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The first of a new generation of high performance X-ray computed tomographic (CT) machines, the Dynamic Spatial Reconstructor, imposes a requirement for digital signal processing rates which are 3–4 orders of magnitude greater than the capability of current X-ray computed tomography processors. To solve the large-scale computational problems for this and similar CT units which are currently under development, three candidate arithmetic implementations of ultra-high-speed convolutional filtering and weighted linear summation algorithms have been developed and compared. Since both convolution and weighted summation are performed via the inner product operation, which is the basis for most digital signal processing algorithms, the results are widely applicable. The three arithmetic approaches are a two's complement modular array, a merged arithmetic module, and a sign/logarithm convolver. A figure of merit, which relates processing speed to complexity, is used to compare the three arithmetic approaches. It is demonstrated that processing rates in the billions of multiply-add operations per second may be realized with special-purpose processors of moderate complexity.