Multidimensional Logarithmic Number System:Binary to MDLNS Conversion

Binary to MDLNS Conversion

A logarithmic representation, a, of the number, x, is given by the relationship in the following equation, where s is the sign of the number and the base is r (usually 2):

The MDLNS provides more degrees of freedom than the LNS by use of the orthogonal bases and the ability to use multiple digits. However, these extra features introduce new complexities in the binary conversion process. The binary-to-LNS conversion process is simplified owing to the monotonic relationship between x and a. Unfortunately, this solution is not applicable to the MDLNS since there is no monotonic relationship between x and the multiple digits/bases.

The technique initially proposed for binary-to-MDLNS conversion used simple LUTs [14]; however, the input data range for the target application area of video processing was only 8 bits. Although an LUT offers a simple and fast binary-to-MDLNS conversion scheme, the size of the LUTs required is exponentially dependent on the input binary dynamic range. The LUT sizes further depend on the number of digits and bases in the MDLNS representation. For example, an error-free (where the absolute representation error is <0.5) 12-bit unsigned range with 3-digits and 2-bases [2] would require a direct mapping LUT of the size 4096 X 33 or 136 kbit, a reasonably sized VLSI component. However, if an error-free 23-bit unsigned range were needed, the mapping LUT would be 8388608 X 48 or 403 Mbit, which is not reasonable at all for VLSI fabrication.

In this section we will describe several hardware implementation techniques for converting a binary representation into a 2DLNS representation. The techniques are based on the reversal of a previously published MDLNS-to-binary converter [14] with the aid of a new memory device [29].

To simplify the presentation of the binary-to-MDLNS process we will restrict ourselves to a subset of the MDLNS with only two bases (an n-digit 2DLNS representation) and we will assume that the exponent of the second (or nonbinary) base has a predefined finite precision (equivalent to limiting the number of bits of precision in a classical LNS). The simplified representation of an input, x, as an n-digit 2DLNS is shown in the following equation:

The second base, D, is a suitably chosen number (relatively prime to 2), si Î {-1, 0, + 1}, and the exponents are integers. R is the number of bits of the second base exponent (i.e., bi Î {–2R–1,…, 2R–1 – 1}) and it directly affects the complexity of the MDLNS system. The precision of the binary exponent is B bits (i.e., ai Î {–2B–1,…, 2B–1 – 1}). Unlike R, B does not directly affect the complexity of the system.