Block-based eigensystem of the 1±D and 1-D2 partial response channels

We find analytical expressions for the block-based input and output eigenvectors and eigenvalues of the systems with responses 1 D and 1 0'. The input eigenvectors form an orthonormal basis which is the optimum modulator for a channel with that transfer function. The output eigenvectors form an orthonormal basis with the same spectral nulls as the corresponding system. This basis can be used to produce line codes with spectral nulls. The eigenvectors are sinusoids. Manuscript received November 14, 1991; revised September 20, 1993. This work was supported by the Natural Sciences and Engineering Research Council of Canada. This paper was presented in part at the Twenty-ninth Annual Allerton Conference, October 1991. A. K. Khandani is with the Department of electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. P. Kabal is with the Department of Electrical Engineering, McGill University, 3480 University, Montreal, Quebec, Canada H3A 2A7. IEEE Log Number 9404920. 0018-9448/94$04.00 O 1994 IEEE 1646 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5, SEPTEMBER 1994 Fig. 1. Block diagram of the transmission system. M C This reduces the computational complexity by allowing for fast trans(b) form algorithms to perform the modulation for a block of data. Fig. 2. (a) Block diagram of the line coder. (b) An equivalent form for (a).