PUCCH Format 3 Report (3 PRBs, QPSK)

1. Real-World Execution Configuration

Physical layer parameters configured for high-capacity UCI transmission using 3 PRB DFT-s-OFDM transform precoding.

Parameter	C++ Config Key	Value	3GPP Specification / DSP Note
Channel Format	`formatType`	PUCCH Format 3	Long PUCCH format without time-domain OCC block spreading
Modulation Scheme	`m_modulationType`	QPSK (2 bits/symbol)	High payload efficiency for large UCI bit sequences
C-RNTI	`m_nrnti`	0x3B21 (15137)	Radio Network Temporary Identifier used for scrambling sequence $c_{\text{init}}$
Allocated PRBs	`m_numPrbs` / `m_M_rb`	3 PRBs ($M_{\text{sc}} = 36$)	Prime factor rule verified: $M_{\text{RB}} = 2^0 \cdot 3^1 \cdot 5^0 = 3$ PRBs
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_cs_idx`	0	Initial cyclic shift index $m_0 = 0$
Payload Capacity	`M_coded_bits`	432 Coded Bits	$3 \text{ PRBs} \times 12 \text{ REs} \times 6 \text{ Data Symbols} \times 2 \text{ bits/symbol}$

2. Physical Layer Mathematical Checkpoints

3GPP TS 38.211 verification criteria for sequence generation and transform precoding.

3GPP Prime Factor Rule: $M_{\text{RB}} = 2^{\alpha_2} \cdot 3^{\alpha_3} \cdot 5^{\alpha_5} = 2^0 \cdot 3^1 \cdot 5^0 = 3 \text{ PRBs}$ ($M_{\text{sc}} = 36$ subcarriers) DM-RS Base Sequence Type: $M_{\text{sc}} = 36 \ge 30 \implies$ Type 1 Low-PAPR sequence ($\alpha = 0$) Transform Precoding: 36-point DFT applied across each of the 6 QPSK UCI data symbols

3. Time & Frequency Domain Signals

Generated time-domain waveform envelope and power spectral density across 3 PRBs.

Figure 1: Time Domain Waveform Envelope

Figure 2: Power Spectral Density (PSD) Across 36 Subcarriers

4. Physical Layer IQ Constellations

Unified constellation plot showcasing pre-DFT QPSK symbols ($d_m$), DM-RS pilot sequence ($dmrs\_export$), and post-36pt DFT frequency-domain grid ($R_{\text{mat}}$).

Figure 3: Combined IQ Constellation Analytics (Pre-DFT QPSK, DM-RS Pilots, and Post-DFT Precoded 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:     2.3222
--- 3GPP DM-RS Mathematical Verification (N_RE = 36) ---

            [Test 1] Constant Modulus Check:

              Mean Magnitude:     1.000000

              Standard Deviation: 8.777084e-17

              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 -2.8798  2.3562  2.8798]

              Phase Steps (rad):    [ 0.5236 -1.0472  5.2360  0.5236]

              Estimated Cyclic Shift α: 6.10106 rad

              • Note: Computer-Generated Base Sequence (Msc = 36 > 30) verified with dynamic symbol-dependent cyclic shift phase ramp.