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3GPP TR 38.901 Deep Dive

5G Channel Models Explained

3GPP TR 38.901 — From Room Acoustics to Linear Algebra

0.5 – 100 GHz
Frequency range
5
Deployment scenarios
UMa · UMi · RMa · InH · InF
N=20, M=20
Cluster model
clusters · rays per cluster
10
TDL/CDL profiles
reference models
TR 38.901 TR 38.900 TR 36.873 ITU-R M.2412 WINNER+

3GPP TR 38.901 is the authoritative channel model specification for 5G New Radio (NR), covering the full frequency range from 0.5 GHz to 100 GHz across all deployment environments from dense urban streets to factory floors. The standard adopts a geometry-based stochastic channel model (GSCM) that randomly generates scatterer clusters in 3D space, producing a time-varying, spatially consistent multi-antenna channel. From this full stochastic model, TR 38.901 also extracts a set of simplified TDL (Tapped Delay Line) and CDL (Clustered Delay Line) reference profiles widely used in 3GPP link-level simulations for fair vendor comparison. This notebook explains the model in both intuitive physical terms and rigorous linear-algebraic notation, taking you from path loss basics all the way to full MIMO channel matrices.

§1 What is a Wireless Channel?

Before diving into the mathematics of TR 38.901, it is worth building physical intuition for what a wireless channel actually is and does. A channel is not just a wire with noise — it is a complex physical medium that stretches, reflects, attenuates, and time-warps the transmitted signal in ways that depend on the environment, the carrier frequency, and the relative motion of transmitter and receiver.

1.1 The Room Acoustics Analogy

🎵 Everyday Analogy — Sound in a Room

When you speak in a large empty room, your friend hears three distinct phenomena simultaneously:

  1. The direct sound — the line-of-sight (LOS) path that travels straight from your mouth to your friend's ear with a single propagation delay.
  2. Echoes from walls, floor, and ceiling — reflected paths that each travel a longer distance, arriving later and with reduced amplitude. In a reverberant room you may hear dozens of overlapping echoes.
  3. Slight pitch variations as either of you moves — the Doppler effect: motion compresses or stretches the wave, shifting its apparent frequency.

A wireless radio channel is exactly the same physics. The "room" is the physical environment (city streets, office corridors, factory floor). The "sound" is the radio wave at gigahertz frequencies. The "echoes" are multipath reflections off buildings, cars, furniture, and the human body. The only difference is that the mathematics replaces air pressure oscillations with complex baseband amplitude — a phasor that encodes both the magnitude and phase of the carrier-frequency wave.

1.2 The Linear System View

A wireless channel is a linear time-varying (LTV) system. In the simplest single-antenna case, the input–output relationship in continuous time is the convolution integral:

\[ y(t) = \int_{-\infty}^{\infty} h(\tau,\, t)\, x(t - \tau)\, d\tau \;+\; n(t) \]

where:

  • \(y(t)\) — received baseband signal
  • \(x(t)\) — transmitted baseband signal
  • \(h(\tau, t)\) — channel impulse response (CIR): response at delay \(\tau\), varying with absolute time \(t\)
  • \(n(t)\) — additive white Gaussian noise (AWGN), \(\mathcal{CN}(0, N_0)\)

In practice, 5G NR uses CP-OFDM, which converts this time-domain convolution into a simple per-subcarrier multiplication. For OFDM subcarrier index \(k\) and time-slot index \(n\):

\[ y[k,n] \;=\; H[k,n]\, x[k,n] \;+\; w[k,n] \]

\(H[k,n]\) is a complex scalar — a single complex number that simultaneously scales the amplitude and rotates the phase of the transmitted symbol. The cyclic prefix (CP) absorbs all multipath echoes shorter than its duration, so the channel appears flat within each subcarrier bandwidth. With multiple transmit and receive antennas, this scalar generalises to a channel matrix \(\mathbf{H}[k,n]\) (covered in §7).

Four things a channel does to your signal

Effect Symbol Physical cause Remedy in NR
Attenuation \(|H| < 1\) Distance, atmospheric absorption, diffraction loss Power control, beamforming gain
Phase rotation \(\angle H \neq 0\) Propagation path length (multiples of wavelength) Coherent equalization via pilot-based CSI
Delay spread \(\tau_{\text{rms}}\) Multiple paths with different travel times Cyclic prefix in CP-OFDM
Doppler shift \(\nu_D\) Relative motion of TX, RX, or scatterers Pilot-based tracking, PTRS for phase noise

1.3 Why Modeling Matters

Channel models let us test receivers, link adaptation algorithms, and MIMO precoding schemes before building any hardware. A software simulation using TR 38.901 can evaluate thousands of parameter combinations overnight that would take months in a real-world field trial. TR 38.901 provides a standardised synthetic channel that all 3GPP vendors use for fair, reproducible comparison — ensuring that a receiver algorithm verified against these models will work across real-world 5G deployments.

1.3 The Doppler-Delay (Spreading Function) Domain

Beyond the time-delay CIR \(h(\tau, t)\), a doubly-dispersive channel is fully characterized by its delay-Doppler spreading function \(S(\tau, \nu)\) — the 2D Fourier transform of \(h(\tau, t)\) in the time direction: TR 38.901 §5.1

Delay-Doppler Spreading Function
\[ S(\tau, \nu) = \int_{-\infty}^{\infty} h(\tau, t)\, e^{-j2\pi\nu t}\, dt \]
What the three domains represent:
  • Time-delay \(h(\tau,t)\): how the channel impulse response evolves in time — natural for OFDM with CP
  • Delay-Doppler \(S(\tau,\nu)\): sparse representation — each physical scatterer appears as a distinct point \((\tau_l, \nu_l)\) — natural for high-mobility and OTFS modulation
  • Time-frequency \(H(f,t)\): what pilots directly observe — the 2D OFDM channel grid \(H[k,n]\)
In a OFDM system, inter-carrier interference (ICI) arises when the channel changes within one OFDM symbol — i.e., when the Doppler spread exceeds the subcarrier spacing: \(\nu_{\max} \gt \Delta f\). For 5G NR at \(\mu=1\) (30 kHz SCS), this occurs at velocities above ~375 km/h at 3.5 GHz.
§2 Deployment Scenarios

TR 38.901 defines five canonical deployment scenarios, each capturing a fundamentally different propagation environment. Each scenario comes with its own path loss model (§3), large-scale parameter (LSP) tables (§4), and typical inter-site distance (ISD). All five scenarios span the full 0.5–100 GHz frequency range unless stated otherwise. TR 38.901 §7.2

2.1 Outdoor Scenarios

🏙️
UMa
Urban Macrocell
  • BS height: 25 m (above rooftop)
  • ISD: 200–500 m
  • UE height: 1.5–22.5 m
  • Frequency: 0.5–100 GHz
  • O2I penetration loss included
💡 Analogy

The tall rooftop antenna broadcasting to many floors and streets — the workhorse of macro 5G coverage in city centres.

🏪
UMi-SC
Urban Microcell — Street Canyon
  • BS height: 10 m (below rooftop)
  • ISD: 200 m
  • UE height: 1.5 m (pedestrian)
  • Strong NLOS: buildings on both sides form a "canyon"
  • High angular spread at low elevation
💡 Analogy

A small cell mounted on a lamppost in a busy shopping street — signals bounce repeatedly between shopfronts before reaching a pedestrian UE.

🌾
RMa
Rural Macrocell
  • BS height: 35 m
  • ISD: 1732–5000 m
  • Mostly LOS, lower path loss exponent
  • Terrain undulation modelled explicitly
  • Valid 0.5–7 GHz (sub-6 focus)
💡 Analogy

A tall mast on a hill covering farmland and highways — few obstacles, near free-space propagation, but terrain causes slow shadowing.

Outdoor Scenario Comparison

Scenario BS Height ISD Typical PL exponent (NLOS) Dominant propagation effect
UMa 25 m 200–500 m ~3.5 Dense building clutter, O2I loss
UMi-SC 10 m 200 m ~3.7 Street-canyon reflections, high angular spread
RMa 35 m 1732–5000 m ~2.8 Free-space + terrain shadowing

UMi-Open Square TR 38.901 Table 7.2-1 also defines a UMi-Open Square variant for outdoor pedestrian areas (plazas, campuses). Its LOS probability is higher than UMi-Street Canyon due to the absence of building walls on both sides. The path loss model uses the same coefficients as UMi-SC but with a different LOS probability function: \(P_{\text{LOS}} = \min(18/d_{2D}, 1)(1-e^{-d_{2D}/36}) + e^{-d_{2D}/36}\). TR 38.901 Table 7.4.2-1

2.2 Indoor Scenarios

🏢
InH
Indoor Hotspot
  • BS height: 3 m (ceiling-mounted)
  • Room size: 120×50 m typical
  • Frequency: 0.5–100 GHz
  • Office, shopping mall, airport environments
  • Very relevant for mmWave Wi-Fi-like deployments
🏭
InF
Indoor Factory — 5 sub-variants
  • InF-SL — Sparse clutter, Low BS height
  • InF-DL — Dense clutter, Low BS height
  • InF-SH — Sparse clutter, High BS height
  • InF-DH — Dense clutter, High BS height
  • InF-HH — High TX + High RX (conveyor lines)
🏢 Everyday Analogy — Office NLOS

Imagine shouting from your desk to a colleague three offices away. Your direct path is blocked by walls (NLOS), but your voice bounces through doorways, corridors, and open-plan spaces before it arrives — with reduced clarity and a slight echo. InH models exactly this geometry: many short reflective surfaces at frequencies from ~3 GHz up to 60 GHz, where walls that are transparent at sub-6 GHz become nearly opaque at mmWave. The model captures both the dense multipath of corridor reflections and the sharp NLOS attenuation through drywall partitions.

InF was added in Release 16 (TR 38.901 v16.1) to model private 5G (campus network) deployments inside large manufacturing plants. Metal machinery, robot arms, conveyor belts, and metal shelving create uniquely high scattering, increased delay spread, and elevated attenuation compared to an office environment. The five InF sub-variants capture the full range from a near-empty warehouse (InF-SL) to a densely packed factory floor with ceiling-mounted access points (InF-DH/InF-HH).
§3 Path Loss and LOS/NLOS

Path loss is the dominant signal attenuation mechanism over distance — it determines the received signal power budget before any small-scale fading is considered. TR 38.901 provides separate empirical path loss models for LOS (Line-of-Sight) and NLOS (Non-Line-of-Sight) conditions in each deployment scenario, each validated against measurement campaigns from sub-1 GHz up to 100 GHz. TR 38.901 §7.4

3.1 The Log-Distance Model

📢 Everyday Analogy — Shouting Across a Field

If you shout from 1 m away, your voice is loud. From 10 m, it is much quieter — not 10 times quieter in pressure, but 100 times quieter in power, which is 20 dB. Every time you double the distance, you lose roughly 6 dB of received power. This is the inverse-square law for free space: radiated power spreads over a sphere of surface area 4πd², so intensity falls as 1/d². In denser environments (urban NLOS), the effective exponent rises above 2, meaning signal falls off faster than in free space.

