Cedron Dawg Blog on DSPRelated.com

Off Topic: The True Gravitational Geodesic

Cedron Dawg — Tue, 20 May 2025 11:45:25 +0000

Introduction

This is a followup on my "Refraction in a Varying Media" article [1] taking it from a refractive scale to a gravitational scale. It is the answer to my quest in the "Speculations" section to find a vector differential equation for the trajectory of a photon in a gravity field. It is also a correction to my earlier assertion that the natural log of the effective index of...

Frequency Formula for a Pure Complex Tone in a DTFT

Cedron Dawg — Sun, 12 Nov 2023 23:24:49 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving frequency formula for the closely related Discrete Time Fourier Transform (DTFT). The distinction between the two is the latter has a domain on the integers from negative infinity to positive infinity and can be evaluated for any frequency. The DFT has a finite...

Pentagon Construction Using Complex Numbers

Cedron Dawg — Fri, 13 Oct 2023 13:11:48 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by showcasing a special case of the Roots of Unity, which underly the DFT. Admittedly, five bin DFTs aren't used too often, so in actuality you can consider this article another exercise of using complex numbers in context. The complex plane and the corresponding Cartesian...

Overview of my Articles

Cedron Dawg — Sat, 10 Dec 2022 17:08:22 +0000

Introduction

This article is a summary of all the articles I've written here at DspRelated. The main focus has always been an increased understanding of the Discrete Fourier Transform (DFT). The references are grouped by topic and ordered in a reasonable reading order. All the articles are meant to teach math, or give examples of math, in context within a specific application. Many of...

Candan's Tweaks of Jacobsen's Frequency Approximation

Cedron Dawg — Fri, 11 Nov 2022 18:50:52 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by explaining how a tweak to a well known frequency approximation formula makes it better, and another tweak makes it exact. The first tweak is shown to be the first of a pattern and a novel approximation formula is made from the second. It only requires a few extra...

A Recipe for a Basic Trigonometry Table

Cedron Dawg — Tue, 04 Oct 2022 14:43:48 +0000

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...

Off-Topic: A Fluidic Model of the Universe

Cedron Dawg — Wed, 02 Feb 2022 17:30:44 +0000

Introduction

This article is a followup to my previous article "Off Topic: Refraction in a Varying Medium"[1]. Many of the concepts should be quite familiar and of interest to the readership of this site. In the "Speculations" section of my previous article, I mention the goal of finding a similar differential equation as (18) of [1] for light traveling in gravity. It turns out it is...

The Zeroing Sine Family of Window Functions

Cedron Dawg — Sun, 16 Aug 2020 17:43:21 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by introducing a class of well behaved window functions that the author believes to be previously unrecognized. The definition and some characteristics are displayed. The heavy math will come in later articles. This is an introduction to the family, and a very special member...

A Two Bin Solution

Cedron Dawg — Fri, 12 Jul 2019 17:07:54 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by showing an implementation of how the parameters of a real pure tone can be calculated from just two DFT bin values. The equations from previous articles are used in tandem to first calculate the frequency, and then calculate the amplitude and phase of the tone. The approach...

Angle Addition Formulas from Euler's Formula

Cedron Dawg — Sat, 16 Mar 2019 12:42:41 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT), but only indirectly. The main intent is to get someone who is uncomfortable with complex numbers a little more used to them and relate them back to already known Trigonometric relationships done in Real values. It is essentially a followup to my first blog article "

Off Topic: Refraction in a Varying Medium

Cedron Dawg — Wed, 11 Jul 2018 20:55:22 +0000

Introduction

This article is another digression from a better understanding of the DFT. In fact, it is a digression from DSP altogether. However, since many of the readers here are Electrical Engineers and other folks who are very scientifically minded, I hope this article is of interest. A differential vector equation is derived for the trajectory of a point particle in a field of...

Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins

Cedron Dawg — Wed, 14 Mar 2018 18:22:14 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from several DFT bin values and knowing the frequency. This article is functionally an extension of my prior article "Phase and Amplitude Calculation for a Pure Complex Tone in a DFT"[1]...

Phase and Amplitude Calculation for a Pure Complex Tone in a DFT

Cedron Dawg — Sat, 06 Jan 2018 14:40:23 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from a DFT bin value and knowing the frequency. This is a much simpler problem to solve than the corresponding case for a pure real tone which I covered in an earlier blog article[1]. In the...

An Alternative Form of the Pure Real Tone DFT Bin Value Formula

Cedron Dawg — Sun, 17 Dec 2017 17:53:33 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving alternative exact formulas for the bin values of a real tone in a DFT. The derivation of the source equations can be found in my earlier blog article titled "DFT Bin Value Formulas for Pure Real Tones"[1]. The new form is slighty more complicated and calculation...

