Quantization and Roundoff Abstract science or problems of everyday life

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Quantization and RoundoffAbstract science or
problems of everyday life?

• István Kollár
• Budapest University of Technology and Economics
• Department of Measurement and Information Systems

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Outline

• Quantization in life
• Sampling vs. Quantization
• Basic results for uniform quantization
• Dithering
• A few paradoxes
• Quantization in floating-point
• Analysis of algorithms

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Quantization everywhere

• What time is it? 1124
• I buy 100 grams of ham
• I am 55 years old (NOT 54.8 years)
• Easter
• Easter Sunday is the Sunday following the full
  moon after the March Equinox (not quite true, but
  approximately)
• The quantum size is inherently given
• Relative vs. absolute error

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Quantized vs. Continuous

• We are not interested here in quantized nature of
  time or of space
• Consider quantized quantity representing a
  continuous one
• Look closer than good enough
• Sampling vs. Quantization

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Sampling vs. Quantization II.
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Sampling
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Condition never fulfilled!

• No signal is time and bandlimited
• Transients
• Periodic signals (coherent sampling?)
• Stochastic signals
• The key look for information content in sampled
  signal

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Quantizer
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Quantizing Theorem

• If the CF of x is bandlimited, so that

then the replicas contained in will
not overlap the CF of x can be derived from the
CF of x the PDF of x can be derived from the
PDF of x.
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Fulfilment

• No, in theory never
• Amplitude-limited usually it is
• CF-limited never
• Approximations smooth enough, to provide
  approximate CF-limitedness
• No perfect reconstruction of signal!

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Counterexample
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Quantization Noise

• Noise model
• Independent
• Uniformly distributed
• White

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Quantizing Theorem III/A

• If the CF of x fulfils

then the quantization error will be uniform. Is
this enough for us?
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Noise modulation
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Dither
Anti-alias filtering bandlimit input
signal Dither bandlimit characteristic function
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Subtractive vs. Non-subtractive dither
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Surprize (?)
Three mutually independent variables can be NOT
independent as 3 variables
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Uniform dither (Fleetwood Mac Tusk)
Power changes unpleasant
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Solution

• Add triangularly-ditributed noise as dither

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Marker-based movement analysis (pictures after
Jobbágy, Ákos)
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Measurement by using CCD camera
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Determine position of centre
Defocusing of the camera improves position
estimate
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Block-float (fixed-point) FFT errors
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8-point FFT
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Fixed-point FFT with noise model
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Welchs figure for FFT of sine
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Why coherently sampled sine is special
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Maximum error (after Vilmos Pálfi)
7 dB loss in dynamic range
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Error behavior for incoherenly sampled sine
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Signals with the same PDF
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Possibility to measure open-loop error
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Correction use of inverse of algorithm
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Example condition number OK, result not (after
Bart de Moor)
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Evaluation
Theoretically
Numerical results
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Conclusions

• BE CAREFUL!
• Common sense works well except for us (Murphy)
• Tools are available to investigate