Companion to the Open Ultrasound Lab · Parts I–V · Modules 01–33
Ultrasound Formula Sheet
The transform pairs, theorems, numbers and rules of thumb an ultrasound signal-processing expert answers from memory, organized by the five parts of the course. Each part opens with the facts to know cold, then lists the key equations of every module in that part, with their equation numbers in the module.
1Conventions used on this sheet
\[ X(f) = \int x(t)\, e^{-j2\pi f t}\, dt \qquad x(t) = \int X(f)\, e^{+j2\pi f t}\, df \]
Ordinary frequency f in Hz, so no 2π factors in front. With angular frequency ω the inverse gains 1/(2π). \(\operatorname{sinc}(x) = \sin(\pi x)/(\pi x)\), so sinc has zeros at every nonzero integer. \(\operatorname{rect}(t) = 1\) for \(\lvert t\rvert < \tfrac12\). Decibels: amplitude ratios use \(20\log_{10}\), power ratios use \(10\log_{10}\). Equation numbers such as (04.3) refer to the module and equation in the course.
Part I · Modules 01–07
2Fourier transform pairs
Modules 02 (DFT), 03 (LTI systems), 04 (sampling and Fourier)
A comb with spacing T transforms to a comb with spacing 1/T and weight 1/T. Narrow spacing in one domain means wide spacing in the other.
Sampling
Sampled signal \(x_s(t) = x(t)\operatorname{III}_T(t)\) ⇒ \(X_s(f) = X(f) \ast \tfrac{1}{T}\operatorname{III}_{1/T} = f_s\sum_k X(f - k f_s)\). The spectrum repeats every \(f_s\). Replicas overlap (alias) unless \(f_s > 2f_{\max}\).
Arrays
An array is a continuous aperture multiplied by a comb at pitch d. Its beam is the continuous beam convolved with a comb in \(\sin\theta\) at spacing λ/d. Those replicas are the grating lobes: \(\sin\theta_g = \sin\theta_0 \pm m\lambda/d\). Finite length turns each delta into a Dirichlet lobe of width ≈ λ/(Nd).
Pulsed Doppler
Slow-time sampling at the PRF is a comb in time, so the Doppler spectrum repeats every PRF. Velocities with \(\lvert f_D\rvert > \mathrm{PRF}/2\) wrap around: \(v_{\mathrm{Nyq}} = c\cdot\mathrm{PRF}/(4f_0\cos\theta)\).
Periodic ↔ discrete
Periodic in one domain means discrete (line spectrum) in the other. Sampled in time means periodic in frequency. That is why the DFT is both discrete and periodic in both domains.
Windowing causes leakage; apodization shapes the beam; sampling replicates
Correlation
\(\int x^*(t)\, y(t + \tau)\, dt\)
\(X^*(f)\, Y(f)\)
Fast cross-correlation for delay estimation
Wiener–Khinchin
\(R_{xx}(\tau)\)
\(S_{xx}(f)\)
Speckle autocorrelation ↔ system power spectrum
Parseval
\(\int \lvert x\rvert^2 dt\)
\(= \int \lvert X\rvert^2 df\)
Energy is the same in both domains
Duality
\(X(t)\)
\(x(-f)\)
Get the sinc ↔ rect pair from rect ↔ sinc
Differentiation
\(dx/dt\)
\(j2\pi f X(f)\)
Derivatives boost high frequencies (strain from displacement)
Real signal
\(x\) real
\(X(-f) = X^*(f)\)
Negative frequencies are redundant, so keep the analytic signal
Uncertainty
\(\sigma_t\) (rms width)
\(\sigma_t\sigma_f \ge 1/(4\pi)\)
Axial resolution vs bandwidth; equality only for a Gaussian
Watch the uncertainty constant
\(\sigma_t\sigma_f \ge 1/(4\pi)\) uses rms widths of \(\lvert x\rvert^2\) and \(\lvert X\rvert^2\) (energy densities). For the amplitude Gaussian above, the amplitude-function widths satisfy \(\sigma_t\sigma_f = 1/(2\pi)\). Both statements describe the same pulse, so say which width you mean.
5Convolution and correlation, visually
Modules 03 (convolution), 11 (correlation and delay estimation)
Both operations slide one function past the other, multiply, and integrate. Convolution flips g first; correlation does not. That single flip is why convolution models a system (output = input ∗ impulse response) while correlation measures similarity (where does g best match f?).
Cross-correlation: shift g right by t without flipping. Same convention as MATLAB xcorr(f, g). Not commutative: \((g \star f)(t) = (f \star g)^*(-t)\). Correlation = convolution with the conjugated, time-reversed g.
Some texts (and the common Wikipedia figure) define \((f \star g)(t) = \int f^*(\tau)\, g(\tau + t)\, d\tau\), which is the curve above mirrored in t and conjugated. Check the convention before reading a sign off a lag: it decides whether a positive lag means "g is late" or "g is early".
Bin spacing \(\Delta f = f_s/N\). True resolution ≈ \(1/T_{\mathrm{obs}} = f_s/N\). Zero padding interpolates the spectrum but adds no resolution. Multiplying DFTs gives circular convolution, so pad to \(N \ge L + M - 1\) for linear convolution. The FFT costs about \(N\log_2 N\) operations instead of \(N^2\), and fftshift puts 0 Hz in the middle.
Window
Peak sidelobe
−3 dB width (bins)
ENBW (bins)
Sidelobe roll-off
When to use it
Rectangular
−13.3 dB
0.89
1.00
−6 dB/oct
Best resolution, poor dynamic range
Hann
−31.5 dB
1.44
1.50
−18 dB/oct
General purpose; common apodization
Hamming
−42.7 dB
1.30
1.36
−6 dB/oct
Low near sidelobes, slow far roll-off
Blackman
−58.1 dB
1.64
1.73
−18 dB/oct
High dynamic range spectra
7Sampling and aliasing
Module 04, Sections 04.1–04.2
Fact
Formula
Example
Nyquist (real, lowpass)
\(f_s > 2 f_{\max}\)
5 MHz, 100 % BW (\(f_{\max}\) 7.5 MHz): \(f_s\) > 15 MHz; scanners use \(4f_0\) to 62.5 MHz
Alias of a tone
\(f_a = \left\lvert f - f_s\operatorname{round}(f/f_s)\right\rvert\)
For the tone at \(k = k_0\): \(X[k_0] = (AN/2)(\cos\varphi + j\sin\varphi)\) \(= (AN/2)\,e^{j\varphi}\). The magnitude \(AN/2\) no longer depends on \(\varphi\), and the angle is \(\varphi\) itself.symbols
\(X[k]\)
DFT bin k, k = 0 … N − 1 (complex, same units as x times samples)
\(x[n]\)
signal samples, n = 0 … N − 1
\(N\)
record length in samples (FFT length)
\(k\)
bin index: k cycles per record, frequency \(k f_s/N\) for \(k < N/2\)
If d is a multiple of N, r = 1, every term is 1 and the sum is N. Otherwise \(r \ne 1\) and the sum is zero. So the N probes are mutually orthogonal, each with energy N.symbols
\(d = k - l\)
difference of the two probe indices, integer
\(r\)
\(e^{j2\pi d/N}\), one step of d/N turn on the unit circle
\(\delta[\cdot]\)
1 when the argument is a multiple of N, otherwise 0
Equivalently \(\frac{2}{N}\lvert X[k]\rvert\cos(2\pi kn/N + \angle X[k])\) for each \(1 \le k \le N/2 - 1\). This is the precise statement of what Re and Im mean.symbols
\(X[0]/N\)
mean (DC) of x
\(\tfrac{2}{N}\mathrm{Re}\,X[k]\)
cosine amplitude at frequency \(k f_s/N\)
\(-\tfrac{2}{N}\mathrm{Im}\,X[k]\)
sine amplitude at frequency \(k f_s/N\)
\(X[N/2]/N\)
amplitude of the Nyquist alternation \((-1)^n\) (even N only)
\(D_N = 0\) at every non-zero integer \(\delta\) (the bins of an on-bin tone); \(D_N(\delta) \approx N\,\mathrm{sinc}(\delta)\) for \(\lvert\delta\rvert \ll N\). Every bin now receives some of the tone: that is leakage.symbols
\(\delta\)
\(k_0 - k\), offset of the tone from bin k, in bins
\(D_N(\delta)\)
Dirichlet kernel (periodic sinc), \(D_N(0) = N\)
\(e^{j\pi\delta(N-1)/N}\)
linear phase: the record is centred at n = (N − 1)/2
Input-side reading: input sample \(x[k]\) launches a copy of \(h\), scaled by \(x[k]\), starting at \(n = k\). The output is the sum of all the copies. One sequence, \(h\), describes the system completely.symbols
\(y[n]\)
output sample at index \(n\)
\(x[k]\)
input sample at index \(k\), length \(L\)
\(h[n-k]\)
impulse response delayed by \(k\) samples, length \(M\)
The factor \(e^{j\omega n}\) does not depend on \(k\), so it leaves the sum. What remains is one complex number per frequency: the frequency response, the correlation of \(h\) with a probe phasor (Module 02).symbols
\(e^{j\omega n}\)
input phasor: unit length, turning \(\omega\) rad per sample
\(\omega\)
digital frequency, rad/sample (\(\omega = 2\pi f/f_s\))
\(h[k]\)
impulse response taps
\(H(e^{j\omega})\)
frequency response: one complex number per \(\omega\), dimensionless gain
\(y[n]\)
output for the complex exponential input
\(n,\ k\)
sample indices
\(h[k]\)
impulse response taps
\(j\)
imaginary unit, \(j^2=-1\)
\[ \cos\omega n \;\longrightarrow\; y[n] = \mathrm{Re}\,H \cdot \cos\omega n \;-\; \mathrm{Im}\,H \cdot \sin\omega n \]
(03.13)
The output is a real signal made of two real pieces: an in-phase copy of the input with gain \(\mathrm{Re}\,H\), and a quadrature copy, a quarter period late, with gain \(-\mathrm{Im}\,H\).symbols
\(\mathrm{Re}\,H\)
in-phase gain at \(\omega\), dimensionless
\(\mathrm{Im}\,H\)
quadrature gain at \(\omega\), dimensionless (its negative multiplies the late copy)
\(\sin\omega n\)
\(= \cos(\omega n - \pi/2)\): the input a quarter period late
For a narrowband pulse at \(\omega_0\), the carrier is delayed by \(\tau_p(\omega_0)\) and the envelope by \(\tau_g(\omega_0)\). Linear phase \(\varphi = -\omega d\) gives \(\tau_p = \tau_g = d\) at every \(\omega\), so the pulse shape is preserved.symbols
\(\varphi(\omega)\)
unwrapped phase of \(H(e^{j\omega})\), rad
\(\tau_p\)
phase delay: delay of the carrier at \(\omega\), samples (divide by \(f_s\) for seconds)
\(\tau_g\)
group delay: delay of the envelope of a narrowband pulse centered at \(\omega\), samples
Split \(e^{-j\omega n} = e^{-j\omega k}\, e^{-j\omega(n-k)}\). The double sum factors because each exponential depends on only one index. Convolution in time is multiplication in frequency.symbols
\(X, H, Y\)
DTFTs of input, impulse response and output, complex
\(n = 1\) is ordinary Nyquist \((f_{s} \ge 2f_{H})\). The largest n gives the lowest legal rate, approaching 2B only when \(f_{H}\) is an integer multiple of B. For even n, the band arrives spectrally inverted.symbols
\(f_L,\ f_H\)
band edges (here at −6 dB), MHz
\(B\)
bandwidth, MHz
\(n\)
Nyquist zone that holds the whole band; n = 1 is ordinary Nyquist
time-bandwidth product (dimensionless) of the Gaussian pulse
\(\tau,\ B\)
shorthand for \(\tau_{-6\,\mathrm{dB}}\) (µs) and \(B_{-6\,\mathrm{dB}}\) (MHz)
\[ \begin{gathered} \sin\theta_g = \sin\theta_0 \pm m\,\frac{\lambda}{d}, \qquad m = 1, 2, \ldots \\ \text{grating-lobe-free steering to } \pm\theta_{\max} \text{ requires } d < \frac{\lambda}{1 + \sin\theta_{\max}} \end{gathered} \]
(04.14)
A grating lobe exists when \(|\sin \theta _{g}| \le 1\) (the visible region). At \(d = \lambda /(1 + \sin \theta _{\max})\) it sits exactly at endfire \((\theta _{g} = \mp 90^{\circ})\), and \(d \le \lambda /2\) keeps it out for every steering angle.symbols
\(\theta_g\)
grating-lobe direction (exists when \(\lvert\sin\theta_g\rvert \le 1\))
Focusing brings the far field to the focus. For a sampled array, A becomes the array factor of 04.7: the DTFT of the weights, periodic in \(\sin \theta\) with period \(\lambda /d\).symbols
κ is the worst-case relative-error amplification. Rule of thumb: you lose about \(\log_{10}\kappa\) significant digits, of data precision, not just of floating point.symbols
\(\delta\mathbf{b}\)
error in the data (noise, delay error)
\(\delta\mathbf{x}\)
resulting error in the solution of \(A\mathbf{x} = \mathbf{b}\)
\(\sigma_{\max},\ \sigma_{\min}\)
largest and smallest singular values of A (longest and shortest ellipse axes)
\(\kappa\)
2-norm condition number, dimensionless; about \(\log_{10}\kappa\) digits are lost
\(\mathbf{x},\ \mathbf{b}\)
solution and data vectors of \(A\mathbf{x}=\mathbf{b}\)
\(\lVert\cdot\rVert\)
2-norm; the ratios are relative (dimensionless) errors
Each SVD component is multiplied by a filter factor \(f_{i}\). Strong components (\(\sigma_{i}^{2} \gg \lambda\)) pass untouched; weak ones are suppressed. Noise std along \(v_{i}\) drops from \(\sigma /\sigma_{i}\) to \(\sigma \sigma_{i}/(\sigma_{i}^{2} + \lambda)\) at the price of a bias \(-\lambda /(\sigma_{i}^{2} + \lambda) \cdot v_{i}^{T}x\).symbols
\(\mathbf{u}_i,\ \mathbf{v}_i\)
left and right singular vectors of \(A = U\Sigma V^{T}\)
\(\sigma_i\)
singular values of A
\(\lambda\)
regularization weight, same units as \(\sigma_i^{2}\)
\(f_i\)
filter factor: ≈ 1 for \(\sigma_i^{2} \gg \lambda\), ≈ 0 for \(\sigma_i^{2} \ll \lambda\), ½ at \(\sigma_i^{2} = \lambda\)
5 MHz, c = 1540 m/s (λ = 0.308 mm), p = 0.3 mm, N = 128: Δsin θ = 0.0080 (0.46° near broadside) and steering beyond ±30.9° aliases to another bin, which is the grating lobe seen in the angular domain. At 3 MHz (λ = 0.513 mm) the limit is ±58.8°.symbols
\(N\)
number of elements (128)
\(p\)
pitch, mm (0.3 mm)
\(\lambda\)
wavelength c/f, mm (0.308 mm at 5 MHz)
\(\theta\)
plane-wave angle from broadside
\(k_\theta\)
DFT bin across the aperture where the plane wave concentrates
\(\bmod N\)
result is taken modulo \(N\) (bins wrap around)
\(\Delta(\sin\theta)\)
change in \(\sin\theta\) per DFT bin (dimensionless)
\(\lvert\sin\theta\rvert < \lambda/(2p)\)
range with a unique bin (no wrap); \(p\) above \(\lambda/2\) aliases (grating lobes)
\[ \mathbf{d} = H\mathbf{\gamma} + \mathbf{n}, \qquad \text{column } j \text{ of } H = \text{channel data of a unit scatterer at pixel } j \]
(05.20)
\(\mathbf{\gamma}\) is the reflectivity per pixel. For a 0° plane wave the column is the pulse placed on the hyperbola \(\tau_{e}(x, z) = (z + \sqrt{(x - x_{e})^{2} + z^{2}})/c\).symbols
Spectral theorem: a Hermitian matrix is a weighted sum of orthogonal rank-1 projectors. Repeated eigenvalues still admit an orthonormal eigenbasis (choose one inside the eigenspace). If additionally \(\mathbf{A} \succeq 0\) (all \(\lambda_{i} \ge 0\)), as for \(\mathbf{A}^{H}\mathbf{A}\) or any covariance, the eigenvalues are variances along \(\mathbf{q}_{i}\).symbols
\(\mathbf{A} = \mathbf{A}^{H}\)
Hermitian n × n matrix (e.g. a channel covariance)
\(\mathbf{q}_i\)
orthonormal eigenvectors (columns of \(\mathbf{Q}\))
\(\lambda_i\)
real eigenvalues; variances along \(\mathbf{q}_i\) for a covariance
\(\mathbf{U} (m\times m)\) and \(\mathbf{V} (n\times n)\) are unitary; \(\Sigma\) is m×n diagonal with non-negative entries. \(r = \operatorname{rank} \mathbf{A}\). The thin form keeps only the first min(m,n) columns.symbols
