The Power Spectrum
You can get absolute power from a DFT, not just relative spectra. In this post Neil Robertson shows how to convert FFT outputs into watts per bin using Parseval's theorem, how to form one-sided spectra, and how to normalize windows so power is preserved. Matlab examples demonstrate bin-centered and between-bin sinusoids, leakage, scalloping, and how to recover component power by summing bins.
Wavelets I - From Filter Banks to the Dilation Equation
Starting from a practical cascaded FIR filter bank, this post derives the key equations behind the Fast Wavelet Transform. It shows how conjugate-quadrature analysis and synthesis filters give perfect reconstruction and how iterating the cascade produces the scaling function, leading to the dilation equation. DB4 coefficients are used as a concrete example and a linear-system trick yields exact integer-sample values of the scaling function.
An s-Plane to z-Plane Mapping Example
A misleading online diagram prompted Rick Lyons to reexamine how s-plane points map to the z-plane. He spotted apparent errors in the original figure, drew a corrected mapping, and invites readers to inspect both diagrams and point out any remaining mistakes. The short post is a quick visual primer for engineers who rely on accurate s-plane to z-plane mappings in analysis and design.
Digital PLL's -- Part 2
Neil Robertson builds a Z-domain model of a second-order digital PLL with a proportional-plus-integral loop filter, then derives closed-form formulas for KL and KI from the desired loop natural frequency and damping. The post explains the s → (z - 1)/Ts approximation, shows how to form the closed-loop IIR CL(z) for step and frequency responses, and highlights when the linear Z-domain model falls short of nonlinear acquisition behavior.
Digital PLL's -- Part 1
A hands-on introduction to time-domain digital phase-locked loops, Neil Robertson builds a simple DPLL model in MATLAB and walks through the NCO, phase detector, and PI loop filter implementations. The post uses phase-in-cycles arithmetic to show how the phase accumulator, detector wrapping, and loop filter interact, and it contrasts linear steady-state behavior with the nonlinear acquisition seen when initial frequency error is large. Part 2 will cover frequency-domain tuning of the loop gains.
Peak to Average Power Ratio and CCDF
Setting digital modulator levels depends on peak-to-average power ratio, because random signals produce occasional high peaks that cause clipping. This post shows how to compute the CCDF of PAPR from a signal vector, with MATLAB code and examples for a sine wave and Gaussian noise. The examples reveal the fixed 3.01 dB PAPR of a sine and the need for large sample counts to capture rare AWGN peaks.
Filter a Rectangular Pulse with no Ringing
You can filter a rectangular pulse with no ringing simply by using an FIR whose coefficients are all positive, and make them symmetric to get identical leading and trailing edges. This post walks through a MATLAB example that convolves a normalized Hanning window with a 32-sample rectangular pulse, showing that window length controls edge duration and that shorter windows widen the spectrum. It also notes this is not a QAM pulse-shaping solution.
Digital Envelope Detection: The Good, the Bad, and the Ugly
Envelope detection sounds simple, but implementation choices change everything. Rick Lyons gathers common digital detectors, including half-wave, full-wave, square-law, Hilbert-based complex, and synchronous coherent designs, and explains how harmonics, filtering, and carrier recovery change results. He ranks detectors by output SNR from a representative simulation and offers practical tips on filter cutoff, Hilbert transformer bandwidth, and when a simple detector is good enough.
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.
Discrete-Time PLLs, Part 1: Basics
In this series of tutorials on discrete-time PLLs we will be focusing on Phase-Locked Loops that can be implemented in discrete-time signal proessors such as FPGAs, DSPs and of course, MATLAB.
Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals
Textbooks rarely give ready formulas for tracking where individual spectral lines land after bandpass sampling or decimation. Rick Lyons provides three concise equations, with Matlab code, that compute translated frequencies for analog bandpass sampling, real digital downsampling, and complex downsampling. Practical examples show how to place the sampled image at fs/4 and how to translate a complex bandpass to baseband for efficient demodulation.
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.
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.
DFT Bin Value Formulas for Pure Real Tones
Cedron Dawg derives a closed-form expression for the DFT bin values produced by a pure real sinusoid, then uses that formula to explain well known DFT behaviors. The post walks through the algebra from Euler identities to a compact computational form, highlights the integer versus non-integer frequency cases, and verifies the result with C code and printed numeric output.
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.
Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka
For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2]. In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file.
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.
Candan's Tweaks of Jacobsen's Frequency Approximation
Cedron Dawg shows how small tweaks to Jacobsen's three-bin frequency estimator turn a popular approximation into an exact formula, and how a modest cubic correction yields a near-exact, low-cost alternative. The article derives an arctan/tan exact recovery, relates it to Candan's 2011/2013 tweaks, and supplies reference C code and numerical tables so engineers can see when each formula is sufficient.
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.
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.
Filter a Rectangular Pulse with no Ringing
You can filter a rectangular pulse with no ringing simply by using an FIR whose coefficients are all positive, and make them symmetric to get identical leading and trailing edges. This post walks through a MATLAB example that convolves a normalized Hanning window with a 32-sample rectangular pulse, showing that window length controls edge duration and that shorter windows widen the spectrum. It also notes this is not a QAM pulse-shaping solution.
How the Cooley-Tukey FFT Algorithm Works | Part 3 - The Inner Butterfly
At the heart of the Cooley-Tukey FFT algorithm lies a butterfly, a simple yet powerful image that captures the recursive nature of how the FFT works. In this article we discover the butterfly’s role in transforming complex signals into their frequency components with efficiency and elegance. Starting with the 2-point DFT, we reveal how the FFT reuses repeated calculations to save time and resources. Using a divide-and-conquer approach, the algorithm breaks signals into smaller groups, processes them through interleaving butterfly diagrams, and reassembles the results step by step.
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.
FIR Filter to Match Any Magnitude and Phase Response
This post details a technique for designing high quality FIR filters that match arbitrary magnitude and phase responses.
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.
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.
The Discrete Fourier Transform of Symmetric Sequences
Symmetric sequences arise often in digital signal processing. Examples include symmetric pulses, window functions, and the coefficients of most finite-impulse response (FIR) filters, not to mention the cosine function. Examining symmetric sequences can give us some insights into the Discrete Fourier Transform (DFT). An even-symmetric sequence is centered at n = 0 and xeven(n) = xeven(-n). The DFT of xeven(n) is real. Most often, signals we encounter start at n = 0, so they are not strictly speaking even-symmetric. We’ll look at the relationship between the DFT’s of such sequences and those of true even-symmetric sequences.
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.
Approximating the area of a chirp by fitting a polynomial
Once in a while we need to estimate the area of a dataset in which we are interested. This area could give us, for example, force (mass vs acceleration) or electric power (electric current vs charge).
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.
