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What Can We Resolve?

The k-space coverage analysis tells us which spatial frequencies we can measure. Now we translate this into concrete resolution limits: how close can two scatterers be and still be distinguished? We derive the classical Rayleigh and Sparrow criteria for RF imaging and connect them to the familiar radar resolution formulas.

Definition:
Point-Spread Function (PSF)

The point-spread function of the imaging system is the image formed when the scene contains a single point scatterer: $c(\mathbf{p}) = \delta(\mathbf{p} - \mathbf{p}_{0})$ . In the discrete model, the PSF is the column of $\mathbf{A}^{H}\mathbf{A}$ corresponding to the voxel at $\mathbf{p}_{0}$ :

$\text{PSF}(\mathbf{p}_{q}) = [\mathbf{A}^{H}\mathbf{A}]_{q_0, q} = \sum_{m=1}^{MNK} [\mathbf{A}]^*_{m,q_0} [\mathbf{A}]_{m,q}.$

In the wavenumber domain, the PSF is the inverse Fourier transform of the k-space coverage indicator:

$\text{PSF}(\Delta\mathbf{p}) \propto \sum_{i,j,k} e^{j\boldsymbol{\kappa}_{i,j,k}^\mathsf{T} \Delta\mathbf{p}}.$

The width of the PSF mainlobe determines the resolution; the sidelobes determine the artifact level.

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Theorem: Range Resolution

For an imaging system with bandwidth $W$ , the range (radial) resolution is

$\boxed{\Delta r = \frac{\text{c}}{2W}}.$

Two scatterers separated by less than $\Delta r$ along the range direction cannot be distinguished.

The bandwidth $W$ determines the radial extent of k-space coverage: $\Delta\kappa_{\text{radial}} = 4\piW/\text{c}$ . The corresponding spatial resolution is $\Delta r = 2\pi/\Delta\kappa_{\text{radial}} = \text{c}/(2W)$ . This is the same formula as the radar range resolution — because it derives from the same physics.

Proof

From k-space extent to resolution

The radial extent of k-space is $\Delta\kappa_{\text{radial}} = 4\piW/\text{c}$ (from Tk-Space Extent from System Parameters). By the Fourier uncertainty principle, the spatial resolution in the corresponding direction is $\Delta r = 2\pi/\Delta\kappa_{\text{radial}} = 2\pi/(4\piW/\text{c}) = \text{c}/(2W)$ . $\blacksquare$

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Theorem: Cross-Range Resolution

For an imaging system with maximum angular aperture $\theta_{\max}$ (the widest angle subtended by the Tx-Rx directions as seen from the target), operating at wavelength $\lambda = \text{c}/f_0$ , the cross-range resolution is

$\boxed{\Delta x = \frac{\lambda}{2\sin(\theta_{\max}/2)}}.$

For small apertures ( $\theta_{\max} \ll 1$ ), this simplifies to $\Delta x \approx \lambda/\theta_{\max}$ . For a full $360°$ aperture, $\Delta x = \lambda/2$ — the classical diffraction limit.

Angular diversity provides tangential extent in k-space. Wider angular aperture → more tangential coverage → finer cross-range resolution. The diffraction limit $\lambda/2$ is reached when we have full angular coverage (all directions).

Proof

Tangential k-space extent

The tangential extent is $\Delta\kappa_{\text{tang}} = 4\pif_0\sin(\theta_{\max}/2)/\text{c} = (4\pi/\lambda)\sin(\theta_{\max}/2)$ .

Spatial resolution

$\Delta x = 2\pi/\Delta\kappa_{\text{tang}} = 2\pi\lambda/(4\pi\sin(\theta_{\max}/2)) = \lambda/(2\sin(\theta_{\max}/2))$ .

Diffraction limit

At $\theta_{\max} = \pi$ (full aperture), $\sin(\pi/2) = 1$ and $\Delta x = \lambda/2$ . This is the fundamental diffraction limit — the finest detail resolvable with waves of wavelength $\lambda$ . $\blacksquare$

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Resolution Limits Explorer

Explore how range and cross-range resolution depend on bandwidth, carrier frequency, and angular aperture. The PSF mainlobe width is shown for the selected parameters.

Parameters

W

(MHz)200

f_0

(GHz)10

\\theta_{\\max}

(degrees)60

Example: Resolution for a Typical Indoor ISAC System

Compute the range and cross-range resolution for a system with $f_0 = 10$ GHz, $W = 200$ MHz, and three nodes arranged in an equilateral triangle (120-degree angular span).

