PBCH DMRS is a special type of physical layer signal which functions as a reference signal for decoding PBCH. In LTE (at least in TM1, 2, 3, 4), we don't need this kind of special DMRS for PBCH because we can use CRS(Cell Specific Reference Signal) for PBCH decoding.  However, in 5G/NR there is no CRS. That's why we need the DMRS dedicated for PBCH decoding.

• Sequence Generation
• Physical Location (Resource Element Mapping)
• Location Shift by Physical Cell ID
• Reference

Sequence Generation

Following is the equation to generate PBCH DMRS. As in many other Physical Layer Signal, it is generated by Pseudo Random Sequence, but the part unique for PBCH DMRS is c_init(initialization value). As you see below, the initialization value is made up of various components like Physical Cell ID, SSB Index and Half Frame Number. That is, by decoding this DMRS UE can figure out SSB Index and Half Frame.

Figure 1. Both inputs to c_init are things the UE is still establishing. The cell identity has just come from PSS and SSS. The block index is the very thing the UE is trying to confirm, so it has to try each candidate.

• i_SSB is the two least significant bits of the SS/PBCH block index when L_max is 4, and the three least significant bits when L_max is 8 or 64.
• n_hf is 0 for the first half frame and 1 for the second when L_max is 4, and it is always 0 when L_max is 8 or 64.
• The two combine as i_SSB_bar = i_SSB + 4 n_hf, so the half frame bit extends the index rather than replacing part of it.
• c_init is 2^11 (i_SSB_bar + 1) (floor(N_ID_cell / 4) + 1) + 2^6 (i_SSB_bar + 1) + (N_ID_cell mod 4).
• x1 and x2 are the usual length 31 generators, combined into c(n) = (x1(n + Nc) + x2(n + Nc)) mod 2.
• r(m) = (1 - 2 c(2m)) / sqrt(2) + j (1 - 2 c(2m+1)) / sqrt(2). Every symbol therefore lands on a QPSK constellation point.

The sequence generation begins with the determination of the SS/PBCH block index i_SSB, which is either the two least significant bits (LSB) when L_max equals 4, or the three LSB when L_max is 8 or 64.

The half-frame number n_hf is considered next, which is dependent on L_max. For L_max equals 4, n_hf changes value within the frame from 0 to 1. For L_max equals 8 or 64, n_hf is always 0. The value i_SSB is then calculated by adding 4n_hf to the initial l_SSB.

The initialization sequence c_init is computed using the formula combining i_SSB, the physical cell ID N_ID^cell, and constants in a specified manner involving bitwise shifts and additions.

Two sequences x₁(n) and x₂(n) are created using a pseudo-random number generation method defined by the given polynomials. The sequences are then combined to produce the scrambling sequence c(n), which is used to scramble the DMRS symbols.

Finally, the complex DMRS symbols r(m) are generated using the scrambling sequence c(n), involving complex number calculations with real and imaginary components derived from the sequence c(n).

Physical Location (Resource Element Mapping)

The physical location (Resource Elements) of PBCH DMRS is determined as below. As you see here, the location shifts in frequency domain according to Physical Cell ID.

Following diagram illustrates the resource allocation within an SS/PBCH block which includes signals such as the Primary Synchronization Signal (PSS), Secondary Synchronization Signal (SSS), Physical Broadcast Channel (PBCH), and the Demodulation Reference Signals (DM-RS) for PBCH, as specified in the 3GPP TS 38.211.

Figure 2. The DM-RS row is the only one in the table with a v in it. Every other signal in the block sits at a fixed subcarrier, and the reference signal is the one thing that moves.

• The table is 38.211 Table 7.4.3.1-1. It has a column for the OFDM symbol number l and a column for the subcarrier number k, both relative to the start of the SS/PBCH block.
• PSS is in symbol 0 on subcarriers 56 to 182, and SSS is in symbol 2 on the same range.
• PBCH occupies symbols 1 and 3 across subcarriers 0 to 239, and symbol 2 across 0 to 47 and 192 to 239.
• The DM-RS row for symbols 1 and 3 reads 0+v, 4+v, 8+v, ..., 236+v, so it is every fourth subcarrier across the whole symbol.
• The DM-RS row for symbol 2 reads 0+v, 4+v, 8+v, ..., 44+v and 192+v, 196+v, ..., 236+v, because SSS occupies the middle.
• v = N_ID_cell mod 4 is annotated under the table, with a note that the DMRS location moves along the frequency domain as the physical cell ID changes.
• The grid on the left is colour coded, with PSS in green, SSS in blue, PBCH in red and PBCH DMRS in yellow.

