LTE Physical Resource Block (PRB) & Transmission Grid Calculator

Architecture of the LTE Physical Resource Block (PRB) Grid

In the 3GPP Long Term Evolution (LTE) and LTE-Advanced specifications (3GPP TS 36.211 and TS 36.101), the air interface is governed by Orthogonal Frequency Division Multiple Access (OFDMA) in the downlink and Single-Carrier Frequency Division Multiple Access (SC-FDMA) in the uplink. The fundamental atomic unit of resource allocation within the eNodeB MAC scheduler is the Physical Resource Block (PRB).

1. Time-Frequency Structure of a Resource Block

Unlike 5G New Radio (NR) which introduces scalable numerologies with flexible subcarrier spacings ($\Delta f = 15 \times 2^\mu\text{ kHz}$), 4G LTE fixes its subcarrier spacing strictly at:

In the frequency domain, one Physical Resource Block spans exactly $N_{\text{sc}}^{\text{RB}} = 12$ consecutive subcarriers. Thus, the physical bandwidth of an LTE PRB is universally invariant across all channel configurations:

In the time domain, LTE transmission is organized into hierarchical radio frames:

• Radio Frame: Total duration of $T_{\text{frame}} = 10\text{ ms}$, comprising 10 subframes.
• Subframe: Total duration of $T_{\text{subframe}} = 1\text{ ms}$, comprising 2 consecutive time slots.
• Slot: Total duration of $T_{\text{slot}} = 0.5\text{ ms}$.

Because resource grants cannot span less than one full subframe in LTE scheduling, the minimum allocatable data payload block is termed a PRB Pair—spanning 12 subcarriers over two consecutive 0.5 ms slots ($180\text{ kHz} \times 1.0\text{ ms}$).

2. The 90% Spectrum Utilization Rule and Fixed Guardbands

A central engineering hallmark of LTE is its standardized 90% spectral utilization rule. In the 3GPP Release 8 design phase (2007–2008), consumer-grade RF surface acoustic wave (SAW) and bulk acoustic wave (BAW) duplexer filters suffered from finite filter roll-off skirts. To prevent adjacent channel leakage (ACLR) into neighboring spectrum without requiring prohibitively expensive analog filters, 3GPP engineers reserved approximately 10% of every channel allocation for symmetric guardbands:

Examining the 20 MHz commercial profile:

• $N_{\text{RB}} = 100 \implies BW_{\text{trans}} = 100 \times 180\text{ kHz} = 18.0\text{ MHz}$.
• Remaining guardband: $20.0\text{ MHz} - 18.0\text{ MHz} = 2.0\text{ MHz}$ total, or exactly $1.0\text{ MHz}$ on each carrier edge ($GB_{\text{edge}} = 1,000\text{ kHz}$).
• Spectrum utilization efficiency: $\frac{18.0}{20.0} = 90.00\%$.

The only exception is the narrowest $1.4\text{ MHz}$ channel profile ($N_{\text{RB}} = 6$), which occupies $1.08\text{ MHz}$, yielding $77.14\%$ efficiency with $160\text{ kHz}$ edge guardbands to comfortably fit inside legacy $1.25\text{ MHz}$ cdma2000 carrier boundaries.

3. Resource Element (RE) Granularity: Normal vs. Extended Cyclic Prefix

The atomic building block of an LTE frame is the Resource Element (RE), defined as one single subcarrier during one OFDM symbol period. The gross RE capacity per PRB pair depends on the Cyclic Prefix (CP) selection:

• Normal Cyclic Prefix (7 symbols/slot): In standard urban and suburban deployments, delay spread is modest ($< 4.7\ \mu\text{s}$). Symbol 0 has a cyclic prefix of $5.2\ \mu\text{s}$, while Symbols 1 through 6 have a CP of $4.69\ \mu\text{s}$. Each slot carries 7 symbols, giving: $$N_{\text{RE\_gross}}^{\text{PRB}} = 12\text{ subcarriers} \times 14\text{ symbols} = 168\text{ Resource Elements per ms}$$
• Extended Cyclic Prefix (6 symbols/slot): In extreme rural environments, mountainous terrain, or large Single Frequency Networks (MBSFN) where multipath delay spread exceeds $5\ \mu\text{s}$, an extended CP of $16.67\ \mu\text{s}$ is deployed. This reduces the slot capacity to 6 symbols: $$N_{\text{RE\_gross}}^{\text{PRB}} = 12\text{ subcarriers} \times 12\text{ symbols} = 144\text{ Resource Elements per ms}$$ Note: Extended CP incurs a permanent 14.29% throughput capacity penalty compared to Normal CP.

4. Physical Downlink Overheads: CRS and PDCCH Control Footprint

Not all Resource Elements within a PRB carry PDSCH user data. Two mandatory physical layer signals consume dedicated resource elements:

1. Cell-Specific Reference Signals (CRS): Broadcast continuously across the entire operating bandwidth to enable coherent demodulation, channel estimation, and CQI/RSRP/RSRQ measurements:
   • 1 Antenna Port (Port 0): 8 REs per PRB pair (Symbols 0, 4 in Slot 0; Symbols 7, 11 in Slot 1).
   • 2 Antenna Ports (Ports 0, 1): 16 REs per PRB pair (8 REs for Port 0 + 8 REs for Port 1, staggered in frequency by 3 subcarriers).
   • 4 Antenna Ports (Ports 0, 1, 2, 3): 24 REs per PRB pair (16 REs for Ports 0/1 + 8 REs for Ports 2/3 located on Symbols 1 and 8).
2. Physical Downlink Control Channel (PDCCH): Occupies the first 1, 2, or 3 OFDM symbols of each subframe (commanded dynamically by the PCFICH channel). In a 20 MHz cell with 2 CRS ports and $\text{CFI} = 2$:
   • Total symbols for PDCCH: Symbols 0 and 1 ($2 \times 12 = 24\text{ REs/PRB}$).
   • Less CRS embedded in Symbol 0 ($4\text{ REs}$): Net PDCCH REs = $24 - 4 = 20\text{ REs/PRB}$.

Subtracting CRS ($16\text{ REs}$) and PDCCH ($20\text{ REs}$) leaves 132 PDSCH User Data REs per PRB pair, or $13,200\text{ REs/ms}$ across the entire 100 PRB carrier ($78.57\%$ payload efficiency).