Chapter 2.2, Problem 2.24PP

Program Plan Intro

Truncation for unsigned number and two’s complement number:

When truncating a w-bit number $\overrightarrow{\text{x}}\text{=}\left[{\text{x}}_{\text{w-1}}\text{,}{\text{x}}_{\text{w-2}}\text{,}\mathrm{.......}\text{,}{\text{x}}_{\text{0}}\right]$ to a k-bit number, user drop the high-order “w – k” bits, given a result as a bit vector $\overrightarrow{\text{x}}\text{'}\text{=}\left[{\text{x}}_{\text{k-1}}\text{,}{\text{x}}_{\text{k-2}}\text{,}\mathrm{.........}{\text{,x}}_{\text{0}}\right]$

From the equation 2.9, the truncation formula for unsigned number is given below,

${\text{B2U}}_{\text{k}}\left(\left[{\text{x}}_{\text{k-1}}\text{,}{\text{x}}_{\text{k-2}}\text{,}\mathrm{.........}{\text{,x}}_{\text{0}}\right]\right)\text{=}{\text{B2U}}_{\text{w}}\left(\left[{\text{x}}_{\text{w-1}}\text{,}{\text{x}}_{\text{w-2}}\text{,}\mathrm{.........}{\text{,x}}_{\text{0}}\right]\right)\text{mod}{\text{2}}^{\text{k}}$

From the equation 2.10, the truncation formula for two’s-complement number is given below,

${\text{B2T}}_{\text{k}}\left(\left[{\text{x}}_{\text{k-1}}\text{,}{\text{x}}_{\text{k-2}}\text{,}\mathrm{.........}{\text{,x}}_{\text{0}}\right]\right)\text{=}{\text{U2T}}_{\text{k}}\left({\text{B2U}}_{\text{w}}\left(\left[{\text{x}}_{\text{w-1}}\text{,}{\text{x}}_{\text{w-2}}\text{,}\mathrm{.........}{\text{,x}}_{\text{0}}\right]\right)\text{mod}{\text{2}}^{\text{k}}\right)$

“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

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

• From the above equation,
  • The notation $\doteq$ denotes that the side of left-hand is defined to be equal to the side of right-hand.
  • The function “B2U_w” maps string of 0’s and 1’s of length “w” to nonnegative integers.
  • The bit length is “w”.
  • A bit vector is $\overrightarrow{\text{x}}$  which is denoted by entire vector or ${\text{[x}}_{\text{w-1}}\text{,}{\text{x}}_{\text{w-2}}\text{,}\mathrm{........}{\text{,x}}_{\text{0}}\text{]:}$
  • Each bit “x_i” contains value “0” or “1”.

“U2T” function:

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

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