Application of Orthonormalized M-Sequences for Data Reduced and Error Protected Transmission of Pictures

Abstract

With help of n-stage feedback shiftregisters it is possible to generate periodical pulse-trains of maximum length. By orthonormalizing these pulse' trains you get a deterministic ortho­ normal binary function-system, which allows to realize a signaltransform (M-transform), similiar to the Fourieror Walshtransform. Some details of the m-functions and the M-transform are represented in the following. The M-transform is applied to non-moving greylevel-pictures. Procedures for data reduction in the M-spectrum and consequences of channel noise are discussed. The technical realization of the M-transform is very easy with help of new integrated circuits. 1. ONE-DIMENSIONAL m-FUNCTTONS 1.1 Generation of m-Sequences With help of n-stage feedback shiftregisters it is possible to generate periodical two-level pulse-trains with maximum lenght /1/. The pulse-trains are called m-sequences (maximum-lengthsequences). Their period is p = 2n1. (1.1) These m-sequences are described by Golomb for the first time in 1962 /1/. In the meantime they are applied to coding of signals /2/ and to stochastic computing /3/. Figure 1 shows a four-stage feedback shiftregister, which produces m-sequences of period p = 2^-1 = 15. clock Fig. 1: Four-stage feedback shiftregister for the generation of m-sequences. The m-sequence, generated by the register in fig. 1, is 1 o o o 1 o o 1 1 o 1 o 1 1 1 1 o ( 1 . 2 ) To realize a transform similiar to Fourieror Walshtransform it is neces­ sary to orthonormalize the m-sequences. 1.2 One-Dimensional m-Functions The orthonormalization of m-sequences with p = 15 is described in the follow­ ing. The proof for orthonormalization of m-sequences with any possible period is shown in /4/. A system of orthonormalized m-sequences (m-functions) is developed by carry­ ing out the following points: a) By shifting a m-sequence cyclically for p times you get a set of p pulse-trains with amplitudes 0 and 1 (fig. 2) b) The amplitudes of the m-sequences are now called A and B (1 -*• A, 0 B) (fig. 3) c) By orthonormalizing this functionsystem (with amplitudes A and B) it is possible to calculate the ampli­ tudes A and B. The condition for orthonormalization reads: CH1538-8/80/0000-0282$00.75 © 1980 IEEE 282 / ^ (0) ml (0)d0= { 0 k + 1 (1l3) Then the equations for calculating A and B in fig. 3 run as follows: 8_ A 2 + 7_ 2 = 1 15 A + 15 B n 3 B 2 + 8 A B + 4 A 2 = 0 (1.4) . I 0 0 0 I 0 0 I I 0 1 0 I I I i n n n 0 / / / _L Ji_ru i