ECE 375: Digital Design II

Class 25

Radix-4 Multiplication (Unsigned)

Previously, we had $P^{j+1}=[p^j+x_ja2^k]2^{-1}$ as the formula for multiplication.

Now, we use $P^{j+1}=[p^j+x_jaR^k]R^{-1}$ where $R$ is your radix. ($R$ should be a power of 2)

For example, if $R=4$, the equation will be $P^{j+1}=[p^j+x_ja4^k]4^{-1}$

Multiplication example:

$X=(1110)_2=(3,2)4=6{10}\$ $K=2\$

$A=(1011)2=14{10}\$ $2A=1100\$ $3A=10010\$

Value $P_9$ $P_8$ $P_7$ $P_6$ $P_5$ $P_4$ $P_3$ $P_2$ $P_1$ $P_0$
$P^0$ 0 0 0 0 0 0 0 0 0 0
$0+2A4^2$ 1 1 0 0 0 0 0 0
$4P^1$ 1 1 0 0 0 0 0 0
$P^1$ 1 1 0 0 0 0
$0+3A4^2$ 1 0 0 1 0 0 0 0 0
$4P^2$ 1 0 1 0 1 0 0 0 0
$P^2$ 1 0 1 0 1 0 0

$P^2=10101002=84{10}$

Now we are going to add booth encoding to this

Radix-4 Multiplication (Signed) with Booth Encodings

You can convert a Booth encoded number into a Radix-4 encoded number. You do like normal but you have to account for negative one.

For example: $[-1,0] → (-1)2^1+(0)2^0 = -2$

X =  10011 101 10 1011 10
Y = -10100-110-11-1100-10

So...

$Y_4$ -2 2 -1 2 -1 -1 0 -2

Will continue with example in next class