Chapter 2.2, Problem 2.19PP

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

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

• 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 $\overrightarrow{x}$  which is denoted by entire vector or $[x_{w-1},x_{w-2},\mathrm{........},x_0]:$
  • Each bit “x_i” contains 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 $B2U_4\left(0100\right)$

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

“T2U” function:

• The function “T2U” means conversion of “Two’s complement to unsigned” with “w” length bits.
• This function is computed by the below formula

  ${\text{T2U}}_{\text{w}}\left(\text{x}\right)\text{=}{\text{B2U}}_{\text{w}}\left({\text{T2B}}_{\text{w}}\left(\text{x}\right)\right)$

• From the above equation, first convert the two’s complement to binary and then convert the binary to unsigned.

Example:

The example of “T2U_w” is shown below:

Consider, value of x is “-6” and bit length is “4”.

To find the two’s complement, first need to take 1’s complement to given binary and then add one with it.

The formula for find the value of “T2U_w” is given below

${\text{T2U}}_{\text{4}}\left(\text{-6}\right)\text{=}{\text{B2U}}_{\text{4}}\left({\text{T2B}}_{\text{4}}\left(\text{-6}\right)\right)$

Step 1: convert the two’s complement to binary

Binary value of “6”: 0110

1’s complement of “0110”: 1001

Add 1:       1

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2’s complement of binary: 1010 (-6)

Therefore, $T2B_4\left(-6\right)=1010$

Step 2: convert the given binary to unsigned.

$\begin{array}{l}{\text{B2U}}_{\text{4}}\left(\text{1010}\right)\text{=}{\text{1×2}}^{\text{3}}\text{+}{\text{0×2}}^{\text{2}}\text{+}{\text{1×2}}^{\text{1}}\text{+}{\text{0×2}}^{\text{0}}\\ \text{=}\text{1×8}\text{+}\text{0}\text{+}\text{1×2}\text{+}\text{0}\\ \text{=}\text{8}\text{+}\text{0}\text{+}\text{2}\text{+}\text{0}\\ \text{=}\text{10}\end{array}$

${\text{T2U}}_{\text{4}}\left(\text{-6}\right)\text{=}{\text{B2U}}_{\text{4}}\left(\text{1010}\right)\text{=}\text{10}$

Therefore, ${\text{T2U}}_{\text{4}}\left(\text{-6}\right)\text{=}\text{10}$