Periodogram and Windowed Periodogram

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

 

Power Spectral Density (PSD)

The PSD characterizes how the power of a signal is distributed across different frequency components. For a discrete-time signal, the PSD is defined as the Fourier Transform of the signal’s autocorrelation function:

S_x(f) = FT{R_x(τ)}

Here, R_x(τ)}is the autocorrelation function.

FT : Fourier Transform

 

Classical Periodogram

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

P_x(f) = \(\frac{1}{N}\) X(f)²

Here:

• X(f): DFT of the signal x(n)

• N: Signal length

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

 

Windowing to Mitigate Spectral Leakage

Spectral leakage can be minimized by applying a window function to the signal before computing the DFT. The resulting PSD estimate is called the windowed periodogram:

P_w(f) = \(\frac{1}{NW}\) X_w(f)²

Here:

• w(n): Window function

• W: Window normalization factor

Common Window Functions

• Rectangular Window: Equivalent to the classical periodogram.

w[n]=1, 0≤n≤N−1

w[n]=0, otherwise

Where, N is the window length

• Hamming Window: Reduces sidelobe amplitudes, improving frequency resolution.

w[n]=0.5(1−cos(\(\frac{\ 2\pi n}{N - 1}\ \))), 0≤n≤N−1

Where, N is the window length

• Hanning Window: Similar to Hamming but with less sidelobe attenuation.

w[n]=0.54 – 0.46cos(\(\frac{\ 2\pi n}{N - 1}\ \)), 0≤n≤N−1

Where, N is the window length

• Blackman Window: Offers even greater sidelobe suppression but at the cost of wider main lobes.

w[n]=0.42 – 0.5(cos(\(\frac{\ 2\pi n}{N - 1}\ \)) + 0.08(cos(\(\frac{\ 4\pi n}{N - 1}\ \)), 0≤n≤N−1

Where, N is the window length

 

Implementation Steps

1. Segment the Signal: Divide the signal into overlapping or non-overlapping segments of length N.

2. Apply a Window Function: Multiply each segment by a window function w(n).

3. Compute the DFT: Calculate the DFT of the windowed segments.

4. Average the Periodograms: For overlapping segments, average the periodograms to reduce variance.

    

Properties of the Windowed Periodogram

• Bias: Windowing introduces bias in the PSD estimate as the window modifies the signal spectrum.

• Variance: Averaging periodograms (Welch method) reduces variance but decreases frequency resolution.

• Trade-Off: The choice of window affects the trade-off between spectral resolution and leakage suppression.

   

Applications

• Signal Processing: Analyzing frequency content of time-varying signals.

• Communications: Evaluating spectrum occupancy in wireless systems.

• Bioinformatics: Investigating periodicities in biological signals (e.g., EEG, ECG).

• Seismology: Characterizing seismic wave frequencies.

   

  Further Reading

  1. Periodogram in MATLAB