Optimization of calculation of VRMS for real-time applications in embedded systems within Industry 4.0 framework

This project demonstrates and compares three algorithms for computing the velocity RMS (VRMS) value from a synthetic acceleration signal. The signal consists of a single sine component at 159.2 Hz, sampled at 32 kHz.

Signal Description

• Sampling frequency: 32,000 Hz
• Signal duration: 1 second
• Frequency component: 159.2 Hz
• Theoretical velocity amplitude:
  [ A_v = \frac{9810}{2\pi f} = 9.8072 \, \text{mm/s} ]
• Theoretical VRMS value:
  [ VRMS{theoretical} = \sqrt{\frac{1}{2}} \cdot Av = 6.9348 \, \text{mm/s} ]

Script 1: Classical Time-Domain Integration

This approach uses time-domain numerical integration of the acceleration signal.

Key Steps

1. High-pass filtering of the signal (Butterworth, order 4, 10 Hz).
2. Time-domain integration to obtain velocity.
3. DC offset removal post-integration.
4. Calculation of RMS.

Output

• Analytical VRMS: 6.934754288239017 mm/s
• Calculated VRMS: 6.980024376200330 mm/s

Accuracy

• Error vs Theoretical: +0.65%

Script 2: Frequency-Domain Algorithm

This method transforms the acceleration signal using FFT, scales it to velocity, and integrates in the frequency domain.

Key Steps

1. FFT of the signal.
2. One-sided scaling and conversion to velocity spectrum.
3. Band-pass selection from 10 Hz to 1000 Hz.
4. RMS computation using Parseval’s theorem.

Output

• Analytical VRMS: 6.934754288239017 mm/s
• Calculated VRMS: 6.958540507182517 mm/s

Accuracy

• Error vs Theoretical: +0.34%

Script 3: Proposed Robust Algorithm

A robust frequency-domain method that includes: - Downsampling, - Fragmentation, - Windowing, - Min selection of VRMS across overlapping windows.

Parameters

• Downsampling factor: 4
• Window length: 4096 samples
• Fragments: 4
• Overlap: applied manually
• Windowing: Tukey, α = 0.2

Output

• Analytical VRMS: 6.934754288239017 mm/s
• VRMS (no window): 6.944205362809169 mm/s
• VRMS (with Tukey window): 6.487143417476240 mm/s
• Final VRMS (minimum): 6.487143417476240 mm/s

Accuracy

• Error vs Theoretical:
  • No window: +0.14%
  • With window: -6.46%

Summary Table

| Method | VRMS [mm/s] | Error [%] | |-----------------|-------------|-----------| | Time-domain | 6.9800 | +0.65 | | Frequency-domain| 6.9585 | +0.34 | | Proposed (raw) | 6.9442 | +0.14 | | Proposed (Tukey)| 6.4871 | −6.46 | | Theoretical | 6.9348 | – |

Conclusions

• All methods produce results close to the theoretical VRMS.
• The time-domain and frequency-domain methods perform similarly.
• The proposed algorithm offers robustness via windowing and fragment selection, potentially at a slight cost to accuracy due to windowing loss.

Notes

• All scripts assume the signal is ideal and noise-free.
• The Tukey window smooths spectral leakage but reduces amplitude slightly.