**Baugh-Wooley
Multiplier **

In case of signed or 2’s complement floating point multiplication the multiplicand and multiplier is represented as follows. Considering the sign magnitude, then the numbers can be represented as

*X=( - x _{0 }+
∑_{i=1}^{n-1} x_{i} 2^{-I } ) and Y=(
- y_{0 }+ ∑_{j=1}^{n-1} x_{j} 2^{-j
} )*

Now, the multiplication of X & Y gives:

*P = XY =( - x _{0 }+
∑_{i=1}^{n-1} x_{i} 2^{-i} )( - y_{0
}+ ∑_{j=1}^{n-1} x_{j} 2^{-j } )*

* = - x _{0}y_{0}
+ ∑_{i=1}^{n-1} ∑_{j=1}^{n-1 }x_{i
}y_{j }2^{-i-j } – x _{0}∑_{j=1}^{n-1}
y_{j }2^{-j } – y_{0} ∑_{i=1}^{n-1}
x^{i }2^{-i}*

While generating P by combining
the partial products the sign of the partial products must be taken into
consideration. Here (– x_{0}∑_{j=1}^{n-1} y_{j}2^{-j})
and (– y_{0} ∑_{i=1}^{n-1} x_{i}2^{-i})
are two partial product terms that represents negative quantities. Each of two
negative terms can be rewritten as 2’s complement form. The partial products
with negative terms are written at the end rows as shown in Fig. 1. The partial
product row in Fig. 1 (5 x 5 floating point multiplication) containing the row
{1 1 x_{0}y_{1 }x_{0}y_{2 }x_{0}y_{3
}x_{0}y_{4 }1 1 1 1} is added with ‘1’ to
get the 2’s complement form of (– x_{0 }∑_{j=1}^{n-1}
y_{j }2^{-j}) Similarly to get the 2’s complement form of (– y_{0}
∑_{i=1}^{n-1} x_{i }2^{-i}) the row of
the partial product terms {1 1 x_{1}y_{0 }x_{2}y_{0
}x_{3}y_{0 }x_{4}y_{0 }1 1
1 1} shown in Fig. 4.1,is again added with ‘1’.

The Baugh-Wooley’s algorithm is relatively straight forward way of doing signed digit floating multiplier. Fig. 4.2 representing the algorithm for an 5 bit case, (For simplification we are considering 5 bit cases, higher number of bit array can be designed from the similar manner) where the partial product bits have been recognized according to Hatamain’s scheme. The creation of the partial product array comprises two steps.

I. The most significant bit (MSB) of the first N-1 partial and all bits

of the last partial product rows except its MSB, are inverted.

II. ‘1’ is added to the (N+1) th and 2N th column.

Fig.4.1 Conventional approach 2’s complements floating point multiplication