The general free-space path loss from first principles:

\[ \text{PL}_{\text{free}}(d) \;=\; 20\log_{10}(d) \;+\; 20\log_{10}(f_c) \;+\; 20\log_{10}\!\left(\frac{4\pi}{c}\right) \]

Evaluating the constant term and expressing \(d\) in metres and \(f_c\) in GHz:

TR 38.901 Eq. (7.4-1) \[ \text{PL}_{\text{FS}}(d, f_c) \;=\; 32.44 \;+\; 20\log_{10}(d) \;+\; 20\log_{10}(f_c) \quad \text{[dB]} \]

At 3.5 GHz and 500 m this gives 32.44 + 54.0 + 10.9 = 97.34 dB — the minimum possible path loss in free space. Real urban environments will exceed this by 20–50 dB.

3.2 UMa LOS and NLOS Path Loss

The UMa LOS model uses a dual-slope formulation with a breakpoint distance \(d'_{BP}\) where the effective path loss exponent transitions from 2.2 (near-field) to 4.0 (far-field, waveguide-like propagation between buildings):

\[ d'_{BP} \;=\; \frac{4\, h'_{BS}\, h'_{UE}\, f_c}{c} \]

where \(h'_{BS} = h_{BS} - 1\) m and \(h'_{UE} = h_{UE} - 1\) m are the effective antenna heights above the effective environment height. TR 38.901 Table 7.4.1-1

UMa LOS — Region 1   \((10\text{ m} \le d_{2D} \le d'_{BP})\)

\[ \text{PL}_{1}^{\text{UMa-LOS}} \;=\; 28.0 \;+\; 22\log_{10}(d_{3D}) \;+\; 20\log_{10}(f_c) \]

UMa LOS — Region 2   \((d'_{BP} \le d_{2D} \le 5000\text{ m})\)

\[ \text{PL}_{2}^{\text{UMa-LOS}} \;=\; 28.0 \;+\; 40\log_{10}(d_{3D}) \;+\; 20\log_{10}(f_c) \;-\; 9\log_{10}\!\left((d'_{BP})^2 \;+\; (h_{BS} - h_{UE})^2\right) \]

UMa NLOS

TR 38.901 Table 7.4.1-1 \[ \text{PL}^{\text{UMa-NLOS}} \;=\; 13.54 \;+\; 39.08\log_{10}(d_{3D}) \;+\; 20\log_{10}(f_c) \;-\; 0.6\,(h_{UT} - 1.5) \]

Shadow fading is modelled as a zero-mean log-normal random variable added to the deterministic path loss: \(\xi \sim \mathcal{N}(0,\,\sigma_{SF}^2)\) with \(\sigma_{SF} = 6\text{ dB}\) (UMa NLOS). The total received power becomes \(P_r = P_t - \text{PL} - \xi\) [dBm].

Path Loss Formula Parameters

Symbol Meaning Typical range / value
\(d_{2D}\) 2D ground-plane distance BS–UE (horizontal) 10–5000 m
\(d_{3D}\) 3D Euclidean distance including height difference \(\approx d_{2D}\) when height diff is small
\(f_c\) Carrier frequency 0.5–100 GHz
\(h_{BS}\) BS antenna height above ground 25 m (UMa), 10 m (UMi), 35 m (RMa)
\(h_{UE}\) UE antenna height above ground 1.5–22.5 m (outdoor UE)
\(d'_{BP}\) Breakpoint distance (dual-slope transition) \(\approx 4 \times 24 \times 0.5 \times 3.5\text{ GHz} / c \approx 560\text{ m}\)
\(\sigma_{SF}\) Shadow fading standard deviation 4–8 dB (scenario and LOS/NLOS dependent)

3.3 RMa and InH Path Loss TR 38.901 Table 7.4.1-1/2

RMa LOS (Rural Macrocell)

Two-region model with breakpoint \(d_{\text{BP}} = 2\pi h_{BS} h_{UT} f_c / c\):

RMa LOS Region 1 (10 m ≤ d₂D ≤ d_BP)
\[ \text{PL}^{\text{RMa-LOS}}_1 = 20.0 + 26.9\log_{10}(d_{3D}) + 20\log_{10}(f_c) \]
RMa LOS Region 2 (d_BP ≤ d₂D ≤ 10 km)
\[ \text{PL}^{\text{RMa-LOS}}_2 = 20.0 + 40\log_{10}(d_{3D}) + 20\log_{10}(f_c) - 9\log_{10}(d_{\text{BP}}^2 + (h_{BS}-h_{UT})^2) \]
RMa NLOS (10 m ≤ d₂D ≤ 5 km)
\[ \text{PL}^{\text{RMa-NLOS}} = 161.04 - 7.1\log_{10}(W) + 7.5\log_{10}(h) - (24.37 - 3.7(h/h_{BS})^2)\log_{10}(h_{BS}) + (43.42 - 3.1\log_{10}(h_{BS}))(\log_{10}(d_{3D}) - 3) + 20\log_{10}(f_c) - (3.2(\log_{10}(17.625))^2 - 4.97) - 0.6(h_{UT}-1.5) \]

Where \(W\) = street width (default 20 m), \(h\) = average building height (default 5 m). \(\sigma_{SF}^{\text{RMa-NLOS}} = 8\) dB.

InH-Office Path Loss TR 38.901 Table 7.4.1-2

InH-Office LOS (1 m ≤ d₃D ≤ 100 m)
\[ \text{PL}^{\text{InH-LOS}} = 32.4 + 17.3\log_{10}(d_{3D}) + 20\log_{10}(f_c) \]
InH-Office NLOS (1 m ≤ d₃D ≤ 86 m)
\[ \text{PL}^{\text{InH-NLOS}} = 38.3\log_{10}(d_{3D}) + 17.30 + 24.9\log_{10}(f_c) \]

\(\sigma_{SF}^{\text{InH-LOS}} = 3\) dB, \(\sigma_{SF}^{\text{InH-NLOS}} = 8.03\) dB. Valid for 0.5–100 GHz.

3.4 Outdoor-to-Indoor (O2I) Penetration Loss TR 38.901 §7.4.3

Everyday Analogy — Radio Through Walls
A Wi-Fi signal drops sharply when you walk from the garden into a concrete-walled building. The penetration loss depends on the building material and frequency — glass is nearly transparent at 1 GHz but blocks 30+ dB at 28 GHz.

TR 38.901 §7.4.3 defines two building types:

Building typePenetration lossFrequency rangeDominant material
Low-loss\(PL_{O2I} = 5 - 10\log_{10}(0.3 \cdot 10^{-L_{glass}/10} + 0.7 \cdot 10^{-L_{concrete}/10})\)0.5–100 GHzModern glass facade
High-loss\(PL_{O2I} = 5 - 10\log_{10}(0.7 \cdot 10^{-L_{IRR\_glass}/10} + 0.3 \cdot 10^{-L_{concrete}/10})\)0.5–100 GHzIRR glass + concrete

Material loss at frequency \(f_c\) [GHz] per TR 38.901 Table 7.4.3-1:

MaterialLoss formulaLoss @ 3.5 GHzLoss @ 28 GHz
Standard glass\(2 + 0.2f_c\)2.7 dB7.6 dB
IRR glass\(23 + 0.3f_c\)23.9 dB31.4 dB
Concrete\(5 + 4f_c\)19 dB117 dB
Concrete wall penetration loss at 28 GHz exceeds 100 dB — mmWave indoor coverage must rely on dedicated indoor small cells or leaky-cable DAS, not outdoor macro coverage.

3.5 LOS Probability

Because the LOS/NLOS state of a link depends on whether any building blocks the direct path, TR 38.901 defines a probabilistic LOS model: given only the 2D distance, what is the probability that the link is in LOS? The model is fitted to urban building databases. TR 38.901 Table 7.4.2-1

For UMa:

\[ P_{\text{LOS}}(d_{2D}) \;=\; \min\!\left(\frac{18}{d_{2D}},\; 1\right) \left(1 - e^{-d_{2D}/63}\right) \;+\; e^{-d_{2D}/63} \]
Worked example at \(d_{2D} = 100\) m:
\(\min(18/100,\,1) = 0.18\),   \(e^{-100/63} = e^{-1.587} \approx 0.204\)

\[ P_{\text{LOS}} = 0.18 \times (1 - 0.204) + 0.204 = 0.18 \times 0.796 + 0.204 \approx 0.143 + 0.204 \approx 0.35 \] At 100 m in an urban macro cell, roughly 1 in 3 UEs are in LOS conditions. By 500 m this drops to around 10%, reflecting the reality that most city-centre links at macro-cell range are obstructed by buildings.

3.6 Path Loss vs Distance — Interactive Chart

Path Loss vs Distance @ 3.5 GHz (TR 38.901)
At 500 m, UMa NLOS path loss exceeds LOS by approximately 15 dB — the difference between a satisfactory link budget and an outage. The dual-slope kink in the UMa LOS curve near the breakpoint distance (~560 m at 3.5 GHz with h_BS=25 m, h_UE=1.5 m) is visible as the slope steepens from 22-log to 40-log attenuation.
Note on applicability: All TR 38.901 path loss models are empirical curve fits to measurement data. They are valid over the distance ranges stated in the specification tables. Extrapolating outside these ranges (e.g., d < 10 m or d > 5000 m for UMa) may produce physically unrealistic values and should be avoided in system simulations.
§4 Large-Scale Fading (Shadow Fading)

Spec refs: TR 38.901 §7.4 TR 38.901 §7.6.3

4.1 What is Large-Scale Fading?

☁️ Everyday Analogy — Cloud Passing the Sun

Imagine the sun as the base station and you as the UE. On a sunny day (LOS), the irradiance is strong. When a large cloud (a building, a hill) moves between you and the sun, the irradiance drops slowly and stays low for minutes. This slow, large-scale variation — caused by obstructions — is shadow fading. It is log-normally distributed because signal strength is measured in dB, and random obstacles combine multiplicatively in linear scale but additively in log scale.

