Mathuranathan

Mathuranathan Viswanathan, is an author @ gaussianwaves.com that has garnered worldwide readership. He is a masters in communication engineering and has 12 years of technical expertise in channel modeling and has worked in various technologies ranging from read channel, OFDM, MIMO, 3GPP PHY layer, Data Science & Machine learning.

Cramér-Rao Lower Bound (CRLB)-Scalar Parameter Estimation

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Key focus: Discuss scalar parameter estimation using CRLB. Estimate DC component from observed data in the presence of AWGN noise. Consider a set of observed data samples and is the scalar parameter that is to be estimated from the observed samples. The accuracy of the estimate depends on how well the observed data is influenced … Read more

Cramér-Rao Lower Bound: Introduction

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Key concept: Cramér-Rao bound is the lower bound on variance of unbiased estimators that estimate deterministic parameters. Introduction The criteria for existence of having an Minimum Variance Unbiased Estimator (MVUE) was discussed in a previous article. To have an MVUE, it is necessary to have estimates that are unbiased and that give minimum variance (compared … Read more

Score, Fisher Information and Estimator Sensitivity

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As we have seen in the previous articles, that the estimation of a parameter from a set of data samples depends strongly on the underlying PDF. The accuracy of the estimation is inversely proportional to the variance of the underlying PDF. That is, less the variance of PDF more is the accuracy of estimation and vice … Read more

Theoretical derivation of MLE for Gaussian Distribution:

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As a pre-requisite, check out the previous article on the logic behind deriving the maximum likelihood estimator for a given PDF. Let X=(x1,x2,…, xN) are the samples taken from Gaussian distribution given by Calculating the Likelihood The log likelihood is given by, Differentiating and equating to zero to find the maxim (otherwise equating the score … Read more

Theoretical derivation of MLE for Exponential Distribution:

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As a pre-requisite, check out the previous article on the logic behind deriving the maximum likelihood estimator for a given PDF. Let X=(x1,x2,…, xN) are the samples taken from Exponential distribution given by Calculating the Likelihood The log likelihood is given by, Differentiating and equating to zero to find the maxim (otherwise equating the score … Read more

Theoretical derivation of Maximum Likelihood Estimator for Poisson PDF:

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Suppose X=(x1,x2,…, xN) are the samples taken from a random distribution whose PDF is parameterized by the parameter . If the PDF of the underlying parameter satisfies some regularity condition (if the log of the PDF is differentiable) then the likelihood function is given by Here is the PDF of the underlying distribution. Hereafter we … Read more

Maximum Likelihood Estimation (MLE) : Understand with example

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Key focus: Understand maximum likelihood estimation (MLE) using hands-on example. Know the importance of log likelihood function and its use in estimation problems. Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model. The core idea behind MLE is to find the parameter values that maximize the likelihood … Read more

Estimator Bias

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Estimator bias: Systematic deviation from the true value, either consistently overestimating or underestimating the parameter of interest. Estimator Bias: Biased or Unbiased Consider a simple communication system model where a transmitter transmits continuous stream of data samples representing a constant value – ‘A’. The data samples sent via a communication channel gets added with White … Read more

QAM modulation: simulate in Matlab & Python

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A generic complex baseband simulation technique, to simulate all M-ary QAM modulation techniques is given here. The given simulation code is very generic, and it plots both simulated and theoretical symbol error rates for all M-QAM modulation techniques. Rectangular QAM from PAM constellation There exist other constellation shapes (like circular, triangular constellations) that are more … Read more

Natural Binary Codes and Gray Codes

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In a given communication system, we always want to send data that represent real world data representing some physical quantity (be it speech, temperature, etc..,) .The real world physical quantity exist in analog domain and it becomes imperative to convert it to digital domain if we want to send it via a digital communication system. … Read more