OFDM Symbol & Slot Duration Calculator

3GPP Basic Time Units: Derivation of T_c and T_s

In digital cellular radio communications, physical layer sample clocks, guard intervals, and propagation delays are referenced to fundamental discrete time units defined by the 3rd Generation Partnership Project (3GPP).

In 4G LTE (3GPP TS 36.211 Section 4), the basic time unit is designated as T_s. It was derived from the legacy 20 MHz carrier sampling rate, operating with a reference subcarrier spacing of Δf_ref = 15 kHz and a reference Fast Fourier Transform (FFT) size of N_f,ref = 2048:

In 5G New Radio (3GPP TS 38.211 Section 4.1), the air interface was expanded to accommodate massive carrier bandwidths (up to 400 MHz per component carrier), subcarrier spacings up to 480 kHz, and FFT sizes up to 4096. Consequently, 3GPP established a finer basic time unit designated as T_c, calculated using the maximum nominal subcarrier spacing Δf_max = 480 kHz:

Because both architectures are harmonized, there exists an exact integer ratio linking the 4G and 5G reference clocks:

Every timing parameter in 5G NR—from timing advance increments to cyclic prefix samples—is quantized as an integer multiple of T_c.

Why Symbol 0 Has a Longer Cyclic Prefix (Normal CP)

Engineers examining 3GPP timing tables frequently observe an asymmetrical anomaly: under Normal Cyclic Prefix, the first OFDM symbol of every half-subframe (symbol index l = 0 and l = 7·2^μ) has a slightly longer cyclic prefix duration than the subsequent six symbols.

This asymmetry is an inescapable mathematical requirement for aligning discrete digital samples into a continuous 0.5 ms half-subframe:

1. For baseline 15 kHz SCS (μ=0), the useful symbol time is exactly T_u = 2048 T_s (66.67 μs).
2. Seven useful symbols require 7 × 2048 T_s = 14,336 T_s.
3. A 0.5 ms half-subframe contains exactly 0.5 × 30,720,000 = 15,360 T_s.
4. Subtracting the useful symbol duration leaves exactly 15,360 − 14,336 = 1,024 T_s dedicated to the 7 cyclic prefixes.
5. Dividing 1,024 T_s equally across 7 symbols yields 146.2857 T_s per symbol. Because hardware digital-to-analog converters (DACs) require an integer sample count, equal distribution is impossible.

To solve this without sample drift, 3GPP assigned 160 T_s (5.208 μs) to symbol 0, and 144 T_s (4.688 μs) to symbols 1 through 6:

In 5G NR, this exact relationship is generalized for any numerology μ in TS 38.211 Section 5.3:

• For l = 0 or l = 7·2^μ: N_cp,μ = 144·κ·2^−μ + 16·κ = 9,216·2^−μ + 1,024 [in units of T_c].
• For all other symbols: N_cp,μ = 144·κ·2^−μ = 9,216·2^−μ [in units of T_c].

Slot-Level vs. Mini-Slot Scheduling for URLLC

In legacy cellular networks, the transmission scheduling boundary was bound to the slot or subframe. In 5G NR, two distinct scheduling allocation schemes are standardized:

• Slot-Based Scheduling (Mapping Type A): Transmissions span the full 14 symbols of a slot. Demodulation Reference Signals (DMRS) are positioned early in symbol 2 or 3. This format is optimized for high-throughput enhanced Mobile Broadband (eMBB).
• Non-Slot / Mini-Slot Scheduling (Mapping Type B): Transmissions can span 2, 4, or 7 OFDM symbols, beginning at any arbitrary symbol position within a slot. DMRS is located in the very first symbol of the allocation.

Mini-slots are the cornerstone of 5G Ultra-Reliable Low-Latency Communication (URLLC). When a critical vehicular safety alert or industrial emergency stop command arrives at the gNodeB scheduler, the radio does not wait for the next 0.5 ms slot boundary. Instead, it instantly punctures (preempts) ongoing eMBB transmissions to dispatch a 2-symbol mini-slot. At 60 kHz SCS (μ=2), a 2-symbol mini-slot duration is just 35.7 μs, allowing over-the-air physical latency to meet strict sub-millisecond budgets.

Radio Frame & Subframe Hierarchical Architecture

To maintain cross-RAT backward compatibility and uniform network synchronization, 3GPP preserves two fundamental time constants:

• Radio Frame: Always exactly 10.0 milliseconds (307,200 T_s or 19,660,800 T_c).
• Subframe: Always exactly 1.0 millisecond (30,720 T_s or 1,966,080 T_c).

While the subframe duration remains immutable at 1 ms, the number of slots nested inside each subframe scales dynamically as N_slot^subframe,μ = 2^μ. For μ=0, 1 subframe contains 1 slot; for μ=1 (C-Band), 1 subframe contains 2 slots (0.5 ms each); for μ=3 (mmWave), 1 subframe houses 8 slots (125 μs each).

3GPP TS 38.211 & TS 36.211 Timing Reference Table

The lookup table below compiles exact OFDM symbol, cyclic prefix, slot, and subframe timing values across standardized 3GPP configurations: