Ferkans — Interactive Telecom Tutor

What We Need to Replace

In the far field, the channel from a single LoS source at angle $\theta$ to a uniform linear array of $N_t$ elements is captured by the classical plane-wave steering vector $\mathbf{a}(\theta) = [1,\, e^{-j\kappa d\sin\theta},\,\ldots]^T$ . Everything downstream — MRT, ZF, RZF, codebook design, DoA estimation — is written in terms of such vectors. We now replace that formula with the exact spherical-wave response and look at the structure of the resulting array response vector. The upshot is that the single parameter $\theta$ is replaced by a pair of parameters (angle and range), and the Fourier structure of the far-field case becomes a chirp-Fourier structure in the near field.

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Definition:
Near-Field (Spherical-Wavefront) Array Response Vector

Consider a base-station array with $N_t$ elements at known 3-D positions $\mathbf{p}_n \in \mathbb{R}^3$ , $n = 1, \ldots, N_t$ , and a single point source (a user, a scatterer, or a RIS element) at position $\mathbf{p}_k$ . The exact path length from the source to the $n$ -th element is $r_{k,n} \triangleq \|\mathbf{p}_k - \mathbf{p}_n\|.$ Ignoring element patterns, polarisation and atmospheric loss, the free-space LoS channel coefficient is $h_{k,n} = \frac{\lambda}{4\pi\, r_{k,n}}\,e^{-j\kappa r_{k,n}},$ with $\kappa = 2\pi/\lambda$ . When the amplitude variation across the aperture is small (which holds whenever the user is outside the reactive near field), one typically absorbs the common $\lambda/(4\pi r_k)$ term (using any reference distance $r_k$ , e.g. the array centre) into a scalar path-loss $\sqrt{\beta_{k}}$ and writes the near-field array response vector as $\mathbf{a}_{\text{NF}}(\mathbf{p}_k) = \frac{1}{\sqrt{N_t}} \begin{bmatrix} e^{-j\kappa r_{k,1}} \\ e^{-j\kappa r_{k,2}} \\ \vdots \\ e^{-j\kappa r_{k,N_t}} \end{bmatrix}.$ The LoS near-field channel vector is then $\mathbf{h}_k = \sqrt{\beta_{k}}\,\mathbf{a}_{\text{NF}}(\mathbf{p}_k)$ .

Contrast this with the far-field steering vector. In the far field, $r_{k,n} \approx r_k - \mathbf{a}^{T} \mathbf{d}_n$ (a linear function of the element position $\mathbf{d}_n$ ), so the array response factorises into a common phase times a linear-phase steering vector. In the near field, the $r_{k,n}$ are genuine nonlinear functions of $\mathbf{p}_k$ , and the array response depends on the full position, not only the direction.

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Theorem: Quadratic-Phase Form of the Fresnel-Region Steering Vector

Let a ULA along the $y$ -axis have elements at $\mathbf{p}_n = (0, y_n, 0)$ with $y_n = (n - (N_t-1)/2)\,d$ , $d = \lambda/2$ , and let a source be at polar coordinates $(r_k, \theta_k)$ in the $xy$ -plane: $\mathbf{p}_k = (r_k\cos\theta_k,\, r_k\sin\theta_k, 0)$ . Assume $r_k \gg d$ but not necessarily $r_k \geq d_F$ . Then the per-element path length admits the quadratic Fresnel expansion $r_{k,n} = r_k - y_n\sin\theta_k + \frac{y_n^2\cos^2\theta_k}{2 r_k} + \mathcal{O}\!\left(\frac{y_n^3}{r_k^2}\right).$ In the Fresnel (radiating near-field) region, truncating at the quadratic term gives $\mathbf{a}_{\text{NF}}(r_k,\theta_k)_n \;\approx\; \frac{1}{\sqrt{N_t}}\, e^{-j\kappa r_k}\, e^{+j\kappa y_n\sin\theta_k}\, e^{-j\kappa\,y_n^2\cos^2\theta_k/(2 r_k)}.$ The first phase factor is a common range phase, the second is the classical far-field linear steering, and the third is the Fresnel quadratic-phase correction.

The linear phase $e^{+j\kappa y_n\sin\theta_k}$ is what the far-field model keeps — it encodes the direction $\theta_k$ alone. The quadratic phase $e^{-j\kappa y_n^2\cos^2\theta_k/(2 r_k)}$ encodes the range $r_k$ . In the far field, $r_k \to \infty$ and the quadratic term vanishes; at finite $r_k$ , it contributes a chirp across the array — and it is precisely this chirp that allows a near-field beamformer to focus to a point rather than to a direction.

