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.