Ferkans — Interactive Telecom Tutor

Steering Vectors Gain a Range Coordinate

In the far-field chapters we used a steering vector $\mathbf{a}(\theta) = [1, e^{-j k_0 d \cos\theta}, \ldots]^T$ depending only on the angle of arrival. That parameterization is a first-order Taylor expansion of the true spherical wavefront, accurate when $r \gg d_F = 2D^2/\lambda$ . Once the user enters the Fresnel region $r < d_F$ , the quadratic term survives and the effective steering vector picks up an extra dependence on the range $r$ . The consequence is both a burden and a blessing: the channel has one more degree of freedom to estimate, but that degree of freedom has a sparse representation on a well-chosen polar grid. Section 18.4 turns this sparsity into an estimator.

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Definition:
Near-Field Steering Vector

Consider a ULA of $N_t$ antennas with inter-element spacing $d = \lambda/2$ centered at the origin, so the position of antenna $n$ is $x_n = (n - (N_t-1)/2)\,d$ . A point source at $(r, \theta)$ radiates to antenna $n$ a wave with amplitude $\propto 1/r_n$ and phase $-k_0 r_n$ , where $k_0 = 2\pi / \lambda$ and $r_n = \sqrt{r^2 + x_n^2 - 2 r x_n \sin\theta}.$

Expanding $r_n$ to second order in $x_n/r$ : $r_n \approx r - x_n \sin\theta + \frac{x_n^2 \cos^2\theta}{2 r}.$

The far-field approximation keeps only the linear term; the near-field steering vector keeps the quadratic too: $\mathbf{a}(\theta, r) = \frac{1}{\sqrt{N_t}} \left[\, e^{-j k_0 (r_0 - r)} ,\, e^{-j k_0 (r_1 - r)} ,\, \ldots,\, e^{-j k_0 (r_{N_t-1} - r)} \right]^T.$ When $r \to \infty$ , the quadratic term vanishes and $\mathbf{a}(\theta, r)$ reduces to the familiar far-field steering vector $\mathbf{a}(\theta)$ .

The polar parameterization $(r, \theta)$ is more convenient than Cartesian $(x,y)$ because the channel's sparsity in $\{(r_q, \theta_q)\}$ reflects the sparsity in scatterer locations, which is the physically meaningful notion.

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Definition:
Fraunhofer Distance and the Near-Field Regime

For an antenna aperture of size $D$ and wavelength $\lambda$ , the Fraunhofer distance is $d_F = \frac{2 D^2}{\lambda}.$ A source at range $r$ is said to be in the far field if $r \geq d_F$ and in the near field (Fresnel region) if $r < d_F$ but $r$ is larger than a few wavelengths. In the near field, the maximum phase error incurred by the far-field Taylor truncation exceeds $\pi/8$ rad and the spherical curvature of the wavefront becomes observable across the array.

For $\lambda = 8.6$ cm ( $f_c = 3.5$ GHz) and $D = 1$ m (a modest XL-MIMO panel), $d_F = 2 \cdot 1^2 / 0.086 \approx 23.3$ m. Any user within roughly 20 m of the array is in the near field. At sub-THz frequencies the distances are even larger in wavelengths: $D = 30$ cm at $f_c = 140$ GHz ( $\lambda = 2.1$ mm) gives $d_F \approx 86$ m.

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Polar-domain dictionary

A matrix $\mathbf{A}_{\text{polar}} \in \mathbb{C}^{N_t \times G}$ whose columns are near-field steering vectors $\mathbf{a}(\theta_q, r_q)$ at a grid of $G$ polar points. Used as a sparse-representation basis for near-field channel estimation: a physical channel with $L$ scatterers has at most $L$ non-zero coefficients when expressed on a fine-enough polar grid.

Theorem: Polar-Domain Sparsity of Near-Field Channels

A near-field multipath channel with $L$ scatterers at locations $\{(r_\ell, \theta_\ell)\}_{\ell=1}^L$ and complex gains $\{\alpha_\ell\}$ is $\mathbf{H}_{k} = \sum_{\ell=1}^L \alpha_\ell\, \mathbf{a}(\theta_\ell, r_\ell).$ If $\mathbf{A}_{\text{polar}}$ is a polar dictionary with grid spacing $(\Delta\theta, \Delta r)$ matched to the near-field resolution limits $(1/\sqrt{N_t}\,\text{rad},\; r^2/D)$ , there exists a sparse vector $\mathbf{z} \in \mathbb{C}^G$ with $\|\mathbf{z}\|_0 \leq L$ such that $\|\mathbf{H}_{k} - \mathbf{A}_{\text{polar}} \mathbf{z}\|_2 \leq \epsilon_{\text{grid}}$ where $\epsilon_{\text{grid}} \to 0$ as the polar grid is refined.

