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
Outline of the presentation
•
Motivations
•
Theoretical concepts
•
Accuracy evaluation methodology
• Overview of the tool
• Description of the different steps
•
Results
•
Conclusion
D. Menard, O. Sentieys University of Rennes 1
2
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 3.1416
main()
{
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-point
description
Fixed-point
coding
#define pi 3.1416
#define pi 3.1416
main()
{
float x,h,z
for(i=1;i<n;i++)
{
*z= *y++ + *h++
}
Precision
evaluation
Optimization
D. Menard, O. Sentieys University of Rennes 1
for(i=1;i<n;i++)
{
*z= *y++ + *h++
}
VIRGULE FLOTTANTE.C
Fixed-point
description
3
Motivations
•
Accuracy evaluation metric :
• Signal to Quantization Noise Ratio (SQNR) :
SQNR 
Py
Pby
by  yinfinite precision  yfinite precision
•
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
D. Menard, O. Sentieys University of Rennes 1
4
Theoretical concepts
•
Linear time-invariant systems :
• The output quantization noise by(n) is the sum
of filtered noise sources b’(n) [Menard02]
Output signal
hj
Input signal x j (n)
(generated during
a cast operation)
bej (n)
hj
bg 1 ( n )
bg 1 ( n )
hg 1
bgi (n)
Error due to
coefficient
quantization
bhj (n)
h j
Input noise bej (n)
Noise sources
y (n)
+
b y (n)
Output noise
bgi (n)
hgi


E (by )  f  b ' ,  b ' , H b ( z ),  x j ,  x j , H j ( z ) .
2
D. Menard, O. Sentieys University of Rennes 1
5
SQNR Computation Tool
Source
algorithm C
Front-end stages :
CDFG generation
SUIF
Front
End
SFG generation
Intermediate
representation
Back
End
Gs
Analytical
method
 CDFG generation (SUIF)
 CDFG to DFG transformation
 DFG to SFG transformation
Signal flow graph (SFG) +
fixed-point specifications
Back-end stages :
 T1 : Quantization noise modelisation
 T2 : Transfer function determination
 T3 : SQNR computation
SQNR
D. Menard, O. Sentieys University of Rennes 1
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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
Signal
node
u

v
uS
Signal

node
zS
uN
z

T1
vS
Noise
node
vN
zN
+

Noise
node
Multiplication noise model
D. Menard, O. Sentieys University of Rennes 1
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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
D. Menard, O. Sentieys University of Rennes 1
8
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
– Z 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
D. Menard, O. Sentieys University of Rennes 1
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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
D. Menard, O. Sentieys University of Rennes 1
10
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 fixedpoint data format optimization
– Hardware synthesis : minimization of the chip
area under SQNR constraint
Min S (bk )  such as SQNR(b k )  SQNRseuil
bk Z
D. Menard, O. Sentieys University of Rennes 1
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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.
D. Menard, O. Sentieys University of Rennes 1
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