Ultrasound RF & QUS — Nomenclature and Concepts

A reference document for RF-domain ultrasound signal processing, beamforming, and quantitative ultrasound (QUS). Written as a compact, working manual for research engineers moving between the envelope and RF signal domains.

Companion document to qus-python-refresher, a five-notebook Python refresher that implements the RF pulse → echo → moving-window spectral analysis → Lizzi-Feleppa pipeline this document describes. I wrote up the repo itself in a separate post — A Bug That Doesn't Crash Is Worse Than One That Does — if you want the worked code and a genuinely silent bug alongside this reference.

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1. Fundamental principles and trade-offs

1.1 Time-frequency uncertainty (Heisenberg-Gabor)

$$\Delta t \cdot \Delta f \geq \frac{1}{4\pi}$$

Time and frequency are dual domains; sharpness in one implies uncertainty in the other. This physical constraint underlies most ultrasound design trade-offs:

• Short pulse ↔ wide bandwidth. Broadband transducers require short pulses at the cost of transmitted energy and penetration.
• Long spectral window ↔ coarse axial localization. FFT bin width is fs / N_window. Shorter windows yield finer axial resolution but coarser spectral resolution.
• High center frequency ↔ shallow penetration. Frequency-dependent attenuation limits depth reach at higher f₀.

1.2 Nyquist and aliasing

Sampling rate must exceed 2 × f_max. RF signals typically sampled at 40-60 MHz; IQ (baseband) sampling requires only > BW due to complex sampling. Aliasing manifests as spectral folding and is the most common silent error in prototyping — the code runs, the result is wrong.

1.3 Born approximation (single scattering)

Most QUS models assume single scattering. Multiple scattering regimes (fatty liver, tissues with many interfaces) violate this assumption and complicate clinical interpretation.

1.4 Ergodicity assumption

Regional statistics computed from a single image assume time-averaging equals ensemble-averaging. Rarely validated in practice.

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2. Spatial resolution

2.1 Axial resolution

Resolution along the beam axis, set by pulse duration:

$$R_{axial} = \frac{c \cdot \tau}{2}$$

where τ is pulse duration. Short pulses improve axial resolution. Typical values: ~0.6 mm at 5 MHz with a 2-cycle pulse.

2.2 Lateral resolution

Resolution perpendicular to the beam axis, set by beam width:

$$R_{lat} \approx \frac{\lambda \cdot z}{D} = \lambda \cdot F_{number}$$

where z is focal depth and D is aperture width. F-number is defined as z/D. Small F-number implies wide aperture and improved lateral resolution.

Figure 1: Lateral beam profile narrowing toward the focus and widening beyond it, shown at 10/20/30/40 mm depth, with the −6 dB level marked.

2.3 Elevational resolution

Slice thickness of the imaging plane. In 1D arrays, determined by physical lens design (typically 3-5 mm). 2D matrix probes improve this through electronic focusing.

2.4 Resolution cell (resolution volume)

Intersection of axial, lateral, and elevational resolutions. The number of scatterers within the resolution cell determines the local speckle statistics regime:

• Many scatterers (>10) → fully developed speckle, Rayleigh (Nakagami m ≈ 1)
• Few scatterers + specular components → sub-Rayleigh, Nakagami m < 1

Figure 2: Ultrasound image axes, resolution cell, and full-width-half-maximum (FWHM). Previously published in Cüneyitoğlu Özkul & Mumcuoğlu, 2018.

2.5 Full Width Half Maximum (FWHM)

Standard measure of beam profile or PSF width, taken at half of maximum amplitude (approximately −6 dB). Used for cross-comparison of resolution across systems.

2.6 Depth of field (DOF)

Length of the focal zone. Scales with F-number squared:

$$DOF \approx 7.1 \lambda F_{number}^2$$

This approximation is for a linear (1D) array; curvilinear and phased arrays have different beam geometry (diverging aperture, sector scanning) and DOF does not follow the same F-number scaling without modification.

Small F-number improves lateral resolution but reduces DOF — a fundamental trade-off in fixed-focus systems and one motivation for plane-wave imaging.

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3. Transmit chain

3.1 Center frequency (f₀)

Peak frequency of the transducer response. Typical clinical range 2-15 MHz. Higher f₀ yields better resolution but reduced penetration; lower f₀ is used for deep applications (cardiac, obstetric).