The total path loss including shadow fading is modelled as a path-loss exponent term plus a zero-mean Gaussian random variable in dB:

(4.1) \[ \text{PL}_{\text{total}}(d) = \text{PL}(d) + X_{\sigma} \]

where \(X_{\sigma} \sim \mathcal{N}(0,\,\sigma_{\text{SF}}^2)\) is the zero-mean Gaussian shadow fading term in dB. In linear scale this becomes a log-normal random variable:

(4.2) \[ L_{\text{linear}} = 10^{(X_\sigma/10)} \sim \text{LogNormal} \]

TR 38.901 specifies scenario-dependent shadow fading standard deviations: TR 38.901 Table 7.4.1-1

Scenario LOS \(\sigma_{\text{SF}}\) (dB) NLOS \(\sigma_{\text{SF}}\) (dB)
UMa 4 6
UMi-Street Canyon 4 7.82
RMa 4 8
InH-Office 3 8.03
InF-SL 4 7.2
Why log-normal? Each obstacle reduces signal by a random fraction. A signal passing through \(N\) obstacles has power proportional to \(\prod_i a_i\). Taking the log: \(\log(\text{power}) = \sum_i \log(a_i)\) — a sum of independent random variables — which tends to Gaussian by the Central Limit Theorem. Hence dB-domain shadow fading is Gaussian, and linear-scale fading is log-normal.

4.2 Spatial Correlation of Shadow Fading

Shadow fading is spatially correlated — two UEs close together experience similar shadowing from shared obstructions. TR 38.901 models the correlation as an exponential function of separation distance:

(4.3) \[ C(\Delta d) = e^{-\Delta d \,/\, d_{\text{corr}}} \]

Decorrelation distances per scenario TR 38.901 Table 7.6.3.1-2:

Scenario LOS \(d_{\text{corr}}\) (m) NLOS \(d_{\text{corr}}\) (m)
UMa 37 50
UMi-SC 10 13
RMa 37 120
InH 10 13
Spatial correlation matters for network simulations: if two UEs share the same large obstruction (building corner), their fading is correlated. TR 38.901 uses a 2D correlated random field generated via a filtered Gaussian process to capture this correctly. The RMa NLOS decorrelation distance of 120 m reflects how rural terrain features (hills, forests) create long-range correlated shadowing.

Generating Spatially Correlated Shadow Fading Fields TR 38.901 §7.4.3.2

A spatially correlated Gaussian random field \(X_\sigma(\mathbf{r})\) is generated by filtering spatially white Gaussian noise \(N(\mathbf{r}) \sim \mathcal{N}(0,1)\) with a 2D Gaussian filter whose bandwidth matches \(d_{\text{corr}}\):

2D Correlated Shadow Field (Gaussian filter)
\[ X_\sigma(\mathbf{r}) = \sigma_{\text{SF}} \cdot \frac{F(\mathbf{r}) * N(\mathbf{r})}{\|F\|} \] where \(F(\mathbf{r}) = e^{-\|\mathbf{r}\|^2/(2d_{\text{corr}}^2)}\) is the 2D Gaussian filter kernel.

In practice, this is implemented as:

  1. Generate a 2D grid of independent \(\mathcal{N}(0,1)\) samples at the simulation grid resolution.
  2. Apply a 2D Gaussian filter with \(\sigma_{\text{filter}} = d_{\text{corr}} / \Delta_{\text{grid}}\) (in grid samples).
  3. Normalize to unit variance; scale by \(\sigma_{\text{SF}}\).
  4. Sample at UE positions by bilinear interpolation.

4.3 Shadow Fading PDF — Gaussian in dB Domain

Shadow Fading PDF — Gaussian in dB Domain
Shaded area beyond ±2σ = 5% outage region. UMa NLOS σ = 6 dB means 5% of locations have >12 dB excess attenuation relative to the median path loss.
§5 Small-Scale Fading

Spec refs: TR 38.901 §5.4 TR 38.901 Table 7.7.3-6

5.1 Multipath, Delay Spread, and Doppler

🚿 Everyday Analogy — Bathroom Echo

When you sing in the bathroom, every wall reflects your voice. You hear the original sound plus multiple echoes arriving a few milliseconds later. In a wireless channel, radio waves arrive via dozens of paths with different delays. The range of these delays is the delay spread \(\tau_{\text{rms}}\). If \(\tau_{\text{rms}}\) is larger than the symbol period, each symbol overlaps the next — inter-symbol interference (ISI). OFDM solves this by making symbol periods very long (IFFT output) and adding a cyclic prefix.

The RMS delay spread is defined as the square root of the second central moment of the power delay profile:

(5.1) \[ \tau_{\text{rms}} = \sqrt{ \frac{\sum_l |h_l|^2 \tau_l^2}{\sum_l |h_l|^2} - \left(\frac{\sum_l |h_l|^2 \tau_l}{\sum_l |h_l|^2}\right)^2 } \]

The maximum Doppler frequency due to UE velocity \(v\) at carrier frequency \(f_c\):

(5.2) \[ f_D = \frac{v}{\lambda} = \frac{v \cdot f_c}{c} \]

At \(v = 120\) km/h \(= 33.3\) m/s, \(f_c = 3.5\) GHz: \(f_D = (33.3 \times 3.5 \times 10^9) \,/\, (3 \times 10^8) \approx\) 389 Hz

The channel coherence time (time over which the channel is approximately constant):

(5.3) \[ T_c \approx \frac{1}{f_D} \approx 2.5 \text{ ms at 389 Hz} \]

More precisely, using Clarke's model 50% autocorrelation level: \(T_c = 0.423/f_D\). At \(f_D = 389\) Hz: \(T_c = 0.423/389 \approx\) 1.09 ms

The coherence bandwidth (frequency range over which channel response is correlated):

(5.4) \[ B_c \approx \frac{1}{5\,\tau_{\text{rms}}} \]
Two common approximations exist: \(B_c \approx 1/(5\tau_{\text{rms}})\) (90% coherence threshold, used in most textbooks) and \(B_c \approx 1/(2\pi\tau_{\text{rms}})\) (50% threshold from the Fourier uncertainty principle). The former gives a ~3× wider estimate. For 5G NR CP design, the 90% threshold is conservative and preferred.

For \(\tau_{\text{rms}} = 100\) ns (typical urban): \(B_c \approx 2\) MHz — subcarriers within 2 MHz see correlated (near-flat) fading.

TR 38.901 specifies delay spread as a log-normal random variable. \(\mu_{\lg DS}\) is the mean of \(\log_{10}(\tau_{\text{rms}}/1\text{ s})\): TR 38.901 Table 7.7.3-6

Scenario Condition \(\mu_{\lg DS}\) (log\(_{10}\) mean, s) \(\sigma_{\lg DS}\)
UMa LOS −7.03 0.66
UMa NLOS −6.44 0.39
UMi-SC LOS −7.19 0.40
UMi-SC NLOS −6.89 0.54
InH-Office LOS −7.70 0.18
InH-Office NLOS −7.41 0.14

Reading \(\mu_{\lg DS}\): UMa LOS has \(\mu_{\lg DS} = -7.03\), meaning the median delay spread is \(10^{-7.03} \approx 93\) ns. UMa NLOS gives \(10^{-6.44} \approx 363\) ns — nearly 4× larger due to richer multipath in obstructed conditions.

5.2 Rayleigh and Rician Fading

🪨 Everyday Analogy — Pebbles in a Pond

Throw 20 pebbles into a pond. Each creates ripples. At your finger's location, all ripples add up with different phases. Sometimes they add constructively (big wave), sometimes destructively (calm water). This random addition of waves with random phases is exactly what makes a Rayleigh-faded channel — the amplitude fluctuates wildly as you move even a few centimetres.

Rayleigh fading applies when there is no dominant path (NLOS). The envelope amplitude follows:

(5.5) \[ |H| \sim \text{Rayleigh}(\sigma) \implies p(r) = \frac{r}{\sigma^2}\, e^{-r^2/(2\sigma^2)}, \quad r \geq 0 \]

Rician fading applies when a dominant LOS component is present. The envelope PDF is:

(5.6) \[ |H| \sim \text{Rician}(K) \implies p(r) = \frac{2r(K+1)}{\Omega}\, e^{-K - \frac{(K+1)r^2}{\Omega}}\, I_0\!\left(2r\sqrt{\frac{K(K+1)}{\Omega}}\right) \]

The K-factor is the ratio of dominant (LOS) path power to the total scattered power:

(5.7) \[ K = \frac{|h_{\text{LOS}}|^2}{2\sigma^2} \quad [\text{dB}] \]

TR 38.901 models the K-factor in UMa LOS as a distance-dependent quantity TR 38.901 Table 7.7.3-6:

(5.8) \[ K = 9 - 0.35\!\left(\frac{d_{2D}}{1\,\text{m}} - 1\right) \; [\text{dB}], \quad 0 \leq d_{2D} \leq 1000\,\text{m} \]
K (dB) Physical Meaning Fading Severity
\(-\infty\) No LOS, pure scatter Rayleigh — deep fades up to −20 dB
0 dB Equal LOS and scatter power Moderate fades
9 dB Moderate LOS dominance Typical UMa LOS at 100 m
>15 dB Strong LOS dominance Indoor LOS, corridor
\(+\infty\) No scatter, pure AWGN No fading
A key insight: the Rician K-factor collapses to Rayleigh when \(K \to 0\) and to AWGN when \(K \to \infty\). All TR 38.901 LOS scenarios use positive K, while NLOS scenarios use \(K = -\infty\) (Rayleigh). The smooth continuum between these extremes is captured by the Rician model.

5.3 Fading Amplitude PDF: Rayleigh vs Rician

Fading Amplitude PDF: Rayleigh → Rician (increasing K)
Higher K-factor = stronger LOS = less fading variation. RMa LOS at 200 m: K ≈ 8 dB, amplitude varies by ±2 dB only. As K → ∞ the PDF narrows to a delta function at the mean amplitude.
§6 Geometry-Based Stochastic Channel Model (GSCM)

Spec refs: TR 38.901 §7.5 TR 38.901 §7.7

6.1 The Core Idea — Clusters of Scatterers

🏙️ Everyday Analogy — City Block Reflections

In a city, radio signals bounce off groups of objects: a cluster of parked cars, the glass facade of an office building, a row of trees. Each group (cluster) reflects waves from multiple slightly different angles and with slightly different delays. TR 38.901 models this as N clusters (scenario-dependent) each containing M = 20 individual rays. The cluster determines the rough direction and delay; the rays add random spread within each cluster.

