5G NR Frame Structure: Numerology & Slot Scaling

1. The Mathematical Foundation: Numerology ( $\mu$ ) Scaling

The defining characteristic of 5G NR is Flexible Numerology. In legacy 4G LTE, the Subcarrier Spacing (SCS) was locked at $15\,\text{kHz}$ . This was optimal for sub-3GHz coverage but fundamentally limited for high-frequency or low-latency operations. 5G NR introduces the concept of an exponential scaling factor $\mu$ .

The Scaling Law

\Delta f = 15 \times 2^\mu\,\text{kHz}

As $\mu$ increases, the subcarrier spacing doubles, and the symbol duration $T_s$ shrinks by half. This relationship is governed by the Inverse Fourier Transform physics: $T_s = 1/\Delta f$ .

Index ( $\mu$ )	SCS ( $\text{kHz}$ )	Slot Time	Primary Spectrum	Use Case
0	15	1000 μs	Sub-3 GHz	Coverage / FDD Coexistence
1	30	500 μs	3.5 GHz (n78)	eMBB Standard
2	60	250 μs	5 GHz / URLLC	Industrial Automation
3	120	125 μs	28 GHz (FR2-1)	Fixed Wireless Access
4	240	62.5 μs	71 GHz (FR2-2)	Intra-Data Center

2. The Resource Grid Physics: RE, PRB & REG

The 5G NR resource grid is a 2D mapping of time and frequency. Every unit of allocation must be precisely defined to ensure orthogonality and prevent inter-carrier interference.

Resource Element (RE)

The smallest physical unit. 1 Subcarrier x 1 OFDM Symbol. In a 256-QAM scheme, a single RE carries 8 bits of information.

RE = (k, l) \quad \text{where } k \in \text{Freq}, l \in \text{Time}

Physical Resource Block (PRB)

Consists of 12 consecutive subcarriers. Unlike LTE, there is no fixed time-domain length for a PRB; it spans the duration of the allocation (typically 1 slot).

PRB_{BW} = 12 \cdot 15 \cdot 2^\mu\,\text{kHz}

For a $100\,\text{MHz}$ carrier at $\mu=1$ (30kHz), the number of PRBs is calculated as:

N_{PRB} = \frac{100,000\,\text{kHz} - \text{Guard Band}}{30\,\text{kHz} \times 12} \approx 273\,\text{PRBs}

This density allows for massive multi-user multiplexing (MU-MIMO) where dozens of users share the same grid.

Flexible Numerology Simulator

Subframe (1ms) Slot Scaling Analysis

Subcarrier Spacing

30 kHz

Slot Duration

500 µs

Slots / 1ms

START (0ms)END (1ms)

SLOT 0

SLOT 1

Mid-band / Urban

Scientific Context: As Carrier Spacing (SCS) doubles, the slot duration halves. In mmWave ($\mu=3$), we have 8 slots per millisecond, allowing the gNodeB to make scheduling decisions every 125 microseconds—essential for self-driving vehicles and high-speed industrial robotics.

Fig 2.1: Dynamic 5G NR Slot Structure Simulation

3. Channel Coding Evolution: LDPC & Polar Codes

5G NR abandons the Turbo Codes used in 3G/4G in favor of a more modern, hardware-parallelizable coding suite. This shift was necessary to handle the 20Gbps+ peak rates of 5G.

LDPC for Data (PDSCH/PUSCH)

Low-Density Parity-Check (LDPC) codes are used for all shared data channels. Unlike Turbo codes, which are inherently serial, LDPC decoding can be fully parallelized in silicon (ASICs). 3GPP defines two base graphs (BG1 for large blocks/high rates, BG2 for small blocks/low rates) to optimize the parity matrix.

H \mathbf{c}^T = 0 \quad \text{(Parity Check Consistency)}

Polar Codes for Control (PDCCH/PBCH)

For short blocks like control signaling, Polar codes are superior. They "polarize" the bit channels into perfectly reliable or completely unreliable states. By only transmitting information on the "frozen" reliable channels, Polar codes can reach the Shannon Limit for short block lengths.

4. Synchronization & Initial Access: The SSB Beam-Sweep

In a beamformed network, a base station doesn't just broadcast; it "sweeps." The Synchronization Signal Block (SSB) is the mechanism for this. An SSB consists of the PSS, SSS, and PBCH (Master Information Block).

The SSB Burst Pattern

Up to $L$ number of SSBs are transmitted in a $5\,\text{ms}$ window. Each SSB is mapped to a different spatial beam.

Sub-3 GHzL = 4 beams
3-6 GHzL = 8 beams
mmWave (FR2)L = 64 beams

A UE measures the RSRP of each beam and reports back the best SSB index during the RACH procedure, allowing the gNodeB to "lock on" to the user's location.

5. Bandwidth Parts (BWP): Dynamic Power Forensics

In LTE, devices had to monitor the entire carrier. In 5G, with 400MHz carriers, this would drain a battery in minutes. Bandwidth Parts (BWP) solve this by allowing the UE to see only a slice of the spectrum.

Initial BWP

Used for RACH and RRC connection setup. Usually narrow (e.g., 20MHz) for compatibility.

Active BWP

The current operating slice. Can be switched via DCI signaling in under $1\,\text{ms}$ .

Default BWP

The "fallback" slice. If no data activity occurs, the UE retunes here to save energy.

6. HARQ Timing Forensics: K0, K1, K2 Logic

Timing in 5G is no longer deterministic. It is Dynamic. The relationship between control, data, and feedback is defined by three primary offsets.

