Ferkans — Interactive Telecom Tutor

The Sub-THz Frontier

Beyond the mmWave bands explored in 5G, frequencies from 100 to 300 GHz — the sub-terahertz (sub-THz) range — promise unprecedented bandwidths of 10–50 GHz per carrier. At these frequencies, a single link can theoretically achieve data rates exceeding 100 Gbps, enabling applications such as holographic communications, wireless backhaul/fronthaul, data centre interconnects, and sensing-communication convergence.

However, sub-THz communications face formidable challenges:

Atmospheric absorption: Molecular resonances of O $_2$ and H $_2$ O create strong absorption peaks that limit range.
Extreme path loss: FSPL at 140 GHz is 14 dB higher than at 28 GHz; at 300 GHz it is 21 dB higher.
Near-field operation: Large arrays at short wavelengths push the Fraunhofer distance to tens of metres, meaning many communication links operate in the radiative near field.
Hardware limitations: Power amplifier efficiency drops below 5% above 100 GHz, phase noise increases as $f^2$ , and mixer conversion loss is severe.

This section develops the key physical models for sub-THz communications and examines the near-field MIMO paradigm that emerges at these frequencies.

Atmospheric Absorption Spectrum

The atmosphere is not transparent at sub-THz frequencies. Molecular resonances create frequency-dependent absorption:

Oxygen (O $_2$ ) absorption:

Broad complex of lines centred at 60 GHz (the "oxygen absorption band"), causing $\sim$ 15 dB/km peak attenuation.
Isolated line at 119 GHz with $\sim$ 1.5 dB/km.

Water vapour (H $_2$ O) absorption:

Strong line at 22 GHz: $\sim$ 0.2 dB/km.
Very strong line at 183 GHz: $\sim$ 30 dB/km at sea level, 50% relative humidity.
Additional lines at 325 GHz and above.

Between these resonances lie transmission windows where atmospheric loss is manageable:

Window	Centre (GHz)	Bandwidth	Loss (dB/km) at 50% RH
W-band	75–110	35 GHz	0.5–1.5
D-band	130–175	45 GHz	0.5–3.0
G-band upper	200–240	40 GHz	2–10
250–310 GHz	275	~60 GHz	5–20

The total path loss including atmospheric absorption is:

$PL_{\text{total}}(f, d) = PL^{CI}(f, d) + \alpha_{\text{atm}}(f) \cdot d \times 10^{-3}$

where $\alpha_{\text{atm}}(f)$ is the specific attenuation in dB/km and $d$ is in metres. For links below 100 m (typical indoor or short-range outdoor), atmospheric absorption adds less than 0.3 dB — negligible compared to spreading loss. For backhaul links at 200–500 m, it becomes significant and dictates the choice of frequency window.

Atmospheric Absorption Spectrum Animation

Watch a frequency sweep from 1 to 400 GHz revealing the O

_2

and H

_2

O absorption peaks and the transmission windows between them. The W-band (75-110 GHz), D-band (130-175 GHz), and G-band (200-240 GHz) windows are highlighted as viable frequency ranges for sub-THz communications.

Atmospheric specific attenuation (dB/km) at 50% relative humidity, sea level. Absorption peaks at 60, 119, 183, and 325 GHz constrain sub-THz system design to transmission windows between the peaks.

Atmospheric Absorption Spectrum

Visualise the atmospheric specific attenuation $\alpha_\text{atm}(f)$ in dB/km as a function of frequency from 1–400 GHz. Adjust the relative humidity to see how water vapour absorption intensifies at higher moisture levels. The lower panel shows the total path loss (spreading + absorption) for a given link distance, revealing the optimal transmission windows.

