Binary sequence

Digital Signal Processor board fPSP) it is hosted into the PC and processes in real time the binary sequences stored into the acquisition board FIFO memories. The board processes arrival times and extracts the correlated AT generated by AE events. The PC picks up the data stored into the DSP memories and calculates the position of the AE sources. 

In Pig. 4-lc, both the input and the output from the coder are binary sequences, but if planes are spotted only rarely, then the output will contain many fewer binary digits than the input. The theory developed later will show exactly how many binary digits are required from the coder. The important point here is that a reduction is possible and that it depends on the frequency (or probability) of l s in the input data. 

Now let Nb be the smallest integer greater than or equal to N[H(X) + 8]. Since there are 2y binary sequences of length Nb, each of the L sequences satisfying Eq. (4-13) can be coded into a distinct binary sequence of length Nb. Since 8 was arbitrary, Nb/N can be made as close to H(U) as desired. Thus we have proved the following fundamental theorem. 

Theorem 4-2. Given an arbitrary discrete memoryless source, V, of entropy H U), and given any e > 0 and 8 > 0, it is possible to find an N large enough so that all but a set of probability less than e of the N length sequences can be coded into unique binary sequences of length at most [H(U) + h]N. 

An obvious restriction in selecting a set of variable length binary sequences as a code is that the sequence of source symbols must be uniquely specified by a sequence of code words. If, for example, we attempted to code the letters A, B, and C into the binary sequences 0, 1, and 00, it would be impossible to reconstruct the source letters. The binary sequence 00 could represent either AA or C. 

Necessity Let vlt- -,vy be a set of binary sequences satisfying the prefix condition, and consider these sequences as binary fraction expansions of real numbers between 0 and 1 (i.e., 1011 corresponds to the number 1 x 2-1 + 1 x 2-3 + 1 x 2-4). Then if vy has length np no other code word, can fall in the interval... 

As stated in the previous section, the major reactant feed was chosen as the manipulated variable. In the trial this feed was subjected to a pseudo-random binary sequence (PRBS) signal in an open loop operation of the process. The results of the trial, plotted in Fig. 2, show a strong -- but delayed -- cross-correlation between the manipulated feed rate and the reactor temperature. Using techniques described by Box and Jenkins (2), a transfer function relating the manipulated variable to the control variable of interest can be developed. The general form of this transfer function is... 

A very popular sequence of inputs is the pseudorandom binary sequence (PRBS). It is easy to generate and has some attractive statistical properties. See System Identification For Self-Adaptive Control, W. D. T. Davies, London, Wiley-Iflterscience, 1970. 

Now let us take a time period of detector signals large enough to encompass the length of the pseudo random binary sequence Injection code which produced It, and cross-correlate It with this Injection code of -1 and 1. 

A special kind of random noise, pseudo random noise, has the special property of not being really random. After a certain time interval, a sequence, the same pattern is repeated. The most suitable random input function used in CC is the Pseudo Random Binary Sequence (PRBS). The PRBS is a logical function, that has the combined properties of a true binary random signal and those of a reproducible deterministic signal. The PRBS generator is controlled by an internal clock a PRBS is considered with a sequence length N and a clock period t. It is very important to note that the estimation of the ACF, if computed over an integral number of sequences, is exactly equal to the ACF determined over an infinite time. 

The mapping of a watermark message m onto a sequence of watermark letter depends on the coded modulation technique. However, in all schemes considered, the message m is first mapped onto a binary sequence b, with one element hn G 0,1 for each host-data element. Fig. 8 depicts the block diagrams for the encoding of b into d for 4-aiy-CC-TCM,... 

E. Marinari, G. Parisi, and F. Ritort, Replica field theory for deterministic models. 1. Binary sequences with low autocorrelation. J. Phys. A 27, 7615-7646 (1994). 

Encode the binary sequence B in the coefficient sequence S using the hardcopy watermark procedure described in Section 3.1 below, and obtain the watermarked coefficients sequence X=(xi,..., X2n)- The procedure encodes binary sequences of length n in a coefficient sequences of length 2n until the sequence S=(si,..., S2n) is exhausted. 

Extract the binary sequence B from Y using the ML decoding procedure described in Section 4.2 below. The procedure extracts a binary sequence of length n from a coefficient sequence of length 2n. It is performed repeatedly for each block of 2n coefficients until the sequence Y is exhausted. 

Decode the binary sequence B, using the ECC decoding scheme, and obtain the estimated decoded message m. ... 

Binary sequences are encoded by their decimal equivalents ... 

Schroeder, 1970a] Schroeder, M. (1970a). Synthesis of low-peak-factor signals and binary sequences with low autocorrelation. IEEE Trans. Information Theory, IT-16 85-89. 

In this contribution, the experimental concept and a phenomenological description of signal generation in TDFRS will first be developed. Then, some experiments on simple liquids will be discussed. After the extension of the model to polydisperse solutes, TDFRS will be applied to polymer analysis, where the quantities of interest are diffusion coefficients, molar mass distributions and molar mass averages. In the last chapter of this article, it will be shown how pseudostochastic noise-like excitation patterns can be employed in TDFRS for the direct measurement of the linear response function and for the selective excitation of certain frequency ranges of interest by means of tailored pseudostochastic binary sequences. 

Pseudostochastic random binary sequences in combination with fast Fourier transform and correlation techniques avoid these problems and allow for a direct measurement of g(t) with high spectral power density and frequency multiplexing (stochastic TDFRS). Tailoring of the pseudostochastic sequences even allows for a selective enhancement and suppression of certain frequencies and, hence, of certain molecular species [74]. 

Pseudostochastic random binary sequences are noise-like time patterns. They are defined at times n A t and assume only two different values, corresponding to the grating amplitudes -1 and +1, if 180°-phase modulation is used for switching off the optical grating. Only software modifications, and no changes in the hardware of the TDFRS setup, are necessary in order to utilize pseudostochastic excitation sequences. The timing for heterodyne/homodyne separation is identical to the one already described for pulsed excitation. 

In the following, a general treatment of arbitrary binary excitation sequences will be given. Since the proper definition of the excitation and the response function is not unambiguously possible, a problem-independent notation will first be given, which will later be mapped to the actual experiment. For the moment, it is sufficient to picture a linear system with an input x(t), an output y(t) and a linear response function h(t), as sketched in Fig. 22. The input x(t) may be a pulse of finite duration, as discussed in the previous sections, or a pseudostochastic random binary sequence as in Fig. 22. 

In Ref. [75], it is discussed in more detail why it is advantageous to convolute the response of the temperature grating into the excitation and how to treat systematic errors arising from this approximation and from imperfections of the components in the setup. Especially the switching properties of the Pockels cell require careful analysis, since the switching number increases from 2 in case of pulsed excitation to approximately N in case of pseudostochastic binary sequences. 

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]...

Not every arbitrary random binary sequence is well suited. In the following, design criteria for sequences that serve particular requirements will be given. 

Maximum length binary sequences (MLBSs) of length N = 2l-l, where I is a positive integer, have a perfectly flat power spectrum [77]. The deconvolution in Eq. (61) can be computed very efficiently by means of a fast Hadamard transform, and they have, for example, been employed for Hadamard NMR spectroscopy [78]. 

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

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). 

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