Signal Processing for Communication Systems
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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
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
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
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
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
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: 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
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
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
If squares of k independent standard normal random variables are added, it gives rise to central Chi-squared distribution with ‘k’ degrees of freedom. Instead, if squares of k independent normal random variables with non-zero means are added, it gives rise to non-central Chi-squared distribution. Non-central Chi-square distribution is related to Ricean distribution, whereas the central … Read more