Windowed Periodogram

The windowed periodogram is a widely used technique for estimating the Power Spectral Density (PSD) of a discrete-time signal. It improves the classical periodogram by mitigating spectral leakage through the application of a window function. This is essential for accurate frequency-domain analysis.

Classical Periodogram

The periodogram is a non-parametric PSD estimation method based on the Discrete-Time Fourier Transform (DTFT):

\[ P_x(f) = \frac{1}{N} \left| \sum_{n=0}^{N-1} x[n] e^{-j 2 \pi f n} \right|^2 \]

Where:

• \(x[n]\) : Discrete-time signal
• \(N\) : Signal length

The classical periodogram suffers from spectral leakage due to abrupt truncation of the signal.

Windowing to Mitigate Spectral Leakage

Apply a window function \(w[n]\) to the signal before computing the DTFT:

\[ P_x(f) = \frac{1}{N \cdot U} \left| \sum_{n=0}^{N-1} x[n] w[n] e^{-j 2 \pi f n} \right|^2 \]

Where:

• \(w[n]\) : Window function
• \(U = \frac{1}{N} \sum_{n=0}^{N-1} |w[n]|^2\) : Normalization factor to preserve signal power

Common Window Functions

• Rectangular Window: Equivalent to no window, sharp edges \[ w[n] = \begin{cases} 1, & 0 \le n \le N-1 \\ 0, & \text{otherwise} \end{cases} \]
• Hamming Window: Reduces sidelobe amplitudes \[ w[n] = 0.54 - 0.46 \cos\left(\frac{2 \pi n}{N-1}\right), \quad 0 \le n \le N-1 \]
• Hann Window: Smooth transitions at edges \[ w[n] = 0.5 \left(1 - \cos\left(\frac{2 \pi n}{N-1}\right)\right), \quad 0 \le n \le N-1 \]
• Blackman Window: Further reduces sidelobes at the cost of main-lobe width \[ w[n] = 0.42 - 0.5 \cos\left(\frac{2 \pi n}{N-1}\right) + 0.08 \cos\left(\frac{4 \pi n}{N-1}\right), \quad 0 \le n \le N-1 \]

Implementation Steps

1. Segment the signal into overlapping or non-overlapping segments of length \(N\).
2. Multiply each segment by a window function \(w[n]\).
3. Compute the DTFT or FFT of the windowed segments.
4. Average the periodograms to reduce variance.

Applications

• Signal Processing: Analyzing frequency content of time-varying signals.
• Communications: Evaluating spectrum occupancy in wireless systems.
• Biomedical Signal Analysis: Investigating periodicities in physiological signals (EEG, ECG).
• Seismology: Characterizing seismic wave frequencies.

Correlogram Method

Estimates PSD from the DTFT of the estimated autocorrelation function.

PSD via Autocorrelation

\[ P_x(f) = \sum_{k=-(N-1)}^{N-1} R_x[k] \, e^{-j 2 \pi f k} \]

Where \(R_x[k]\) is the autocorrelation function of \(x[n]\) and \(k\) is the lag. In practice, FFT can be used to compute discrete frequency samples.

Autocorrelation Function

For a discrete-time signal \(x[n]\), the biased estimate of autocorrelation is:

\[ R_x[k] = \begin{cases} \frac{1}{N} \sum_{n=0}^{N-1-k} x[n] \, x^*[n+k], & k \ge 0 \\ R_x^*[-k], & k < 0 \end{cases} \]

Here, \(k\) is the lag, \(N\) is the number of samples, and \(R_x^*[-k]\) ensures symmetry for negative lags.

Note on Biased Estimate: Dividing by \(N\) for all lags makes this a biased estimate. It slightly underestimates autocorrelation for large lags but ensures the PSD is always non-negative. An unbiased estimate divides by \(N-k\), correcting the bias at the cost of possibly introducing negative PSD values.

Implementation Steps

1. Estimate autocorrelation.
2. Apply a window to the autocorrelation sequence.
3. Compute DTFT (or FFT) to estimate PSD.

Advantages

• Simple to implement.
• Provides insight into frequency-domain characteristics of signals.

Limitations

• Limited frequency resolution due to finite data length.
• Potential for spectral leakage without windowing.

Applications

• Stationary time-series analysis.
• Frequency-domain analysis in communication systems.
• Study periodic patterns in physiological signals.

Bartlett Method

Estimate PSD by segmenting the signal into \(M\) non-overlapping segments, computing periodograms, and averaging:

\[ P_x(f) = \frac{1}{M \cdot N} \sum_{m=0}^{M-1} \left| \sum_{n=0}^{N-1} x_m[n] e^{-j 2 \pi f n} \right|^2 \]

Where \(x_m[n]\) is the m-th segment, \(M\) is number of segments, and \(N\) is the segment length.

Implementation Steps

1. Segment signal into M non-overlapping parts.
2. Compute periodogram of each segment.
3. Average all periodograms.

Advantages

• Reduces variance vs single periodogram by a factor of \(M\).
• Simple to implement.

Limitations

• Loss of frequency resolution due to shorter segment lengths.
• Bias if signal non-stationary within segments.

Applications

• Stationary signal frequency analysis.
• Communication system PSD estimation.

Blackman-Tukey Method

The Blackman-Tukey method estimates the PSD by applying a window to the autocorrelation and computing its DTFT:

\[ P_x(f) = \sum_{k=-K}^{K} R_x[k] \, w[k] \, e^{-j 2 \pi f k} \]

Where:

• \(R_x[k]\) is the autocorrelation of the signal for lag \(k\).
• \(w[k]\) is the window applied to the autocorrelation to reduce spectral leakage.

Implementation Steps

1. Compute autocorrelation.
2. Apply window function to autocorrelation.
3. Compute DTFT (or FFT) to estimate PSD.

Advantages

• Reduces spectral leakage.
• Smoothens PSD, reducing variance.
• Flexible choice of windows (Hamming, Hann, Blackman).

Limitations

• Lower frequency resolution due to windowing.
• Computationally expensive for large signals.
• Accuracy depends on window type and length.

Applications

• Radar and sonar analysis.
• Audio and speech processing.
• PSD estimation in communication systems.

Welch Method

Improved periodogram by segmenting with overlap, windowing, and averaging:

\[ P_x(f) = \frac{1}{K \cdot L \cdot U} \sum_{k=0}^{K-1} \left| \sum_{n=0}^{L-1} x_k[n] w[n] e^{-j 2 \pi f n} \right|^2 \]

Where:

• \(K\) : Number of segments
• \(L\) : Segment length
• \(U = \frac{1}{L} \sum_{n=0}^{L-1} |w[n]|^2\) : Normalization factor
• \(x_k[n]\) : k-th segment, length \(L\)
• \(w[n]\) : Window applied to each segment

Implementation Steps

1. Divide signal into overlapping segments (typically 50%).
2. Apply window (Hamming, Hann, etc.) to each segment.
3. Compute DTFT (or FFT) of each windowed segment.
4. Average all periodograms to obtain final PSD.

Advantages

• Reduces variance significantly by averaging.
• Flexible segment length, overlap, and window choice.
• Minimizes spectral leakage compared to Bartlett.

Limitations

• Lower frequency resolution due to segment length \(L < \text{Total Length}\).
• Higher computational cost for large signals.

Applications

• Communications and wireless systems.
• Biomedical signals (EEG, ECG, EMG).
• Audio and speech processing.
• Mechanical and vibration analysis.
• Radar and sonar.