Parameters

Humidity (%)50

Distance (km)1

Definition:
Fraunhofer Distance and Near-Field Transition

The Fraunhofer distance (far-field distance) is the distance beyond which the spherical wavefront arriving at an aperture is well approximated by a plane wave. For an antenna array of physical aperture $D$ , the Fraunhofer distance is:

$d_F = \frac{2D^2}{\lambda}$

For a uniform linear array (ULA) with $N$ elements spaced at $d_s = \lambda/2$ , the aperture is $D = (N-1)\lambda/2 \approx N\lambda/2$ for large $N$ , giving:

$d_F = \frac{2(N\lambda/2)^2}{\lambda} = \frac{N^2\lambda}{2}$

At sub-6 GHz with $N = 64$ and $f = 3$ GHz ( $\lambda = 0.1$ m):

$d_F = \frac{64^2 \times 0.1}{2} = 204.8\;\text{m}$

At 140 GHz with $N = 256$ and $\lambda = 2.14$ mm:

$d_F = \frac{256^2 \times 0.00214}{2} = 70.1\;\text{m}$

At 300 GHz with $N = 1024$ and $\lambda = 1.0$ mm:

$d_F = \frac{1024^2 \times 0.001}{2} = 524.3\;\text{m}$

For sub-THz systems with large arrays, $d_F$ can reach hundreds of metres — meaning that most practical communication links operate in the near field. This fundamentally changes the MIMO signal model, beamforming design, and spatial multiplexing capabilities.

Theorem: Near-Field Spherical-Wave Channel Model

In the near field ( $d < d_F$ ), the plane-wave assumption breaks down and the channel between antenna element $m$ at position $\mathbf{p}_m^t$ and element $n$ at position $\mathbf{p}_n^r$ must be modelled using the exact spherical-wave propagation:

$[\mathbf{H}]_{n,m} = \sum_{\ell=1}^{L} \frac{\alpha_\ell}{\|\mathbf{p}_n^r - \mathbf{s}_\ell\| \cdot \|\mathbf{s}_\ell - \mathbf{p}_m^t\|}\, e^{-j\frac{2\pi}{\lambda}\left(\|\mathbf{p}_n^r - \mathbf{s}_\ell\| + \|\mathbf{s}_\ell - \mathbf{p}_m^t\|\right)}$

where $\mathbf{s}_\ell$ is the position of the $\ell$ -th scatterer. For a single LOS path, this simplifies to:

$[\mathbf{H}_{\text{LOS}}]_{n,m} = \frac{\alpha_0}{d_{n,m}}\, e^{-j\frac{2\pi}{\lambda} d_{n,m}}$

where $d_{n,m} = \|\mathbf{p}_n^r - \mathbf{p}_m^t\|$ is the exact distance between the $m$ -th transmit and $n$ -th receive element.

The key consequence is that the channel matrix $\mathbf{H}$ has higher rank in the near field than in the far field, because different antenna elements see the scatterer (or the other array) at genuinely different angles and distances. This enables spatial multiplexing without scattering — even in a pure LOS environment.

In the far field, all array elements see essentially the same angle to a scatterer, so the channel is rank-1 per path. In the near field, the wavefront curvature means that elements at different positions on the array see different angles and path lengths, creating natural spatial diversity. A large near-field array can "focus" energy at a specific 3D point (not just a direction), enabling spatial multiplexing even to a single-antenna user at different distances.

Proof

Rank Analysis of the LOS Near-Field Channel

Consider a Tx ULA with $N_t$ elements and an Rx ULA with $N_r$ elements, both with half-wavelength spacing, separated by distance $d_0$ . The $(n,m)$ entry of the LOS channel is $h_{n,m} \propto e^{-j2\pi d_{n,m}/\lambda}$ .

Expanding $d_{n,m}$ to second order around the array centres:

$d_{n,m} \approx d_0 + \frac{(p_n^r - p_m^t)^2}{2d_0} - \frac{(p_n^r - p_m^t)\sin\theta}{1} + \cdots$

The quadratic phase term $e^{-j\pi(p_n^r - p_m^t)^2/(\lambda d_0)}$ creates a Fresnel-type phase structure across the channel matrix that is not captured by the far-field (linear phase only) model.

This quadratic phase variation across antenna pairs makes rows and columns of $\mathbf{H}$ linearly independent, increasing the matrix rank from $\min(L, N_s)$ in the far field to $\min(N_t, N_r, N_t N_r \lambda / (4 d_0))$ in the near field. For sufficiently large arrays and short distances, the LOS channel alone can support multiple spatial streams. $\blacksquare$

,

Beam Focusing vs. Beam Steering

In the far field, beamforming steers a beam toward a direction $(\phi, \theta)$ using a linear phase gradient across the array. The beam illuminates all points along the direction, regardless of distance — the array has no range resolution.

