# 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?

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NAJIM ALDEEN el 1 de En. de 2020
Respondida: Sudarsanan A K el 2 de Mayo de 2024
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?
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### Respuestas (1)

Sudarsanan A K el 2 de Mayo de 2024
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:
I hope this helps!
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