Periodogram.... Which best method does reduce variance and bias of periodogram in these two states : finite length of noisy sound signal and several pure and noisy signals?

Which best method does reduce variance and bias of periodogram in these two states :

finite length of noisy sound signal and several pure and noisy signals?

Answer 1

[Sudarsanan A K]

Hi Najim,

To reduce the variance and bias of the periodogram in the two states you mentioned, you can use the Welch's method or the multitaper method.

1. Welch's method: This method divides the signal into overlapping segments and computes the periodogram for each segment. The individual periodograms are then averaged to obtain a smoother estimate with reduced variance. You can use the "pwelch" function in MATLAB to implement Welch's method.
2. Multitaper method: This method uses a set of orthogonal tapers (also known as windows) to compute multiple periodograms. The individual periodograms are then combined using a weighted average to reduce bias and variance. MATLAB provides the "pmtm" function to implement the multitaper method.

Here is an example to visualize the performance of these methods:

% Simulation Parameters
fs = 1000;            % Sampling frequency (Hz)
t = 0:1/fs:1-1/fs;    % Time vector for 1 second
f = 50;               % Frequency of sine wave (Hz)
nfft = 1024;          % Number of FFT points

% Generate a noisy signal
signal = sin(2*pi*f*t) + 0.5*randn(size(t));

% Periodogram
[pxx_periodogram, f_periodogram] = periodogram(signal, [], nfft, fs);

% Welch's Method - Using a Hamming window of 256 samples and 50% overlap
[pxx_welch, f_welch] = pwelch(signal, hamming(256), 128, nfft, fs);

% Multitaper Method - Using time-halfbandwidth product of 3.5 and 5 tapers
[pxx_multitaper, f_multitaper] = pmtm(signal, 3.5, nfft, fs);

% Plot the PSD estimates
figure;
plot(f_periodogram, 10*log10(pxx_periodogram), 'LineWidth', 1, 'DisplayName', 'Periodogram');
hold on;
plot(f_welch, 10*log10(pxx_welch), 'LineWidth', 1, 'DisplayName', 'Welch');
plot(f_multitaper, 10*log10(pxx_multitaper), 'LineWidth', 1, 'DisplayName', 'Multitaper');
hold off;
xlabel('Frequency (Hz)');
ylabel('Power/Frequency (dB/Hz)');
title('PSD Estimates Comparison');
legend('show');
grid on;

Welch's and Multitaper methods generally offer better bias and variance than the Periodogram due to averaging and multiple tapers, respectively. However, the best choice depends on signal characteristics, accuracy-complexity trade-offs, and specific application requirements. Experimentation with various methods and settings on representative data is advised to identify the optimal approach. Here are some resources that you may find useful:

• "pwelch": https://www.mathworks.com/help/signal/ref/pwelch.html
• "pmtm": https://www.mathworks.com/help/signal/ref/pmtm.html
• Bias and Variability in the Periodogram: https://www.mathworks.com/help/signal/ug/bias-and-variability-in-the-periodogram.html

I hope this helps!