New Lagrange Multipliers for the Blind Adaptive Deconvolution Problem Applicable for the Noisy Case

Abstract

Abstract:Recently, a new blind adaptive deconvolution algorithm was proposed based on a new closed-form approximated expression for the conditional expectation (the expectation of the source input given the equalized or deconvolutional output) where the output and input probability density function (pdf) of the deconvolutional process were approximated with the maximum entropy density approximation technique. The Lagrange multipliers for the output pdf were set to those used for the input pdf. Although this new blind adaptive deconvolution method has been shown to have improved equalization performance compared to the maximum entropy blind adaptive deconvolution algorithm recently proposed by the same author, it is not applicable for the very noisy case. In this paper, we derive new Lagrange multipliers for the output and input pdfs, where the Lagrange multipliers related to the output pdf are a function of the channel noise power. Simulation results indicate that the newly obtained blind adaptive deconvolution algorithm using these new Lagrange multipliers is robust to signal-to-noise ratios (SNR), unlike the previously proposed method, and is applicable for the whole range of SNR down to 7 dB. In addition, we also obtain new closed-form approximated expressions for the conditional expectation and mean square error (MSE).

A New Efficient Expression for the Conditional Expectation of the Blind Adaptive Deconvolution Problem Valid for the Entire Range ofSignal-to-Noise Ratio

“…In the following, the Bussgang-type blind equalization algorithms are considered, where the conditional expectation (the expectation of the source input given the equalized or deconvolutional output) is derived for estimating the desired response. In the literature, we can find several approximated expressions for the conditional expectation related to the blind adaptive deconvolutional problem [ 20 , 43 , 44 , 45 , 46 , 47 , 48 , 49 ]. However, References [ 43 , 44 , 45 , 46 ] are valid only for a uniformly distributed source input and References [ 20 , 47 , 48 ] were designed only for the noiseless case.…”

“…However, References [ 43 , 44 , 45 , 46 ] are valid only for a uniformly distributed source input and References [ 20 , 47 , 48 ] were designed only for the noiseless case. Recently [ 49 ], a new blind adaptive equalization method was proposed based on Reference [ 47 ] that is applicable for signal-to-noise ratio (SNR) values down to 7 dB. However, the computational burden of the method in Reference [ 49 ] is relative high.…”

“…Recently [ 49 ], a new blind adaptive equalization method was proposed based on Reference [ 47 ] that is applicable for signal-to-noise ratio (SNR) values down to 7 dB. However, the computational burden of the method in Reference [ 49 ] is relative high. The closed-form approximated expression proposed in Reference [ 49 ] for the conditional expectation with Lagrange multipliers up to order four (thus applicable for the 16QAM input case) is composed of a polynomial function of the equalized output of order twenty one.…”

Inspection of the Output of a Convolution and Deconvolution Process from the Leading Digit Point of View—Benford’s Law

“…where µ is the step-size parameter, ⋅ is the absolute value of ( ) [26]. Next, we introduce Benford's Law.…”

“…It is well known that ISI is a limiting factor in many communication environments where it causes an irreducible degradation of the bit error rate thus imposing an upper limit on the data symbol rate [1] [4]. In order to overcome the ISI problem, a blind adaptive equalizer can be implemented in those systems [2]- [26]. But, since no training symbols are used in the deconvolution process and the channel coefficients are unknown to the receiver, no indication can be made (via the ISI or MSE expressions) during the deconvolution process whether the blind adaptive equalizer succeeded to remove the heavy ISI from the transmitted symbols or not.…”