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.
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.
Interpolator Design: Get the Stopbands Right
In this article, I present a simple approach for designing interpolators that takes the guesswork out of determining the stopbands.
The Discrete Fourier Transform as a Frequency Response
Neil Robertson shows that the discrete frequency response H(k) of an FIR filter is exactly the DFT of its impulse response h(n). He derives the continuous H(ω) and discrete H(k) using complex exponentials for a four-tap FIR, then replaces h(n) with x(n) to recover the general DFT formula. The post keeps the math simple and calls out topics left for separate treatment, such as windowing and phase.
Add the Hilbert Transformer to Your DSP Toolkit, Part 2
This post shows a simple practical route to a Hilbert transformer by starting from a half-band FIR filter and tweaking its symmetry. It walks through a 19-tap example synthesized with Matlab's firpm (Parks-McClellan), explains the required frequency scaling, and shows how even-numbered taps become (or can be forced) zero through symmetry and coefficient quantization. Useful design rules are summarized for choosing ntaps.
Add the Hilbert Transformer to Your DSP Toolkit, Part 1
Learn how the Hilbert transformer creates a 90-degree phase-shifted quadrature component without down-conversion, and why it is simply a special FIR filter. Part 1 defines the transformer, derives its ideal frequency response H(ω)=j for ω<0 and -j for ω≥0, and walks through Matlab examples that demonstrate phase shifting and image attenuation for bandpass signals.
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.
A Recipe for a Basic Trigonometry Table
Cedron Dawg walks through building a degree-based sine and cosine table from first principles, showing both recursive and multiplicative complex-tone generators. The article highlights simple drift-correction tricks such as mitigated squaring and compact normalization, gives series methods to compute one-degree and half-degree values, and includes practical C code that ties the table to DFT usage and frequency estimation.
Learn About Transmission Lines Using a Discrete-Time Model
A simple discrete-time approach makes lossless transmission-line behavior easy to simulate and visualize. The post introduces MATLAB functions tline and wave_movie to model uniform lossless lines with resistive terminations, compute time and frequency responses, and animate travelling waves. A microstrip pulse example shows how reflections produce ringing and how source matching nearly eliminates it, making this a practical learning tool.
The Discrete Fourier Transform and the Need for Window Functions
The FFT alone can mislead: capturing a finite-length signal with a rectangular window smears energy across frequency, producing spectral leakage that hides real components. This post explains the origin of leakage, shows how tapered windows such as the Hanning window suppress sidelobes, and demonstrates the tradeoff between sidelobe suppression and mainlobe widening while covering practical tips on zero-padding and record length.
Use Matlab Function pwelch to Find Power Spectral Density -- or Do It Yourself
Neil Robertson walks through using Matlab's pwelch and shows how to implement PSD estimation yourself with fft. The post uses concrete examples and complete m-files to demonstrate window selection, converting pxx (W/Hz) to W/bin, Welch DFT averaging, and a worked C/N0 calculation. Readers get practical, runnable recipes for accurate spectrum units, variance reduction with averaging, and peak-power extraction.
A Lesson in Statistics Using Random Sequences
Statistics may come naturally to some people, but it is a difficult subject for many of us. Using simulations helps to remove some of the mystery of statistics. Luckily, it is trivial to produce a pseudo-random Gaussian or uniform sequence: a single command in Matlab (or Python) does it. In this article, I try to explain a few concepts using Gaussian and uniform random sequences generated in Matlab. First, we’ll generate a Gaussian sequence, calculate its variance, and plot a histogram and probability density function (PDF). Next, we’ll simulate the roll of a single die to illustrate a uniform distribution. Then we’ll simulate rolling two dice and finally, as an illustration of the Central Limit Theorem, we’ll sum the total when rolling several dice to obtain an approximately Gaussian distribution.
Design IIR Filters Using Cascaded Biquads
High-order IIR filters are numerically sensitive, especially at low cutoff frequencies. This article shows how to implement a Butterworth lowpass as a cascade of second-order biquads, deriving the per-section coefficient formulas and giving a Matlab biquad_synth example. It explains computing denominator coefficients from pole pairs, using b = [1 2 1] with K = sum(a)/4 for unity DC gain, and highlights reduced quantization sensitivity.
