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.
IIR Bandpass Filters Using Cascaded Biquads
This post provides a Matlab function that builds Butterworth bandpass IIR filters by cascading second-order biquad sections. The biquad approach, implemented in Direct Form II, reduces sensitivity to coefficient quantization, which matters most for narrowband filters. The included biquad_bp function computes each section's feedforward and feedback coefficients plus gains from a lowpass prototype order, center frequency, bandwidth, and sampling rate.
Generating Partially Correlated Random Variables
Designing signals to match a target covariance is simpler than it sounds. This post shows how to build partially correlated complex signals by hand for the two-signal case, then generalizes to N signals using the Cholesky decomposition. Short MATLAB examples demonstrate the two-line implementation and the article highlights numerical caveats when a covariance is only positive semidefinite.
Angle Addition Formulas from Euler's Formula
Complex numbers are rotations and scalings in the plane, and Cedron Dawg walks through polar and Cartesian representations to make that concrete. Using Euler's formula, the article shows how multiplying complex numbers multiplies magnitudes and adds angles, and how that directly yields the sine and cosine angle-addition formulas. Practical notes cover using atan2/arg and a brief Gambas example to verify results.
Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT
This post makes a basic DSP principle tangible by computing the DFT over an extended set of bins and plotting the results. It demonstrates that a sampled signal's spectrum repeats every sampling rate, explains the k-to-frequency mapping, and contrasts common bin ranges such as 0..N-1 and -N/2..N/2-1. The write-up also highlights symmetry for real sequences and recommends using the FFT for efficiency.
Compute the Frequency Response of a Multistage Decimator
This post shows a practical way to compute the full frequency response of a multistage decimator by representing every stage at the input sample rate. The author walks through upsampling lower-rate FIR coefficients, convolving to form the overall impulse response, and taking a DFT, then demonstrates how aliasing and stopband placement affect the aliased components. Example Matlab code and plots illustrate each step.
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.
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.
Design a DAC sinx/x Corrector
Neil Robertson provides a compact Matlab function and coefficient tables for designing linear-phase FIR sinx/x correctors to undo the DAC sinc roll-off. The post explains the sinc_corr(ntaps,fmax,fs) call, shows worked examples with ntaps=5 and different fmax values, and demonstrates fixed-point quantization including a k=512 example and CSD digit guidance. Practical notes cover corrector gain and input back-off to avoid clipping.
Digital PLL's, Part 3 -- Phase Lock an NCO to an External Clock
Phase-locking a numerically controlled oscillator to an external clock that is unrelated to system clocks is practical and largely unexplored. Neil Robertson presents a time-domain digital PLL that converts the ADC-sampled clock into I/Q with a Hilbert transformer and measures phase error with a compact complex phase detector. The post shows loop-filter coefficient formulas and simulations that reveal how ADC quantization and Gaussian clock noise map into NCO phase noise and how loop bandwidth shapes the result.
Modeling Anti-Alias Filters
Modeling anti-alias filters brings textbook aliasing examples to life. This post shows how to build discrete-time models G(z) for analog Butterworth and Chebyshev lowpass anti-alias filters, compares bilinear transform and impulse invariance, and simulates ADC input/output including aliasing of sinusoids and Gaussian noise. It concludes that impulse invariance gives better stopband accuracy and includes Matlab helper functions.
IIR Bandpass Filters Using Cascaded Biquads
This post provides a Matlab function that builds Butterworth bandpass IIR filters by cascading second-order biquad sections. The biquad approach, implemented in Direct Form II, reduces sensitivity to coefficient quantization, which matters most for narrowband filters. The included biquad_bp function computes each section's feedforward and feedback coefficients plus gains from a lowpass prototype order, center frequency, bandwidth, and sampling rate.
ADC Clock Jitter Model, Part 1 -- Deterministic Jitter
Clock jitter on ADC sample clocks corrupts high-frequency signals, and this post builds a practical MATLAB model to show exactly how deterministic (periodic) jitter maps into phase modulation and discrete sidebands. The author explains a parabolic-interpolation approach using twice-rate samples, demonstrates examples from single tones to pulses, and matches simulation spectra to closed-form sideband formulas so engineers can predict jitter effects.
