Floating-Point to Fixed-Point Transformation Toolbox

Kyungtae Han and Brian L. Evans
Embedded Signal Processing Laboratory
Department of Electrical and Computer Engineering
The University of Texas at Austin, Austin, TX 78712-1084

05/22/06

This freely distributable toolbox automates conversion of floating-point programs to fixed-point programs and quantifies the tradeoff in signal quality vs. implementation complexity in fixed-point wordlength choices. This toolbox can automatically generate optimized fixed-point MATLAB programs in terms of wordlength (bit width), if floating-point programs and desired error specification are given. This software can be useful to implement digital signal processing algorithms on hardware for lower power consumption [11] in digital signal processors (DSPs), field programmable gate arrays (FPGAs), and application-specific integrated circuits (ASICs), and for lower hardware complexity in FPGA and ASIC.

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Demo

Filter design (Block design)

This demo demonstrates the automating floating-point to fixed-point transformation of a 10-variable infinite impulse response (IIR) filter. The transformation includes fixed-point conversion and wordlength optimization [1]. The given floating-point IIR filter program ( iirfilter.m) will be converted into a fixed-point IIR filter program. Then, data wordlengths to optimize the signal quality vs. implementation complexity are searched by the search algorithm selected by the user.

>> demo_afx

Then press 'enter' key four times. You can see search procedure. After finishing simulation, you can see the difference between floating-point output and optimized fixed-point output in a plot. You can also find some area reduction of optimized fixed-point code ( iirfilter_fixed.m) in the command window. The Genetic algorithm option provides a Pareto front ( 500th generation). In the GA results, the horizontal axis is the signal distortion and the vertical axis is implementation complexity (as measured in lookup table area on an FPGA).

BPSK system design (System design)

This demo demonstrates the automating floating-point to fixed-point transformation of a 4-variable BPSK system. The transformation includes fixed-point conversion and wordlength optimization [1]. The given floating-point BPSK system program ( bpsksystem.m) will be converted into a fixed-point BPSK system program. Then, data wordlengths to optimize the signal quality (bit error rate) vs. implementation complexity are searched by the search algorithm selected by the user.

>> demo_afxbpsk

Select 'system', then press 'enter' key three times. You can see search procedure. After finishing simulation, you can see some area reduction of optimized fixed-point code ( bpsksystem_fixed.m) in the command window. The Genetic algorithm option provides a Pareto front ( 50th generation) and a selection to choose a particular tradeoff point on the curve after simulations. In the GA results, the horizontal axis is the signal distortion and the vertical axis is implementation complexity (as measured in lookup table area on an FPGA).

Tutorial

Gradient-based search approach

Gradient-based wordlength search algorithms utilizing gradient information can provide faster ways to find data wordlengths but can get caught in local optima [1-3].
  1. Open iirfilter.m and look at the given floating point program.

    >> open iirfilter

  2. Run a fixed-point code generator. It generates fixed-point program file (iirfilter_fx.m) as well as auxiliary files.

    >> fxcodegen
    or
    >> fxcodegen('iirfilter.m')

  3. Select search algorithm and options, or just press 'enter' key for default option. Setting of weight of 1 utilizes only hardware complexity information (Local search) [4,5]. Setting of weight of 0 utilizes only error information [3]. A fraction value such as 0.5 utilizes complexity information and error information [1-2].
  4. Run wordlength optimization algorithm with input data.

    >> in = rand(1,100)
    >> iirfilter_top(in)

    You can see the search results. An optimized fixed-point file is stored in iirfilter_fixed.m

  5. Open the optimized fixed-point program

    >> open iirfilter_fixed

    Optimum wordlength is embedded in the fixed-point file. You can replace the floating program with the automatically generated and optimized fixed-point program.

You can also change and add configurations for general search algorithm in config.m. This file is copied to TOP file during code generation. Thus, after changing configuration, code generation should be run again.

Genetic algorithm approach

Wordlength search algorithms utilizing genetic and evolutionary algorithms provide Pareto optimum set of data wordlength to optimize the signal quality vs. implementation complexity tradeoffs [1]. Multiple solutions for optimum wordlength will be provided after simulation. More information about Genetic algorithm is available in [6-8].
  1. Open iirfilter.m and look at the given floating-point program

    >> open iirfilter

  2. Run fixed-point code generator. It generates fixed-point program file (iirfilter_fx.m) as well as auxiliary files.

    >> fxcodegen
    or
    >> fxcodegen('iirfilter.m')

  3. Select genetic algorithm.
  4. Run wordlength optimization algorithm with input data

    >> in = rand(1,100)
    >> iirfilter_top(in)

    You can see the graphical results of 100 generations. Simulation results are stored in iirfilter_result.mat. You can load the result and look the Pareto Optimal set.

    >> load iirfilter_result
    >> pareto_fr

The Pareto front variable has the last generation's Pareto optimal set with the following format [Error, Area, WL1, WL2, WL3, ...., WL10]. (Option) You can change and add configurations for general search algorithm in config.m. You can also change and add configurations for Genetic algorithm in configgea.m. These files are copied to TOP file during code generation. Thus, after changing configuration, code generation should be run again.

