Above Image is from Arts, L.P.A., van den Broek, E.L. The fast continuous wavelet transformation (fCWT) for real-time, high-quality, noise-resistant time–frequency analysis. Nat Comput Sci 2, 47–58 (2022). https://doi.org/10.1038/s43588-021-00183-z
How does wavelet transform differ from spectrogram when used on the same time series data ?
ChatGPT : Both wavelet transform and spectrograms are time-frequency analysis tools, but they differ significantly in how they analyze and represent the frequency content of time series data. Here's a breakdown of the differences:
1. Spectrogram (based on Short-Time Fourier Transform - STFT)
How it works: - Splits the signal into short, fixed-length time windows. - Applies the Fourier Transform to each window. - Gives a time vs frequency plot showing how frequency content changes over time.
Key Characteristics: - Fixed time-frequency resolution: The window length is constant, so you have a trade-off: - Good time resolution → poor frequency resolution. - Good frequency resolution → poor time resolution. - Best for stationary or slowly varying signals.
Analogy: Like taking snapshots of a signal using the same size lens—details get lost either in time or frequency depending on the window size.
How it works: - Decomposes the signal using scaled and shifted versions of a wavelet (a small waveform). - Automatically adjusts the resolution: - High frequencies → short time windows (good time resolution, poor frequency resolution). - Low frequencies → long time windows (good frequency resolution, poor time resolution).
Key Characteristics: - Multi-resolution analysis: Adapts to different scales. - Better suited for non-stationary or transient signals. - Captures short-lived features like bursts, edges, or spikes better than STFT.
Analogy: Like zooming in and out on a signal with a flexible lens—details appear at multiple scales.
Comparison Table
| Feature | Spectrogram (STFT) | Wavelet Transform (CWT) | |--------|---------------------|-------------------------| | Resolution | Fixed | Adaptive (multi-resolution) | | Time-Frequency Tradeoff | Yes (fixed window) | No (scale-dependent) | | Frequency Analysis | Global (within each window) | Localized (scale-wise) | | Best for | Stationary signals | Non-stationary/transient signals | | Window Function | Fixed (e.g., Hamming, Hann) | Wavelet (e.g., Morlet, Mexican Hat) | | Output | Time-Frequency plot | Time-Scale plot (can be converted to frequency) |
In Practice
- Spectrograms are simpler and computationally cheaper, good for general overview. - Wavelet transforms give richer detail and are often used in applications like: - Biomedical signal analysis (e.g., ECG, EEG) - Seismic analysis - Fault detection in mechanical systems
If you're working with signals where the frequency content changes quickly (like engine vibrations, fault signatures, or speech), wavelets often reveal more subtle features than spectrograms.
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Wavelets in Signal Processing
Amardip Ghosh #Advanced Propulsion Systems (APSYS) Lab
Above Image is from Arts, L.P.A., van den Broek, E.L. The fast continuous wavelet transformation (fCWT) for real-time, high-quality, noise-resistant time–frequency analysis. Nat Comput Sci 2, 47–58 (2022). https://doi.org/10.1038/s43588-021-00183-z
How does wavelet transform differ from spectrogram when used on the same time series data ?
ChatGPT : Both wavelet transform and spectrograms are time-frequency analysis tools, but they differ significantly in how they analyze and represent the frequency content of time series data. Here's a breakdown of the differences:
1. Spectrogram (based on Short-Time Fourier Transform - STFT)
How it works:
- Splits the signal into short, fixed-length time windows.
- Applies the Fourier Transform to each window.
- Gives a time vs frequency plot showing how frequency content changes over time.
Key Characteristics:
- Fixed time-frequency resolution: The window length is constant, so you have a trade-off:
- Good time resolution → poor frequency resolution.
- Good frequency resolution → poor time resolution.
- Best for stationary or slowly varying signals.
Analogy: Like taking snapshots of a signal using the same size lens—details get lost either in time or frequency depending on the window size.
2. Wavelet Transform (especially Continuous Wavelet Transform - CWT)
How it works:
- Decomposes the signal using scaled and shifted versions of a wavelet (a small waveform).
- Automatically adjusts the resolution:
- High frequencies → short time windows (good time resolution, poor frequency resolution).
- Low frequencies → long time windows (good frequency resolution, poor time resolution).
Key Characteristics:
- Multi-resolution analysis: Adapts to different scales.
- Better suited for non-stationary or transient signals.
- Captures short-lived features like bursts, edges, or spikes better than STFT.
Analogy: Like zooming in and out on a signal with a flexible lens—details appear at multiple scales.
Comparison Table
| Feature | Spectrogram (STFT) | Wavelet Transform (CWT) |
|--------|---------------------|-------------------------|
| Resolution | Fixed | Adaptive (multi-resolution) |
| Time-Frequency Tradeoff | Yes (fixed window) | No (scale-dependent) |
| Frequency Analysis | Global (within each window) | Localized (scale-wise) |
| Best for | Stationary signals | Non-stationary/transient signals |
| Window Function | Fixed (e.g., Hamming, Hann) | Wavelet (e.g., Morlet, Mexican Hat) |
| Output | Time-Frequency plot | Time-Scale plot (can be converted to frequency) |
In Practice
- Spectrograms are simpler and computationally cheaper, good for general overview.
- Wavelet transforms give richer detail and are often used in applications like:
- Biomedical signal analysis (e.g., ECG, EEG)
- Seismic analysis
- Fault detection in mechanical systems
If you're working with signals where the frequency content changes quickly (like engine vibrations, fault signatures, or speech), wavelets often reveal more subtle features than spectrograms.