GATE CS 2017 Set 1 — Question 7
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Digital Logic → Number Systems & Arithmetic
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Question
The -bit fixed-point representation of an unsigned real number uses bits for the fraction part. Let . The range of decimal values for in this representation is
Correct answer
(D) 0 to (2ⁱ - 2^(-f))
Solution
In an unsigned fixed-point representation with bits total and fractional bits:
With integer bits and fractional bits:
Max value = (Max integer value of bits) (weight of LSB)
Max integer value of bits = .
Weight of LSB = .
Max value = .Thus, the range is to .
- The smallest value is when all bits are 0, which represents .
- The largest value is when all bits are 1.
With integer bits and fractional bits:
- The maximum integer part (all 1s) is .
- The maximum fractional part (all 1s) is (geometric series sum).
Max value = (Max integer value of bits) (weight of LSB)
Max integer value of bits = .
Weight of LSB = .
Max value = .Thus, the range is to .
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