c - understanding fixed point bit model

Question

So I'm learning about binary values and I'm trying to understand how to calculate the negative version of a number using the fixed point bit model.

In my textbook it says 000111.01 = 7.25 which I understand. But it's telling me -7.25 is 111000.11 which I don't understand how that's possible.

I know it's using the two's complement rule where it flips all the bits and adds a 1 at the end. But if I did the calculation: (1 * 2^5) + ( 1 * 2^4) + (1 & 2^3) + 0 + 0 + 0 + (1 * 2^-1) + ( 1 * 2^-2) it gives me 56.75.

I tried looking up tutorials online but I still can't figure out why -7.25 is 111000.11. I would appreciate it if someone could explain the mathematical process behind getting -7.25.

Answer 1

You have an signed 8-bit 6Q2 format fixed point representation having a resolution of 2^-2 (or 0.25). You can look at this purely arithmetically:

The int8_t value of 11100011 is -29, but with the fixed point scaling that is -29 x 0.25 = -7.25.

Or then in terms of 2's compliment bit values which for an 8 bit integer are:

-2⁷ (-128₁₀)
2⁶ (64₁₀)
2⁵ (32₁₀)
2⁴ (16₁₀)
2³ (8₁₀)
2² (4₁₀)
2¹ (2₁₀)
2⁰ (1₁₀)

but the Q2 fixed point shifts those place values by 2 as follows:

-2⁵ (-32₁₀)
2⁴ (16₁₀)
2³ (8₁₀)
2² (4₁₀)
2¹ (2₁₀)
2⁰ (1₁₀)
2^-1 (0.5₁₀)
2^-2 (0.25₁₀)