A linear algebra approach to minimal convolutional encoders

This semitutorial paper starts with a review of some of Fomey’s contributions on the algebraic structure of convolutional encoders on which some new results on minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Fomey’s definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic e n d k g mutrix (minimal overall constraint length over equivalent basic e n c d i matrices), minimal encoding mutrix (minimal number of abstract states over equivalent encoding matrices), and minimal encoder (realization of a minimal encoding matrix with a minimal number of memory elements over all realizations). Among other results, it is shown that all minimalbasic encoding matrices are minimal, but that there exist (basic) minimal encoding matrices that are not minimal-basic! Several equivalent conditions are given for an encoding matrix to be minimal. It is also proven that the constraint lengths of two equivalent minimal-basic encoding matrices are equal one by one up to a rearrangement. All results are proven using only elementary h e a r algebra. Most important among the new results are a simple minimality test, the surprising fact that there exist basic encoding matrices that are minimal but not minimal-basic, the existence of basic encoding matrices that are nonminimal, and a recent result, due to Fomey, that states exactly when a basic encoding matrix is minimal.