Chapter 2.3, Problem 2.34PP

Program Plan Intro

Unsigned and two’s-complement multiplication:

The bit-level representation of the product operation is identical for both unsigned and two’s-complement multiplication.

Assume the integers “x” and “y” in the range “0 ≤ x, y ≤ 2^w – 1” but their product “x * y” can range between “0” and “${\left({\text{2}}^{\text{w}}\text{-}\text{1}\right)}^{\text{2}}\text{=}{\text{2}}^{\text{2w}}\text{-}{\text{2}}^{\text{w-1}}\text{+}\text{1}$ ”.

Using the below formula, truncating an unsigned number to “w” bits is given below

$\text{x}{\text{* }}_{\text{w}}^{\text{u}}\text{y}\text{=}\left(\text{x}\text{.y}\right)\text{mod}{\text{2}}^{\text{w}}$

Using the below formula, truncating an two’s complement number to “w” bits is given below

$\text{x}{\text{* }}_{\text{w}}^{\text{t}}\text{y}\text{=}{\text{U2T}}_{\text{w}}\left(\left(\text{x}\text{.}\text{y}\right)\text{mod}{\text{2}}^{\text{w}}\right)$

“B2U” function:

• The unsigned encoding is represented by the interpretation function of “B2U_w”.
  • The function “B2U_w” means “binary to unsigned” of length “w”.
• The equation for “B2U_w” is

${\text{B2U}}_{\text{w}}\left(\overrightarrow{\text{x}}\right)\text{ = }{\displaystyle \sum _{\text{i}\text{=}\text{0}}^{\text{w}\text{-}\text{1}}{\text{x}}_{\text{i}}{\text{2}}^{\text{i}}}$

“B2T” function:

• The two’s complement encoding is represented by the interpretation function of “B2T_w”.
  • The function “B2T_w” means “binary to two’s-complements” of length “w”.
• The equation for “B2T_w” is

${\text{B2T}}_{\text{w}}\left(\overrightarrow{\text{x}}\right)\text{=}{\text{-x}}_{\text{w-1}}{\text{2}}^{\text{w-1}}\text{+}{\displaystyle \sum _{\text{i}\text{=}\text{0}}^{\text{w}\text{-}\text{2}}{\text{x}}_{\text{i}}{\text{2}}^{\text{i}}}$

“U2T” function:

• This function is used to convert the unsigned number to two’s-complement number.
• The equation for computing this function is

${\text{U2T}}_{\text{w}}\left(\text{u}\right)\text{=}{\text{-u}}_{\text{w-1}}{\text{2}}^{\text{w}}\text{+}\text{u}$