The Real Star of Star Trek
Rick Lyons argues the real star of Star Trek is not an actor but the USS Enterprise, whose image drove much of the franchise's power. He traces the ship from two 1966 scale models through Smithsonian restoration, NASA naming influence, global architecture, and magazine art to show how an engineered prop became a worldwide cultural icon. The piece mixes nostalgia with concrete examples and a hands-on modeler lesson.
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.
Should DSP Undergraduate Students Study z-Transform Regions of Convergence?
Rick Lyons argues z-transform regions of convergence are mostly a classroom abstraction with little practical use for real-world DSP engineers. For all stable LTI impulse responses encountered in practice the ROC includes the unit circle, so DTFT and DFT exist and ROC analysis rarely affects implementation. He notes digital oscillators are a notable exception, and suggests reallocating classroom time to more practical engineering topics.
Implementing Impractical Digital Filters
Some published IIR block diagrams are impossible to implement because they contain delay-less feedback paths, and Rick Lyons shows how simple algebra fixes that. He works through two concrete examples—a bandpass built from a FIR notch and a narrowband notch using a feedback loop—and derives equivalent, implementable second-order IIR transfer functions. The post emphasizes spotting problematic loops and replacing them with practical block diagrams.
An Astounding Digital Filter Design Application
Rick Lyons was astonished by the ASN Filter Designer, a hands-on filter design tool that makes tweaking frequency responses as simple as dragging markers with your mouse. The software updates magnitude plots, z-plane pole/zero locations, and filter coefficients in real time, and it also includes a signal analyzer plus a MATLAB-like scripting language for custom coefficient generation. The post links to a demo and user guides so you can try it yourself.
The Swiss Army Knife of Digital Networks
A single discrete-signal network can masquerade as a comb filter, a recursive section, or something much more versatile. Rick Lyons shows how this seven-coefficient structure can be reconfigured to realize a wide range of DSP functions, with tables of impulse responses, pole-zero plots, and frequency responses to illustrate each case. The full explanations live in the downloadable PDF, but the post gives a strong feel for why this is such a handy building block.
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.
A Useful Source of Signal Processing Information
A surprisingly handy web tool turned up for finding signal processing material in PDF and PowerPoint form. Rick Lyons shows how a plain-looking site can surface lots of topic-specific documents, using FM demodulation as the example. If you often hunt for reference slides and papers, this is a quick source worth bookmarking.
Optimizing the Half-band Filters in Multistage Decimation and Interpolation
Multistage decimation and interpolation by powers of two get a lot cheaper if you size each half-band filter differently. Rick Lyons walks through spectra for three-stage examples that show why early stages can use narrower filters for decimation while interpolation reverses the order, and how aliasing and images are handled by later stages. Learn a simple rule to cut multipliers without sacrificing performance.
Implementing Simultaneous Digital Differentiation, Hilbert Transformation, and Half-Band Filtering
Recently I've been thinking about digital differentiator and Hilbert transformer implementations and I've developed a processing scheme that may be of interest to the readers here on dsprelated.com.
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.
Accurate Measurement of a Sinusoid's Peak Amplitude Based on FFT Data
Measuring a sinewave's peak from FFT data can be severely biased by scalloping loss, producing errors up to 36.3 percent. Rick Lyons demonstrates how to apply a flat-top window via frequency-domain convolution to the FFT bins, cutting maximum amplitude error to about 0.02 dB compared with 3.9 dB for rectangular windows. The post includes Matlab code and practical caveats for reliable use.
A Brief Introduction To Romberg Integration
Romberg integration delivers dramatic accuracy gains for definite integrals by combining multiple trapezoidal approximations into a single highly accurate result. Rick Lyons demonstrates how just five samples can achieve 0.0038% error versus a trapezoidal rule needing 100 samples, and a 17-sample example hits 3.6×10−4% error. The post outlines the N-segment procedure, cost scaling, and links to MATLAB code.
Reduced-Delay IIR Filters
Rick Lyons investigates a simple 2nd-order IIR modification that reduces passband group delay by just under one sample, inspired by Steve Maslen's reduced-delay concept. He walks through the conversion steps and compares z-plane, magnitude, and group-delay plots for Butterworth, elliptic, and Chebyshev prototypes, showing how zeros shift and stopband attenuation degrades. A linked PDF extends the study to 1st-, 3rd-, and 4th-order cases so you can follow the tradeoffs.
An Astounding Digital Filter Design Application
Rick Lyons was astonished by the ASN Filter Designer, a hands-on filter design tool that makes tweaking frequency responses as simple as dragging markers with your mouse. The software updates magnitude plots, z-plane pole/zero locations, and filter coefficients in real time, and it also includes a signal analyzer plus a MATLAB-like scripting language for custom coefficient generation. The post links to a demo and user guides so you can try it yourself.
The DFT Magnitude of a Real-valued Cosine Sequence
Rick Lyons proves a simple but often-missing result: the N-point DFT peak magnitude of a real cosine with an integer number of cycles equals A·N/2. He uses Euler's formula and geometric-series summation, shows a neat shortcut that avoids l'Hôpital's rule, and connects the math to practical fixed-point FFT sizing and overflow prevention on two's-complement hardware. The post also notes conjugate symmetry and the same result for sine inputs.
