Signal Processing for Communication Systems
Mathuranathan Key focus: AR, MA & ARMA models express the nature of transfer function of LTI system. Understand the basic idea behind those models & know their frequency responses. How do you describe a complex, random signal—like the sound of a human voice or the fluctuating power of a fading channel—using only a few numbers? The … Read more
Key focus: Applying Cramér-Rao Lower Bound (CRLB) for vector parameter estimation. Know about covariance matrix, Fisher information matrix & CRLB matrix. CRLB for Vector Parameter Estimation CRLB for scalar parameter estimation was discussed in previous posts. The same concept is extended to vector parameter estimation. Consider a set of deterministic parameters $latex \mathbb{\theta}=[ \theta_1, \theta_2, … Read more
This post contains interactive python code which you can execute in the browser itself. 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 … Read more
Mathematical details of convolution, its relationship to polynomial multiplication and the application of Toeplitz matrices in computing linear convolution are discussed in the previous article. A short survey of different techniques to compute discrete linear convolution (with Matlab code) is given here. Definition Given an LTI (Linear Time Invariant) system with impulse response \(h[n]\) and … Read more
Convolution operation is ubiquitous in signal processing applications. The mathematics of convolution is strongly rooted in operation on polynomials. The intent of this text is to enhance the understanding on mathematical details of convolution. Polynomial functions: Polynomial functions are expressions consisting of sum of terms, where each term includes one or more variables raised to … Read more
A random variable is always associated with a probability distribution. When the random variable undergoes mathematical transformation the underlying probability distribution no longer remains the same. Consider a random variable $latex Z$ whose probability distribution function (PDF) is a standard normal distribution ($latex \mu=0$ and $latex \sigma^2=1$). Now, if the random variable is squared (a … Read more
Calculating the energy and power of a signal was discussed in one of the previous posts. Here, we will verify the calculation of signal power using Discrete Fourier Transform (DFT) in Matlab. Check here to know more on the concept of power and energy. The total power of a signal can be computed using the … Read more
Key focus: Clearly understand the terms: power and energy of a signal, their mathematical definition, physical significance and computation in signal processing context. Energy of a signal: Defining the term “size”: In signal processing, a signal is viewed as a function of time. The term “size of a signal” is used to represent “strength of … Read more
Uniform random variables are used to model scenarios where the expected outcomes are equi-probable. For example, in a communication system design, the set of all possible source symbols are considered equally probable and therefore modeled as a uniform random variable. The uniform distribution is the underlying distribution for an uniform random variable. A continuous uniform … Read more
Definition A random process (or signal for your visualization) with a constant power spectral density (PSD) function is a white noise process. Power Spectral Density Power Spectral Density function (PSD) shows how much power is contained in each of the spectral component. For example, for a sine wave of fixed frequency, the PSD plot will … Read more