Signal Processing for Communication Systems
Key focus: Know the expressions to solve triangular matrix using forward and backward substituting techniques and the FLOPS required for solving it. Forward Substitution: Consider a set of equations in a matrix form $latex Ax=b $, where A is a lower triangular matrix with non-zero diagonal elements. The equation is re-written in full matrix form … Read more
Fourier series fundamentals: representing periodic signals with harmonics, computing coefficients, and links to Gibbs phenomenon and Parseval.
Test if a matrix is positive definite: leading minors, eigenvalues > 0, and Cholesky success — with Matlab eig/chol and NumPy equivalents for covariance checks.
Matrix inversion is seen ubiquitously in signal processing applications. For example, matrix inversion is an important step in channel estimation and equalization. For instance, in GSM normal burst, 26 bits of training sequence are put in place with 114 bits of information bits. When the burst travels over the air interface (channel), it is subject … Read more
Key focus : Essential matrix algebra: formation of matrices, determinants, rank, inverse & transpose of matrix and solving simultaneous equations. I thought of making a post on Cholesky Decomposition, which is a very essential technique in digital signal processing applications like generating correlated random variables, solving linear equations, channel estimation etc.., But jumping straight to … Read more
Key focus: Know how to estimate unknown parameters using Ordinary Least Squares (OLS) method. As mentioned in the previous post, it is often required to estimate parameters that are unknown to the receiver. For example, if a fading channel is encountered in a communication system, it is desirable to estimate the channel response and cancel … Read more
“Mean Square Error”, abbreviated as MSE, is an ubiquitous term found in texts on estimation theory. Have you ever wondered what this term actually means and why is this getting used in estimation theory very often ? Any communication system has a transmitter, a channel or medium to communicate and a receiver. Given the channel … Read more
Key focus: Normalized CRLB (Cramér-Rao Lower bound) is an alternate form of CRLB. Let’s explore how normalized CRLB is related to estimator sensitivity. The variance of an estimate is always greater than or equal to Cramér-Rao Lower Bound of the estimate. The CRLB is in turn given by inverse of Fisher Information. The following equation … Read more
Key focus: Derive the Cramer-Rao lower bound for phase estimation applied to DSB transmission. Find out if an efficient estimator actually exists for phase estimation. Problem formulation Consider the DSB carrier frequency estimation problem given in the introductory chapter to estimation theory. A message is sent across a channel modulated by a sinusoidal carrier with … Read more
It has been reiterated that not all estimators are efficient. Even not all the Minimum Variance Unbiased Estimators (MVUE) are efficient. Then how do we quantify whether the estimator designed by us is efficient or not? An efficient estimator is defined as the one that is* Unbiased (mean of the estimate = true value of … Read more