In this paper we provide a rigorous proof that feedback cannot increase the capacity of the channel with additive colored gaussian noise by more than a factor of two. We also give a tighter bound showing that any increase in capacity is less than the normalized correlation between the signal and noise. It is further shown that gaussian signals and linear feedback processing will achieve capacity. The practical implications are that (i) feedback should be used to simplify encoding and decoding since there is little to be gained in the way of increased capacity and (ii) the various proposed schemes which use linear feedback are doing the correct thing.

The cascade and parallel realizations of an arbitrary digital filter are both formed using second order sections as building blocks. This simple recursive filter is commonly implemented using 2's complement arithmetic for the addition operation. Overflow can then occur at the adder and the resulting nonlinearity causes self-oscillations in the filter. The character of the resulting oscillations for the second order section are here analyzed in some detail. A simple necessary and sufficient condition on the feedback tap gains to insure stability, even with the presence of the nonlinearity, is given although for many desired designs this will be too restrictive. A second question studied is the effect of modifying the “arithmetic” in order to quench the oscillations. In particular it is proven that if the 2's complement adder is modified so that it “saturates” when overflow occurs, then no self-oscillations will be present.

We derive a new family of convolutional character-error-correcting codes which are a convolutional form of the Reed-Solomon block codes and as such have nonbinary symbols. We also derive a bound on the error correcting capabilities of these codes in which the error-correcting capability per constraint length grows approximately with the square root of the constraint length. When these codes are used on a binary channel they are effective for both random and burst error correction because a single character spans several channel digits. These codes have greater error-correcting capabilities than the Robinson-Bernstein self-orthogonal codes but are harder to decode. The single-character-error-correcting codes, when interleaved, are shown to be more powerful than the equivalent Hagelbarger code and appear to be simpler to implement. They are also slightly better than the interleaved version of Berlekamp's code. We discuss encoding and decoding algorithms and illustrate a simple decoding algorithm for some of the codes. These codes are closely related to the Bose-Chaudhuri-Hocquenghem block codes and share with them the decoding simplification for character erasures in place of errors. Any Bose-Chaudhuri-Hocquenghem decoding algorithm can be used to decode these codes.

A simple proof of the fact that a frequency modulation process can not be bandlimited is given.