Adding signed integers

For integer values $x$ and $y$ in the range $- 2^{w - 1} \leq x, y \leq 2^{w - 1} - 1$

Visual demonstration

Case 1 and Case 4 demonstrate overflow situations

Examples

$x$	$y$	$x + y$	$x +_{4}^{t} y$	Case
-8 [`1000`]	-5 [`1011`]	-13 [10011]	3 [`0011`]	1
-8 [`1000`]	-8 [`1000`]	-16 [`10000`]	0 [0000]	1
-8 [`1000`]	5 [`0101`]	-3 [`11101`]	-3 [`1101`]	2
2 [`0010`]	5 [`0101`]	7 [`00111`]	7 [`0111`]	3
5 [`0101`]	5 [`0101`]	10 [`01010`]	-6 [`1010`]	4

The column “Case” corresponds to the “Case” shown in the figure above

Detecting overflow in two’s complement addition

Assume $s = x +_{w}^{t} y$
Then $s$ is a positive overflow if and only if $x > 0$ and $y > 0$ but $s \leq 0$ .
The computation is a negative overflow if and only if $x < 0$ and $y < 0$ but $s \geq 0$

Sources

Computer Systems - A Programmer’s Perspective