PUCCH Format 4 Report (1 PRB, QPSK, SF=4)

1. Real-World Execution Configuration

Physical layer parameters configured for block-spread long PUCCH transmission with 1 PRB FD-OCC UE multiplexing.

Parameter	C++ Config Key	Value	3GPP Specification / DSP Note
Channel Format	`formatType`	PUCCH Format 4	Long PUCCH with frequency-domain block spreading (FD-OCC)
Modulation Scheme	`m_modulationType`	QPSK (2 bits/symbol)	Enables cyclic shift phase ramp evaluation ($\alpha$)
Spreading Factor	`m_sf`	N_SF = 4	Block-wise frequency-domain spreading ($N_{\text{SF}}^{\text{PUCCH,4}} \in \{2, 4\}$)
Orthogonal Seq Index	`m_n`	n = 1	Selects FD-OCC vector $w_n(i)$ for orthogonal UE multiplexing
C-RNTI	`m_nrnti`	0x3B21 (15137)	Radio Network Temporary Identifier for sequence group generation
Allocated PRBs	`m_numPrbs` / `m_M_rb`	1 PRB ($M_{\text{sc}} = 12$)	3GPP TS 38.211 constraint: Format 4 is strictly limited to 1 PRB
Symbol Allocation	`m_numSymbols_pucch`	8 Symbols (Symbols 6–13)	2 DM-RS pilot symbols + 6 UCI data symbols
Starting PRB (Hop 0)	`m_startPrb`	PRB 5	Intra-slot frequency hopping enabled (`ISFH_ENABLE`)
Cyclic Shift Index	`m_cs.m_o`	6	Initial cyclic shift index $m_O = 6$ ($\alpha$ hop dynamic)
Payload Capacity	`M_coded_bits`	36 Coded Bits	$(12 \text{ REs} / N_{\text{SF}}) \times 6 \text{ Data Symbols} \times 2 \text{ bits/symbol}$

2. Physical Layer Mathematical Checkpoints

3GPP TS 38.211 verification criteria for computer-generated base sequence and block-spreading transform precoding.

Bandwidth Constraint: $M_{\text{sc}} = 12 < 30 \implies$ Computer-Generated Low-PAPR Base Sequence ($u=11, v=0$) Frequency-Domain OCC: $w_n(i)$ vector applied across subcarriers prior to 12-point DFT transform precoding Dynamic Cyclic Shift: $\alpha = \frac{2\pi \cdot n_{\text{cs}}(n_{\text{s,f}}^{\mu}, l)}{12}$ hopping across symbol boundaries

3. Time & Frequency Domain Signals

Generated time-domain waveform envelope and power spectral density across 1 PRB (12 subcarriers).

Figure 1: Time Domain Waveform Envelope

Figure 2: Power Spectral Density (PSD) Across 12 Subcarriers

4. Physical Layer IQ Constellations

Unified constellation plot showcasing pre-DFT QPSK symbols ($d_m$), DM-RS pilot sequence ($dmrs\_export$), and post-12pt DFT block-spread frequency grid ($R_{\text{mat}}$).

Figure 3: Combined IQ Constellation Analytics (Pre-DFT QPSK, DM-RS Pilots, and Post-DFT Block-Spread Grid)

5. Automated DSP & Mathematical Verification

Runtime verification metrics executing Parseval's power conservation, FD-OCC block spreading, and 3GPP TS 38.211 DM-RS sequence validation tests.

--- DFT Transform Precoding & FD-OCC Power Conservation ---
Pre-DFT Average Power:  1.0000
Post-DFT Average Power: 1.0000
Max Peak Magnitude:     2.4495

--- 3GPP DM-RS Mathematical Verification (N_RE = 12) ---

            [Test 1] Constant Modulus Check:

              Mean Magnitude:     1.000000

              Standard Deviation: 1.570092e-16

              RESULT: PASSED (Sequence has perfectly uniform envelope!)
        
            [Test 2] PAPR Performance:

              Peak-to-Average Ratio: 0.0000 dB (Linear: 1.0000)

              RESULT: PASSED (Low-PAPR criteria satisfied!)
        
            [Test 3] Phase Linearity & Cyclic Shift Check:

              First 5 Phases (rad): [-2.3562 -1.8326  1.8326  2.3562  1.3090]

              Phase Steps (rad):    [ 0.5236  3.6652  0.5236 -1.0472]

              Estimated Cyclic Shift α: 6.18799 rad

              • Note: Computer-Generated Base Sequence (Msc = 12 < 30) verified with dynamic symbol-dependent cyclic shift phase ramp.