Sampling bandpass signals
Bandpass signals can be sampled at rates below the usual Nyquist limit, and this note shows how the band-limited spectrum appears in baseband after sampling. Using a simple example figure, it defines the center frequency fc = (fmax + fmin)/2 and bandwidth Δf = fmax - fmin, and highlights that choosing fs less than twice the signal's highest frequency violates the sampling theorem.
Simulink-Simulation of SSB demodulation
This post walks through Simulink models that implement SSB demodulation and modulation, using Richard Lyons' phasing method as a foundation. It shows practical models for simple carrier multiplication and for the phasing method with cosine and -sin paths plus Hilbert filtering, and it highlights sampling, decimation, filter choices, and delay alignment to make the techniques work in simulation.
Compute Images/Aliases of CIC Interpolators/Decimators
CIC filters provide multiplier-free interpolation and decimation for large sample-rate changes, but their images and aliases can trip up designs. This post supplies two concise Matlab functions and hands-on examples to compute interpolator images and decimator aliases, showing spectra and freqz plots. Readers will learn how interpolation ratio and number of stages alter passband, stopband, and aliasing behavior.
Exploring Human Hearing Range
Audacity makes it simple to explore the limits of human hearing by generating and inspecting single-tone audio. This post walks through creating a 9 kHz sine tone, noticing the default 44,100 Hz sample rate, and verifying the result with Audacity's Plot Spectrum tool. Follow the steps and use low playback volume to safely try higher or lower test frequencies yourself.
The Zeroing Sine Family of Window Functions
A previously unrecognized family of DFT window functions is introduced, built from products of shifted sines that deliberately zero out tail samples and control nonzero support. Cedron Dawg presents recursive and semi-root constructions, runnable code, and numerical examples, and shows that the odd-N member L=(N-1)/2 numerically matches a discrete Hermite-Gaussian DFT eigenvector. The post highlights practical properties, an even-N fix, and applications to spectrograms and tone decomposition.
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.
Digging into an Audio Signal and the DSP Process Pipeline
In this post, I'll look at the benefits of using multiple perspectives when handling signals.A Pre-existing Audio FileLet's say we have an audio file of interest. Let's load it into Audacity and zoom in a little (using View → Zoom → Zoom In, multiple times). The figure illustrates the audio signal: just a basic single-tone signal.
By continuing to zoom into the signal, we eventually get to the point of seeing individual samples as illustrated below. Notice that I've marked one...
A Free DSP Laboratory
You don't need expensive gear to start exploring audio DSP, free open-source tools are enough. This post shows how to build a simple audio DSP laboratory with Audacity, covering signal generation, playback, waveform zooming, exporting to WAV/MP3/OGG, and viewing spectra. It's a short, practical intro to inspecting signals in both time and frequency domains with minimal setup.
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.
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.
The Exponential Nature of the Complex Unit Circle
Euler's equation links exponential scaling and rotation by translating a distance along the unit-circle circumference into a complex value. Cedron Dawg develops an intuitive geometric view, using integer and fractional powers of i to show how points, roots of unity, and multiplication behave as additive moves along that circumference. The article also connects this picture to radians and the conventional Taylor-series proof for broader perspective.
Polyphase filter / Farrows interpolation
Markus Nentwig shows how polyphase filtering and the Farrow interpolator provide a practical, computation‑efficient way to realize sub-sample delays and variable resampling. He starts from the upsample-filter-decimate view, explains how polyphase decomposition reduces per-phase work, then describes how the Farrow structure fits polynomials to coefficient banks for continuous fractional-delay control. The post includes warnings about filter choices and links to code and references.
Dealing With Fixed Point Fractions
Fixed-point fractional math is easy to botch, and this post lays out pragmatic ways to avoid those mistakes. It clarifies the difference between integer and fractional overflow, shows how Q notation helps track binary-point scaling, and explains why multiplies add sign bits that may require shifting. Read for concrete FPGA strategies: keeping bit growth, selective shifts, or aggressive normalization, plus testing tips.
Pulse Shaping in Single-Carrier Communication Systems
Eric Jacobsen clears up common confusion around pulse shaping in single-carrier communications, focusing on matched filtering, Nyquist filtering, and related terminology. He uses the NRZ rectangular pulse as a concrete example to show how the transmit spectrum becomes a sinc envelope when the bitstream has enough randomness, and he highlights how bit patterns and context-sensitive terms can change the observed behavior.
