Multi-Sine Excitation Analysis: Superposition of Discrete Harmonic Sinusoids¶

Executive Summary¶

This document analyzes the feasibility and potential benefits of using a custom multi-sine excitation signal (superposition of harmonic sinusoids at discrete frequencies) for far-field simulations.

Key Findings¶

Bottom Line¶

For far-field: 432 simulations → 48 simulations (89% reduction, ~4× speedup)

For near-field: Not recommended due to different antenna models per frequency and grid penalties.

Context: Your Current Problem¶

What You Have Now¶

• Harmonic simulations: Single frequency per run, ~5-10 ns simulation time
• Multiple configurations: Different phantoms, scenarios, frequencies
• Bottleneck: Many simulations × each frequency = long total campaign time

Why Gaussian is Appealing But Problematic¶

From your existing analysis (gaussian_pulse_timing_analysis.md): - The promise: Single simulation → full frequency response via FFT - The reality: For 5 MHz frequency resolution, you need ~200 ns simulation time (vs ~5-10 ns for harmonic) - The premium: ~20-40× longer per simulation to get wideband data

Your Multi-Sine Idea¶

Instead of: - Gaussian (continuous spectrum, requires long tail for FFT resolution) - Multiple harmonic runs (one frequency each)

Why not: - Sum of N sinusoids at discrete frequencies you care about - Each sine reaches steady-state (no decay tail) - Extract each frequency via DFT at those specific frequencies

Mathematical Foundation¶

Multi-Sine Signal Definition¶

Your proposed excitation would be:

s(t) = Σᵢ Aᵢ · cos(2π·fᵢ·t + φᵢ)

Where: - Aᵢ = amplitude of each tone (typically equal for equal weighting) - fᵢ = discrete frequency of interest (e.g., 700, 900, 1800 MHz) - φᵢ = phase offset (can be zero, but see "crest factor" below)

Key Properties¶

1. Steady-State Behavior ✅ - Unlike Gaussian, sinusoids don't decay to zero - No "tail problem" - the signal is perpetually oscillating - Sim4Life can use convergence to steady-state criteria

2. Discrete Spectrum ✅ - Energy exists ONLY at the N chosen frequencies - No spectral leakage to intermediate frequencies - Perfect extraction at known frequencies (if done correctly)

3. No DC Component ✅ - Sum of sines has no DC (assuming no constant offset) - Avoids numerical issues with DC in FDTD

The Critical Question: Simulation Time Requirements¶

Why Warmup is Still Needed (But Finite)¶

Even though sinusoids are steady-state signals, two time scales matter:

1. Transient Propagation Time (T_prop)

T_prop = L_bbox / c ≈ 1-5 ns (typical)

2. Beat Period Time (T_beat) - THE KEY ISSUE

When you superimpose multiple sinusoids, their sum creates beat patterns:

f₁ = 700 MHz, f₂ = 900 MHz
Beat frequency = |f₂ - f₁| = 200 MHz
Beat period = 1/200 MHz = 5 ns

For the system to reach a true steady-state where DFT extraction is valid, you need to simulate at least a few beat periods.

But here's the catch with widely spaced frequencies:

If you have: - f₁ = 700 MHz - f₂ = 900 MHz
- f₃ = 1800 MHz

The beat frequencies are: - |900-700| = 200 MHz → T = 5 ns - |1800-700| = 1100 MHz → T = 0.9 ns - |1800-900| = 900 MHz → T = 1.1 ns

These are all fast beat frequencies because your frequencies are widely spaced!

But if frequencies are close (e.g., for antenna detuning detection): - f₁ = 700 MHz - f₂ = 710 MHz - f₃ = 720 MHz

Beat frequencies: - |710-700| = 10 MHz → T = 100 ns - |720-710| = 10 MHz → T = 100 ns - |720-700| = 20 MHz → T = 50 ns

For close frequencies, you need longer simulation times for beat pattern to establish!

The Periodicity Constraint¶

A crucial mathematical fact: The sum of sinusoids is only periodic if all frequency ratios are rational.

