Rounding in Fixed Point Number Conversions
When converting from one fixed point representation to another, there is often a right shift operation to eliminate bits. (Or higher order bits are just stored without keeping the lower order bits.) This occurs when converting from a Q31 to a Q15 format number for example, since 16 bits need to be eliminated. Before throwing away the unused bits, sometimes it is desirable to perform a rounding operation first. This can improve the accuracy of results, and can prevent the introduction of a bias during conversion of a signal. Rounding is also an important operation when generating fixed point filter coefficients from floating point values, but that is not the subject of this post.
To illustrate rounding, I will use an example where six different signed Q7.8 numbers are converted to a signed Q15.0 number (a regular 16 bit integer). I will illustrate truncation (throwing away the least significant eight bits) and rounding. Recall that a Q7.8 number has seven integer bits and eight fractional bits. For the example, the six numbers will be 1.25, 1.5, 1.75, -1.25, -1.5 and -1.75.
The first thing to determine is how these numbers will be represented in a 16 bit integer register. Multiplying each by 256 (which is two to the power eight) gives the following result (in hexadecimal):
1.25 = 0x0140
1.5 = 0x0180
1.75 = 0x01C0
-1.25 = 0xFEC0
-1.5 = 0xFE80
-1.75 = 0xFE40
Now if the numbers are truncated, the result is found by shifting right by eight. Here are the results:
truncate(1.25) = 0x0001 = 1
truncate(1.5) = 0x0001 = 1
truncate(1.75) = 0x0001 = 1
truncate(-1.25) = 0xFFFE = -2
truncate(-1.5) = 0xFFFE = -2
truncate(-1.75) = 0xFFFE = -2
For the positive numbers, the result of truncation is that the fractional part is discarded. The negative number results are more interesting. The result is that the fractional part is lost, and the integer part has been reduced by one. If a series of these numbers had a mean of zero before truncation, then the series would have a mean of less than zero after truncation. Rounding is used to avoid this problem of introduced bias and to make results more accurate.
Truncation is not really the correct term for the example above. More accurately, a “floor” operation is being executed. A floor operation returns the greatest integer that is not greater than the operand.
In a common method of rounding, a binary one is added to the most significant bit of the bits that are to be thrown away. And then a truncation is performed. In the current example, we would add 0.5, represented as 128 decimal or 0x0080 in our 16 bit integer word. So the results in our example are as follows:
round(1.25) = (0x0140 + 0x80) >> 8 = 0x0001 = 1
round(1.5) = (0x0180 + 0x80) >> 8 = 0x0002 = 2
round(1.75) = (0x01C0 + 0x80) >> 8 = 0x0002 = 2
round(-1.25) = (0xFEC0 + 0x80) >> 8 = 0xFFFF = -1
round(-1.5) = (0xFE80+ 0x80) >> 8 = 0xFFFF = -1
round(-1.75) = (0xFE40 + 0x80) >> 8 = 0xFFFE = -2
These results are less problematic than using simple truncation, but there is still a bias due to the non-symmetry of the 1.5 and -1.5 cases. The amount of bias depends on the data set. Even if a set of data to be converted contained only positive values, there is still a bias introduced, because all of the values that end in exactly .5 are rounded to the next highest integer. One way to eliminate this bias is to round even and odd values differently (even and odd to the left of the rounding bit position).
For the more common conversion of Q31 to Q15 numbers, the rounding constant is one shifted left by fifteen, or 32768 decimal, or 0x8000 hexadecimal.
Some of the Texas Instrument DSPs have rounding instructions that can be performed on the accumulator register prior to saving a result to memory. For example, the TMS320C55x processor includes the ROUND instruction (full name is “round accumulator content”). The instruction has two different modes. The “biased” mode adds 0x8000 to the 40 bit accumulator register. The “unbiased” mode conditionally adds 0x8000 based on the value of the least significant 17 bits. It is designed to address the bias problems I described above. Wikipedia has a good discussion of rounding and bias errors (http://en.wikipedia.org/wiki/Rounding). The TMS320C55x is using the “round half to even” method of rounding for the unbiased mode, and “round half up” for the biased mode.
Although it seems simple on the surface, rounding in fixed point conversions has some important effects on the bias of resulting computations.