3.2 Fractional bandwidth

Spectral width around f₀ expressed as a fraction. BW = 0.6 means −6 dB points span 60% of f₀. Broadband transducers (BW > 0.7) benefit two independent things, not one dependent on the other: harmonic imaging (needs enough sensitivity at both f₀ and 2f₀ to transmit and receive on the same element) and reliable QUS spectral fitting (needs enough usable bandwidth for a stable slope/intercept/midband estimate — see section 7.1-7.3). QUS itself is ordinarily performed entirely at the fundamental frequency; it does not require harmonic content.

3.3 Pulse duration

Clinical pulses typically span 1-3 cycles; τ ≈ n_cycles / f₀. Short duration yields wide bandwidth and improved axial resolution.

3.4 Transmit apodization

Weighting of array elements during transmission (Hamming, Hann, Tukey windows) to shape the beam pattern and suppress side lobes.

3.5 Transmit focus

• DAS (Delay-and-Sum): Focused to a specific depth via TX delay pattern — see section 4.1 for the full beamforming description.
• Plane wave imaging: No transmit focus; the entire field of view is insonified by a plane (or diverging) wavefront. Focusing is deferred to receive-side software beamforming.

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4. Beamforming approaches

4.1 Delay-and-Sum (DAS)

Classical approach. Applies element-wise delays and sums the received signals per image line. Focused TX and RX give line-by-line images. Frame rate is limited by the number of scan lines and pulse repetition time.

4.2 Plane wave / Synthetic aperture

TX is unfocused (plane or diverging wave); the entire volume is reconstructed simultaneously via receive-side software beamforming. Frame rates exceed 5000 Hz. Enables ultrafast imaging modalities like shear wave elastography.

Coherent compounding. Combining RF responses from multiple plane-wave angles by complex (pre-envelope) summation, preserving phase. Introduced to clinical practice by SuperSonic Imagine.

4.3 Adaptive beamforming

Data-dependent element weights instead of uniform apodization. Improves contrast and resolution at higher computational cost.

• MVDR / Capon: Minimum variance criterion.
• DAX: Dual apodization with cross-correlation.

Predominantly research-domain at present.

4.4 Receive apodization

Element weighting on reception to shape the acceptance pattern. Trade-off between resolution and sensitivity to off-axis signals.

4.5 F-number

F# = focal_depth / aperture_width. Standard descriptor of beam narrowness.

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5. Signal chain (RF to B-mode)

5.1 RF (radiofrequency)

Raw ADC output. Real-valued, zero-mean, carries the carrier frequency and phase information. Typically sampled at 40-60 MHz. Phase carries diagnostic information that is lost in later stages of the chain.

5.2 IQ / baseband / analytic signal

Complex baseband representation:

• Z(t) = I(t) + jQ(t)
• Envelope: |Z|
• Phase: angle(Z)

Obtained via:

• Quadrature demodulation: Multiply by cos(2πf₀t) and sin(2πf₀t), then low-pass filter.
• Hilbert transform: Z = RF + j·H{RF}, then frequency shift to baseband.

Complex sampling reduces required rate to > BW, decreasing data volume by a factor of 3-4.

5.3 Envelope detection

Magnitude of the analytic signal. Discards phase, retaining only amplitude modulation. Envelope statistics (Rayleigh, Rician, Nakagami) are defined at this stage.

5.4 Log compression

Logarithmic mapping of envelope for display:

bmode = 20·log₁₀(envelope / max_envelope), clipped to [-60, 0] dB

Compresses the 60+ dB dynamic range for visualization. Must generally be inverted for downstream QUS analysis.

5.5 Scan conversion

Coordinate transformation from acquisition geometry (polar for sector/curvilinear probes) to Cartesian display grid.

5.6 Time Gain Compensation (TGC)

Depth-dependent gain applied at reception to compensate for attenuation. Operator-adjustable in clinical systems. Must be characterized and removed for quantitative measurements.

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6. Attenuation and tissue properties

6.1 Attenuation coefficient (α)

Units: dB/cm/MHz. Approximate values in soft tissue:

• Fat: 0.6
• Liver: 0.5-0.7
• Muscle: 1.0-1.5
• Bone: >10

Attenuation increases approximately linearly with frequency, producing a depth-dependent spectral downshift that must be compensated in QUS.

