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A METHODOLOGY FOR EVALUATING THE PRECISION OF
FIXED-POINT SYSTEMS
.
Abstract The minimization of cost, power
consumption and time to market of DSP
applications requires the development of
methodologies for the automatic implementation of
floating-point algorithms in fixed-point
architectures. In this paper, a new methodology
for evaluating the quality of an implementation
through the automatic determination of the Signal
to Quantization Noise Ratio (SQNR) is under
consideration. The modelization of the system at
the quantization noise level and the expression
of the output noise power have been detailed for
linear systems. Then, the different phases of the
methodology are explained and the ability of our
approach for computing the SQNR efficiently is
shown.

• Embedded Digital Signal Processing (DSP) systems
  
• Specification with floating-point data types
• Implementation in fixed-point architectures
• Precision evaluation based on simulation
  Coster98, Keding01, Kim98
• Long simulation time Coster98
• Optimization process requires multiple
  simulations Sung95
• Definition of a new methodology based on an
  analytical approach

Linear Time-Invariant System Model
SQNR Computation Methodology
Introduction

Source C algorithm
Signal

System noise model determination


DFG Generation
System Inputs
Fixed-point coding
Precision evaluation

• Noise source detection and insertion in the SFG
• Replacement of operators by their propagation
  noise model

Floating-point description
Fixed-point specification
Front End
Output signal
SUIF

SFG Generation
Optimization

Error due to coefficient quantization
Input noises
Signal Flow Graph fixed-point specifications
Gs

Noise modelization

• SFG to DAG transformation
• Detection of cycles in the SFG
• Enumeration of the cycles
• Dismantling of the cycles
• DAG linear function computation
• Partial T.F. determination
• Global T.F. determination

Transfer Function (T.F.) determination

Gsn

• Propagation noise models
• Addition z u v
• Multiplication z u ? v
• Quantization noise model
• bgi(n) additive random variable
• Stationary and uniformly distributed white noise
  
• Uncorrelated with y(n)
• First and second-order moments

Noise models
H(z) Determination
Output noise
Back End
Noise sources (generated during a cast operation)

GH
SQNR Determination
SQNR
Output quantization noise
Experimentation, Results and Perspectives
Q

• Test of the tool on classical DSP algorithms
• FFT, FIR and IIR filters
• Precision of the estimation
• Measurement of the relative error between our
  estimation and the one obtained by simulation
• IIR 2 lt 8.2
• FIR 16 lt 1.5
• FFT 16 lt 2.3
• Execution time of the tool
• Perspectives
• Hardware synthesis minimization of the chip
  area under SQNR constraint

Quantization noise
Noise due to coefficient quantization

• Most of the time is consumed by the cycle
  enumeration stage