## Overflow Handling in Fixed Point Computations

Overflow handling is an important consideration when implementing signal processing algorithms. If overflow is not controlled appropriately it can lead to problems such as detection errors, or poor quality audio output. Typical digital signal processing CPUs include hardware support for handling overflow. Some RISC processors may include these modes as well. (In fact I helped define and implement such modes for the 32 bit MIPS processor core used in many Broadcom products). These processors often have a “saturating” mode that sets an instruction result to a minimum or maximum value on an overflow condition. (The term “saturating” comes from analog electronics, in which an amplifier output will be limited, or clipped, between fixed values when a large input is applied.) Commonly the CPU will limit the result to a 32 bit twos complement integer (0x7FFFFFFF or 0x80000000). For unsigned operations, the result would be limited to 0xFFFFFFFF. There are a number of situations in which overflow can occur, and I will discuss some of them below.

#### Addition and Subtraction

Overflow with twos complement integers occurs when the result of an addition or subtraction is larger the largest integer that can be represented, or smaller than the smallest integer. In fixed point representation, the largest or smallest value depends on the format of the number. I will assume Q31 in a 32 bit register for any examples that follow. In this case, a CPU with saturation arithmetic would set the result to -1 or (just below) +1 on an overflow, corresponding to the integer values 0x80000000 and 0x7FFFFFFF.

Overflow in addition can only occur when the sign of the two numbers being added is the same. Overflow in subtraction can occur only when a negative number is subtracted from a positive number, or when a positive number is subtracted from a negative number.

#### Negation

There is one case where negation of a number causes an overflow condition. When the smallest negative number is negated, there is no way to represent the corresponding positive value in twos complement. For example, the value -1 in Q31 is 0x80000000. When this number is negated (flip the bits and add one) the result is again -1. If the saturation mode is set, then the CPU will set the result to 0x7FFFFFFF (just less than +1).

#### Arithmetic Shift

Overflow can occur when shifting a number left by 1 to n bits. In fixed point computations, left shifting is used to multiply a fixed point value by a power of two, or to change the format of a number (Q15 to Q31 for example). Again, many CPUs have saturation modes to set the output to the minimum or maximum 32 bit integer (depending on whether the original number was positive or negative). Furthermore, a common feature is an instruction that counts the number of leading ones or zeros in a number. This helps the programmer avoid overflow since the number of leading sign bits determines how large a shift can be done without causing overflow.

Overflow will not occur when right shifting a number.

#### Multiplication

Overflow doesn’t really occur during multiplication if the result register has enough bits (32 bits if two 16 bit numbers are multiplied). But it is partly a matter of interpretation. When multiplying a fixed point value of -1 by -1 (0x8000 by 0x8000 using Q15 numbers), the result is +1. If the result is interpreted as a Q1.30 number (one integer bit and 30 fractional bits) then there is no problem. If the result is to be a Q30 number (no integer bits) then an overflow condition has occurred. And if the number was to be converted to Q31 (by shifting the result left by 1) then an overflow would occur during the left shift. The overall affect would be that -1 times -1 equals -1.

I have used a CPU that handles this special case with saturation hardware. Some CPUs have a multiplication mode that shifts the product left by one bit after a multiply operation. The reason for doing so is to create a Q31 result when two Q15 numbers are multiplied. Then if a Q15 result is desired, it can be found by storing the upper 16 bits of the result register (if the register is only 32 bits). The saturating mode automatically sets the result to 0x7FFFFFFF when the number 0x8000 is multiplied by itself, and the “shift left by one” multiplication mode is enabled.

A very often used operation in DSP algorithms is the “multiply accumulate” or “MAC”, where a series of numbers is multiplied and added to a running sum. I would recommend not using the “left shift by one” mode if possible when doing MACs, since this only increases the chance for overflow. A better technique is to keep the result as Q1.30, and then handle overflow if converting the final result to Q31 or Q15 (or whatever). This is also a good technique to use on CPUs without saturation modes, since the number of overflow checks can be greatly reduced in some cases.

#### Division

Overflow in division can occur when the result would have more bits than was calculated. For example, if the magnitude of the numerator is several times larger than that of the denominator, than the result must have enough bits to represent numbers larger than one. Overflow can be avoided by carefully considering the range of numbers being operated on, and calculating enough bits for the result. I have not seen a CPU that implements a saturation mode for division.

Division by 0 is undefined, and not really an overflow case.

#### Conclusion

Many CPUs include hardware supported handling of overflow using saturation modes. These modes are useful, but it is better to avoid overflow in the first place if possible. This can lead to more accurate results in computations. And when using a CPU without saturation arithmetic, it is best to design the arithmetic operations so that the number of overflow checks is minimized.

January 22, 2017 at 5:14 am

Dear Shawn,

Thanks for understanding me the overflow in different arithmetic operations. If I wanted to multiply Q.25 and Q.15 then my result will be Q2.40. Please explain me how many maximum LSB bits I am allowed to truncate ?.

As I understand I can keep the full bit precision. But that is not possible because of my minimal resources.

Cheers,

Ravi.

January 22, 2017 at 10:05 am

Ravi,

The number of bits to keep depends upon your particular application, and how much precision is needed in subsequent operations. You need to analyze your algorithm to see how small the numbers need to be. Maybe it makes sense to keep 15 or 25 bits? I hope that helps.

-Shawn

January 24, 2017 at 2:53 am

Dear Shawn,

Thank you for the quick reply, here I kept the 25 bits and truncated 15 LSB. It is working.

Cheers,

Ravi.