Show Hint

Start from the identity $r_{k,n} = \|\mathbf{p}_k - \mathbf{p}_n\|$ and expand the square.

Use $\sqrt{1 + x} \approx 1 + x/2 - x^2/8$ on the dimensionless quantity $(-2 y_n\sin\theta_k + y_n^2)/r_k^2$ .

Drop terms beyond second order in $y_n/r_k$ .

Proof

Expand the square of the distance

$r_{k,n}^2 = r_k^2 - 2 r_k y_n\sin\theta_k + y_n^2 = r_k^2\left(1 + \frac{-2 y_n\sin\theta_k + y_n^2/r_k}{r_k}\right).$ $Set$ \epsilon_n = (-2 y_n\sin\theta_k + y_n^2/r_k)/r_k $. Then$ r_{k,n} = r_k\sqrt{1 + \epsilon_n}$.

Apply the square-root Taylor expansion

Using $\sqrt{1 + \epsilon} \approx 1 + \epsilon/2 - \epsilon^2/8$ and dropping cubic and higher orders in $y_n/r_k$ , $r_{k,n} \approx r_k + \frac{r_k\epsilon_n}{2} - \frac{r_k\epsilon_n^2}{8}.$ The leading correction is $r_k\epsilon_n/2 = -y_n\sin\theta_k + y_n^2/(2 r_k)$ , and the $\epsilon_n^2$ term contributes $-(y_n\sin\theta_k)^2/(2 r_k) + \mathcal{O}(y_n^3/r_k^2)$ . Collecting, $r_{k,n} \approx r_k - y_n\sin\theta_k + \frac{y_n^2\cos^2\theta_k}{2 r_k}.$

Turn path length into phase

Plugging into $e^{-j\kappa r_{k,n}}$ and extracting the common $e^{-j\kappa r_k}$ phase, $e^{-j\kappa r_{k,n}} \approx e^{-j\kappa r_k}\cdot e^{+j\kappa y_n\sin\theta_k} \cdot e^{-j\kappa y_n^2\cos^2\theta_k/(2 r_k)}.$ The first factor is absorbed in the reference channel phase; the second is the ordinary far-field steering vector evaluated at angle $\theta_k$ ; the third is the Fresnel quadratic phase. $\blacksquare$

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The Near-Field Basis Is Chirp-Fourier

Because of the quadratic phase $e^{-j\kappa y_n^2\cos^2\theta_k/(2 r_k)}$ , the set of near-field array responses $\{\mathbf{a}_{\text{NF}}(r_k,\theta_k)\}$ is no longer a DFT/Fourier basis. It is a chirp basis — the same structure one meets in radar and in linear FM ranging. This has three immediate consequences:

Codebook design is no longer a plane wave at every angle; it must also sample ranges. The near-field codebook is a 2-D grid in $(\theta, r)$ rather than a 1-D grid in $\theta$ .
Channel estimation cannot rely on simple FFT-domain sparsity; chirp-matched dictionaries (à la LFM radar) recover the point sources correctly (see Cui & Dai, 2022).
Orthogonality between two channel vectors requires that they differ in either angle or range (or both); two users at the same angle but different ranges are now separable by the array, which is impossible in the far field.

Near-Field vs Far-Field Beamformer Pattern

A ULA forms two matched filters toward the same target: one that uses the exact spherical-wave delays (near-field matched), and one that uses the far-field plane-wave steering vector at the same angle (far-field matched). The plot shows the normalised gain each beamformer delivers when a scatterer is swept in angle at a fixed range $d$ . When the target range is inside $d_F$ , the far-field beamformer is not matched to any scatterer — its response is misaligned and leaks energy into the sidelobes.

Parameters

N_t

64

Carrier

f_0

(GHz)28

Target range

d

(m)5

Focus angle (deg)0

Example: How Large Is the Plane-Wave Phase Error?

A ULA has $N_t = 128$ elements spaced by $\lambda/2$ at $f_0 = 28$ GHz, steering broadside. A user is at range $d = 5$ m. Compute the worst-case quadratic-phase error across the array (in radians) that the far-field steering vector ignores. Is this error tolerable?