Example: - f₁ = 700 MHz, f₂ = 1400 MHz → ratio 2:1 → periodic ✅ - f₁ = 700 MHz, f₂ = 900 MHz → ratio 9:7 → periodic ✅ (period = 7 × T₁ = 9 × T₂) - f₁ = 700 MHz, f₂ = 700.001 MHz → ratio irrational → never exactly periodic ❌

For practical frequencies like telecom bands (700, 900, 1800, 2100, 3500 MHz), the ratios ARE rational (since frequencies are integers in Hz), so the combined waveform is periodic, just with a potentially very long period.

Simulation Time Requirements: Multi-Sine vs Alternatives¶

Scenario A: Widely Spaced Frequencies (Different Bands)¶

Goal: Simulate at 700, 900, 1800 MHz in one run

Analysis: For widely-spaced frequencies, multi-sine is promising because: - Beat frequencies are high (short periods) - You only need a few periods of the slowest beat to establish steady-state - Extraction is trivial (lock-in at known frequencies)

Scenario B: Closely Spaced Frequencies (Detuning Detection)¶

Goal: Detect resonance shift ±25 MHz around 700 MHz (e.g., 675-725 MHz in 5 MHz steps = 11 frequencies)

Analysis: For closely-spaced frequencies, multi-sine loses advantage because: - Beat frequency = 5 MHz → requires ~200 ns for clean extraction - You're back to Gaussian-like time requirements!

The ExtractedFrequencies Feature in Sim4Life¶

You mentioned Sim4Life's ExtractedFrequencies variable. Let me clarify how this works:

# From your code (source_setup.py):
far_field_sensor_settings.ExtractedFrequencies = (
    extracted_frequencies_hz,  # List of frequencies
    self.units.Hz,
)

What this does: - Tells Sim4Life to compute frequency-domain data at those specific frequencies - Uses DFT (Discrete Fourier Transform) on the time-domain results - Works with ANY excitation type (Gaussian, Harmonic, or custom)

Key insight: You can use ExtractedFrequencies with a multi-sine source AND get per-frequency results!

This is exactly what you need - you excite with a multi-sine signal, and Sim4Life's post-processing DFT extracts the response at each of your chosen frequencies.

Practical Implementation Considerations¶

1. Crest Factor Problem¶

When you add multiple sinusoids, their peaks can add constructively, creating very high instantaneous amplitudes:

N sinusoids in phase: peak amplitude = N × individual amplitude

Issue: This can cause numerical issues or require lower per-tone amplitudes

Solution: Use phase offsets (Schroeder phases) to minimize crest factor:

# Schroeder phase sequence minimizes peak amplitude
phases = [np.pi * k * (k+1) / N for k in range(N)]

2. Warmup Ramp Still Needed¶

Even for multi-sine, you need to ramp the source gradually:

# Instead of sudden turn-on:
s(t) = ramp(t) × Σᵢ Aᵢ cos(2πfᵢt + φᵢ)

# Where ramp(t) transitions smoothly from 0 to 1

3. Frequency Separation and Orthogonality¶

For clean DFT extraction, the simulation time should be an integer multiple of the period at each frequency:

T_sim = K / f_lowest = K × T_period_longest

If frequencies don't share a common period, DFT leakage can occur, but this is usually minor for well-separated frequencies.

4. Memory and Computational Cost¶

The FDTD computational cost per timestep is the same regardless of excitation type. What matters is: - Total timesteps = T_sim / Δt - Post-processing: DFT at N specific frequencies is O(N × M) where M = number of time samples - This is MUCH faster than full FFT O(M log M) for the typical case where N << M

Decision Matrix: When to Use Which Approach¶

Potential Issues (Resolved via API Verification)¶

1. Inter-Frequency Coupling in Nonlinear Systems¶

• If your phantom/antenna system has any nonlinearity, multi-sine excitation can create intermodulation products
• For linear SAR calculations (which yours are), this is NOT an issue
• FDTD with linear dispersive materials is inherently linear

2. SAR at Each Frequency¶

• SAR is power normalized
• With multi-sine, the total input power is distributed across frequencies
• Need to normalize SAR per frequency by the power at that specific frequency

3. Sim4Life Compatibility: VERIFIED ✅¶

• UserDefined excitation with custom expression is supported (you already use it for Gaussian)
• Multi-sine expression would be:

  expression = " + ".join([
      f"{amp} * cos(2 * pi * {freq} * _t + {phase})" 
      for amp, freq, phase in zip(amplitudes, frequencies, phases)
  ])
  