\(\mathbf{A} \in \mathbb{C}^{m\times n}\)
any matrix, r = rank A
\(\mathbf{U},\ \mathbf{V}\)
unitary, m × m and n × n; columns \(\mathbf{u}_i\) (output) and \(\mathbf{v}_i\) (input) singular vectors
\(\Sigma\)
m × n diagonal, singular values \(\sigma_i \ge 0\)
\(\mathbf{A}\mathbf{v}_i = \sigma_i\mathbf{u}_i\)
the defining relation
\(\sigma_i\)
singular values, ordered, \(\sigma_i\ge 0\)
\(\mathbf{u}_i,\ \mathbf{v}_i\)
left and right singular vectors (columns of \(\mathbf{U}\), \(\mathbf{V}\))
\(r\)
rank of \(\mathbf{A}\), number of nonzero \(\sigma_i\)
Both are spectral filters: TSVD uses a 0/1 filter factor, Tikhonov a smooth one. The discrete Picard condition says a meaningful solution exists only while \(\lvert \mathbf{u}_{i}^{H}\mathbf{b}\rvert\) decays faster than \(\sigma_{i}\); once \(\lvert \mathbf{u}_{i}^{H}\mathbf{b}\rvert\) hits the noise floor, stop.symbols
\(\mathbf{b}\)
measured data (signal + noise)
\(\mathbf{u}_i^{H}\mathbf{b}\)
data coefficient i (Picard plot)
\(k\)
truncation index (0/1 filter)
\(\alpha\)
Tikhonov parameter, same units as \(\sigma_i\); filter ½ at \(\sigma_i = \alpha\)
\(\mathbf{x}_k\)
truncated-SVD solution (keeps the first \(k\) terms)
\(\mathbf{x}_\alpha\)
Tikhonov solution
\(\sigma_i,\ \mathbf{u}_i,\ \mathbf{v}_i\)
SVD quantities, as in (06.7)
\(\sigma_i^{2}/(\sigma_i^{2}+\alpha^{2})\)
Tikhonov filter factor (smooth 0-to-1 version of truncation)
A projection of every pixel's time series onto a data-derived temporal subspace. Contrast this with a fixed temporal high-pass, which projects onto a fixed band of Doppler frequencies regardless of what the tissue is doing.symbols
\(k_{\mathrm{lo}},\ k_{\mathrm{hi}}\)
first and last kept rank (tissue below, noise above)
\(\mathbf{V}_b\)
n × (k_hi − k_lo + 1) matrix of the kept temporal singular vectors
\(\mathbf{S}_f\)
filtered ensemble (pixels × frames), input to power Doppler or ULM
\(\sigma_k,\ \mathbf{u}_k,\ \mathbf{v}_k\)
singular values and vectors of \(\mathbf{S}\), as in (06.14)
Derivation: \(\operatorname{var} (\bar{x}) = (1/N^{2}) \sum _{i}\sum _{j} \operatorname{cov} (x_{i}, x_{j})\) and there are \(N - |k|\) pairs at separation k. If \(\rho (k) \approx 1\) over a correlation length of \(\ell\) samples and 0 beyond, \(N_{\mathrm{eff}} \approx N/\ell\): the number of independent speckle cells, not the number of pixels.symbols
\(\bar{x}\)
sample mean of N samples of a WSS sequence
\(\sigma^2\)
variance of one sample
\(\rho(k)\)
correlation coefficient between samples k apart
\(N_{\mathrm{eff}}\)
effective number of independent samples (\(\approx N/\ell\) for a correlation length of \(\ell\) samples)
Approximately normal with a variance that does not depend on \(\rho\). A 95% interval is \(\tanh ( z \pm 1.96/\sqrt{N - 3} )\).symbols
\(\hat{\rho}\)
sample correlation coefficient from N independent bivariate-normal pairs
\(\rho\)
true correlation coefficient
\(z\)
Fisher's z, approximately normal
\(\mathcal{N}(m, v)\)
normal distribution with mean m and variance v
\(\operatorname{atanh}\)
inverse hyperbolic tangent
\(\ln\)
natural logarithm
\(N\)
number of pairs, (07.9)
\(\approx\)
approximately distributed as
\[ \begin{gathered} \rho = \frac{\mathrm{SINR}(\hat{\mathbf{R}}^{-1}\mathbf{s})}{\mathrm{SINR}(\mathbf{R}^{-1}\mathbf{s})} \sim \mathrm{Beta}(K - N + 2,\ N - 1) \;\Rightarrow\; \mathrm{E}[\rho] = \frac{K - N + 2}{K + 1} \\ \mathrm{E}[\rho] = \tfrac{1}{2}\ (-3\ \mathrm{dB}) \text{ at } K = 2N - 3 \approx 2N \\ K = N{:}\ \mathrm{E}[\rho] = \frac{2}{N + 1}\ (-12.2\ \mathrm{dB} \text{ for } N = 32); \qquad K = 4N{:}\ -1.2\ \mathrm{dB} \end{gathered} \]
(07.18)
Verified by Monte Carlo (\(N = 32,\ 2000\) trials): \(K = 61\) gives \(0.500, K = 128\) gives 0.762 against 0.760 predicted. The result does not depend on \(\mathbf{R}\) or on the steering vector \(\mathbf{s}\). It is the origin of the "K ≈ 2N snapshots for 3 dB" rule. The demo below checks it with its own Monte Carlo button.symbols
\(N\)
number of channels (array elements)
\(K\)
number of i.i.d. complex Gaussian training snapshots, free of the desired signal
The power spectral density is the Fourier transform of the autocorrelation. Non-negativity of S is the continuous version of "R is PSD" (Bochner's theorem): a Toeplitz covariance matrix is diagonalized, approximately, by the DFT, with the PSD samples as eigenvalues.symbols
A triangle that depends only on the lag: \(\mathbf{R}\) is Toeplitz. It reaches zero at a lag equal to the transmit aperture width. The receive aperture only samples the curve: its width \(N_{r}\) sets the largest lag available \((N_{r} - 1)\), and receive apodization cancels in the per-pair normalization. A 32-element transmit received on 64 elements gives \(1 - |m|/32\), zero for \(m \ge 32\). With transmit apodization, C(m) is the autocorrelation of that apodization: a 64-element Hann transmit gives \(C(16) = 0.67,\) \(C(32) = 0.18, C(48) = 0.01\). At the focus the curve does not depend on frequency when expressed in elements, so the broadband result is the same triangle. In the demo \(N_{t} = N_{r} = N = 64\).symbols
\(t \star t\)
autocorrelation of the transmit apodization
\(m\)
element lag (separation in elements)
\(N_t\)
number of transmit elements
\(C(m)\)
spatial coherence, per-pair normalized covariance averaged along a diagonal of R
Noise adds only to the diagonal, so it creates a spike at lag 0 and scales every other lag by \(1/(1 + 1/\mathrm{SNR})\). At 0 dB, lag-1 coherence halves. This is why SLSC suppresses incoherent clutter and noise: it ignores lag 0.symbols
\(\mathbf{R}_s\)
speckle (signal) covariance
\(\sigma_n^2\)
noise power per element, uncorrelated between elements
\(\mathrm{SNR}\)
per-element signal-to-noise power ratio
\(C_s(m)\)
noise-free coherence
\(\mathbf{R}\)
measured covariance, (07.14)
\(\mathbf{I}\)
identity matrix
\(C(m)\)
coherence at lag m, (07.28)
\(m\)
element lag
Part II · Modules 08–14
11IQ and the analytic signal
Modules 01 (phasors, I and Q), 08 (analytic signal and IQ demodulation)
Fact
Formula
Example
Complex (IQ) sampling
\(f_{s,\mathrm{IQ}} \ge B\)
A 3 MHz-wide band needs ≥ 3 MHz complex rate, i.e. 6 M real numbers/s
Write \(1 + \operatorname{sgn} f = 1 + j\,(-j\operatorname{sgn} f)\). The imaginary part of \(x_a\) is \(x\) filtered by \(H(f)\): an all-pass filter that shifts every positive frequency by −90° and every negative frequency by +90°.symbols
\(x_a(t)\)
analytic signal (complex), same units as \(x\)
\(\mathcal{H}\{x\}\)
Hilbert transform of \(x\): the quadrature component
\(H(f)\)
Hilbert transfer function: unit magnitude, −90° for \(f > 0\), +90° for \(f < 0\)
\(\mathrm{p.v.}\)
Cauchy principal value of the integral at \(\tau = t\)
With an ideal LPF, IQ is the analytic signal with its carrier removed. Hence \(\lvert\mathrm{IQ}\rvert = \lvert x_a\rvert\) exactly, and the IQ phase is the analytic phase minus \(2\pi f_d t\).symbols
\(f_d\)
demodulation (mixing) frequency, MHz; stored with the data
\(\Delta\)
offset of the echo center from \(f_d\), MHz
\(\mathrm{LPF}\)
low-pass filter that passes the baseband lobe and rejects the image at \(-(f_0+f_d)\)
\(\mathrm{IQ}\)
\(I + jQ\), with \(I = \mathrm{Re}\{\mathrm{IQ}\}\), \(Q = \mathrm{Im}\{\mathrm{IQ}\}\)
5 MHz, 80 % (−6 dB) bandwidth: \(B_{-6} = 4\) MHz, but the Gaussian −20 dB width is 7.3 MHz and the −40 dB width 10.3 MHz. A practical IQ rate is 8 to 10 MHz (2 reals per sample → 16 to 20 Mreal/s), versus ≥ 17 MHz real for RF at the −20 dB edge (\(f_{\max} = 8.6\) MHz). \(M = 4\) from 40 MHz is the common choice.symbols
\(f_s\)
RF sampling rate, MHz
\(M\)
decimation factor (integer)
\(f_{s,\mathrm{IQ}}\)
complex (IQ) sample rate, MHz
\(B\)
full width of the echo spectrum at the level you must preserve (−20 to −40 dB, not −6 dB), MHz
\(f_{\max}\)
highest RF frequency at that level, MHz
\(\lvert\Delta\rvert\)
magnitude of the offset in (08.10), MHz
\(\gt\)
requirement: rate must exceed the right-hand side to avoid aliasing
The rotation undoes the demodulator's own phase. Then \(y_{\mathrm{IQ}}\) = analytic signal of the RF-beamformed output, demodulated: \(\lvert y_{\mathrm{IQ}}\rvert\) equals the RF-DAS envelope.symbols
\(y_{\mathrm{IQ}}(\mathbf{r})\)
beamformed complex pixel at \(\mathbf{r} = (x, z)\)
\(w_n\)
apodization weight of channel \(n\)
\(\mathrm{IQ}_n(\tau_n)\)
channel IQ interpolated at the focusing delay (slow envelope, so linear interpolation at \(f_{s,\mathrm{IQ}}\) is adequate)
\(e^{+j2\pi f_d\tau_n}\)
phase rotation by the full delay times the demodulation frequency \(f_d\), not \(f_0\)
The \(\sigma_p\) form is common in the spectral-shift literature (e.g. Kuc, 1984). Units: Np/(cm·MHz) × MHz² × cm = MHz. The shift is linear in depth and proportional to bandwidth squared, so broadband pulses downshift more. The spectrum stays Gaussian with the same width, and the amplitude factor is \(e^{-2\alpha_0 f_0 z + 2\alpha_0^2 \sigma_f^2 z^2}\).symbols
\(f_c(z)\)
center frequency of the echo from depth \(z\), MHz
\(\alpha_0\)
amplitude attenuation slope, Np/(cm·MHz) (divide dB values by 8.686)
\(\sigma_f\)
standard deviation of the amplitude spectrum, MHz
\(\sigma_p\)
\(\sigma_f/\sqrt{2}\): standard deviation of the power spectrum \(\lvert S\rvert^2\), MHz
\(z\)
one-way depth, cm
\(f_0\)
centre frequency of the transmitted pulse, MHz
\(\Longleftrightarrow\)
equivalent form in terms of the power-spectrum width
For \(r = 1 - \varepsilon\), \(\lvert e^{j\omega} - p\rvert^2 \approx \varepsilon^2 + (\omega - \theta)^2\), so the peak falls 3 dB at \(\lvert\omega - \theta\rvert = \varepsilon\). The impulse response decays as \(r^n = e^{n\ln r}\), time constant \(-1/\ln r \approx 1/(1 - r)\). r = 0.95: width 0.1 rad/sample, delay 19 samples at the peak, ringing ≈ 20 samples. The one-pole formula (03.17) of Module 03 is the θ = 0 case.symbols
\(\omega\)
digital frequency, rad/sample
\(r,\ \theta\)
radius and angle of the root (zero or pole), \(r\) dimensionless, \(\theta\) rad
Kaiser (1974). The window method cannot give a looser passband than stopband. The power series \(I_0(x) = \sum_k (x/2)^{2k}/(k!)^2\) reaches double precision within 30 terms for \(\beta \le 10\).symbols
\(w[n]\)
Kaiser window, \(n = 0,\dots,M\)
\(I_0\)
zeroth-order modified Bessel function of the first kind
\(\beta\)
Kaiser shape parameter
\(A\)
attenuation \(-20\log_{10}\delta\), dB, with \(\delta = \min(\delta_p, \delta_s)\)
The left half-plane maps inside the unit circle (stability preserved) and the whole jΩ axis maps once onto the circle, so there is no aliasing, but the frequency axis is compressed: Ω = ∞ lands at Nyquist. Prewarp each critical edge, design the analog filter at the warped edges, transform: the digital edges land exactly. Analog zeros at infinity map to z = −1. Low-pass to band-pass, \(s \to (s^2 + \Omega_0^2)/(B s)\) with \(\Omega_0^2 = \Omega_1\Omega_2\), \(B = \Omega_2 - \Omega_1\), doubles the order.symbols
\(s,\ \Omega\)
Laplace variable and analog frequency, rad/s
\(z\)
z-transform variable
\(T\)
sampling interval \(1/f_s\), s
\(\omega\)
digital frequency, rad/sample
\(\Omega_p,\ \omega_p\)
analog (prewarped) and digital passband edge
\(j\)
imaginary unit
\[ \lvert H(f)\rvert^2 = \frac{\lVert \mathbf{A}\,\mathbf{e}_f \rVert^2}{N_{\text{out}}}, \qquad e_f[n] = e^{\,j2\pi f n/\mathrm{PRF}}, \quad n = 0, \dots, N - 1 \]
(09.11)
For a time-invariant filter in steady state this is the usual \(\lvert H\rvert^2\). It is the definition used by Bjaerum, Torp & Kristoffersen (2002) to compare FIR, IIR and regression filters on equal terms.symbols
\(\lvert H(f)\rvert^2\)
average power gain of the filter at Doppler frequency \(f\)
\(\mathbf{A}\)
wall-filter matrix, \(N_{\text{out}} \times N\)
\(N\)
ensemble length (slow-time samples per pixel)
\(N_{\text{out}}\)
number of valid outputs
\(\mathbf{e}_f,\ e_f[n]\)
unit-amplitude complex exponential at Doppler frequency \(f\) and its \(n\)th sample
\(\lVert\cdot\rVert\)
Euclidean norm
\(n\)
slow-time sample index
\(f,\ \mathrm{PRF}\)
Doppler frequency and pulse repetition frequency, Hz
The matched filter is the time-reversed, conjugated echo. Its output is the cross-correlation of the record with the template: \(y(t) = \int x(u)\, s^*(u - t + t_0)\,du\), the correlation receiver.symbols
\(\kappa\)
arbitrary nonzero gain (does not change SNR)
\(H(f),\ S(f)\)
filter response and echo spectrum, as in (10.1)
\(S^*\)
complex conjugate of \(S\)
\(h(t)\)
matched-filter impulse response
\(s(t)\)
echo shape; \(s^*\) is its complex conjugate (equal to \(s\) for real RF)
\(t_0,\ j\)
arrival time and imaginary unit, as in (10.1)
\(\iff\)
if and only if
\[ \begin{gathered} \text{rect reference of length } 1/B:\ G = TB; \\ \text{Gaussian reference, same −6 dB band:}\ G = \frac{T}{\sqrt{\pi}\,\sigma_t} = \frac{2\sqrt{\pi}}{2.3548}\,TB = 1.505\,TB \end{gathered} \]
(10.6)
Constant-envelope code of amplitude \(A\): energy \(A^2T\) (complex envelope). Gaussian envelope \(A\,e^{-t^2/2\sigma_t^2}\): energy \(\sqrt\pi\,\sigma_t A^2\). TB = 50 gives 18.77 dB; a Monte Carlo with \(10^4\) trials gives 18.70 dB (m10_chirp_compression.m).symbols
\(G\)
compression gain, as in (10.5)
\(T,\ B\)
code duration (µs) and bandwidth (MHz), as in (10.5)
Exact for the rect-envelope complex chirp (m10_chirp_compression.m matches it to \(3\times10^{-5}\)). As \(TB \to \infty\) it becomes \(\mathrm{sinc}(B\tau)\): −6 dB width \(1.207/B\), first null \(1/B\), peak sidelobe −13.26 dB. At TB = 10: \(1.234/B\) and −14.9 dB; at TB = 200: \(1.208/B\) and −13.3 dB.symbols
\(R(\tau)\)
matched-filter output (autocorrelation) at lag \(\tau\); \(R(0)\) is its peak
\(\tau\)
lag, µs
\(T,\ B\)
chirp duration (µs) and bandwidth (MHz)
\(\mathrm{sinc}(x)\)
\(\sin(\pi x)/(\pi x)\)
\(w_{-6\,\mathrm{dB}}\)
full width at half amplitude of the compressed envelope, µs
