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Section Roadmap: Ultrasound Imaging and Beamforming

Ultrasound imaging operates in the near-field regime with wavelengths of 0.1--1.5 mm (1--15 MHz), producing wavelength-to-aperture ratios comparable to RF imaging at mmWave frequencies. The pulse-echo model is a time-domain analogue of the RF matched filter, and beamforming in ultrasound is the direct counterpart of delay-and-sum imaging (Ch 13). This section develops the ultrasound forward model, connects beamforming to matched filtering, and introduces plane-wave imaging and coherent compounding — techniques with immediate RF imaging analogues.

Definition:
Pulse-Echo Model for Ultrasound

A phased-array transducer with $N_e$ elements transmits a focused pulse and records echoes from scatterers in the medium. The signal received by element $i$ from a point scatterer at position $\mathbf{p}$ is

$r_i(t) = \sigma(\mathbf{p})\, a\!\left(t - \frac{d_{\mathrm{tx}}(\mathbf{p}) + d_i(\mathbf{p})}{c_s}\right),$

where $\sigma(\mathbf{p})$ is the scattering amplitude, $a(t)$ is the transmitted pulse waveform, $d_{\mathrm{tx}}(\mathbf{p})$ is the one-way distance from the transmit focal point to $\mathbf{p}$ , $d_i(\mathbf{p})$ is the distance from $\mathbf{p}$ to receive element $i$ , and $c_s \approx 1540$ m/s is the speed of sound in tissue.

Parallel to RF: Compare with the RF round-trip delay $\tau_{i,j,q} = (d(\ntn{tx_pos}_i, \mathbf{p}_{q}) + d(\mathbf{p}_{q}, \ntn{rx_pos}_j)) / c$ from Ch 09. The ultrasound model is a single-frequency, near-field, scalar version of the RF sensing model.

Definition:
Delay-and-Sum (DAS) Beamforming

To form an image at pixel position $\mathbf{p}$ , the delay-and-sum (DAS) beamformer aligns and sums the received signals:

$I_{\mathrm{DAS}}(\mathbf{p}) = \sum_{i=1}^{N_e} w_i\,r_i\!\left(\frac{d_{\mathrm{tx}}(\mathbf{p}) + d_i(\mathbf{p})}{c_s}\right),$

where $w_i$ are apodization weights (e.g., Hanning window to suppress sidelobes).

This is the matched filter: DAS correlates the received data with the expected response from a point scatterer at $\mathbf{p}$ . In matrix form, $\mathbf{I}_{\mathrm{DAS}} = \mathbf{A}_{\mathrm{US}}^H \mathbf{r}$ , which is the backpropagation image $\hat{\mathbf{c}}^{\text{BP}}$ of Ch 13 applied to the ultrasound forward model.

Theorem: DAS Beamforming as Matched Filtering

Let $\mathcal{A}_{\mathrm{US}}: \sigma \mapsto \mathbf{r}$ denote the linear ultrasound forward operator mapping the reflectivity function $\sigma(\mathbf{p})$ to the received channel data $\mathbf{r}$ . The DAS beamformer output at $\mathbf{p}$ satisfies

$I_{\mathrm{DAS}}(\mathbf{p}) = [\mathcal{A}_{\mathrm{US}}^* \mathbf{r}](\mathbf{p}),$

where $\mathcal{A}_{\mathrm{US}}^*$ is the adjoint (matched filter) of the forward operator. This is a special case of the backpropagation image from Ch 13: $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}$ .

DAS beamforming compensates for the round-trip travel time to each pixel and coherently sums across elements. This is exactly what the adjoint operator does: it correlates the data with the "steering vector" to each pixel. In radar, the same operation is called matched filtering; in seismology, it is migration; in radio astronomy, it is dirty imaging.