4. ExtractedFrequencies with Multi-Sine: VERIFIED ✅¶

From Sim4Life Python API Reference:

# API confirms this works with any excitation type:
settings.ExtractedFrequencies = [450e6, 700e6, 835e6, 1450e6, 2140e6, 2450e6, 3500e6, 5200e6, 5800e6]
settings.OnTheFlyDFT = True
settings.RecordingDomain = 'RecordInFrequencyDomain'

5. Dispersive Materials: VERIFIED ✅¶

From Sim4Life Python API Reference (EmFdtdMaterialSettings.eMaterialModel):

kConst             # Constant (frequency-independent)
kLinearDispersive  # ← IT'IS database uses this (frequency-dependent)
kMetaMaterial      # Metamaterial

• IT'IS tissues use kLinearDispersive with Debye parameters
• FDTD solver handles all frequencies correctly via ADE method
• Each frequency component sees correct ε(f) and σ(f) automatically

Recommendation: A Hybrid Strategy¶

Based on this analysis, I recommend a pragmatic hybrid approach:

For Your Main Workload (Different Frequencies×Phantoms×Scenarios):¶

Group by frequency separation:

1. Widely separated frequencies in same simulation (700, 900, 1800 MHz)
2. Use multi-sine with 3-5 tones per group
3. Expect ~15-30 ns simulation time
4. Extract SAR at each frequency via DFT

5. Potential speedup: 3-5× fewer simulations

6. Closely spaced frequency sweeps (for detuning or fine analysis)

7. Stick with Gaussian (you need ~200 ns anyway)

8. Or run multiple harmonics if Gaussian overhead is too high

9. Single critical frequency (final validation)

10. Use harmonic for maximum accuracy
11. No inter-frequency artifacts

Implementation Roadmap (If You Proceed):¶

Phase 1: Proof of Concept (4-8 hours) 1. Create a test config with 3 widely-spaced frequencies 2. Implement multi-sine expression in source_setup.py 3. Run single test, compare each frequency's SAR with harmonic baseline 4. Validate power normalization is correct

Phase 2: Full Integration (if Phase 1 succeeds) 1. Add config options for multi-sine 2. Handle extracted frequencies based on source specification 3. Update power/SAR extraction for multi-frequency normalization 4. Create user documentation

Mathematical Appendix: Beat Period Calculation¶

For N frequencies {f₁, f₂, ..., fₙ}, the beat frequencies are all pairwise differences:

f_beat(i,j) = |fᵢ - fⱼ|

The overall period of the combined waveform (if all frequencies are rationally related) is:

T_total = LCM(T₁, T₂, ..., Tₙ)

Where LCM is the Least Common Multiple and Tᵢ = 1/fᵢ.

Example Calculation: - f₁ = 700 MHz → T₁ = 1.429 ns - f₂ = 900 MHz → T₂ = 1.111 ns

Integer multiple relationship: - 700 MHz × 9 = 6300 MHz - 900 MHz × 7 = 6300 MHz - Period of sum = 9 × T₁ = 7 × T₂ = 12.86 ns

So you'd need at least ~13 ns to see one complete cycle of the combined waveform, plus propagation and warmup → ~30-50 ns total should be safe.

Conclusion¶

Your multi-sine idea is physically sound and highly valuable for far-field simulations:

For Far-Field (RECOMMENDED) ✅¶

✅ All 9 frequencies in one simulation
✅ No antenna model constraint (plane wave)
✅ No grid penalty (already using 1.0 mm for 5800 MHz)
✅ Dispersive materials work automatically
✅ ExtractedFrequencies API supports multi-frequency extraction

For Near-Field (NOT RECOMMENDED) ⚠️¶

• Different antenna models per frequency (PIFA vs IFA)
• Different grid requirements per frequency
• Limited to grouping only highest 3 frequencies (22% reduction)

Implementation Priority¶

1. Far-field multi-sine: High priority, ~4× speedup
2. Near-field: Keep using harmonic simulations
3. Detuning detection: Keep Gaussian (need continuous spectrum)

Document created: 2025-12-12
Updated: 2025-12-12 with API verification
Related documents: multi_sine_grouping_math.md, gaussian_pulse_timing_analysis.md