Fixed point multiplication with negative numbers

Question

I am stumped on a simple problem. Let's say I have two 4 bit numbers in Q0.3 format. One sign bit and three fractional bits. So I can represent $-1$ through to $0.875$.

Let's now say I wish to do this calculation: $-0.25 \times 0.875$. Which is:

$$ \frac{-2}{2^3} \times \frac{7}{2^3} $$

Which means I am multiplying $1110$ ($-2$) by $0111$ ($7$). Of course the answer is $-0.21875$ or $-0.25$ using the closest Q0.3 number.

Let's do the working.

$$ 1110 \times 0111 = 01100010 $$

which when viewed as a Q0.6 number is $1.100010$, which is $-0.46875$ by my books. Why is this incorrect? I expect an answer of $1.110010$ ($-0.21875$).

What have I done wrong?

Answer 1

When multiplying two's complement numbers, you have to perform sign extensions to the operands to meet the number of digits your multiplication will yield, i.e., in your case $4 + 4 = 8$ digits.

$$ 11111110_2 \times 00000111_2 = 11110010_2 $$

As there are $2 * 3$ fractional bits, the result is $1.110010_2 = \frac{-14}{2^6} = -0.21875$. Normalizing this number to $3$ fractional bits in Q0.3 format yields $1.110_2 = -0.25$.