Automatic Floating-Point to Fixed-Point Transformations Kyungtae Han, Alex G. Olson, Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin 2006 Asilomar Conference on Signals, Systems, and Computers October 30th, 2006

Outline Introduction Background Fixed-point wordlength optimizations Automate transformations of systems Conclusion

Implementing Digital Signal Processing Algorithms Hardware Price Power* Floating- Point Processor $ Floating-Point Program Code Conversion H L Digital Signal Processing Algorithms Fixed- Point Processor Fixed Point (Uniform Wordlength) $ Wordlength Optimization L H Fixed- Point ASIC Fixed Point (Optimized Wordlength) $ L H ASIC: Application Specific Integrated Circuit * Power consumption

Transformations to Fixed Point Advantages Lower hardware complexity Lower power consumption Faster speed in processing Disadvantages Introduces distortion due to quantization error Search for optimum wordlength by trial & error is time-consuming Research goals Automate transformations to fixed point Control distortion vs. complexity tradeoffs Floating-Point Program Code Conversion Transformation Wordlength Optimization Fixed Point (Optimized Wordlength)

Distortion vs. Complexity Tradeoffs Shorter wordlength may increase application distortion and decrease implementation complexity Application distortion d(w) c(w) Implementation cost function d(w) Application distortion function Feasible region Optimal tradeoff curve Implementation complexity c(w) Minimize implementation cost Minimize application distortion

Search for Optimum Wordlength Complete search Search whole space Impractical in systems with many variables Gradient-based search Utilizes gradient information to determine next candidates Complexity measure (CM) [Sung and Kum, 1995] Distortion measure (DM) [Han et al., 2001] Complexity-and-distortion measure (CDM) [Han and Evans, 2004] Guided random search Genetic algorithm for single objective [Leban and Tasic, 2000] Multiple objective genetic algorithm

Complexity-and-Distortion Measure Weighted combination of measures Single objective function: Gradient-based search Initialization Iterative greedy search based on complexity and distortion gradient information c(w) Complexity function d(w) Distortion function Dmax Constant for maximum distortion Cmax Constant for maximum complexity Wordlength lower bound upper bound

Genetic Algorithm Evolutionary algorithm Inspired by Holland 1975 New Gene Pool Function Evaluation Mutation Selection Mating Child Genes Parental Genes w/ Measure Evolutionary algorithm Inspired by Holland 1975 Mimic processes of plant and animal evolution Find optimum of a complex function [From Greg Rohling’s Ph.D Defense 2004]

Case Study: Filter Design Infinite impulse response (IIR) filter Complexity measure: Area model of field-programmable gate array (FPGA) [Constantinides, Cheung, and Luk 2003] Distortion measure: Root mean square (RMS) error Seven fixed-point variables (indicated by slashes) Delay b0 b1 -a1 x[n] y[n]

Case Study: Gradient-Based Search CDM could lead to lower complexity and lower number of simulations compared to DM and CM Search Method Gradient Measure Number of Simulations Complexity Estimate (LUT) Distortion (RMS)* Complete DM CDM CM - 316 145 417 167 ** 51.05 49.85 51.95 0.0981 0.0992 0.0986 * Maximum distortion measured by root mean square (RMS) error is 0.1 ** 167 = 268,435,456 (8.5 years, if 1 second per 1 simulation)

Case Study - IIR: Genetic Algorithm Search Pareto optimal set (nondominated) Handles multiple objectives: Error and Area Pareto Front 9,000 simulations 22,500 simulations 45,000 simulations 100th Generation 250th Generation 500th Generation * Population for one generation: 90 LUT: Lookup table

Case Study: Comparison Contribution #1 Case Study: Comparison Superpose gradient-based search (GS) results on GA results 50th Generation (4500 simulations) 500th Generation (45000 simulations) * Required RMSmax for gradient-based search are Dmax {0.12, 0.1, 0.08} GS methods can get stuck in a local minimum GS methods reduce running time (CDM: 145 simulations)

Automating Transformations from Floating Point to Fixed Point Existing fixed-point tools Support fixed-point simulation Convert floating-point code to raw fixed-point code Manually find optimum wordlength by trial and error Automating transformations Fully automate conversion and wordlength optimization process (Proposed) SNU gFix, Autoscaler CoWare SPW HDS Synopsys CoCentric MATLAB Fixed-point toolbox MATLAB Fixed-point blockset AccelChip DSP synthesis Catalytic RMS, MCS Fixed-point tools Floating-Point Program Code Conversion Wordlength Optimization Wordlength-Optimized Fixed-Point Program

