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When the Far-Field Assumption Breaks

The forward model developed in Chapter 7 assumes far-field conditions: planar wavefronts, paraxial angles, and narrowband signals. When these assumptions fail — due to large arrays (XL-MIMO), short ranges, or wide angular coverage — the sensing matrix $\mathbf{A}$ must be corrected. This section quantifies when corrections are needed and how they modify the forward model.

Definition:
Fresnel and Fraunhofer Regions

The boundary between near-field and far-field is characterized by the Fraunhofer distance:

$d_{\text{FF}} = \frac{2D^2}{\lambda},$

where $D$ is the largest dimension of the antenna aperture and $\lambda$ is the wavelength.

Region	Distance $d$	Wavefront	Phase error
Reactive near-field	$d < 0.62\sqrt{D^3/\lambda}$	Evanescent	Not applicable
Radiating near-field (Fresnel)	$0.62\sqrt{D^3/\lambda} < d < 2D^2/\lambda$	Spherical	Quadratic
Far-field (Fraunhofer)	$d > 2D^2/\lambda$	Planar	Negligible

The far-field forward model is valid only in the Fraunhofer region. In the Fresnel region, the wavefront curvature introduces a quadratic phase term that must be included in $\mathbf{A}$ .

Theorem: Near-Field Phase Correction

The exact distance from source $\mathbf{s}$ to point $\mathbf{p}$ is $d(\mathbf{s}, \mathbf{p}) = \|\mathbf{p} - \mathbf{s}\|$ . The far-field (first-order Taylor) approximation uses:

$d(\mathbf{s}, \mathbf{p}) \approx d(\mathbf{s}, \mathbf{p}_{0}) + \nabla_{\mathbf{p}} d(\mathbf{s}, \mathbf{p})\big|_{\mathbf{p}_{0}}^T (\mathbf{p} - \mathbf{p}_{0}).$

The near-field (Fresnel) correction adds the second-order term:

$d(\mathbf{s}, \mathbf{p}) \approx d_0 + \hat{\mathbf{s}}^T \tilde{\mathbf{p}} + \frac{1}{2d_0}\left(\|\tilde{\mathbf{p}}\|^2 - (\hat{\mathbf{s}}^T \tilde{\mathbf{p}})^2\right),$

where $d_0 = d(\mathbf{s}, \mathbf{p}_{0})$ , $\hat{\mathbf{s}} = (\mathbf{p}_{0} - \mathbf{s})/d_0$ is the unit direction vector, and $\tilde{\mathbf{p}} = \mathbf{p} - \mathbf{p}_{0}$ .

The quadratic correction produces a position-dependent phase:

$\Delta\phi_{\text{NF}} = -\kappa\frac{\|\tilde{\mathbf{p}}_\perp\|^2}{2d_0},$

where $\tilde{\mathbf{p}}_\perp$ is the component of $\tilde{\mathbf{p}}$ perpendicular to $\hat{\mathbf{s}}$ .

The far-field model treats the wavefront as a plane. In the near field, the wavefront is spherical, and the curvature introduces a quadratic phase across the scene. This is exactly the same phenomenon as the Fresnel diffraction integral in optics.

Proof

Second-order Taylor expansion

Write $d(\mathbf{s}, \mathbf{p}_{0} + \tilde{\mathbf{p}}) = \|(\mathbf{p}_{0} - \mathbf{s}) + \tilde{\mathbf{p}}\|$ . Let $\mathbf{u} = \mathbf{p}_{0} - \mathbf{s}$ , so $d_0 = \|\mathbf{u}\|$ and $\hat{\mathbf{s}} = \mathbf{u}/d_0$ . Then:

$d = \sqrt{d_0^2 + 2\mathbf{u}^T\tilde{\mathbf{p}} + \|\tilde{\mathbf{p}}\|^2} = d_0\sqrt{1 + \frac{2\hat{\mathbf{s}}^T\tilde{\mathbf{p}}}{d_0} + \frac{\|\tilde{\mathbf{p}}\|^2}{d_0^2}}.$

Expand the square root

Using $\sqrt{1 + x} \approx 1 + x/2 - x^2/8 + \cdots$ with $x = 2\hat{\mathbf{s}}^T\tilde{\mathbf{p}}/d_0 + \|\tilde{\mathbf{p}}\|^2/d_0^2$ :

$d \approx d_0 + \hat{\mathbf{s}}^T\tilde{\mathbf{p}} + \frac{\|\tilde{\mathbf{p}}\|^2 - (\hat{\mathbf{s}}^T\tilde{\mathbf{p}})^2}{2d_0}.$

The last term is the Fresnel correction: it depends on the transverse displacement $\|\tilde{\mathbf{p}}_\perp\|^2$ .

