Ferkans — Interactive Telecom Tutor

The Golden Thread of Chapter 22

Parts I-IV of this book established a theoretical object: a base station with a large antenna array, a reciprocity-based TDD operation, and a linear precoder that is near-optimal when $N_t \gg K$ . 5G NR takes this object and stuffs it into a standardized air interface whose fundamental unit of time is a slot of about $1$ ms, whose frequency dimension is partitioned into subcarriers via OFDM, and whose CSI feedback is quantized into a codebook index of tens to hundreds of bits.

The question Part V takes up is not whether massive MIMO can work in theory — Chapters 1-10 settled that — but how much of the theoretical gain survives the engineering realities: a pilot budget that competes with data symbols for slot resources, a codebook that discretizes the precoder, a beam management procedure that must track users in mobility, and TRPs that coordinate with each other across a non-ideal backhaul. Section 22.1 builds the scaffolding — the frame structure — and subsequent sections hang the pieces on it.

Definition:
NR Numerology and Slot Structure

5G New Radio (3GPP Release 15 and later) supports a family of OFDM numerologies indexed by an integer $\mu \in \{0, 1, 2, 3, 4\}$ . For numerology $\mu$ , the subcarrier spacing is $\Delta f(\mu) = 15\,\text{kHz} \cdot 2^{\mu},$ and the slot duration is $T_{\text{slot}}(\mu) = \frac{1\,\text{ms}}{2^{\mu}} = \frac{14}{\Delta f(\mu) \cdot 14} = 14 \cdot T_{\text{OFDM}}(\mu),$ where every slot contains exactly 14 OFDM symbols regardless of $\mu$ . The table below summarizes the five numerologies:

$\mu$	$\Delta f$ (kHz)	$T_{\text{slot}}$	Typical band
0	15	$1$ ms	FR1 sub-3 GHz
1	30	$0.5$ ms	FR1 3-6 GHz
2	60	$0.25$ ms	FR1 / FR2
3	120	$0.125$ ms	FR2 mmWave
4	240	$0.0625$ ms	FR2 (SSB only)

The parameter $\mu$ is chosen to trade off inter-carrier interference (from Doppler or phase noise, which push toward large $\Delta f$ ) against inter-symbol interference (from large $\sigma_\tau$ , which pushes toward small $\Delta f$ and long cyclic prefix).

The common choices in commercial deployments are $\mu = 0$ at sub-1 GHz, $\mu = 1$ at mid-band (3.5 GHz), and $\mu = 3$ at mmWave (28 GHz and above). Numerology $\mu = 4$ is reserved for initial access signals.

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Definition:
Frequency Range 1 and Frequency Range 2

3GPP partitions the NR spectrum into two non-overlapping ranges:

FR1: $410$ MHz to $7.125$ GHz. Sub-6 GHz. Low path loss, good diffraction, near-far-field transition rarely relevant. Typical $N_t \in \{32, 64, 128\}$ in a panel-like active antenna unit; typical $W \in \{10, 20, 40, 100\}$ MHz per component carrier.
FR2: $24.25$ GHz to $71$ GHz (FR2-1) and up to $114$ GHz (FR2-2, added in Rel-17). Millimeter-wave. Severe path loss ( $\sim 20\log_{10}(f_0)$ extra Friis loss relative to FR1), sparse channels, directional beams mandatory. Typical $N_t \in \{64, 128, 256\}$ in a hybrid-beamforming module; typical $W \in \{100, 200, 400, 800\}$ MHz.

The operating-point difference is stark: at FR1 an omnidirectional receive pattern is viable and digital per-antenna processing is the default; at FR2 the system must first acquire a beam direction before any useful link budget exists, and the RF chain count is a hard budget driven by cost and DC power (see Chapter 20 for hybrid beamforming).

