. Shift multiplicand register left one bit: This step is simply update the multiplicand register by its data that has been shifted left by 1. Simply use a Shifter provided by logisim
under Arithmetic. Note at the first step before the algorithm starts, you need to update
multiplicand register by the input Multiplicand. So, you need a multiplexer to select which
data should go to the multiplicand register (Multiplicand input or multiplicand << 1.
The block diagram of the circuit is shown below:
u
m
x
Multiplicand
Multiplicand
0
16
16
32
1
3. Shift multiplier register right one bit: This step is pretty much the same as in previous step.
You need to be able to load the content of the multiplier or update it with multiplier >> 1
Note that we need an ability to control what to do at each clock cycle. For example, in the
first clock cycle, we need to load contents of all registers. The next clock cycle, we need to
perform product = product + (multiplicand ∗ multiplier0
). The third clock cycle, we need
to perform multiplicand = multiplicand << 1. The fourth clock cycle, we need to perform
multiplier = multiplier >> 2, and so on. To be able to control each clock cycle, we will use a
combination of counter and Read Only Memory (ROM) as shown below:
Clock
Mul
Counter ROM
M Ready
When Mul is 1, it will clear the Counter to 0. At the same time, it will allow the clock signal to
go to the Counter. So, the Counter will start counting up until its desired maximum value which
can be set. When it reaches its maximum value, its Carry signal will be 1 which can be used for
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the signal M Ready. The output of the Counter will be use as the address of a ROM. The content
of the ROM will be control signal for each clock cycle. In other words, you can program what do
you want to do at each clock cycle by content of the ROM.
Note that you MUST set the maximum value of the counter to stay at a specific value based
on the number of clock cycles that your hardware uses. MAKE SURE that the last output value
of the ROM should maintain the output of your product register. When we grade your circuit, we
will simply put value of multiplicand and multiplier, and let the clock tick until M Ready turn green
without stopping the clock and check the result.
Introduction to the Division Hardware
The 16-bit division hardware that we discussed in class is shown below:
16
16
1
Clock
Hardware
Divisor
Dividend Quotient
1
16
16
Remainder
D Ready
Division
Div
You can consider the above circuit as a sub-circuit named division which contains the following
input/output:
• Dividend: a 16-bit input
• Divisor: a 16-bit input
• Div (1-bit input): This input will be one if the instruction is the division instruction
• Clock (1-bit input)
• Quotient (16-bit output)
• Remainder (16-bit output)
• D Ready (1-bit output): This output will be 1 if the product is ready
The division hardware that we discussed in class is shown below:
Control test
32 bits
32 bits
16 bits
Quotient
Divisor
Write
Remainder
Shift right
Shift left
32−bit Adder
Again, the above hardware is for dividing two 16-bit numbers and produce a 16-bit quotient and
16-bit remainder. The flowchart of this hardware is shown below:
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Start
1. Subtract the Divisor register from the
Remainder register and place the
result in the Remainder register
Remainder
Test
2b. Restore the original value by adding
the Divisor register to the Remainder
register and placing the sum in the
Remainder register. Also shift the
Quotient register to the left, setting the
new least significant bit to 0
3. Shift the Divisor register right 1 bit
Done
2a. Shift the Quotient register to the left,
setting the new least significant bit 1
Remainder >= 0 Remainder < 0
No: < 17 repetitions
Yes: 17 repetitions
17rd
repetition?
The design concept of this division circuit will be pretty much the same as in multiplication circuit
but it requires more steps. For example, when the subtraction result is less than 0, you have to
restore to its original value by adding it back. Another different is the quotient, sometime we shift
it left and insert a 0 but sometime we insert a 1.
What to Do?
For this project, start with the given starter file named muldiv.circ. This starter file contains
two sub-circuits, 16-bit multiplication and 16-bit division. In both sub-circuit, the counter
and ROM are provided. Simply build your multiplication and division circuits there. Once you are
finish, put your circuits in the main and connect them with appropriate input/output. We will test
your circuit from the main circuit.
Again, you MUST set the maximum value of the counter to stay at a specific value based on
the number of clock cycles that each of your hardwares use. We will not stop the clock when we
check your results.
Submission
The due date of this project is stated in the CourseWeb under this project. Late submissions will
not be accepted. You should submit the file muldiv.circ via CourseWeb.
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