Complex Numbers Archives

February 7, 2022 Wireless/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 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

Intuition behind multiplying two complex numbers

March 14, 2025 Wireless/SDR

Intuitive Reason behind Multiplication and Division of Complex Numbers

Any introduction to complex numbers and their operations follows a common pattern: the formulas are given without building any intuition. If you have ever wondered about why $j=\sqrt{-1}$, you can read about the origin of complex numbers. In this article, I will explain the intuitive reason behind why a product of two complex numbers multiplies their magnitudes and adds their angles, and a division of two complex numbers divides their magnitudes and subtracts their angles. Let us start with the multiplication. The division scenario can be analogously derived with inverse operations. Multiplication of Complex Numbers The intuition behind multiplying two