Key Takeaway

One slot, 14 symbols, fixed. The NR slot is the atom of scheduling. Whatever the pilot, the CSI feedback, the data transmission, and the control signaling need to do, they must fit into 14 OFDM symbols. Any additional overhead beyond data comes directly out of data capacity. The engineering game of 5G NR massive MIMO is to squeeze the CSI acquisition and beam management into a small fraction of these 14 symbols while retaining enough accuracy for near-optimal precoding.

Theorem: TDD Slot-Budget Inequality

Consider a single-cell massive MIMO system in TDD mode at numerology $\mu$ , serving $K$ users in a coherence block of $\tau_c$ symbols. Let $\tau_p$ be the number of symbols allocated to orthogonal uplink SRS (pilot) transmission. The ergodic sum rate is upper-bounded (up to the UatF pre-log factor) by $\bar{C}_{\text{sum}} \leq \left(1 - \frac{\tau_p}{\tau_c}\right) \sum_{k=1}^{K} \log_2\!\left(1 + \text{SINR}_k(\tau_p)\right),$ where $\tau_p \geq K$ is required for orthogonal pilot assignment and $\text{SINR}_k(\tau_p)$ is monotonically non-decreasing in $\tau_p$ .

Pilots and data compete for symbols in the coherence block. Short pilots leave more slots for data but give noisier channel estimates that hurt precoding; long pilots leave fewer data slots but give cleaner estimates. The optimal $\tau_p$ is found by balancing the pre-log $(1 - \tau_p/\tau_c)$ against the SINR improvement, and at fixed $K$ the sum rate grows only as $\log \tau_c$ — the coherence block sets a hard ceiling.

Show Hint

Write the uplink SRS signal model as $\mathbf{S}_{i,k} = \sqrt{\text{SNR}\,\tau_p}\,\mathbf{H} + \mathbf{w}$ after pilot despreading.

Apply the MMSE estimate variance formula to obtain $\text{Var}(\hat{\mathbf{H}}) = \text{SNR}\,\tau_p\,\mathbf{R}_k / (1 + \text{SNR}\,\tau_p\,\text{tr}(\mathbf{R}_k))$ .

Plug the MMSE estimate into the use-and-then-forget (UatF) rate formula.

Proof

Pilot-phase signal model

In a TDD slot, the first $\tau_p$ symbols are allocated to uplink SRS. After despreading against orthogonal pilot sequences, the BS receives $\sqrt{\tau_p P_p}\,\mathbf{H}_{k} + \mathbf{w}_{k}$ for each user, where $P_p$ is the per-symbol pilot power. Only the fraction $(\tau_c - \tau_p)/\tau_c$ of symbols carries data.

MMSE channel estimate

The MMSE estimate $\hat{\mathbf{H}}_k$ of $\mathbf{H}_{k}$ has error covariance $\mathbb{E}[\|\mathbf{H}_{k} - \hat{\mathbf{H}}_k\|^2] = \text{tr}(\mathbf{R}_k) - \frac{\tau_p\,\text{SNR}\,\text{tr}(\mathbf{R}_k^2)}{1 + \tau_p\,\text{SNR}\,\text{tr}(\mathbf{R}_k)}.$ The estimation error is monotonically non-increasing in $\tau_p$ , so $\text{SINR}_k$ is non-decreasing.

Sum rate with pre-log

Assigning the useful data fraction $1 - \tau_p/\tau_c$ to each user's $\log_2(1 + \text{SINR}_k)$ bound and summing gives the statement. The $\tau_p \geq K$ constraint ensures orthogonal pilot assignment. $\blacksquare$

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NR Slot Budget vs Numerology and Pilot Overhead

Explore how the effective data fraction $(1 - \tau_p/\tau_c)$ varies with the NR numerology $\mu$ , the user count $K$ , and the CSI-RS periodicity. Longer slots (small $\mu$ ) give more headroom for pilots at low mobility; short slots (large $\mu$ ) are needed at high Doppler but hurt the pilot fraction.