Identify the quadratic phase

Multiplying by $-\kappa$ to get the phase:

$\phi \approx -\kappa d_0 - \kappa\hat{\mathbf{s}}^T\tilde{\mathbf{p}} - \frac{\kappa\|\tilde{\mathbf{p}}_\perp\|^2}{2d_0}.$

The first two terms are the far-field model. The third is the Fresnel correction $\Delta\phi_{\text{NF}}$ . $\blacksquare$

Definition:
Near-Field Sensing Matrix

The near-field sensing matrix replaces the far-field steering vectors with exact spherical wavefront phases. For each voxel $q$ , the $(i,j,k)$ -th entry of $\mathbf{A}_{\text{NF}}$ is:

$[\mathbf{A}_{\text{NF}}]_{(i,j,k),q} = \frac{\sqrt{G^{\text{tx}}_{i} G^{\text{rx}}_{j}}\,e^{-j\kappa_{k}(d(\mathbf{s}_{i}, \mathbf{p}_{q}) + d(\mathbf{p}_{q}, \mathbf{r}_{j}))}}{d(\mathbf{s}_{i}, \mathbf{p}_{q})\,d(\mathbf{p}_{q}, \mathbf{r}_{j})},$

computing the exact distances $d(\mathbf{s}_{i}, \mathbf{p}_{q})$ and $d(\mathbf{p}_{q}, \mathbf{r}_{j})$ instead of using the Taylor approximation.

Cost: The far-field model has Kronecker structure ( $\mathbf{A} \approx \mathbf{A}_{\text{rx}} \otimes \mathbf{A}_{\text{tx}} \otimes \mathbf{A}_{\text{freq}}$ ), enabling efficient $O(M + Q)$ matrix-vector products. The near-field model loses this structure, requiring explicit $O(MQ)$ computation. For large $M$ and $Q$ , this can be orders of magnitude slower.

Example: When Does Near-Field Matter? A Numerical Example

A MIMO radar operates at $f_0 = 28$ GHz ( $\lambda = 10.7$ mm) with a ULA of $N = 64$ elements at half-wavelength spacing.

(a) Compute the Fraunhofer distance $d_{\text{FF}}$ .

(b) At range $R = 2$ m, compute the maximum near-field phase error across the scene (scene width = 1 m centered on boresight).

(c) Is the far-field model adequate at this range?

Solution

Fraunhofer distance

Array aperture: $D = (N-1) \times \lambda/2 = 63 \times 5.35\text{ mm} \approx 0.337$ m.

$d_{\text{FF}} = \frac{2D^2}{\lambda} = \frac{2 \times 0.337^2}{0.0107} \approx 21.2\text{ m}.$

Maximum phase error at 2 m

At range $R = 2$ m (well inside the Fraunhofer distance), the maximum transverse displacement is $\tilde{p}_\perp = 0.5$ m (edge of the 1 m scene). The Fresnel phase error is:

$\Delta\phi_{\text{NF}} = \frac{\kappa\,\tilde{p}_\perp^2}{2R} = \frac{2\pi \times 0.5^2}{0.0107 \times 2 \times 2} = \frac{2\pi \times 0.25}{0.0214} \approx 73.4\text{ rad} \approx 11.7\text{ cycles}.$

This is an enormous phase error.

Assessment

The far-field model is completely inadequate at $R = 2$ m for this array. The phase error of $\sim 12$ full cycles means the far-field $\mathbf{A}$ would produce a completely defocused image. The near-field (exact distance) model is mandatory.

The far-field model becomes adequate at $R \gg d_{\text{FF}} = 21.2$ m. At intermediate ranges ( $5$ - $20$ m), the Fresnel correction (second-order Taylor) may suffice without computing exact distances.

Wide-Angle Scattering Beyond Paraxial

The paraxial (small-angle) approximation assumes that all scatterers subtend small angles relative to the array boresight. This approximation breaks when:

Wide-aperture arrays observe targets at large off-boresight angles.
Distributed MIMO systems have Tx/Rx at diverse positions around the scene.
360-degree imaging (e.g., automotive surround radar).

Consequences of wide-angle scattering:

The spatial frequency $\boldsymbol{\kappa}_{s,r}$ is no longer approximately linear in angle — the Fourier relationship between scene and measurements becomes non-uniform.
Aspect-dependent scattering (DAspect-Dependent Scattering) becomes significant: the same target looks different from different angles.
The Kronecker structure of $\mathbf{A}$ may break for very large angular spans.

