beta encoder
A beta encoder is an analog-to-digital conversion (A/D) system in which a real number in the unit interval is represented by a finite representation of a sequence in base beta, with beta being a real number between 1 and 2. Beta encoders are an alternative to traditional approaches to pulse-code modulation.
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|last1=Du |first1= Ke-Lin
|last2=Swamy |first2= M. N. S.
|year=2010
|isbn=978-0-521-11403-5
|page=483
|publisher=Cambridge University Press |doi=
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|last1= Daubechies |first1=I.
|author-link=Ingrid Daubechies
|last2= Yilmaz |first2= O.
|title= Robust and Practical Analog-to-Digital Conversion With Exponential Precision
|journal= IEEE Transactions on Information Theory
|year= 2006
|volume= 52 |issue= 8 |pages= 3533–3545
|doi=10.1109/TIT.2006.878220
|s2cid=15166166
}}
As a form of non-integer representation, beta encoding contrasts with traditional approaches to binary quantization, in which each value is mapped to the first N bits of its base-2 expansion. Rather than using base 2, beta encoders use base beta as a beta-expansion.London Mathematical Society lecture note series, Volume 312 by Felipe Cucker, London Mathematical Society 2004 {{ISBN|0-521-54253-7}} page 23.
In practice, beta encoders have attempted to exploit the redundancy provided by the non-uniqueness of the expansion in base beta to produce more robust results. An early beta encoder, the Golden ratio encoder{{cite journal|last1=Daubechies|first1=Ingrid|last2=Gunturk|first2=C. Sinan|last3=Wang|first3=Yang|last4=Yilmaz|first4=Özgür|title=The Golden Ratio Encoder|journal=IEEE Transactions on Information Theory|volume=56|issue=10|year=2010|pages=5097–5110|issn=0018-9448|doi=10.1109/TIT.2010.2059750|arxiv=0809.1257|s2cid=8513029}} used the golden ratio base for its value of beta, but was susceptible to hardware errors. Although integrator leaks in hardware elements make some beta encoders imprecise, specific algorithms can be used to provide exponentially accurate approximations for the value of beta, despite the imprecise results provided by some circuit components.
{{Citation
|last= Ward |first=Rachel
|year=2008
|title= On Robustness Properties of Beta Encoders and Golden Ratio Encoders
|journal=IEEE Transactions on Information Theory
|volume=54 |issue=9 |pages= 4324–4334
|doi=10.1109/TIT.2008.928235
|arxiv=0806.1083|s2cid=12926540
}}
An alternative design called the negative beta encoder (called so due to the negative eigenvalue of the transition probability matrix) has been proposed to further reduce the quantization error.
{{cite arXiv
|first1=Tohru |last1= Kohda
|first2= Satoshi |last2= Hironaka
|first3= Kazuyuki |last3= Aihara
|date=2009
|title= Negative Beta Encoder
|class=cs.IT
|eprint=0808.2548
}}