OFDM, Numerology & Resource Grid

#1Why OFDM?

Traditional single-carrier systems transmit one modulated symbol per time slot. In wideband channels, this leads to Inter-Symbol Interference (ISI) because the channel's multipath delay spread causes adjacent symbols to overlap.

  Single-Carrier (Wide Symbol)                    Multi-Carrier / OFDM (Narrow Symbols)
  ────────────────────────────                    ─────────────────────────────────────

  │╲───╱╲───╱╲───╱│        Multipath              │╲╱╲╱╲╱╲╱│      Each subcarrier
  │  ╲ ╱  ╲ ╱  ╲ ╱ │  ────► ISI overlap            │  sub    sub    │  sees flat fading
  │   ╳    ╳    ╳   │                               │  carrier carrier│  (no ISI within
  │  ╱ ╲  ╱ ╲  ╱ ╲ │                               │               │   one symbol)
  │ ╱   ╲╱   ╲╱   ╲│                               └───────────────┘
  └─────────────────┘ t                             └───────────────┘ t
  One wide symbol                                   Many narrow symbols

OFDM solves this by splitting the data across N parallel subcarriers, each with a much longer symbol duration relative to the channel's delay spread.

Key Benefits

#2Orthogonality Principle

The word Orthogonal in OFDM means that subcarriers are designed so they do not interfere with each other, even though they overlap in frequency.

How Orthogonality Works

Two signals are orthogonal if their inner product over a given interval is zero:

  ∫₀ᵀˢ cos(2πf₁t) · cos(2πf₂t) dt = 0    when f₁ ≠ f₂ and f₁, f₂ are integer multiples of 1/Ts

This means we can pack subcarriers very close together — separated by exactly Δf = 1/Ts — and still perfectly separate them at the receiver.

At the receiver, multiplying the composite signal by a reference subcarrier and integrating (or equivalently, taking the FFT) isolates that subcarrier's data — all others cancel to zero.

#3OFDM System Architecture

Transmitter Chain

  Bits ──► QAM      ──► Serial-to-  ──► IFFT  ──► CP      ──► DAC ──► RF
          Mapper         Parallel         (N-pt)   Insert         Up-conv
                                                                  │
                                                            Transmit

1. QAM Mapping: Input bits are grouped and mapped to complex QAM symbols (e.g., 64-QAM → 6 bits per symbol).
2. Serial-to-Parallel: N QAM symbols are collected for one OFDM symbol.
3. IFFT: The N frequency-domain symbols are transformed to N time-domain samples.
4. Cyclic Prefix Insertion: The last N_cp samples are copied to the front, creating a guard interval.
5. DAC: Digital-to-analog conversion produces the continuous-time baseband signal.
6. RF Up-conversion: Multiply by cos(2πf_c t) to shift to the carrier frequency.

Receiver Chain

  RF ──► RF       ──► ADC ──► CP       ──► FFT   ──► Parallel-to- ──► QAM
  Rx      Down-conv           Remove       (N-pt)      Serial         Demapper
                                                                  │
                                                              Bits out

The receiver reverses the process. After FFT, each subcarrier's complex value is divided by the estimated channel response (equalization), then demapped back to bits.

#4Subcarrier Spacing & Numerology

In 5G NR, the term numerology refers to the set of physical-layer parameters defined by the subcarrier spacing.

The Numerology Formula

  Δf = 15 × 2^μ   (kHz)

  where μ ∈ {0, 1, 2, 3, 4} is the numerology index

As μ increases, the subcarrier spacing doubles, and the symbol duration halves.

