When a mobile device (UE) powers on, it must synchronize with a cell tower (eNodeB) to detect timing, frame structure, and Physical Cell ID (PCI). The PCI identifies the cell and is derived from two downlink synchronization signals: PSS (Primary Synchronization Signal) and SSS (Secondary Synchronization Signal).

PCI Formula (3GPP TS 36.211 §6.11)

PCI = 3 × N_ID⁽¹⁾ + N_ID⁽²⁾

• N_ID⁽²⁾ (0–2): From PSS, identifies the cell sector within a group.
• N_ID⁽¹⁾ (0–167): From SSS, identifies the PCI group. There are 168 groups.
• Total: 3 × 168 = 504 unique PCIs (0–503).

PSS — Zadoff-Chu Sequence

PSS is a length-63 Zadoff-Chu (ZC) sequence mapped to the center 62 subcarriers (DC subcarrier is null). The root index u depends on N_ID⁽²⁾:

• N_ID⁽²⁾ = 0 → u = 25
• N_ID⁽²⁾ = 1 → u = 29
• N_ID⁽²⁾ = 2 → u = 34

Formula: d_u(n) = e^{−jπu·n·(n+1)/63} for n = 0, 1, ..., 61.

The complex constellation (I/Q plane) shows how different roots produce different phase patterns.

SSS — Interleaved m-Sequences (TS 36.211 §6.11.2)

The 62-length SSS sequence d(n) is built from base sequences s̃, c̃, and z̃ with different polynomials:

• ˜s, ˜c: x(i+5) = (x(i+3) + x(i)) mod 2
• ˜z: x(i+5) = (x(i+4) + x(i+2) + x(i+1) + x(i)) mod 2
• m₀, m₁: Derived from N_ID⁽¹⁾ via q′, q, m′
• d(2n): s × c₀; d(2n+1): s × c₁ × z₁

Subframe 0 vs 5: The roles of m₀ and m₁ swap between subframes, enabling the UE to detect the 10 ms frame boundary.

Resource Grid Mapping

PSS and SSS occupy the center 6 Resource Blocks (72 subcarriers) in subframes 0 and 5:

• PSS: Last symbol of slot 0 and slot 10 (OFDM symbol 6) — 62 subcarriers k = 5–66.
• SSS: Second-to-last symbol (OFDM symbol 5) — same 62 subcarriers.

UE first detects PSS for slot timing, then SSS for frame timing, N_ID⁽¹⁾, and subframe (0 vs 5) by correlating with both hypotheses. The subframe is identified because m₀/m₁ swap between subframes. Finally, PCI is computed.

 

Usage

1. N_ID⁽¹⁾ (SSS): Slider 0–167 (default 50). Changes the PCI group; observe the SSS m-sequence stem plot shifting.
2. N_ID⁽²⁾ (PSS): Dropdown 0–2. Changes the sector ID; observe the PSS Zadoff-Chu constellation rotating.
3. Subframe: Toggle Subframe 0 vs 6 to set the transmitted signal. The SSS d(n) differs between subframes (m₀/m₁ swap roles).
4. SNR: Slider 0–40 dB (default 40). Adds AWGN to the PSS constellation and SSS stem plot. Lower SNR increases noise spread.
5. Resource Grid: Blue = PSS (symbol 6), Magenta = SSS (symbol 5). Hover over PSS/SSS REs for tooltips.
6. PCI Correlation Heatmap: PSS bar chart shows match strength for each N_ID⁽²⁾. SSS heatmaps (SF0/SF5) show correlation across all 168 N_ID⁽¹⁾ candidates. The peak (green glow) indicates the detected PCI. At low SNR, heatmaps appear grainy.
7. Sync Simulation: Simulates UE cell search. Correlates with each PSS candidate (N_ID⁽²⁾=0,1,2), then each SSS candidate (N_ID⁽¹⁾ with Subframe 0 and 5 hypotheses). Shows correlation values and detects subframe from the winning hypothesis. History lists every correlation attempt until match.

Key 3GPP References

• TS 36.211 §6.11 — Primary and secondary synchronization signals.
• TS 36.300 — Physical cell identity (0–503).