Complex Sinusoids Archives | Page 2 of 3

Multiple objects at different speeds for an FMCW radar

November 6, 2023 DSP / Localization

FMCW Radar Part 2 – Velocity, Angle and Radar Data Cube

In Part 1 of FMCW radar series, we described how a radar estimates the range of one or more 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 a wonderful 1991 paper "Wireless Digital Communication: A View Based on Three Lessons Learned", Andrew Viterbi summarizes the Shannon theory for digital communications in the form of 3 lessons, the first of which was the following. "Never discard information prematurely that may be

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

Symbolic representation of linear phase rotation that changes with frequency index

May 14, 2019 DSP

Effect of Time Shift in Frequency Domain

Children usually ask questions like “How many hours have passed?” And they have no idea about the start time to be taken as a reference. Just like the zero of a measuring tape, a zero reference for time plays a crucial role in analyzing the signal behaviour in time and frequency domains. Until now, we assumed that reference time $0$ coincides with the start of a sine and a cosine wave to understand the frequency domain. Later, we will deal with symbol timing synchronization problem in single-carrier systems and carrier frequency synchronization problem in multicarrier systems, both of which address

December 22, 2017 DSP

Interpreting Time Domain Derivative in Frequency Domain

Although this article explains the concepts in terms of mathematical constants e and j as well as integration, my book on SDR steers clear of the complex notation and integrals to describe the underlying concepts from the ground up to an advanced level. One of the properties of Fourier Transform is that the derivative of a signal in time domain gets translated to multiplication of the signal spectrum by $j2\pi f$ in frequency domain. This property is usually derived as follows. For a signal $s(t)$ with Fourier Transform $S(f)$ \begin{equation*} s(t) = \frac{1}{2\pi}\int \limits _{-\infty}^{+\infty} S(f) e^{j2\pi ft}df, \end{equation*} we

November 30, 2023 DSP / Localization

FMCW Radar Part 3 – Design Guidelines

The Bloom’s Taxonomy describes the levels of mastery one attains in a field. Its last two stages are Synthesis and Evaluation. This is where the masters can be differentiated from the experts. In a job interview, for example, a good technique to judge a candidate’s ability is to ask them where the system in question breaks. A little learning is a dangerous thing Drink deep, or taste not the Pierian spring There shallow draughts intoxicate the brain And drinking largely sobers us again While the first two parts of the FMCW radar series addressed the lower levels, Part 3 is