Solution

Range resolution

$\Delta r = \text{c}/(2W) = (3 \times 10^8)/(2 \times 200 \times 10^6) = 0.75$ m.

Cross-range resolution

The angular aperture from the triangular arrangement is approximately $\theta_{\max} = 120°$ . $\Delta x = \lambda/(2\sin(60°)) = 0.03/(2 \times 0.866) = 0.0173$ m $\approx 1.7$ cm.

Interpretation

The cross-range resolution (1.7 cm) is excellent — far finer than the range resolution (75 cm). This is typical in RF imaging: angular diversity provides fine cross-range resolution at the cost of requiring multi-static deployment, while range resolution is limited by the available bandwidth. In this example, we would need $W \approx 8.7$ GHz to match the cross-range resolution in range — impractical for most systems.

Beyond the Diffraction Limit: Super-Resolution Preview

The resolution limits derived above are classical limits assuming no prior information about the scene. If the scene is sparse (only a few scatterers), compressed sensing and sparse recovery algorithms can resolve features finer than the diffraction limit — this is super-resolution. The idea is that sparsity provides additional constraints that compensate for the limited k-space coverage.

We develop sparse recovery methods in detail in Ch 13-14, and show that for scenes with $s$ scatterers, the effective resolution can be improved by a factor of $\sqrt{MNK/s}$ compared to the classical limit. However, super-resolution comes at the cost of increased sensitivity to noise and model mismatch.

Definition:
Rayleigh and Sparrow Criteria for RF Imaging

The Rayleigh criterion states that two point scatterers are just resolved when the PSF peak of one falls on the first null of the other's PSF. For a sinc-like PSF (arising from rectangular k-space coverage), this gives the classical resolutions $\Delta r$ and $\Delta x$ .

The Sparrow criterion is more stringent: two points are resolved when the combined PSF has a detectable dip between the two peaks. The Sparrow limit is typically about 80% of the Rayleigh limit.

In RF imaging, the PSF is generally not a simple sinc (because k-space coverage is irregular), so these criteria are applied to the numerically computed PSF of the specific system.

PSF from k-Space Coverage

See the point-spread function that results from different k-space coverage patterns. The PSF is computed as the inverse Fourier transform of the coverage indicator. Observe how gaps in k-space coverage create sidelobes and artifacts.

Parameters

Configuration

W

(MHz)200

f_0

(GHz)10

Common Mistake: Resolution Is Not Accuracy

Mistake:

Confusing resolution (the ability to distinguish two nearby scatterers) with localization accuracy (the precision with which a single scatterer's position can be estimated).

Correction:

Resolution is limited by the PSF width (diffraction limit). But the position of a single isolated scatterer can be estimated to much finer precision than the resolution, by fitting a model to the PSF peak. This is the super-localization phenomenon, analogous to sub-pixel estimation in optical microscopy (STORM, PALM). With high SNR, the Cramér-Rao bound on localization can be orders of magnitude smaller than the resolution.

Diffraction Limit

The fundamental limit on spatial resolution imposed by the wave nature of the probing signal. For a system with wavelength $\lambda$ and full ( $360°$ ) angular aperture, the diffraction limit is $\lambda/2$ . In practice, limited angular aperture degrades the cross-range resolution beyond this limit. Range resolution is limited by bandwidth: $\text{c}/(2W)$ .

Point-Spread Function (PSF)

The image of a single point scatterer, characterizing the spatial impulse response of the imaging system. The PSF depends on the k-space coverage: its mainlobe width determines resolution, and its sidelobes determine artifact levels. For the discrete model, the PSF is encoded in the Gram matrix $\mathbf{A}^{H}\mathbf{A}$ .

Resolution Limits

What Can We Resolve?

Definition: Point-Spread Function (PSF)

Theorem: Range Resolution

From k-space extent to resolution

Theorem: Cross-Range Resolution

Tangential k-space extent

Spatial resolution

Diffraction limit

Resolution Limits Explorer

Parameters

Example: Resolution for a Typical Indoor ISAC System

Range resolution

Cross-range resolution

Interpretation

Beyond the Diffraction Limit: Super-Resolution Preview

Definition: Rayleigh and Sparrow Criteria for RF Imaging

PSF from k-Space Coverage

Parameters

Common Mistake: Resolution Is Not Accuracy

Diffraction Limit

Point-Spread Function (PSF)

Definition:
Point-Spread Function (PSF)

Definition:
Rayleigh and Sparrow Criteria for RF Imaging