Beta encoder
Encyclopedia
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.
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.
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
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 of the transition probability matrix) has been proposed to further reduce the quantization error.
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
in the unit interval
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...
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.
As a form of non-integer representation, beta encoding contrasts with traditional approaches to binary quantization
Quantization (signal processing)
Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...
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
Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary...
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 of the transition probability matrix) has been proposed to further reduce the quantization error.
See also
- Pulse code modulation
- Quantization (signal processing)Quantization (signal processing)Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...
- Sampling (signal processing)Sampling (signal processing)In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....