Comparison of Periodogram Techniques
Overview of spectral estimation methods and their characteristics.
Periodogram
- Fast but high variance.
- Take the raw time-domain signal x[n]x[n] of length NN.
- Optionally apply a window (e.g., Hann, Hamming) to reduce spectral leakage.
- Compute the DFT or FFT of the (optionally windowed) signal.
- Estimate the power spectral density (PSD) from the squared magnitude of the FFT.
Welch’s Method
- Averaged periodogram with reduced variance.
- Divide the signal into overlapping segments.
- Apply a window function to each segment.
- Compute the FFT and PSD of each windowed segment.
- Average the PSDs across all segments.
Correlogram
- Estimate the autocorrelation function (ACF) of the full signal.
- Apply a window if desired.
- Compute the Fourier Transform of the autocorrelation to obtain the PSD.
Blackman–Tukey Method
- Estimate the autocorrelation function (ACF) of the signal for a limited range of lags (typically from −M−M to +M+M, where M<NM<N).
- Apply a window to the truncated ACF to smooth the estimate.
- Compute the Fourier Transform of the windowed ACF to obtain the PSD.
Bartlett’s Method
- Divide the signal into non-overlapping segments of equal length.
- Optionally apply a window to each segment.
- Compute the FFT and PSD for each segment.
- Average the PSDs across all segments to reduce variance.
Summary Table
| Method | Description | Key Characteristics | Variance | Frequency Resolution | Computational Complexity |
|---|---|---|---|---|---|
| Classical (Standard) Periodogram | Estimates the Power Spectral Density (PSD) using the squared magnitude of the Discrete Fourier Transform (DFT) of the signal. | Simple implementation, no averaging, prone to spectral leakage, noisy estimates | High | High | Low |
| Bartlett Method | Divides the signal into non-overlapping segments and averages their individual periodograms to reduce variance. | Reduced variance, improved stability, non-overlapping segments | Medium | Medium-Low | Low |
| Welch Method | Improved Bartlett method using overlapping segments and windowing before averaging. | Better variance reduction, reduced leakage, overlapping segments | Low | Medium | Medium |
| Blackman–Tukey Method | Computes PSD via Fourier Transform of the autocorrelation function. | Flexible smoothing, reduced leakage, autocorrelation-based | Low-Medium | Adjustable | High |
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