Computer Arithmetic

Which gives the resolution and range of the 16-bit signed fixed-point Q15 format (15 fraction bits)?

  1. AResolution 2⁻¹⁶, range 0 to 1
  2. BResolution 2⁻¹⁵, range −1 to 1 − 2⁻¹⁵
  3. CResolution 2⁻¹⁵, range −32768 to 32767
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Answer: B. Resolution 2⁻¹⁵, range −1 to 1 − 2⁻¹⁵

Q15 uses 1 of 16 bits for sign and 15 for the fraction, representing the integer reading divided by 2¹⁵ = 32768. Dividing the integer range −32768 to 32767 by 32768 gives −1 to 32767/32768 = 1 − 2⁻¹⁵, with resolution 1/32768 = 2⁻¹⁵ ≈ 3.05 × 10⁻⁵. Note that exactly +1 cannot be represented. −32768 to 32767 is the range when read as an integer, ignoring the binary point. Q15 is common in audio and other signal processing.

  • A. Being signed, it covers negatives, and the resolution is 2⁻¹⁵.
  • B. Correct. Resolution 2⁻¹⁵, from −1 to 1 − 2⁻¹⁵.
  • C. −32768 to 32767 is the range when read as an integer.
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More Computer Arithmetic questions

  1. When two fixed-point numbers with 8 fraction bits are multiplied using an integer multiplier, how should the result be handled?
  2. Which describes saturating arithmetic used in signal processing?
  3. In a carry-lookahead adder, which gives the definitions of each bit's generate signal g and propagate signal p (a and b are that bit's inputs) and the carry into the next bit, c(i+1)?
  4. Why are carry-save adders used to sum the many partial products in a multiplier?
  5. Which describes a Wallace tree used in fast multipliers?