\(\mathrm{PSL}\)
peak sidelobe level relative to the mainlobe, dB (20 log)
\(N\) = 16 (\(\max\lvert R_A\rvert\) = 5) at 5 MHz: \(d\) = 2 µm → −37.9 dB, 5 µm → −29.9 dB, 20 µm → −17.4 dB. At a 5 kHz PRF, 2 µm per firing is 10 mm/s of axial velocity. The 1-cycle-chip RF simulation agrees within 0.5 dB.symbols
\(\mathrm{PSL}_{\mathrm{residual}}\)
residual peak sidelobe level relative to the peak, linear
\(R_A(k),\ N\)
autocorrelation and code length, as in (10.13)
\(k\)
lag in chips
\(d\)
axial displacement between the two firings, µm
\(f_0\)
centre frequency, MHz
\(c\)
sound speed, 1540 m/s
\(\varphi\)
round-trip phase shift of echo B, rad
\[ \Delta z = \frac{G}{2\,\alpha\, f_0} \quad [\mathrm{cm}] \]
(10.20)
15 dB at 0.5 dB/cm/MHz and 5 MHz: 3.0 cm. TB-50 chirp (18.77 dB): 3.75 cm at 5 MHz, 6.26 cm at 3 MHz, 2.50 cm at 7.5 MHz. Barker-13 (11.14 dB): 2.23 cm at 5 MHz.symbols
The kernel \(x\) stays fixed; \(y\) slides. Because \(R_{ss}(\tau)\) is maximal at \(\tau = 0\) (Cauchy–Schwarz), the peak of \(R_{xy}\) sits at \(\tau = D\).symbols
\(\hat{\rho}(k)\)
normalized cross-correlation (NCC) at lag \(k\), \(\lvert\hat\rho\rvert \le 1\)
\(x[n],\ y[n]\)
sampled reference and delayed records, as in (11.1)
This is the Walker–Trahey bound: the Carter form evaluated with flat coherence across the band, derived by (11.10)–(11.12). \(f_0\) in Hz and \(T\) in s give \(\sigma_D\) in s. Example: \(f_0\) = 5 MHz, \(B\) = 0.6, \(T\) = 2 µs, SNR = 30 dB, \(\rho\) = 1 gives \(\sigma_D \ge\) 0.41 ns (0.31 µm); \(\rho\) = 0.99 gives 1.35 ns.symbols
This is the Kasai lag-one autocorrelation estimator (1985). \(\angle R(1)/(2\pi T_{\mathrm{PRF}})\) is the mean Doppler frequency (the centroid of the slow-time power spectrum, exactly for a symmetric spectrum). It costs one complex multiply–add per sample. It is not the ML frequency estimator: for a stationary tone at high SNR the sum telescopes, so it effectively uses only the first and last pulses, and its variance exceeds the CRLB by 1.8× at \(M\) = 8 and 14× at \(M\) = 64 (m11_kasai_doppler.m, 20 dB; the high-SNR limit is \(M(M+1)/(6(M-1))\), 1.7× and 11×).symbols
\(\hat{R}(1)\)
lag-one slow-time autocorrelation
\(z_g[m]\)
slow-time samples of the \(g\)th gate, as in (11.14)
\((\cdot)^*\)
complex conjugate
\(M\)
ensemble length (pulses)
\(g\)
index of the averaged gates (spatial or channel)
\(\hat{v}\)
axial velocity estimate, m/s
\(c,\ f_0,\ \mathrm{PRF}\)
sound speed, centre frequency and pulse repetition frequency, as in (11.14)
\(\angle\)
phase angle, rad, wrapped to \((-\pi, \pi]\)
\[ v_{\mathrm{Nyq}} = \frac{c \cdot \mathrm{PRF}}{4 f_0} \qquad \text{(displacement per pulse limited to } \pm\lambda/4) \]
(11.16)
\(f_0\) = 5 MHz, PRF = 5 kHz: \(v_{\mathrm{Nyq}} = 1540 \cdot 5000 / (2\times10^7)\) = 0.385 m/s. The same \(\pm\lambda/4\) wrap limits phase-based displacement tracking between any two acquisitions, so ARFI uses PRFs of 5–10 kHz.symbols
\(v_{\mathrm{Nyq}}\)
Nyquist (aliasing) velocity, m/s
\(c,\ f_0,\ \mathrm{PRF}\)
as in (11.14)
\(\lambda\)
wavelength \(c/f_0\), m
\[ \hat{\tau}_n = \sum_{i=0}^{n-1} \Delta\hat{\tau}_i \quad\Rightarrow\quad \mathrm{Var}\bigl(\hat{\tau}_n - \tau_n\bigr) = n\, \sigma_\Delta^2 \qquad \text{(independent errors: a random walk)} \]
(11.17)
With \(\sigma_\Delta\) = 2 ns per pair, the error at element 64 is about 16 ns (≈ 29° at 5 MHz). A single peak hop adds a \(1/f_0\) step to every element after it. Cures: correlate each element against a beamsum reference (errors no longer accumulate), least-squares integration of multiple lags (\(\Delta\tau\) from \(i\) to \(i+1\), \(i+2\), …), and NCC-weighted integration.symbols
\(\hat{\tau}_n\)
accumulated delay estimate at element \(n\), s
\(\tau_n\)
true delay at element \(n\), s
\(\Delta\hat{\tau}_i\)
estimated delay between neighbouring elements \(i\) and \(i+1\), s
Bias and variance trade against each other. An RMSE of 3.6 µm can be 2 µm of bias plus 3 µm of jitter: \(\sqrt{2^2 + 3^2} = 3.6\) µm. Report both parts, because only the variance averages down with more data.symbols
\(\mathrm{MSE}\)
mean-squared error of the estimator, units of \(\theta^2\)
Equality holds if and only if the score factors as \(I(\theta)\,(g(\mathbf{y}) - \theta)\); then \(g\) is the efficient estimator. Information from independent samples adds.symbols
\(p = p(\mathbf{y}; \theta)\)
data density (likelihood)
\(I(\theta)\)
Fisher information, 1/units² of \(\theta\)
\(\hat\theta\)
estimator of \(\theta\)
\(\operatorname{var}(\hat\theta)\)
estimator variance, units² of \(\theta\)
\(\mathrm{E}[\cdot]\)
expectation over the data
\(b(\theta),\ b'(\theta)\)
estimator bias and its slope with respect to the true value (dimensionless)
Information is the energy of the signal's derivative, measured against the noise: a parameter is well determined when changing it changes the signal a lot compared with the noise. The bound depends on the noise only through its covariance, and the parameters only through \(\mathbf{J}\).symbols
\(I_{ij},\ \mathbf{I}\)
Fisher information entry and matrix, as in (12.5)
\(\sigma^2\)
total noise power per complex sample (\(\sigma^2/2\) per quadrature)
\(N\sum n^2 - (\sum n)^2 = N^2(N^2 - 1)/12\) for \(n = 0,\dots,N-1\). The variance falls as \(N^{-3}\): N samples of SNR each, times an \(N^2\) lever arm, because a frequency error produces a phase error that grows linearly along the record.symbols
\(\mathbf{I}\)
Fisher matrix for the unknowns \((\omega, \varphi)\)
\(A,\ \varphi\)
tone amplitude and unknown phase, rad
\(\omega,\ \hat\omega\)
true and estimated frequency, rad/sample (Doppler: \(2\pi f_D/\mathrm{PRF}\))
For noise white over a rectangular band whose edges sit where each 1D PSF energy spectrum falls to −40 dB (\(\lvert k\rvert \le 0.483/w\) cycles per unit length on each axis), the peak-pixel SNR is \(10\log_{10}(\ln 10^4/\pi)\) = 4.7 dB below the ENR in the continuous limit (4.8–5 dB on the discrete grids used here). The offset depends on the band shape: the elliptical −40 dB contour of the 2D spectrum gives 3.6 dB. In real noise with \(\mathrm{ENR}_r = \sum s^2/\sigma^2\), the same steps give \(\sigma_x \ge \sqrt{2}\,w_x/\sqrt{\mathrm{ENR}_r}\).symbols
\(I_{x_0x_0}\)
Fisher information for the lateral position, 1/µm²
\(a,\ g,\ w_x,\ w_z\)
amplitude and PSF \(g = g_x g_z\), as in (12.10)
\(N_0\)
noise power density per unit image area
\(\iint\)
integral over the image plane
\(\partial g/\partial x\)
lateral derivative of the PSF
\(\mathrm{ENR}\)
energy-to-noise ratio \(\lvert a\rvert^2\pi w_x w_z/N_0\): matched-filter output SNR (linear, not dB)
\(\sigma_x,\ \sigma_z\)
lower bounds on the position standard deviation, µm
Equivalent stacked least squares: \(\min \lVert[\mathbf{A};\ \lambda\mathbf{I}]\,\mathbf{x} - [\mathbf{y};\ 0]\rVert\), which you solve by QR without forming \(\mathbf{A}^{T}\mathbf{A}\) (Module 05). \(f_i \approx 1\) for \(\sigma_i \gg \lambda\), \(f_i \approx \sigma_i^2/\lambda^2\) for \(\sigma_i \ll \lambda\), and exactly ½ at \(\sigma_i = \lambda\): λ is a soft singular-value threshold. Truncated SVD is the hard version.symbols
\(\lambda\)
regularization parameter, units of a singular value (Hansen convention)
The first term is the regularization error (a bias: \(\mathrm{E}[\mathbf{x}_\lambda] = \mathbf{R}\mathbf{x}\), not \(\mathbf{x}\)); the second is the perturbation error. Raising λ trades the second for the first. The cap \(1/(2\lambda)\), reached at \(\sigma_i = \lambda\), is the most any component of the noise can be amplified.symbols
\(\mathbf{R}\)
model-resolution matrix \(\mathbf{A}^{\#}\mathbf{A}\); its rows are the reconstruction's point-spread functions
\(\mathbf{A}^{\#}\)
the regularized inverse, \((\mathbf{A}^{T}\mathbf{A} + \lambda^2\mathbf{I})^{-1}\mathbf{A}^{T}\)
\(\mathbf{x}_\lambda,\ \mathbf{x},\ \mathbf{n}\)
regularized estimate, true unknowns and noise, as in (13.1) and (13.3)
\(\mathbf{I}\)
identity matrix
\(f_i,\ \sigma_i,\ \lambda\)
as in (13.3)
\(\lvert\cdot\rvert,\ \le\)
noise gain per component is bounded by \(1/(2\lambda)\), peak at \(\sigma_i = \lambda\)
Soft thresholding is the proximal map of the L1 norm: the exact minimizer of \(\tfrac12(x - v)^2 + \tau\lvert x\rvert\). It sets \(\lvert v\rvert \le \tau\) to zero and shrinks the rest by \(\tau\).symbols
\(\mu\)
L1 weight, units of \(\mathbf{A}^{T}\mathbf{y}\)
\(\lVert\mathbf{x}\rVert_1\)
\(\sum_j \lvert x_j\rvert\)
\(S_\tau\)
soft-thresholding operator with threshold \(\tau\)
Tikhonov = Gaussian prior; Tikhonov with \(\mathbf{D}_1\) = Gaussian prior on increments (a random walk); L1 = Laplace prior; TV = Laplace prior on increments. Sparsity is a property of the MAP estimator, not of the prior: a Laplace draw is never exactly zero, and only \(1 - e^{-0.1}\) = 9.5 % of its samples have \(\lvert x\rvert < 0.1\beta\).symbols
Linear in slowness, not in c: \(\Delta c \approx -c_0^2\,\Delta s\), so 1 ns/mm of slowness is 2.37 m/s at 1540 m/s, and a +50 m/s inclusion is −20.4 ns/mm (10 mm of it advances an echo by 204 ns, a full period at 5 MHz). Refraction and the position shift of the misfocused echo are neglected.symbols
\(\mathbf{r}\)
pixel position \((x, z)\)
\(\theta_t,\ \theta_r\)
transmit and receive steering angles from vertical
\(\Delta s\)
slowness deviation, ns/mm (1 ns/mm = 10⁻⁶ s/m)
\(c_0\)
beamforming sound speed, m/s
\(\tau\)
delay error of the echo, ns
\(L_\theta(\mathbf{r})\)
delay accumulated along the one-way path through \(\mathbf{r}\) at angle \(\theta\), ns
\(\lambda \to 0\) recovers Gauss–Newton; \(\lambda \to \infty\) gives a short step along the (diagonally scaled) negative gradient. Divide \(\lambda\) by 10 after a successful step, multiply by 10 after a failed one (Levenberg 1944, Marquardt 1963). At the solution, \(\sigma^2(\mathbf{J}^{T}\mathbf{J})^{-1}\) is the Gaussian-noise CRLB of Module 12 evaluated at the estimate: the optimizer hands you the error bars.symbols
\(\mathbf{\delta}\)
parameter update
\(\lambda\)
Levenberg–Marquardt damping, dimensionless
\(\sigma^2\)
noise variance per data sample
\(\mathrm{cov}(\hat{\mathbf{\theta}})\)
linearized covariance of the estimate at convergence
Exact up to a term \(e^{-(2\pi f_0\sigma)^2} \approx 2 \times 10^{-7}\) for \(f_0\) = 5 MHz, \(\sigma\) = 0.125 µs (−6 dB bandwidth 3 MHz, 60 %). Local minima sit just inside multiples of \(T_0\): at ±194, ±388, ±583 ns. The maxima between them, at ±97 ns, bound the basin of the true minimum: start within about \(T_0/2\) or converge to the wrong cycle.symbols
\(\sigma\)
envelope standard deviation, µs (0.125 µs here)
\(f_0\)
centre frequency, MHz
\(\tau\)
delay error, ns
\(T_0 = 1/f_0\)
carrier period, 200 ns at 5 MHz
\(s(t)\)
Gaussian-windowed cosine pulse
\(t\)
time, µs
\(R_{ss}(0)\)
pulse energy \(E\), as in (14.12)
\(R_{ss}(\tau)\)
pulse autocorrelation, as in (14.12)
\(e\)
exponential function
\[ \text{fixed-depth pixel: } t(\hat c) = \frac{2z}{\hat c} \;\Rightarrow\; \Delta t = \frac{2z\,\Delta\hat c}{\hat c^{2}} \;\Rightarrow\; \text{one carrier period per } \Delta\hat c = \frac{\hat c^{2}}{2 z f_0} \]
(14.19)
Tying the pixel to depth turns a sound-speed error into a bulk time shift of the echo through the pixel, so the RF value oscillates exactly like the delay misfit of Section 14.4. Tying the ROI to two-way time removes the bulk shift: what remains is defocus, which varies slowly (Module 07: edge phase \(\approx (\pi D^2/(2\lambda z))(\Delta c/c)\), 1.6 rad for 40 m/s at F/1.6).symbols
\(z\)
pixel depth, m
\(f_0\)
centre frequency, Hz
\(\Delta\hat c\)
sound-speed change that slides the echo by one period, m/s: 7.9 m/s at 1540 m/s, 30 mm, 5 MHz
No subtraction, so no cancellation: \(h = 10^{-20}\) gives full machine precision (Lyness & Moler, 1967; Squire & Trapp, 1998). It needs a real-analytic implementation: abs, conj, the ' operator in MATLAB, max/min branches and comparisons break it silently, and it cannot be used on code that already computes with complex IQ data.symbols
\(i\)
imaginary unit
\(h\)
imaginary step; any \(h \le 10^{-8}\), even \(10^{-20}\) or \(10^{-100}\)
\(f\)
real-analytic function implemented with complex-safe operations
The steepest-descent direction in \(\mathbb{C}^N\) (viewed as \(\mathbb{R}^{2N}\)) is \(-\partial\mathcal{L}/\partial\mathbf{z}^*\), not \(-\partial\mathcal{L}/\partial\mathbf{z}\) (Brandwood, 1983).symbols
\(z = x + iy\)
complex parameter (IQ weight, focusing-law entry)
\(z^*\)
complex conjugate, treated as an independent variable
\(Z = \rho_0 c_0\) is the specific acoustic impedance (Rayl = kg m⁻² s⁻¹). For a plane progressive wave p and u are in phase; close to a point source (\(kr \lesssim 1\)) they are not, and the field stores reactive energy. 1 MPa in tissue (Z = 1.62 MRayl) carries \(I\) = 31 W/cm² while the pulse lasts.symbols
\(f,\ g\)
arbitrary waveforms travelling in +x and −x
\(A\)
source strength of the spherical wave, Pa·m
\(r\)
distance from the source, m
\(k\)
wavenumber \(\omega/c_0\), rad/m
\(Z = \rho_0 c_0\)
specific acoustic impedance, Rayl (kg m⁻² s⁻¹)
\(p_0\)
pressure amplitude, Pa
\(I\)
time-averaged intensity, W/m²
\(t,\ x\)
time (s) and propagation coordinate (m)
\(p,\ u\)
acoustic pressure (Pa) and particle velocity along \(x\) (m/s)
Fat → muscle: R = 0.093 (−20.7 dB); liver → muscle: R = −0.004 (−48 dB); soft tissue → skull: R = 0.54, 29 % of the energy; soft tissue → air: R = −0.9995, so only 0.1 % (−30 dB) of the energy crosses. Soft-tissue contrast in B-mode comes from interfaces 20–50 dB below a perfect reflector and from diffuse scattering, not from large impedance jumps.symbols
\(Z_1,\ Z_2\)
acoustic impedances of the incident and transmitting media, MRayl
\(p_i,\ p_r,\ p_t\)
incident, reflected and transmitted pressure amplitudes, Pa
\(R,\ T\)
pressure reflection and transmission coefficients