Improved Three Bin Exact Frequency Formula for a Pure Real Tone in a DFT

Cedron Dawg — Mon, 06 Nov 2017 22:24:01 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by extending the exact two bin formulas for the frequency of a real tone in a DFT to the three bin case. This article is a direct extension of my prior article "Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT"[1]. The formulas derived in the previous article are...

Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT

Cedron Dawg — Wed, 04 Oct 2017 23:19:23 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the frequency of a real tone in a DFT. This time it is a two bin version. The approach taken is a vector based one similar to the approach used in "Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT"[1]. The real valued formula...

Exact Near Instantaneous Frequency Formulas Best at Zero Crossings

Cedron Dawg — Thu, 20 Jul 2017 20:10:39 +0000

Introduction

This is an article that is the last of my digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). It is along the lines of the last two.

In those articles, I presented exact formulas for calculating the frequency of a pure tone signal as instantaneously as possible in the time domain. Although the formulas work for both real and...

Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 2)

Cedron Dawg — Sun, 11 Jun 2017 22:45:08 +0000

Introduction

This is an article that is a continuation of a digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). It is recommended that my previous article "Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 1)"[1] be read first as many sections of this article are directly dependent upon it.

A second family of formulas for...

Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 1)

Cedron Dawg — Fri, 12 May 2017 20:41:59 +0000

Introduction

This is an article that is a another digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). Although it is not as far off as the last blog article.

A new family of formulas for calculating the frequency of a single pure tone in a short interval in the time domain is presented. They are a generalization of Equation (1) from Rick...

A Recipe for a Common Logarithm Table

Cedron Dawg — Sat, 29 Apr 2017 20:42:21 +0000

Introduction

This is an article that is a digression from trying to give a better understanding to the Discrete Fourier Transform (DFT).

A method for building a table of Base 10 Logarithms, also known as Common Logarithms, is featured using math that can be done with paper and pencil. The reader is assumed to have some familiarity with logarithm functions. This material has no...

Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT

Cedron Dawg — Thu, 13 Apr 2017 19:19:31 +0000

Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving exact formulas for the frequency of a complex tone in a DFT. This time it is three bin versions. Although the problem is similar to the two bin version in my previous blog article "A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT"[1], a slightly...

A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT

Cedron Dawg — Mon, 20 Mar 2017 22:11:47 +0000

Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a complex tone in a DFT. It is basically a parallel treatment to the real case given in Exact Frequency Formula for a Pure Real Tone in a DFT. Since a real signal is the sum of two complex signals, the frequency formula for a...

DFT Bin Value Formulas for Pure Complex Tones

Cedron Dawg — Fri, 17 Mar 2017 22:32:42 +0000

Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an analytical formula for the DFT of pure complex tones and an alternative variation. It is basically a parallel treatment to the real case given in DFT Bin Value Formulas for Pure Real Tones. In order to understand how a multiple tone signal acts in a DFT it...

Exponential Smoothing with a Wrinkle

Cedron Dawg — Thu, 17 Dec 2015 20:05:19 +0000

Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by providing a set of preprocessing filters to improve the resolution of the DFT. Because of the exponential nature of sinusoidal functions, they have special mathematical properties when exponential smoothing is applied to them. These properties are derived and explained in...

Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1

Cedron Dawg — Thu, 21 May 2015 20:04:23 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found...

Exact Frequency Formula for a Pure Real Tone in a DFT

Cedron Dawg — Mon, 20 Apr 2015 14:24:06 +0000

Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a real tone in a DFT. According to current teaching, this is not possible, so this article should be considered a major theoretical advance in the discipline. The formula is presented in a few different formats. Some sample...

DFT Bin Value Formulas for Pure Real Tones

Cedron Dawg — Fri, 17 Apr 2015 13:21:07 +0000

Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an analytical formula for the DFT of pure real tones. The formula is used to explain the well known properties of the DFT. A sample program is included, with its output, to numerically demonstrate the veracity of the formula. This article builds on the ideas...

DFT Graphical Interpretation: Centroids of Weighted Roots of Unity

Cedron Dawg — Fri, 10 Apr 2015 16:26:22 +0000

Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by framing it in a graphical interpretation. The bin calculation formula is shown to be the equivalent of finding the center of mass, or centroid, of a set of points. Various examples are graphed to illustrate the well known properties of DFT bin values. This treatment will...

The Exponential Nature of the Complex Unit Circle

Cedron Dawg — Tue, 10 Mar 2015 15:11:37 +0000

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...