An Algorithm for Converting Floatingpoint Computations to Fixedpoint in MATLAB Based FPGA Design

1
An Algorithm for Converting Floating-point
Computations to Fixed-point in MATLAB Based FPGA
Design

• Sanghamitra Roy
• Prith Banerjee
• June 9, 2004

Department of Electrical and Computer Engineering
2
Introduction

• FPGA designs of DSP applications limited to
  fixed-point owing to cost and complexity of
  floating point hardware
• First step in the flow to map MATLAB applications
  to hardware conversion of
  floating-point MATLAB algorithm to fixed-point
  using MATLAB quantizers
• Automated method for floating to fixed point
  conversion reduces design time considerably,
  bridges gap between algorithm and hardware design
• Error-constrained area minimization problem is
  NP-hard

3
MATLAB Quantizers

• MATLAB quantization functionality comes withthe
  filter design and analysis toolbox
• quantizer object with two method functions
• quantizer() constructor for quantizer object
• quantize() applies quantizer to numbers

4
Autoquantization Algorithm

• We use profiling of expected inputsto estimate
  errors
• The algorithm attempts to minimize hardware
  resources while constraining quantization error
  within a specified limit
• Consists of the following passes
• Levelization
• Scalarization
• Computation of ranges of variables
• Generate fixed-point MATLAB code
• Autoquantization

5
Autoquantization Algorithm

• Levelization Take complex expressions on right
  side, convert to single operator form
• Scalarization Take a vectorized statement,
  convert it to scalar form using FOR loops
• Computation of Ranges Assign max, min values to
  input variables from input files Forward
  propagate ranges to LHS variables

6
Generation of Fixed-point MATLAB Code

• Replace each arithmetic and assignment operation
  with a quantized computation
• Set each quantizer to a default maximum precision
  quantizer(fixed, floor, wrap, 40,32)

• q1quantizer(fixed, floor, wrap, 40, 32)
• .
• q5quantizer(fixed, floor, wrap, 40, 32)
• bquantize(q1,load(file1))
• cquantize(q2,load(file2))
• dquantize(q3,load(file3))
• for i11100
• temp1(i)quantize(q4,(c(2i-1)d(i)))
• end
• for i11100
• a(i)quantize(q5,(b(i) temp1(i)))
• end

7
Autoquantization Step

• Fix Integral ranges of quantizers
• For each quantizer qi the integral range mi is
  calculated using
• mi floor(log2(maxi))2,
• where maxi max(abs(Maxi), abs(Mini)
• Coarse Optimize to get coarse precisions of
  quantizers
• Fine Optimize to get fine precisions of quantizers

8
Coarse Optimization Step

• Follows a divide-and-conquer approach
• Starts with lowest precision (0/4) bit and
  highest precision32 bit quantizers
• Moves quickly to the Coarse Optimal Point where
  all quantizers have equal precisions

9
Fine Optimization Step

• Starts from Coarse Optimal Point , moves to Fine
  Optimal Point
• Find a quantizer whose impact on quantization
  error is smallest, and reduce its precision by 1
  bit as long as error is within EMmax
• Find a quantizer whose impact on error is largest
  and increase its precision by 1 bit, while
  simultaneously reducing 2 bits in a quantizer
  with smallest impact on the error, as long as
  error is within EMmax

10
Autoquantization in Sample Code
EMmax0.1

• q1quantizer(fixed, floor, wrap, 8p1,
  p1) m8
• q2quantizer(fixed, floor, wrap, 10,
  p2) m10
• q3quantizer(fixed, floor, wrap, 9p3,
  p3) m9
• q4quantizer(fixed, floor, wrap, 18p4,
  p4) m18
• q5quantizer(fixed, floor, wrap, 18p5,
  p5) m18

• q1quantizer(fixed, floor, wrap, 11, 3)
• q2quantizer(fixed, floor, wrap, 13, 3)
• q3quantizer(fixed, floor, wrap, 12, 3)
• q4quantizer(fixed, floor, wrap, 21, 3)
• q5quantizer(fixed, floor, wrap, 21, 3)

• q1quantizer(fixed, floor, wrap, 8,0)
• q2quantizer(fixed, floor, wrap, 13, 3)
• q3quantizer(fixed, floor, wrap, 12, 3)
• q4quantizer(fixed, floor, wrap, 18, 0)
• q5quantizer(fixed, floor, wrap, 18, 0)
  

11
Use of Training Sets

• A small percentage of the actual inputs is used
  by our algorithm to generate optimal set of
  quantizers Training
• A larger set of actual inputs is used to test our
  system and verify that actual quantization error
  is within acceptable limit Testing
• Factor of safety for EM constraint

12
Experimental Results on int_fir Benchmark in
MATLAB
13
Conclusions and Future Work

• We presented an automatic quantization algorithm
  for MATLAB based FPGA design
• Based on input profiling
• Future Work
• Full Automation
• Autoquantization with power optimization
• Autoquantization in non-MATLAB environments
• Improved search algorithms for faster convergence
• Compiler approaches for precision propagation