Wordlength Optimization with Complexity-and-Distortion Measure and Its Application to Broadband Wireless Demodulator Design

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Wordlength Optimization withComplexity-and-Distor
tion Measure andIts Application to Broadband
Wireless Demodulator Design

• Kyungtae Han and Brian L. Evans
• Embedded Signal Processing Laboratory
• Wireless Networking and Communications Group
• The University of Texas at Austin

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Fixed-Point Design
Introduction

• Digital signal processing algorithms
• Often developed in floating point
• Later mapped into fixed point for digital
  hardware realization
• Fixed-point digital hardware
• Lower area
• Lower power
• Lower per unit production cost

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Fixed-Point Design
Introduction

• Float-to-fixed point conversion required to
  target
• ASIC and fixed-point digital signal processor
  core
• FPGA and fixed-point microprocessor core
• All variables have to be annotated manually
• Avoid overflow
• Minimize quantization effects
• Find optimum wordlength
• Manual process supported by simulation
• Time-consuming
• Error prone

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Optimum Wordlength
Background
Optimum wordlength

• Longer wordlength
• May improve applicationperformance
• Increases hardware cost
• Shorter wordlength
• May increase quantization errorsand overflows
• Reduces hardware cost
• Optimum wordlength
• Maximize application performanceor minimize
  quantization error
• Minimize hardware cost

Distortion d(w) 1/performance
Cost c(w)
Wordlength (w)
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Wordlength Optimization
Background

• Express wordlengths in digital system as vector
• Wordlength range for kth wordlength
• Cost function c

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Wordlength Optimization
Background

• Application performance function p
• Wordlength optimization problem
• Iterative update equation
• Good choice of update direction can reduce number
  of iterations to find optimum wordlength

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Sequential Search K. Han et al. 2001
Search Methods

• Greedy search based on sensitivity information
  (gradient)
• Example
• Minimum wordlengths 2,2
• Direction of sequential search
• Optimum wordlengths 5,5
• 12 iterations
• Advantage Fewer trials
• Disadvantage Could miss global optimum point

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Measures for Optimum Wordlength
Measures

• Complexity measure method W.Sung and K.Kum 1995
• Minimize complexity c(w) subject to constraint on
  distortion d(w)
• Update direction uses complexity sensitivity
  information
• Distortion measure K. Han et al. 2001
• Minimize distortion d(w) subject to constraint on
  complexity c(w)
• Update direction uses distortion sensitivity
  information

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Complexity-and-Distortion Measure
Measures

• Combine complexity measure with distortion
  measure by weighting factor (0a1)
• Tradeoffs between measuresby changing weighting
  factor
• Update direction uses both sources of sensitivity
  information

Objective function
Optimization problem
Update direction
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Broadband Wireless Access(IEEE 802.16a)
Demodulator
Case Study
w0 Input wordlength of orthogonal frequency
division multiplex (OFDM) demodulator
which performs a fast Fourier transform (FFT) w1
Input wordlength of equalizer w2 Input
wordlength of channel estimator w3 Output
wordlength of channel estimator
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Simulations
Case Study

• Assumptions
• Internal wordlengths of blocks have been decided
• Complexity increases linearly as wordlength
  increases
• Required application performance
• Bit error rate of 1.5 x 10-3 (without error
  correcting codes)
• Simulation tool
• LabVIEW 7.0

Input Weight
FFT 1024
Equalizer (right) 1
Estimator 128
Equalizer (upper) 2
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Minimum Wordlengths
Case Study

• Change one wordlength variable while keeping
  other variables at high precision
• 1,16,16,16,2,16,16,16,...
• 16,1,16,16,16,2,16,16,...
• 16,16,16,15,16,16,16,16
• Minimum wordlength vector is 5,4,4,4

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Number of Trials
Case Study

• Start at 5,4,4,4 wordlength
• Next wordlength combination for complexity
  measure (a 1.0)
• 5,4,4,4,
• 5,5,4,4,
• Increase wordlength one-by-one until satisfying
  required application performance