Add the Hilbert Transformer to Your DSP Toolkit, Part 2
This post shows a simple practical route to a Hilbert transformer by starting from a half-band FIR filter and tweaking its symmetry. It walks through a 19-tap example synthesized with Matlab's firpm (Parks-McClellan), explains the required frequency scaling, and shows how even-numbered taps become (or can be forced) zero through symmetry and coefficient quantization. Useful design rules are summarized for choosing ntaps.
Add the Hilbert Transformer to Your DSP Toolkit, Part 1
Learn how the Hilbert transformer creates a 90-degree phase-shifted quadrature component without down-conversion, and why it is simply a special FIR filter. Part 1 defines the transformer, derives its ideal frequency response H(ω)=j for ω<0 and -j for ω≥0, and walks through Matlab examples that demonstrate phase shifting and image attenuation for bandpass signals.
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.
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.
DAC Zero-Order Hold Models
This article provides two simple time-domain models of a DAC’s zero-order hold. These models will allow us to find time and frequency domain approximations of DAC outputs, and simulate analog filtering of those outputs. Developing the models is also a good way to learn about the DAC ZOH function.
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.
Design IIR Bandpass Filters
Designing Butterworth IIR bandpass filters is easier than it looks when you start from a lowpass prototype. This post walks through the s-domain lowpass-to-bandpass transform, bilinear digital mapping, and the bp_synth.m Matlab implementation that produces scaled numerator and denominator coefficients. Practical pole-zero intuition and Matlab examples help you verify magnitude and group-delay behavior for real sampling rates and bandwidths.
Simplest Calculation of Half-band Filter Coefficients
Half-band FIR filters put the cutoff at one-quarter of the sampling rate, and nearly half their coefficients are exactly zero, which makes them highly efficient for decimation-by-2 and interpolation-by-2. This post shows the straightforward window-method derivation of half-band coefficients from the ideal sinc impulse response, providing a clear, hands-on explanation for engineers learning filter design. It also points to equiripple options such as Matlab's firhalfband and a later Parks-McClellan implementation.
Interpolation Basics
Neil Robertson demonstrates interpolation by an integer factor using a frequency-domain approach, showing how zero-insertion followed by an FIR low-pass filter reconstructs a higher-rate signal. The article walks through spectra, passband and stopband selection, and a 41-tap Parks-McClellan filter example applied to a Chebyshev-window test signal. Matlab code and percent-error plots are included so engineers can reproduce and evaluate the method.
How to Find a Fast Floating-Point atan2 Approximation
This post shows how a compact, fast atan2 can be built from a Remez-derived arctangent approximation and a matching 3rd-order polynomial. It walks through using Boost's remez_minimax to recover coefficients 0.97239411 and -0.19194795, integrating the polynomial into an atan2 with quadrant reduction, and applying branch reduction, bit tricks, and SSE2 SIMD to cut runtime while keeping max error under about 0.005 radians.
Second Order Discrete-Time System Demonstration
Want a hands-on way to see how continuous second-order dynamics appear in discrete time? Neil Robertson converts a canonical H(s) to H(z), shows z-plane pole mapping for different damping ratios, and walks through impulse-invariance scaling and zero placement. The post includes a MATLAB function so_demo.m that computes numerator and denominator coefficients, plots poles, and compares impulse and frequency responses so you can experiment with sampling effects.
Plotting Discrete-Time Signals
Neil Robertson demonstrates a practical interpolate-by-8 FIR approach to make sampled signals look like their continuous-time counterparts when plotted. The post explains a 121-tap filter designed for signals up to 0.4*fs, shows Matlab examples for a sinusoid and a filtered pulse, and highlights the transient and design trade-offs so you can reproduce clean plots with the supplied interp_by_8.m code.
Phase or Frequency Shifter Using a Hilbert Transformer
A Hilbert transformer converts a real input into an analytic I+jQ pair, enabling phase shifts and frequency shifts while keeping real inputs and outputs. This article shows Matlab implementations (31-tap FIR with Hamming or Blackman windows), derives y = I cosθ - Q sinθ for phase and frequency shifting, and highlights practical limits from finite taps and coefficient/NCO quantization.
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.
Evaluate Window Functions for the Discrete Fourier Transform
Spectral leakage makes DFTs of continuous sinewaves misleading, and windowing is the practical workaround. This post supplies Matlab code to plot spectra of windowed sinewaves and compute figures of merit, so you can compare windows such as flattop and Chebyshev. See how sidelobe level, mainlobe bandwidth, processing loss, noise bandwidth, and scallop loss trade off to guide your window choice.
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.
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.