\(R_I,\ T_I\)
reflected and transmitted fractions of the intensity
\[ \begin{aligned} &\alpha(f) = \alpha_0 f^y, \quad y \approx 1\text{–}1.3 \text{ in soft tissue},\ y = 2 \text{ in water} \quad (1\ \mathrm{Np} = 8.686\ \mathrm{dB}) \\ &\text{round-trip loss} = 2\, \alpha_0 f z\text{:}\quad 0.5\ \mathrm{dB/cm/MHz},\ 5\ \mathrm{MHz},\ 10\ \mathrm{cm} \to 50\ \mathrm{dB} \end{aligned} \]
(15.6)
With a 60 dB dynamic range budget and 0.5 dB/cm/MHz the usable depth is 20 cm at 3 MHz, 12 cm at 5 MHz and 6 cm at 10 MHz. Water at 20 °C: \(2.2 \times 10^{-3}\) dB/cm/MHz², 0.02 dB/cm at 3 MHz.symbols
Compressions catch up with the zero crossing ahead of them; the slope at the zero crossing becomes infinite at \(\bar{x}\). 3 MHz, 1 MPa: water 49.2 mm, liver 49.1 mm, fat 25.6 mm. \(\bar{x}\) halves when either \(f\) or \(p_0\) doubles.symbols
Energy flows from the fundamental to the harmonics; at σ = 1 the fundamental has lost 1.1 dB to them. After the shock the waveform is a sawtooth whose amplitude decays because energy is dissipated at the shock front.symbols
\(\sigma = x/\bar{x}\)
shock parameter (distance over shock distance), dimensionless
−6 dB widths in \(\lambda_0 F/D\): fundamental one-way 1.207, harmonic transmit (\(\operatorname{sinc}^2\)) 0.886 (× 0.734), fundamental two-way 0.886, harmonic two-way 0.511, a probe driven linearly at \(2f_0\) 0.443. First sidelobes: −13.3, −26.5, −26.5, −27.9, −26.5 dB. The harmonic beam is almost as narrow as a \(2f_0\) beam while the transmit stays at \(f_0\), which penetrates (attenuation halves in dB) and keeps the array's grating lobes away.symbols
\(p_2\)
local second-harmonic source strength
\(P_{\mathrm{tx}},\ P_{\mathrm{rx}}\)
transmit and receive lateral (focal-plane) patterns
Free (air-loaded on both sides), lossless plate. Well above resonance (between harmonics) it behaves as the capacitor \(C_0\); well below, \(Z \to (1 - k_t^2)/(j\omega C_0)\), the capacitance \(C_{\mathrm{LF}}\) of (16.6). \(\lvert Z\rvert \to \infty\) at \(x = \pi /2\): the antiresonance \(f_a\), where the plate is a half wavelength thick at the stiffened speed. \(Z = 0\) at the resonance \(f_r\), where \(k_t^2 \tan (x)/x = 1\).symbols
IEEE Std 176-1987 relation, obtained by setting \(Z = 0\) above \((\tan (\pi /2 - x) = \cot x)\). Strictly \(f_r\), \(f_a\) are the series and parallel frequencies \(f_s\), \(f_p\) (maximum conductance and maximum resistance); for low loss they equal the minimum and maximum of \(\lvert Z\rvert\). Small-gap approximation: \(k_t^2 \approx (\pi^2 /4)(f_a - f_r)/f_a\), 6 % high in \(k_t\) at \(k_t = 0.5\).symbols
antiresonance (parallel, \(f_p\)): maximum \(\lvert Z\rvert\), Hz
\(x_r\)
normalized resonance frequency, rad
\(k_t\)
thickness coupling factor, as in (16.5)
\(\tan,\ \cot\)
tangent and cotangent (argument in rad)
\[ \begin{gathered} \text{KLM:}\quad C_0 \text{ in series with } jX_1 \\ X_1 = \frac{k_t^2}{\omega C_0}\,\operatorname{sinc}\!\left(\frac{f}{f_a}\right) \\ \varphi = k_t\,\sqrt{\frac{\pi}{\omega_0 C_0 Z_0}}\;\operatorname{sinc}\!\left(\frac{f}{2 f_a}\right) \\ \omega_0 = 2\pi f_a \end{gathered} \]
(16.11)
\(\operatorname{sinc} (x) = \sin (\pi x)/(\pi x)\). Equivalently \(X_1 = (h_{33}^2 /\omega^2 Z_0) \sin (kt)\) and \(\varphi = (2h_{33} /\omega Z_0) \sin (kt/2)\) in V/N: \(V = \varphi F\) and \(I = u/\varphi\), so an acoustic impedance Z at the centre node appears at the electrical side as \(\varphi^2 Z\). The plate itself is two line sections of length t/2, impedance \(Z_0\), joined at the transformer.symbols
Receive uses a Thevenin acoustic source \(2F_{\mathrm{inc}}\) behind \(Z_L\) and \(\det M = 1\). H assumes a plane wave, normal incidence on a perfect planar reflector, no diffraction or attenuation in the medium. Insertion loss \(\mathrm{IL} = 20 \log_{10} \lvert 2H\rvert\): 0 dB means all the available power came back. The impulse response is the inverse FFT of H.symbols
\(A,\ B,\ C,\ D\)
elements of the chain matrix \(\mathbf{M}\)
\(Z_L\)
load (tissue) mechanical impedance, kg/s
\(R_s,\ R_r\)
source and receiver resistance, Ω
\(F_L,\ F_{\mathrm{inc}}\)
force on the load and incident echo force, N
\(V_s,\ V_{\mathrm{out}}\)
source and received voltage, V
\(H\)
pulse-echo transfer function, dimensionless
\(Z_{\mathrm{in}}\)
impedance seen at the electrical port, Ω
\(\Delta(R)\)
common denominator of the transfer functions for terminating resistance \(R\), Ω
Open-circuit (or high-impedance) plate resonating at \(f_a\); −3 dB fractional bandwidth ≈ 1/Q. PZT-5H (34.5 MRayl) into water with air backing: \(Q = 36\) (KLM, open circuit: 36.1); with \(Z_B = 5\) MRayl: \(Q = 8.3\) (8.2); with 12 MRayl the formula is 10 % high. Valid for \(Z_B + Z_F \ll Z_p\). Under voltage drive (\(R_s \to 0\)) the response peaks near \(f_r\) and the Q is ≈ 20 % lower; a finite source resistance adds electrical damping.symbols
\(T_I\) is the intensity (power) transmission. \(T_p\) can exceed 1 going into a stiffer medium; energy is still conserved because intensity is \(p^2 /Z\).symbols
\(Z_1,\ Z_2\)
acoustic impedances of the incident and transmitting media, MRayl
Amplitude falls as \(e^{-\pi n/Q}\) after n periods. Air backing: \(Q \approx 36 (r = 0.916\), −0.76 dB per cycle, 26 cycles to −20 dB). Backing 5 MRayl: \(Q \approx 8.2 (r = 0.682\), −3.3 dB per cycle, 6 cycles to −20 dB). Mechanical (acoustic) Q only; the electrical port adds its own filtering.symbols
\(r\)
amplitude ratio per round trip, \(R_{\mathrm{front}} R_{\mathrm{back}}\)
Choose \(Z_1^2 /Z_L = Z_p\), i.e. \(Z_1 = \sqrt{Z_p Z_L } = 7.16\) MRayl, and the plate sees a matched load at \(f_0\): power transmission 1. Same algebra as the microwave quarter-wave transformer.symbols
The optical lensmaker's \(1/f = (n - 1)/R\) with \(n = c_t /c_l\). F is measured from the lens apex into tissue. Because the speed contrast is large, F depends strongly on \(c_t\): the same lens focuses at 48.0 mm in 1450 m/s fat-like tissue instead of 40.0 mm.symbols
Distributed plate (this module, FD). The piston spring \(k = 192\pi D/a^{2}\) is the force \(\pi a^{2}P\) divided by the average deflection \(Pa^{4}/(192D)\) of a clamped plate under uniform pressure \(P\). The distributed constant 27.8 comes from the finite-difference solution below (27.76; unchanged to 0.02 % between 60, 120 and 240 radial intervals). It holds for any \(a\), \(h\), \(g\) as long as the plate stays in small deflection (\(w \ll h\)) and the gap is uniform. Default cell (Si, \(a\) = 20 µm, \(h\) = 1 µm, \(g_0\) = 0.2 µm, 0.1 µm nitride): \(V_{\mathrm{PI}}\) = 53.1 V, against 76.0 V from the piston.symbols
\(C = \sum \varepsilon_0 W_i/(g_{\mathrm{eff}} - w_i)\). For a piston the rank-one term exactly cancels the softening (\(K_Q = k\)) and \(k^{2} = 1 - k_{\mathrm{eff}}/k\), the Module 19 formula. For a plate it does not cancel fully, and \(k^{2}\) is lower: 0.04, 0.16, 0.26 and 0.37 at 0.5, 0.8, 0.9 and 0.95 \(V_{\mathrm{PI}}\) for the default cell (piston: 0.08, 0.28, 0.43, 0.56). Bayram et al. (IEEE TUFFC 2003) found the same character with FEM: \(k_T^{2}\) exceeds 0.35 only above 85 % of the collapse voltage.symbols
\(\eta\) is 0.8 here. \(k_{a,\mathrm{eff}}\) includes spring softening, so \(f_{\mathrm{low}}\) falls as the bias rises. The second mass term is the water added mass of an isolated baffled cell, \((8/3)\rho_w a^{3}\) per cell; in an array moving in phase it is smaller, so the true \(f_{\mathrm{high}}\) lies between the two estimates. For the 20 µm cell at 0.9 \(V_{\mathrm{PI}}\): \(f_{\mathrm{low}} \approx\) 0.8 MHz, \(f_{\mathrm{high}} \approx\) 14 MHz with added mass. A PZT element, in contrast, is a half-wave resonator with mechanical impedance ≈ 34 MRayl and \(Q \approx (\pi/2)Z_p/(Z_B + Z_L)\) that needs matching layers to broaden (Modules 16–17).symbols
\(\eta\)
fraction of the element area covered by cells
\(k_{a,\mathrm{eff}}\)
spring per unit area including bias softening, Pa/m
First axisymmetric mode of a clamped circular plate in vacuum (Leissa, Vibration of Plates, NASA SP-160, 1969). The constant is verified here twice: bisection on the Bessel characteristic equation gives \(\lambda_1\) = 3.1962, and the finite-difference eigen-solve used for the CMUT converges to 10.2113, 10.2147, 10.2155 with 40, 80, 160 intervals. \(f \propto h/a^{2}\): halve the radius and the frequency quadruples.symbols
\(\lambda_1\)
first root of \(J_0(\lambda)I_1(\lambda) + I_0(\lambda)J_1(\lambda) = 0\)
\(a\)
plate radius, m
\(D\)
flexural rigidity, N·m
\(\bar\rho h\)
mass per unit area of the plate, kg/m²
\(f_{\mathrm{vac}}\)
fundamental resonance in vacuum, Hz
\(J_0,\ J_1,\ I_0,\ I_1\)
Bessel and modified Bessel functions of the first kind, orders 0 and 1
Lamb's added virtual mass for a clamped plate with water on one side (H. Lamb, Proc. R. Soc. Lond. A 98, 1920; the form used by Ayela and Nicu to validate PMUT measurements). For \(a\) = 50 µm and 3 µm of AlN-on-Si, \(\beta\) = 4.2: water lowers the resonance by a factor 2.3.symbols
\(\beta\)
added-mass ratio (water on one side)
\(\rho_w\)
water density, kg/m³
\(f_{\mathrm{water}},\ f_{\mathrm{vac}}\)
fundamental resonance with water on one side and in vacuum (18.9), Hz
\(a,\ \bar\rho h\)
plate radius (m) and mass per unit area (kg/m²), as in (18.9)
The film stress \(e_{31,f}V/h_p\) acting over \(h_p\) at lever arm \(z_p\) (film mid-plane to neutral axis) is a moment per unit length \(M_p\), independent of \(h_p\). Its virtual work on the mode shape \(\varphi = (1 - r^{2}/a^{2})^{2}\) over an inner electrode of radius \(\beta_e a\) is the generalized force \(Q_{\mathrm{gen}}\) (one-term Ritz, quasi-static). It vanishes for a full electrode, because the curvature changes sign near \(r = 0.71a\), and is maximal at \(\beta_e = 1/\sqrt{2} = 0.707\): the origin of the familiar 70 % electrode.symbols
Rigid baffle: the strip sits in an infinite rigid plane (Rayleigh integral). Soft baffle: the surrounding surface is pressure-release, which adds the obliquity factor \(\cos \theta\). Measurements on kerfed arrays usually sit closer to the soft-baffle curve (Selfridge, Kino & Khuri-Yakub, Appl. Phys. Lett. 1980). As w → 0 the rigid model goes omnidirectional; the soft model still falls to −6 dB at 60°.symbols
\(E(\theta)\)
element (directivity) factor, amplitude relative to broadside
\(w\)
element width, mm
\(\theta\)
angle from the element normal, rad
\(\lambda\)
wavelength in the medium, mm
\(\operatorname{sinc}(x)\)
normalized sinc, \(\sin(\pi x)/(\pi x)\); \(x\) is its dimensionless argument
The acceptance angle is the half-angle at which the element amplitude has fallen 6 dB. An aperture edge at F# subtends atan(1/(2F#)); below \(F\#_{\min}\) the edge elements receive at less than half amplitude and add noise faster than signal. Example: \(w = 0.27\) mm at 5 MHz \((w/\lambda = 0.88)\) gives 43.5° rigid / 36.0° soft, \(F\#_{\min} 0.53 / 0.69\); at 7.5 MHz 27.3° / 25.4°, \(F\#_{\min} 0.97 / 1.05\).symbols
\[ \begin{gathered} P(\theta) = E(\theta)\cdot \mathrm{AF}(u - u_0), \qquad u = \sin\theta \\ \mathrm{AF}(u) = \frac{\sin(N\pi d u/\lambda)}{N\sin(\pi d u/\lambda)} \end{gathered} \]
(19.3)
AF peaks at \(u - u_0 = m\lambda /d\). Grating lobes appear at \(\sin \theta_g = \sin \theta_0 \pm m\lambda /d\); they stay invisible (\(\lvert \sin \theta_g \rvert\) > 1) for every steering angle up to \(\theta_{\mathrm{max}}\) if \(d \le \lambda /(1 + \sin \theta_{\mathrm{max}})\): \(0.67\lambda\) for 30°, \(0.59\lambda\) for 45°, \(0.5\lambda\) for 90°.symbols
\(P(\theta)\)
one-way array pattern
\(\mathrm{AF}\)
array factor
\(N\)
number of elements
\(d\)
element pitch, mm
\(u_0\)
\(\sin\theta_0\), the steering direction
\(u,\ \theta\)
direction sine \(u = \sin\theta\); angle from the array normal in the scan plane
PZT with \(\varepsilon_r^S \approx 1500\), \(t = 0.3\) mm. A 1D element 0.27 × 5 mm: 60 pF, so 100–200 pF of cable costs 8.5–12.8 dB. A 2D element 0.2 × 0.2 mm: 1.8 pF (18 kΩ at 5 MHz), and the same cable costs 35–41 dB. The noise of the receiver behind the cable then dominates; the fix is an amplifier or beamformer within millimetres of the element.symbols
\(V_{\mathrm{out}},\ V_{\mathrm{open}}\)
voltage at the cable end and open-circuit element voltage, V
\(\lvert\mathrm{AF}\rvert = 1\) at every replica centre, so a replica inside the unit circle is a grating lobe at full main-lobe height, reduced only by the element factor, \(E(u_{pq}, v_{pq})/E(u_0, v_0)\). Replicas outside the circle are evanescent, but a replica within about one main-lobe width of the edge still leaks its skirt into visible space. In 2D the 1D rule \(\sin\theta_g = \sin\theta_0 \pm \lambda/d\) becomes "translate a lattice and look through a round window".symbols
\(u,\ v\)
direction cosines, \(u = \sin\theta\cos\varphi\), \(v = \sin\theta\sin\varphi\)
\(\theta,\ \varphi\)
polar angle from the array normal and azimuth from the x axis
Cone 30° / 45° / 60° / 90°: square 0.667 / 0.586 / 0.536 / 0.500 \(\lambda\); triangular 0.770 / 0.676 / 0.619 / 0.577 \(\lambda\). The triangular grid allows a 15.5 % larger spacing and needs \(\sqrt{3}/2 = 0.866\) times the elements (13.4 % fewer) for the same aperture and cone, the classic result for planar phased arrays (Sharp, IRE Trans. Antennas Propag. 1961). Seen the other way, at 0.7\(\lambda\) a square grid is lobe-free only to 25.4° in every azimuth, a triangular grid with the same spacing to 40.5° (m19_matrix_gratinglobe_uv.m).symbols
\(\theta_{\max}\)
half-angle of the steering cone (all azimuths)
\(d\)
nearest-neighbour spacing of the triangular grid, mm
\(d_x,\ d_y,\ \lambda\)
as in (19.7)