Model a Sigma-Delta DAC Plus RC Filter
Sigma-delta digital-to-analog converters (SD DAC’s) are often used for discrete-time signals with sample rate much higher than their bandwidth. For the simplest case, the DAC output is a single bit, so the only interface hardware required is a standard digital output buffer. Because of the high sample rate relative to signal bandwidth, a very simple DAC reconstruction filter suffices, often just a one-pole RC lowpass. In this article, I present a simple Matlab function that models the combination of a basic SD DAC and one-pole RC filter. This model allows easy evaluation of the overall performance for a given input signal and choice of sample rate, R, and C.
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.
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.
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.
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.
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.
Digital PLL's, Part 3 -- Phase Lock an NCO to an External Clock
Phase-locking a numerically controlled oscillator to an external clock that is unrelated to system clocks is practical and largely unexplored. Neil Robertson presents a time-domain digital PLL that converts the ADC-sampled clock into I/Q with a Hilbert transformer and measures phase error with a compact complex phase detector. The post shows loop-filter coefficient formulas and simulations that reveal how ADC quantization and Gaussian clock noise map into NCO phase noise and how loop bandwidth shapes the result.
Angle Addition Formulas from Euler's Formula
Complex numbers are rotations and scalings in the plane, and Cedron Dawg walks through polar and Cartesian representations to make that concrete. Using Euler's formula, the article shows how multiplying complex numbers multiplies magnitudes and adds angles, and how that directly yields the sine and cosine angle-addition formulas. Practical notes cover using atan2/arg and a brief Gambas example to verify results.
Digital PLL's, Part 3 -- Phase Lock an NCO to an External Clock
Phase-locking a numerically controlled oscillator to an external clock that is unrelated to system clocks is practical and largely unexplored. Neil Robertson presents a time-domain digital PLL that converts the ADC-sampled clock into I/Q with a Hilbert transformer and measures phase error with a compact complex phase detector. The post shows loop-filter coefficient formulas and simulations that reveal how ADC quantization and Gaussian clock noise map into NCO phase noise and how loop bandwidth shapes the result.
Compute the Frequency Response of a Multistage Decimator
This post shows a practical way to compute the full frequency response of a multistage decimator by representing every stage at the input sample rate. The author walks through upsampling lower-rate FIR coefficients, convolving to form the overall impulse response, and taking a DFT, then demonstrates how aliasing and stopband placement affect the aliased components. Example Matlab code and plots illustrate each step.
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.
Canonic Signed Digit (CSD) Representation of Integers
Canonic Signed Digit (CSD) encoding slashes the number of nonzero bits in integer coefficients, enabling multiplierless FIR filters implemented with shifts and adds. This post uses MATLAB code to demonstrate CSD rules, show how negative values work, and plot the distribution of signed digits as bit width changes. It finishes with practical techniques to minimize signed digits per coefficient for area and power efficient filter designs.
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.
How precise is my measurement?
Precision is quantifiable, not guesswork. This post walks through practical, measurement-oriented statistics you can apply to static or dynamic signals to answer the question, "How precise is my measurement?" It focuses on using multiple samples, checking distribution assumptions, and constructing confidence intervals and levels so you can trade measurement time for a desired precision.
Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT
This post makes a basic DSP principle tangible by computing the DFT over an extended set of bins and plotting the results. It demonstrates that a sampled signal's spectrum repeats every sampling rate, explains the k-to-frequency mapping, and contrasts common bin ranges such as 0..N-1 and -N/2..N/2-1. The write-up also highlights symmetry for real sequences and recommends using the FFT for efficiency.
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.
Compute Modulation Error Ratio (MER) for QAM
Neil Robertson shows how to define and compute Modulation Error Ratio (MER) for QAM using a simplified baseband model and decision-slice errors. The post derives per-symbol and averaged MER formulas, explains when MER tracks carrier-to-noise ratio under AWGN and matched root-Nyquist filters, and provides example Pav values for QAM-16 and QAM-64 plus a Matlab script and practical tips.