In the near field, beamforming can focus energy at a specific 3D point $\mathbf{p} = (x_0, y_0, z_0)$ by applying a quadratic (spherical) phase profile across the array elements:

$w_m = \frac{1}{\sqrt{N}}\, e^{j\frac{2\pi}{\lambda}\|\mathbf{p}_m - \mathbf{p}\|}$

This creates a focal point with energy concentrated both angularly and in range. The depth of focus (range resolution) is approximately:

$\Delta r \approx \frac{8\lambda}{\pi} \left(\frac{d_0}{D}\right)^2$

where $d_0$ is the focal distance and $D$ is the array aperture.

Implications for multi-user MIMO:

Users at the same angle but different distances can be spatially separated through beam focusing (impossible in the far field)
The effective number of spatial degrees of freedom increases
Distance-aware precoding is required: the precoder must know both the angle and distance to each user, not just the angle

Distance-aware precoding modifies the standard far-field beamforming vector by incorporating the spherical phase correction:

$\mathbf{f}_\text{NF}(\phi, r) = \frac{1}{\sqrt{N}} \left[e^{j\frac{2\pi}{\lambda}(\|\mathbf{p}_1 - \mathbf{p}\| - r)}, \ldots, e^{j\frac{2\pi}{\lambda}(\|\mathbf{p}_N - \mathbf{p}\| - r)}\right]^T$

where $r = \|\bar{\mathbf{p}} - \mathbf{p}\|$ is the distance from the array centre $\bar{\mathbf{p}}$ to the focal point $\mathbf{p}$ .

Sub-THz Hardware Limitations and State of the Art

Sub-THz hardware is in a rapid but still early state of development. Key limitations include:

Power amplifier (PA) efficiency: CMOS PAs at 140 GHz achieve saturated output power of 5–15 dBm with power-added efficiency (PAE) of 3–8%. InP and SiGe BiCMOS technologies reach 15–20 dBm at $\sim$ 10% PAE. This is far below the 30–40% PAE typical at sub-6 GHz, requiring either many more PA elements (with complex power combining) or acceptance of shorter link ranges.

Phase noise: Local oscillator (LO) phase noise scales approximately as $20\log_{10}(f/f_\text{ref})$ when frequency-multiplied from a lower reference. At 140 GHz, a 10 GHz reference oscillator with $-$ 110 dBc/Hz at 100 kHz offset yields:

$\mathcal{L}(140\,\text{GHz}) \approx -110 + 20\log_{10}(14) = -110 + 23 = -87\;\text{dBc/Hz}$

This level of phase noise significantly degrades high-order constellations (64-QAM and above), requiring either phase-noise compensation algorithms or accepting lower spectral efficiency.

ADC/DAC: Wideband ADCs for 10+ GHz bandwidth require sampling rates of $\geq$ 20 GS/s. State-of-the-art ADCs at this speed achieve 5–6 effective number of bits (ENOB), compared to 10–12 ENOB at lower bandwidths. This motivates research on low-resolution (1–4 bit) ADC architectures and hybrid analog-digital signal processing.

Antenna arrays: The small wavelength ( $\lambda \approx 1$ – $3$ mm) enables massive arrays in compact form factors — a 1024-element array at 140 GHz fits in $\sim$ 5 cm $\times$ 5 cm. However, feed network losses, mutual coupling, and thermal management become critical challenges.

Example: Near-Field Spatial Degrees of Freedom

A sub-THz base station at 140 GHz has a $32 \times 32$ UPA ( $N = 1024$ elements) with half-wavelength spacing ( $d_s = \lambda/2 = 1.07$ mm).

(a) Compute the Fraunhofer distance.

(b) For a single-user at $d = 5$ m (near field), estimate the number of spatial degrees of freedom (DoF) available in the LOS channel.

(c) Compare with the far-field case at $d = 100$ m.

Solution

Array aperture and Fraunhofer distance

Physical aperture: $D = 32 \times 1.07\;\text{mm} = 34.2\;\text{mm} = 0.0342$ m.

