Accelerating seismic computations using customized number representations on FPGAs |

Precision and range are key resources to be traded off against the performance of a computation. We looked at three different types of number representation: fixed-point, floating-point and logarithmic. Consider the case when is represented as a fixed-point number, with an integer part which is bits in length, and a fraction part which is bits in length.

The integer bit-width, which represents the dynamic range of the number, is calculated according to equation (1):

(1) |

For the floating-point number system, let represent a floating-point number , where is the sign bit, is the mantissa with a bit-width of bits, and is the exponent with a bit-width of bits.

The value of the mantissa is expressed as:

(2) |

where .

It is possible to relate the bit-width of the mantissa of the node to the error when representing the mantissa by a finite bit-width , as follows:

where is the value of the exponent at the node.

Since there is no standard to encode logarithmic numbers, in this report we use a fixed-point format to store the logarithmic value.

Accelerating seismic computations using customized number representations on FPGAs |

2007-09-18