DCI to DL Data

Offset between PDCCH (Control) and PDSCH (Data). In high-performance 5G, $K0=0$ (Self-contained slot).

DL Data to ACK/NACK

The "Processing Budget." Defines how many slots the UE has to decode data and send feedback on the PUCCH.

DCI to UL Data

Offset between the UL Grant and the actual PUSCH transmission from the UE.

7. TDD Slot Formats: The 61 Flavors

5G NR doesn't just have UL/DL; it has Flexible (F) symbols. This allows for Dynamic TDD, where a cell can change its uplink/downlink ratio based on demand.

8. Phase Tracking Reference Signals (PTRS)

At high frequencies (mmWave), Phase Noise becomes a critical bottleneck. PTRS is a new reference signal introduced in 5G NR to track the common phase error (CPE) in the time domain. Unlike DMRS which is sparse in time, PTRS is inserted into every OFDM symbol to allow the receiver to compensate for oscillator jitter.

9. 5G NR Technical Encyclopedia

Numerology

The configuration of subcarrier spacing (SCS) and cyclic prefix (CP).

SCS

Subcarrier Spacing; the frequency distance between adjacent subcarriers.

PRB

Physical Resource Block; 12 consecutive subcarriers.

CORESET

Control Resource Set; time-frequency regions where the UE monitors for control signaling.

BWP

Bandwidth Part; a contiguous slice of the carrier bandwidth.

SSB

Synchronization Signal Block; contains sync signals and broadcast channel.

DMRS

De-Modulation Reference Signal; used for channel estimation.

PTRS

Phase Tracking Reference Signal; combats phase noise in high-order QAM.

CSI-RS

Channel State Information Reference Signal; used for beam management.

LDPC

Low-Density Parity-Check; channel coding for data channels.

Polar Codes

Channel coding for control and broadcast channels.

HARQ

Hybrid Automatic Repeat Request; fast retransmission protocol.

Frequently Asked Questions

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Beam Management in Millimeter-Wave NR

At frequencies above 24 GHz (mmWave), 5G NR relies on narrow directional beams formed by phased array antennas to overcome the high path loss. The beam management procedure — the process by which the gNB and UE establish, maintain, and switch beams — is one of the most complex aspects of the NR physical layer. The 3GPP specification defines three phases: initial beam acquisition, beam refinement, and beam recovery.

In initial beam acquisition, the gNB sweeps a set of Synchronization Signal Block (SSB) beams across the cell sector, each beam transmitted in a different time slot. The SSB burst set consists of up to 64 beams in FR2 (24–52 GHz), each occupying 4 OFDM symbols. The UE measures the received signal power (SS-RSRP) on each SSB index and reports the best beam to the gNB. The number of beams and their angular width depend on the antenna array configuration: a 4×4 array with 16 elements can form beams with approximately 15° half-power beamwidth, requiring 24 beams to cover a 120° sector. An 8×8 array with 64 elements achieves 7° beamwidth, requiring 48 beams for the same sector coverage.

Beam refinement uses Channel State Information Reference Signals (CSI-RS) with a finer angular granularity than SSB. The gNB transmits CSI-RS beams on a subset of candidate directions around the current serving beam, and the UE reports the best refined beam. This process can be repeated periodically (every 20–80 ms) to track the UE as it moves. The tracking range depends on the beamwidth: for a 15° beam, the UE can move approximately 5–7 meters before the beam must be switched at a 100 meter cell radius. Beam switching latency — the time from the UE crossing the beam boundary to the gNB switching to the new beam — is typically 1–3 ms excluding MAC layer scheduling delays, achieved by maintaining multiple candidate beam hypotheses in the gNB scheduler.

CSI Acquisition and Precoding in Frequency Division Duplex

While TDD (Time Division Duplex) benefits from channel reciprocity — the downlink channel can be estimated from uplink Sounding Reference Signals — FDD (Frequency Division Duplex) systems require explicit Channel State Information (CSI) feedback from the UE because the uplink and downlink frequencies are separated by several hundred MHz, making the channels uncorrelated. The CSI feedback framework in 5G NR is a multi-stage process designed to balance reporting overhead against channel accuracy.

The gNB transmits CSI-RS on configured resources (up to 32 ports in NR Release 15, expanding to 64 in Release 17). The UE measures the channel matrix on each port and computes the preferredprecoding matrix indicator (PMI), rank indicator (RI), and channel quality indicator (CQI). The PMI is selected from a predefined codebook of precoding matrices. In NR Type I codebook (single-panel), the UE reports a single preferred beam with a co-phasing coefficient between polarizations. Type II codebook (dual-panel) provides higher resolution by reporting a linear combination of multiple beams with amplitude and phase coefficients — requiring 5–10x more feedback bits but providing 1–2 dB of precoding gain improvement.

W = \frac{1}{\sqrt{p}} \sum_{i=1}^{L} a_i b_i v_i^T

The Type II precoder structure as a linear combination of $L$ beams, each with amplitude $a_i$ , spatial beam direction $b_i$ , and polarization coefficient $v_i$ .

The CSI reporting periodicity is configurable from 5 ms to 160 ms. Aperiodic CSI (triggered by DCI format 0_1) is used for fast channel tracking during UE mobility. The trade-off between CSI accuracy and overhead is managed through CSI compression using discrete Fourier transform (DFT) basis sets — the UE reports only the dominant coefficients in the angular-delay domain, achieving up to 10x compression with less than 0.5 dB of precoding loss. For FDD massive MIMO systems with 64 antennas, this compression is essential to keep the uplink feedback overhead below 1% of the total uplink capacity while maintaining the spatial multiplexing gain of 4–8 streams per UE.