Automatic Evaluation of the Accuracy of Fixed-point

Algorithms

Daniel MENARD1, Olivier SENTIEYS1,2

1 LASTI, University of Rennes 1

Lannion, FRANCE

2 IRISA/INRIA

Rennes, FRANCE

2D. Menard, O. Sentieys University of Rennes 1

Outline of the presentation • Motivations

• Theoretical concepts

• Accuracy evaluation methodology

• Overview of the tool

• Description of the different steps

• Results

• Conclusion

3D. Menard, O. Sentieys University of Rennes 1

Motivations • Embedded Digital Signal Processing (DSP) systems

• Specification with floating-point data types

• Implementation in fixed-point architectures

Development of new methodologies for the automatic transformation of floating-point descriptions into fixed-point specifications

#define pi 3.1416#define pi 3.1416main(){float x,h,z for(i=1;i<n;i++) { *z= *y++ + *h++ }

for(i=1;i<n;i++) { *z= *y++ + *h++ }

VIRGULE FLOTTANTE.C

Fixed-point coding

Precisionevaluation

#define pi 3.1416#define pi 3.1416main(){float x,h,z for(i=1;i<n;i++) { *z= *y++ + *h++ }

for(i=1;i<n;i++) { *z= *y++ + *h++ }

VIRGULE FLOTTANTE.C

Floating-pointdescription

Fixed-pointdescriptionOptimization

4D. Menard, O. Sentieys University of Rennes 1

Motivations• Accuracy evaluation metric :

• Signal to Quantization Noise Ratio (SQNR) :

• Methodologies for the SQNR evaluation are based on simulation [Coster98], [Keding01], [Kim98]

• Drawbacks : – Long simulation time [Coster98]

– Fixed-point format optimization process requires multiple simulations [Sung95]

• Presentation of a new methodology based on an analytical approach

precision finiteprecision infinite yybP

PSQNR y

b

y

y

5D. Menard, O. Sentieys University of Rennes 1

Theoretical concepts • Linear time-invariant systems :

• The output quantization noise by(n) is the sum of filtered noise sources b’(n) [Menard02]

+

)(nbej

)(nx j

)(nbej

)(nbhj

)(1 nbg )(1 nbg

)(nbgi )(nbgi

1gh

gih

jh

jh

jh

)(nby

)(ny

Input signal

Noise sources(generated

during a cast operation)

Output noise

Output signal

Input noise

Error due to coefficient

quantization

.)(,,),(,,)( ''2 zHzHfbE jxxbbby jj

.)(,,),(,,)( ''2 zHzHfbE jxxbbby jj

6D. Menard, O. Sentieys University of Rennes 1

SQNR Computation Tool

FrontEnd

BackEnd

Intermediaterepresentation

Analytical method

Analytical method

SQNR

GsSignal flow graph (SFG) +fixed-point specifications

Back-end stages :

T1 : Quantization noise modelisation T2 : Transfer function determination T3 : SQNR computation

Front-end stages :

CDFG generation (SUIF) CDFG to DFG transformation DFG to SFG transformation

Source algorithm C

CDFG generationCDFG generation

SFG generationSFG generation

SU

IF

7D. Menard, O. Sentieys University of Rennes 1

Transformation T1 : Gs Gsn

• Goal : specify the system with a Signal Flow Graph Gsn at the quantization noise level

• Steps :

– Detection and insertion of the noise sources

– Introduction of the operator noise models

zS

uS

uN

vS

vN

zN

+

Multiplication noise model

u

v

z

T1

Noise node

Signal node

Signal node

Noise node

8D. Menard, O. Sentieys University of Rennes 1

Transformation T2 : Gsn GH (1)

• Stage T21 : Gsn Gk

• Goal : transform the Signal Flow Graph Gsn into several directed acyclic graphs (DAG)

• Steps : – Detection of the cycles : linear complexity algorithm

– Enumeration of the cycles : polynomial complexity

– Dismantle of the cycles in order to obtain DAG

• Stage T22 : Gk Geq

• Goal : specify the system with a set of linear functions

• Step : depth-first traversal of the graph with a post-order recursive algorithm

9D. Menard, O. Sentieys University of Rennes 1

Transformation T2 : Gsn GH (2)

• Stage T23 : Geq GHi

• Goal : specify the system with a set of partial transfer functions

• Steps :

– Application of a set of variable substitutions

– transformation of the linear functions

• Stage T24 : GHi GH

• Goal : specify the system with a set of global transfer functions

• Step : Computation of the global transfer functions from the partial transfer functions

10D. Menard, O. Sentieys University of Rennes 1

Results • Test of the tool on classical DSP algorithms : FIR

and IIR filters, FFT

• Precision of the estimation :

• Measurement of the relative error between our estimation and the one obtained by simulation

– IIR 2 < 8.2 %

– FIR 16 < 1.5 %

– FFT 16 < 2.3 %

• Execution time :

• Most of the time is consumed by the stage T2– FIR 256 : 0.86 s

– IIR 4 : 0.65 s

11D. Menard, O. Sentieys University of Rennes 1

Conclusion • Definition of a new methodology for computing the

SQNR based on an analytical approach

• Development of a tool for implementing this methodology

• Limitation : the method can not cope with recursive non-linear systems

• Advantages :

• Smaller optimization time for the process of fixed-point data format optimization

– Hardware synthesis : minimization of the chip area under SQNR constraint

seuilkkb

SQNR)SQNR(bbSk

assuch )(Min

12D. Menard, O. Sentieys University of Rennes 1

References• [Coster98] L. D. Coster, M. Ade, R. Lauwereins, and J. Peperstraete. Code

Generation for Compiled Bit-True Simulation of DSP Applications. In Proceedings of ISSS’98, Taiwan, December 1998.

• [Johnson75] D. B. Johnson. Finding All the Elementary Circuits of a Directed Graph. SIAM Journal on Computing, 4(1):77–84, March 1975.

• [Keding01] H. Keding, M. Coors, O. Luthje and H. Meyr. FRIDGE: Fast Bit True simulation. In Design Automation Conference 2001 (DAC 01), June 2001, Las Vegas.

• [Kim98] S. Kim, K. Kum, and S. Wonyong. Fixed-Point Optimization Utility for C and C++ Based Digital Signal Processing Programs. IEEE Transactions on Circuits and Systems–II: Analog and Digital Signal Processing, 45(11), November 1998.

• [Menard02] D. Menard and O. Sentieys. A methodology for evaluating the precision of fixed-point systems. ICASSP 2002, May 2002, Orlando

• [Sung95] W. Sung and K. Kum. Simulation-Based Word-Length Optimization Method for Fixed-Point Digital Signal Processing Systems. IEEE Transactions on Signal Processing, 43(12), December 1995.