Parameters

Numerology

\mu

1

Users

K

8

UE speed (km/h)30

Carrier

f_0

(GHz)3.5

Example: Slot Budget for $\mu = 1$ at 3.5 GHz

A 3.5 GHz cell uses numerology $\mu = 1$ ( $\Delta f = 30$ kHz, $T_{\text{slot}} = 0.5$ ms) and serves $K = 10$ users at $3$ km/h pedestrian mobility. The UE has a cyclic-prefix overhead of $\approx 7\%$ . Compute the CSI-RS periodicity that keeps the channel estimate within $0.5$ symbols of the slot boundary.

Solution

Doppler

At $f_0 = 3.5$ GHz and $v = 3$ km/h $= 0.83$ m/s, the maximum Doppler is $f_D = v/\lambda = v\,f_0/c \approx 9.7$ Hz.

Coherence time

$T_c \approx 0.423/f_D \approx 44$ ms, equivalent to $88$ slots at $\mu = 1$ .

Periodicity choice

The standard CSI-RS periodicities are $\{5, 10, 20, 40, 80\}$ slots. Choosing $T_{\text{CSI}} = 40$ slots gives a CSI refresh every $20$ ms — well inside $T_c$ — and leaves room for other control signals. With $K = 10$ and $40$ data slots between CSI updates, the pilot fraction is $10/(14 \cdot 40) \approx 1.8\%$ . The UatF rate loses about $0.03$ bits/s/Hz per user from this overhead alone. $\blacksquare$

Definition:
TDD and FDD Modes in NR

5G NR supports both time-division duplex (TDD) and frequency-division duplex (FDD) operation. In TDD, uplink and downlink share the same carrier and are separated in time via a slot format containing D (downlink), U (uplink), and F (flexible) symbols; reciprocity can be exploited for CSI acquisition. In FDD, uplink and downlink occupy separate paired carriers and no reciprocity exists; CSI must be fed back explicitly by the UE.

TDD: All 5G NR mid-band deployments (n77, n78, n79 at 3.5 GHz) and all mmWave deployments use TDD. SRS-based reciprocity provides precoder CSI for the downlink; the pilot budget scales with $K$ , not $N_t$ .
FDD: Most sub-3 GHz deployments (n1, n3 at 2.1 GHz) use FDD for backward-compatibility with LTE. CSI must be fed back via codebooks (Section 22.3); the feedback overhead scales with $N_t$ , which forces Type II codebooks and limits effective port counts.

This duplexing choice drives nearly every other design decision in the NR MIMO framework, and is why the Type II codebook exists at all.

TDD vs FDD for Massive MIMO

Aspect	TDD	FDD
CSI source for precoding	SRS (uplink, reciprocity)	Downlink CSI-RS + UE feedback
Pilot overhead scaling	$\tau_p \sim K$	$\tau_p \sim N_t$ without structure
Limit as $N_t \to \infty$	Feasible	Infeasible without JSDM
Codebook needed?	No	Yes (Type I / Type II)
Commercial NR bands	n77/n78/n79 (3.5 GHz), mmWave	n1/n3 (2.1 GHz), n41 (2.5 GHz FDD pair)
Calibration needed?	Yes (TX/RX RF mismatch)	No (RX chain only)
Backhaul latency sensitivity	Low	Medium (CSI ageing)

Common Mistake: Confusing OFDM Symbol with NR Slot

Mistake:

A common confusion is to treat the NR slot as a fundamental time unit independent of numerology, or to equate one slot with one OFDM symbol.

Correction:

An NR slot is always 14 OFDM symbols, regardless of numerology. What changes with $\mu$ is the length of each symbol: $T_{\text{OFDM}}(\mu) \propto 2^{-\mu}$ . Slot duration shrinks from $1$ ms at $\mu = 0$ to $62.5$ µs at $\mu = 4$ , but the symbol count stays at 14. Pilot and data resource allocation is counted in symbols within the slot, not in the slot itself. When comparing overhead across numerologies, always normalize to the coherence block length $\tau_c$ in symbols.

Historical Note: From LTE to NR: Why 14 Symbols per Slot?

