Bits fractional part

bits

#1

The n - bit fixed point representation of an unsigned real number x uses f bits for the fraction part . Let i= n- f. The range of decimal values for x in this representation is?


#2

Since given number is in unsigned bit representation, its decimal value starts with 0.
We have i bit in integral part so maximum value will be 2^i
Thus integral value will be from 0 to 2^i – 1

We know fraction part of binary representation are calculated as (1/0)*2^-f

Thus with f bit maximum number possible = sum of GP series with a = 1/2 and r = 1/2

Thus fmax = {1/2(1 – (1/2)^f}/(1 – 1/2)
= 1 – 2^-f

Thus maximum fractional value possible = 2i – 1 + (1 – 2^-f )
= 2i - 2^-f

Source: https://www.geeksforgeeks.org/gate-gate-cs-2017-set-1-question-32/