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Complexity and Number of Iterations
Case Study

• Each iteration computes complexity distortion
  measures
• Distortion measure high cost, low iterations
• Complexity-distortion medium cost, fewer
  iterations
• Complexity measure low cost, more iterations
• Full search low cost, more iterations

Method a w Complexity Iterations
Distortion Only Complexity-Distortion Complexity Only Full Search 0 0.5 1 n/a 10,9,4,10 7,10,4,6 7,7,4,6 7,7,4,6 10781 7702 7699 7699 16 15 69 210
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Conclusion

• Summary
• Fixed-point conversion requires wordlength
  optimization
• Develop complexity-and-distortion measure
• Complexity-and-distortion method finds optimal
  solution in one-third the time that full search
  takes for case study
• Future extensions for wordlength optimization
• Automate selection of wordlength range
• Combine simulation-based and analytical
  approaches
• Employ genetic algorithms

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Fixed-Point Representation
Introduction

• Fixed point type
• Wordlength
• Integer wordlength
• Quantization modes
• Round
• Truncation
• Overflow modes
• Saturation
• Saturation to zero
• Wrap-around

SystemC format www.systemc.org
Back
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Full Search W. Sung and K. Kum 1995
Search Methods

• Exhaustive search of all possible wordlengths
• Advantages
• Does not miss optimum points
• Simple algorithm
• Disadvantage
• Many trials (experiments)
• Distance
• Expected number of iterations

Direction of full searchminimum wordlengths
2,2optimum wordlengths 5,5d 6trials
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Back
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FFT Cost

• N Tap FFT cost
• 256 Tap FFT cost

Back
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Complexity Measure W.Sung and K.Kum 1995
Measures

• Uses complexity sensitivity information as
  direction to search for optimum wordlength
• Advantage minimizes complexity
• Disadvantage demands large number of iterations

Objective function
Optimization problem
Update direction
Back
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Distortion Measure K. Han et al. 2001
Measures

• Applies the application performance information
  to search for the optimum wordlengths
• Advantage Fewer number of iterations
• Disadvantage Not guaranteed to yield optimum
  wordlength for complexity

Objective function
Optimization problem
Update direction
Back
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Broadband Wireless Access Demodulator Simulation
Case Study
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Top-Level Simulation
Case Study
Back
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Tools for Fixed-Point Simulation
Introduction

• gFix (Seoul National University)
• Using C, operator overloading
• Simulink (Mathworks)
• Fixed-point block set 4.0
• SPW (Cadence)
• Hardware design system
• CoCentric (Synopsys)
• Fixed-point designer

gFix a(12,1) gFix b(12,1) gFix c(13,2) c a
b
float a float b float c c a b
Wordlengths determined manually Wordlength
optimization tool needed
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Wordlength Optimization Methods
Background

• Analytical approach
• Quantization error model
• Overestimates signal wordlength
• For feedback systems, instability and limit
  cycles can occur
• Difficult to develop analytical quantization
  error model of adaptive or non-linear systems
• Simulation-based approach
• Wordlengths chosen while observing error criteria
• Repeated until wordlengths converge
• Long simulation time

?
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Optimum Wordlength Search Methods
Search Methods

• Full search W. Sung and K. Kum 1995
• Min b bit search W. Sung and K. Kum 1995
• Max b bit search M. Cantin et al. 2002
• Hybrid search M. Cantin et al. 2002
• Sequential search K. Han et al. 2001
• Preplanned search K. Han et al. 2001
• Branch and bound search H. Choi and W.P.Burleson
  1994
• Simulated annealing search P.D. Fiore and L. Lee
  1999

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Procedure
Case Study

• Range estimation
• Find maximum and minimum values for each
• Find minimum wordlengths
• Defines starting wordlength values to use
• Iterative search
• Increase/decrease wordlengths one-by-one until
  meeting specification using one of the measures