Power Spectral Density (PSD) Estimation Techniques
| Method | PSD Formula / Definition | Key Features / Windowing | Advantages | Limitations | Applications |
|---|---|---|---|---|---|
| Classical Periodogram | \( P_x(f) = \frac{1}{N} |X(f)|^2 \) | No windowing (Rectangular by default) | Simple, conceptually clear | High spectral leakage, high variance | Frequency content analysis, basic signal processing |
| Windowed Periodogram | \( P_w(f) = \frac{1}{N \cdot U} |X_w(f)|^2 \), \(U = \frac{1}{N} \sum_{n=0}^{N-1} |w[n]|^2\) | Rectangular, Hamming, Hanning | Reduces spectral leakage, preserves power | Slightly reduced frequency resolution | Signal processing, communications, biomedical signals, seismology |
| Correlogram | \( S_x(f) = \sum_{k=-(N-1)}^{N-1} R_x[k] e^{-j 2 \pi f k} \) \( R_x[k] = \frac{1}{N} \sum_{n=0}^{N-1-k} x[n] x^*[n+k] \) |
Window applied to autocorrelation (optional: Rect/Hamming/Hanning) | Simple, shows frequency-domain characteristics | Limited frequency resolution, spectral leakage if no window | Stationary signals, communication systems, biomedical signals |
| Bartlett | \( P_x(f_k) = \frac{1}{M} \sum_{m=0}^{M-1} |X_m[k]|^2 \) | Segment signal into M non-overlapping parts (window optional) | Reduces variance vs classical periodogram, simple | Loss of frequency resolution due to segmenting | Stationary signal analysis, PSD estimation in communications |
| Blackman-Tukey | \( P_x[m] = \text{FFT}\{ R_x[k] \cdot w[k] \} \) | Window applied to autocorrelation \(R_x[k]\) | Reduces spectral leakage, smooth PSD, flexible window choice | Lower frequency resolution, computational cost, depends on window choice | Radar/sonar analysis, audio/speech, communications |
| Welch | \( P_x[m] = \frac{1}{K} \sum_{k=0}^{K-1} | \text{FFT}\{ x_k[n] \cdot w[n] \} |^2 \) | Segment length \(L\), overlap (typically 50%), windowing (Hamming/Blackman) | Reduces variance, flexible segment/windowing, minimizes leakage | Slightly lower frequency resolution, higher computation | Communications/wireless systems, biomedical signals (EEG/ECG/EMG), audio, vibration, radar/sonar |