This section considers design of bit-serial radix-2 hybrid redundant number based on-line multiplication. These redundant multipliers process the msd of the input data first and generate the msd of the result first. On-line arithmetic leads to low−latency architectures and is useful for operations such as division and square root which are inherently msd-first operations.

Consider multiplication of 2 numbers, *A* = *a*_{3}.*a*_{2}*a*_{1}*a*_{0} and *B* = *b*_{3}.*b*_{2}*b*_{1}*b*_{0}, using the 2 multiplication dependence graphs (DGs) shown in Fig. 14.11 and Fig. 14.12. The basic processing element (the black solid circle in the DGs) carries out an add operation defined by the following equation :

where the dashed lines represent *carry* and the solid lines represent *sum*. The elliptic circles on the last row are dummy operations used only to group the bits together to form the product digits. In the DG of Fig. 14.11, the carry moves vertically down whereas it moves diagonally in the DG of Fig. 14.12 [19].

*Fig. 14.9* Minimally Redundant HYbrid radix-4 Adder cell (*mrHY*4*A*).

*Fig. 14.10* Four-digit mrHY4A.

Using systolic design methodology (see Chapter 7), and the following ...

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