$d_F = \frac{2D^2}{\lambda} = \frac{2 \times 0.0342^2}{0.00214} = \frac{2.34 \times 10^{-3}}{2.14 \times 10^{-3}} = 1.09\;\text{m}$

Wait — let us recompute more carefully. With $N = 1024$ in a $32 \times 32$ UPA, the aperture along one dimension is $D_1 = 31 \times 1.07\;\text{mm} = 33.2\;\text{mm}$ . The diagonal aperture is $D = \sqrt{2} \times 33.2 = 46.9\;\text{mm}$ .

$d_F = \frac{2 \times (0.0469)^2}{0.00214} = \frac{4.40 \times 10^{-3}}{2.14 \times 10^{-3}} = 2.06\;\text{m}$

So $d = 5$ m is in the far field for this relatively compact array. For near-field operation, we need a larger array. Consider instead a $128 \times 128$ UPA ( $N = 16384$ , $D = 127 \times 1.07 = 136\;\text{mm}$ ):

$d_F = \frac{2 \times 0.136^2}{0.00214} = \frac{0.0370}{0.00214} = 17.3\;\text{m}$

Now $d = 5$ m is well within the near field.

Near-field spatial DoF (d = 5 m)

For a 2D planar array of aperture $D \times D$ communicating with a point at distance $d_0$ in the near field, the number of LOS spatial degrees of freedom is approximately (Miller, 2000; Pizzo et al., 2022):

$\text{DoF}_\text{NF} \approx \frac{D^4}{4\lambda^{2} d_0^2}$

for a single-antenna receiver, or more precisely, the number of resolvable focal spots within the receiver's extent. For a single-antenna UE, this gives the number of independent spatial channels the Tx array can create at different focal points:

$\text{DoF}_\text{NF} \approx \min\!\left(N_t,\; \frac{D^2}{4\lambda d_0}\right) = \min\!\left(16384,\; \frac{0.136^2}{4 \times 0.00214 \times 5}\right) = \min(16384,\; 432)$

So the near-field array can resolve approximately 432 independent spatial modes at 5 m distance, even in pure LOS.

Far-field comparison (d = 100 m)

At $d = 100$ m $\gg d_F = 17.3$ m, the channel is in the far field. A pure LOS far-field channel with a single-antenna UE has:

$\text{DoF}_\text{FF} = 1$

The LOS channel is rank-1: the array can beamform (array gain) but cannot spatially multiplex. To achieve $\text{DoF} > 1$ in the far field requires scattering (NLOS multipath).

This dramatic difference — from 1 DoF in the far field to potentially hundreds in the near field — is the key motivation for near-field MIMO at sub-THz frequencies. $\blacksquare$

Near-Field Beam Focusing Animation

Compare far-field beam steering (parallel rays, direction-only control) with near-field beam focusing (converging rays, energy concentrated at a 3D point). The animation shows how the quadratic phase profile creates a focal spot with finite depth of focus

\Delta r

.

Left: far-field beam steering sends a plane wave in a fixed direction. Right: near-field beam focusing concentrates energy at a specific point

(r, \phi)

, enabling distance-based user separation.

Fraunhofer Distance and Near-Field Beam Patterns

Explore how the Fraunhofer distance $d_F = 2D^2/\lambda$ scales with array size and frequency. The upper panel shows $d_F$ as a function of the number of antennas for several frequencies. The lower panel illustrates the beam pattern: in the far field, the beam has infinite depth of focus (plane-wave steering); in the near field, the beam focuses to a spot with finite range resolution.

Parameters

N

antennas256

f

(GHz)140

d/\lambda

0.5

Quick Check

A 256-element ULA at 300 GHz with half-wavelength spacing has a Fraunhofer distance of approximately:

0.33 m

3.3 m

33 m

330 m

Correction:

33 m

Correct. $\lambda = c/f = 3 \times 10^8 / 300 \times 10^9 = 1.0$ mm. $d_F = N^2\lambda/2 = 256^2 \times 10^{-3}/2 = 65536 \times 10^{-3}/2 = 32.8$ m $\approx 33$ m. Most indoor and many outdoor links at 300 GHz with 256 elements operate in the near field.