A far-field channel is sparse in angle; a near-field channel is sparse in (angle, range). The extra range dimension costs an additional polar grid axis, but the sparsity level $L$ — the number of scatterers — stays the same. Compressed-sensing guarantees therefore apply with roughly the same measurement budget as in far-field angular estimation, at the price of a larger dictionary.

Proof

Decomposition

Each $\mathbf{a}(\theta_\ell, r_\ell)$ lies in the polar dictionary column space, up to the grid mismatch. Round each $(r_\ell, \theta_\ell)$ to the nearest grid point $q(\ell)$ and let $\mathbf{z}_{q(\ell)} = \alpha_\ell$ , all other entries zero.

Bound the grid error

The perturbation from rounding is bounded by a first-order Taylor expansion of the steering vector with respect to $(\theta, r)$ . The angular gradient has norm $\leq \pi D/\lambda$ and the range gradient has norm $\leq \pi D^2 / (\lambda\, r^2)$ . Matching the grid spacings to these sensitivities makes the per-scatterer rounding error bounded and summable.

Sparsity count

At most $L$ of the $\mathbf{z}$ entries are non-zero, proving the sparsity claim. The $\ell_2$ tail $\epsilon_{\text{grid}}$ vanishes in the limit of infinite grid density (one grid point per physical scatterer). $\blacksquare$

Polar-Domain Sparsity of a Near-Field Channel

Generate a near-field channel with $L$ scatterers and display its energy distribution on (i) the far-field angular DFT basis and (ii) the polar (angle, range) dictionary. The far-field DFT spreads each near-field scatterer over many bins because the curvature is not captured; the polar dictionary concentrates each scatterer into a single cell.

Parameters

N_t

128

Scatterers

L

3

Min range (m)5

Max range (m)40

Carrier

f_c

(GHz)28

Polar-OMP for Near-Field Channel Estimation

Complexity:

\mathcal{O}(L \cdot G \cdot N_t \cdot \tau_p)

— dominated by the correlation step 3.

Input: Pilot observation

\mathbf{y}_p = \mathbf{S}_{i,k}^{H} \mathbf{H}_{k} + \mathbf{w}

with

\mathbf{S}_{i,k} \in \mathbb{C}^{N_t \times \tau_p}

(effective measurement

matrix

\boldsymbol{\Phi} = \mathbf{S}_{i,k}^{H}

), polar dictionary

\mathbf{A}_{\text{polar}} \in \mathbb{C}^{N_t \times G}

, target sparsity

L

.

Output: Near-field channel estimate

\hat{\mathbf{H}}_k = \mathbf{A}_{\text{polar}} \hat{\mathbf{z}}

.

1. Initialize residual

\mathbf{r} \leftarrow \mathbf{y}_p

, support

\mathcal{I} \leftarrow \emptyset

.

2. for

\ell = 1, \ldots, L

do

3.

\quad

g_q \leftarrow |(\boldsymbol{\Phi} \mathbf{A}_{\text{polar}})[:, q]^H \mathbf{r}|

for

q = 1, \ldots, G

4.

\quad

q^* \leftarrow \arg\max_q g_q

5.

\quad

\mathcal{I} \leftarrow \mathcal{I} \cup \{q^*\}

6.

\quad

Solve least squares on the active support:

\hat{\mathbf{z}}_\mathcal{I} \leftarrow (\boldsymbol{\Phi} \mathbf{A}_{\text{polar},\mathcal{I}})^\dagger \mathbf{y}_p

7.

\quad

\mathbf{r} \leftarrow \mathbf{y}_p - \boldsymbol{\Phi} \mathbf{A}_{\text{polar},\mathcal{I}} \hat{\mathbf{z}}_\mathcal{I}

8. end for

9.

\hat{\mathbf{z}}

on

\mathcal{I}

from step 6, zero elsewhere.

10. return

\hat{\mathbf{H}}_k \leftarrow \mathbf{A}_{\text{polar}} \hat{\mathbf{z}}

.

Polar-OMP is the simplest dictionary-based estimator; more sophisticated variants use weighted $\ell_1$ regularization or simultaneous orthogonal matching pursuit (SOMP) when multiple pilot snapshots share the same scatterer geometry.

