Study the following 9 bit number 010011000 The number format is fixed-point with 3 bit integer bits, unsigned. Suppose you add this number to itself. What is the actual round-off error?
Question
Study the following 9 bit number 010011000 The number format is fixed-point with 3 bit integer bits, unsigned. Suppose you add this number to itself. What is the actual round-off error?
Solution
The given 9-bit number is 010011000.
The number format is fixed-point with 3-bit integer and 6-bit fraction, unsigned.
The 3-bit integer part is 010, which is 2 in decimal.
The 6-bit fraction part is 011000. To convert this to decimal, we consider each bit as a negative power of 2. So, 011000 in binary is 0*(1/2) + 1*(1/4) + 1*(1/8) + 0*(1/16) + 0*(1/32) + 0*(1/64) = 0.25 + 0.125 = 0.375 in decimal.
So, the given 9-bit number 010011000 in fixed-point with 3-bit integer and 6-bit fraction, unsigned is 2.375 in decimal.
If we add this number to itself, we get 2.375 + 2.375 = 4.75 in decimal.
However, since we only have 3 bits for the integer part, the maximum integer we can represent is 7 (111 in binary). So, 4.75 can be represented accurately with 3 integer bits and 6 fraction bits.
Therefore, there is no round-off error when adding this number to itself.
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