Complex Sinusoids Archives

September 30, 2023 DSP / Localization

FMCW Radar Part 1 – Ranging

This is Part 1 of a 3-Part series in which we describe how an FMCW radar finds the range of multiple stationary targets. In Part 2, we talk about estimating the velocities of several moving targets and their directions through forming a structure known as the radar cube. Part 3 presents system design guidelines for an FMCW radar. In his book Multirate Signal Processing, Fred Harris mentions a great problem solving technique: "When faced with an unsolvable problem, change it into one you can solve, and solve that one instead." We will see in this article how an FMCW radar

August 1, 2022 DSP

Spectral Shift without any Multiplications

One of the great advantages of Digital Signal Processing (DSP) is an unexpected simplification of operations in seemingly complicated scenarios. See the Cascade Integrator Comb (CIC) filters for how to accomplish the task of sample rate conversion along with filtering with minimal resources. As another example, in wireless communications and many other applications, a frequency translation is often required in which the spectrum of a signal centered at a particular frequency needs to be moved to another frequency. From the properties of Fourier Transform, a shift by frequency $\omega_0=2\pi F_0$ requires sample-by-sample multiplication with a complex sinusoid $e^{j\omega_0 t}$. \[

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

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