Linear analog codes: The good and the bad

Analog error correction codes, by mapping real-/complex-valued source data to real-/complex-valued codewords, present generalization of the conventional digital error correction codes. This paper studies the theory of linear analog error correction coding. Since classical concepts of minimum Hamming distance and minimum Euclidean distance fail in the analog context, a new metric, termed the “minimum (squared Euclidean) distance ratio,” is defined. It is shown that the linear analog code that achieves the biggest possible minimum distance ratio also achieves the smallest possible mean square error (MSE) performance. Based on this achievability, a concept of “maximum (squared Euclidean) distance ratio expansible (MDRE)” is established for analog codes, similar in spirit to “maximum distance separable (MDS)” for digital codes. Code design and analysis reveal that the criteria of MDRE and MDS, although evaluated against different distance metrics, need not conflict each other, but can be effectively unified in the same code design. It is shown that unitary codes are best linear analog codes that simultaneously achieves MDRE and MDS; at the same time, however, nonlinear analog codes appear to outperform the best linear analog codes.