Multidimensional Logarithmic Number System:Applying an Optimal Base to Input Data

Applying an Optimal Base to Input Data

We have seen how applying an optimal base to the coefficients of a digital filter can significantly increase the accuracy of the mapping to 2DLNS. This same improvement can be seen when applied to the input

data of the filter; however, the nonbinary bases must be the same for both data and coefficients for the 2DLNS arithmetic to operate properly. For implementation efficiency the sum of the corresponding coefficient and input data indices should never overflow while performing a 2DLNS multiplication. We can guarantee that this situation will never happen by limiting the range on both the coefficient and data exponents. For example, with the earlier one-digit optimal base case, the nonbinary exponent range is 9 bits or [-256, 255]. To avoid overflow we could limit the range on the two-digit input data mapping nonbinary exponent from [-32, 31] (or 6 bits), which would require a change to the range of the coefficients to [-224, 224] to prevent overflow on the sum (9 bits only). Changing these limits requires that we recalculate the optimal base, which results in slightly lowering the stop-band attenuation of 77.741 dB from the original value of 80.315 dB. Our range of the input data nonbinary exponent influences the input data mapping. In this case we are unable to achieve a complete error-free mapping. Table 84.12 summarizes the results from three different input ranges for an optimal base selected from the coefficients. Another approach to limit the number of nonerror-free input representations is to find the optimal base for the inputs themselves. Doing this will significantly limit the stop-band attenuation as the coefficients are single digits and mapped using the optimal base from the input data. Table 84.12 also includes results from this approach. Clearly, for the one-digit coefficient/two-digit input data case one has to select what should be prioritized.

We will explore the same scenario with the two-digit case where the coefficient nonbinary exponent range is [-4, 3]. Since this range is very small, we need only limit the input data nonbinary exponent ranges, initially, to [-28, 28] so that the range of the sum is [-32, 31] (or 6 bits). Our stop-band attenuation does not change from 81.562 dB; however, the data mapping is far from being error-free. Table 84.13 summarizes the results from the same three input ranges as in the previous case. To try to improve the input data mapping, the nonbinary exponent ranges can be increased to [-60, 60] which results in better mapping (see Table 84.13), but larger LUTs in the inner product processor. The alternative, again, is to optimize the base for the input data. In the single-digit case earlier, the stop-band attenuation suffered as a result of the coefficients not being mapped as well as before. To avoid this, we can increase the range on the coefficients to [-8, 7]; therefore, providing more possible representations and reducing

the input data nonbinary exponent range to [-24, 24] to maintain the same dynamic range at the output of the adder as before. This approach allows a completely error-free mapping for an input range of [–8192, 8191] with a simpler converter, but limits the size of the inner product LUTs. Table 84.13 summarizes the results for other ranges. For two-digit input data and coefficient architectures, this appears to be the best approach for minimizing hardware and computational error.