Proof

Write the forward operator explicitly

Discretize the scene on a grid of $Q$ pixels. The received signal at element $i$ is

$r_i(t) = \sum_{q=1}^{Q} \sigma_q\, a\!\left(t - \frac{d_{\mathrm{tx}}(\mathbf{p}_q) + d_i(\mathbf{p}_q)}{c_s}\right).$

In matrix form: $\mathbf{r} = \mathbf{A}_{\mathrm{US}}\boldsymbol{\sigma}$ where $[\mathbf{A}_{\mathrm{US}}]_{(i,t),q}$ encodes the delayed pulse.

Compute the adjoint

The adjoint $\mathbf{A}_{\mathrm{US}}^H$ time-reverses the data and sums across elements, which is exactly the DAS operation:

$[\mathbf{A}_{\mathrm{US}}^H \mathbf{r}]_q = \sum_{i=1}^{N_e} w_i\,r_i\!\left(\frac{d_{\mathrm{tx}}(\mathbf{p}_q) + d_i(\mathbf{p}_q)}{c_s}\right) = I_{\mathrm{DAS}}(\mathbf{p}_q). \quad \blacksquare$

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Definition:
Plane-Wave Imaging and Coherent Compounding

Instead of transmitting a focused beam (one focal point per transmission), plane-wave imaging transmits an unfocused plane wave across all elements simultaneously. A single plane-wave transmission illuminates the entire field of view, enabling ultrafast imaging (>10,000 frames/sec vs ~30 fps for focused mode).

The image quality of a single plane wave is poor due to the lack of transmit focusing. Coherent compounding recovers the quality: transmit $N_{\mathrm{pw}}$ plane waves at different angles $\alpha_1, \ldots, \alpha_{N_{\mathrm{pw}}}$ , beamform each independently, and sum coherently:

$I_{\mathrm{comp}}(\mathbf{p}) = \sum_{n=1}^{N_{\mathrm{pw}}} I_{\mathrm{DAS}}^{(n)}(\mathbf{p}).$

Parallel to RF imaging: Coherent compounding is the ultrasound analogue of multi-view coherent imaging in RF — combining images from different illumination angles to improve the PSF.

Example: DAS Resolution and the Diffraction Limit

A linear ultrasound array has $N_e = 128$ elements with pitch $d = 0.3$ mm, operating at $f_0 = 5$ MHz (wavelength $\lambda = c_s / f_0 = 0.308$ mm, speed of sound $c_s = 1540$ m/s).

Tasks:

Compute the lateral resolution at focal depth $z = 40$ mm.
Compute the axial resolution for a pulse bandwidth of $B = 3$ MHz.
Compare with the resolution of an RF imaging system at $f_0 = 77$ GHz with $B = 4$ GHz and a 64-element ULA.

Solution

Lateral resolution (ultrasound)

The aperture is $D = N_e \cdot d = 128 \times 0.3 = 38.4$ mm. The f-number is $F = z / D = 40 / 38.4 \approx 1.04$ . Lateral resolution: $\delta_{\mathrm{lat}} \approx \lambda F = 0.308 \times 1.04 \approx 0.32$ mm.

Axial resolution (ultrasound)

Axial resolution depends on the pulse bandwidth: $\delta_{\mathrm{ax}} = c_s / (2B) = 1540 / (2 \times 3 \times 10^6) \approx 0.257$ mm.

Comparison with RF imaging

At 77 GHz: $\lambda = c / f_0 = 3 \times 10^8 / 77 \times 10^9 \approx 3.9$ mm. With 64 elements at half-wavelength spacing, $D = 64 \times 1.95 \approx 125$ mm. At range $z = 1$ m: $F = 1000 / 125 = 8$ , $\delta_{\mathrm{lat}} = 3.9 \times 8 = 31.2$ mm. Axial: $\delta_{\mathrm{ax}} = c / (2B) = 3 \times 10^8 / (2 \times 4 \times 10^9) = 37.5$ mm.

Observation: Ultrasound achieves sub-mm resolution at cm-range depth because its wavelength-to-range ratio is much smaller. RF imaging at 77 GHz achieves cm-level resolution at m-range depth. The physics is the same; the scales differ.

