Off Topic: Refraction in a Varying Medium
Cedron Dawg derives a compact vector differential equation for a point particle moving through a smoothly varying refractive medium using the Euler-Lagrange variational method. By introducing a log refractive index called "fluff density," the paper expresses acceleration purely in terms of the fluff gradient and velocity, then explores curvature, superposition, and point-source capture radii with simple closed-form results.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins
Cedron presents exact, closed-form formulas to extract the phase and amplitude of a pure complex tone from multiple DFT bin values, using a compact vector formulation. The derivation introduces a delta variable to simplify the sinusoidal bin expression, stacks neighboring bins into a basis vector, and solves for the complex amplitude q by projection. The phase and magnitude follow directly from q, and extra bins reduce leakage when the tone falls between bins.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT
Cedron Dawg derives compact, exact formulas to recover the phase and amplitude of a single complex tone from a DFT bin when the tone frequency is known. The paper turns the complex bin value into closed-form expressions using a sine-fraction amplitude correction and a simple phase shift, and includes working code plus a numeric example for direct implementation.
An Alternative Form of the Pure Real Tone DFT Bin Value Formula
Cedron Dawg derives an alternative exact formula for DFT bin values of a pure real tone, sacrificing algebraic simplicity for better numerical behavior near integer-valued frequencies. By rewriting cosine differences as products of sines and shifting to a delta frame of reference, the derivation avoids catastrophic cancellation and preserves precision for near-integer tones. The analysis also shows the integer-frequency case is a degenerate limit that yields the familiar M/2 e^{iφ} bin value.
Improved Three Bin Exact Frequency Formula for a Pure Real Tone in a DFT
Cedron Dawg extends his two-bin exact frequency formulas to a three-bin DFT estimator for a pure real tone, and presents the derivation in computational order for practical use. The method splits complex bin values into real and imaginary parts, forms vectors A, B, and C, applies a sqrt(2) variance rescaling, and computes frequency via a projection-based closed form. Numerical tests compare the new formula to prior work and show improved accuracy when the tone lies between bins.
Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT
Cedron Dawg derives exact, closed-form frequency formulas that recover a pure real tone from just two DFT bins using a geometric vector approach. The method projects bin-derived vectors onto a plane orthogonal to a constraint vector to eliminate amplitude and phase, yielding an explicit cos(alpha) estimator; a small adjustment improves noise performance so the estimator rivals and slightly betters earlier two-bin methods.
Exact Near Instantaneous Frequency Formulas Best at Zero Crossings
Cedron Dawg derives time-domain formulas that yield near-instantaneous frequency estimates optimized for zero crossings of pure tones. Complementing his earlier peak-optimized results, these difference-ratio formulas work for real and complex signals, produce four-sample estimators similar to Turners, and cancel amplitude terms, making them attractive low-latency options for clean tones while warning they degrade in noise and at peaks.
Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 2)
Cedron Dawg derives a second family of exact time domain formulas for single-tone frequency estimation that trade a few extra calculations for improved noise robustness. Built from [1+cos]^k binomial weighting of neighbor-pair sums, the closed-form estimators are exact and are best evaluated at signal peaks for real tones, while complex tones do not share the zero-crossing limitation. Coefficients up to k=9 are provided.
Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 1)
Cedron Dawg presents a new family of exact time-domain formulas to estimate the instantaneous frequency of a single pure tone. The methods generalize a known one-sample formula into k-degree neighbor-pair sums with spacing d, giving exact results in the noiseless case and tunable robustness in noise. The paper explains why real-tone estimates must be taken at peaks and shows the formulas also work for complex tones.
A Recipe for a Common Logarithm Table
Cedron Dawg shows how to construct a base-10 logarithm table from scratch using only pencil-and-paper math. The recipe combines simple series for e and ln(1+x) with clever factoring and neighbor-based recurrences so minimal square-root work is required. Along the way the post explains a practical algorithm, high-accuracy interpolation and inverse-log reconstruction so you can reproduce published log tables by hand.
Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT
Cedron Dawg derives closed-form three-bin frequency estimators for a pure complex tone in a DFT using a linear algebra view that treats three adjacent bins as a vector. He shows any vector K orthogonal to [1 1 1] yields a = (K·Z)/(K·D·Z) and derives practical K choices including a Von Hann (Pascal) kernel and a data-driven projection. The post compares estimators under noise and gives simple selection rules.
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
Cedron Dawg derives an exact two-bin frequency formula for a pure complex tone in the DFT, eliminating amplitude and phase to isolate frequency via a complex quotient and the complex logarithm. He presents an adjacent-bin simplification that replaces a complex multiply with a bin offset plus an atan2 angle, and discusses integer-frequency handling and aliasing. C source and numerical examples show the formula working in practice.
DFT Graphical Interpretation: Centroids of Weighted Roots of Unity
DFT bin values can be seen as centroids of weighted roots of unity, a geometric picture that makes many DFT properties immediate. Cedron Dawg uses the geometric-series identity and polar plots of integer and fractional tones to show why constants appear only at DC, how wrapping relates to bin index, and how phase, scaling, offsets, and real-signal symmetry affect bin magnitudes and angles.
Exponential Smoothing with a Wrinkle
Cedron Dawg shows how pairing forward and backward exponential smoothing produces exact, frequency-dependent dampening for sinusoids while canceling time-domain lag. The average of the two passes scales the tone by a closed-form factor, and their difference acts like a first-derivative with a quarter-cycle phase shift. The post derives the analytic dampening formulas, compares them to the derivative, and includes a Python demo for DFT preprocessing.