It outlines the OFDM symbol numbers and the corresponding subcarrier numbers for each channel or signal:

• The PSS is mapped to OFDM symbol number 0 and subcarrier numbers 56, 57, ..., 182.
• The SSS is mapped to OFDM symbol number 2 and the same subcarrier range as PSS.
• Regions set to 0 are used for OFDM symbol number 0 and a range of subcarrier numbers.
• The PBCH is mapped to OFDM symbol numbers 1,3 and 2, with various subcarrier number ranges.
• The DM-RS for PBCH is mapped to OFDM symbol numbers 1 and 3 on subcarriers 0+v, 4+v, 8+v, ..., 236+v.
• In OFDM symbol number 2 the DM-RS is mapped to subcarriers 0+v, 4+v, 8+v, ..., 44+v, and to 192+v, 196+v, ..., 236+v. The SSS in the middle of that symbol is what forces the split.

The diagram also shows that the location of the PBCH DMRS moves along the frequency domain as the Physical Cell ID changes. This is indicated by the variable 'v' in the DM-RS resource mapping, which is calculated as a function of the physical cell ID modulo 4.

Location Shift by Physical Cell ID

As mentioned above, the array of PBCH DMRS shifts vertically (i.e, in frequency domain) with Physical Cell ID in a manner shown below. The shift is small and it repeats quickly, so it is worth being clear about what it achieves. Four positions is enough to keep the reference signals of neighbouring cells apart, and it is not enough to identify a cell on its own.

Following diagram depicts how the PBCH DMRS array shifts vertically, that is, in the frequency domain, with different Physical Cell IDs. This shift occurs in a pattern as explained below.

Figure 3. Four columns and four values of v is the whole cycle. Cell 4 has the same pattern as cell 0, so the shift repeats every four identities rather than being unique to a cell.

• Four grids are shown, headed N_ID_cell = 0, 1, 2 and 3, each with v worked out as N_ID_cell mod 4.
• A second identity is given under each heading, 4, 5, 6 and 7, with the same value of v. That is what makes the repetition explicit.
• Subcarrier numbers run up the left of each grid, from 0 at the bottom to 239 at the top.
• The yellow DM-RS cells move toward higher subcarrier numbers as v increases, by one subcarrier per step.
• The green PSS and blue SSS blocks stay in the same rows in every column, because only the DM-RS position depends on v.

Four positions is a deliberate choice rather than a large one. The DM-RS carries the block index, so it already has to be decoded rather than merely detected, and a longer shift cycle would buy little. What the four positions do achieve is that two neighbouring cells whose identities differ by anything other than a multiple of four put their reference signals on different subcarriers.

• The shift cycle is four, not 1008 : v is N_ID_cell mod 4, so identities 0, 4, 8 and so on share a pattern.
• Only the DM-RS moves : PSS, SSS and the PBCH data occupy fixed subcarriers, so the shift changes the reference signal alone.
• The shift is not an identifier : four positions cannot distinguish 1008 cell identities, so v narrows the search rather than resolving it.

For each Physical Cell ID N_ID^cell, the value of 'v' is determined by taking N_ID^cell modulo 4. Depending on the result, which ranges from 0 to 3, the PBCH DMRS shifts its position vertically in the grid. The diagrams show the shift for Physical Cell IDs from 0 to 7, illustrating the following patterns:

• When N_ID^cell modulo 4 equals 0, the value of 'v' is 0, indicating no shift from the original position.
• When N_ID^cell modulo 4 equals 1, the value of 'v' is 1, resulting in a shift down by one position.
• When N_ID^cell modulo 4 equals 2, the value of 'v' is 2, resulting in a shift down by two positions.
• When N_ID^cell modulo 4 equals 3, the value of 'v' is 3, resulting in a shift down by three positions.

Each column in the image corresponds to a specific Physical Cell ID and demonstrates the corresponding shift in the PBCH DMRS location. This shifting mechanism is crucial for the UE to identify the correct Physical Cell ID during cell search and synchronization processes.

Reference

[1] 3GPP TS 38.211 - NR; Physical channels and modulation. Clause 7.4.1.4 for the DM-RS sequence, and Table 7.4.3.1-1 for the resource elements it occupies.

[2] 3GPP TS 38.213 - NR; Physical layer procedures for control. Clause 4.1 for how the UE uses the block index the DM-RS carries.