By Gary Gygax
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Extra resources for Dungeon Geomorph Assortment: Set 1-3 Basic Dungeons, Caves & Caverns, Lower Dungeons (Advanced Dungeons & Dragons AD&D)
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Though this approach is common, it is not without difficulties, as additions can still cause overflows and sometimes filter coefficients require representations of values larger than one. 15, its value only determines the sign of the represented number, so it is essentially only a sign bit. It is somewhat of a reversal of intuition to think in terms of keeping the signal near a magnitude of 1 (but not over) to preserve precision in the fraction bits when you compare this to the traditional integer approach taken in a typical C program, where lower bits are never lost, but care must be taken to avoid producing values too large for the datatype, and effectively, the most significant bits are always discarded.
It is exactly this behavior in C that makes it difficult to express a multiply operation that starts with two low precision operands and produces a double precision result. The definition of C [14] requires the operands to be cast to the type of the result before the multiply occurs, which is quite inefficient in hardware. 31 format would be sufficient. 31 requires only a left shift of one bit, and the operation may be required often, some DSP μP’s provide a mode or instruction allowing the left shift by one bit of every product.
Since many signal processing tasks are computed on blocks of data to minimize loop overhead, significant improvements in computational accuracy can be obtained by determining the maximum value in a block of data, and then scaling all the values up (usually by a left shift) to the maximum range without overflow. The shift represents a power of 2, and also a scaling exponent, just as in floating–point, but there is only a single exponent for the entire block. Therefore, the values in the block can still be used in computations as fixed–point values and many more bits of precision are available for small values when they are scaled up to occupy the most significant bits of the data word.