Chapter 2.2, Problem 2.22PP

A.

Program Plan Intro

Two’s-Complement encodings:

• The two’s complement encoding is represented by the interpretation function of “B2T_w”.
• The equation 2.3 is given below

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

• From the above equation,
  • The function “B2T_w” means “binary to two’s-complements” of length “w”.
  • 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 $\overrightarrow{\text{x}}$  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”.

Example:

The example of “B2T_w” is shown below:

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

Therefore using the above equation, user can compute the ${\text{B2T}}_{\text{4}}\left(\text{1111}\right)$

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

B.

Program Plan Intro

Two’s-Complement encodings:

• The two’s complement encoding is represented by the interpretation function of “B2T_w”.
• The equation 2.3 is given below

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

• From the above equation,
  • The function “B2T_w” means “binary to two’s-complements” of length “w”.
  • 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 $\overrightarrow{\text{x}}$  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”.

Example:

The example of “B2T_w” is shown below:

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

Therefore using the above equation, user can compute the ${\text{B2T}}_{\text{4}}\left(\text{1111}\right)$

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

C.

Program Plan Intro

Two’s-Complement encodings:

• The two’s complement encoding is represented by the interpretation function of “B2T_w”.
• The equation 2.3 is given below

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

• From the above equation,
  • The function “B2T_w” means “binary to two’s-complements” of length “w”.
  • 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 $\overrightarrow{\text{x}}$  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”.

Example:

The example of “B2T_w” is shown below:

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

Therefore using the above equation, user can compute the ${\text{B2T}}_{\text{4}}\left(\text{1111}\right)$

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