Open Ultrasound Lab

Open Ultrasound Lab

The mathematics, physics and signal processing behind modern ultrasound imaging, from phasors and the FFT to transducers, beamforming, localization microscopy, sound-speed estimation and aberration correction. Every idea comes with figures you can change.

Free, open, interactive.

M. Nguyen, with Claude (Anthropic) · version 1.0, last reviewed 2026-10-16

Start with module 01Choose a learning pathFormula sheet

Why this exists

I am an ultrasound engineer who is still learning every day. For years I wished the fundamentals, from phasors and the FFT to transducers, beamforming and the newest imaging methods, were gathered in one place and explained with figures you can change. This is my attempt, built with the help of AI and shared freely, especially for students and young engineers who are just getting started.

It is not a product, and there is nothing to buy. It certainly contains mistakes. If you find one, please tell me; if you can explain a topic better, please contribute. I hope it saves you some of the time it would have saved me.

M. Nguyen

The course as a 33-element array, one element per module, grouped into five parts. Pick a learning path below to light up its active aperture; select an element to open that module.

010203040506070809101112131415161718192021222324252627282930313233Part I · Math & signal foundationsPart II · Ultrasound signal processing & estimationPart III · Physics & hardwarePart IV · Image formation & qualityPart V · Imaging modes & capstones

Learning paths

Each path is an ordered subset of modules for one goal. Earlier modules are prerequisites for later ones, so read them in the order shown.

Transducer and hardware engineer

The signal basics, then wave physics, piezoelectrics, stacks, CMUT/PMUT, arrays, front ends, characterization and acoustic output. 12 modules, in order.

  1. 01Complex numbers & phasors
  2. 02The DFT and FFT by hand
  3. 03Convolution, LTI & filters
  4. 04Sampling, Fourier, LTI
  5. 15Wave physics & harmonics
  6. 16Piezoelectrics & KLM
  7. 17Matching, backing, lens, FEA
  8. 18CMUT, PMUT & new materials
  9. 19Arrays: 1D to 2D, RCA, CMUT
  10. 20Front-end electronics
  11. 21Simulation & characterization
  12. 22Acoustic output & safety

Interview preparation

Every module’s interview and qualifying-exam questions with model answers, on one page, grouped by module.

Open the 368 questions

The five parts

33 modules, 233 interactive demos, 607 numbered equations and 159 MATLAB/Octave scripts. Each module follows the same arc: derivations with live figures, a visual summary, how experts think about it, a cheat sheet, interview questions, a MATLAB lab, a self-check, figures from the literature and further reading.

Part I · Math & signal foundations

Modules 01–07

Seven modules that rebuild the mathematics every later module leans on.

  1. 01Complex numbers and phasorsj, Euler, rotation, I/Q
  2. 02The DFT and FFT by handcorrelation with phasors, bins, butterflies
  3. 03Convolution, LTI systems and filtersimpulse response, z-domain, eigenfunctions
  4. 04Sampling, Fourier & LTI systemsaliasing, leakage, PSF ↔ transfer function
  5. 05Matrices as operatorssubspaces, projections, least squares
  6. 06Eigendecomposition & SVDlow rank, clutter filtering, distortion matrix
  7. 07Random vectors, covariance & correlationchannel covariance, VCZ coherence

Part II · Ultrasound signal processing & estimation

Modules 08–14

Seven modules on turning RF and IQ samples into estimates.

  1. 08Analytic signal & IQ demodulationHilbert, envelope, demodulation, decimation
  2. 09Digital filter designFIR vs IIR, windows, Parks–McClellan, zero-phase, wall filters
  3. 10Matched filtering & pulse compressionchirps, Golay codes, range sidelobes, SNR gain
  4. 11Correlation & time-delay estimationsubsample bias, jitter, Kasai, block matching
  5. 12Estimation theory & the Cramér–Rao boundMLE, Fisher information, ULM localization limits
  6. 13Inverse problems and regularizationill-posedness, Tikhonov, TV, sound-speed tomography
  7. 14Optimization & automatic differentiationGauss–Newton, Adam, differentiable beamforming

Part III · Physics & hardware

Modules 15–22

Eight modules on the physical system under every algorithm.

  1. 15Wave physics, nonlinear propagation and harmonic imagingwave equation, interfaces, diffraction, B/A, harmonics
  2. 16Piezoelectric transduction and equivalent circuitsconstitutive equations, Mason and KLM models, impedance
  3. 17Matching layers, backing, lenses and finite-element designquarter-wave layers, backing, lenses, finite elements
  4. 18CMUT, PMUT and new transducer materialsclamped-plate CMUT, PMUT, single crystals, lead-free, films
  5. 19Array architecturespitch, elevation control, 2D channel count, row-column
  6. 20Front-end electronics and system architecturepulser, T/R, LNA, TGC, ADC, link budget
  7. 21Simulation and characterization toolsspatial impulse response, full-wave codes, the bench
  8. 22Acoustic output, bioeffects and regulatory measurementMI, TI, derating, FDA/IEC output table

Part IV · Image formation & quality

Modules 23–27

Five modules on how a beamformed image is formed and how its quality is measured.

  1. 23Beam patterns, k-space and compoundingaperture ↔ beam, grating lobes, plane-wave compounding
  2. 24Fourier-domain beamformingf-k (Stolt) migration, plane-wave Fourier imaging, k-space
  3. 25Adaptive and coherence-based beamformingMV, ESBMV, CF, GCF, PCF, DMAS, SLSC, FXPF
  4. 26Speckle statistics and image-quality metricsRayleigh to K, speckle size, CNR and gCNR
  5. 27Spectral estimation & QUSWelch, multitaper, AR, attenuation and BSC

Part V · Imaging modes & capstones

Modules 28–33

Six modules that combine everything before them into imaging modes and research targets.

  1. 28Microbubble physics, contrast imaging and the ULM signal sourcebubble resonance, shells, CEUS pulse sequences, perfusion
  2. 29Doppler and flow imagingCW/PW/color/power, ultrafast Doppler, vector flow
  3. 30Elastography: strain, ARFI and shear-wave imagingstrain, ARFI, shear-wave elastography, dispersion
  4. 31ULM end to end (capstone)acquisition to super-resolved maps
  5. 32Sound-speed estimation (capstone)global, local, tomographic, differentiable
  6. 33Aberration correction (capstone)phase screens to the distortion matrix

What makes it different

Every figure says where it comes from

Each plot carries one of four badges: Exact formulaSimulationApprox. modelSchematic. Simulations name the solver that actually ran; approximate models state their assumptions beside the figure.

Runnable MATLAB and Octave

159 scripts in base MATLAB with no toolboxes. Every one runs unchanged in GNU Octave 8 and is executed headless on each change; the numbers it prints match the page text.

Literature you can check

156 "Figures from the literature" entries point to a specific figure in a published paper, with citation and DOI, each looked up before it was included. Unverifiable details are removed or listed on the errata page.

Checked, and still learning

Every number on these pages was checked by running code. Some modules (01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33) have also had a careful second review, in which a separate AI model re-derived the key results and looked up the citations; the others will follow. Anything that could not be confirmed is listed openly on the errata page. The most valuable review of all comes from readers, so corrections are always welcome.

Formula sheet

All the key equations of the course, with symbols defined, on one page: open the formula sheet.

How to cite

Nguyen, M. (2026). Open Ultrasound Lab (Version 1.0) [Interactive educational resource]. Zenodo. https://doi.org/10.5281/zenodo.0000000

BibTeX and CITATION.cff are on the citation page. Found a mistake? Open an issue or read how to contribute. Please read the disclaimer before relying on anything here.