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.
A Fast Guaranteed-Stable Sliding DFT Algorithm
Rick Lyons presents a compact, computationally efficient sliding DFT that computes a single N-point DFT bin output for each input sample in real time. The design replaces the traditional complex resonator with a 2nd-order real resonator and uses pole/zero cancellation to match the DFT bin response. Crucially, the resonator poles remain on the z-plane unit circle even with quantized coefficients, guaranteeing numerical stability.
A New Contender in the Quadrature Oscillator Race
Rick Lyons highlights a compact quadrature oscillator introduced by A. David Levine and Martin Vicanek, offering guaranteed stability, accurate low-frequency tuning, and modest computational cost. The post walks through the simple u, v, w recurrences used for software implementation. Appendices provide transfer functions and an algebraic stability proof for engineers who want formal verification before deployment.
A DSP Quiz Question
A short visual puzzle from Rick Lyons shows how a common plotting trick can fool even experienced DSP engineers. He presents a 3D circular plot that looks like a triangular window but is actually a 32-point hann window, then explains why the circular projection distorts the view. The post highlights the importance of checking equations and 2D plots before naming a window by sight.
An Efficient Full-Band Sliding DFT Spectrum Analyzer
Rick Lyons shows two compact sliding DFT networks that compute the 0th bin and all positive-frequency outputs for even and odd N, running sample-by-sample on real input streams. The designs reduce computational workload versus a prior observer-based sliding DFT by using fewer parallel paths, while remaining guaranteed stable and avoiding the traditional comb delay-line. A simple initialization and streaming procedure makes them practical for real-time spectrum analysis.
A Simpler Goertzel Algorithm
Rick Lyons presents a streamlined Goertzel algorithm that simplifies computing a single DFT bin by removing the textbook method's extra shift and zero-input steps. The proposed network changes the numerator so you run the main stage N times then perform one final output stage, making the implementation cleaner and slightly cheaper computationally. Rick also points out that common textbook forms differ from Gerald Goertzel's 1958 original.
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 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.
A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters
Rick Lyons breaks down cascaded integrator-comb filters into clear, practical terms, showing why they are the efficient choice for high-rate decimation and interpolation in hardware. The post explains CIC structure, its sinc-like frequency response, multistage tradeoffs, register-bit-width rules, and simple FIR compensation tricks. Hands-on tips and references make it easy for engineers to design and implement robust CIC-based decimators and interpolators.
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.
A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters
Rick Lyons breaks down cascaded integrator-comb filters into clear, practical terms, showing why they are the efficient choice for high-rate decimation and interpolation in hardware. The post explains CIC structure, its sinc-like frequency response, multistage tradeoffs, register-bit-width rules, and simple FIR compensation tricks. Hands-on tips and references make it easy for engineers to design and implement robust CIC-based decimators and interpolators.
A Quadrature Signals Tutorial: Complex, But Not Complicated
Quadrature signals are essential in modern communications, yet complex numbers and the j operator intimidate many engineers. In this tutorial Rick Lyons uses phasor geometry, three-dimensional time and frequency plots, and practical I/Q sampling examples to demystify complex exponentials, negative frequency, and how to generate baseband complex signals. Read to get physical intuition and hands-on rules you can apply to modulation, demodulation, and DSP implementations.
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.
Understanding the 'Phasing Method' of Single Sideband Demodulation
Rick Lyons explains how the phasing method separates overlapping single sideband transmissions using quadrature processing and the Hilbert transform, making SSB demodulation practical in crowded RF environments. After reviewing simple synchronous detection, he walks through spectra and block diagrams that show how complex downconversion produces i and q paths which reinforce the desired sideband and cancel the other. The post also covers DSP implementation tips and BFO error effects.
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.
Computing FFT Twiddle Factors
Rick Lyons gives two compact algorithms to compute individual twiddle factors for radix-2 DIF and DIT FFTs, handy when you need only a subset of outputs such as in pruned FFTs. He explains stage indexing, provides closed-form formulas including the bit-reversal step for DIT, and walks through N=8 examples so you can implement the twiddle-angle calculations directly.
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.
Controlling a DSP Network's Gain: A Note For DSP Beginners
Rick Lyons calls out a simple but costly mistake beginners make when normalizing digital networks, scaling the input instead of the output. Using fixed-point examples he shows that pre-multiplying an A/D output by 1/8 throws away bits and costs about 18 dB of SQNR. The practical guidance is to place gain control as the final multiplication stage and beware a faulty Simpson's 1/3 integrator example.
Computing the Group Delay of a Filter
Rick Lyons presents a neat, practical way to get a filter's group delay directly from its impulse response using only DFTs. The method computes an N-point DFT of h(n) and of n·h(n), divides them in the frequency domain, and takes the real part to obtain group delay in samples, avoiding phase unwrapping. The post includes MATLAB code, a zero-division warning, and a caution that the method is reliable for FIR filters but not always for IIRs.
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.
A Quadrature Signals Tutorial: Complex, But Not Complicated
Quadrature signals are essential in modern communications, yet complex numbers and the j operator intimidate many engineers. In this tutorial Rick Lyons uses phasor geometry, three-dimensional time and frequency plots, and practical I/Q sampling examples to demystify complex exponentials, negative frequency, and how to generate baseband complex signals. Read to get physical intuition and hands-on rules you can apply to modulation, demodulation, and DSP implementations.
A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters
Rick Lyons breaks down cascaded integrator-comb filters into clear, practical terms, showing why they are the efficient choice for high-rate decimation and interpolation in hardware. The post explains CIC structure, its sinc-like frequency response, multistage tradeoffs, register-bit-width rules, and simple FIR compensation tricks. Hands-on tips and references make it easy for engineers to design and implement robust CIC-based decimators and interpolators.
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.
Understanding the 'Phasing Method' of Single Sideband Demodulation
Rick Lyons explains how the phasing method separates overlapping single sideband transmissions using quadrature processing and the Hilbert transform, making SSB demodulation practical in crowded RF environments. After reviewing simple synchronous detection, he walks through spectra and block diagrams that show how complex downconversion produces i and q paths which reinforce the desired sideband and cancel the other. The post also covers DSP implementation tips and BFO error effects.
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.
Computing FFT Twiddle Factors
Rick Lyons gives two compact algorithms to compute individual twiddle factors for radix-2 DIF and DIT FFTs, handy when you need only a subset of outputs such as in pruned FFTs. He explains stage indexing, provides closed-form formulas including the bit-reversal step for DIT, and walks through N=8 examples so you can implement the twiddle-angle calculations directly.
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.
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.
Free DSP Books on the Internet
Finding reliable DSP textbooks online is hit-or-miss. Rick Lyons assembled a curated list of over forty legally downloadable signal processing books, organized by topic from theory and communications to audio, image processing, and implementation. The post points to vendor manuals, MATLAB and algorithm resources, and clear copyright guidance so engineers can grab useful references without breaking licensing rules.
FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending
Rick Lyons follows a numerical mismatch from a published astronomy paper into a short detective story about FFT interpolation. He shows a commonly published interpolation formula produces large errors, explains why the algebraic approximations fail, and presents several correct alternatives with algebraic simplifications that greatly reduce computation. Engineers get both the debugging lesson and practical, lower‑cost formulas for evaluating X(k) between FFT bins.
