ECE 375: Digital Design II
Class 25Radix-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: $[-1,0] â (-1)2^1+(0)2^0 = -2$
X = 10011 101 10 1011 10
Y = -10100-110-11-1100-10
So...
$Y_4$ | -2 | 2 | -1 | 2 | -1 | -1 | 0 | -2 |
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Will continue with example in next class