Code Generation for Fixed-Point Program Adder function in MATLAB Function [c] = adder_fx(a, b) c = 0; a = fi (a, 1,32,16); b = fi (b, 1,32,16); c = fi (c, 1,32,16); c(:) = a + b; Function [c] = adder(a, b) c = 0; c = a + b; Determined by designers with trial and error (a) Floating point program for adder (b) Raw fixed-point program Function [c] = adder_fx(a, b, numtype) c = 0; a = fi (a, numtype.a); b = fi (b, numtype.b); c = fi (c, numtype.c); c(:) = a + b; WL S FWL fi(a, S,WL,FWL) is a constructor function for a fixed-point object in fixed-point toolbox [S: Signed, WL: Wordlength, FWL: Fraction length] (c) Converted fixed-point program for automating optimization (Proposed)

Automating Transformation Environment for Wordlength Optimization Input Data Top Program Floating-Point Program Optimum Wordlength Evaluation Program (Objectives) Search Engine Fixed-Point Program Gradient-based or Genetic algorithm Range Estimation Complexity Estimation Error Estimation Given floating-point program and options, auxiliary programs are automatically generated Given input data, optimum wordlength is searched

Demo of Released Software

Conclusion Search for optimum wordlength Gradient-based search reduces execution time with complexity-and-distortion measure method while solutions could be trapped in local optimum Genetic algorithm can find distortion vs. complexity tradeoff curve, but it requires longer execution time Automate transformations from floating-point programs to fixed-point programs Free software release is available at www.ece.utexas.edu/~bevans/projects/wordlength/converter/

End Thank you!

Backup Slides Backup Slides

Case Study- Receiver: Gradient-Based Search Demodulate Integrate & Dump Search Method Gradient Measure Number of Simulations Complexity Estimate (LUT) Distortion (RMS)* Complete DM CDM CM - 66 65 195 164 40.65 43.65 41.95 0.083 0.085 0.081 * Maximum distortion measured by bit error rate (BER) is 0.1

Case Study - Receiver: Genetic Algorithm Population for one generation: 90 25th Generation 50th Generation 100th Generation 200th Generation

Fixed-Point Data Format Integer wordlength (IWL) Number of bits assigned to integer representation Fractional wordlength (FWL) Number of bits assigned to fraction Wordlength (WL) SystemC format www.systemc.org S X Wordlength Integer wordlength Fractional (Binary point) π = 3.14159…(10) [Floating Point] 3.140625(10) = 011.001001(2) [WL=9; IWL=3; FWL=6] 3.141479492(10) = 011.00100100001110(2) [WL=16; IWL=3; FWL=13]

Wordlength Optimization Constraints Distortion constraint Complexity constraint Application-specific distortion d(w) Application-specific distortion d(w) Dmax Cmax Implementation Complexity c(w) Implementation Complexity c(w) Enforcing both constraints bounds the search to a finite area region

Wordlength Optimization Wordlengths of signals (variables) in digital system as vector Single objective optimization Multiple objective optimization

Pareto Optimality Pareto optimality: “best that could be achieved without disadvantaging at least one group” [Allan Schick 1970] Pareto optimal set is set of nondominated solutions E is dominated by C as all objectives for C are less than corresponding objectives for E Solutions A, B, C, D are nondominated (not dominated by any solution) Pareto front is boundary (tradeoff curve) that connects Pareto optimal set solutions Pareto Front I A G Objective 2 H B E C F D Objective 1 : Nondominated : Dominated

Comparison of Proposed Methods Gradient-based search Genetic algorithm Type of Solution One point Family of points Tradeoff Curve Found No Yes Execution Time Short Long Amount of Computation Low High Parallelism

Automatic Transformation Flow Code generation Parse floating-point program Generate a raw fixed-point program and auxiliary programs (top, objective, cost, etc.) Range estimation Estimate range to avoid overflow (Analytical/Simulation) Determine integer wordlength (IWL) Wordlength optimization Optimize wordlength according to given input, and error specification (Analytical/Simulation) Determine fractional wordlength (FWL) Code Generation Range Estimation Wordlength Optimization

Code Generations <Run Code Generation> <Floating-point Program>