Complex Numbers Archives

March 23, 2025 DSP / Wireless and SDR

Two Birds with One Tone: I/Q Signals and Fourier Transform – Part 1

This article is a deep and highly intuitive breakdown of the fundamental concept behind I/Q signals. If you want to see how these concepts function inside actual communication receivers and hardware, you can see read about I/Q signals from a communication systems perspective. When a new member arrives at the Signal Processing Club, this is what they find at the club gate: I/Q signals. Perhaps a secret plot to keep most people out of the party? Some return from here to try another area (e.g., machine learning, which pays more and is easier to understand but less interesting than signal

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

February 7, 2022 Wireless and SDR

Coordinate Rotation Digital Computer (CoRDiC)

Digitial Signal Processing (DSP) plays a crucial role in algorithm implmentation for building digital and wireless communication systems. A common theme in all those algorithms is that they can be implemented with the following simple operations: addition multiplication shift In fact, these are the basic principles on which a digital signal processor is constructed. However, when it comes to implementation of real-time systems in hardware such as FPGAs, we find ways to reduce the complexity even further. Which operation (out of the above three) do you think is the most demanding in computations? It is the multiplications. Therefore, it is

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

(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 If you are an absolute beginner looking for a really intuitive understanding of I/Q signals, read the introductory guide on I/Q signals from the ground up. The article below explains this idea from a communication systems perspective. 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