The total channel impulse response is a double sum over clusters and rays:

(6.1) \[ h(\tau, \theta, \phi) = \sum_{n=1}^{N} \sum_{m=1}^{M} c_{n,m}\; \delta(\tau - \tau_n - \tau_{n,m})\; \delta(\theta - \theta_{n,m})\; \delta(\phi - \phi_{n,m}) \]

where:

  • \(N\) = number of clusters (scenario-dependent: N=12 for UMa LOS, N=20 for UMa NLOS, N=12 for UMi LOS, N=7 for InH-Office LOS — see TR 38.901 Table 7.7.3-6)
  • \(M\) = number of rays per cluster (20)
  • \(\tau_n\) = cluster delay, \(\tau_{n,m}\) = intra-cluster ray delay offset
  • \(\theta_{n,m},\,\phi_{n,m}\) = elevation and azimuth angles of departure/arrival
  • \(c_{n,m}\) = complex coefficient (amplitude × phase × polarisation)
ScenarioConditionN (clusters)M (rays)
UMaLOS1220
UMaNLOS2020
UMi-SCLOS1220
UMi-SCNLOS1920
RMaLOS1120
RMaNLOS1020
InH-OfficeLOS720
InH-OfficeNLOS720

TR 38.901 Table 7.7.3-6

The GSCM produces a random channel. Every simulation run draws new cluster positions, angles, and delays from the scenario's statistical distributions. This is what makes it "stochastic" — but the statistics (mean delay spread, angle spread) match measured propagation data from real deployments.

6.2 Large-Scale Parameter Generation

TR 38.901 §7.5 Step 4 requires drawing seven large-scale parameters (LSPs) jointly as correlated Gaussian random variables:

(6.2) \[ \mathbf{z} = \bigl[DS,\; ASD,\; ASA,\; ZSD,\; ZSA,\; K,\; SF\bigr]^T \]

These 7 parameters are jointly Gaussian with a scenario-specific cross-correlation matrix \(\mathbf{C}_{\text{LSP}}\):

(6.3) \[ \mathbf{z} \sim \mathcal{N}\!\left(\boldsymbol{\mu}_z,\; \mathbf{C}_{\text{LSP}}\right) \]

Key cross-correlations for UMa LOS TR 38.901 Table 7.5-6:

LSP Pair Correlation Coefficient Interpretation
DS vs ASD +0.4 More delay spread → slightly wider departure angle
DS vs ASA +0.8 Strong: rich multipath → wide arrival angle spread
DS vs K −0.4 Stronger LOS → less multipath → smaller delay spread
ASD vs ASA 0 Departure and arrival angular spreads are independent
SF vs K 0 Shadow fading and K-factor are independent
Physical insight on correlations: DS and ASA are positively correlated (0.8) — rich multipath in delay also means wide spread in arrival angles. DS and K are negatively correlated (−0.4) — a stronger LOS component (higher K) means less multipath scatter and therefore smaller delay spread. These correlations are derived from measurement campaigns, not assumed theoretically.

6.3 Channel Impulse Response Construction

TR 38.901 §7.5 defines a step-by-step procedure for generating the full MIMO channel matrix. The key steps are:

Step 1 — Generate cluster delays (exponentially distributed, scaled by delay spread DS):

(6.4) \[ \tau_n' \sim -r_\tau\, DS\, \ln(X_n), \quad X_n \sim \mathcal{U}(0,1) \quad \Rightarrow \quad \text{normalise, sort ascending} \]

Step 2 — Generate cluster powers (exponential decay in delay, plus per-cluster log-normal shadow term \(Z_n \sim \mathcal{N}(0,3\,\text{dB})\)):

(6.5) \[ P_n \propto e^{-\tau_n / DS} \cdot 10^{-Z_n/10} \]

Step 3 — Generate angles (AOD/AOA/ZOD/ZOA). Ray angles within each cluster are the cluster mean angle plus fixed ray offsets \(\Delta_{m,n}\) tabulated in TR 38.901 Table 7.5-3. The cluster angle spread (CAS) is scenario-specific.

Step 4 — Compute per-ray channel coefficient for UE antenna \(u\) and BS antenna \(s\):

(6.6) \[ h_{u,s,n,m}(t) = \sqrt{\frac{P_{n,m}}{M}}\; \mathbf{F}_{\text{rx},u}^T\; \mathbf{\Phi}_{n,m}\; \mathbf{F}_{\text{tx},s}\; e^{\,j2\pi(v_{n,m}\,t + \phi_0)} \]

where:

  • \(\mathbf{F}_{\text{rx},u},\,\mathbf{F}_{\text{tx},s}\) = receive/transmit antenna element field patterns (2×1 vectors for dual polarisation)
  • \(\mathbf{\Phi}_{n,m}\) = 2×2 cross-polarisation random phase matrix
  • \(v_{n,m}\) = Doppler frequency for ray \((n,m)\) determined by arrival angle and UE velocity vector
  • \(\phi_0\) = initial random phase, \(\phi_0 \sim \mathcal{U}(0, 2\pi)\)

Step 5 — Sum over rays then clusters to obtain the full time-variant MIMO channel matrix:

(6.7) \[ \mathbf{H}(t,\tau) = \sum_{n=1}^{N} \left[\sum_{m=1}^{M} h_{u,s,n,m}(t)\right] \delta(\tau - \tau_n) \]
Complexity note: The 20-cluster × 20-ray structure gives 400 terms in the channel sum. For a 64 × 16 MIMO system (64 UE antennas × 16 BS antennas), each channel realisation is a 64×16 matrix evaluated at each delay tap — roughly 64 × 16 × 20 = 20,480 complex numbers per snapshot. At 1 μs resolution over a 10 ms drop, this is ~200 million complex multiplications — motivating hardware accelerators for system-level simulations.

Step 6 — Sub-cluster Splitting for the 2 Strongest Clusters TR 38.901 §7.5 Step 11

The two clusters with the largest power are each split into 3 sub-clusters to better model the intra-cluster delay spread. Sub-cluster delays are offset from the parent cluster delay \(\tau_n\) by:

Sub-clusterRays includedDelay offsetPower fraction
1 (early)Rays 1,2,3,4,5,6,7,8\(\tau_n + 0\)10/20 = 50%
2 (mid)Rays 9,10,11,12,13,14\(\tau_n + 1.28\,c_{\text{DS}}\)6/20 = 30%
3 (late)Rays 15,16,17,18,19,20\(\tau_n + 2.56\,c_{\text{DS}}\)4/20 = 20%

where \(c_{\text{DS}} = \max(0.25\tau_{\text{cluster}}, 0.25\,\text{ns})\) is the intra-cluster delay spread constant (TR 38.901 Table 7.5-5). This splitting means the effective number of delay taps for CDL models is \(N_{\text{clusters}} - 2 + 2\times 3 = N + 4\) (the 2 strongest clusters each become 3). For UMa NLOS: 20 clusters → 24 CDL taps.

Sub-cluster splitting ensures the CDL/TDL power delay profile accurately captures the intra-cluster angular and delay spread. Without it, the GSCM would produce clusters that are unrealistically point-like in delay.

6.4 Angle Spread Parameters

The azimuth angle spreads of departure (ASD) and arrival (ASA) are log-normally distributed. \(\mu_{\lg ASD}\) is the mean of \(\log_{10}(\text{ASD}/1°)\). TR 38.901 Table 7.7.3-6

Scenario Condition \(\mu_{\lg ASD}\) \(\sigma_{\lg ASD}\) \(\mu_{\lg ASA}\) \(\sigma_{\lg ASA}\)
UMa LOS 1.06 0.28 1.81 0.20
UMa NLOS 1.52 0.31 1.80 0.18
UMi-SC LOS 1.20 0.43 1.75 0.19
UMi-SC NLOS 1.53 0.23 1.68 0.18
InH-Office LOS 1.60 0.18 1.62 0.22
ASD \(\approx 1.52\) for UMa NLOS means \(10^{1.52} \approx 33°\) azimuth spread of departure — the signal leaves the BS spread over \(\pm 33°\). This determines how many spatial beams the gNB can form independently (spatial degrees of freedom \(\approx \text{array aperture} / \text{ASD}\)). A narrow ASD in InH LOS (\(\mu_{\lg ASD} = 1.60\) but tight \(\sigma = 0.18\)) allows very precise beamforming in indoor corridors.
Elevation vs Azimuth: TR 38.901 also specifies zenith angle spreads (ZSD, ZSA). For massive MIMO with vertical sectorisation (3D beamforming), ZSD is critical — UMa NLOS has \(\mu_{\lg ZSD} \approx 0.9\) (i.e., ~8° median elevation spread), enabling 3–4 independent vertical beams in a 64T64R panel.

6.5 Spatial Consistency TR 38.901 §7.6.3.2

For mobility simulations, cluster positions must remain correlated as the UE moves — a cluster does not suddenly disappear as the UE takes one step. TR 38.901 §7.6.3.2 defines the spatial consistency procedure:

  1. Initialize cluster positions, powers, and angles at the UE's starting location.
  2. As the UE moves by \(\Delta\mathbf{r}\), update the cluster parameters using a correlated random walk with decorrelation distance \(d_{\text{corr}}\).
  3. New clusters appear ("birth") and old clusters fade ("death") according to a Poisson process with rate \(\lambda = 1/d_{\text{corr}}\).
ParameterUMaUMi-SCInH-OfficeSpec ref
Cluster birth/death distance12 m15 m6 mTR 38.901 Table 7.6.3.2-2
LSP update distance50 m (NLOS)13 m (NLOS)13 m (NLOS)TR 38.901 Table 7.6.3.1-2
Spatial consistency is required for: beam management simulation (handover, beam switching), V2X channel modeling, massive MIMO systems where adjacent UEs share correlated channels, and dual-mobility (both BS and UE moving) scenarios like drone-to-ground.
§7 The MIMO Channel Matrix H

TR 38.901 §7.7 TS 38.211 §7.3

7.1 From Scalar to Matrix — Why MIMO?

🎻 Everyday Analogy — Multiple Microphones

A single microphone in a noisy room picks up all sounds mixed together. Three directional microphones pointed at different parts of the room can separate the violin, piano, and cello — this is beamforming. MIMO antennas do the same for radio: multiple receive antennas observe the transmitted signal from different spatial perspectives, giving us enough equations to solve for multiple simultaneous data streams.

The SISO channel is a single complex scalar \(H[k,n] \in \mathbb{C}\) per subcarrier \(k\) and OFDM symbol \(n\). Once we add multiple transmit and receive antennas, the scalar becomes a matrix:

(7.1) \[ \mathbf{y}[k,n] = \mathbf{H}[k,n]\,\mathbf{x}[k,n] + \mathbf{w}[k,n] \]

where the dimensions are:

Physical meaning of each \(H_{ij}\): The entry at row \(i\), column \(j\) is the complex channel gain between TX antenna \(j\) and RX antenna \(i\) — a single complex number encoding amplitude attenuation and phase shift for that pair of antennas.