Algebra's Laws of Powers and Roots: Handle With Care
Rick Lyons shows that familiar power and root rules from algebra can break down when exponents are complex. He tests common identities for two scenarios, real and fully complex exponents, with positive and negative mantissas, and compiles a table of cases that sometimes fail. The post includes MATLAB examples that reproduce counterexamples and a clear warning to numerically verify algebraic steps involving complex powers.
Should DSP Undergraduate Students Study z-Transform Regions of Convergence?
Rick Lyons argues z-transform regions of convergence are mostly a classroom abstraction with little practical use for real-world DSP engineers. For all stable LTI impulse responses encountered in practice the ROC includes the unit circle, so DTFT and DFT exist and ROC analysis rarely affects implementation. He notes digital oscillators are a notable exception, and suggests reallocating classroom time to more practical engineering topics.
Beat Notes: An Interesting Observation
Rick Lyons overturns a common intuition about beat notes, showing that adding two nearby audio tones yields an average-frequency tone whose amplitude fluctuates, rather than a separate low-frequency sinusoid. He contrasts multiplication and summation of sines, provides simple trigonometric insight, and includes Matlab audio demos to explain why aircraft engine "whump" sounds are amplitude fluctuations of the average engine frequency.
Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images
Zero stuffing in the time domain creates spectral copies, and Rick Lyons walks through why that happens using DFT and DFS viewpoints. He shows that inserting L-1 zeros between samples yields a longer DFT with replicated spectral blocks, and that true interpolation requires lowpass filtering to remove those images. The post uses a concrete L=3 example and an inverse-DFT summation proof to make the effect intuitive.
Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images
Zero stuffing in the time domain creates spectral copies, and Rick Lyons walks through why that happens using DFT and DFS viewpoints. He shows that inserting L-1 zeros between samples yields a longer DFT with replicated spectral blocks, and that true interpolation requires lowpass filtering to remove those images. The post uses a concrete L=3 example and an inverse-DFT summation proof to make the effect intuitive.
An Astounding Digital Filter Design Application
Rick Lyons was astonished by the ASN Filter Designer, a hands-on filter design tool that makes tweaking frequency responses as simple as dragging markers with your mouse. The software updates magnitude plots, z-plane pole/zero locations, and filter coefficients in real time, and it also includes a signal analyzer plus a MATLAB-like scripting language for custom coefficient generation. The post links to a demo and user guides so you can try it yourself.
A Simple Complex Down-conversion Scheme
Rick Lyons shows a compact way to turn a real bandpass signal centered at ±fs/4 into a complex, zero-centered analytic signal. The trick uses a delay, a Hilbert transform filter, and a 4:1 downsample, with a small compensation filter to widen the usable passband. He also points out a no-multiplier implementation using shift-and-add coefficients, or a higher-attenuation version with two multiplies per output sample.
Correcting an Important Goertzel Filter Misconception
A common claim says the Goertzel algorithm is marginally stable and prone to numerical errors. Rick Lyons shows that the usual second-order Goertzel filter has conjugate poles exactly on the unit circle, so pole placement alone does not make it unstable. The practical limits are coefficient quantization, which reduces frequency precision, and accumulator overflow for very large N.
Should DSP Undergraduate Students Study z-Transform Regions of Convergence?
Rick Lyons argues z-transform regions of convergence are mostly a classroom abstraction with little practical use for real-world DSP engineers. For all stable LTI impulse responses encountered in practice the ROC includes the unit circle, so DTFT and DFT exist and ROC analysis rarely affects implementation. He notes digital oscillators are a notable exception, and suggests reallocating classroom time to more practical engineering topics.
The DFT of Finite-Length Time-Reversed Sequences
Rick Lyons digs into a surprisingly under-documented corner of DSP, showing how finite-length time reversal changes a sequence's DFT. The post distinguishes flip and circular time-reversal, gives closed-form DFT relationships, and explains why modulo N arithmetic matters. Engineers get ready-to-use tables and derivations that clarify when and how time reversal affects spectral analysis.
Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved
Rick Lyons resolves a long-standing confusion about the coupled-form 2nd-order IIR resonator by deriving its correct z-domain transfer function and explaining why textbooks can appear to contradict pole plots. He shows that with infinite precision the coupled and standard denominators match, but finite-bit quantization of rcos(Θ) and rsin(Θ) changes the z^-2 coefficient and shifts pole positions. Read to learn the correct H(z) to predict quantized behavior and when the coupled form outperforms the standard design.
Is It True That j is Equal to the Square Root of -1 ?
A viral YouTube video claimed that saying j equals the square root of negative one is wrong. Rick Lyons shows the apparent paradox comes from misusing square-root identities with negative arguments, not from the usual definition of j. He argues it is safer to define j by j^2 = -1 and illustrates how careless root operations produce contradictions in two appendices.
Frequency Translation by Way of Lowpass FIR Filtering
Rick Lyons shows how you can translate a signal down in frequency and lowpass filter it in a single operation by embedding cosine mixing values into FIR coefficients. The post explains how to build the translating FIR, how to choose the number of coefficient sets, and how decimation can dramatically reduce storage needs while noting practical constraints like the requirement that ft be an integer submultiple of fs.
Improved Narrowband Lowpass IIR Filters
Rick Lyons presents a practical trick from his DSP book that makes narrowband lowpass IIR filters usable in fixed-point systems. By replacing unit delays with M-length delay lines to form an interpolated-IIR, pole radii and angles are transformed so desired poles fall into quantizer-friendly locations without wider coefficient words or extra multiplies. A following CIC image-reject stage removes replicated passbands to meet tight stopband specs.
