Overflow Detection in Verilog

Posted on Edited on In
  • Arithmetic operations have a potential to run into a condition known as overflow.
  • Overflow occurs with respect to the size of the data type that must accommodate the result.
  • Overflow indicates that the result was too large or too small to fit in the original data type.

Overflow when adding unsigned number

When two unsigned numbers are added, overflow occurs if

  • there is a carry out of the leftmost bit.

Overflow when adding signed numbers

When two signed 2's complement numbers are added, overflow is detected if:

  1. both operands are positive and the result is negative, or
  2. both operands are negative and the result is positive.

Notice that when operands have opposite signs, their sum will never overflow. Therefore, overflow can only occur when the operands have the same sign.

A B carryout_sum overflow
011 (3) 011 (3) 0_110 (6) overflow
100 (-4) 100 (-4) 1_000 (-8) underflow
111 (-1) 110 (-2) 1_101 (-3) -

carryout information ISN'T needed to detect overflow/underflow for signed number addition

EXTBIT:MSB

extended 1bit and msb bit can be used to detect overflow or underflow

1
2
3
4
5
6
7
8
9
10
reg signed  [1:0]      acc_inc;
reg signed  [10-1:0]   acc;
wire signed [10  :0]   acc_w;  // extend 1b for saturation
wire signed [10-1:0]   acc_stat;

assign acc_w = acc + acc_inc;   // signed arithmetic

assign acc_stat = (acc_w[10-1 +: 2] == 2'b01) ? {1'b0, {(10-1){1'b1}}} : // up saturation
    (acc_w[10-1 +: 2] == 2'b10) ? {1'b1, {(10-1){1'b0}}}  :   			 // down saturation
    acc_w[10-1:0];

2'b01 : overflow, up saturation

2'b10: underflow, down saturation

2's complement negative number

  1. Flip all bits
  2. Add 1.

N-bit signed number \[ A = -M_{N-1}2^{N-1}+\sum_{k=0}^{N-2}M_k2^k \] Flip all bits \[\begin{align} A_{flip} &= -(1-M_{N-1})2^{N-1} +\sum_{k=0}^{N-2}(1-M_k)2^k \\ &= M_{N-1}2^{N-1}-\sum_{k=0}^{N-2}M_k2^k -2^{N-1}+\sum_{k=0}^{N-2}2^k \\ &= M_{N-1}2^{N-1}-\sum_{k=0}^{N-2}M_k2^k -1 \end{align}\]

Add 1 \[\begin{align} A_- &= A_{flip}+1 \\ &= M_{N-1}2^{N-1}-\sum_{k=0}^{N-2}M_k2^k \\ &= -A \end{align}\]

reference

Overflow Detection: http://www.c-jump.com/CIS77/CPU/Overflow/lecture.html