A \(3 \times 2\) example (\(N_r=3\), \(N_t=2\)):

TX ant 1 TX ant 2 row meaning
RX ant 1 \(H_{11}\) \(H_{12}\) RX ant 1 sees both TX antennas
RX ant 2 \(H_{21}\) \(H_{22}\) RX ant 2 sees both TX antennas
RX ant 3 \(H_{31}\) \(H_{32}\) RX ant 3 sees both TX antennas
For a 64-antenna gNB (\(N_t=64\)) and 4-antenna UE (\(N_r=4\)), the channel matrix \(\mathbf{H}\) is \(4 \times 64\). This means the system has \(4 \times 64 = 256\) complex-valued "links" per subcarrier — yet in practice, only \(\text{rank}(\mathbf{H}) \leq \min(4,64) = 4\) independent data streams can be sent simultaneously.

7.2 SVD — The Skeleton of the MIMO Channel

🌀 Everyday Analogy — Finding the Natural Directions

Imagine a funnel with an oval opening. You can pour water through it most efficiently if you pour in the direction of the longer axis. SVD finds the "natural directions" of data flow through the MIMO channel — the directions that experience the least and most attenuation, independently of each other. These are the eigenmodes (or spatial layers).

The Singular Value Decomposition (SVD) of \(\mathbf{H}\):

(7.2) \[ \mathbf{H} = \mathbf{U}\,\boldsymbol{\Sigma}\,\mathbf{V}^H \]

Optimal transmission — eigenmode beamforming: Pre-code the transmit signal with \(\mathbf{V}\) and post-combine at the receiver with \(\mathbf{U}^H\):

(7.3) \[ \tilde{\mathbf{y}} = \mathbf{U}^H \mathbf{y} = \boldsymbol{\Sigma}\,(\mathbf{V}^H \mathbf{x}) + \mathbf{U}^H \mathbf{w} \]

Because \(\boldsymbol{\Sigma}\) is diagonal and \(\mathbf{U}\) is unitary (so \(\mathbf{U}^H\mathbf{w}\) is still white noise), the MIMO channel decomposes into \(r\) independent SISO channels with gains \(\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_r\).

What SVD tells you about a MIMO channel

SVD property Physical meaning Design implication
\(\text{rank}(\mathbf{H})\) Number of independent streams Max spatial multiplexing order
\(\sigma_1^2 / \sigma_r^2\) Condition number How spread out stream SNRs are
Columns of \(\mathbf{V}\) Precoding beamforming weights TX beamforming codebook
Columns of \(\mathbf{U}\) Combining weights RX combining filter
\(\sigma_i^2\) Power of \(i\)-th stream Water-filling power allocation

7.3 MIMO Capacity

Single-user MIMO capacity with perfect CSI at both ends Shannon 1948 / Telatar 1999:

(7.4) \[ C = \sum_{i=1}^{r} \log_2\!\left(1 + \frac{p_i\,\sigma_i^2}{\sigma_w^2}\right) \quad [\text{bps/Hz}] \]

With total power constraint \(\sum_i p_i = P\), optimal per-stream powers come from water-filling:

(7.5) \[ p_i = \left(\mu - \frac{\sigma_w^2}{\sigma_i^2}\right)^{\!+} \]

where \(\mu\) is the "water level" chosen so that \(\sum_i p_i = P\) and \((x)^+ \equiv \max(x,0)\) (channels too weak below the noise floor get zero power).

Without CSI at the transmitter (equal power \(P/N_t\) per antenna):

(7.6) \[ C = \log_2 \det\!\left(\mathbf{I}_{N_r} + \frac{\text{SNR}}{N_t}\,\mathbf{H}\mathbf{H}^H\right) \]
The determinant formula reveals why MIMO is powerful: \[ \det\!\left(\mathbf{I} + \tfrac{\text{SNR}}{N_t}\mathbf{H}\mathbf{H}^H\right) = \prod_{i=1}^{r}\!\left(1 + \frac{\text{SNR}\,\sigma_i^2}{N_t}\right) \] Each eigenmode contributes a multiplicative factor. With 4 equal-strength streams at SNR = 10 dB: \(C = 4\log_2(1+10) \approx 13.8\) bps/Hz — versus \(\log_2(11) \approx 3.5\) bps/Hz for SISO. A \(\mathbf{4\times}\) spectral-efficiency gain!

7.4 Singular Value Profile — MIMO Channel Conditioning

Normalized Singular Values for 4×4 MIMO (Three Propagation Scenarios)

Rich NLOS scattering gives near-equal singular values — all 4 streams are usable. Strong LOS concentrates power in one dominant eigenmode, reducing effective rank.

7.5 Channel Reciprocity (TDD)

In 5G NR TDD deployments, UL and DL channels occupy the same frequency band in alternating time slots. Provided the interval between UL and DL transmissions is less than the coherence time \(T_c\), the channels are reciprocal:

(7.7) \[ \mathbf{H}_{\text{DL}}[k,n] \approx \mathbf{H}_{\text{UL}}^T[k,n] \]

Note: it is the transpose, not the conjugate transpose — reciprocity holds in an isotropic medium because the propagation path is reversible but the antenna element patterns and RF chains introduce conjugation symmetry.

The gNB measures \(\mathbf{H}_{\text{UL}}\) from the UL pilot (SRS — Sounding Reference Signal), computes the SVD, and uses the right singular vectors as the DL precoder \(\mathbf{V}\) without any feedback from the UE. This is implicit beamforming, the cornerstone of massive MIMO in gNBs with 32–192 antennas. This approach assumes the UL-DL time gap is within the channel coherence time \(T_c\). For a vehicle at 120 km/h, \(T_c \approx 1.09\) ms — NR TDD must complete the UL SRS measurement and DL beam application within ~1 slot (0.5 ms at μ=1) to remain within this budget.

Calibration requirement: Reciprocity holds for the propagation channel but not necessarily for the measured channel. RF chain differences between transmit and receive paths (IQ imbalance, phase noise, different LNA/PA gain, antenna coupling asymmetry) break reciprocity. A calibration procedure is required to align the TX and RX chain responses before using \(\mathbf{H}_{\text{UL}}\) as a proxy for \(\mathbf{H}_{\text{DL}}\).

In 5G NR, uplink SRS-based downlink precoding (implicit CSI) relies on this reciprocity. TS 38.214 §5.2.2.6 defines the SRS resource configuration used for DL precoding. Hardware calibration error is modelled as a diagonal mismatch matrix: \(\mathbf{H}_{\text{meas}} = \mathbf{D}_{\text{rx}}\, \mathbf{H}_{\text{true}}\, \mathbf{D}_{\text{tx}}^{-T}\) where \(\mathbf{D}_{\text{rx}}, \mathbf{D}_{\text{tx}}\) are diagonal complex matrices representing per-antenna chain responses. TS 38.214 §5.2.2.6
TR 38.901 §7.7.2 provides a spatial-consistency model: \(\mathbf{H}\) at position \(\mathbf{x}+\Delta\mathbf{x}\) is correlated with \(\mathbf{H}\) at \(\mathbf{x}\), with coherence distance on the order of a few metres. This is why the gNB needs fresh SRS every few slots as UEs move — the precoder \(\mathbf{V}\) must track the time-varying channel.
§8 Antenna Arrays and Steering Vectors

TR 38.901 §7.3 TR 38.901 Annex A

8.1 Uniform Linear Array (ULA)

🪖 Everyday Analogy — Line of Soldiers Listening

Imagine 8 soldiers standing in a straight line, spaced 1 metre apart. A distant helicopter is approaching at 30° from the front. Each soldier hears the helicopter slightly later than the one before — because the helicopter is off-axis, the sound wavefront hits each soldier at a slightly different time. If the soldiers electronically combine their microphone signals with the right delays (phase shifts), they can amplify the helicopter signal and suppress other directions. This is exactly how a ULA beamforms.

Consider an \(N\)-element ULA with inter-element spacing \(d = \lambda/2\) (half-wavelength). A plane wave arriving at azimuth angle \(\theta\) (measured from broadside) travels an extra path \(d\sin\theta\) to each successive element. The resulting phase increment per element is \(2\pi d\sin\theta/\lambda = \pi\sin\theta\). The array steering vector is:

(8.1) \[ \mathbf{a}(\theta) = \frac{1}{\sqrt{N}} \begin{bmatrix} 1 \\[2pt] e^{\,j\pi\sin\theta} \\[2pt] e^{\,j2\pi\sin\theta} \\[2pt] \vdots \\[2pt] e^{\,j(N-1)\pi\sin\theta} \end{bmatrix} \in \mathbb{C}^N \]

The \(n\)-th element (0-indexed) accumulates phase \(e^{\,jn\pi\sin\theta}\) relative to the reference element at \(n=0\). The \(1/\sqrt{N}\) normalisation ensures \(\|\mathbf{a}(\theta)\| = 1\).

The steering vector \(\mathbf{a}(\theta)\) is a unit-norm vector: \(\|\mathbf{a}(\theta)\| = 1\). The inner product between two steering vectors at different angles is: \[ \mathbf{a}^H(\theta_1)\,\mathbf{a}(\theta_2) = \frac{1}{N}\sum_{n=0}^{N-1} e^{\,jn\pi(\sin\theta_2 - \sin\theta_1)} \] This is a Dirichlet kernel — it equals 1 when \(\theta_1=\theta_2\) and decreases as the angles separate. The array resolves two directions when \(|\sin\theta_1 - \sin\theta_2| > 2/N\), giving an angular resolution \(\Delta\theta \approx 2/(N\cos\theta)\) radians.

8.2 Beamforming as Inner Product Projection

A beamforming weight vector \(\mathbf{w} \in \mathbb{C}^N\) linearly combines the \(N\) antenna outputs into a scalar:

(8.2) \[ z = \mathbf{w}^H \mathbf{y} = \mathbf{w}^H \bigl(\mathbf{a}(\theta)\,s + \mathbf{n}\bigr) = \underbrace{\mathbf{w}^H \mathbf{a}(\theta)}_{\text{array gain}}\,s + \mathbf{w}^H \mathbf{n} \]

Maximum Ratio Combining (MRC) / matched-filter beamforming sets \(\mathbf{w}_{\text{MF}} = \mathbf{a}(\theta)\):

(8.3) \[ \mathbf{w}_{\text{MF}}^H\,\mathbf{a}(\theta) = \mathbf{a}^H(\theta)\,\mathbf{a}(\theta) = \|\mathbf{a}(\theta)\|^2 = 1 \]

The signal power scales as \(|z_s|^2 = 1\), but the noise power \(\mathbb{E}[|\mathbf{w}^H\mathbf{n}|^2] = \sigma_n^2 \|\mathbf{w}\|^2 = \sigma_n^2\) — unchanged after normalisation. However, compared to a single antenna that observes only \(1/N\) of the coherent signal, the coherent combining of \(N\) antennas increases the effective SNR by factor \(N\):

(8.4) \[ \text{SNR}_{\text{array}} = N \times \text{SNR}_{\text{single}} \]
For a 64-element array: +18 dB array gain over a single antenna. This is the fundamental reason why massive MIMO can serve UEs at the cell edge that a single-antenna base station could not reach.

