Add a Power Marker to a Power Spectral Density (PSD) Plot
Read absolute power directly from a PSD plot with a simple MATLAB helper. The author presents psd_mkr, a function that computes the PSD with pwelch and overlays a power marker in three modes: normal for narrowband tones, band-power for integrated power over a specified bandwidth, and 1 Hz for noise density readings. Examples show how bin summing, window loss, and scalloping are handled for accurate measurements.
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.
Find Aliased ADC or DAC Harmonics (with animation)
If a sinewave drives an ADC or DAC, device nonlinearities create harmonics that can fold back as aliases above Nyquist. This post shows a simple Matlab model, using an NCO, a static nonlinearity, and a DFT to generate spectra and reveal aliased harmonics, with animated illustrations to make aliasing intuitive. The approach works for both ADC and DAC measurement setups and highlights realistic effects like quantization noise.
Adaptive Beamforming is like Squeezing a Water Balloon
Think of adaptive beamforming as squeezing a water balloon, a simple analogy that reveals how combining multiple antennas creates focused gains and deep nulls. This post walks through the MVDR (Wiener-filter–based) solution, explains steering and scanning vectors, and shows how array geometry and known signal direction control what you can and cannot cancel. Practical tips highlight limits like the N-1 interferer rule.
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 DSP Online Conference - Right Around the Corner!
Three months after a forum post, Stephane Boucher and Jacob Beningo pulled together the DSP Online Conference, a two-day virtual event featuring 14 talks from leading DSP experts. Most sessions are 30 to 60 minutes with a 30-minute Zoom Q&A, while extended deep dives from speakers like fred harris are included. Registered attendees get one-year on-demand access, and free or reduced passes are available.
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.
Design Square-Root Nyquist Filters
A multirate signal processing textbook presents a neat method for designing square-root Nyquist FIR filters that combine zero ISI with strong stopband attenuation. This post walks through the principle that matched transmit and receive filters need square-root Nyquist responses, gives the key design relations for excess bandwidth and stopband edge, and includes a Matlab implementation to produce practical FIR matched filters for QAM-style systems.
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.
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.
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.
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.
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.
Multiplierless Half-band Filters and Hilbert Transformers
This article provides coefficients of multiplierless Finite Impulse Response 7-tap, 11-tap, and 15-tap half-band filters and Hilbert Transformers. Since Hilbert transformer coefficients are simply related to half-band coefficients, multiplierless Hilbert transformers are easily derived from multiplierless half-bands.
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.
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.
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.
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.
Frequency-Domain Periodicity and the Discrete Fourier Transform
Sampling turns a continuous spectrum into an infinite set of replicas, and this article explains why the DFT and DTFT inevitably show periodic, circular spectra. Eric Jacobsen combines rigorous math with a geometric, wagon-wheel intuition to clarify aliasing, bandlimited sampling, and sampled-IF techniques. Read it to see when center frequency doesn't matter, how cyclic baseband shifts behave, and why bandwidth, not absolute frequency, determines alias-free sampling.
Computing Large DFTs Using Small FFTs
Rick Lyons demonstrates a practical trick for computing large N-point DFTs by combining multiple smaller radix-2 FFTs when only limited FFT sizes are available. He walks through 16-point and 24-point examples using two and three 8-point FFTs, shows how to assemble outputs with twiddle factors, and explains a symmetry that reduces twiddle storage to N/4 values. The method supports non-power-of-two DFT lengths.
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.
Padé Delay is Okay Today
High-order Padé approximations for time delays break in surprising ways, but the failure is not magic. Jason Sachs walks through why coefficient-based transfer functions and companion-form state-space are numerically fragile, shows how to compute poles and zeros directly from the hypergeometric form with Newton iteration, and demonstrates building modal or block-diagonal state-space realizations to make high-order Padé delays practical while noting remaining limits.
Time Machine, Anyone?
Causal filters can look like time machines, but they do not break physics. Andor Bariska reproduces a classic electronic experiment in MATLAB, showing how a minimum-phase peaking filter and its FDLS biquad approximation produce negative group delay bands that make predictable, bandlimited signals appear to emerge early. The post walks through group delay, discretization, pulse and random-signal tests, and why unpredictability restores causality.
Oscilloscope Dreams
Jason Sachs walks through practical oscilloscope buying criteria for embedded engineers, focusing on bandwidth, channel count, hi-res acquisition, and probing. He explains why mixed-signal scopes and hi-res mode matter, when a 100 MHz scope is sufficient and when to keep a higher-bandwidth instrument, and how probe grounding and waveform export can ruin measurements. Real-world brand notes and try-before-you-buy advice round out the guidance.
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.
Noise shaping
Markus Nentwig presents a compact, practical introduction to noise shaping by treating quantization error as the first sample of a designed impulse response. He shows how to derive a noise shaper from a target spectrum, demonstrates the tradeoff between in-band noise reduction and total noise increase, and includes a Matlab example while highlighting clipping and stability caveats for sigma-delta contexts.
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.
