{
  "title": "Repetition, noise and the energy budget",
  "kind": "Synthetic ideal coherent BPSK in independent additive white Gaussian noise",
  "seed": 240927,
  "rng": "numpy default_rng PCG64",
  "numpy_version": "2.5.3",
  "bits_per_condition": 1000000,
  "snr_grid_dB": [
    -6,
    -3,
    0,
    3,
    6
  ],
  "formulas": {
    "fixed_per_use": "BER=Q(sqrt(2*r*E_use/N0))",
    "fixed_total": "BER=Q(sqrt(2*E_total/N0)); E_use=E_total/r",
    "noise": "Per-use Gaussian noise variance=N0/2; unit per-use amplitude for Monte Carlo",
    "confidence": "Pointwise 95% Wilson binomial interval; not a simultaneous confidence band"
  },
  "assumptions": [
    "Known symbol timing, known phase, ideal coherent detection and equiprobable independent bits.",
    "Each retransmission has independent noise and the receiver adds analog-valued samples (soft combining).",
    "At fixed per-use energy, r repeats cost r times the energy and r channel uses per information bit.",
    "At fixed total energy, divide per-use energy by r; analytical BER is unchanged in this ideal channel.",
    "Replaying the same stored noisy sample multiplies both signal and noise and cannot change its sign decision."
  ],
  "limitations": [
    "Not measured performance, speech intelligibility, room acoustics or a human listener.",
    "No fading, interference, timing errors, phase errors, power limits, overhead or modern error-correcting code comparison.",
    "Paired repetition conditions share samples; confidence intervals describe each BER, not independent between-condition differences.",
    "Zero observed errors do not imply zero probability: retain the Wilson upper limit."
  ],
  "validation": {
    "max_Q_quadrature_abs_error": 2.7755575615628914e-16,
    "fixed_total_normalized_margin_invariant": true,
    "same_sample_decisions_identical": true,
    "monte_carlo_checks": [
      {
        "db": -6,
        "repetitions": 1,
        "binomial_standardized_difference": 1.9415487896611143
      },
      {
        "db": -6,
        "repetitions": 2,
        "binomial_standardized_difference": 2.260351990974916
      },
      {
        "db": -6,
        "repetitions": 3,
        "binomial_standardized_difference": -0.04679937600186755
      },
      {
        "db": -3,
        "repetitions": 1,
        "binomial_standardized_difference": -1.9867066678038077
      },
      {
        "db": -3,
        "repetitions": 2,
        "binomial_standardized_difference": -1.2671864536748503
      },
      {
        "db": -3,
        "repetitions": 3,
        "binomial_standardized_difference": 0.5385895623350866
      },
      {
        "db": 0,
        "repetitions": 1,
        "binomial_standardized_difference": -0.7674976333495528
      },
      {
        "db": 0,
        "repetitions": 2,
        "binomial_standardized_difference": 0.7502591570232963
      },
      {
        "db": 0,
        "repetitions": 3,
        "binomial_standardized_difference": 0.6059042791154624
      },
      {
        "db": 3,
        "repetitions": 1,
        "binomial_standardized_difference": 1.9034266961175865
      },
      {
        "db": 3,
        "repetitions": 2,
        "binomial_standardized_difference": -0.9983263982150105
      },
      {
        "db": 3,
        "repetitions": 3,
        "binomial_standardized_difference": -0.1881349369911657
      },
      {
        "db": 6,
        "repetitions": 1,
        "binomial_standardized_difference": -0.25179964178507586
      },
      {
        "db": 6,
        "repetitions": 2,
        "binomial_standardized_difference": -1.9096101154653276
      },
      {
        "db": 6,
        "repetitions": 3,
        "binomial_standardized_difference": -0.7148062971483458
      }
    ]
  }
}