The Logarithmic Number System (LNS) has area and power advantages over fixed-point and floating-point number systems in some applications that tolerate moderate precision. LNS multiplication/division require only addition/subtraction of logarithms. Normally, LNS is implemented with ripple-carry binary arithmetic for manipulating the logarithms; however, this paper uses carry-free residue arithmetic instead. The Residue Logarithmic Number System (RLNS) has the advantage of faster multiplication and division. In contrast, RLNS addition requires table-lookup, which is its main area and delay cost. The bipartite approach, which uses two tables and an integer addition, is introduced here to optimize RLNS addition. Using the techniques proposed here, RLNS with dynamic range and precision suitable for MPEG applications can be synthesized. Synthesis results show that bipartite RLNS achieves area savings and shorter delays compared to naive RLNS.
The Logarithmic Number System (LNS) has lower power and larger dynamic range than fixed point, which makes LNS suitable for designing low-power, portable devices. Motion estimation is a key part of the MPEG encoding system. This paper introduces LNS into motion estimation for the MPEG encoding system. The block matching technique is the most commonly used motion-estimation method in MPEG encoding. The Mean Absolute Difference (MAD) is an inexpensive fixed-point cost function, which uses the sum of the absolute difference of the pixel values in the reference and encoded frames. Since LNS addition and subtraction are expensive, we propose the quotient of the two pixels' values instead of the difference. LNS division only needs a fixed-point subtractor. Similar to the absolute difference, we take the quotient of the larger value over the smaller value. We call this new cost function Mean Larger Ratio (MLR). The product of such ratios is calculated for each of the macroblocks in MPEG frames. Using MLR, LNS has approximately the same hardware as MAD for fixed point. Example videos show MLR provides a practical cost function to perform motion estimation with LNS.