Off Topic: Refraction in a Varying Medium
Cedron Dawg derives a compact vector differential equation for a point particle moving through a smoothly varying refractive medium using the Euler-Lagrange variational method. By introducing a log refractive index called "fluff density," the paper expresses acceleration purely in terms of the fluff gradient and velocity, then explores curvature, superposition, and point-source capture radii with simple closed-form results.
Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT
Cedron Dawg derives exact, closed-form frequency formulas that recover a pure real tone from just two DFT bins using a geometric vector approach. The method projects bin-derived vectors onto a plane orthogonal to a constraint vector to eliminate amplitude and phase, yielding an explicit cos(alpha) estimator; a small adjustment improves noise performance so the estimator rivals and slightly betters earlier two-bin methods.
Pentagon Construction Using Complex Numbers
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.
Exact Frequency Formula for a Pure Real Tone in a DFT
Cedron Dawg derives an exact closed form formula to recover the frequency of a pure real sinusoid from three DFT bins, challenging the usual teaching that it is impossible. The derivation solves for cos(alpha) in a bilinear form and gives a computationally efficient implementation (eq.19), with practical notes on implicit Hann-like weighting and choosing the peak bin for robustness.
A Recipe for a Common Logarithm Table
Cedron Dawg shows how to construct a base-10 logarithm table from scratch using only pencil-and-paper math. The recipe combines simple series for e and ln(1+x) with clever factoring and neighbor-based recurrences so minimal square-root work is required. Along the way the post explains a practical algorithm, high-accuracy interpolation and inverse-log reconstruction so you can reproduce published log tables by hand.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT
Cedron Dawg derives compact, exact formulas to recover the phase and amplitude of a single complex tone from a DFT bin when the tone frequency is known. The paper turns the complex bin value into closed-form expressions using a sine-fraction amplitude correction and a simple phase shift, and includes working code plus a numeric example for direct implementation.
A Two Bin Solution
Cedron Dawg shows how a real sinusoid's frequency, amplitude and phase can be recovered from only two adjacent DFT bins. The article derives exact two-bin formulas, gives a clear Gambas reference implementation, and demonstrates that accurate parameters can be obtained with very few samples when the tone lies between the bins. It also explains when the method breaks down and how the real-valued unfurling improves robustness.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT
Cedron Dawg derives compact, exact formulas to recover the phase and amplitude of a single complex tone from a DFT bin when the tone frequency is known. The paper turns the complex bin value into closed-form expressions using a sine-fraction amplitude correction and a simple phase shift, and includes working code plus a numeric example for direct implementation.
Off Topic: Refraction in a Varying Medium
Cedron Dawg derives a compact vector differential equation for a point particle moving through a smoothly varying refractive medium using the Euler-Lagrange variational method. By introducing a log refractive index called "fluff density," the paper expresses acceleration purely in terms of the fluff gradient and velocity, then explores curvature, superposition, and point-source capture radii with simple closed-form results.
Frequency Formula for a Pure Complex Tone in a DTFT
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.
Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT
Cedron Dawg derives closed-form three-bin frequency estimators for a pure complex tone in a DFT using a linear algebra view that treats three adjacent bins as a vector. He shows any vector K orthogonal to [1 1 1] yields a = (K·Z)/(K·D·Z) and derives practical K choices including a Von Hann (Pascal) kernel and a data-driven projection. The post compares estimators under noise and gives simple selection rules.
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
Cedron Dawg derives an exact two-bin frequency formula for a pure complex tone in the DFT, eliminating amplitude and phase to isolate frequency via a complex quotient and the complex logarithm. He presents an adjacent-bin simplification that replaces a complex multiply with a bin offset plus an atan2 angle, and discusses integer-frequency handling and aliasing. C source and numerical examples show the formula working in practice.
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins
Cedron presents exact, closed-form formulas to extract the phase and amplitude of a pure complex tone from multiple DFT bin values, using a compact vector formulation. The derivation introduces a delta variable to simplify the sinusoidal bin expression, stacks neighboring bins into a basis vector, and solves for the complex amplitude q by projection. The phase and magnitude follow directly from q, and extra bins reduce leakage when the tone falls between bins.
Overview of my Articles
Cedron presents a guided tour of his DSPRelated articles that teach the discrete Fourier transform through derivations, numerical examples, and sample code. The collection centers on novel "bin value" formulas and exact frequency estimators for complex and real tones, with methods for phase and amplitude recovery and iterative multitone resolution. The overview also points to a zeroing-sine window family and an integer pseudo-differentiator for efficient peak and zero-crossing detection.
DFT Bin Value Formulas for Pure Complex Tones
Cedron Dawg derives closed-form DFT bin formulas for single complex exponentials, eliminating the need for brute-force summation and showing how phase acts as a uniform rotation of all bins. He also gives a Dirichlet-kernel form that yields the magnitude as (M/N)|sin(δN/2)/sin(δ/2)|, explains the large-N sinc limit, and includes C code to verify the results.
Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT
Cedron Dawg derives exact, closed-form frequency formulas that recover a pure real tone from just two DFT bins using a geometric vector approach. The method projects bin-derived vectors onto a plane orthogonal to a constraint vector to eliminate amplitude and phase, yielding an explicit cos(alpha) estimator; a small adjustment improves noise performance so the estimator rivals and slightly betters earlier two-bin methods.