6.2 Backscatter coefficient (BSC)

Scattered energy per unit volume per unit solid angle, as a function of frequency. The central quantity in QUS, extracted via reference phantom normalization.

6.3 Speed of sound (c)

Clinical convention: 1540 m/s. Actual values vary:

• Fat: ~1450 m/s
• Soft tissue: 1540 m/s
• Bone: ~4000 m/s

Deviations impact beamforming accuracy and elastography interpretation.

6.4 Scattering regimes (ka dependence)

Ratio of scatterer size (a) to wavelength (λ) governs the scattering behavior:

• ka << 1 (Rayleigh): σ_bs ∝ f⁴, isotropic scattering.
• ka ~ 1 (Mie / resonance): Oscillatory frequency dependence.
• ka >> 1 (geometric): Specular reflection, tissue interfaces.

Most soft-tissue scatterers fall in the Rayleigh regime for clinical ultrasound frequencies.

6.5 Acoustic impedance mismatch

Z = ρ·c. Reflection coefficient at an interface:

$$R = \frac{Z_2 - Z_1}{Z_2 + Z_1}$$

Typical values: ~1% at soft tissue interfaces, ~50%+ at soft tissue/bone, ~100% at soft tissue/air.

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7. QUS parameters (Lizzi-Feleppa framework)

7.1 Spectral slope (dB/MHz)

Slope of a linear fit to the calibrated log-power spectrum within the usable transducer bandwidth. Sensitive to scatterer size, attenuation, and depth.

7.2 Spectral intercept (dB)

Extrapolated fit value at 0 MHz. Sensitive to effective scatterer concentration.

7.3 Midband fit (dB)

Fit value at f₀. Reflects total backscattered energy in the band.

7.4 Effective scatterer diameter (ESD)

Scatterer size estimate from spectral shape, using Gaussian or spherical form-factor models. Associated with the Insana-Wagner framework.

7.5 Effective scatterer concentration (ESC)

Number of scatterers per unit volume, derived from spectral amplitude.

7.6 Reference phantom method

To isolate tissue backscatter from system response (transducer response, TGC, beamforming), the in vivo spectrum is normalized by that of a phantom with known backscatter properties, imaged with identical acquisition parameters. This is a required step in quantitative QUS.

Key references: Yang & Zagzebski 1990; Yao & Zagzebski 1990.

7.7 Attenuation compensation

Correction for depth-dependent spectral downshift. Common methods:

• Spectral shift method
• Spectral difference method
• Hybrid method (Kim & Varghese 2007)

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8. Envelope statistics

8.1 Rayleigh distribution

Envelope of a zero-mean complex Gaussian process. Corresponds to fully developed speckle from many independent scatterers:

$$p(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$

8.2 Rician (Rice) distribution

Rayleigh with an added coherent component. Characterized by the K-factor:

K = coherent_power / diffuse_power

8.3 K distribution

Envelope model for sub-Rayleigh regimes with few effective scatterers per resolution cell.

8.4 Nakagami distribution

Generalized envelope model. The shape parameter m spans:

• m < 1: pre-Rayleigh clustering (sub-Rayleigh)
• m = 1: Rayleigh (fully developed)
• m > 1: post-Rayleigh (coherent structures)

8.5 Homodyned-K distribution

Two-parameter model capturing coherent, diffuse, and sub-Rayleigh regimes. Preferred in modern QUS envelope statistics.

8.6 Speckle

Coherent interference pattern from unresolved scatterers within the resolution cell. Not stochastic noise: repeat imaging of the same region reproduces the same pattern. Carries diagnostic information exploitable via statistical modeling.

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9. Simulation tools

9.1 Field II (Jensen 1996)

Approach: Linear systems using spatial impulse response (Tupholme-Stepanishen).

Covers: Transducer geometry, focused/phased-array beamforming, point scatterer response, B-mode imaging simulation.

Does not cover: Nonlinearity, refraction, reflection at heterogeneous boundaries, wave-medium interaction beyond linear scattering.

Typical runtime: Seconds to minutes.

Use for: B-mode phantom simulation, beamformer development, QUS pipeline testing, teaching.

9.2 k-Wave (Treeby & Cox 2010)

Approach: Full-wave PDE solver on a k-space pseudospectral grid.

