Shift register sequences

There is a wealth of research that has been done on shift transformations. One of the best references is Golomb s classic Shift Register Sequences [golom67]. 

S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, 1967. 

The shift-register sequences can be parameterized through the choice of a,. One can systematically assign the values of a, to the processors to produce distinct maximal period shift-register sequences [24]. This scheme can be justified based on exponential sum bounds, as in the case of the prime modulus LCG. This similarity is no accident, and is based on the fact that both generators are maximal period linear recursions over a finite field... 

How are such masks constructed There are three known ways to construct a cyclic S-matrix. They are described in detail in Section A.2 of ref. 2. Here we shall just sketch the most important construction, which is based on maximal length shift-register sequences. This produces cyclic S-matrices of orders n = " - 1 for all integers m, i.e., orders n = 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, . ... 

S.W. Golomb, Shift-Register Sequences and Spread-Spectrum Communications, in Third International Symposium on Spread Spectrum Techniques and Applications, 1994. IEEE ISSSTA 94, IEEE, 1994, pp. 14-15. 

MLBSs are generated by a shift register of length L with feedback as shown in Fig. 27. The bit position to which the output is added depends on L. Initially, all bits are loaded with 1. At every clock cycle, the register contents is shifted to the right, the rightmost bit is applied to the output and the result of the modulo 2 addition is fed into the leftmost bit. The output sequence is repeated after 2L-l steps. For practical purposes, the elements bj. of the MLBS can be computed from a recursion formula ... 

Fig. 27.3-bit shift register to generate a maximum length binary sequence of length 23-1=7. The summation is modulo 2... 

An alternative method for excitation of nuclei over a range of chemical shifts is by irradiation with a weak, noise-modulated radio-frequency, instead of with strong r.f. pulses. In one realization of this method, protons were irradiated with repetitive sequences of noise that was truly random,162 and, in another,163 fluorine nuclei were excited by pseudo-random noise generated by amplitude modulation of the r.f. with maximum-length sequences of pulses from a computer or shift register (a series of flip-flop devices connected by feedback loops). With the carrier wave suppressed, the latter process is equivalent to phase modulation of the r.f. by+7r/2 radians when the pulse is turned on, and by —ir/2 radians when it is turned off. This method is identical with that used in most broadband, heteronuclear, noise decouplers, except that greater power is required for decoupling. 

Parameterized Iteration. Just as the disjoint cycles can be parameterized, so can many of the iteration functions for the internal state as with the generalized feedback shift register described in the next section. Here, sequence i gets iteration function 7]. 

Incidentally the output sequence from one of these shift registers is called a pseudo-noise sequence or m-sequence. These have been extensively studied and have many applications--see for example refs. 46-49. 

A binary pseudorandom, excitation is the simplest to generate signal [7]. It can be formed directly from a clock signal with frequency /elk using the sequence of k trigger circuits (oo to flk i) constituting a shift register... 

Fig. 27a). The shift register is feedbacked through an exclusive OR (XOR) logic element. The number of produced random pulses is N = 2 — 1. This pattern is a period of the maximum length sequence (MLS) with duration Tmls = N//cik (Fig. 27b). The MLS has a continuous RMS spectral density expressed via the sinus cardinalis function (lsin(pf)/(pf)l) see Fig. 27c. It has enough power up to the frequency, at which the power spectral density PSD still retains its 50 % level ( 3 dB RMS level). 

FIGURE 12.79 Binary shift register PN sequence generator. 

Convolutional codes Codes generated by a linear shift-register circuit that performs a convolution operation on the information sequence. 

Figure 5.9 (a) Shift register in a While loop (b) Sequence local in a stacked sequence structure. 

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