By Richard W. Hamming
This introductory textual content examines electronic filtering — the procedures of refining signals— and its relevance to many functions, relatively computer-related services. Assuming just a wisdom of calculus and a few data, it concentrates on linear sign processing, with a few attention of roundoff results and Kalman filters. quite a few examples and workouts.
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Additional info for Digital Filters (3rd Edition)
On the other hand, one does not have to draw all the inference arrows and intermediate terms that are possible, but only those that are relevant for the ﬁnal result. The problem with too much detail should be handled at a different level. The diagrammatic proof should be seen for what it is, as a proof, so it must be unambiguous and show all the information that is necessary to easily convince the observer that the stated fact do indeed hold. After the proof is done, then one need only to display the part of the diagram that is interesting for the present purpose.
J. (1980). Correctness Preserving Program Reﬁnements: Proof Theory and Applications, volume 131 of Mathematical Center Tracts. Mathematical Centre, Amsterdam. -J. (1988). A calculus of reﬁnements for program derivations. Acta Informatica, 25:593–624. [Back, 1991] Back, R. J. (1991). Reﬁnement diagrams. In Morris, J. M. and Shaw, R. C. , editors, Proceedings of the 4th Reﬁnement Workshop, Workshops in Computer Science, pages 125–137, Cambridge, England. Springer-Verlag. -J. (2002). Software construction by stepwise feature introduction.
Thus, subset inclusion is the pointwise extension of implication, and relation inclusion is the pointwise extension of subset inclusion. Reﬁnement is also the pointwise extension of subset inclusion, to a different domain than relations. Finally, contract ordering is the pointwise extension of the reﬁnement ordering. The reﬁnement calculus hierarchy also contains other, more exotic lattices, which we will not describe in detail here. The reﬁnement calculus hierarchy contains, in addition to the lattice operations, also other operations that are deﬁned on these domains.