This VHDL package for floating-point arithmetic was originally developed at Johns Hopkins University. We expect anyone using this material already understands floating-point arithmetic and the IEEE 32-bit format, and we rely on the documentation in the VHDL file itself to explain the details. The package was developed and tested in our FPGA lab using a XSA-3S1000 development board supplied by the XESS corporation. This boards contains the XILINX XC3S1000-FT256 Spartan3 chips. We have not tested with any other chips or boards.
The FloatPt.vhd file contains all the components used to implement arithmetic operations with 32-bit IEEE standard floating-point numbers, along with the FloatPt package which contains all the declarations and functions to use the components. The components include FPP_MULT (for multiplication), FPP_ADD_SUB (for addition and subtraction) FPP_DIV (for division), and MantissaDivision (mantissa non-restoring division used in the FPP_DIV component). The package contains two functions: SIGNED_TO_FPP and FPP_TO_SIGNED for converting N-bit signed vectors to and from floating-point numbers, respectively.
The FPP_MULT, FPP_ADD_SUB and FPP_DIV components use state machines to implement the required arithmetic sequences on two floating-point numbers. We tested these with a 50 MHz clock, and (to be conservative) the mantissa division component at 25 MHz. All 3 components wait for an input request signal to go high, carry out their operation, set an output "done" signal high, then wait till the request line goes low to return to the idle state and wait for the next request. This hand-shaking transaction coordinates them with a higher level process and avoids a race condition if they finish quickly.
Floating-point numbers are simply std_logic_vectors that the components interpret as a sign bit, followed by an 8-bit exponent in excess-128 encoding, and a 23-bit mantissa with a leading 1 understood, but not present. Inside the components, the fields are typecast to std_logic unsigned vectors to carry out the necessary arithmetic, and then pasted back together for a final std_logic_vector result. The addition component only performs addition and expects the higher level process supplying the numbers to change the sign of one of the inputs for a subtraction.
Each of the floating-point components uses the following resources:
| Resource | FPP_ADD_SUB | FPP_MULT | FPP_DIV |
| Flip Flops | 131 | 133 | 277 |
| LUTs | 1279 | 171 | 603 |
| MULT 18x18 | 0 | 4 | 0 |
| BUFGMUX | 1 | 1 | 2 |
Particularly in the FP_ADD_SUB component, a good deal of space was used to trade off for a better execution time. This was in the exponent alignment step, and the post-normalization step. Rather than iterate the mantissa shifts needed for these operations, we implemented a single-stage shifter controlled by expanded logic to determine its value (Something like using a barrel shifter). Because of the number of logic levels needed, performing the single shift for each of these steps in a single state machine clock puts some stress on the 20ns clock. So implementations may need some specific path delay constraints. We managed to have it work OK in our filter example by specifying only a 20ns global period constraint.
The original VHDL code for the FPP_ADD_SUB that iterated the mantissa shifts was commented out, and left in the package for interest. With iteration the FPP_ADD_SUB can take fifty or more clocks depending on the difference in the magnitude of the two arguments. With barrel shifter type logic, the FPP_ADD_SUB completes in 4 clocks like the FPP_MULT. Of course the multiplier speed requires the use of the Spartan 18x18 multiplier primitives for the 24-bit multiplies. Without these, there's no way for the FPP_MULT state machine can run at 50 MHz. Since division is inherently slow and iterative, and we were mainly interested in addition and multiplication, we didn't put much effort into speeding it up. This is a possible area of improvement, as well as introducing a multiply-accumulate (MAC) operation.
To be useful, an FPGA system doing floating-point arithmetic needs to be able to convert between floating-point numbers and std_logic_vector fixed-point numbers (which are typically used when interfacing with an ADC or DAC). This requires assuming something about the position of the decimal point in an N-bit signed number. In the interest of simplicity, we just assume the signed numbers are integers and convert them into normalized floating-point numbers. We have tested this with 16-bit integers, and we believe it will work for any integer less than 24 bits. Also, using our convention, converting a floating-point number that is less than one will result in a fixed-point number truncated to zero.
We left hooks for an overflow output signal in our design, but did not use it in our tests even though we recognize its importance for system debug. We simply decided that we would make the result be zero if there was an underflow and make the result be the largest possible number with the correct sign for overflow. This is another area of possible future improvement
LowPassFP3.vhd contains an example of using the FloatPt package to build a third-order low-pass filter. We used it with our XESS development board to create a system that generated a square wave at selectable frequency ranging from 250 Hz up to 8 KHZ. Then we filtered the square wave 16-bit samples at 43 KHZ, with a 1KHz cutoff. The filtered 16-bit samples were sent out to the codec DAC on the XESS XST-4 board, and we observed the analog speaker output from the codec chip. It was very obvious from this how well the filter was performing.
The filter component is a good illustration of how to interact with the FloatPt package. It instantiates both FPP_ADD_SUB and FPP_MULT components, each driven by a 50 MHz clock. Then a 43 KHz state machine converts the 16-bit signed samples to floating-point format and requests the adds and multiplies needed to compute the Y0 recursive filter output. With seven multiplies and six adds per Y0 output sample, and four 50 MHz clocks for each operation, it should take about 52 or 53 clocks per Y0. This is what we observed in simulations. We were able to implement this in hardware by specifying only a 20 ns global clock period constraint, and a "normal" effort in synthesis and place-and-route.
We tested the FPP_DIV operation in hardware, but only with a 25 MHz master clock.
FloatPt.vhd
Floating-point components and package definition.
LowPassFP3.vhd
Design example that uses the floating-point components to build a third-order low-pass filter.
This example design was tested with the following version of software:
Xilinx WebPACK : 13.1Ryan Fay - JHU (student)
Alex Hsieh - JHU (student)
David Jeang - JHU (student)
Bob Jenkins - JHU (faculty adviser)
Send bug reports to [email protected].
Copyright 2011 by the Johns Hopkins University ECE department.
This application can be freely distributed and modified as long as you do not remove the attributions to the author or his employer.
07/28/2011 - Initial release.