5G NR Numerologies

#5Frame Structure

5G NR organizes time into a hierarchical structure:

  Radio Frame (10 ms)
  ┌─────────────────────────────────────────────────────────────────┐
  │ Subframe (1 ms) × 10 per frame                                  │
  │ ┌───────────────────────────────────────────────────────────┐   │
  │ │ Slot × (14 symbols for normal CP)                         │   │
  │ │ ┌─────────────────────────────────────────────────────┐   │   │
  │ │ │ OFDM Symbol × 14 per slot (normal CP)               │   │   │
  │ │ │ ┌───────────────────────────────────────────────┐   │   │   │
  │ │ │ │ Resource Element (RE) = 1 subcarrier × 1 symbol │   │   │   │
  │ │ │ └───────────────────────────────────────────────┘   │   │   │
  │ │ └─────────────────────────────────────────────────────┘   │   │
  │ └───────────────────────────────────────────────────────────┘   │
  └─────────────────────────────────────────────────────────────────┘

Slots per Subframe

  μ=0 (15 kHz, 1 slot/subframe):
  ┌─────────────────────────────────────────────┐
  │  14 symbols (1 ms slot)                     │
  └─────────────────────────────────────────────┘

  μ=2 (60 kHz, 4 slots/subframe):
  ┌─────┬─────┬─────┬─────┐
  │slot0│slot1│slot2│slot3│  (0.25 ms each)
  └─────┴─────┴─────┴─────┘

#6Resource Grid

The resource grid is the 2D time-frequency plane on which all physical-layer signals and channels are mapped.

Key Resource Grid Concepts

#7Physical Resource Block (PRB)

A PRB is the fundamental scheduling unit in 5G NR.

  PRB Structure (12 subcarriers × 14 symbols = 168 REs)

  Frequency ▲
            │
   SC 11    │ ┌───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┐
   SC 10    │ │ D │ D │ D │ D │ D │ D │ D │ D │ D │ D │ D │ D │
    ...     │ ├───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
   SC  1    │ │ D │ D │ R │ D │ D │ R │ D │ D │ R │ D │ D │ R │  ← DM-RS
   SC  0    │ ├───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
            │ │ C │ C │ C │ D │ D │ D │ D │ D │ D │ D │ D │ D │  ← PDCCH
            └─┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴─► Time
              Sym0 Sym1 Sym2 Sym3 ...                         Sym13

  D = PDSCH data, R = DM-RS reference, C = PDCCH control

PRB Size Across Numerologies

#8Cyclic Prefix (CP)

The Cyclic Prefix is a guard interval inserted at the beginning of each OFDM symbol to absorb multipath delay spread and prevent ISI.

How CP Works

  Without CP:
  ┌──────────────────────┐
  │   OFDM Symbol (Ts)   │◄── multipath echoes overlap next symbol → ISI
  └──────────────────────┘

  With CP:
  ┌──────┬──────────────────────┐
  │  CP  │   OFDM Symbol (Ts)  │◄── CP absorbs multipath; no ISI into useful part
  └──────┴──────────────────────┘
   copy of
   last N_cp samples

5G NR CP Lengths

Trade-off: Longer CP → better multipath immunity but lower spectral efficiency (~7% overhead for normal CP, ~25% for extended CP).

#9PAPR — Peak-to-Average Power Ratio

One of OFDM's main drawbacks is its high PAPR. When many subcarriers add constructively, the instantaneous power can spike far above the average:

  PAPR = Peak Power / Average Power (linear)
  PAPR_dB = 10 · log₁₀(PAPR)

  Typical OFDM PAPR: 8–12 dB

Why High PAPR Is a Problem

PAPR Mitigation Techniques

5G NR uplink supports both CP-OFDM (high throughput, high PAPR) and DFT-s-OFDM (lower PAPR, coverage-limited scenarios).

#10OFDM vs Single-Carrier

#11Numerology Selection & Use Cases

  Numerology Selection Decision Tree

  Sub-6 GHz, wide area, low mobility?
      │ Yes → μ=0 (15 kHz) — maximum coverage, longest symbol
      │ No  ↓
  Urban macro, balanced coverage/throughput?
      │ Yes → μ=1 (30 kHz) — common for C-band deployments
      │ No  ↓
  Indoor hotspot, high throughput needed?
      │ Yes → μ=2 (60 kHz) — shorter symbols, lower latency
      │ No  ↓
  mmWave, high mobility or ultra-low latency?
      │ Yes → μ=3 (120 kHz) — tracks fast channel changes
      │ No  ↓
  mmWave, extreme low latency (URLLC)?
      └───→ μ=4 (240 kHz) — shortest symbols, mini-slots

Practical Deployment Examples

#12Bandwidth Parts (BWP)

A Bandwidth Part is a contiguous subset of PRBs within a carrier's total bandwidth.