Solution

Compute aperture and $d_F$

$\lambda = c/f_0 = 3\times 10^8 / 2.8\times 10^{10} \approx 1.07$ cm. Aperture: $D = (N_t-1)\lambda/2 = 127 \cdot 5.36$ mm $\approx 0.68$ m. $d_F = 2\cdot 0.68^2 / 0.0107 \approx 86.6$ m. The user at $d = 5$ m is well inside $d_F$ — $d/d_F \approx 0.058$ .

Quadratic phase at the array edge

At the edge $y_n = D/2 = 0.34$ m, broadside ( $\theta = 0$ , so $\cos^2\theta = 1$ ): $\Delta\phi_{\max} = \kappa\frac{(D/2)^2}{2 d} = \frac{2\pi}{\lambda}\cdot\frac{(D/2)^2}{2 d} = \frac{2\pi}{0.0107}\cdot\frac{0.34^2}{10} \approx 6.8\ \text{rad}.$

Interpretation

Six radians is more than a full wavelength of phase error across the aperture edge — a catastrophic mismatch for any matched-filter beamformer. The far-field model would insist that the entire array sees the same plane wave; in reality, the edge elements are phase- shifted by more than $2\pi$ relative to the centre. A plane-wave matched filter targeting this user loses essentially all coherent gain. The near-field array response is mandatory here.

Common Mistake: Angle-Only Codebooks Are Inadequate in the Near Field

Mistake:

A common shortcut is to reuse the DFT codebook from 5G NR (which indexes beams by angle only) and apply it verbatim in XL-MIMO or near-field deployments. "We already have a codebook — the hardware just needs to be bigger."

Correction:

A DFT codebook samples directions but not ranges. In the near field, two users at the same angle but different ranges have nearly orthogonal channel vectors — an angle-only codebook picks a single beam for both and enjoys no spatial separation. The correct near-field codebook is a two-dimensional grid in $(\theta, r)$ , typically called a "polar-domain codebook" (Cui, Dai et al. 2022). The effective size grows from $\mathcal{O}(N_t)$ angular bins to $\mathcal{O}(N_t)$ angular bins $\times$ $\mathcal{O}(\sqrt{N_t})$ range bins, i.e. $\mathcal{O}(N_t^{3/2})$ codewords.

Near-Field Array Response Vector

The unit-norm vector $\mathbf{a}_{\text{NF}}(\mathbf{p}_k)_n = e^{-j\kappa r_{k,n}}/\sqrt{N_t}$ whose $n$ -th entry encodes the exact spherical-wave phase from a source at $\mathbf{p}_k$ to the $n$ -th array element. In the Fresnel region it admits a linear-plus-quadratic phase expansion and depends on both angle and range, unlike the plane-wave steering vector, which depends on angle alone.

Chirp (Fresnel) Basis

The collection of near-field array responses $\{\mathbf{a}_{\text{NF}}(r_k,\theta_k)\}$ parameterised by angle and range. Unlike the DFT basis of the far field, it is overcomplete and range-dependent. Recovery of a small number of sources from near-field measurements is a sparse-chirp problem, closely related to the LFM (linear-FM) ranging formulation used in pulse-compression radar.

🎓CommIT Contribution(2023)

Spatial Non-Stationarity Priors for XL-MIMO Channel Estimation

Y. Xu, G. Caire — IEEE Transactions on Wireless Communications (preprint)

The spherical-wave channel model of this section is the starting point for Xu and Caire's XL-MIMO channel estimation framework, which we treat in detail in Chapter 18. Their contribution is a 2-D Markov random field prior over per-antenna visibility, plus a chirp-dictionary model for the near-field LoS component, which together allow a tractable Bayesian estimator whose complexity is $\mathcal{O}(N_t\logN_t)$ rather than $\mathcal{O}(N_t^{2})$ . The spherical-wavefront array response vector defined here is the atom in their dictionary — an example of how near-field physics and sparse-recovery algorithms meet.

xl-mimochannel-estimationnear-fieldsparse-recoveryView Paper →

Key Takeaway

The near-field array response is parameterised by position, not direction. The exact phase $e^{-j\kappa \|\mathbf{p}_k - \mathbf{p}_n\|}$ admits a Fresnel expansion in which the linear-phase term recovers the classical plane-wave steering vector and the quadratic-phase term is the extra information that locks the beam to a range. Everything that follows — beam focusing, depth of focus, near-field DoF — is a consequence of this one quadratic term.

The Spherical Wavefront Model