05_03_04_numerical_specifications - Estado del Proyecto¶

✅ COMPLETADO¶

Fecha de completación: 2025-10-10

📊 Métricas Finales¶

Tests: 38/38 pasando ✅
Cobertura: 100% ✅
Líneas de código: ~1,600 LOC
Ejemplos funcionales: 2
Documentación: Completa

🎯 Objetivos Alcanzados¶

1. Sistema de Formatos Numéricos ✅¶

FixedPointFormat con Q notation
FloatingPointFormat (float32, float64, bfloat16)
Cálculo de rangos y resolución
SNR teórico automático
Recomendaciones de accumulator width

2. Especificación de Rangos Dinámicos ✅¶

Clase ValueRange para señales y parámetros
Validación de valores contra rangos
Cálculo de headroom en dB
Soporte para escalas lineales y logarítmicas

3. Sistema de Tolerancias ✅¶

4. Manejo de Overflow/Underflow ✅¶

Enums para comportamientos
Saturation, wrapping, warning
Flush-to-zero vs gradual underflow
Detección de riesgo de overflow
Verificación de headroom

5. Especificación Completa ✅¶

Clase NumericalSpecification unificada
Validación de señales y coeficientes
Cálculo de SNR teórico
Cálculo de ENOB
Generación de reportes
Detección de riesgos de overflow

6. Factory Functions ✅¶

create_standard_audio_spec(bit_depth)
create_fixed_point_biquad_spec(q_format)
Specs predefinidos para 16/24/32-bit audio
Notas de implementación automáticas

7. Suite de Tests ✅¶

TestFixedPointFormat (6 tests): - Q1.15 format, range, resolution - Q1.23 format - Unsigned formats - String representation

TestFloatingPointFormat (4 tests): - float32/float64 formats - Dynamic range calculation - String representation

TestValueRange (4 tests): - Normalized audio range - Validation - Headroom calculation - String representation

TestTolerance (4 tests): - Absolute tolerance - Relative percent - Relative dB - String representation

TestNumericalSpecification (10 tests): - Basic specification - Signal/coefficient validation - Theoretical SNR (fixed/float) - ENOB calculation - Accumulator width recommendation - Overflow risk detection - Headroom checking - Report generation

TestFactoryFunctions (6 tests): - 16/24/32-bit audio specs - Biquad spec creation - Implementation notes - Invalid bit depth handling

TestEdgeCases (4 tests): - Zero tolerance checking - Negative headroom - Missing range validation - Overflow with no ranges

8. Ejemplos Funcionales ✅¶

format_comparison.py - Comparación de formatos numéricos
overflow_analysis.py - Análisis de overflow y headroom

🏗️ Arquitectura Implementada¶

numerical_spec.py
├── Enums
│   ├── PrecisionType (FIXED_POINT, FLOATING_POINT, INTEGER)
│   ├── OverflowBehavior (SATURATION, WRAPPING, WARNING)
│   ├── UnderflowBehavior (FLUSH_TO_ZERO, GRADUAL)
│   ├── ToleranceType (ABSOLUTE, RELATIVE_PERCENT, RELATIVE_DB, LSB)
│   └── ScalingStrategy (L1_NORM, L2_NORM, INFINITY_NORM)
│
├── FixedPointFormat
│   ├── integer_bits, fractional_bits, is_signed
│   ├── total_bits, q_notation
│   ├── range_min, range_max, resolution
│   └── __str__()
│
├── FloatingPointFormat
│   ├── mantissa_bits, exponent_bits
│   ├── total_bits, precision_bits
│   ├── dynamic_range_db
│   └── __str__()
│
├── ValueRange
│   ├── min_value, max_value, units, scale
│   ├── contains(value)
│   ├── validate(value)
│   ├── headroom_db(nominal_max)
│   └── __str__()
│
├── Tolerance
│   ├── value, tolerance_type
│   ├── check(expected, actual)
│   └── __str__()
│
├── NumericalSpecification
│   ├── precision_type
│   ├── signal_format, coefficient_format, accumulator_format
│   ├── input_range, output_range, internal_range
│   ├── overflow_behavior, underflow_behavior
│   ├── coefficient_tolerance, output_tolerance
│   ├── scaling_strategy, headroom_db
│   ├── expected_snr_db, max_thd_n_percent
│   │
│   ├── validate_signal_range(value, stage)
│   ├── validate_coefficient(value)
│   ├── calculate_theoretical_snr()
│   ├── calculate_enob(measured_sinad_db)
│   ├── recommend_accumulator_width()
│   ├── check_overflow_risk(max_gain)
│   └── generate_report()
│
└── Factory Functions
    ├── create_standard_audio_spec(bit_depth)
    └── create_fixed_point_biquad_spec(q_format)