Example: Sparsity Level vs Measurement Budget

A mmWave XL-MIMO uplink uses $N_t = 256$ antennas at $f_c = 28$ GHz ( $\lambda \approx 1.07$ cm), aperture $D = 1.28$ m. A user is at range $r = 15$ m, in the near field ( $d_F = 306$ m). The physical channel has $L = 4$ dominant scatterers. (a) How many polar grid points $G$ does a grid spacing of $\Delta\theta = 2/\sqrt{N_t}$ rad and $\Delta r = r^2/D$ need to cover an azimuth range $[-60^\circ, 60^\circ]$ and a radial range $[5, 30]$ m? (b) What is the minimum number of pilot samples $\tau_p$ required by the compressed-sensing guarantee $\tau_p \geq 2 L \log G$ ?

Solution

Angular grid size

$\Delta\theta = 2/\sqrt{256} = 0.125$ rad. Range $[-60^\circ, 60^\circ] = [-\pi/3, \pi/3]$ rad has total width $2\pi/3 \approx 2.09$ rad. Number of angular bins $G_\theta = \lceil 2.09 / 0.125 \rceil = 17$ .

Range grid size

$\Delta r = r^2 / D = 15^2 / 1.28 \approx 176$ m — larger than the whole range window! This tells us that at $r = 15$ m the range resolution is worse than the entire operating interval, so a single range bin suffices: $G_r = \lceil 25 / 176 \rceil = 1$ . Recompute at $r = 5$ m: $\Delta r = 5^2/1.28 \approx 19.5$ m, still coarse. The polar grid is logarithmically spaced in $r$ , and for this geometry $G_r \approx 3$ is sufficient.

Total grid

$G = G_\theta \cdot G_r = 17 \cdot 3 = 51$ grid points.

Pilot budget

$\tau_p \geq 2 L \log_2 G = 2 \cdot 4 \cdot \log_2(51) \approx 2 \cdot 4 \cdot 5.67 \approx 45$ pilot symbols. Compare with a full DFT-based far-field estimator which needs $\tau_p \geq N_t = 256$ . The sparse polar estimator saves $\sim 5.7$ x in pilot overhead. $\blacksquare$

Range Resolution Is a Function of $r^2/D$

Unlike angular resolution which scales as $1/D$ , near-field range resolution scales as $r^2/D$ — it worsens quadratically with distance and improves only linearly with aperture. This is why the polar grid is coarse in $r$ at large ranges: there is simply not enough phase curvature across the array to distinguish $r = 100$ m from $r = 110$ m. Conversely, at short ranges near the array the range axis becomes finely resolvable, which is where near-field beam focusing (Chapter 17) earns its keep.

Common Mistake: Do Not Use the Far-Field DFT Dictionary in the Near Field

Mistake:

Reuse the same far-field DFT dictionary and just wait for the estimator to "figure out" the near-field structure.

Correction:

A single near-field scatterer, when expanded on the angular DFT basis, produces a spreading pattern across many DFT bins whose width scales as $D^2/(\lambda\, r)$ . A sparsity-based estimator looking for isolated spikes in the DFT basis will either miss the true scatterer (low correlation with any single bin) or declare many false scatterers (one per spread bin). The polar dictionary fixes this by accounting for the curvature in each column. In code terms: the dictionary is the only thing that changes — the OMP / LASSO engine is the same.

⚠️Engineering Note

Designing the Polar Grid in Practice

Designing the polar grid is the main implementation choice of a near-field sparse estimator. Three rules of thumb:

Logarithmic range sampling. Because range resolution scales as $r^2/D$ , sample $r$ logarithmically: $r_q \in \{r_{\min}, \alpha r_{\min}, \alpha^2 r_{\min}, \ldots\}$ with $\alpha \in [1.2, 1.5]$ . A typical sub-6 GHz panel needs 6-12 range bins between $r_{\min} = 2$ m and $r_{\max} = d_F$ .
Uniform angular sampling. Sample the angle uniformly with spacing $\Delta\theta = 2/(N_1-1)$ for a $N_1$ -element horizontal axis. This matches the far-field angular resolution and keeps the dictionary coherence bounded.
Merge the far-field boundary. For $r > d_F$ all range bins degenerate to a single "far-field" column — the polar dictionary should smoothly merge into the far-field DFT as $r \to d_F$ .

The final grid size is $G \approx N_1 \cdot \lceil \log_\alpha(d_F / r_{\min}) \rceil$ , typically $G = 5$ – $10 N_t$ . That sounds expensive but OMP only touches the grid once per iteration, so runtime is dominated by the $L \cdot G$ inner product and stays well below the subarray MMSE budget of Section 18.3.

Practical Constraints

•
Logarithmic range factor $\alpha \in [1.2, 1.5]$
•
Grid size $G \in [5N_t, 10N_t]$
•
Runtime per user: $< 0.1$ ms per scatterer on commodity CPUs

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Near-Field Channel Estimation