2015-2018

The choice of 14 OFDM symbols per slot in NR is inherited from LTE's normal-cyclic-prefix subframe (1 ms, two slots of 7 symbols). Keeping the symbol count the same allowed existing LTE channel coding, HARQ, and MAC procedures to be adapted rather than redesigned. What NR changed was the scalable numerology: LTE was locked at $\Delta f = 15$ kHz, while NR allows $\Delta f$ to scale by powers of two, enabling the same frame structure to operate at $f_0$ from $410$ MHz to $114$ GHz. The decision was crystallized at the March 2017 3GPP RAN meeting (Dubrovnik) after months of debate over whether to retain 14 symbols or switch to a shorter "mini-slot" as the default.

NR Frame, Subframe, Slot, and Symbol Hierarchy — The NR frame (10 ms) contains 10 subframes (1 ms each). Each subframe contains $2^{\mu}$ slots of 14 OFDM symbols. Numerology $\mu$ controls both the subcarrier spacing and the slot duration, but always preserves the 14-symbol structure.

NR Numerology

The integer $\mu \in \{0,1,2,3,4\}$ that selects the subcarrier spacing $\Delta f = 15\,\text{kHz}\cdot 2^{\mu}$ and, correspondingly, the slot duration $1\,\text{ms}/2^{\mu}$ . Larger $\mu$ gives shorter slots and wider subcarriers, suited to mmWave bands with high phase noise and Doppler.

Frequency Range (FR1, FR2)

3GPP's partition of the 5G spectrum into FR1 (sub-7.125 GHz) and FR2 (24.25-71 GHz for FR2-1, up to 114 GHz for FR2-2 in Rel-17). FR1 deployments use digital or fully-digital massive MIMO; FR2 deployments use hybrid beamforming with codebook-based beam management.

Quick Check

At numerology $\mu = 3$ ( $\Delta f = 120$ kHz), how long is one NR slot?

$1$ ms

$0.5$ ms

$0.125$ ms

$14$ µs

Correction:

0.125

ms

$T_{\text{slot}}(\mu) = 1\,\text{ms}/2^{\mu} = 1/8 = 0.125$ ms at $\mu = 3$ . The slot still contains 14 OFDM symbols, each of duration $\approx 8.3$ µs.

Why This Matters: From NR to 6G

The 5G NR framing is already showing strain at the top of FR2: at $f_0 = 100$ GHz, phase noise argues for $\Delta f > 1$ MHz, which would require numerologies beyond $\mu = 4$ . Current 6G research (Chapters 25 and 27) is proposing new numerologies up to $\mu = 6$ ( $\Delta f = 960$ kHz) together with shorter slots. The NR frame structure is the last universal anchor before these extensions fork.

The NR MIMO Framework: Numerology, FR1/FR2, TDD/FDD

The Golden Thread of Chapter 22

Definition: NR Numerology and Slot Structure

Definition: Frequency Range 1 and Frequency Range 2

Key Takeaway

Theorem: TDD Slot-Budget Inequality

Pilot-phase signal model

MMSE channel estimate

Sum rate with pre-log

NR Slot Budget vs Numerology and Pilot Overhead

Parameters

Example: Slot Budget for μ=1\mu = 1μ=1 at 3.5 GHz

Doppler

Coherence time

Periodicity choice

Definition: TDD and FDD Modes in NR

TDD vs FDD for Massive MIMO

Common Mistake: Confusing OFDM Symbol with NR Slot

Historical Note: From LTE to NR: Why 14 Symbols per Slot?

NR Frame, Subframe, Slot, and Symbol Hierarchy

NR Numerology

Frequency Range (FR1, FR2)

Quick Check

Why This Matters: From NR to 6G

Definition:
NR Numerology and Slot Structure

Definition:
Frequency Range 1 and Frequency Range 2

Example: Slot Budget for $\mu = 1$ at 3.5 GHz

Definition:
TDD and FDD Modes in NR