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The Exponential Nature of the Complex Unit Circle

Cedron Dawg March 10, 20155 comments
Introduction

This is an article to hopefully give an understanding to Euler's magnificent equation:

$$ e^{i\theta} = cos( \theta ) + i \cdot sin( \theta ) $$

This equation is usually proved using the Taylor series expansion for the given functions, but this approach fails to give an understanding to the equation and the ramification for the behavior of complex numbers. Instead an intuitive approach is taken that culminates in a graphical understanding of the equation.

Complex...

Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals

Rick Lyons October 31, 20131 comment

In digital signal processing (DSP) we're all familiar with the processes of bandpass sampling an analog bandpass signal and downsampling a digital bandpass signal. The overall spectral behavior of those operations are well-documented. However, mathematical expressions for computing the translated frequency of individual spectral components, after bandpass sampling or downsampling, are not available in the standard DSP textbooks. The following three sections explain how to compute the...


A Quadrature Signals Tutorial: Complex, But Not Complicated

Rick Lyons April 12, 201365 comments

Introduction Quadrature signals are based on the notion of complex numbers and perhaps no other topic causes more heartache for newcomers to DSP than these numbers and their strange terminology of j operator, complex, imaginary, real, and orthogonal. If you're a little unsure of the physical meaning of complex numbers and the j = √-1 operator, don't feel bad because you're in good company. Why even Karl Gauss, one the world's greatest mathematicians, called the j-operator the "shadow of...


A Fixed-Point Introduction by Example

Christopher Felton April 25, 201122 comments
Introduction

The finite-word representation of fractional numbers is known as fixed-point.  Fixed-point is an interpretation of a 2's compliment number usually signed but not limited to sign representation.  It extends our finite-word length from a finite set of integers to a finite set of rational real numbers [1].  A fixed-point representation of a number consists of integer and fractional components.  The bit length is defined...


A Recipe for a Basic Trigonometry Table

Cedron Dawg October 4, 2022
Introduction

This is an article that is give a better understanding to the Discrete Fourier Transform (DFT) by showing how to build a Sine and Cosine table from scratch. Along the way a recursive method is developed as a tone generator for a pure tone complex signal with an amplitude of one. Then a simpler multiplicative one. Each with drift correction factors. By setting the initial values to zero and one degrees and letting it run to build 45 values, the entire set of values needed...


New Video: Parametric Oscillations

Tim Wescott January 4, 2017

I just posted this last night.  It's kinda off-topic from the mission of the channel, but I realized that it had been months since I'd posted a video, and having an excuse to build on helped keep me on track.


Pentagon Construction Using Complex Numbers

Cedron Dawg October 13, 2023

A method for constructing a pentagon using a straight edge and a compass is deduced from the complex values of the Fifth Roots of Unity. Analytic values for the points are also derived.


Frequency Formula for a Pure Complex Tone in a DTFT

Cedron Dawg November 12, 2023

The analytic formula for calculating the frequency of a pure complex tone from the bin values of a rectangularly windowed Discrete Time Fourier Transform (DTFT) is derived. Unlike the corresponding Discrete Fourier Transform (DFT) case, there is no extra degree of freedom and only one solution is possible.