Chapter 2.2, Problem 2.17PP

Program Plan Intro

Unsigned encoding:

• 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

  $B2U_w\left(\overrightarrow{x}\right)\doteq {\displaystyle \sum _{i=0}^{w-1}{x}_i{2}^i}$

• 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 $\overrightarrow{x}$  which is denoted by entire vector or $[x_{w-1},x_{w-2},\mathrm{........},x_0]:$
  • Each bit “x_i” contain value “0” or “1”.

Example:

The example of “B2U_w” is shown below:

Consider, value of $\overrightarrow{x}$ is “0100” which contains bits length “4”.

Therefore using the above equation, user can compute the $B2U_4\left(0100\right)$

$\begin{array}{l}{\text{B2U}}_{\text{4}}\left(\text{0100}\right)\text{=}{\text{0×2}}^{\text{3}}\text{+}{\text{1×2}}^{\text{2}}\text{+}{\text{0×2}}^{\text{1}}\text{+}{\text{0×2}}^{\text{0}}\\ \text{=}\text{0}\text{+}\text{4}\text{+}\text{0}\text{+}\text{0}\\ \text{=}\text{4}\end{array}$

Two’s-Complement encodings:

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

  $B2T_w\left(\overrightarrow{x}\right)\doteq -x_{w-1}{2}^{w-1}+{\displaystyle \sum _{i=0}^{w-2}{x}_i{2}^i}$

• 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.
  • “x_w-1” is the most significant bit. It is also named as “sign bit”.
    • It weight “-2^w-1” means the weight is in negation for representation of an unsigned.
      • When the sign bit is set to “1”, then the value is denoted by “negative”.
      • When the sign bit is set to “0”, then the value is denoted by “nonnegative”.
  • The bit length is “w”.
  • A bit vector is $\overrightarrow{x}$  which is denoted by entire vector or $[x_{w-1},x_{w-2},\mathrm{........},x_0]:$
  • Each bit “x_i” contain value “0” or “1”.

Example:

The example of “B2T_w” is shown below:

Consider, value of $\overrightarrow{x}$ is “1111” which contains bits length “4”.

Therefore using the above equation, user can compute the $B2T_4\left(1111\right)$

$\begin{array}{l}{\text{B2T}}_{\text{4}}\left(\text{1111}\right)\text{=}{\text{-1×2}}^{\text{3}}\text{+}{\text{1×2}}^{\text{2}}\text{+}{\text{1×2}}^{\text{1}}\text{+}{\text{1×2}}^{\text{0}}\\ \text{=}\text{-1×8}\text{+}\text{1×4}\text{+}\text{1×2}\text{+}\text{1×1}\\ \text{=}\text{-8}\text{+}\text{4}\text{+}\text{2}\text{+}\text{1}\\ \text{=}\text{-1}\end{array}$