Signal processing methods
Welch's Method for Smooth Spectral Decomposition
Welch's method is a technique for estimating the power spectral density (PSD) of a signal. It improves upon the periodogram method by dividing the signal into overlapping segments, applying a window function to each segment, and averaging the resulting periodograms to obtain a smoother PSD estimate.
Steps:
-
Segmentation:Divide the signal into overlapping segments to preserve information. -
Windowing:Apply a window function (e.g., Hamming) to each segment to reduce spectral leakage. -
Computing Periodograms:Calculate the squared magnitude of the discrete Fourier transform (DFT) for each windowed segment. -
Averaging Periodograms:Average the individual periodograms to obtain the final smoothed PSD estimate.
If$P_i(f)$ is the periodogram of the$i^{th}$ segment and N is the total number of segments, then the smoothed PSD estimate, S(f), is given by:$S(f) = \frac{1}{N} \sum_{i=1}^{N} P_i(f)$
| Aspect | Welch's Method | FFT (Fast Fourier Transform) |
|---|---|---|
| Purpose | Estimate Power Spectral Density (PSD) | Compute the Discrete Fourier Transform (DFT) |
| Output | Smoothed PSD estimate | Complex-valued spectrum with amplitudes and phases |
| Segmentation | Divides signal into overlapping segments | Analyzes the entire signal without segmentation |
| Windowing | Applies window function to each segment | Typically applied to reduce spectral leakage |
| Averaging | Averages periodograms from different segments | No explicit averaging, provides global frequency view |
| Resolution | Provides smoother representation of spectral content | High-frequency resolution, but may suffer from leakage |
| Use Case | Effective for non-stationary signals, reduces variance | General-purpose for frequency analysis of a signal |
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