\[ \begin{gathered} \rho(\Delta u) = \frac{\lvert\mathrm{SF}(\Delta u - \lambda/D)\rvert}{\lvert\mathrm{SF}(\Delta u)\rvert} = \frac{\sin(\pi d\,\Delta u/\lambda)}{\sin\!\left(\frac{\pi}{m} - \frac{\pi d\,\Delta u}{\lambda}\right)} \\ \Delta u = u_s - u_0, \qquad D = m d \\ \Delta u_{\max} = \frac{\lambda}{\pi d}\arctan\frac{\rho_{\max}\sin(\pi/m)}{1 + \rho_{\max}\cos(\pi/m)} \\ N_{\mathrm{cab}} = \frac{N^2}{m^2} \end{gathered} \]
(19.10)
Small errors: \(\rho \approx \pi d\,\Delta u/(\lambda\sin(\pi/m))\), linear in the pre-steer error. 3 × 3 at \(\lambda/2\), 10° off broadside: \(\rho = -8.3\) dB, the table value. The window is symmetric in \(u\), not in angle: 3 × 3 pre-steered to 30° keeps lobes below −20 dB from 26.6° to 33.5°. In 2D the subarray factor is separable, so steering off by \(\Delta u\) and \(\Delta v\) gives axis lobes \(\rho(\Delta u)\), \(\rho(\Delta v)\) and a diagonal lobe \(\rho(\Delta u)\rho(\Delta v)\): −15.2 dB and −30.5 dB for 5° in both.symbols
\(\rho\)
amplitude of the worst coarse (subarray) grating lobe relative to the main lobe, valid for \(0 \le \Delta u < \lambda/D\)
\(\Delta u\)
pre-steer error in direction sine, \(\sin\theta_s - \sin\theta_0\)
\(D\)
subarray pitch, mm
\(m,\ d\)
subarray size (m × m elements) and element pitch, mm
\(\rho_{\max}\)
allowed lobe amplitude (0.1 = −20 dB)
\(\Delta u_{\max}\)
half-width of the steering window around the pre-steer, in direction sine
\(N,\ N_{\mathrm{cab}}\)
elements per side of an \(N \times N\) matrix; system channels (cable conductors) after micro-beamforming
\(\mathrm{SF},\ u_s,\ u_0,\ \lambda\)
as in (19.9)
\[ \begin{aligned} k x &= \frac{\varepsilon_0 A V^2}{2(g_0 - x)^2} \\ \to\ k_{\mathrm{eff}} &= k - \frac{\varepsilon_0 A V^2}{(g_0 - x)^3} = k\left(1 - \frac{2x}{g_0 - x}\right) \end{aligned} \]
(19.13)
Spring force balances the electrostatic force; the electrostatic force grows faster with x than the spring, so it acts as a negative spring ("spring softening"). \(k_{\mathrm{eff}}\) reaches 0 at \(x = g_0 /3\): pull-in (collapse), at \(V_{\mathrm{PI}} = \sqrt{8 k g_0^3 / (27\,\varepsilon_0 A)}\).symbols
Normalized forms. The coupling coefficient \(k_T^2 = 1 - k_{\mathrm{eff}} /k\) rises from 0 to 1 as the bias approaches pull-in (Yaralioglu et al., IEEE TUFFC 2003); the resonance falls with it. At 90 % of \(V_{\mathrm{PI}}\): \(\xi = 0.177\), \(k_T^2 = 0.43\), \(f = 0.75 f_0\). This is why CMUTs are biased at 70–90 % of pull-in.symbols
A positive half-cycle that is \(\Delta = 1\) ns too long at 5 MHz (\(T = 200\) ns, \(\varepsilon = 0.005\)) gives a second harmonic at −36.1 dBc; 2 ns gives −30.1 dBc. A −45 dBc second-harmonic spec at 5 MHz is a duty error below 0.36 ns. Unequal rise and fall times break the symmetry the same way. An amplitude mismatch \(+V \ne -V\) alone does not: it adds a step at the burst envelope, which is low-frequency.symbols
\(c_1,\ c_2\)
fundamental and second-harmonic amplitudes, V
\(D\)
duty cycle of the positive half, \(D = \tfrac{1}{2} + \varepsilon\), dimensionless
Purely capacitive limit: \(\lvert H \rvert = C_0/(C_0 + C_{\mathrm{cable}} + C_{\mathrm{in}}) = 30/195\) = −16.3 dB for a 150 pF cable and 15 pF LNA input. At resonance the motional branch lowers the source impedance, and the BVD model gives −12.0 dB. Either way the cable is a large signal loss.symbols
\(V_{\mathrm{oc}}\)
open-circuit echo voltage of the element, V
\(V_{\mathrm{LNA}}\)
voltage at the LNA input, V
\(Z_e\)
element impedance (20.6), Ω
\(Z_L\)
load seen by the element, Ω
\(C_{\mathrm{cable}},\ C_{\mathrm{in}}\)
cable and LNA input capacitance, pF
\(R_{\mathrm{term}}\)
termination resistance (if fitted), Ω
\(L_{\mathrm{tune}}\)
shunt tuning inductance (if fitted), H
\(\omega,\ j\)
angular frequency (rad/s) and imaginary unit
\(\parallel\)
parallel combination, as in (20.6)
\[ F = 1 + \frac{e_n^2 + i_n^2 \lvert Z_s \rvert^2}{4kT \operatorname{Re} Z_s}, \qquad \mathrm{NF} = 10 \log_{10} F \]
(20.9)
Noise figure is the SNR degradation relative to the source's own thermal noise at \(T_0\) = 290 K. For a resistive source and uncorrelated \(e_n\), \(i_n\), \(F\) is minimum at \(R_{\mathrm{opt}} = e_n/i_n\) with \(F_{\mathrm{min}} = 1 + e_n i_n/(2kT)\). 0.8 nV/√Hz and 1.5 pA/√Hz: \(R_{\mathrm{opt}} = 533\) Ω, \(\mathrm{NF}_{\mathrm{min}} = 0.61\) dB.symbols
\(e_n\)
LNA input voltage-noise density, nV/√Hz
\(i_n\)
LNA input current-noise density, pA/√Hz
\(Z_s\)
source impedance seen by the LNA (element + cable), Ω
\(\alpha = 0.5\) dB/cm/MHz at 5 MHz: 5 dB/cm, 0.385 dB/µs. At 20 cm depth that is 100 dB of attenuation, which no single VGA covers; in practice the gain saturates and deep echoes become noise-limited (Module 10 for the SNR-versus-depth view).symbols
From the slope error of a full-scale sine sampled at a time error \(\sigma_t\) (rms). 1 ps at 10 MHz: 84.0 dB; 1 ps at 15 MHz: 80.5 dB; 5 ps at 10 MHz: 70.1 dB. Noise powers add: \(1/\mathrm{SNR} = 1/\mathrm{SNR}_q + 1/\mathrm{SNR}_j + 1/\mathrm{SNR}_{\mathrm{thermal}}\).symbols
\(h\) has units of m/s. It is the potential produced by an impulsive velocity \(\delta(t)\): the aperture's arrival-time histogram, weighted by \(1/(2\pi R)\) per unit area. Pressure is the time derivative of the velocity convolved with h.symbols
Jensen's model (JASA 1991) behind Field II: one time derivative from radiation, two from Rayleigh scattering by a small inhomogeneity, \(E_m\) the electromechanical response, and the transmit and receive spatial impulse responses. The three derivatives tilt the spectrum by \(f^3\) (card in 21.1).symbols
\(y(t)\)
received echo voltage, V
\(E_m(t)\)
electromechanical impulse response of the transducer
\(v\)
surface velocity waveform, m/s
\(h_t,\ h_r\)
transmit and receive spatial impulse responses, m/s
\(\propto\)
proportional to
\(t,\ \mathbf{r}\)
time (s) and field point (m)
\(*\)
convolution in time
\(\partial^3/\partial t^3\)
third time derivative of the velocity: one from radiation, two from Rayleigh scattering by a small inhomogeneity
\[ \begin{aligned} &\text{disc, on axis, CW:}\quad \lvert p(z) \rvert = 2\rho_0 c\, v_0 \left\lvert \sin\left[ \frac{\pi}{\lambda} \left( \sqrt{z^2 + a^2} - z \right) \right] \right\rvert \\ &\text{last maximum at } z = \frac{a^2}{\lambda} - \frac{\lambda}{4} \approx \frac{a^2}{\lambda} \\ &\text{square of side } 2a\text{: last maximum near } 1.36\, a^2/\lambda \\ &\text{(exact Rayleigh integral; the Fresnel two-slit approximation gives } 1.37\, a^2/\lambda\text{)} \end{aligned} \]
(21.5)
Beyond the last maximum the on-axis pressure falls as \(1/z\): the far field. Pulsed excitation averages the near-field interference over the band, so the deep CW nulls fill in.symbols
\(v_0\)
surface velocity amplitude, m/s
\(a\)
disc radius, or half the side of the square, m
\(\lambda\)
wavelength, m
\(z\)
distance along the axis, m
\(p(z)\)
on-axis pressure amplitude, Pa
\(\rho_0,\ c\)
density (kg/m³) and sound speed (m/s)
\(\lvert\cdot\rvert\)
magnitude
\[ \begin{aligned} &\sin(\omega \Delta t/2) = \mathrm{CFL} \cdot S(k \Delta x) \\ &\text{2nd-order:}\quad S = \sin(k\Delta x/2) \\ &\text{4th-order:}\quad S = (9/8) \sin(k\Delta x/2) - (1/24) \sin(3k\Delta x/2) \\ &\text{pseudospectral (leapfrog):}\quad S = k\Delta x/2 \\ &\text{k-space corrected:}\quad \partial/\partial x \to jk \cdot \operatorname{sinc}(c_{\mathrm{ref}} k \Delta t/2) \;\Rightarrow\; \omega = c_{\mathrm{ref}} k \text{ exactly} \end{aligned} \]
(21.7)
The numerical phase speed is \(\omega/k\). Second-order FD2 is slow by \(\approx (k\Delta x)^2 (1 - \mathrm{CFL}^2)/24\), so 1.5 % at 10 points per wavelength (PPW) and CFL 0.3; over 100 wavelengths that is 1.5 cycles of phase error. Stability needs \(\mathrm{CFL} \cdot \max S \le 1\) in 1D (\(\mathrm{CFL} \le 1\), 6/7 and \(2/\pi\) for the three schemes), divided by \(\sqrt{d}\) in \(d\) dimensions (for all three schemes).symbols
\(\omega\)
numerical angular frequency, rad/s
\(k\)
wavenumber, rad/m
\(\Delta x,\ \Delta t\)
grid spacing (m) and time step (s)
\(\mathrm{CFL}\)
Courant number \(c\Delta t/\Delta x\), dimensionless
\(S\)
the scheme's discrete spatial-derivative operator, dimensionless
\(c_{\mathrm{ref}}\)
reference sound speed of the k-space operator, m/s
\(j\)
imaginary unit
\(\operatorname{sinc}\)
\(\sin(\pi u)/(\pi u)\)
\(x\)
position on the grid, m
\[ \begin{aligned} &\text{derating:}\quad p_{\mathrm{r.3}}(z) = p_r(z) \cdot 10^{-0.3 f z / 20} \quad (f \text{ in MHz},\ z \text{ in cm}) \\ &\mathrm{MI} = p_{\mathrm{r.3}} / \sqrt{f_{\mathrm{awf}}} \quad (p \text{ in MPa},\ f \text{ in MHz}) \\ &\text{focus } p(r) = \operatorname{jinc}(r/\lambda F\#),\ \text{hydrophone radius } b\text{:} \\ &\langle p \rangle / p(0) \approx 1 - (\pi b / \lambda F\#)^2 / 16 \end{aligned} \]
(21.9)
The averaging law follows from \(p(r) \approx 1 - (\pi r/\lambda F\#)^2/8\) near the peak and \(\langle r^2 \rangle = b^2/2\) over a disc; it is within 0.1 dB of the exact average for \(b < 0.3\, \lambda F\#\). IEC 62359 evaluates MI at the depth defined by the derated pulse-intensity integral, which is shallower than the water focus.symbols
\(p_r\)
peak rarefactional pressure measured in water, MPa
3 MHz at 5 cm: 4.5 dB, pressure × 0.596, intensity × 0.355. 5 MHz at 5 cm: 7.5 dB, × 0.422 and × 0.178. 7.5 MHz at 4 cm: 9.0 dB, × 0.355 and × 0.126. Each derated quantity has its own maximum depth: the water focus, the derated \(p_r\) peak and the derated \(I_{\mathrm{ta}}\) peak need not coincide.symbols
\(p\)
pressure in water, MPa
\(I,\ \mathrm{PII},\ P\)
intensity, pulse-intensity integral and power in water
Bone at surface. \(C_{\mathrm{TIS}}\) = 210 mW·MHz, \(C_{\mathrm{TIC}}\) = 40 mW/cm, \(C_{\mathrm{MI}}\) and the break-point depth \(1.69\sqrt{A}\) (= \(1.5\,D_{\mathrm{eq}}\)) are confirmed in public sources (Health Canada guidance, ITRUSST consensus, Nowicki 2020); the diameter form of TIB with 40 mW/cm is Nowicki 2020, Eq. (6). The large-aperture TIS and the TIB forms follow the ODS lineage that IEC 62359 kept; the standard computes non-scanning TIB from \(P_{\alpha}\) and \(I_{\mathrm{ta},\alpha}\) at the depth where their product peaks, \(\sqrt{P_{\alpha} I_{\mathrm{ta},\alpha}}/C_{\mathrm{TIB}}\). Flag: \(C_{\mathrm{TIB}}\) = 50 mW/cm is the value commonly quoted from the ODS, but no public source consulted for this module (October 2026) states it, so treat it as unverified. If \(C_{\mathrm{TIB}}\) = 50 mW/cm, a round Gaussian beam gives \(\sqrt{P I_0} = 1.329\,P/d_{-6}\), i.e. \(\mathrm{TIB} = P/(37.6\, d_{-6})\), 6.3 % above the diameter form (m22_ti_estimates.m prints the same ratio, 1.063). IEC 62359 Ed. 2 (2010) added at-surface and below-surface summations for combined modes. Use the standard's own text for submission numbers.symbols
Exact for a round Gaussian focal spot and a circular, uniformly sensitive hydrophone element centred on the axis. Diameter = \(W_{-6}/4\): −0.19 dB, i.e. pressure 2.1 % and intensity 4.2 % low. Diameter = \(W_{-6}/2\): −0.74 dB, pressure 8.2 % and intensity 15.7 % low. The dB loss is the same for pressure and intensity (\(10\log_{10} p^{2} = 20\log_{10} p\)); only the percentage doubles.symbols
\(w\)
Gaussian focal spot parameter: amplitude \(e^{-r^{2}/w^{2}}\), −6 dB width \(W_{-6} = 1.665\,w\), mm
\(b\)
hydrophone radius, mm
\(\langle p\rangle\)
pressure averaged over the hydrophone face, MPa
\(p(0)\)
true on-axis peak pressure, MPa
\(r\)
radial distance from the beam axis, mm
\(e\)
exponential function
\(w,\ b\)
Gaussian spot parameter and hydrophone radius, mm
\(\text{intensity error}\)
fractional intensity (pressure squared) reading relative to the true peak; below 1 means under-reading
Large \(N_F\): focusing removes many waves of phase error and the focus is sharp and in place. \(N_F \lesssim 1\): there is little to remove, and the intensity peak slides toward the aperture, where the 1/r amplitude is larger.symbols
\(N_F\)
Fresnel number: half-period Fresnel zones across the half-aperture (dimensionless)
\(d \le \lambda/2\): no grating lobe at any steering (phased arrays; 2.5 MHz, ±45° needs \(d < 0.586\lambda\) = 0.36 mm). The reference linear array (0.3 mm) is \(0.974\lambda\) at 5 MHz, so it is grating-lobe free only up to 1.5° of steering; at 7.5 MHz it is \(1.46\lambda\) and has a lobe at ±43.2° even unsteered. At \(d = \lambda/(1+\lvert\sin\theta_{\max}\rvert)\) the lobe sits at endfire, \(u = \mp 1\).symbols
\(d\)
pitch, mm
\(\lambda\)
wavelength at the highest frequency that matters, mm
Valid while \(M < N\). The two-way energy is lower by 2/3 (−1.8 dB) because the triangular coarray weights give \(\sum_k c_k^2 \approx 2N^3/3\) instead of \(N^3\). Check: N = 128, FBW 60 % (M = 1.47): −38.2 dB peak, −17.6 dB energy, against 0 dB in CW.symbols
\(\sigma_t\)
standard deviation of the Gaussian envelope, µs
\(f_0\)
center frequency, MHz
\(M\)
pulse length in cycles, \(f_0\tau\), with \(\tau\) the −6 dB envelope duration
\(\mathrm{FBW}\)
−6 dB fractional bandwidth
\(N\)
number of elements
\(\mathrm{GL}\)
grating-lobe level relative to the main lobe: peak is an amplitude ratio (20 log), energy a power ratio (10 log)
Collecting pairs with the same sum gives the coarray \(c = a * b\) (a convolution), so \(P_2 = \mathcal{F}\{a * b\}\). Walker & Trahey (1998) develop this k-space view for pulse-echo; m23_kspace_coarray.m checks the identity to \(2\times 10^{-15}\).symbols
\(a_t,\ b_r\)
transmit and receive element weights
\(\xi_t,\ \xi_r\)
transmit and receive element positions, mm
\(x,\ z\)
lateral position in the focal plane and its depth, mm
\(T,\ R\)
transmit and receive focal-plane factors
\(k_x = (\xi_t + \xi_r)/(\lambda z)\)
lateral spatial frequency measured by one Tx/Rx pair, cycles/mm
With \(F_{\#,\mathrm{rx}} = 1.5\) and ±15°, the narrowband width is 0.456 mm and the broadband simulation gives 0.459 mm, between the receive-only 0.56 mm and the two-way F# 1.5 value 0.41 mm.symbols
