MSE vs RMSE in Signal Processing
In signal processing, evaluating the accuracy of signal reconstruction, filtering, or prediction is critical. Two widely used metrics are MSE (Mean Squared Error) and RMSE (Root Mean Squared Error).
1. MSE (Mean Squared Error)
Definition:
MSE = (1/N) Σ (s[n] - ŝ[n])²
where s[n] is the original signal sample, ŝ[n] is the estimated or filtered signal, and N is the total number of samples.
Characteristics:
- Squares the errors → penalizes large deviations more heavily, which is useful when large spikes/noise are critical.
- Unit is squared relative to the original signal amplitude.
- Mathematically convenient for designing optimal filters (e.g., Wiener filters) since it’s differentiable.
Use Cases in Signal Processing:
- Filter design: Minimizing MSE is common in FIR/IIR filter optimization to reduce noise while preserving signal.
- Signal prediction: Evaluating predictive models (like AR, LSTM, or Transformer) on signals such as ECG, speech, or stock signals.
- Noise reduction: Quantifying reconstruction error in denoising methods.
2. RMSE (Root Mean Squared Error)
Definition:
RMSE = √MSE = √((1/N) Σ (s[n] - ŝ[n])²)
Characteristics:
- Same units as the signal amplitude → easier to interpret.
- Still penalizes large errors, but provides an intuitive scale.
- Often used for reporting filter/prediction performance rather than optimization.
Use Cases in Signal Processing:
- Performance reporting: Evaluating the effectiveness of filters, reconstruction algorithms, or predictive models.
- Comparing methods: For example, comparing two denoising algorithms using RMSE gives interpretable magnitude of residual noise.
- Quality assessment: In audio, video, or sensor signals, RMSE indicates how close the processed signal is to the original.
Summary
| Metric | Penalizes Large Errors? | Units | Signal Processing Use |
|---|---|---|---|
| MSE | Yes, squared | Squared amplitude | Filter optimization, noise reduction, predictive modeling |
| RMSE | Yes, then rooted | Same as amplitude | Performance reporting, signal reconstruction quality, interpretability |
Rule of Thumb:
- Use MSE when designing filters or training models for signal prediction.
- Use RMSE when reporting results to show interpretable error magnitudes in the same units as the signal.