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Isaac
George

ECE 375: Digital Design II

Class 26

Radix-4 Multiplication (Signed) with Booth Encodings

a = 0110   =  6
x = 1010   = -6
y = -11-10 // Booth Encoded X
  a = 00110
 2a = 01100
 -a = 11010
-2a = 10100

$P^{j+1}=[p^j+x_ja4^2]4^{-1}$

$Z_4$ -1 -2
$P^0$ 0 0 0 0 0 0 0 0 0 0
$Z_0a4^2$ 1 1 0 1 0 0 0 0 0 0
$4P^1$ 1 1 0 1 0 0 0 0 0 0
$P^1$ 1 1 1 1 0 1 0 0 0 0
$Z_1a4^2$ 1 1 1 0 1 0 0 0 0 0
$4P^2$ 1 1 1 0 1 1 1 0 0 0
$P^2$ 1 1 1 1 1 0 1 1 1 0

$P^2 = 1101,11002 = -36{10}$

Radix-4 Booth Recoding Hardware Design

You need 2 symbols for binary, 3 for Booth, and 5 for Radix-4. So how do we store this?

$X_{i+1}$ $X_i$ $X_{i-1}$ $Y_{i+1}$ $Y_i$ $Z_i$
0 0 0 0 0 0
0 0 1 0 1 1
0 1 0 1 -1 1
0 1 1 1 0 2
1 0 0 -1 0 -2
1 0 1 -1 1 -1
1 1 0 0 -1 -1
1 1 1 0 0 0