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September 5, 2022 DSP

Specifications of a Radio Receiver

When designing a radio receiver, a system architect has to deal with the issues such as dynamic range, noise floor and sensitivity of a radio receiver [1]. The ultimate purpose is computing a power budget to ensure that a minimum amount of signal power is available at the receiver during operation. This is not much different than how a country assigns an available budget into different sectors such as defense, education and health. Dynamic Range Dynamic range is the ratio of the largest signal level to the smallest signal level that the system can process in analog and digital stages.

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

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

(Top) An 8-PSK waveform. (Bottom) Two constellation diagrams: one at the Tx shown by thick red lines and the other at the Rx for a phase offset of 17 degrees shown by dotted purple lines

December 8, 2020 DSP

I/Q Signals 101: Neither Complex Nor Complicated

Dec 04, 2020 There was a recent discussion on GNU Radio mailing list in regards to the simplest possible intuition behind I/Q signals. Why is I/Q sampling required? Question: The original question from Kristoff went like this: “… when you mention `GNU Radio complex numbers’, you also have to mention I/Q signals, which is a topic that is very difficult to explain in 10 seconds to an audience who has never seen anything about I/Q sampling before.” Comment: According to Jeff Long: “This is a great thing to try to figure out. If we can come up with an answer

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