1. 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

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

3. 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

4. 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)

5. 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

6. 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

7. 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

8. 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]

9. 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]

10. 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)

11. 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

12. 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)

13. 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

14. 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)

15. 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

16. Demo of Released Software

17. 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

18. End
Thank you!

19. Backup Slides
Backup Slides

20. 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

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

22. 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 S X
Wordlength
Integer
wordlength
Fractional
(Binary point)
π = …(10) [Floating Point]
(10) = (2) [WL=9; IWL=3; FWL=6] (10) = (2) [WL=16; IWL=3; FWL=13]

23. 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

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

25. 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

26. 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

27. 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

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