A bright target at \(x_0\) also appears at \(x_0 \pm x_{\mathrm{rep}}\), attenuated only by the receive pattern there. Broadband pulses help further: at \(x_{\mathrm{rep}}\) the angles arrive one period apart, so the virtual array has the broadband grating-lobe factor of Section 23.3. This estimate tracks the simulated ghost within 1.5 dB at 3° and 15° steps and within about 8 dB elsewhere; it fails near narrowband receive nulls.symbols
\(x_{\mathrm{rep}}\)
lateral distance of the transmit replica from the target, mm
\(\Delta(\sin\theta)\)
step in the sine of the steering angle (\(\Delta\theta\) in rad for small angles)
\(M\)
pulse length in cycles, 0.883/FBW
\(N\)
number of angles
\(\lambda\)
center wavelength, mm
\(R(x_{\mathrm{rep}})\)
receive pattern (23.12) at the replica, amplitude
\[ \sin\theta_g = \sin\theta_{\mathrm{tx}} - m\,\frac{\lambda}{d}, \qquad \text{a second plane wave exists when } d > \frac{\lambda}{1 + \lvert\sin\theta_{\mathrm{tx}}\rvert} \]
(23.16)
This is (23.6) applied to transmit. Reference array (0.3 mm) at 15°: threshold \(0.794\lambda\) (0.245 mm at 5 MHz, 0.163 mm at 7.5 MHz). At 5 MHz (\(d = 0.974\lambda\)) the second wave leaves at −50.2°; at 7.5 MHz (\(1.46\lambda\)) at −25.2°. Unsteered, 7.5 MHz already launches a pair at ±43.2°. The element factor weights it by \(\lvert E(\theta_g)/E(\theta_{\mathrm{tx}})\rvert\).symbols
\(\theta_g\)
propagation direction of the grating-lobe plane wave, from the array normal
\(\theta_{\mathrm{tx}}\)
steering angle of the transmitted plane wave
\(m\)
replica order, \(\pm 1, \pm 2, \dots\) (for \(\theta_{\mathrm{tx}} > 0\) the first to appear is \(m = +1\), \(\theta_g < 0\))
One angle: \(0.603\,\lambda \cdot 2F_\# = 1.207\,\lambda F_\#\), the receive-only width. Equal supports give the focused two-way value \(0.886\,\lambda F_\#\). At 5 MHz, F# 1.5: 0.557, 0.544, 0.507, 0.456, 0.404 mm for 0°, ±5°, ±10°, ±15°, ±20°. Resolution improves with the span; the number of angles only has to sample the span densely enough (next equation). Narrowband, focal plane, uniform weights.symbols
\(w_{-6\,\mathrm{dB}}\)
two-way lateral −6 dB width at the target, mm
\(\kappa\)
shape factor of the trapezoidal k-space support, computed numerically; 0.603 for one angle (or \(F_{\#,\mathrm{rx}} \to \infty\)), 0.886 when \(\sin\theta_{\max} = 1/(2F_{\#,\mathrm{rx}})\); 0.742, 0.834, 0.877, 0.886 for ±5°, ±10°, ±15°, ±20° at F# 1.5
\(\theta_{\max}\)
half-span of the angle set
\(F_{\#,\mathrm{rx}}\)
receive F-number; \(1/(2F_\#)\) is the paraxial receive numerical aperture
\(k_z\) is real for \(\lvert k_x\rvert \le \lvert f\rvert/c\) (propagating) and imaginary otherwise (evanescent). A sample at \((k_x, f)\) is a plane wave arriving from angle α; the cone edge \(\lvert k_x\rvert = f/c\) is a grazing arrival. At 5 MHz, \(f/c = 3.25\) cycles/mm.symbols
\(d(x, t)\)
channel data: element position x (mm), fast time t (µs)
The weight is the Jacobian: the cosine of the propagation angle of the plane wave. Since \(f \ge v\lvert k_x\rvert\) on the output grid, the evanescent region is never read. Only \(f > 0\) is used, so the image is analytic and \(\lvert r\rvert\) is the envelope.symbols
\(R(k_x, k_z)\)
image spectrum on the output wavenumber grid
\(v k_z/\sqrt{k_x^2 + k_z^2}\)
Jacobian \(\lvert\partial f/\partial k_z\rvert = v\cos\varphi\), φ the propagation angle
\(\mathrm{IFFT}_2\)
inverse 2D FFT over \((k_x, k_z)\)
\(f,\ k_x,\ k_z\)
as in (24.1) and (24.5)
\(D\)
data spectrum, as in (24.1)
\(df\)
frequency element mapped to \(dk_z\) by the change of variables (Stolt mapping)
Each sample measures the object at the transmit wavevector plus a receive wavevector of length \(f/c\). For fixed f these points lie on a circle of radius \(f/c\) centred on the transmit wavevector: the Ewald circle of diffraction tomography (Kak & Slaney, 1988; Figure 24.B). Lu's Fourier method (Cheng & Lu, 2006) maps plane-wave data the same way.symbols
\[ \begin{aligned} f &= \frac{c\,(K_x^{2} + K_z^{2})}{2M} \\ M &= K_x\sin\theta + K_z\cos\theta \\ \left\lvert\frac{\partial f}{\partial K_z}\right\rvert &= \frac{c\,\lvert 2K_z M - (K_x^{2} + K_z^{2})\cos\theta\rvert}{2M^{2}} \end{aligned} \]
(24.15)
θ = 0: \(f = c(K_x^2 + K_z^2)/(2K_z)\). For small \(K_x\), \(K_z \approx 2f/c - cK_x^2/(2f)\): exactly the second-order behaviour of the ERM, whose curve \(\sqrt{4(f/c)^2 - 2K_x^2}\) is an ellipse that osculates the Ewald circle at α = 0. They part at large receive angles, which is the ERM error of Section 24.3 seen in k-space.symbols
\(f\)
data frequency to read for output sample \((K_x, K_z)\), MHz
\(M\)
projection of \(\mathbf{K}\) on the transmit direction, cycles/mm
validity
\(K_z \ge (f/c)\cos\theta\) (upper half of the circle)
After the delays a focused echo is the same on every element, so the steering vector is \(\mathbf{a} = \mathbf{1}\). Everything off focus (off-axis echoes, clutter, noise) is a departure from a constant vector: a phase ramp, a random phase, or an amplitude pattern.symbols
\(s_i(\mathbf{r})\)
delayed, phase-rotated, angle-compounded sample of element i at pixel \(\mathbf{r}\)
\(\mathrm{IQ}_{i,\theta}(t)\)
IQ channel data of element i for transmit angle θ, interpolated at time t
\(\tau_{\theta,i}(\mathbf{r})\)
round-trip delay: plane-wave transmit plus receive path to element \(x_i\), µs
\(f_0\)
centre frequency, 5 MHz
\(N\)
number of receive elements, 64
\(\mathbf{w}\)
receive weights; DAS uses \(\mathbf{1}/N\)
\(x,\ z\)
lateral and axial pixel coordinates, mm
\(c\)
sound speed, 1540 m/s
\(\theta\)
plane-wave transmit angle (summed over all angles), rad
\(j\)
imaginary unit
\(\mathbf{s},\ y_{\mathrm{DAS}}\)
vector of the \(s_i\) over the aperture; DAS output
Lagrangian \(\mathbf{w}^H\mathbf{R}\mathbf{w} + \lambda(1 - \mathbf{w}^H\mathbf{a}) + \lambda^*(1 - \mathbf{a}^H\mathbf{w})\); setting the gradient with respect to \(\mathbf{w}^*\) to zero gives \(\mathbf{R}\mathbf{w} = \lambda\mathbf{a}\) (Capon, Proc. IEEE 1969). With \(\mathbf{R} = \sigma^2\mathbf{I}\) the weights are \(\mathbf{1}/N\): DAS is MV in white noise.symbols
\(\mathbf{w}\)
adaptive receive weights (N, or L with subarrays)
\(\mathbf{R}\)
covariance of the aperture vector, \(\mathrm{E}[\mathbf{s}\mathbf{s}^H]\)
\(\mathbf{a}\)
steering vector toward the focus; all ones after the delays
\(P_{\mathrm{MV}}\)
minimum output power
\((\cdot)^{H},\ (\cdot)^{-1}\)
conjugate transpose, matrix inverse
\(\mathbf{1}\)
vector of ones
\(\mathrm{E}[\cdot]\)
expectation
\(\min,\ \text{s.t.}\)
minimize over \(\mathbf{w}\), subject to the distortionless constraint \(\mathbf{w}^H\mathbf{a} = 1\)
The identity \((1/N)\sum\lvert s_i\rvert^2 = \lvert\bar{y}\rvert^2 + \hat{\sigma}^2\) turns CF into a signal-to-(signal + noise) ratio. The Wiener postfilter of Nilsen & Holm (IEEE TUFFC 2010) has the same estimates but divides the channel noise by N, because DAS averages N independent noise samples. CF therefore overestimates the output noise by a factor N and suppresses far harder: for pure noise CF = 1/N (−36 dB on an amplitude image), \(H = N/(2N - 1) \approx 0.5\) (−6 dB).symbols
\(\mathrm{CF}\)
coherence factor, in [0, 1]; its expected value for white noise is 1/N
\(\bar{y}\)
aperture mean, equal to the DAS output
\(\hat{\sigma}^2\)
spread of the channels about the mean (channel "noise" estimate)
\(H_{\mathrm{Wiener}}\)
Wiener postfilter gain applied to \(y_{\mathrm{DAS}}\)
The second form costs O(N) instead of \(N(N - 1)/2 = 2016\) pair products for N = 64. \(s_i\) is real RF here, sampled every 18.75 µm ≈ λ/16 in depth (41 MHz equivalent) so that the \(2f_0\) band is not aliased. For an aligned echo every pair term equals \(\lvert \mathrm{rf}(t)\rvert\), whose spectrum has a DC term, a \(2f_0\) band and weaker \(4f_0\): the band-pass (here Gaussian, centred at \(2f_0\) with \(\sigma = \sqrt{2}\,\sigma_f\)) keeps only the \(2f_0\) part.symbols
\(s_i\)
delayed real RF sample of element i (Re of the aperture vector)
M is commonly 5–30 % of N. A point gives \(R(m) \approx 1\), so \(R_{\mathrm{sl}} \approx M\); fully developed speckle follows the VCZ curve (\(R_{\mathrm{sl}} \approx \sum(1 - m/m_{\mathrm{tx}}) = 5.7\) for M = 8; 6.0 in the noise-free simulation); noise and spatially incoherent clutter give \(R(m \ge 1) \approx 0\). Bell, Dahl & Trahey (IEEE TUFFC 2015) analysed SLSC resolution and brightness versus M.symbols
\(R(m)\)
normalized spatial coherence at element lag m
\(\rho_i(m)\)
normalized correlation of elements i and i + m over the kernel
\(n\)
sample index over an axial kernel (here ±0.15 mm, 17 RF samples)
Clipping at −DR is monotone but not strictly, so it can only lower gCNR (ties merge the two distributions at the floor). CF behaves roughly like DAS with a pixel-dependent power law: its CR in Demo 25.6 (52 dB) is close to that of DAS² (46 dB), but its gCNR is lower than DAS's because its noisy weights reorder tissue pixels.symbols
\(D_W, D_{\mathrm{DAS}}\)
displayed dB images (before clipping) with and without the weight W
\(A\)
linear output amplitude
\(\mathrm{CR}_{\mathrm{dB}}\)
difference of the mean dB values of tissue and cyst, before clipping
\(T\)
any strictly increasing map (power law, log compression, S-curve)
\(\mathbf{r}\)
pixel position
\(W(\mathbf{r})\)
adaptive weight map (e.g. CF or PCF), dimensionless
The ratio does not depend on \(\langle I\rangle\), so gain, TGC level and transmit power cancel. Wagner, Smith, Sandrik and Lopez (1983) introduced \(\mathrm{SNR}_0\) = 1.91 as the benchmark for fully developed speckle.symbols
\(z\)
beamformed IQ sample (circular complex Gaussian)
\(R = \lvert z\rvert\)
envelope, linear units
\(\langle I\rangle\)
mean intensity \(\mathrm{E}\lvert z\rvert^2\)
\(\mathrm{SNR}_0\)
envelope mean over envelope std (the "speckle SNR")
The log-intensity follows a (reversed) Gumbel / Fisher–Tippett law: skewed, with a long tail toward dark values and the mode at 0 dB. Gain shifts \(D\), so the 5.57 dB spread is gain independent. Module 08 quotes the 5.57 dB; the −2.51 dB offset is the one that biases contrast measurements.symbols
\(D\)
log-compressed pixel, dB re \(10\log_{10}\langle I\rangle\)
\(\gamma\)
Euler's constant, 0.5772
4.343
\(10/\ln 10\), dB per neper of intensity
\(\mathrm{E}D,\ \mathrm{std}\,D\)
mean and standard deviation of the log-compressed pixel, dB
Speckle size here = FWHM of the normalized intensity autocovariance \(\lvert\mu\rvert^2\). Other papers use the correlation length \(\int\lvert\mu\rvert^2\) or the envelope ACF; state yours.symbols
\(\Delta\)
spatial (or time) lag
\(\mu(\Delta)\)
normalized field autocorrelation of the PSF
\(\mathrm{cov}_I(\Delta)\)
intensity autocovariance
\(H\)
pulse-echo transfer function (k-space support of the PSF)
\(\mathrm{E}[\cdot]\)
expected value over the speckle ensemble
\(I(x)\)
intensity at position x
\(x\)
position, mm
\(\langle I\rangle\)
mean intensity, as in (26.1)
\(\lvert\mu\rvert^2\)
squared magnitude of the normalized field autocorrelation
Example: \(N\) = 9 looks with \(\rho_I\) = 0.25 give \(N_{\mathrm{eff}}\) = 3, not 9.symbols
\(I_k\)
intensity image of look \(k\)
\(N\)
number of looks
\(r_{kl}\)
complex field correlation between looks \(k\) and \(l\)
\(\rho_{I,kl}\)
intensity correlation, \(\lvert r_{kl}\rvert^2\)
\(C_S\)
speckle contrast of the compound
\(N_{\mathrm{eff}}\)
effective number of independent looks
\(S\)
sum (compound) intensity
\(k,\ l\)
look indices
\(\langle I\rangle\)
mean intensity of each look, as in (26.1)
\(\rho_I\)
intensity correlation between looks, as in (26.13)
\(C\)
speckle contrast of independent looks
\(\equiv\)
defined as
\[ \begin{aligned} C &= 10\log_{10}\frac{\langle I\rangle_{\mathrm{in}}}{\langle I\rangle_{\mathrm{out}}}\ \ \mathrm{dB} \quad (\text{ratio of mean intensities}) \\ \mathrm{CNR} &= \frac{\lvert\mu_{\mathrm{in}} - \mu_{\mathrm{out}}\rvert}{\sqrt{\sigma_{\mathrm{in}}^2 + \sigma_{\mathrm{out}}^2}} \quad \text{on the linear envelope } A = \lvert z\rvert \\ \mathrm{gCNR} &= 1 - \mathrm{OVL}, \qquad \mathrm{OVL} = \int \min\!\left(p_{\mathrm{in}}(v),\, p_{\mathrm{out}}(v)\right) dv \end{aligned} \]
(26.14)
Common variants of CNR: the same ratio on intensity, on the dB image, or with \(\sigma_{\mathrm{out}}\) alone in the denominator. Contrast is also often computed as \(20\log_{10}(\mu_{\mathrm{in}}/\mu_{\mathrm{out}})\) of envelope means, which equals \(C\) only when both regions have the same envelope distribution shape.symbols
Bins spaced \(f_s/N\) are approximately independent (rectangular window). A longer record gives more bins, not less noise per bin: the periodogram is not a consistent estimator.symbols
\(P(f)\)
one periodogram bin
\(S(f)\)
true spectrum at that frequency
\(\chi^2_2\)
chi-squared variable with 2 degrees of freedom (real and imaginary parts); \(\chi^2_2/2\) is a unit-mean exponential
\(\mathrm{E},\ \mathrm{var},\ \mathrm{std}\)
mean, variance and standard deviation over realizations
\(\approx\)
valid away from the band edges and for a smooth \(S\)
Because \(\partial S/\partial z = -4\alpha_0 f S\). For a Gaussian power spectrum \(\sigma^2 = \sigma_p^2\) is constant, so the downshift is linear: \(f_c = f_0 - 4\alpha_0\sigma_p^2 z\), the same as Module 08's \(-2\alpha_0\sigma_f^2 z\) with \(\sigma_f = \sqrt{2}\,\sigma_p\) the amplitude-spectrum std. For other shapes \(\sigma^2\) changes with depth and the downshift curves (card 4 below).symbols
Taking logs, \(\ln[\mathrm{BSC}_s/\mathrm{BSC}_r]\) (with a Rayleigh reference, \(\mathrm{BSC}_r \propto f^4\)) is linear in \(k^2\) with slope \(-0.827a^2\): an ordinary least-squares problem. The intercept is proportional to the effective acoustic concentration (number density × squared impedance contrast).symbols
\(k\)
wavenumber, rad/µm (with \(c\) = 1540 µm/µs and \(f\) in MHz)
\(a\)
effective scatterer radius, µm
\(\mathrm{ESD}\)
effective scatterer diameter, µm
0.827
constant of the Gaussian form factor \(F = \exp(-0.827\,k^2a^2)\)
