It also has the advantage of being repeatable. Practically speaking, the sequence output by this completely deterministic device does a pretty good job of mimicking the behavior of an IID binary sequence. Second, the autocorrelation function of this sequence is nearly equal to that of a truly random binary IID sequence (again, it is as close as we can possibly get with a sequence of period 15 see Exercise 12.1).
First, the number of ones and zeros is almost equal (8 ones and 7 zeros is as close as we can get to “equally likely” with a sequence of length 15). The sequence generated by our LFSR does possess several desirable properties. If we are interested in generating longer sequences, we could construct a shift register with more stages so that the period of the resulting sequence would be sufficiently long that its periodicity would not be a concern. While periodicity is not a desirable property for a pseudorandom number generator, if we are interested in generating short sequences (of length less than 15 bits), then the periodicity of this sequence generator would not come into play. If the shift register were clocked longer, it would become apparent that the output sequence would be periodic with a period of 15 bits. For example, in addition to having the various bits in the sequence be statistically independent, we might also want 0s and 1s to be equally likely. So, to simulate an IID sequence of random variables, essentially we would like to create a computer program that will output a binary sequence with the desired statistical properties. Naturally, we seek to assign this task to a computer. If our application demanded a sequence of length 1 million, not many of us would have the patience to flip the coin that many times. One drawback to this approach is that it is very time consuming. To start with, suppose we would like to simulate a sequence of independent and identically distributed (IID) Bernoulli random variables, x 1, x 2, x 3, One way to do this would be to grab a coin and start flipping it and observe the sequence of heads and tails, which could then be mapped to a sequence of 1s and 0s. Miller, Donald Childers, in Probability and Random Processes, 2004 12.1.1 Binary Pseudorandom Number Generators This correlation process effectively filters out noise present in the data. The output of the random test procedure is a cross-correlation function corresponding to the unit impulse response of the system. Test duration must, however, be sufficiently long for the time averaging in the autocorrelation function to be accurate. Random signals represent the simultaneous injection of all frequencies into a system and thus lead to shorter test times than harmonic signals. White noise analysis has found application particularly in the electrophysiology domain for example, in the work of Marmarelis and Marmarelis (1978). The sequence of zero and unity pulses, which is cyclically repeatable, is such that over one cycle it has white noise properties. They often take the form of pseudorandom binary sequences, a sequence of pulses of equal duration and taking the values of zero (no disturbance for the pulse duration) or one (a pulse of fixed amplitude). Hence, in general, it is necessary to generate the random test signals specially. Such a situation is convenient, but in practice does not often occur. Random fluctuations approximating to white noise may occur spontaneously on one of the variables of the physiological system and they may be used as a test signal without introducing an external stimulus. The cross-correlation function then gives the unit impulse response. Random signal testing consists then of applying to the system a test signal approximating to white noise and then cross-correlating input and output.
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