Blogs    Cedron Dawg  

Exponential Smoothing with a Wrinkle

Cedron Dawg ● December 17, 2015●4 comments
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 this blog article.

Basic Exponential Smoothing

Exponential smoothing is also known as...

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

Cedron Dawg ● May 21, 2015●1 comment
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 coordinates are then used to calculate the phase and amplitude of the pure real tone in the DFT. This article...

Exact Frequency Formula for a Pure Real Tone in a DFT

Cedron Dawg ● April 20, 2015●2 comments
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 calculations are provided to give a numerical demonstration of the formula in use. This article is...

DFT Bin Value Formulas for Pure Real Tones

Cedron Dawg ● April 17, 2015●1 comment
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 developed in my previous two blog articles:

DFT Graphical Interpretation: Centroids of Weighted Roots of Unity

Cedron Dawg ● April 10, 2015
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 only consider real valued signals. Complex valued signals can be analyzed in a similar manner with...

The Exponential Nature of the Complex Unit Circle

Cedron Dawg ● March 10, 2015●3 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...

Off Topic: Refraction in a Varying Medium

Cedron Dawg ● July 11, 2018●1 comment
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 varying index of refraction. This applies to light, of course, but since it is a purely theoretical...

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

Cedron Dawg ● June 11, 2017●4 comments
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 calculating the frequency of a single pure tone in a short interval in the time domain is presented. It...

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

Cedron Dawg ● December 17, 2017
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 intensive, but it is more computationally accurate in the vicinity of near integer frequencies. This...

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...

Overview of my Articles

Cedron Dawg ● December 10, 2022
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 the articles also have sample programs which demonstrate the equations derived in the articles. My...

Candan's Tweaks of Jacobsen's Frequency Approximation

Cedron Dawg ● November 11, 2022
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 calculations beyond the original approximation to come up with an approximation suitable for most...