📈 Resultados de Ejemplo¶

Standard Audio Formats¶

16-bit Audio:
  Format:         Q1.15 (16-bit signed)
  Accumulator:    Q1.31 (32-bit signed)
  SNR (expected): 96.0 dB
  Resolution:     0.0000305176 (LSB)

24-bit Audio:
  Format:         Q1.23 (24-bit signed)
  Accumulator:    Q1.47 (48-bit signed)
  SNR (expected): 144.0 dB
  Resolution:     0.0000001192 (LSB)

32-bit Audio:
  Format:         float32 (23m + 8e)
  SNR (expected): 144.0 dB
  Dynamic range:  ~192 dB

Q Format Comparison¶

Format    Bits    Resolution              Range                 SNR (dB)
Q1.7      8       0.007812500000          [-2.0, 1.992188]      49.9
Q1.15     16      0.000030517578          [-2.0, 1.999969]      98.1
Q1.23     24      0.000000119209          [-2.0, 2.000000]      146.2
Q1.31     32      0.000000000466          [-2.0, 2.000000]      194.4

Overflow Risk Analysis¶

Standard Audio (6 dB headroom):
  Gain 0.5x: ✓ OK    No overflow risk detected
  Gain 1.0x: ✓ OK    No overflow risk detected
  Gain 1.5x: ⚠️  RISK  Overflow risk: 3.5 dB above maximum
  Gain 2.0x: ⚠️  RISK  Overflow risk: 6.0 dB above maximum

Headroom Recommendations¶

Application                  Headroom    Linear    Rationale
Simple gain/filter           3.0 dB      1.41x     Minimal processing
EQ chain (3-5 bands)         6.0 dB      2.00x     Multiple filters may peak-align
Compressor/Limiter           12.0 dB     3.98x     Handles extreme dynamics
Reverb/Delay feedback        12.0 dB     3.98x     Multiple taps sum together
Complex multi-FX             18.0 dB     7.94x     Many processing stages

🔧 Tecnologías Utilizadas¶

Python 3.12 con type hints
dataclasses para estructuras
Enum para tipos
math para cálculos
unittest para testing

📝 Fórmulas Implementadas¶

SNR Teórico¶

# Fixed-point (N-bit quantization):
SNR = 6.02 * N + 1.76 dB

# Floating-point (M-bit mantissa):
SNR = 6.02 * (M + 1) dB  # +1 for implicit bit

ENOB (Effective Number of Bits)¶

ENOB = (SINAD - 1.76) / 6.02

Headroom¶

headroom_dB = 20 * log10(max_value / nominal_max)

Q-Format Range¶

# For Qm.n (m integer, n fractional):
range_min = -2^m (if signed)
range_max = 2^m - 2^(-n)
resolution = 2^(-n)  # LSB

🎓 Conceptos DSP Demostrados¶

Fixed-Point vs Floating-Point
Fixed: Uniform resolution, limited dynamic range
Float: Adaptive resolution, wide dynamic range
Headroom Allocation
Reserve bits/range for peaks
Tradeoff: headroom vs SNR in fixed-point
"Free" in floating-point
Accumulator Sizing
Rule: 2x signal width for MAC operations
Prevents intermediate overflow
Critical for filters/convolution
Tolerance Specifications
Absolute: ±0.001
Relative: ±1%
dB: ±0.5 dB
Application-specific
Overflow Handling
Saturation: Clamp to range (introduces distortion)
Wrapping: Modulo arithmetic (catastrophic if unexpected)
Warning: Detect and report
SNR Limitations
16-bit: ~96 dB theoretical
24-bit: ~144 dB theoretical
Actual < theoretical due to implementation

🔗 Integración¶

Este módulo se integra con: - 05_03_01_equation_database: Numerical specs en algorithm entries - 05_03_03_transfer_functions: Coefficient validation - 05_03_05_algorithm_templates (siguiente): Template specs - 05_03_06_validation_criteria: Numerical validation rules

🚀 Próximos Pasos¶

Módulo completado. Continuar con: - TAREA 6: 05_03_05_algorithm_templates - TAREA 7: 05_03_06_validation_criteria

📚 Referencias¶

Fixed-Point Arithmetic: An Introduction to Digital Signal Processors, Texas Instruments
Q Number Format: Wikipedia, Q (number format)
Audio Precision: Audio EQ Cookbook
IEEE 754: Standard for Floating-Point Arithmetic