  Carrier Bandwidth: 100 MHz (μ=1, 273 PRBs)

  ┌─────────────────────────────────────────────────────────────┐
  │                    Full Carrier (273 PRBs)                  │
  └─────────────────────────────────────────────────────────────┘

  ┌──────────┐         ┌──────────────┐         ┌──────────┐
  │ BWP #0   │         │   BWP #1     │         │ BWP #2   │
  │ 20 MHz   │         │   50 MHz     │         │ 30 MHz   │
  │ μ=1      │         │   μ=1        │         │ μ=2      │
  │ (idle)   │         │  (active)    │         │ (hotspot)│
  └──────────┘         └──────────────┘         └──────────┘

A UE can be configured with up to 4 BWPs per carrier and switches between them via RRC signaling or DCI.

#13OFDM Implementation: IFFT/FFT

Building N separate oscillators for N subcarriers is impractical. The IFFT elegantly generates all subcarriers simultaneously.

  Tx:  [Q₀, Q₁, ..., Q_{N-1}]  ──►  IFFT  ──►  [s₀, s₁, ..., s_{N-1}]
        frequency domain                         time domain

  Rx:  [r₀, r₁, ..., r_{N-1}]  ──►  FFT   ──►  [R₀, R₁, ..., R_{N-1}]
        time domain                          frequency domain

  N = IFFT/FFT size (e.g., 2048 for LTE, up to 4096 for 5G NR)

#14OFDMA — Multi-User Access

OFDMA extends OFDM by allowing multiple users to transmit simultaneously on different subsets of subcarriers.

  OFDMA Resource Allocation (Time-Frequency)

  Frequency ▲
            │  UE-A  ┌───────┐       ┌───────┐
            │        │ PRB 0 │       │ PRB 5 │
            │  UE-B  │ PRB 1 │ PRB 2 │       │ PRB 6 │
            │        │ PRB 3 │       │ PRB 7 │
            │  UE-C  │       │ PRB 4 │       │ PRB 8 │
            └────────┴───────┴───────┴───────┴───────┴──► Time
                    Slot 0  Slot 1  Slot 2  Slot 3

#15Practical Considerations

Common Pitfalls

Numerology & Latency

  Scheduling Latency (approximate, one-way):

  μ=0 (15 kHz):  Slot = 1 ms     → ~1–2 ms
  μ=1 (30 kHz):  Slot = 0.5 ms   → ~0.5–1 ms
  μ=2 (60 kHz):  Slot = 0.25 ms  → ~0.25–0.5 ms
  μ=3 (120 kHz): Slot = 0.125 ms → ~0.125–0.25 ms
  μ=4 (240 kHz): Slot = 62.5 μs  → ~0.06–0.125 ms

  For URLLC: mini-slots (2, 4, or 7 symbols) allow even lower latency

#16LTE vs 5G NR OFDM

#17Further Reading

3GPP Specifications

• TS 38.211 — NR Physical Channels and Modulation
• TS 38.212 — NR Multiplexing and Channel Coding
• TS 38.213 — NR Physical Layer Procedures
• TS 38.331 — NR RRC Protocol (BWP configuration)

Research Papers & Resources

• "OFDM Wireless Communications" — IEEE Communications Surveys & Tutorials
• "5G NR: The Next Generation Wireless Access Technology" — Erik Dahlman et al.
• "Fundamentals of OFDM" — John G. Proakis, Digital Communications

Podcast

Beamforming Deep Dive — Telecom Leaders Podcast

Companion Resources

• Modulation.md — Modulation schemes from analog to 5G/6G QAM
• Beamforming.md — Antenna arrays, beamforming, and spatial processing
• SpatialMultiplexing.md — MIMO layers, precoding, and spatial multiplexing
• UE-RACH-Procedure.md — Random access, SSB association, and power ramping
• ISAC-Aware MAC Scheduling — OFDM waveform for joint sensing & communication (6G ISAC)