CORDIC Trigonometric IP

Rotation Mode – Parameter Sensitivity & Accuracy Characterization

1. Configuration Under Test

2. Error Metrics

• Max Error: Maximum absolute deviation across angle sweep
• RMS Error: Root-mean-square error across angle sweep
• Reference values computed using double-precision math library

3. Iteration Count Sweep (Convergence Depth)

Observations

• Sin/Cos error decreases exponentially with iteration count until convergence
• Beyond ~ITER = 16, improvement is limited by fixed-point quantization
• Tangent error does not improve beyond ~8 iterations
• Tangent error is dominated by 1 / cos(x) behavior near |x| → π/2

4. Angle Fraction Bits Sweep (Phase Quantization)

Observations

• Sin/Cos error scales approximately with 2^(-ANGLE_FRAC_BITS)
• Below ~24 bits, phase quantization dominates
• Tangent accuracy improves only until angle resolution exceeds tangent fractional resolution

5. Internal Width Sweep (Datapath Precision)

Observations

• Internal width below ~28 bits causes noticeable amplitude distortion
• WIDTH ≥ 32 provides stable convergence for sin/cos
• Excessively large width with mismatched output scaling causes catastrophic failure

6. Sin/Cos Output Width & Scaling

6.1 Correct Scaling (Reference)

6.2 Reduced Output Width (Quantization Noise)

6.3 Incorrect Scaling (Failure Region)

Observations

• Incorrect output scaling causes:

  • Sign inversion
  • Amplitude collapse
  • Severe clipping

• RMS error saturates near ~0.7–0.75 in failure regions

• These behaviors are deterministic and reproducible

7. Tangent Accuracy Summary

Observations

• Tangent error is dominated by cos(x) approaching zero
• Large absolute errors near ±1.5 rad are expected
• Tangent behavior is numerically stable but inherently ill-conditioned

8. Key Takeaways (Rotation Mode)

• ITER = 16 is sufficient for full convergence at 30-bit angle precision
• ANGLE_FRAC_BITS ≥ 28 required for high-accuracy sin/cos
• WIDTH ≥ 28 recommended for clean amplitude behavior
• Correct output scaling is mandatory
• Tangent accuracy is fundamentally limited and should be treated as best-effort

CORDIC Trigonometric IP

Vectoring Mode – Parameter Sensitivity & Accuracy Characterization

1. Configuration Under Test

2. Error Metrics

• Max Error: Maximum absolute deviation across input grid
• RMS Error: Root-mean-square error across input grid
• Reference values computed using double-precision sqrt(x²+y²) and atan2(y,x)

3. Iteration Count Sweep (Convergence Depth)

Observations

• Magnitude error decreases exponentially with iteration count until convergence
• Beyond ~ITER = 12–16, magnitude accuracy is limited by output quantization
• ATAN2 error does not improve with additional iterations
• ATAN2 RMS error remains approximately constant at ~0.49 rad
• Maximum ATAN2 error occurs at (x ≈ 0, y ≈ 0) due to undefined phase

4. Angle Fraction Bits Sweep (Phase Quantization)

Observations

• Magnitude error scales approximately with 2^(-ANGLE_FRAC_BITS)
• Below ~24 bits, phase quantization dominates magnitude accuracy
• ATAN2 accuracy does not significantly improve beyond ~20–24 bits
• ATAN2 error floor is dominated by vectoring decision ambiguity near zero magnitude

5. Internal Width Sweep (Datapath Precision)

Observations

• WIDTH < 28 introduces significant magnitude distortion

• WIDTH ≥ 32 provides stable magnitude convergence

• Excessive internal width with mismatched output scaling causes:

  • Severe magnitude collapse
  • Incorrect angle quadrant decisions
  • Catastrophic numerical failure

6. Magnitude Output Width & Scaling

6.1 Correct Scaling (Reference)

6.2 Reduced Output Width (Quantization Noise)

6.3 Incorrect Scaling (Failure Region)

Observations

• Correct output scaling is mandatory for magnitude stability
• Underscaling introduces deterministic amplitude bias
• Overscaling causes catastrophic underflow and sign instability

7. ATAN2 Accuracy Summary

Observations

• ATAN2 error is dominated by:

  • (x,y) → (0,0) singularity
  • Quadrant boundary ambiguity

• Large absolute errors near zero magnitude are expected

• ATAN2 is numerically stable but inherently ill-conditioned

8. Key Takeaways (Vectoring Mode)

• ITER = 12–16 sufficient for full magnitude convergence
• ANGLE_FRAC_BITS ≥ 24 required for accurate magnitude
• WIDTH ≥ 28 recommended; WIDTH = 32 ideal
• Correct output scaling is mandatory
• ATAN2 output should be treated as best-effort phase estimation
• (x,y) = (0,0) must be explicitly handled in system logic