Question

Plot the magnitude of both a signed and unsigned 16-bit number.

Answer

Some may be able to remember the ranges for signed and unsigned easily. If I can’t remember I will try a trivial example to jog my memory. Once the solution is found with this example it can be easily applied to the more elaborate case (16-bit).

Instead of 16-bits, try 2-bits.

Recall how to convert 2s compliment numbers. Invert the bits (not) and add one.

10 (2s) -> 01 + 1 = 10 (-2)

Then create a table of all the cases.

  2-bit
--------
00  0  0
01  1  1
10  2 -2
11  3 -1

Now the equations for the magnitude can be easily found.

unsigned: 0           to  (2^b - 1)
  signed: -(2^(b - 1) to  (2^(b-1) - 1)

Then use these equations for the 16-bit case.

unsigned: 0            to  (2^16 - 1)
  signed: -(2^(16 - 1) to  (2^(16 - 1) - 1)
          -2^15        to  2^15 - 1

And finally, produce a plot:

Background

This question was given during an interview on 1/17/2014 for a firmware test engineer position.