Covers: Nonlinearity (Westervelt equation), heterogeneous attenuation, refraction, reflection, complex boundary geometries, photoacoustic sources.

Trade-off: Computationally intensive; overkill for simple point-scatterer B-mode simulation.

Typical runtime: Minutes to hours (accelerated with GPU).

Use for: Photoacoustic reconstruction, HIFU treatment planning, aberration studies, nonlinear harmonic imaging, situations requiring full wave physics.

9.3 Selection guide

Task	Preferred tool
B-mode phantom simulation	Field II
QUS pipeline testing	Field II
Beamformer development	Field II
HIFU / therapeutic ultrasound	k-Wave
Photoacoustic imaging	k-Wave
Aberration correction	k-Wave
Nonlinear harmonic imaging	k-Wave
Rapid prototyping	Custom NumPy/SciPy pipeline

9.4 Python ecosystem

• k-wave-python — Official Python port of k-Wave (waltsims/k-wave-python). Provides both a pure NumPy/CuPy solver and an interface to the compiled C++/CUDA k-Wave binaries.
• jwave — JAX-based, differentiable acoustic simulation. Suited to machine-learning integrated workflows.
• stride — Medical ultrasound tomography framework.
• USBMD — PyTorch-based differentiable beamforming library.
• PICMUS dataset — Plane-wave imaging benchmark with reference beamformer implementations.

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10. Frequently overlooked considerations

10.1 Log-domain estimation bias

Applying log(S + noise) is nonlinear; bias grows as SNR decreases. The order of averaging and log transformation affects the result. Reference: Alam & Lizzi, UMB 2006.

10.2 Focal zone dependence

Beam width varies with depth. Outside the focal zone, lateral resolution degrades, and QUS parameters become depth-biased — see Figure 1 (section 2.2) for the beam-width-vs-depth behavior this is describing.

10.3 Scattering anisotropy

Most QUS models assume isotropic scatterers. Real tissue (muscle fibers, collagen) is anisotropic, so imaging angle influences spectral slope estimates.

10.4 Noise distribution across the chain

• RF domain: approximately Gaussian
• Envelope: Rayleigh (envelope of Gaussian)
• Log domain: approximately Gumbel

Estimator design should reflect the distribution at the analysis stage.

10.5 Window boundary effects

Rectangular windowing produces spectral leakage. Hann or Hamming windows are generally preferred for QUS spectral estimation.

10.6 Log-domain windowing bias

Shorter windows reduce local SNR, amplifying log-domain bias in slope estimation. Both mean and variance of the estimator can shift with window size.

10.7 Silent IQ demodulation errors

• Incorrect LPF cutoff causing spectral leakage
• Baseband sampling below Nyquist causing aliasing
• Hardware amplitude/phase imbalance between I and Q channels

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11. Reading list (recommended order)

1. Oelze & Mamou (2016). "Review of Quantitative Ultrasound: Envelope Statistics and Backscatter Coefficient Imaging and Contributions to Diagnostic Ultrasound." IEEE TUFFC. Bridges envelope statistics and RF spectral analysis; most efficient starting point.
2. Lizzi, Greenebaum, Feleppa, Elbaum, Coleman (1983). "Theoretical framework for spectrum analysis in ultrasonic tissue characterization." JASA 73:1366-1373. Foundational reference for the Lizzi-Feleppa method.
3. Yao, Zagzebski, Madsen (1990). "Backscatter coefficient measurements using a reference phantom to extract depth-dependent instrumentation factors." Ultrasonic Imaging. Practical protocol for reference phantom normalization.
4. Alam & Lizzi (2006). "On the statistics of ultrasonic spectral parameters." UMB. Log-domain bias analysis; relevant to Bayesian estimation workflows.
5. Bercoff, Tanter, Fink (2004). "Supersonic shear imaging: a new technique for soft tissue elasticity mapping." IEEE TUFFC. Foundational for plane wave imaging with coherent compounding.
6. Mamou & Oelze (eds.) (2013). Quantitative Ultrasound in Soft Tissues. Springer. Comprehensive treatment; chapters 1-2 are the most efficient overview.

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This started as a personal working reference, updated as new concepts and experimental observations get incorporated. The runnable side of it — pulse simulation, the moving-window spectral analysis, the Lizzi-Feleppa fits — lives in qus-python-refresher, written up separately here.