Array gain summary

\(N\) antennas Array gain (linear) Array gain (dB) 5G NR usage
2\(2\times\)3 dBBasic 2T2R
4\(4\times\)6 dBSmall cells
8\(8\times\)9 dBMid-band gNB
16\(16\times\)12 dBFR1 massive MIMO
32\(32\times\)15 dBFR1/FR2 massive MIMO
64\(64\times\)18 dBFR2 mmWave gNB
256\(256\times\)24 dBFuture 6G

8.3 Uniform Planar Array (UPA) — 2D Beamforming

Real gNB radio units mount antennas in a 2D panel with \(M_H\) columns (horizontal) and \(M_V\) rows (vertical). The 2D steering vector is the Kronecker product of independent horizontal and vertical steering vectors:

(8.5) \[ \mathbf{a}_{\text{UPA}}(\theta,\phi) = \mathbf{a}_H(\theta,\phi) \otimes \mathbf{a}_V(\phi) \]

where:

For a typical 64-TXP panel with \(M_H = 8\), \(M_V = 8\): \(\mathbf{a}_{\text{UPA}} \in \mathbb{C}^{64}\).

The Kronecker structure means vertical and horizontal beamforming separate: \(\mathbf{w}_{\text{UPA}} = \mathbf{w}_H \otimes \mathbf{w}_V\). The gNB can codebook-combine an 8-element horizontal beam with an 8-element vertical tilt independently. This is the basis for 3D beamforming (FD-MIMO) standardised in TR 36.873 / TS 38.214 Type II CSI. The DFT-based codebook for NR supports \(O_1 \times O_2\) oversampled UPA grids (e.g., 4×4 oversampling = 256 beams from 64 elements).

8.4 MIMO Channel via Steering Vectors

The physical channel matrix can be written explicitly in terms of TX and RX steering vectors. For a geometric channel model with \(L\) rays (across all clusters):

(8.6) \[ \mathbf{H} = \sqrt{\frac{N_r N_t}{L}} \sum_{l=1}^{L} \alpha_l\; \mathbf{a}_r\!\left(\theta_l^r, \phi_l^r\right) \mathbf{a}_t^H\!\left(\theta_l^t, \phi_l^t\right) \]

where:

This is the rank-\(L\) outer product decomposition of \(\mathbf{H}\). Each ray contributes a rank-1 matrix \(\mathbf{a}_r \mathbf{a}_t^H\). The MIMO channel is therefore a sum of rank-1 matrices — one per ray.

This formula shows why mmWave channels (FR2) have low rank: at 28 GHz there are very few scatterers (\(N_{\text{clusters}} \approx 6\) in TR 38.901 for UMi mmWave, each with a small number of sub-rays), so \(\text{rank}(\mathbf{H}) \approx 6\) even for a 256-antenna array. More antennas give higher SNR per beam (via array gain \(N_t\)) but not more simultaneous spatial streams. This is the fundamental difference between beamforming gain (SNR) and multiplexing gain (capacity/Hz).

8.5 ULA Beam Pattern — Array Resolution vs. Element Count

ULA Array Factor AF(θ) — half-λ spacing, steered to 0°

64-element ULA achieves 18 dB array gain with a very narrow main lobe (~1.8° 3 dB beamwidth). 8 elements gives ~14° beamwidth — suitable for sector-level beamforming. Sidelobes are visible at ~−13 dB relative to the main lobe (uniform aperture); real arrays apply amplitude tapering (e.g. Chebyshev windows) to suppress them.

Key Takeaways — §7 + §8
  • The MIMO channel matrix \(\mathbf{H}\) contains \(N_r \times N_t\) complex gains but only \(\text{rank}(\mathbf{H}) \leq \min(N_r, N_t)\) independent streams exist.
  • SVD reveals the natural transmission directions: transmit along \(\mathbf{V}\), receive along \(\mathbf{U}^H\), and the channel decouples into \(r\) SISO sub-channels.
  • Water-filling maximises capacity by allocating more power to strong eigenmodes and shutting off weak ones.
  • ULA steering vectors are complex exponential sequences; beamforming is a matched-filter projection onto the signal direction, giving \(N\)-fold SNR gain.
  • Real gNB panels use 2D UPAs with Kronecker structure, enabling separable azimuth + elevation beamforming (3D / FD-MIMO).
  • The geometric channel model (Eq. 8.6) connects the physics (rays, AoA, AoD) to the matrix \(\mathbf{H}\): each ray is a rank-1 outer product of TX and RX steering vectors.
Common exam pitfall: Do not confuse array gain (SNR improvement = \(N\)) with multiplexing gain (number of simultaneous streams = \(\text{rank}(\mathbf{H})\)). A 64×1 massive MIMO system achieves 18 dB array gain to a single UE but zero spatial multiplexing — you need both \(N_t > 1\) and \(N_r > 1\) and rich scattering to get multiple independent streams.
§9 TDL and CDL Reference Models

TR 38.901 §7.7.2 TR 38.901 §7.7.3 TR 38.901 Annex B

9.1 Why Simplified Models?

💬 Everyday Analogy — Weather Forecast vs. Climate Model

A full climate model simulates every air molecule. A weather forecast uses a simplified model that captures the key dynamics. TDL and CDL are the "weather forecast" version of the GSCM: they capture the essential delay-and-angle structure of the full stochastic channel in a fixed set of taps, perfect for repeatable link-level simulation and hardware testing.

The full GSCM channel is rich but random. For link-level simulation, TR 38.901 defines two families of simplified reference models:

  • TDL (Tapped Delay Line): frequency-domain SISO model — fixed delays + fixed power fractions + specified fading type (Rayleigh/Rician). No angle information → suitable for SISO/receive diversity evaluation.
  • CDL (Clustered Delay Line): MIMO-capable extension — each tap has azimuth/elevation angles, enabling spatial modeling for beamforming and MIMO evaluation.

When to use each:

Model Antennas Use case Angle info Spec
TDL-A/B/C Any (no spatial) Link budget, BLER vs SNR curves None TR 38.901 Table 7.7.2-1/2/3
TDL-D/E Any (Rician) LOS link simulation None TR 38.901 Table 7.7.2-4/5
CDL-A/B/C Multi-antenna NLOS MIMO precoding, beamforming Yes TR 38.901 Table 7.7.3-1/2/3
CDL-D/E Multi-antenna LOS MIMO with LOS component Yes TR 38.901 Table 7.7.3-4/5

9.2 TDL Model Structure

A TDL model has \(L\) fixed taps:

(9.1) \[ h(t,\tau) = \sum_{l=1}^{L} c_l(t)\,\delta(\tau - \tau_l) \]

where:

TDL-A (23 taps, NLOS, spread delay profile — typical NLOS urban). First 8 taps: TR 38.901 Table 7.7.2-1

Tap # Normalized Delay Power (dB) Fading Type
10.0000−13.4Rayleigh
20.38190.0Rayleigh
30.4025−2.2Rayleigh
50.4610−6.0Rayleigh
60.5375−8.2Rayleigh
80.5750−11.3Rayleigh
40.5868−4.0Rayleigh
70.6708−9.9Rayleigh

Rows sorted by ascending normalized delay for readability. Original tap numbering in TR 38.901 Table 7.7.2-1 follows cluster assignment order.

TDL-D (13 taps, LOS Rician \(K = 13.3\) dB, first tap Rician). First 4 taps: TR 38.901 Table 7.7.2-4

Tap # Normalized Delay Power (dB) Fading Type K-factor (dB)
10.0000−0.2Rician13.3
20.0350−13.5Rayleigh—
30.6120−18.8Rayleigh—
41.6782−21.0Rayleigh—
The normalized delay must be scaled by a delay scaling factor \(DS_{\text{desired}} / DS_{\text{nominal}}\). For TDL-A: \(DS_{\text{nominal}} = 30\) ns. If you want to simulate a channel with 100 ns RMS delay spread, scale all delays by \(100/30 = 3.33\). This makes TDL models parameterizable for any environment.

9.3 CDL Model Structure

CDL adds spatial information to each cluster: angles of departure (AOD/ZOD) and arrival (AOA/ZOA).

Each CDL cluster \(l\) has:

  • Power \(P_l\), delay \(\tau_l\) (same as TDL)
  • Mean AOD \(\bar{\varphi}_l^{\text{AOD}}\), ZOD \(\bar{\theta}_l^{\text{ZOD}}\)
  • Mean AOA \(\bar{\varphi}_l^{\text{AOA}}\), ZOA \(\bar{\theta}_l^{\text{ZOA}}\)
  • Cross-polarization ratio \(\text{XPR}_l\)

The MIMO channel \(\mathbf{H}\) at delay tap \(l\):

(9.2) \[ \mathbf{H}_l = \sqrt{\frac{P_l}{M}} \sum_{m=1}^{M} \mathbf{F}_{\text{rx}}^T(\Omega_l^{rx,m})\, \mathbf{\Phi}_{l,m}\, \mathbf{F}_{\text{tx}}(\Omega_l^{tx,m})\, \mathbf{a}_r(\Omega_l^{rx,m})\, \mathbf{a}_t^H(\Omega_l^{tx,m}) \]

where \(M=20\) sub-rays, \(\mathbf{\Phi}_{l,m}\) is the \(2\times 2\) polarization matrix, and \(\mathbf{F}\) are element patterns.

  • CDL-A (23 clusters, NLOS): wide angle spread (ASD≈65°, ASA≈65°), suitable for rich-scattering NLOS environments.
  • CDL-C (24 clusters, NLOS): wider delay spread (DS≈316 ns), suitable for large cells.
  • CDL-D (13 clusters, LOS Rician): \(K=13.3\) dB LOS component, narrow angle spread.

CDL-A first 5 clusters: TR 38.901 Table 7.7.3-1

Cluster # Delay (ns) Power (dB) AOD (°) AOA (°) ZOD (°) ZOA (°)
10−13.4−178.151.398.581.5
2300.0−4.2−152.798.880.3
338.9−2.2−4.2−152.7100.482.0
456.7−4.0−4.2−152.7100.482.0
544.7−6.090.276.6100.582.0
CDL models enable spatial consistency: as the UE moves, the cluster angles change slowly. TR 38.901 §7.6.3.2 provides a spatial-consistency model for CDL that ensures adjacent time snapshots are correlated — critical for handover and beam management simulations.

