A beta encoder is an

analog-to-digital conversion In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide ...

(A/D) system in which a

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

in the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...

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 Pulse-code modulation (PCM) is a method used to digitally represent analog signals. It is the standard form of digital audio in computers, compact discs, digital telephony and other digital audio applications. In a PCM stream, the amplitud ...

. As a form of

non-integer representation A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix ''β'' > 1, the value of :x = d_n \dots d_2d_1d_0.d_d_\dots d_ is :\begin x &= \beta^nd_n + \cdots + \beta^2d_2 + ...

, 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. 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'' used the

golden ratio base Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number \frac ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean b ...

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. An alternative design called the ''negative beta encoder'' (called so due to the negative

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

of the transition probability matrix) has been proposed to further reduce the quantization error.

References

{{Reflist Digital signal processing Signal processing

See also

References