_{1}

In the communication field, during transmission, a source signal undergoes a convolutive distortion between its symbols and the channel impulse response. This distortion is referred to as Intersymbol Interference (ISI) and can be reduced significantly by applying a blind adaptive deconvolution process (blind adaptive equalizer) on the distorted received symbols. But, since the entire blind deconvolution process is carried out with no training symbols and the channel’s coefficients are obviously unknown to the receiver, no actual indication can be given (via the mean square error (MSE) or ISI expression) during the deconvolution process whether the blind adaptive equalizer succeeded to remove the heavy ISI from the transmitted symbols or not. Up to now, the output of a convolution and deconvolution process was mainly investigated from the ISI point of view. In this paper, the output of a convolution and deconvolution process is inspected from the leading digit point of view. Simulation results indicate that for the 4PAM (Pulse Amplitude Modulation) and 16QAM (Quadrature Amplitude Modulation) input case, the number “1” is the leading digit at the output of a convolution and deconvolution process respectively as long as heavy ISI exists. However, this leading digit does not follow exactly Benford’s Law but follows approximately the leading digit (digit 1) of a Gaussian process for independent identically distributed input symbols and a channel with many coefficients.

We consider a blind deconvolution problem in which we observe the output of an unknown, possibly nonminimum phase, linear system (single-input-single-output (SISO) finite impulse response (FIR) system) from which we want to recover its input (source) using an adjustable linear filter (equalizer). During transmission, a source signal undergoes a convolutive distortion between its symbols and the channel impulse response. This distortion is referred to as ISI [

According to [

In the communication field, the transmitted symbols may belong to a squared constellation input such as the 16QAM or to a PAM constellation where the transmitted symbols are statistically independent and have the same probability to appear for transmission. Thus, the recovered symbols should also appear approximately with equal probability. Up to now, the output of a convolution process such as the output of a channel was mainly investigated from the ISI point of view. In this paper, the output of a convolution and deconvolution process is inspected from the leading digit point of view. Simulation results will indicate that for the 4PAM and 16QAM constellation input, the number 1 is the leading digit at the output of a convolution and deconvolution process respectively as long as heavy ISI exists. However, this leading digit does not follow exactly Benford’s Law but follows approximately the leading digit (digit 1) of a Gaussian process for independent identically distributed input symbols and a channel with many coefficients.

The paper is organized as follows: After having described the system under consideration in Section 2, the inspection of the output of a convolution and deconvolution process from the leading digit point of view is given via simulation results in Section 3. 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

The transmitted sequence

where “

where

where

where

where d is the digit number.

In this section we first start to inspect the output of the convolution process from the leading digit point of view for the noiseless case. After that, we turn to inspect the output of the deconvolution process considering also the noisy situation.

In the following we use the 4PAM constellation input with three different channel cases having real valued coefficients:

Case I: Rayleigh fading channel with variance equal to 0.2.

Case II: Gaussian channel with zero mean and variance equal to 1.

Case III: A channel where the coefficients are uniformly distributed within

Figures 2-8 show the averaged value for the leading-digit distribution for 9000 symbols

(belonging to a 4PAM constellation) sent via the above mentioned channel cases with different values for the channel length, compared with Benford’s law. 100 Monte Carlo trials were used to get the averaged results for the leading-digit distribution for each channel case and channel length, where for each trial, the channel coefficients were randomly selected from the predefined channel case. According to Figures 2-8, the leading digit at the output channel is the number 1. This phenomenon becomes even more evident for higher values for the channel length (

number 1 minus the probability of occurrence of the number “i” (where

Now, we turn to the deconvolution case where the equalized output is of our interest. 9000 symbols belonging to a 16QAM input constellation were sent via the channel used in [

Usually, the equalizer’s coefficients are updated constantly. However, in order to see the behaviour of the leading digit at the equalized output during the deconvolution process, the updating mechanism of the equalizer’s coefficients was stopped at several places during the deconvolution process. Namely, the equalizer’s coefficients were updated as long as we have not reached the desired iteration number. For example, if the desired iteration number was set to 10, then the equalizer’s coefficients were not updated anymore after 10 iterations. Thus, the residual ISI level at the equalizer’s output remained fixed. Figures 17-20 show the averaged value for the leading-digit distribution at the equalized output for SNR of 30 dB compared with Benford’s law. Based on the obtained results for the convolution case described earlier in this section, it was quite expected to get the number 1 as the leading digit at the equalized output at the early stages of the deconvolution process (

not follow Benford’s law. According to

minus the probability of occurrence of the number “i” (where

output for SNR of 20 dB. According to Figures 32-34, the probability of occurrence of the number 1 minus the probability of occurrence of the number 3 is approximately zero only at the latter stages of the deconvolution process (

In this paper, we have shown via simulation results that the number 1 is the leading digit at the output of a convolution and deconvolution process for a 4PAM and 16QAM input constellation respectively, as long as heavy ISI exists. In addition, simulation results have shown that the behaviour of this leading digit does not follow exactly Benford’s Law but follows approximately the leading digit (digit 1) of a Gaussian process for independent identically distributed input symbols and a channel with many coefficients.

We thank the Editor and the referee for their comments.

Pinchas, M. (2016) Inspection of the Output of a Convolution and Deconvolution Process from the Leading Digit Point of View―Benford’s Law. Journal of Signal and Information Processing, 7, 227-251. http://dx.doi.org/10.4236/jsip.2016.74020

ISI: Intersymbol Interference

SNR: Signal to Noise Ratio

SISO: Single Input Single Output

FIR: Finite Impulse Response

QAM: Quadrature Amplitude Modulation

PAM: Pulse Amplitude Modulation

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