\(\mathrm{BSC}(f)\)
backscatter coefficient, as in (27.16)
\(f\)
frequency, MHz
\(c\)
sound speed, µm/µs
\(\propto\)
proportional to
\[ \sigma(\hat\alpha_0) \approx \sqrt{ \frac{4.5\, \gamma_w\, \psi_1(K)\, c}{B^3\, L^3} } \qquad [\mathrm{Np/cm/MHz};\ c \text{ in cm}/\mu\mathrm{s}] \]
(27.26)
The gate duration \(T\) cancels: shorter gates give noisier spectra but more of them. Monte Carlo (2, 4, 8 mm gates, rectangular and Hann) agrees within 5 %. Hann, \(K = 32\), \(B = 4\) MHz, \(L = 3\) cm: 0.041 dB/cm/MHz. Overlapping Hann gates (75 %) recover the tapered edges and do somewhat better (0.030 dB/cm/MHz in m27_attenuation_sld.m for 1.5–4.5 cm). Two gates of length \(L/2\) at the ends give the constant 6 instead of 4.5, and two short gates far apart are worse still.symbols
\(L\)
axial ROI length, cm
\(K\)
independent lines (lateral looks) in the ROI
\(B\)
analysis bandwidth, MHz
\(c\)
sound speed, 0.154 cm/µs
\(\gamma_w\)
1.75 for Hann gates, 0.93 for rectangular
\(\sigma(\hat\alpha_0)\)
standard deviation of the slope estimate, Np/cm/MHz
\(\hat\alpha_0\)
estimated slope, as in (27.20)
\(\psi_1(K)\)
trigamma function, as in (27.22)
\(\approx\)
valid when the ROI length \(L\) replaces the gate separation \(\Delta z\) of (27.22)
Derived by linearizing the Marmottant equation of 28.3 (28.8); for \(\sigma_0 \ll \chi\) it reduces to the \(3\kappa P_0 + 4\chi/R_0\) form usually quoted. With \(\chi\) = 0.5 N/m and \(R_0\) = 2 µm the shell term (1.0 MPa) is three times the gas term (0.33 MPa): \(f_0\) = 2.99 MHz instead of 1.44 (Minnaert, \(\kappa\) = 1.07). Verified against an RK4 ring-down to 0.03 % in m28_resonance_shell.m.symbols
\(\sigma_0\)
\(\sigma(R_0)\), shell surface tension at rest, N/m
Thermal damping is omitted here. At resonance \(\sigma_s = 4\pi R_0^{2}/\delta^{2}\); far above it \(\sigma_s \to 4\pi R_0^{2}\) (four times the geometric area); far below it \(\sigma_s \approx 4\pi R_0^{2}(f/f_0)^{4}\), the compressibility-only Rayleigh law above.symbols
At 5 MHz with \(F_x = \lambda\), \(F_z = 0.75\lambda\) and \(k\) = 1: \(A\) = 0.224 mm², \(n^{*}\) = 4.5 per mm², at most 1.65 isolated bubbles per mm² per frame. A Monte Carlo in m28_ulm_concentration.m agrees within 0.01. At 15 MHz \(A\) shrinks 9 times (\(n^{*}\) = 40 per mm²).symbols
\(P_{\mathrm{iso}}\)
probability that a bubble has no neighbour within the exclusion area
\(n\)
areal bubble density in the image, per mm²
\(A\)
exclusion area, mm²
\(k\)
separation needed, in FWHM (elliptical metric)
\(F_x,\ F_z\)
lateral and axial FWHM, mm
\(e\)
Euler’s number
\(n^{*}\)
density that maximizes the isolated-bubble yield, per mm²
The factor 2 is the round trip. At 5 MHz, 1 m/s along the beam gives 6.49 kHz: Doppler shifts of blood are audio frequencies, \(v/c \approx 10^{-3}\) of the carrier.symbols
\(f_D\)
Doppler shift, Hz (positive for flow toward the probe)
The range–velocity product depends only on \(f_0\) and the angle: 0.0593 m²/s at 5 MHz and \(\theta\) = 0 (0.59 m/s at 10 cm), 0.0988 m²/s at 3 MHz (0.99 m/s at 10 cm). Halving \(f_0\) doubles it; that is why cardiac Doppler of fast jets uses 2–2.5 MHz and still needs CW.symbols
Because \(R(T) = \int S(f)\,e^{j2\pi f T}\,df\), its phase equals \(2\pi \bar f T\) exactly for a symmetric spectrum and approximately otherwise: the Kasai estimate is the mean Doppler frequency, with no FFT. Averaging \(\hat R(1)\) over a few depth samples (and lateral pixels) before taking the phase is cheap and reduces the variance.symbols
\(z_n\)
complex (IQ) sample of one pixel at pulse \(n\), after the wall filter
Sensitivity grows as \(\sqrt{N}\): four times the ensemble doubles the CNR (m29_ultrafast_power_doppler.m measures ×1.98 from 100 to 400 frames). With \(P_b/P_n\) = −6 dB a vessel crosses CNR = 3 near \(N\) = 150. Coherent compounding of \(n_a\) angles adds up to \(10\log_{10} n_a\) dB to \(P_b/P_n\) on top.symbols
\(\hat P\)
power Doppler value of a pixel
\(P_b,\ P_n\)
blood and noise power per sample (after compounding)
\(N - K\)
independent noise samples left after the filter
\(\mathbf{r},\ n\)
pixel position and frame index
\(s_f(\mathbf{r}, n)\)
IQ sample after clutter filtering
\(N,\ K\)
number of frames and clutter rank removed, as in (29.12)
With \(\alpha = \beta = 0\) it reduces to (29.1). For two symmetric pairs at \(\pm\gamma\) (transmit = receive), \(\mathbf{A}^{T}\mathbf{A}\) is diagonal and \(\sigma_{v_x}/\sigma_{v_z} = \cot\gamma\): 5.67 at 10°, 2.75 at 20°. Receiving at 0° only gives \(\cot(\gamma/2)\), twice as bad.symbols
\(\alpha_k,\ \beta_k\)
transmit and receive steering angles of pair \(k\) (from the array normal)
\(v_x,\ v_z\)
lateral and axial velocity (z downward, so \(v_z < 0\) is toward the probe), m/s
\(\mathbf{A}\)
\(K\times 2\) matrix of the bracketed coefficients times \(-f_0/c\)
\(\sigma_f\)
standard deviation of each Doppler estimate, Hz
\(f_0,\ c\)
as in (29.1)
\(f_k\)
Doppler shift measured for pair \(k\), Hz
\(k\)
index of the transmit/receive angle pair, \(1\dots K\)
\(\hat{\mathbf{v}},\ \hat{\mathbf{f}}\)
estimated velocity vector \((v_x, v_z)\) and the vector of measured shifts \(f_k\)
Ultrasound travels on \(K\), elastography on \(\mu\). Because \(K/\mu \sim 10^{6}\), tissue is nearly incompressible and \(E = 3\mu\) to six digits: 1 m/s is 1 kPa of \(\mu\) and 3 kPa of \(E\); 2 m/s is 4 and 12 kPa; 3 m/s is 9 and 27 kPa.symbols
\(c_s\)
shear (transverse) wave speed, m/s
\(c_L\)
longitudinal (ultrasound) speed, ≈ 1540 m/s
\(\mu\)
shear modulus, Pa (soft tissue 0.3–30 kPa)
\(K\)
bulk modulus, Pa (≈ 2.2–2.4 GPa, close to water)
\(E\)
Young's modulus, Pa
\(\nu\)
Poisson's ratio, dimensionless
\(\rho\)
mass density, ≈ 1000 kg/m³ (1050–1060 for liver and muscle)
\(N\) = 2 gives \(\sqrt2\,\sigma_u/\Delta\), the two-point rule. \(N\) = 9 gives \(\sqrt{12/720}\,\sigma_u/\Delta = 0.129\,\sigma_u/\Delta\); at \(\Delta\) = 0.25 mm: with \(\sigma_u\) = 0.5 µm the strain noise is 0.026 %, an SNR\(_e\) of 39 at 1 % strain. Overlapping kernels make the \(\hat u_i\) correlated, so the real gain is smaller than this formula.symbols
\(N\)
number of displacement estimates in the LSQ span
\(\Delta\)
spacing of the displacement estimates (kernel step), m
\(\sigma_u\)
displacement jitter, m (Walker–Trahey bound plus decorrelation, Module 11)
\(\mathrm{SNR}_e\)
elastographic SNR: mean strain over its standard deviation in a uniform region
\(z_i,\ \hat u_i\)
depth (m) and estimated displacement (m) of the \(i\)-th kernel
\(\bar z,\ \bar u\)
their means over the \(N\) kernels
\(\hat\varepsilon,\ \bar\varepsilon\)
least-squares strain estimate and its mean in the region
\(\sigma_\varepsilon\)
standard deviation of the strain estimate, dimensionless
\[ F = \frac{2\alpha I}{c} \]
(30.6)
0.5 dB/cm/MHz at 4 MHz is 2 dB/cm = 23.0 Np/m. With \(I\) = 500 W/cm² in situ, \(F\) = 2 × 23.0 × 5×10⁶ / 1540 = 1.5×10⁵ N/m³. That intensity is about what a linear 4 MHz push at the regulatory MI of 1.9 delivers: \(p\) = 1.9 × √4 = 3.8 MPa, \(I = p^{2}/(2\rho c)\) = 470 W/cm².symbols
\(F\)
body force per unit volume along the beam, N/m³
\(\alpha\)
amplitude absorption coefficient, Np/m (1 Np = 8.686 dB); usually approximated by the attenuation coefficient, which overestimates \(F\) by the scattering fraction
\(I\)
temporal-average intensity during the push, W/m² (1 W/cm² = 10⁴ W/m²)
\(c\) = 2 m/s, \(\sigma_t\) = 50 µs (half a frame at 10 kHz), \(L\) = 10 mm, \(N\) = 51: \(\sigma_c/c\) = 0.48 % for independent errors. Errors between neighbouring traces are correlated (same speckle, same noise realization in overlapping beams), so real precision is several times worse; the scaling is still right: precision improves as \(L^{3/2}\) at fixed \(\Delta x\), and resolution worsens as \(L\). That is the SWE resolution–precision trade-off in one line.symbols
\(t_i\)
arrival time (peak, or a correlation delay) at lateral position \(x_i\), s
\(\sigma_t\)
standard deviation of each arrival-time estimate, s
\(N,\ \Delta x\)
number and spacing of lateral positions in the fit; \(L = (N-1)\Delta x\) is the lateral kernel
\(\hat c\)
estimated speed, m/s (the inverse of the fitted slope)
The Fourier resolution is one bin, \(2\pi/L\): with \(\lambda\) = 4 mm (2 m/s at 500 Hz) and \(L\) = 10 mm the raw resolution is 40 % of \(k\). Peak interpolation on a smooth ridge recovers far better than a bin, but the k–f method needs several wavelengths in the window, so it trades lateral resolution for spectral detail. It is the method of choice for dispersion and guided waves, not for small lesions.symbols
\(V(k,f)\)
2D spectrum of the windowed space–time data; a \(+x\) wave peaks at \(k = +k(\omega)\) with this sign convention
\(w(x)\)
lateral window (Hann) of length \(L\), m
\(k_{\mathrm{peak}}(f)\)
wavenumber of the spectral maximum at frequency \(f\), rad/m, refined by peak interpolation
\(\lambda\)
shear wavelength at \(f\), m
\(x,\ t\)
lateral position (m) and time (s)
\(v(x,t)\)
particle velocity (or displacement) field
\(k,\ f\)
lateral wavenumber (rad/m) and temporal frequency (Hz)
\(j\)
imaginary unit
\(\delta c,\ \delta k\)
resolution in speed (m/s) and in wavenumber (rad/m)
With \(x = \omega\eta/\mu\) small, \(c_{\mathrm{ph}} \approx c_0(1 + 3x^{2}/8)\), \(c_0 = \sqrt{\mu/\rho}\). For \(c_0\) = 1.5 m/s and \(\eta\) = 1 Pa·s: 1.54, 1.66, 2.21 and 3.16 m/s at 100, 200, 500 and 1000 Hz, and \(\alpha\) = 197 Np/m at 200 Hz (the amplitude falls by \(e\) every 5 mm). At high frequency KV predicts \(c_{\mathrm{ph}} \to \sqrt{2\omega\eta/\rho}\), growing without bound; measured tissue disperses much more gently, which is the main argument against KV.symbols
At 30 mm depth \(f_{\mathrm{PRF}} \le\) 25.7 kHz, so 5 angles allow at most 5.1 kHz; most systems run near 1 kHz, limited by data transfer and beamforming. At 1 kHz a 1 mm/s capillary bubble moves 1 µm per frame and a 30 mm/s arteriole bubble 30 µm (λ/10). PALA's default gate of 2 pixels of λ at 15.6 MHz (about 200 µm) tracks up to about 200 mm/s at 1 kHz. Five angles raise the ENR by 7 dB and shrink the localization error by \(\sqrt5\).symbols
\(f_{\mathrm{PRF}}\)
transmit (firing) rate, Hz
\(z_{\max}\)
deepest echo that must return before the next firing, m
\(N_a\)
number of steered transmits compounded into one frame
\(\mathrm{FR}\)
compounded frame rate, frames/s
\(v,\ \delta\)
bubble speed (m/s) and displacement between frames (m)
\(d_{\mathrm{link}}\)
largest displacement the tracker accepts between frames, m
\(\mathrm{ENR}\)
energy-to-noise ratio of one bubble in one frame (Module 12); coherent compounding adds signal in amplitude and noise in power
A 40 µm vessel has a lumen spread of 8.9 µm (8.2 to 10 µm for a cylinder); with \(\sigma_{\mathrm{loc}}\) = 10 µm its localizations spread over 13.4 µm (FWHM 32 µm if Gaussian). An uncorrected 100 µm breathing excursion adds 71 µm rms and the vessel appears 170 µm wide. m31_ulm_pipeline.m checks the quadrature sum without motion: across its 60 µm vessel the localizations spread by 25.3 µm while the true positions of the same bubbles spread by 7.9 µm (a few bubbles only in 0.3 s), implying 24.0 µm of perpendicular localization error against 23.6 µm measured directly.symbols
\(\sigma_{\mathrm{loc}}\)
localization error of one bubble (Section 31.4), m
\(\sigma_{\mathrm{m}}\)
rms of the uncorrected motion along the direction considered, m
\(\sigma_{\mathrm{reg}}\)
error of the motion estimate when correction is applied, m
\(s_{\mathrm{vessel}}\)
standard deviation of localizations across a vessel of radius \(R\) when each crossing bubble is counted once (track rendering) and the profile is a 2D parabola, \(\mathrm{E}[u^{2}] = 1/5\), the model of the simulations here; a cylindrical lumen gives \(R^{2}/6\) for track counts and \(R^{2}/4\) when every frame's localization is counted (slow wall bubbles then appear in more frames and the positions are uniform over the cross-section)
\(P_{\mathrm{FA}} = 10^{-5}\) needs \(\eta^{2}/\sigma_y^{2}\) = 11.5. Detection is then 50 % at an ENR of 10.4 dB, 82 % at 12 dB and 99.9 % at 14.8 dB: the transition is sharp, so the bubbles that matter are lost to merging and filtering, not to the threshold. The 4.9 × 4.9 mm field of Demo 31.1 holds 341 cells of λ × 0.75λ, so noise alone gives 0.003 false alarms per frame; the 0.26 measured in m31_ulm_pipeline.m come from residual clutter and the filter's ghosts.symbols
\(y(\mathbf{x})\)
matched-filter output at pixel \(\mathbf{x}\) (complex)
\(g\)
PSF envelope used as template
\(\eta\)
threshold on \(\lvert y\rvert\)
\(\sigma_y^{2}\)
noise power at the filter output; \(\lvert y\rvert^{2}\) is exponential under noise only, so the median gives \(\sigma_y^{2} = \mathrm{median}\lvert y\rvert^{2}/\ln 2\)
\(Q_1\)
Marcum Q-function: detection probability of a constant-amplitude, random-phase echo
\(\mathrm{ENR}\)
output SNR of the matched filter = energy-to-noise ratio of the bubble
\(\bar N_{\mathrm{FA}}\)
expected false alarms per frame, counting independent resolution cells in the field \(A_{\mathrm{FOV}}\)
\(\mathbf{x},\ \mathbf{p}\)
output pixel and summation pixel, m
\(s_{\mathrm{b}}(\mathbf{p})\)
beamformed bubble-filtered IQ signal at pixel \(\mathbf{p}\)
\(P_{\mathrm{FA}},\ P_{\mathrm{D}}\)
false-alarm and detection probability per resolution cell
At 5 MHz (\(w_x\) = 131 µm, \(w_z\) = 98 µm) an ENR of 20 dB gives 13.1 × 9.8 µm, 30 dB gives 4.1 × 3.1 µm; reaching 10 µm laterally takes 22.3 dB. On λ/4 pixels the peak pixel is 8.3 dB below the ENR. The bound is local: below about 15 dB noise peaks start to win (Section 12.4), and the bound no longer describes the error.symbols
\(w_x,\ w_z\)
lateral and axial standard deviation of the amplitude PSF, m (FWHM/2.3548)
\(a\)
complex bubble amplitude (unknown, nuisance)
\(N_0\)