Sampling bandpass signals
Bandpass signals can be sampled at rates below the usual Nyquist limit, and this note shows how the band-limited spectrum appears in baseband after sampling. Using a simple example figure, it defines the center frequency fc = (fmax + fmin)/2 and bandwidth Δf = fmax - fmin, and highlights that choosing fs less than twice the signal's highest frequency violates the sampling theorem.
Simple Concepts Explained: Fixed-Point
Fixed-point is the bridge between real-world values and integer arithmetic, and this post makes that bridge tangible with a hands-on ADC-to-gain example. It walks through mapping voltages to Q-format integers, choosing gain resolution in bits, and how multiplication adds bit growth and produces quantization error. Read it to build intuition for practical fixed-point choices when implementing DSP on FPGA or ASIC.
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 and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1
Cedron Dawg shows how to get exact amplitude and phase for a real sinusoid whose frequency does not land on an integer DFT bin. The method treats a small neighborhood of DFT bins as a complex vector, builds two basis vectors from the cosine and sine transforms, and solves a 2x2 system using conjugate dot products to recover real coefficients that give amplitude and phase. A C++ example and sample output verify the formulas.
Design study: 1:64 interpolating pulse shaping FIR
Markus Nentwig presents a practical 1:64 root-raised cosine interpolator built from cascaded FIR stages that slashes computational cost. By separating pulse shaping from rate conversion, designing each interpolator to suppress only known alias bands, and equalizing the pulse shape, the design achieves just 4.69 MACs per output, roughly 12 percent of a straight polyphase implementation while meeting EVM targets.
An Efficient Linear Interpolation Scheme
A simple trick slashes the cost of linear interpolation to at most one multiply per output sample, and often to none. The post shows a zero-order-hold based network that preserves input samples, has a short L-1 transient, and lets 1/L scaling be implemented as a binary shift when L is a power of two. It also gives a fixed-point layout that moves scaling to the end to reduce quantization distortion.
Pulse Shaping in Single-Carrier Communication Systems
Eric Jacobsen clears up common confusion around pulse shaping in single-carrier communications, focusing on matched filtering, Nyquist filtering, and related terminology. He uses the NRZ rectangular pulse as a concrete example to show how the transmit spectrum becomes a sinc envelope when the bitstream has enough randomness, and he highlights how bit patterns and context-sensitive terms can change the observed behavior.
Frequency Dependence in Free Space Propagation
Free-space propagation of electromagnetic waves is essentially independent of frequency, a counterintuitive conclusion Eric Jacobsen demonstrates step by step. He shows the λ^2 factor in the Friis transmission equation comes from antenna effective area and gain, not from the space between antennas, explaining why dipoles favor lower bands while dishes improve with frequency. The post also reminds engineers that material penetration and atmospheric absorption remain genuine frequency dependent concerns.
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.
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.
Take Control of Noise with Spectral Averaging
Spectral averaging turns noisy FFT outputs into repeatable, measurable spectra by trading time for noise control. This post explains the practical difference between RMS averaging, which reduces variance without changing the noise floor, and vector averaging, which can lower the noise floor but requires phase-coherent, triggered inputs. It also shows how linear and exponential weighting affect reaction time for live displays and measurement accuracy.
An Efficient Linear Interpolation Scheme
A simple trick slashes the cost of linear interpolation to at most one multiply per output sample, and often to none. The post shows a zero-order-hold based network that preserves input samples, has a short L-1 transient, and lets 1/L scaling be implemented as a binary shift when L is a power of two. It also gives a fixed-point layout that moves scaling to the end to reduce quantization distortion.
Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1
Cedron Dawg shows how to get exact amplitude and phase for a real sinusoid whose frequency does not land on an integer DFT bin. The method treats a small neighborhood of DFT bins as a complex vector, builds two basis vectors from the cosine and sine transforms, and solves a 2x2 system using conjugate dot products to recover real coefficients that give amplitude and phase. A C++ example and sample output verify the formulas.
Sampling bandpass signals
Bandpass signals can be sampled at rates below the usual Nyquist limit, and this note shows how the band-limited spectrum appears in baseband after sampling. Using a simple example figure, it defines the center frequency fc = (fmax + fmin)/2 and bandwidth Δf = fmax - fmin, and highlights that choosing fs less than twice the signal's highest frequency violates the sampling theorem.
Signed serial-/parallel multiplication
Struggling with costly wide adders for signed multiplication on FPGAs? Markus Nentwig unpacks a neat bit-level trick that turns two's-complement signed-signed multiplication into a serial-parallel routine using only a one-bit wider adder. Learn how flipping sign bits and a small, controlled constant cancel lets you avoid full sign-extension, and get a parametrized Verilog RTL plus synthesis notes to try it yourself.
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.