9.4 TDL-A Power Delay Profile

TDL-A Power Delay Profile (23 taps, NLOS) — TR 38.901 Table 7.7.2-1
The dominant tap at normalized delay 0.38 (0 dB) represents the first strong cluster arrival. Scale by \(DS_{\text{desired}} / 30\,\text{ns}\) to obtain absolute delays for your target environment.
§10 Key Parameters Cheat Sheet

10.1 Scenario Parameters Comparison

TR 38.901 Tables 7.4.1-1 through 7.4.2-1 TR 38.901 Tables 7.7.3-6

Scenario LOS PL exp. NLOS PL exp. σSF LOS (dB) σSF NLOS (dB) DS LOS (ns) DS NLOS (ns) Max ISD
UMa 2.2 (R1) / 4.0 (R2) 3.9 4 6 93 (\(10^{-7.03}\)) 363 (\(10^{-6.44}\)) 5 km
UMi-SC 2.1 (R1) / 4.0 (R2) 3.2 4 7.82 65 (\(10^{-7.19}\)) 129 (\(10^{-6.89}\)) 500 m
RMa 2.1 (R1) / 4.0 (R2) 3.8 4 8 32 37 21 km
InH-Office 1.73 3.19 3 8.03 20 (\(10^{-7.70}\)) 39 (\(10^{-7.41}\)) 150 m
InF-SL 1.56 3.3 4 7.2 15 30 300 m
InF-DH 1.56 3.5 4 7.4 15 30 300 m

10.2 Cluster and Ray Parameters

Parameter Value Notes
\(N\) clusters 20 (most scenarios) 6 for mmWave UMi
\(M\) rays/cluster 20 Fixed for all scenarios
AoD offset angles ±0.0447, ±0.1413, ±0.2492… TR 38.901 Table 7.5-3
Cross-polarization XPR 7–12 dB (LOS), 0–9 dB (NLOS) Log-normal distributed
Per-cluster shadowing \(\sigma = 3\) dB All scenarios
Delay scaling \(r_\tau\) 2.5 (UMa), 3.0 (UMi), 1.7 (InH) Controls exponential decay

10.3 Frequency-Dependent Adjustments

TR 38.901 extends to mmWave (FR2: 24.25–52.6 GHz) with key differences:

Parameter Sub-6 GHz (FR1) mmWave FR2
\(N\) clusters 20 6–12 (fewer scatterers)
NLOS PL exponent 3.5–3.9 3.4–4.2
Oxygen absorption Negligible 0.4–10 dB/km at 60 GHz
Building penetration 20–30 dB 40–80 dB (glass/concrete)
Coherence bandwidth ~200 kHz–2 MHz ~50–500 MHz
Typical use Macro cells, coverage Indoor hotspot, backhaul
At 60 GHz, oxygen absorption is 15 dB/km. A 200 m link loses 3 dB to oxygen alone. TR 38.901 §7.4.5 provides specific attenuation models for O2, H2O, rain, and foliage.

10.4 Quick-Reference Implementation Checklist

Implementing TR 38.901 in a simulator — step by step:

Step TR 38.901 Reference What to generate
1. Choose scenario §7.2 UMa / UMi / RMa / InH / InF
2. Draw large-scale params §7.5 Step 4 + Table 7.5-6 DS, ASD, ASA, ZSD, ZSA, K, SF (correlated)
3. Compute LOS probability §7.4.2 + Table 7.4.2-1 \(P_{\text{LOS}}(d_{\text{2D}})\)
4. Generate cluster delays §7.5 Step 5 \(\tau_n\) (exponential distribution, scale by DS)
5. Generate cluster powers §7.5 Step 6 \(P_n\) (from delays + per-cluster shadowing)
6. Generate AOD/AOA/ZOD/ZOA §7.5 Step 7 Cluster angles from Gaussian/Laplacian distribution
7. Generate ray angles §7.5 Step 8 ±offset from cluster angle per Table 7.5-3
8. Compute \(\mathbf{H}(t,\tau)\) §7.5 Step 11 + Eq. 7.5-22 Full MIMO channel tensor
9. Scale to desired DS §7.7.3 Multiply delays by \(DS_{\text{target}} / DS_{\text{nominal}}\)
10. Apply Doppler §7.5 Step 11 Multiply by \(\exp(j 2\pi v_{n,m} t)\)

10.5 Scenario Channel Complexity Comparison

TR 38.901 Scenario Channel Complexity Comparison
Normalized (0–1) comparison of 4 TR 38.901 scenarios across 5 channel complexity dimensions. RMa-LOS dominates mobility (rural high-speed), while UMa-NLOS leads in delay spread and path loss severity. InH-NLOS shows moderate delay/angle spread with very low Doppler owing to pedestrian-speed UEs.
§A Glossary

A.1 Acronyms

Acronym Meaning Acronym Meaning
GSCMGeometry-based Stochastic Channel Model TDLTapped Delay Line
CDLClustered Delay Line LOSLine-of-Sight
NLOSNon-Line-of-Sight UMaUrban Macro
UMiUrban Micro (street canyon) RMaRural Macro
InHIndoor Hotspot InFIndoor Factory
DSDelay Spread (RMS) ASDAzimuth Spread of Departure
ASAAzimuth Spread of Arrival ZSDZenith Spread of Departure
ZSAZenith Spread of Arrival XPRCross-Polarization Ratio
PDPPower Delay Profile PLPath Loss
SFShadow Fading MIMOMultiple-Input Multiple-Output
SVDSingular Value Decomposition ULAUniform Linear Array
UPAUniform Planar Array CSIChannel State Information
SRSSounding Reference Signal SSBSynchronization Signal Block
FR1Frequency Range 1 (sub-6 GHz) FR2Frequency Range 2 (mmWave, 24.25–52.6 GHz)
AWGNAdditive White Gaussian Noise CIRChannel Impulse Response
AoAAngle of Arrival (azimuth) AoDAngle of Departure (azimuth)
ZoAZenith Angle of Arrival ZoDZenith Angle of Departure
K-factorRician K-factor (LOS-to-scatter power ratio) ISDInter-Site Distance
PASPower Angular Spectrum BSBase Station (gNB node)
UEUser Equipment (terminal node)   

A.2 Key Parameters Glossary

Symbol Full Name Spec Ref Typical Value
\(\text{DS}\) Delay Spread (RMS) TR 38.901 §5.4 30–300 ns (sub-6 GHz)
\(\text{ASD}\) Azimuth Spread of Departure TR 38.901 §5.4 10°–65°
\(\text{ASA}\) Azimuth Spread of Arrival TR 38.901 §5.4 20°–75°
\(\text{ZSD}\) Zenith Spread of Departure TR 38.901 §5.4 5°–15°
\(\text{ZSA}\) Zenith Spread of Arrival TR 38.901 §5.4 5°–20°
\(K\) Rician K-factor TR 38.901 §5.4 \(-\infty\) (NLOS), 7–15 dB (LOS)
\(\sigma_{\text{SF}}\) Shadow Fading (std. dev.) TR 38.901 §5.4 4–8 dB
\(N\) Number of clusters TR 38.901 §7.5 20 (sub-6 GHz), 6–12 (mmWave)
\(M\) Rays per cluster TR 38.901 §7.5 20 (fixed)
\(\text{XPR}\) Cross-Polarization Ratio TR 38.901 §7.5 0–12 dB
\(d_{\text{corr}}\) Decorrelation distance TR 38.901 §7.6.3 10–120 m
\(r_\tau\) Delay scaling ratio TR 38.901 §7.5 1.7–3.0
\(d'_{\text{BP}}\) Breakpoint distance TR 38.901 §7.4.1 ~200–500 m (UMa)
\(\sigma_{\text{SF}}\) Shadow fading std. dev. TR 38.901 Table 7.4.1-1 4–8 dB
\(f_D\) Doppler spread (max) TR 38.901 §5.4 5–400 Hz (\(v/\lambda\))
Quick recall: For a 30 km/h UE at 3.5 GHz, the maximum Doppler is \(f_D = v/\lambda = (30/3.6) / (3\times10^8 / 3.5\times10^9) \approx 97\) Hz. The coherence time is approximately \(T_c \approx 0.423 / f_D \approx 4.4\) ms — comfortably longer than a 5G NR slot (0.5 ms at 30 kHz SCS), so the channel is nearly static within a slot.
Delay spread ↔ coherence bandwidth relationship: \[ B_c \approx \frac{1}{2\pi \cdot \text{DS}_{\text{RMS}}} \] (50% coherence threshold; for 90% threshold use \(B_c \approx 1/(5\tau_{\text{rms}})\) — both are valid approximations) For UMa-NLOS (\(\text{DS} = 363\) ns): \(B_c \approx 440\) kHz. A 5G NR subcarrier spacing of 15 kHz (RB bandwidth 180 kHz) sits well within \(B_c\), meaning each OFDM subcarrier experiences flat fading — the core assumption behind OFDM channel estimation.
§11 Channel Patterns in 3D — From Pedestrian to High-Speed Train

Previous sections built the statistical framework for 5G channels — path-loss exponents, shadow-fading margins, cluster geometry. This section brings the channel to life in time. We simulate real fading traces using two classical tools that underpin every 3GPP channel model:

  • Jakes' oscillator model (Jakes 1974, building on Clarke 1968) — generates time-varying Rayleigh fading envelopes by summing sinusoids at angles uniformly distributed around a ring of scatterers. Referenced in TR 38.901 §5.4.2 as the basis for the Doppler power spectrum.
  • TDL-A tap structure — the tapped-delay-line model from TR 38.901 Table 7.7.2-1 provides the delay/power profile for each of the 8 dominant multipath taps. Each tap is independently faded by a Jakes process.

We explore three velocity scenarios that span the full 5G NR deployment range:

  • Pedestrian v = 3 km/h → fD = 9.7 Hz
  • Vehicle v = 120 km/h → fD = 389 Hz
  • High-Speed Train v = 350 km/h → fD = 1134 Hz

All Doppler shifts are computed for fc = 3.5 GHz (mid-band 5G NR): \( f_D = v \cdot f_c / c \).

The Jakes fading model for one multipath tap is:

(11.1) \[ h(t) = \sqrt{\frac{2}{N_{\text{osc}}}} \sum_{n=1}^{N_{\text{osc}}} \cos\!\left(2\pi f_D \cos\!\left(\frac{2\pi n}{N_{\text{osc}}}\right) t + \phi_n \right) \]

where \(N_{\text{osc}}\) is the number of oscillators (typically 8–24), \(f_D\) is the maximum Doppler shift, and \(\phi_n\) are independent random initial phases. Each oscillator represents a scatterer at angle \(2\pi n / N_{\text{osc}}\) around the unit circle. The \(\sqrt{2/N_{\text{osc}}}\) normalisation ensures unit mean power.

The resulting Doppler Power Spectral Density (Clarke's isotropic scattering model) is:

(11.2) \[ S(f) = \frac{1}{\pi f_D \sqrt{1 - (f/f_D)^2}}, \qquad |f| < f_D \]

The characteristic U-shaped spectrum with integrable singularities at \(\pm f_D\) arises from the cosine projection of isotropic scatterer angles onto the velocity axis. Spec ref: TR 38.901 §5.4.2, TR 38.901 §7.7.1.

