Multidimensional Logarithmic Number System:Non Error-Free Integer Representations

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NonError-Free Integer Representations

Clearly, error-free representations are special cases of the MDLNS, but the extra degree of freedom provided by the use of multiple digits can mitigate the nonuniform quantization properties of the classical LNS.

To illustrate this, we present numerical results for mapping 10-bit signed binary input data to the two-digit 2DLNS where we treat the nonbinary base as a parameter. To demonstrate the ability to closely match input data with very small exponents, we have restricted the odd base exponent to 3-bits only. We are allowing the binary exponent to be unrestricted; however, owing to the 10-bit input range, the system automatically limits itself to 6-bits. We will see in the next section that this has very little bearing on the overall complexity of the inner product implementation (i.e., the hardware complexity is mainly driven by the dynamic range of the nonbinary exponents). As stated above, we require quantization errors to be <0.5 to match the quantization error of a binary representation. Table 84.1 shows the number of nonerror-free representations along with the worst quantization error for nonbinary bases in the set {3, 5, 7, 11, 13, 17, 47}.

The goal of applying this approximation scheme is to reduce as much as possible the size of the nonbinary exponent(s). For example, with a nonbinary base of 47, x = 01010011102 = 33410 is represented as 33410 = 01010011102 ® 2947-1 + 22547-3 = 334.082429. In this case we have used only three bits for the nonbinary exponents, that is they are restricted to the set {-4, -3, -2, -1, 0, 1, 2, 3}. Although a base of 47 only has 2 nonerror-free representations for a 10-bit signed range, it is possible to select a noninteger base that will provide completely error-free representations. We will see an example of this in Section 84.5.2.

To compare these results with an implementation using a classical LNS representation, we need to determine the number of bits of the logarithm to produce an absolute error of <0.5. A previous study [4] has found that we require n + log2(n) bits for the logarithm to achieve this accuracy for an n-bit positive number [26]. We have, in fact, checked this for the case of n = 9 (used in our two-digit 2DLNS study) and 12-bits are required for the logarithm to satisfy the same accuracy. If we assume that the hardware complexity of the classical LNS representation is driven by the number of bits in the logarithm, then we can see a potential for an enormous reduction in the implementation complexity of the two-digit 2DLNS versus the classical LNS.

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