Optimized Random Binary Sequences

Truly random binary sequences do not show the time-localized echoes of MLBSs shown in Fig. 28. The perturbation is spread instead as additional noise over the entire time window, which is less critical since there is no systematic deformation of g(t). The noise amplification U of an arbitrary sequence is, however, unacceptably high. In the following, it will be shown how random sequences with low noise amplification and without the problems of MLBSs can be constructed. 

Mapping to Ising model An alternative view of the problem is obtained, when the pseudostochastic random binary sequence is interpreted as a spin-1/2 system. Every element x[k] corresponds to a spin sk that points either up or down (Fig. 29). 

The energy E of this spin system may be defined as the integral noise amplification U as given in Eq. (70), which can be expressed as a function of the pair products of all spins  

The coupling constants are the Fourier coefficients = exp(- 2 nikl/N). The task of finding a sequence with a low noise amplification is equivalent to the one of finding a minimum in the energy of all possible 2N spin configurations. 

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Fig. 26. Memory function g(t) as obtained from stochastic TDFRS with an optimized random binary sequence. The inserts show a short sequence of the excitation and the corresponding heterodyne response. From Ref. [75]...

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