Programmable logic is great for implementing mathematical algorithms for a range of applications from filtering to machine learning. Recently we have examined how mathematical operations are implemented within programmable logic using fixed point implementations. Including a detailed Hackster project.
One of the lessons we have learned as we worked with fixed point numbers is very quickly we can experience significant bit grown when adding or multiplying numbers together.
It is natural in our application that we may want to reduce the number of bits especially in the fractional bits.
However, how we do this may have an impact on our algorithms performance and output. For example if we may introduce offsets or biases within the result.
So in this blog we are going to take a look at how different rounding schemes work when we are working with fixed point maths in FPGAs. Lets look at the different rounding methods we can use in our FPGAs and when we might use them or not. Most of these can be very easily implemented within programmable logic using the VHDL 2008 fixed package types.
Truncation – The simplest approach which just truncates the bit vector, as such it requires no additional logic to implement. It does however introduce a bias towards zero which is not useful for many applications. This is also often called rounding towards zero or fixed rounding.
Algorithm = Truncate fractional LSBsTowards +Inf – This approach rounds the number towards positive infinity, it ensure a number will never decrease when it is rounded. For example 1.375 would round to 2, while -1.375 would round towards -1 this is rounding style is often referred to as taking the ceiling. It does however introduce a upwards bias as the result is never allowed to decrease.
Algorithm = IF fractional !=0
integer = integer + 1;
fractional = 0
ELSE
fractional = 0Towards -Inf – Similar to rounding towards positive infinity, rounding towards negative infinity ensure the value never increases. As such it adds in a downwards bias as the result is never allowed to increase. An example of this is 1.375 would round to 1 while -1.375 rounds to -2. This rounding method is often called talking the floor.
Algorithm = IF fractional !=0
integer = integer - 1;
fractional = 0
ELSE
fractional = 0Round Half Up – This rounds to the nearest integer if the fractional part is equal to or greater than 0.5, otherwise it rounds down. Rounding in this style adds in a upward bias.
Algorithm = IF fractional >= 0.5
integer = integer + 1;
fractional = 0
ELSE
fractional = 0Round Half Down – This rounds down if the fractional part is greater than 0.5, otherwise it rounds down. Rounding in this style adds in a downwards bias.
Algorithm = IF fractional > 0.5
integer = integer + 1;
fractional = 0
ELSE
fractional = 0Convergent Rounding – Unlike all of the other rounding schemes presented convergent rounding is a unbiased rounding scheme. Convergent rounding does this by rounding to the nearest even integer when the value is half way between two integers. For fractions greater or less than 0.5 it rounds up, other wise it rounds down. This adds a little extra complexity however, it produces a result which is unbiased, as such it is popular in DSP and ML/AL applications.
Algorithm = IF fractional > 0.5
integer = integer + 1;
fractional = 0
ELSE IF fractional = 0.5
ELSE IF odd(integer)
Integer = integer +1
fractional = 0
ELSE
Integer = integer
fractional = 0
ELSE
Integer = integer
fractional = 0Implementing these functionalities within a VHDL or Verilog function which should enable reuse of these rounding schemes across several applications as we need them.
In the mean time the table below summarizes the options above to provide a good reference of the different rounding styles, the logic impact and of course any bias.
Scheme | Bias | Best For | Complexity |
Truncate | Zero bias | Fast computations | Very low |
+∞ (Ceil) | Upward | Conservative limits | Low |
-∞ (Floor) | Downward | Safety-critical applications | Low |
Round Half Up | Upward | General computing | Low |
Round Half Down | Downward | General computing | Low |
Convergent | Unbiased | DSP, AI | Moderate |
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