Complex Sinusoids Archives | Page 2 of 4

Discrete Fourier Transform (DFT) of a DFT-even sequence

July 11, 2022 DSP

The Beauty of Symmetry in Fourier Transform

In 1978, Fred Harris was a relatively unknown faculty member at the San Diego State University when he published his landmark paper titled On the use of windows for harmonic analysis with the discrete Fourier transform. That paper made him a superstar in DSP community. It presented a brief overview of signal windows and their impact on the detection of harmonic signals in the presence of broad-band noise and nearby harmonic interference. More importantly, he pointed out several common errors in the application of windows when used in the context of Discrete Fourier Transform (DFT). Today I am going to

Interaction between populations of rabbits and foxes

August 6, 2024 DSP

Rabbits, Foxes and IQ Signals

If we pay attention, each term in a mathematical equation carries a meaning that resonates with common sense. Today I will explain where Lotka-Volterra equations come from. These equations describe the dynamics of a biological interaction in which a predator (e.g., foxes) and a prey species (e.g., rabbits) engage with each other in a continuous struggle for survival. We will see that the math expressions just line up to describe the phenomenon almost as in words. Moreover, they have a little connection to IQ signals, the fundamental concept in digital signal processing, that will also be presented in the article.

March 21, 2022 DSP

A Visualization of Causality and Stability in z-Transform

Most of the books and resources explain the z-Transform as a mathematical concept rather than a signal processing idea. Today I will provide a simple explanation of how the z-Transform helps in determining whether a system is causal and stable. I hope that this visual approach will help my readers learn this concept in a better manner. The z-Transform For a discrete-time signal $h[n]$ (that is the impulse response of a system), the z-Transform is defined as \begin{equation}\label{equation-z-transform} H(z) = \sum _{n=-\infty}^{\infty} h[n]z^{-n} \end{equation} Then, $z$ is a complex number and hence can be written as \[ z = re^{j\omega}

March 25, 2024 DSP / Miscellaneous

Why the Constant e Arises in Complex Plane as a Rotation

In the tutorial on how complex numbers arose, we asked three questions. The first two were answered in the same article while the answer to the third question, repeated below, is explained here. Why is the expression $e^{i \theta}$ a rotation of 1 by $\theta$ radians on a unit circle? Is it possible to make sense out of a number like $2.71828^{\sqrt{-1}\cdot\theta}$? The constant e is a special number discovered by Jacob Bernoulli while studying compound interests. It appears in many other forms as well which are all related to each other but that topic is a complete account in

Plots for positive integer powers of x in 3D

June 26, 2020 DSP / Miscellaneous

A Real-Imaginative Guide to Complex Numbers

June 18, 2020 On a cold morning in August 2015, I narrowly missed a train to my office in Melbourne city. With nothing else to do in the next 20 minutes, my mind wandered towards an intuitive view of complex numbers, something that has puzzled me since long. In particular, I wanted to seek answers to the following questions. (a) What is the role of the number $\sqrt{-1}$ in mathematics? What sets it apart from other impossible numbers, e.g., a number $k$ such that $|k|=-1$? (The origins of this question might lie in how I cut apple slices for my