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.
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.
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.
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.
Coefficients of Cascaded Discrete-Time Systems
Multiplying discrete-time transfer functions is just polynomial multiplication, and polynomial multiplication is convolution. Neil Robertson shows that the numerator and denominator coefficients of cascaded systems come from convolving the individual coefficient vectors, then demonstrates the idea with MATLAB code and a 2nd-order IIR cascade that yields a 4th-order response. The approach makes computing time and frequency responses straightforward.
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.
Design IIR Highpass Filters
Neil Robertson walks through a compact, six-step procedure to synthesize IIR Butterworth highpass filters using pre-warping and the bilinear transform. The post gives the pole transformations, the placement of N zeros at z=1, the scaling to unity gain at fs/2, and a ready-to-run MATLAB hp_synth implementation that reproduces MATLAB's butter results.
Design IIR Band-Reject Filters
This post walks through designing IIR Butterworth band-reject filters and provides two MATLAB synthesis functions, br_synth1.m and br_synth2.m. br_synth1 accepts a null frequency plus an upper -3 dB frequency, while br_synth2 takes lower and upper -3 dB frequencies. The author demonstrates an example where a 2nd-order prototype yields a 4th-order H(z), prints b and a coefficients, and plots the response using freqz.
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.
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.
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.
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.
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.
The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase
Rick Lyons pulls back the curtain on a little-known coefficient constraint that makes complex-coefficient FIR filters exhibit linear phase. Rather than simple symmetry of real coefficients, the key is a conjugate-reflection relation involving the filter phase at DC, which collapses to ordinary symmetry for real taps. The post includes derivations, intuition using the inverse DTFT, and a Matlab example to verify the result.
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.
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.
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.
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.
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.
Sinusoidal Frequency Estimation Based on Time-Domain Samples
Rick Lyons presents three time-domain algorithms for estimating the frequency of real and complex sinusoids from samples. He shows that the Real 3-Sample and Real 4-Sample estimators, while mathematically exact, fail in the presence of noise and can produce biased or invalid outputs. The Complex 2-Sample (Lank-Reed-Pollon) estimator is more robust but can be biased at low SNR and near 0 or Fs/2, so narrowband filtering is recommended.
Design IIR Band-Reject Filters
This post walks through designing IIR Butterworth band-reject filters and provides two MATLAB synthesis functions, br_synth1.m and br_synth2.m. br_synth1 accepts a null frequency plus an upper -3 dB frequency, while br_synth2 takes lower and upper -3 dB frequencies. The author demonstrates an example where a 2nd-order prototype yields a 4th-order H(z), prints b and a coefficients, and plots the response using freqz.
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.
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.
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.
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.
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.
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.
ADC Clock Jitter Model, Part 2 – Random Jitter
Neil Robertson shows how to simulate ADC sample-clock random jitter in Matlab, moving from band-limited Gaussian noise to wideband and close-in phase noise. The post highlights practical artifacts such as aliasing of wideband clock noise, the 20*log10 dependence of jitter sidebands on input frequency, and why cubic interpolation plus a custom noise_filter produces accurate rms and spectral results engineers can trust.
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.
How the Cooley-Tukey FFT Algorithm Works | Part 1 - Repeating Calculations
The Fourier Transform is a powerful tool, used in many technologies, from audio processing to wireless communication. However, calculating the FT can be computationally expensive. The Cooley-Tukey Fast Fourier Transform (FFT) algorithm provides a significant speedup. It exploits the repetitive nature of calculations within the Discrete Fourier Transform (DFT), the mathematical foundation of the FT. By recognizing patterns in the DFT calculations and reusing intermediate results, the FFT vastly reduces the number of operations required. In this series of articles, we will look at how the Cooley-Tukey FFT algorithm works.