Mitigation: Use the exact (non-paraxial) steering vectors in $\mathbf{A}$ , or partition the angular domain into narrow sectors and process each independently (sub-aperture processing).

Near-Field vs. Far-Field Point Spread Function

Compare the point spread function (PSF) of the imaging system under far-field (Kronecker) and near-field (exact distance) models. At close range, the far-field PSF is defocused and asymmetric; the near-field PSF remains sharp.

Parameters

Target range (m)3

Number of array elements32

Carrier frequency (GHz)28

⚠️Engineering Note

Computational Cost of Near-Field Models

The far-field model with Kronecker structure enables efficient matrix-vector products: $\mathbf{A}\mathbf{c}$ can be computed as a sequence of smaller matrix multiplications, reducing the cost from $O(MQ)$ to $O(M + Q)$ (up to logarithmic factors with FFTs).

The near-field model destroys this structure. For a system with $M = N_t N_r N_f = 10^4$ measurements and $Q = 10^4$ voxels, a single $\mathbf{A}\mathbf{c}$ product requires $10^8$ complex multiply-accumulates. Iterative reconstruction (ISTA, ADMM) requires hundreds of such products, making the total cost $\sim 10^{10}$ operations.

Practical approaches:

Fresnel approximation: Add the quadratic phase correction to the Kronecker model ( $O(Q)$ additional cost per product).
Non-uniform FFT (NUFFT): Accelerate the non-uniform Fourier transform to $O(Q \log Q)$ .
Sub-aperture processing: Split the array into sub-apertures where the far-field model holds within each.

Practical Constraints

•
Near-field exact model requires $O(MQ)$ storage for $\mathbf{A}$
•
Iterative algorithms require $O(100)$ matrix-vector products
•
NUFFT provides $O(Q\log Q)$ acceleration for the Fresnel case

Common Mistake: Using Far-Field Model with XL-MIMO Arrays

Mistake:

Applying the standard far-field Kronecker-structured sensing matrix when imaging with extremely large arrays (XL-MIMO, $D \gg \lambda$ ) at ranges $R < 2D^2/\lambda$ .

Correction:

Check the Fraunhofer distance $d_{\text{FF}} = 2D^2/\lambda$ before choosing the forward model. If targets are in the Fresnel region, use the exact-distance or Fresnel-corrected sensing matrix.

Quick Check

A ULA with $N = 128$ elements at $\lambda/2$ spacing operates at $f_0 = 60$ GHz ( $\lambda = 5$ mm). What is the Fraunhofer distance?

$\approx 2$ m

$\approx 10$ m

$\approx 40$ m

$\approx 200$ m

Correction:

\approx 40

m

$D = 127 \times 2.5\text{ mm} = 0.318$ m. $d_{\text{FF}} = 2 \times 0.318^2 / 0.005 = 40.3$ m. Targets closer than 40 m require near-field corrections.

Fraunhofer distance

The distance $d_{\text{FF}} = 2D^2/\lambda$ beyond which the far-field (planar wavefront) approximation introduces less than $\pi/8$ radians of phase error. $D$ is the aperture size, $\lambda$ is the wavelength.

Related: {{Ref:Def Fresnel Fraunhofer}}

Fresnel region

The intermediate range between the reactive near-field and the far-field, where the wavefront is approximately spherical and the phase error is quadratic. Also called the radiating near-field.

Key Takeaway

Near-field corrections are mandatory when $R < 2D^2/\lambda$ . The quadratic phase error from spherical wavefronts defocuses the image. The near-field model loses Kronecker structure, increasing computational cost. NUFFT and sub-aperture processing mitigate the cost. Wide-angle effects break the paraxial approximation and require exact steering vectors.

Near-Field and Wide-Angle Effects

When the Far-Field Assumption Breaks

Definition: Fresnel and Fraunhofer Regions

Theorem: Near-Field Phase Correction

Second-order Taylor expansion

Expand the square root

Identify the quadratic phase

Definition: Near-Field Sensing Matrix

Example: When Does Near-Field Matter? A Numerical Example

Fraunhofer distance

Maximum phase error at 2 m

Assessment

Wide-Angle Scattering Beyond Paraxial

Near-Field vs. Far-Field Point Spread Function

Parameters

Computational Cost of Near-Field Models

Common Mistake: Using Far-Field Model with XL-MIMO Arrays

Quick Check

Fraunhofer distance

Fresnel region

Key Takeaway

Definition:
Fresnel and Fraunhofer Regions

Definition:
Near-Field Sensing Matrix