Systematic Wavelet Codes: Designed and Decoded by DFT Vectors
G. Robert Redinbo · IEEE Transactions on Communications · 2018
A class of complex-valued convolutional codes, wavelet codes, that encode numbers into numbers, is constructed by assigning consecutively indexed discrete Fourier transform (DFT) vectors to the parity-checking requirements. For a rate k/n code with constraint length m, the parameters must satisfy (m + 1)(n-k)<;n. The algebraic structures are very similar to the finite field convolutional codes but the decoding procedures operate over the complex numbers. These wavelet codes are effective against the impulse errors even when round off noise due to computational operations is allowed too. When the groups of impulsive errors are separated by the constraint length m, the errors can be corrected by extracting from the convolutional code's syndromes, new syndromes associated with a DFT block code inherent in the code's algebraic structure. A modified Berlekamp-Massey algorithm computes the error locations and the correcting values are determined by an inverse DFT operation. Systematic forms of these codes are developed by manipulating the original generating transfer function. Each systematic form is a subcode of the original wavelet code from which it was determined. Moreover, this form may be decoded by DFT code methods too. Extensive simulation results verify the power and viability of both forms of the codes and their decoding techniques.