A New Contender in the Digital Differentiator Race
Rick Lyons presents a compact FIR differentiator that widens the usable linear-frequency range while remaining simple to implement. The five-tap impulse response boosts the linear operating band by roughly 33% over his earlier design, offers exact two-sample group delay and linear phase, and can be realized in a folded multiplier-free form using binary right shifts. The design targets signals below pi/2 radians per sample.
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.
Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm
Rick Lyons lays out four practical techniques to get an inverse FFT when you only have forward FFT software or FPGA cores available. The post highlights a classic data-reversal trick, a conjugate-symmetry optimized flow, and two methods that avoid reversals using data swapping or complex conjugation plus scaling. Each method notes when it is preferable so engineers can pick the least costly implementation.
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.
Handy Online Simulation Tool Models Aliasing With Lowpass and Bandpass Sampling
Rick Lyons walks through Analog Devices' Frequency Folding Tool, a hands-on simulator that makes aliasing intuitive. The post shows step-by-step demos for lowpass and bandpass sampling and highlights four key behaviors: all analog components fold below Fs/2, bandpass translation, harmonic bandwidth growth, and aliased harmonics interfering with fundamentals. It’s a practical tutorial for engineers learning sampling effects.
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.
Complex Down-Conversion Amplitude Loss
Rick Lyons shows why a standard complex down-converter seems to halve amplitudes yet only imposes a -3 dB power loss. He walks through mixing math from an RF cosine to i and q paths, demonstrates that each path has peak A/2 but the complex output has half the average power, and offers practical guidance for software modeling and avoiding spectral interpretation traps.
A Complex Variable Detective Story – A Disconnect Between Theory and Implementation
A subtle phase-wrap gotcha turned a clean pencil-and-paper derivation into a software mismatch for a 5-tap FIR filter with complex coefficients. Rick Lyons shows why two algebraically equivalent-looking expressions can disagree in code, and traces the real culprit to angle limits in rectangular-form complex arithmetic. The fix is simple once you see it, but the trap is easy to miss.
The Number 9, Not So Magic After All
Rick Lyons dismantles the mystique around the number 9 by showing its 'magic' stems from our base-10 system rather than any unique numeral power. He walks through classic 9 tricks, including digit-sum divisibility, digital-root behavior, and division patterns, then generalizes them to base-B where digit B-1 plays the same role. The post is a short, playful link between recreational arithmetic and radix thinking.
Sum of Two Equal-Frequency Sinusoids
Rick Lyons exposes a frequent trig mistake and delivers complete closed-form expressions for collapsing two equal-frequency sinusoids into a single sinusoid. Using complex-exponential phasor addition and equating real and imaginary parts, he compiles easy-to-use tables for cosine+cosine, sine+sine, and cosine+sine cases and shows how to derive each form. Engineers get corrected identities and compact derivations useful for analysis and communications.
Computing Chebyshev Window Sequences
Rick Lyons gives a compact, practical recipe for building M-sample Chebyshev (Dolph) windows with user-set sidelobe levels, not just theory. The post walks through computing α and A(m), evaluating the Nth-degree Chebyshev polynomial, doing an inverse DFT, and the simple postprocessing needed to form a symmetric time-domain window. A worked 9-sample example and an implementation caveat for even-length windows make this immediately usable.
Errata for the book: 'Understanding Digital Signal Processing'
Rick Lyons collects all errata for every edition and printing of his book Understanding Digital Signal Processing into one centralized list, with downloadable PDFs for each variant. The post also shows how to identify your book's printing number for American 1st, 2nd, and 3rd editions and flags a few oddball versions that lack errata.
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.
60-Hz Noise and Baseline Drift Reduction in ECG Signal Processing
Rick Lyons shows a very efficient way to clean up ECGs when both baseline drift and 60 Hz power-line interference are getting in the way. He starts from a linear-phase DC removal filter, reshapes it into a notch filter that hits both 0 Hz and 60 Hz, and then tests it on a noisy real-world ECG. The payoff is a practical design that uses only two multiplications and five additions per sample.
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.