complex noise power per unit area (per-pixel variance × pixel area)
\(\Delta\)
pixel size, m
\(\mathrm{SNR}_{\mathrm{peak}}\)
SNR of the brightest pixel
\(\sigma_x,\ \sigma_z\)
lateral and axial localization standard deviation, m (lower bound set by the PSF width and the ENR)
m31_tracking_hungarian.m (60 bubbles on a torus, 40 trials, noise 0.1 d, velocity change 0.2 d): at ρ = 0.1 nearest neighbour is right 72.9 % of the time against 73.0 % predicted; greedy global assignment only 62.7 %, Hungarian 85.6 %, Hungarian with prediction 97.5 %. At ρ = 1 they reach 5.6, 6.0, 12.0 and 80.4 %. With prediction \(d_{\mathrm{eff}}\) = 0.40 d.symbols
\(n\)
areal density of localizations per frame, m⁻²
\(d\)
displacement between frames, \(v/\mathrm{FR}\), m
\(\rho\)
bubbles per displacement squared (dimensionless crowding)
\(P_{\mathrm{c,NN}}\)
probability that no other bubble is closer than the true partner: nearest neighbour is right
\(\pi\)
a one-to-one assignment of frame-\(t\) to frame-\((t+1)\) localizations; the Hungarian algorithm finds the minimum in \(O(N^{3})\)
\(\hat{\mathbf{v}}_i\)
predicted velocity (zero for plain matching)
\(d_{\mathrm{eff}}\)
rms prediction residual: velocity change plus localization noise; a Gaussian residual turns the exponential into \(1/(1+\pi\rho\,d_{\mathrm{eff}}^{2}/d^{2})\)
\(\Delta t\)
time between frames, \(1/\mathrm{FR}\), s
\(\mathbf{x}_i,\ \mathbf{y}_j\)
position of localization \(i\) in frame \(t\) and of candidate \(j\) in frame \(t+1\), m
\(C_{ij}\)
assignment cost (squared distance to the predicted position), m²
With \(C\) = 200 µL⁻¹ (2×10⁵ per mL, the concentration Hingot et al. report), \(v\) = 0.2 mm/s per µm of diameter and \(\eta\) = 1 (m31_saturation_frc.m): \(T\) = 0.6 s for a 100 µm vessel, 15.9 s for 20 µm with 5 µm pixels, and 255 s for a 5 µm capillary whatever the pixel size, since one bubble every 255 s crosses it. Real \(\eta\) is far below 1, which is why Hingot et al. found that vessels below 25 µm may need more than 10 minutes. \(T\) scales as \(d^{-2}\) above the pixel size and \(d^{-3}\) below it (with \(v \propto d\)); in 3D the extra factor \(d/l\) makes small-voxel volumes much slower.symbols
\(T(d)\)
saturation time constant of a vessel of diameter \(d\) imaged with pixels of size \(l\), s
\(C\)
bubble concentration in blood, µL⁻¹
\(Q,\ v\)
volumetric flow (µL/s) and mean speed in the vessel
\(\eta\)
fraction of crossing bubbles that end up in kept tracks (in plane, isolated, detected, linked)
\(S(t)\)
fraction of the vessel's pixels visited after acquiring for \(t\); 90 % at \(t = T\ln 10 = 2.30\,T\)
\(T_{3\mathrm{D}}\)
the same for voxels of size \(l^{3}\) (Hingot et al. Eq. 4′): every bubble now paints one of \((d/l)^{2}\) columns
The tolerance is the λ/8 (π/4 phase) criterion at the aperture edge \(x = X\). At 5 MHz and 30 mm it is 7.9 m/s at F/1, 32 m/s at F/2 and 71 m/s at F/3: a large aperture is both what needs the right speed and what can measure it. The curvature error is twice that of a pixel held at fixed depth, because the time-tied pixel moves by \(\Delta c/c\) as well.symbols
\(\delta\tau(x)\)
receive delay error of the element at offset \(x\) after the pixel is placed at the correct two-way time, s
\(c,\ \hat c,\ \Delta c\)
true, assumed and error of the sound speed, m/s
\(z,\ \hat z\)
true and displayed depth of the pixel, m (\(\hat z/z = \hat c/c\))
\(F\)
receive F-number \(z/D\), with \(D = 2X\) the active aperture
For small offsets the moveout of a reflection under flat layers is \(t^{2}(x) \approx t_0^{2} + x^{2}/V_{\mathrm{rms}}^{2}\) (Dix, Geophysics, 1955), so a curvature fit or a focus maximizer returns \(V_{\mathrm{rms}}\), not \(z/T\). For 10 mm of fat (1450) over liver (1580) at 40 mm: \(V_h\) = 1545.4, \(V_{\mathrm{rms}}\) = 1546.4 m/s. The difference (1.1 m/s) is small; the difference from the liver itself (33 m/s) is not.symbols
\(V_h\)
harmonic (time-average) speed: depth over one-way time, m/s
\(V_{\mathrm{rms}}\)
root-mean-square speed, time-weighted, m/s
\(w_i\)
fraction of the vertical travel time spent in layer \(i\)
Differencing two averages taken 1 mm apart at 30 mm multiplies their noise by 42: 2 m/s on each average becomes about 85 m/s on the local speed (83.5 from the formula; m32_layered_inversion.m predicts 85.8 because the slowness-to-speed conversion adds the factor \((c_{\mathrm{loc}}/\bar c)^{2}\), and finds 87.1 by Monte Carlo; 142 and 148 at 50 mm). Locality costs \(z/\Delta z\) in precision, and no estimator escapes that factor without a prior.symbols
\(\bar s(z) = 1/\bar c(z)\)
average slowness from the surface to depth \(z\), s/m
\(s(z)\)
local slowness at depth \(z\)
\(\Delta z\)
spacing of two average measurements, m
\(\sigma_{\bar c},\ \sigma_{c,\mathrm{loc}}\)
standard deviation of each average and of the local estimate, m/s (independent errors, \(c_{\mathrm{loc}} \approx \bar c\))
Monte Carlo in m32_cute_phase_shift.m: N = 64, ρ = 0.95 gives 0.0295 rad (formula 0.0291), 0.94 ns at 5 MHz; ρ = 0.99 gives 0.41 ns. The formula fails at low ρ and small N (0.355 versus 0.306 rad at ρ = 0.5, N = 16) where the phase distribution grows tails. Correlation is worth more than kernel size: going from ρ = 0.9 to 0.99 is worth a factor 3.4 in σ, the same as 12 times more samples.symbols
\(a_{1,k},\ a_{2,k}\)
complex pixel values of the two beamformed images in the kernel
\(N\)
number of independent speckle cells in the kernel
\(\rho\)
correlation coefficient between the two images (speckle decorrelation plus noise)
\(\mathbf{R}\) does not depend on the data, only on the acquisition design and \(\lambda\). Compute it before acquiring anything: it is the design tool of SoS tomography, as the beam pattern is of an array.symbols
\(\mathbf{d}\)
all delay-shift data (32.8)–(32.9), ns
\(\mathbf{A}\)
differences of ray-weight rows, mm (path length per cell)
\(\Delta\mathbf{s}\)
slowness deviation per cell, ns/mm
\(\mathbf{G}\)
lateral and axial first differences between neighbouring cells
\(\mathbf{R}\)
resolution matrix: row \(k\) is the point-spread function of the reconstruction at cell \(k\); \(\operatorname{tr}\mathbf{R}\) counts the resolved degrees of freedom
m32_sos_error_budget.m: 64 elements (18.9 mm), z = 30 mm, \(\sigma_t\) = 5 ns: exact CRLB 1.69 m/s, closed form 1.62 (1.67 for the continuous aperture), Monte Carlo 1.68 m/s. The closed form grows linearly with depth at a fixed aperture (the exact bound only once F ≳ 1; at F < 1 it grows more slowly) and falls about as \(N^{-2.5}\) at fixed pitch. A 1 ns edge calibration error biases \(c\) by −0.83 m/s, comparable to the CRLB, and it never averages out.symbols
\(\sigma_t\)
per-element arrival-time jitter, s
\(N,\ X\)
number of elements and half-aperture, uniform spacing; \(\operatorname{sd}(x^{2}) = X^{2}\sqrt{4/45}\) for a continuous uniform aperture
\(F\)
F-number \(z/(2X)\)
\(b_{\mathrm{cal}}\)
bias from a systematic delay error that grows quadratically to \(\delta\) at the aperture edge (lens, element delay calibration), m/s
\(c,\ z\)
sound speed (m/s) and depth (m)
\(x_i\)
lateral position of element \(i\) in the aperture, m
At 5 MHz, 10 ns of delay is 0.314 rad. A 40 ns rms screen is 1.26 rad: a fifth of the 200 ns period (72°), enough to remove two thirds of the focal intensity. The model is narrowband (one phase per element) and puts the aberrator at the array, which makes it exactly isoplanatic; Section 33.4 moves it away.symbols
\(\tau(x)\)
residual (aberration) delay at aperture position \(x\) after geometric focusing with \(c_0\), s
\(\phi(x)\)
the same delay as a phase at the centre frequency, rad
\(\sigma_\tau,\ \sigma_\phi\)
rms delay (ns) and rms phase (rad)
\(\rho,\ \ell\)
correlation coefficient of the screen and its correlation length (Gaussian model), mm
The double sum uses \(\mathbb{E}[e^{j(\phi_i-\phi_j)}] = e^{-\frac12\mathrm{Var}(\phi_i - \phi_j)}\) for Gaussian phases. m33_phase_screen_psf.m (128 elements, \(\ell\) = 4 mm, 400 screens): at 40 ns the Monte Carlo Strehl is 0.311, the double sum 0.315, \(e^{-\sigma_\phi^2}\) only 0.206; at 80 ns 0.078, 0.079 and 0.002, where the large-\(\sigma\) form gives 0.073. Two way (same screen on transmit and receive) the Strehl of a realization is exactly \(S^{2}\); its mean \(\mathbb{E}[S^{2}]\) = 0.131 at 40 ns is above \(\mathbb{E}[S]^{2}\) = 0.099 because \(S\) fluctuates between realizations.symbols
\(S\)
Strehl ratio: on-axis intensity with the screen over intensity without, one way
\(N,\ x_i\)
number of elements and their positions
\(D\)
aperture length \(N p\), mm
\(\mathbb{E}[S]\)
ensemble mean over screen realizations
\(\phi_i\)
aberration phase of element \(i\), rad (\(\phi(x_i)\) of (33.1))
\(\sigma_\phi,\ \rho,\ \ell\)
as in (33.1)
\(j\)
imaginary unit
\(i,\ j\)
element indices (in the double sum, \(j\) is an index, not the imaginary unit)
The unconstrained maximizer is the dominant eigenvector (Varslot, Mo, Angelsen and Krogstad, J. Acoust. Soc. Am., 2004, showed that the brightness-maximizing filter focuses on the regions of high insonification of the aberrated transmit beam). Its phase is the screen phase plus the phase of \(\mathbf{u}_1(\mathbf{C}_0)\), which is flat (up to a ramp from the spot's centroid) only when the insonified spot is compact and symmetric. A phase-only search (\(\lvert w_i\rvert = 1\)) is solved by coordinate ascent, \(w_i \leftarrow e^{j\arg(\mathbf{C}\mathbf{w})_i}\), which never decreases \(B\).symbols
\(B\)
mean speckle brightness of the beamformed pixel
\(\mathbf{s}\)
focused (geometrically delayed) channel signals, one per element
\(\mathbf{C},\ \mathbf{C}_0\)
aperture covariance with and without the screen; \(\mathbf{C}_0\) is the van Cittert–Zernike matrix of the transmit focal intensity
\(\mathbf{\Phi}\)
diagonal screen matrix at the array (Module 05)
\(\mathbf{u}_1\)
eigenvector of the largest eigenvalue
\(\mathbf{w}\)
receive weight vector, \(N\) elements
\(\mathbb{E}\)
expectation over speckle
\((\cdot)^{H}\)
conjugate transpose
\(\mathrm{diag}\)
diagonal matrix of the listed entries
\(\phi_i,\ j\)
screen phase of element \(i\) (rad) and imaginary unit
The residual of a borrowed law has variance \(2\sigma_\phi^{2}(1-\rho)\), which exceeds the uncorrected \(\sigma_\phi^{2}\) once \(\rho < 1/2\): outside the patch, a correction is worse than none. m33_isoplanatic_patch.m (40 ns, \(\ell\) = 4 mm, z = 40 mm, 128 elements, finite-aperture exact formula and Monte Carlo): half-widths at \(S = 1/2\) of 16.5, 8.3 and 4.3 mm for d = 5, 10 and 20 mm; the large-aperture form gives 8.0 mm at 10 mm. The borrowed law falls to the uncorrected level at 22.5, 11.0 and 5.1 mm, against \(\Delta x_\times\) = 26.6, 13.3 and 6.7 mm (the finite aperture keeps the uncorrected Strehl above \(e^{-\sigma_\phi^{2}}\)). The exponential in (33.8) is not a small-phase expansion: like \(e^{-\sigma_\phi^{2}}\) in (33.2), it is the exact Gaussian average in the limit \(\ell \ll D\). At finite aperture, elements closer than \(\ell\) add the floor of Section 33.1, which lifts the uncorrected Strehl (0.349 at d = 10 mm, against 0.206) more than the borrowed one (0.349 against 0.304 at 11 mm) and so moves the crossing inward.symbols
\(d,\ z\)
depth of the screen and of the image points, mm
\(\Delta x,\ \delta\)
lateral distance from the guide point, and the resulting slide of the aperture footprint on the screen, mm
\(S(\Delta x)\)
Strehl ratio at \(\Delta x\) with the law of the guide point (large aperture, \(\ell \ll D\))
\(\Delta x_\times\)
distance at which the borrowed correction is no better than no correction (\(\rho = 1/2\))
\(\Delta x_{1/2}\)
isoplanatic half-width at \(S = 1/2\) (defined for \(\sigma_\phi^{2} > \ln 2/2\))
Re-centring makes every focal point in an isoplanatic patch look like the same source at the origin, seen through the same screen; averaging the columns over the patch averages over independent speckle realizations (one per resolution cell). This is (33.6) with the speckle average done by the matrix. \(\hat\phi\) is the receive-side law; by reciprocity (the same screen on both paths) the script and Demo 33.6 apply it to both sides of \(\mathbf{K}\) and recompute: the corrected transmit shrinks the virtual reflector toward a point and the estimate converges. Lambert et al. instead alternate the correction between the transmit and receive sides and shrink the window at each step.symbols
\(\mathbf{R}_{ux}\)
reflection matrix with focused transmit (columns \(x_{\mathrm{in}}\)) and element receive (rows \(u\))
\(\circ\)
Hadamard (element-wise) product
\(\mathbf{D}\)
distortion matrix: each column is a received wavefront re-centred on its own focal point
\(\mathbf{C},\ \mathbf{U},\ \mathbf{\Sigma}\)
its correlation matrix, left singular vectors and singular values
\(\mathbf{C}_{\mathrm{vr}}\)
van Cittert–Zernike covariance of a virtual reflector shaped like the transmit focal intensity, common to all columns within a patch
\(\hat\phi_u\)
estimated aberration law (phase of the first singular vector)
\(\mathbf{K},\ \mathbf{G}_0\)
as in (33.10)
\(\mathbf{\Phi}\)
diagonal screen matrix, as in (33.6)
\(U_{u1}\)
entry \(u\) of the first left singular vector \(\mathbf{U}_{\cdot 1}\)
\(\arg\)
phase angle, rad
\((\cdot)^{H}\)
conjugate transpose
22Rapid fire, with pictures
Each card is a question you should answer in five seconds, the answer, and the picture to keep in mind. Drag the width slider: every transform pair rescales together, so narrowing in time always widens in frequency.
Depth-of-field and attenuation values are common approximations; the exact constant depends on the definition (−3 vs −6 dB, one-way vs two-way) and on the tissue. The modules derive each entry and give the conditions; the key-equation lists are generated from the modules' key-equation boxes when the sheet is built.