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Multiple-Base Number System

Theory and Applications

Multiple-Base Number System by Vassil Dimitrov
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Technology, Applications, and Computation

Ancient Roots

Analog or Digital?

Where Are We Now?

Arithmetic and DSP

Discrete Fourier Transform (DFT)

Arithmetic Considerations

Convolution Filtering with Exact Arithmetic


The Double-Base Number System (DBNS)

Motivation

The Double-Base Number System

The Greedy Algorithm

Reduction Rules in the DBNS

A Two-Dimensional Index Calculus


Implementing DBNS Arithmetic

Arithmetic Operations in the DBNS

Conversion between Binary and DBNS Using Symbolic Substitution

Analog Implementation Using Cellular Neural Networks


Multiplier Design Based on DBNS

Multiplication by a Constant Multiplier

Using the DBNS

DBNS Multiplication with Subquadratic Complexity

General Multiplier Structure

Results and Comparisons

Some Multiplier Designs

Example Applications


The Multidimensional Logarithmic Number System (MDLNS)

The Multidimensional Logarithmic Number System (MDLNS)

Arithmetic Implementation in the MDLNS

Multiple-Digit MDLNS

Half-Domain MDLNS Filter


Binary-to-Multidigit Multidimensional Logarithmic Number System Conversion

Single-Digit 2DLNS Conversion

Range-Addressable Lookup Table (RALUT)

Two-Digit 2DLNS-to-Binary Conversion

Binary-to-Two-Digit 2DLNS Conversion

Multidigit 2DLNS Representation (n > 2)

Extending to More Bases

Physical Implementation

Very Large-Bit Word Binary-to-DBNS Converter


Multidimensional Logarithmic Number System: Addition and Subtraction

MDLNS Representation

Simple Single-Digit MDLNS Addition and Subtraction

Classical Method

Single-Base Domain

Addition in the Single-Base Domain

Subtraction in the Single-Base Domain

Single-Digit MDLNS Addition/Subtraction

Two-Digit MDLNS Addition/Subtraction

MDLNS Addition/Subtraction with Quantization Error Recovery

Comparison to an LNS Case


Optimizing MDLNS Implementations

Background

Selecting an Optimal Base

One-Bit Sign Architecture

Example Finite Impulse Response Filter

Extending the Optimal Base to Three Bases


Integrated Circuit Implementations and RALUT Circuit Optimizations

A 15th-Order Single-Digit Hybrid DBNS Finite Impulse Response (FIR) Filter

A 53rd-Order Two-Digit DBNS FIR Filter

A 73rd-Order Low-Power Two-Digit MDLNS Eight-Channel Filterbank

Optimized 75th-Order Low-Power Two-Digit MDLNS Eight-Channel Filterbank

A RISC-Based CPU with 2DLNS Signal Processing Extensions

A Dynamic Address Decode Circuit for Implementing Range Addressable Look-Up Tables


Exponentiation Using Binary-Fermat Number Representations

Theoretical Background

Finding Suitable Exponents

Algorithm for Exponentiation with a Low Number of Regular Multiplications

Complexity Analysis Using Exponential Diophantine Equations

Experiments with Random Numbers

A Comparison Analysis

Final Comments

CRC Press; January 2012
292 pages; ISBN 9781439830475
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Title: Multiple-Base Number System
Author: Vassil Dimitrov; Graham Jullien; Roberto Muscedere
 
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