Some Thoughts on a German Mathematician
Rick Lyons revisits the remarkable career of Carl Friedrich Gauss, mixing memorable anecdotes with technical highlights. The post links Gauss’s work on the Gaussian curve, complex-plane representation, orbit prediction, and early telegraph experiments to ideas familiar to DSP engineers, and notes historical evidence that he developed trigonometric series before Fourier. It’s a short, engaging reminder of Gauss’s broad influence.
A New Contender in the Digital Differentiator Race
Rick Lyons presents a compact FIR differentiator that widens the usable linear-frequency range while remaining simple to implement. The five-tap impulse response boosts the linear operating band by roughly 33% over his earlier design, offers exact two-sample group delay and linear phase, and can be realized in a folded multiplier-free form using binary right shifts. The design targets signals below pi/2 radians per sample.
Goertzel Algorithm for a Non-integer Frequency Index
Rick Lyons demonstrates how to run the Goertzel algorithm with a non-integer frequency index k, letting you target DTFT frequencies that do not align with DFT bin centers. He interprets Rajmic and Sysel's generalization, provides a simple implementation, and presents a real-valued reformulation that reduces the final multiplies for real inputs. Example Matlab code is included to reproduce and adapt the technique.
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.
Stereophonic Amplitude-Panning: A Derivation of the 'Tangent Law'
Rick Lyons presents a clear geometrical derivation of the stereophonic amplitude-panning Tangent Law, filling a gap left by common references. Using vector components and the equidistant speaker assumption to keep signals in phase, he arrives at the Tangent Law and isolates practical gain formulas gL and gR needed to place an apparent source at a desired panning angle. Engineers can apply Eqs. (12) and (14) directly.
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.
Do Multirate Systems Have Transfer Functions?
Multirate systems can fool you into thinking standard z-domain analysis always applies. Rick Lyons shows why CIC decimation and Hogenauer implementations do not have a single z-domain transfer function from the input to the downsampled output, because downsampling breaks the one-to-one frequency mapping of LTI systems. Use the cascaded-subfilter H(z) up to the decimation point, then explicitly account for aliasing when predicting the decimated spectrum.
A Fast Real-Time Trapezoidal Rule Integrator
Rick Lyons presents a compact, recursive real-time Trapezoidal Rule integrator that computes N-sample discrete integration using only four arithmetic operations per input sample. The proposed network yields a finite-length, linear-phase impulse response with constant group delay (N-1)/2 and cuts substantial computation compared with a tapped-delay implementation, making it useful for speeding Romberg-based digital filters.
Online DSP Classes: Why Such a High Dropout Rate?
Rick Lyons digs into a startling statistic: online DSP courses reported a 97% dropout rate. He argues the main culprits are math-heavy curricula that overwhelm beginners and rigid, non-self-paced schedules that demand sustained 8-10+ hours per week. Rick urges course creators to rethink pacing and mathematical depth to improve completion rates and student engagement.
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.
A New Contender in the Digital Differentiator Race
Rick Lyons presents a compact FIR differentiator that widens the usable linear-frequency range while remaining simple to implement. The five-tap impulse response boosts the linear operating band by roughly 33% over his earlier design, offers exact two-sample group delay and linear phase, and can be realized in a folded multiplier-free form using binary right shifts. The design targets signals below pi/2 radians per sample.
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.
60-Hz Noise and Baseline Drift Reduction in ECG Signal Processing
Rick Lyons shows a very efficient way to clean up ECGs when both baseline drift and 60 Hz power-line interference are getting in the way. He starts from a linear-phase DC removal filter, reshapes it into a notch filter that hits both 0 Hz and 60 Hz, and then tests it on a noisy real-world ECG. The payoff is a practical design that uses only two multiplications and five additions per sample.
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.
Errata for the book: 'Understanding Digital Signal Processing'
Rick Lyons collects all errata for every edition and printing of his book Understanding Digital Signal Processing into one centralized list, with downloadable PDFs for each variant. The post also shows how to identify your book's printing number for American 1st, 2nd, and 3rd editions and flags a few oddball versions that lack errata.