Key Takeaway

Near-field MIMO at sub-THz frequencies is a qualitative paradigm shift, not merely an incremental improvement. In the far field, arrays can only steer beams in direction — range resolution is zero. In the near field, arrays can focus energy at specific 3D points, enabling spatial multiplexing in pure LOS, distance-based user separation, and spatial degrees of freedom that scale as $D^2/(4\lambda d_0)$ rather than being limited to the number of scattering paths.

Why This Matters: Near-Field Arrays in RF Imaging

The near-field spherical-wave channel model developed here is the foundation of the RF imaging forward model (Book RFI, Chapters 3–5). In RF imaging, the transmit and receive arrays are often in the near field of the target region, and the sensing matrix $\mathbf{A}$ encodes the exact spherical-wave propagation between each antenna element and each voxel in the scene. The near-field spatial degrees of freedom directly determine the imaging resolution: more DoF means finer voxel resolution. The beam focusing concept is exploited in RF imaging as matched-filter beamforming (backpropagation), which focuses the received signals to each candidate voxel location.

Why This Matters: XL-MIMO and Near-Field in the MIMO Book

The near-field MIMO theory of this section extends extensively in the MIMO book (Book MIMO, Chapters 19–21). There, the spherical-wave model is developed for extra-large MIMO (XL-MIMO) arrays spanning metres of aperture, the concept of visibility regions (where only a subset of the array elements contribute to the channel) is formalised, and near-field codebook design for practical beam focusing is developed. The Xu/Caire (2023) 2D Markov prior for visibility region detection is a CommIT result that builds directly on this near-field foundation.

Common Mistake: Extrapolating mmWave Range Expectations to Sub-THz

Mistake:

Expecting sub-THz systems (140–300 GHz) to achieve similar outdoor coverage ranges as 28 GHz mmWave systems (100–200 m).

Correction:

Sub-THz systems face compounding challenges that severely limit range compared to 28 GHz:

FSPL increase: 14 dB higher at 140 GHz, 21 dB at 300 GHz
Atmospheric absorption: 0.5–3 dB/km in transmission windows, but 15–30 dB/km at absorption peaks
PA output power: 5–15 dBm at 140 GHz (vs 30 dBm at 28 GHz)

Even with 1024-element arrays (30 dBi gain), the practical outdoor range at 140 GHz is 20–50 m, and at 300 GHz it is 5–20 m. Sub-THz is best suited for indoor short-range (data centres, kiosks), backhaul (point-to-point with high-gain dishes), and sensing (high-resolution imaging) — not wide-area cellular coverage.

Sub-Terahertz (Sub-THz)

Radio frequencies in the range 100–300 GHz, offering tens of GHz of bandwidth but facing severe atmospheric absorption, extreme path loss, and hardware limitations. Targeted for 6G short-range high-capacity links.

Fraunhofer Distance

The distance $d_F = 2D^2/\lambda$ beyond which the radiated field is well approximated by a plane wave. For $d < d_F$ , the spherical wavefront structure must be accounted for (near-field regime).

Beam Focusing

Near-field beamforming that concentrates energy at a specific 3D point (range and angle) rather than just a direction. Achieved by applying a spherical (quadratic) phase profile across the array elements.

Sub-THz Communications (100-300 GHz)

The Sub-THz Frontier

Atmospheric Absorption Spectrum

Atmospheric Absorption Spectrum Animation

Atmospheric Absorption Spectrum

Parameters

Definition: Fraunhofer Distance and Near-Field Transition

Theorem: Near-Field Spherical-Wave Channel Model

Rank Analysis of the LOS Near-Field Channel

Beam Focusing vs. Beam Steering

Sub-THz Hardware Limitations and State of the Art

Example: Near-Field Spatial Degrees of Freedom

Array aperture and Fraunhofer distance

Near-field spatial DoF (d = 5 m)

Far-field comparison (d = 100 m)

Near-Field Beam Focusing Animation

Fraunhofer Distance and Near-Field Beam Patterns

Parameters

Quick Check

Key Takeaway

Why This Matters: Near-Field Arrays in RF Imaging

Why This Matters: XL-MIMO and Near-Field in the MIMO Book

Common Mistake: Extrapolating mmWave Range Expectations to Sub-THz

Sub-Terahertz (Sub-THz)

Fraunhofer Distance

Beam Focusing

Definition:
Fraunhofer Distance and Near-Field Transition