The coherence time — the time interval over which the channel is approximately constant — is derived from the Doppler spread as:

(11.3) \[ T_c \approx \frac{0.423}{f_D} \]

For the pedestrian, \(T_c \approx 43.6\) ms. For the train, \(T_c \approx 0.37\) ms — shorter than a single NR slot at numerology \(\mu = 1\).

3D Time-Varying Channel — TDL-A Structure

3D Channel Impulse Response: Pedestrian (Left) vs Vehicle (Right) — TDL-A @ 3.5 GHz
Top: Pedestrian (v = 3 km/h, fD = 9.7 Hz, Ttotal = 500 ms). Amplitude ridges persist for ~100 ms along the time axis — the channel is nearly frozen. Bottom: Vehicle (v = 120 km/h, fD = 389 Hz, Ttotal = 50 ms). The surface ripples rapidly — a full fade cycle occurs every ~2.5 ms (Tc = 0.423/fD). Each row of the surface is one TDL-A tap; the delay spread structure is visible vertically. Colorscale: Viridis (pedestrian), Plasma (vehicle).
Everyday Analogy — Window on a Train: Look out a train window at night and watch the streetlights. Each light blinks at the same frequency, but as the train speeds up, the lights seem to flash faster (higher Doppler). The 3D surface shows exactly this: the vehicle surface has faster ripples along the time axis than the pedestrian surface. Each horizontal "stripe" in the surface corresponds to one multipath tap — a reflection off a different object at a different distance.

Fading Time Series — Three Velocity Regimes

The dominant tap (tap index 2, 0 dB power) is simulated for each scenario. The absolute time scale differs by two orders of magnitude between pedestrian and train — illustrating why the same NR slot structure behaves very differently depending on UE velocity.

Channel Fading: Pedestrian — 3 km/h, fD = 9.7 Hz
Channel Fading: Vehicle — 120 km/h, fD = 389 Hz
Channel Fading: High-Speed Train — 350 km/h, fD = 1134 Hz
Red shading marks deep fades (amplitude more than 15 dB below mean). Each scenario uses 2000 time points over its respective observation window. Note that the number of fades per unit time scales exactly with velocity — the train experiences ~117× more fades per second than the pedestrian.
ScenarioVelocityfD Tc = 0.423/fDFades/second
Pedestrian3 km/h9.7 Hz43.6 ms~22
Vehicle120 km/h389 Hz1.09 ms~900
High-Speed Train350 km/h1134 Hz0.37 ms~2600
Everyday Analogy — Puddle Reflections: A puddle reflects sunlight unpredictably — sometimes bright, sometimes dark — as your viewing angle changes slightly. A deep radio fade is the same phenomenon: multipath components cancel destructively, causing a sudden -20 to -30 dB drop in received power over just a few centimetres of movement. For a pedestrian at 3 km/h, a deep fade might last 10–50 ms. For a vehicle at 120 km/h, it lasts only 0.2–1 ms — too brief for most retransmission protocols to catch unless the HARQ round-trip time is below 1 ms.

Deep Fade Statistics — Rayleigh Envelope CDF

The Rayleigh distribution governs the amplitude envelope in rich-scattering NLOS environments. Understanding the probability of deep fades is critical for link budget margin setting and diversity order requirements. TR 38.901 §7.7.1 mandates Rayleigh-faded small-scale components for NLOS conditions.

Deep Fade Depth CDF — Pedestrian Channel (Rayleigh Fading)
Empirical CDF from 5000-point Jakes simulation vs. theoretical Rayleigh CDF. Vertical dashed lines mark the -5, -10, -15, -20 dB levels. The log-scale y-axis reveals the tail behaviour critical for outage probability calculations.
Rayleigh fading outage probability: For a Rayleigh envelope with mean power \(\Omega\), the probability that the instantaneous power falls below a threshold \(\gamma_0\) is: \[ P(\text{outage}) = 1 - e^{-\gamma_0 / \Omega} \] At −20 dB below mean: \(\gamma_0 / \Omega = 10^{-20/10} = 0.01\), so \( P = 1 - e^{-0.01} \approx 0.01 \) — meaning 1% of time is in deep fade at any given instant. This is why diversity order (spatial, frequency, or time) is critical: each independent branch has an independent 1% deep-fade probability, and with \(N\) branches the outage drops to \(\approx (0.01)^N\). With 4 Rx antennas: \(10^{-8}\) — negligible.

Delay-Doppler Spreading Function — LOS vs Rich NLOS

The delay-Doppler spreading function \(H(\tau, \nu)\) is the 2D Fourier transform of the time-varying channel impulse response \(h(t, \tau)\). It captures both the delay spread (multipath) and Doppler spread (mobility) in a single 2D representation. TR 38.901 §5.4.2 defines the spreading function formally.

LOS channels appear as sparse, well-defined clusters in this domain. NLOS channels show broad, smeared energy — harder to estimate and equalize. This sparsity motivates OTFS modulation.

Delay-Doppler Spreading Function: LOS (Left) vs Rich NLOS — TDL-A (Right)
LOS: Two point clusters — the dominant LOS component at (τ=0, ν=+fD) and a weak reflection. The delay-Doppler plane is nearly empty. NLOS (TDL-A): Eight clusters spread across 0–250 ns delay, each broadened by ±fD/2 Doppler. The spreading function fills a rectangle in the delay-Doppler plane. Vehicle scenario: fD = 389 Hz.
OTFS Modulation and the Delay-Doppler Domain: The delay-Doppler spreading function is the fundamental basis for OTFS (Orthogonal Time-Frequency Space) modulation, a leading candidate for 6G high-mobility channels. In OFDM, the channel \(H[k,n]\) varies across all time-frequency resource elements — requiring dense pilot grids and per-subcarrier equalization. In OTFS, modulation is performed directly in the delay-Doppler domain: the channel appears as a sparse, near-static 2D map with only a few non-zero entries (one per reflector). This dramatically reduces pilot overhead and simplifies equalization for UEs at 350+ km/h. See TR 38.901 §5.4.2 for the spreading function definition.

Doppler Power Spectral Density — The U-Shaped Clarke Spectrum

The Doppler PSD shape directly determines how pilot subcarriers must be spaced in frequency to track channel variations. For NR, the minimum pilot density is set to satisfy the Nyquist criterion in both time (sampling at ≥ 2fD) and frequency domains. TR 38.901 §5.4.2, TS 38.211 §5.4.

Doppler PSD — Clarke/Jakes Spectrum for Three Velocities
The U-shaped spectrum (integrable singularities at ±fD) arises from the cosine projection of isotropic scatterer angles onto the velocity axis: more scatterers contribute near the ±90° broadside angles. In practice, the singularities are smoothed by finite scatterer density and antenna patterns. The train PSD (red) extends to ±1134 Hz — requiring DMRS pilot density far beyond pedestrian scenarios. Reference: Clarke (1968), Jakes (1974).
NR Implication: 5G NR defines DMRS (DeModulation Reference Signal) patterns in TS 38.211 §7.4.1. DMRS Type 1 places 6 pilots per OFDM symbol (every other subcarrier). For the pedestrian case (fD = 9.7 Hz, Tc = 43.6 ms), one DMRS symbol per 10 slots is adequate. For the train (fD = 1134 Hz, Tc = 0.37 ms), DMRS symbols may be needed every 2–3 OFDM symbols to prevent channel estimation from going stale.
5G NR Slot Duration vs Channel Coherence: The NR frame has 1 ms subframes. At numerology μ = 1 (30 kHz SCS), a slot is 0.5 ms. For a high-speed train at 350 km/h, Tc ≈ 0.37 ms — shorter than a single NR slot. This means the channel may change within a single slot, violating the quasi-static assumption that underpins most NR channel estimation designs. Mitigation requires: (1) DMRS pattern Type 2 with 4 pilot symbols per slot (vs. Type 1 with 2), (2) smaller slot duration via higher numerology (μ = 2, μ = 3), or (3) predictive channel tracking using Kalman filtering over the Jakes Doppler model. Reference: TS 38.211 §7.4.1.

Coherence Map — All TR 38.901 Deployment Scenarios

The coherence bandwidth \(B_c = 1/(5 \cdot \sigma_\tau)\) and coherence time \(T_c = 0.423/f_D\) define the 2D "sweet spot" for OFDM design: the subcarrier spacing must be much less than \(B_c\), and the OFDM symbol duration must be much less than \(T_c\). Plotting all TR 38.901 scenarios on a single log-log coherence map reveals the operating envelope.

Spec refs: TR 38.901 §7.7.1 (scenario definitions), TR 38.901 §5.4.2 (delay/Doppler spread parameters), TS 38.211 §5.4 (NR numerology).

Coherence Time vs Coherence Bandwidth — TR 38.901 Scenarios
Upper-right quadrant (large Bc, large Tc) = OFDM's sweet spot: wide coherence bandwidth allows large subcarrier spacing; long coherence time means few pilot symbols needed. Lower-left quadrant (small Bc, small Tc) = doubly-selective fading — the hardest regime for OFDM. High-speed train in urban NLOS sits here. Marker size scales with inter-site distance (ISD). The vertical dashed line marks 15 kHz subcarrier spacing (NR μ=0 Bc reference).
Design Warning — UMa NLOS + High-Speed Train: The bottom-left point (UMa NLOS, train velocity) has Bc ≈ 138 kHz and Tc ≈ 0.37 ms. The NR μ = 1 subcarrier spacing (30 kHz) is comfortably below Bc, but the coherence time is shorter than a slot. Frequency-domain interpolation of DMRS is safe; time-domain interpolation is not. This motivates per-OFDM-symbol channel tracking for high-speed railway deployments.

Specification References

ReferenceTopicRelevance to §11
TR 38.901 §5.4.2 Doppler power spectrum Clarke/Jakes U-shaped PSD definition; spreading function \(H(\tau,\nu)\)
TR 38.901 §7.7.1 General channel model approach Small-scale fading model; Rayleigh for NLOS, Rician for LOS
TR 38.901 Table 7.7.2-1 TDL-A tap structure Delay/power profile used for all 3D surface plots in this section
TR 38.901 §7.7.3 CDL model (cluster delay line) LOS channel represented in delay-Doppler chart (CDL-D like)
TS 38.211 §5.4 Doppler in NR physical layer Carrier frequency offset; Doppler pre-compensation for HST
TS 38.211 §7.4.1 DMRS patterns Pilot density requirements driven by coherence time analysis
Clarke (1968) Statistical theory of mobile-radio reception Isotropic scattering model; U-shaped Doppler PSD derivation (informative)
Jakes (1974) Microwave Mobile Communications Oscillator-sum simulation model for Rayleigh fading (informative)