Usage

  1. MODIFYING: Please modify your floating point program for this software. Current version has a simple parser. Please, make your program as simple as possible, and follow the rules below. (See example file: iirfilter.m )
    1. The first line of function has an input argument for parameter passing since wordlength is optimized according to input data
      (e.g) function out = example (in) % for block
      (e.g) function error = system (gain) % for system
    2. The first line of function has an output argument for parameter passing since outputs of floating-point and fixed-point will be compared for block design. For system design, output argument should have distortion value such as bit error rate.
      (e.g) function out = example (in)
      (e.g) function error = system (gain) % for system
    3. Each line has only one arithmetic operation.
    4. Comment with 'fx' for desired assigned fixed-point variable
      (e.g) out = a + b; % fx
      (Variable 'out' is changed into fixed-point variable during code generation)
    5. Variables to be fixed point should be initialized with constants such as 0 or 1.
      (e.g) acc = 0 % fx
    6. Array variable to be fixed point should be initialized with array constants such as zeros or ones function.
      (e.g) data = zeros(1,1000) % fx

  2. CODE GENERATION: Run fixed-point code generator with your floating point file. It generates fixed-point program file as well as auxiliary files.

    >> fxcodegen('yourfile.m')

    Warning: The quotation marks are required. Select search algorithm and options, or just press 'enter' key for default option. Options are described in the other section.

  3. WORDLENGTH OPTIMIZATION: The generated TOP file can search wordlength with given input data.
    (e.g.)
    >> in = rand(1,100)
    >> yourfile_top(in)
    You can see the search results. Optimized fixed-point file is stored in yourfile_fixed.m. The optimum wordlength is embedded in the fixed-point file. You can replace your floating program with the automatically generated and optimized fixed-point program in a system.

Option

You can also change and add configurations for general search algorithm in config.m. This file is copied to TOP file during code generation. Thus, after changing configuration, code generation should be run again.
  1. Gradient-based search algorithm
    Weight
    Search algorithm gives weight on the error or hardware complexity for gradient information. Setting of weight of 1 utilizes only hardware complexity information. Setting of weight of 0 utilizes only error information. A fraction value such as 0.5 utilizes both information for search direction.
    RMS error
    Root Mean Square Error of output data between floating-point program and fixed-point program.
  2. Genetic algorithm In configgea.m file, "... 'Termination.MaxGenerations', 50, ..." indicates maximum generations. Please look at http://www.geatbx.com/ for more information for GEATbx.

Hardware Complexity Model

This software estimate hardware area for FPGA. The hardware area model in [9,10] is used. You can modify the model by changing the cost files such as cost_add.m and cost_mul.m.

References

  1. K. Han, "Fixed-point Transformations for Low-Power Embedded Hardware and Software Design", Ph.D. dissertation , Electrical and Computer Engineering, The University of Texas at Austin, August 2006.
  2. K. Han and B. L. Evans, "Optimum Wordlength Search Using Sensitivity Information", EURASIP Journal on Applied Signal Processing, special issue on Design Methods for DSP Systems, vol. 2006, no. 5, pp. 103-116, 2006.
  3. K. Han and B. L. Evans, "Wordlength Optimization with Complexity-And-Distortion Measure and Its Applications to Broadband Wireless Demodulator Design", Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., May 17-21, 2004, vol. 5, pp. 37-40, Montreal, Canada.
  4. W. Sung and K. Kum, "Simulation-based word-length optimization method for fixed-point digital signal processing systems", IEEE Trans. Signal Processing , vol. 43, no. 12, pp. 3087-3090, 1995.
  5. H. Choi and W. P. Burleson, "Search-based wordlength optimization for VLSI/DSP synthesis", Proc. IEEE Workshop on VLSI Signal Processing, vol. VII, Calif, USA, Oct. 1994, pp. 198-207.
  6. K. S. Tang, K. F. Man, S. Kwong and Q. He, "Genetic algorithms and their applications", IEEE Signal Processing Magazine, vol. 13, pp. 22-37, November 1996.
  7. C. M. Fonseca and P. J. Fleming, "Genetic algorithms for multiobjective optimization: Formulation, discussion and generalization", Proc. Int. Conf. Genetic Algorithm, July 1993, pp. 416-423.
  8. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Reading, MA: Addison-Wesley, 1989.
  9. G. A. Constantinides, P. Y. Cheung, and W. Luk, "Wordlength optimization for linear digital signal processing", IEEE Trans. Computer-Aided Design , vol. 22, no. 10, pp. 1432-1442, Oct. 2003.
  10. G. A. Constantinides, P. Y. Cheung, and W. Luk, "Optimum wordlength allocation", Proc. IEEE Sym. on Field-Programmable Custom Computing Machines, April 2002, pp. 219-228.
  11. K. Han, B. L. Evans, and E. E. Swartzlander, Jr., "Low-Power Multipliers with Data Wordlength Reduction", Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Oct. 30-Nov. 2, 2005, pp. 1615-1619, Pacific Grove, CA USA.

Mail comments about this page to Prof. Brian L. Evans at bevans@ece.utexas.edu.