Signal Processing for Communication Systems
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Cholesky decomposition is an efficient method for inversion of symmetric positive-definite matrices. Let’s demonstrate the method in Python and Matlab. Cholesky factor Any $latex n \times n$ symmetric positive definite matrix $latex A $ can be factored as $latex A=LL^T &s=2$ where $latex L$ is $latex n \times n$ lower triangular matrix. The lower triangular … Read more
[ratings] It is often required to check if a given matrix is positive definite or not. Three methods to check the positive definiteness of a matrix were discussed in a previous article . I will utilize the test method 2 to implement a small matlab code to check if a matrix is positive definite.The test … Read more
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
Understand Fourier Series, Fourier Cosine Series, Fourier Sine Series, partial sums, even odd symmetry. Hands-on simulation with Matlab code given. Fourier analysis and Fourier Synthesis: Fourier analysis – a term named after the French mathematician Joseph Fourier, is the process of breaking down a complex function and expressing it as a combination of simpler functions. … Read more
A review of top books on basics of communication systems is given here. The list given is neither comprehensive nor listed based on any ranking. If you feel we have missed out any of your favorite books, please post them in the comment box below. Communication Systems – Simon Haykins The standard text book recommended … Read more
In order to perform Cholesky Decomposition of a matrix, the matrix has to be a positive definite matrix. I have listed down a few simple methods to test the positive definiteness of a matrix. Methods to test Positive Definiteness: Remember that the term positive definiteness is valid only for symmetric matrices. Test method 1: Existence … Read more
[ratings] 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 … 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
[ratings] “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 … Read more
