_{1}

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Due to non-ideal coefficients of the adaptive equalizer used in the system, a convolutional noise arises at the output of the deconvolutional process in addition to the source input. A higher convolutional noise may make the recovering process of the source signal more difficult or in other cases even impossible. In this paper we deal with the fluctuations of the arithmetic average (sample mean) of the real part of consecutive convolutional noises which deviate from the mean of order higher than the typical fluctuations. Typical fluctuations are those fluctuations that fluctuate near the mean, while the other fluctuations that deviate from the mean of order higher than the typical ones are considered as rare events. Via the large deviation theory, we obtain a closed-form approximated expression for the amount of deviation from the mean of those fluctuations considered as rare events as a function of the system’s parameters (step-size parameter, equalizer’s tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur.

In this paper, we deal with the convolutional noise arising at the output from a blind deconvolutional process. A blind deconvolution process arises in many applications such as seismology, underwater acoustic, image restoration and digital communication [

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory [

In this paper we address indirectly those fluctuations of the convolutional noise that are larger than the typical ones, which occur very rare. Namely, we consider the fluctuations of the arithmetic average (sample mean) of the real part of consecutive convolutional noises which deviate from the mean of order higher than the typical fluctuations. As already mentioned, typical fluctuations are those fluctuations that fluctuate near the mean, while the other fluctuations that deviate from the mean of order higher than the typical ones are considered as rare events. Via the large deviation theory, we obtain a closed-form approximated expression for the probability that these rare events may occur as a function of the step-size parameter, equalizers’s tap length, SNR, input signal statistics, characteristics of the chosen equalizer (based on a cost function where the error of the equalized output can be expressed as a polynomial function of order up to three), channel power and the amount of deviation from the mean. Based on this new expression we are able to evaluate approximately the amount of deviation from the mean of those fluctuations considered as rare events as a function of the system’s parameters (step-size parameter, equalizers’s tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur.

The paper is organized as follows: after having described the system under consideration in Section 2 we evaluate in Section 3 approximately the amount of deviation from the mean of those fluctuations considered as rare events as a function of the systems parameters (step-size parameter, equalizerss tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur. Section 4 is our conclusion.

The system under consideration is illustrated in

1) The input sequence

2) The unknown channel

3) The equalizer

4) The noise

where

The transmitted sequence

where “

where

[

where

cost function that characterizes the intersymbol interference, see [

respect to the equalizer parameters will reduce the convolutional error. In the following we deal with those

equalization methods where

three as is in the case of [

In this section we obtain a closed-form approximated expression for the amount of deviation from the mean of those fluctuations considered as rare events as a function of the system’s parameters (step-size parameter, equalizers’s tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur.

Theorem 1. For the following assumptions:

1) The convolutional noise

2) The source signal

zero. Namely,

3) The convolutional noise

4)

order three.

5) The gain between the source and equalized output signal is equal to one.

6) The convolutional noise

The amount of deviation

where the probability that the fluctuations of the arithmetic average (sample mean) of the real part of consecutive convolutional noises which deviate from the mean of order higher than the typical ones is expressed by:

and

x_{r} is the real part of

and

where

Proof. Let us first define:

Next, by using assumption 1 from this section, we may write:

The propability density function (pdf) for

since a sum of Gaussian random variables is also exactly Gaussian-distributed. According to [

The Central Limit Theory (CLT) governs random fluctuations only near the mean-deviations from the mean

of the order of

much bigger: they are large deviations from the mean [

where according to [

the decaying exponential

where

and

Thus from (17) and (13) we have:

Next we turn to find a closed-form approximated expression for

convolutional noise power

In this paper we dealt with the fluctuations of the arithmetic average (sample mean) of the real part of consecutive convolutional noises which deviate from the mean of order higher than the typical ones. Via the large deviation theory, we obtained a closed-form approximated expression for the amount of deviation from the mean of those fluctuations considered as rare events as a function of the system’s parameters (step-size pa- rameter, equalizerss tap length, SNR, input signal statistics, characteristics of the chosen equalizer and channel power), for a pre-given probability that these events may occur.

We thank the Editor and the referee for their comments.

Monika Pinchas, (2015) Convolutional Noise Analysis via Large Deviation Technique. Journal of Signal and Information Processing,06,259-265. doi: 10.4236/jsip.2015.64024