Delay-and-Sum Beamforming for Ultrasound

Simulates a linear ultrasound array imaging point scatterers.

Left: The DAS beamformed image showing the PSF at each scatterer location. Right: A cross-range profile through the brightest scatterer showing the mainlobe width (lateral resolution) and sidelobe level.

Increasing the number of elements narrows the mainlobe (better resolution). Apodization (Hanning window) reduces sidelobes at the cost of a wider mainlobe.

Parameters

Number of elements

N_e

128

Apodization window

Center frequency (MHz)5

Ultrasound vs RF Imaging Parameter Comparison

Parameter	Ultrasound	RF Imaging (mmWave)	RF Imaging (sub-6 GHz)
Carrier frequency	1--15 MHz	60--77 GHz	3.5--6 GHz
Wavelength	0.1--1.5 mm	3.9--5 mm	50--86 mm
Propagation speed	1540 m/s (tissue)	$3 \times 10^8$ m/s (air)	$3 \times 10^8$ m/s (air)
Typical range	1--20 cm	1--50 m	1--100 m
Wavelength/range ratio	$10^{-3}$ to $10^{-2}$	$10^{-4}$ to $10^{-2}$	$10^{-3}$ to $10^{-1}$
Array elements	64--256	16--256	8--64
Beamforming	DAS + apodization	Matched filter / MUSIC	Matched filter / MUSIC
Near-field?	Always	Often (short range)	Sometimes
Main challenge	Tissue attenuation, speckle	Limited angular coverage	Severe ill-conditioning

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Quick Check

In the DAS beamformer $I_{\mathrm{DAS}}(\mathbf{p}) = \sum_i w_i\,r_i(\tau_i(\mathbf{p}))$ , what is the role of the delay $\tau_i(\mathbf{p})$ ?

It compensates for the round-trip propagation time from the transmitter to the point $\mathbf{p}$ to receiver $i$

It applies a frequency-dependent filter to suppress noise

It selects the frequency band of interest

Correction:

It compensates for the round-trip propagation time from the transmitter to the point

\mathbf{p}

to receiver

i

The delay aligns the echoes from $\mathbf{p}$ so they add coherently. This is the time-domain version of phase alignment in the matched filter.

🔧Engineering Note

Clutter and Aberration in Ultrasound vs RF Imaging

Both ultrasound and RF imaging suffer from clutter — unwanted echoes that degrade image quality. In ultrasound, clutter arises from off-axis scattering, multipath reflections from tissue boundaries, and phase aberration due to speed-of-sound inhomogeneity. In RF imaging, clutter comes from multipath propagation, static background reflections, and electromagnetic interference.

Aberration correction in ultrasound (adaptive beamforming, coherence-based methods) has direct analogues in RF imaging: autofocusing in SAR (Ch 12) corrects for phase errors from platform motion, and the model mismatch robustness analysis of Ch 08 addresses the RF version of the same problem.

Practical Constraints

•
Speed-of-sound variations in tissue (1450--1600 m/s) cause wavefront distortion analogous to atmospheric phase errors in RF
•
Speckle noise in ultrasound is the coherent imaging analogue of fading in RF — both arise from interference of unresolved scatterers

Plane-wave imaging

An ultrasound imaging mode where unfocused plane waves are transmitted, enabling ultrafast frame rates (>10 kHz). Multiple plane-wave transmissions at different angles are coherently compounded to recover image quality.

Coherent compounding

Combining beamformed images from multiple transmit events (different angles, frequencies, or focal points) by coherent summation, improving the point spread function and SNR.

Key Takeaway

Ultrasound beamforming (DAS) is the matched filter applied to the pulse-echo forward model — structurally identical to $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}$ in RF imaging. Both modalities operate in the near-field with comparable wavelength-to- aperture ratios. Plane-wave imaging with coherent compounding is the ultrasound analogue of multi-view coherent RF imaging. The physics transfers directly; the scales differ by a factor of $10^5$ in propagation speed.

Ultrasound Imaging and Beamforming