An Analytical Approach for Dynamic Range Estimation

1
An Analytical Approach for Dynamic Range
Estimation

• Bin Wu, Jianwen Zhu, Farid N. Najm
• Dept. of Electrical and Computer Engineering
• University of Toronto

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Motivation

• Data-path bit-width
• An important design decision
• Significant impact on area, speed, and power
• Bit-width determined by signal dynamic range
• Min/Max bound
• Detailed distribution

Probability
Value
Min
Max
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State of the Art

• Profiling (simulation) is the extensively used
  method. Cao01, Kum01
• Some analytical methods proposed, such as
• Lp norm Jackson70, Carletta03
• Moment propagation Ortiz03
• Bitwidth / interval propagation Mahlke01
• Affine arithmetic method Fang03

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Problems of Previous Approaches

• Spatial Correlation

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Problems of Previous Approaches

• Temporal Correlation

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A New Problem Formulation

• Input data stream modeled as discrete timerandom
  process.
• C programs systems
• All program variables random processes(
  response to random input)
• Dynamic range estimation problemsolve the
  random response and obtain its statistics.

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Outline

• Introduction
• Extraction of Random Process Model
• Random Response of Linear System
• Statistics of System Variables
• Experiment Conclusions

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Extraction of Random Process Model

• Use Karhunen-Loeve Expansion (KLE) to extract
  input random process model from sample data.
• Capture temporal correlation
• Dimension reduction
• Computation involved
• Compute autocorrelation matrix fromsample data
• Solve eigensystem problem

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Karhunen-Loeve Expansion

• Discrete-time Karhunen-Loeve Expansion

zero-mean discrete time random process.
eigenvalue and eigenfunction of
autocorrelation function of pk
a set of orthogonal random variables with zero
mean unity variance.
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Karhunen-Loeve Expansion

• can be computed from

obtained from sample data
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Random Response of Linear System

• Observation two parts in KLE
• Deterministic part deterministic
  functions in time domain
• Random part has no dependence in time
  domain
• Superposition property of Linear system
• Solution Transform a random response problem
  into m deterministic response problems

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Random Response of Linear System
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Statistics of System Variables

• System variable X has KLE
• Its second order moments
• Arbitrary N order moments of Xk can be obtained
  from its KLE.

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Statistics of System Variables

• Probability Density Function (PDF) of Random
  Variable X can be constructed from its moments.
• For Gaussian distribution, 2-order moment can
  determine the whole pdf.
• For more general distribution, the pdf can be
  recovered by approximation methods
• such as Edgeworth expansion and generalized
  lambda distribution

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Workflow
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Experiments

• Experiment Setup
• Experiments conducted on a set of benchmarks(
  including FIRs, IIRs, FFT)
• Input sample data from Auto-Regression Moving
  Average (ARMA) model
• Goal
• Computation time, accuracy of KLE methods
• Trade-off between SNR and bitwidth
• Impact of input temporal correlation on the
  dynamic range estimation

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KLE Extraction

• Sample set size 10,000 traces of 100-time-point
• From most noisy sample to highly correlated
• Noisy sample set needs more terms kept

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KLE versus Profiling

• Accuracy (variance comparison, for rp1)

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KLE versus Profiling

• Accuracy (variance comparison, for rp4)

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KLE versus Profiling

• Accuracy (rp1)

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KLE versus Profiling

• Computation time (rp1)

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Bitwidth and SNR Tradeoff

• Signal-to-noise ratio can be computed from
  variable distribution for every specific range

(for benchmark FFT128)
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Impact of Temporal Correlation

• Probability density function from KLE and other
  oversimplified models

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Impact of Temporal Correlation

• Variance from KLE and other oversimplified models

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Conclusions

• Speed of KLE method
• Much faster than profiling
• As fast as other analytical approaches
• KLE method is accurate.
• No oversimplified assumption made, input model
  directly from sample data
• Fully considers both temporal and spatial
  correlation
• KLE provides complete info of dynamic range
• All statistics and pdf available
• Enables the tradeoff between SNR and reliability
• Future work
• Extend this analytic method to nonlinear systems