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One of the biggest and slowest circuits in
an arithmetic and logic unit is the multiplier.

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We'll start by developing a straightforward
implementation and then, in the next section,

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look into tradeoffs to make it either smaller
or faster.

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Here's the multiplication operation for two
unsigned 4-bit operands broken down into its

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component operations.

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This is exactly how we learned to do it in
primary school.

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We take each digit of the multiplier (the
B operand) and use our memorized multiplication

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tables to multiply it with each digit of the
multiplicand (the A operand),

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dealing with any carries into the next column
as we process the multiplicand right-to-left.

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The output from this step is called a partial
product and then we repeat the step for the

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remaining bits of the multiplier.

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Each partial product is shifted one digit
to the left, reflecting the increasing weight

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of the multiplier digits.

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In our case the digits are just single bits,
i.e., they're 0 or 1 and the multiplication

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table is pretty simple!

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In fact, the 1-bit-by-1-bit binary multiplication
circuit is just a 2-input AND gate.

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And look Mom, no carries!

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The partial products are N bits wide since
there are no carries.

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If the multiplier has M bits, there will be
M partial products.

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And when we add the partial products together,
we'll get an N+M bit result if we account

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for the possible carry-out from the high-order
bit.

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The easy part of the multiplication is forming
the partial products - it just requires some

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AND gates.

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The more expensive operation is adding together
the M N-bit partial products.

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Here's the schematic for the combinational
logic needed to implement the 4x4 multiplication,

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which would be easy to extend for larger multipliers
(we'd need more rows) or larger multiplicands

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(we'd need more columns).

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The M*N 2-input AND gates compute the bits
of the M partial products.

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The adder modules add the current row's partial
product with the sum of the partial products

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from the earlier rows.

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Actually there are two types of adder modules.

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The full adder is used when the modules needs
three inputs.

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The simpler half adder is used when only two
inputs are needed.

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The longest path through this circuit takes
a moment to figure out.

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Information is always moving either down a
row or left to the adjacent column.

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Since there are M rows and, in any particular
row, N columns, there are at most N+M modules

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along any path from input to output.

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So the latency is order N, since M and N differ
by just some constant factor.

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Since this is a combinational circuit, the
throughput is just 1/latency.

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And the total amount of hardware is order
N^2.

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In the next section, we'll investigate how
to reduce the hardware costs, or, separately,

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how to increase the throughput.

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But before we do that, let's take a moment
to see how the circuit would change if the

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operands were two's complement integers instead
of unsigned integers.

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With a two's complement multiplier and multiplicand,
the high-order bit of each has negative weight.

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So when adding together the partial products,
we'll need to sign-extend each of the N-bit

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partial products to the full N+M-bit width
of the addition.

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This will ensure that a negative partial product
is properly treated when doing the addition.

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And, of course, since the high-order bit of
the multiplier has a negative weight, we'd

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subtract instead of add the last partial product.

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Now for the clever bit.

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We'll add 1's to various of the columns and
then subtract them later, with the goal of

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eliminating all the extra additions caused
by the sign-extension.

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We'll also rewrite the subtraction of the
last partial product as first complementing

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the partial product and then adding 1.

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This is all a bit mysterious but…

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Here in step 3 we see the effect of all the
step 2 machinations.

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Let's look at the high order bit of the first
partial product X3Y0.

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If that partial product is non-negative, X3Y0
is a 0, so all the sign-extension bits are

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0 and can be removed.

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The effect of adding a 1 in that position
is to simply complement X3Y0.

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On the other hand, if that partial product
is negative, X3Y0 is 1, and all the sign-extension

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bits are 1.

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Now when we add a 1 in that position, we complement
the X3Y0 bit back to 0, but we also get a

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carry-out.

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When that's added to the first sign-extension
bit (which is itself a 1), we get zero with

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another carry-out.

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And so on, with all the sign-extension bits
eventually getting flipped to 0 as the carry

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ripples to the end.

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Again the net effect of adding a 1 in that
position is to simply complement X3Y0.

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We do the same for all the other sign-extended
partial products, leaving us with the results

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shown here.

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In the final step we do a bit of arithmetic
on the remaining constants to end up with

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this table of work to be done.

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Somewhat to our surprise, this isn't much
different than the original table for the

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unsigned multiplication.

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There are a few partial product bits that
need to be complemented, and two 1-bits that

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need to be added to particular columns.

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The resulting circuitry is shown here.

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We've changed some of the AND gates to NAND
gates to perform the necessary complements.

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And we've changed the logic necessary to deal
with the two 1-bits that needed to be added

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in.

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The colored elements show the changes made
from the original unsigned multiplier circuitry.

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Basically, the circuit for multiplying two's
complement operands has the same latency,

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throughput and hardware costs as the original
circuitry.