Power Spectral Density Estimation

 What is Power Spectral Density(PSDs):

In name itself, implies they are used to analyze vibration environment in frequency spectrum. They are used to quantify various vibration environment.Power spectrum reveals the existence, or the absence, of repetitive patterns and correlation structures in a signal process. These structural patterns are important in a wide range of applications such as data forecasting, signal coding, signal detection, radar, pattern recognition, and decision making systems. The most common method of spectral estimation is based on the fast Fourier transform (FFT). For many applications, FFT-based methods produce sufficiently good results. However, more advanced methods of spectral estimation can offer better frequency resolution, and less variance.

Objective:

One purpose of estimating the spectral density is to detect any periodicities in the data by observing peaks at the frequencies corresponding to these periodicities.Some SDE techniques assume that a signal is composed of a limited (usually small number of generating frequencies plus noise and seek to find the location and intensity of the generated frequencies. Others make no assumption on the number of components and seek to estimate the whole generating spectrum.

Advantages: 

Power spectral density function is a very useful tool if you want to identify oscillatory signals in your time series data and want to know their amplitude. 

Power spectral density tells us at which frequency ranges variations are strong and that might be quite useful for further analysis.

If you compute power spectral density of your signal you have estimated its frequency content and that is rather useful. 

For instance, in the analysis of atrial fibrillation (AF) it is important to know what are the dominant frequencies and where are they as they relate to atrial activation 

It is useful to know the path of the abnormal atrial activation to be able to ablate the sites responsible for triggering or maintaining AF. Still in cardiology one very famous QRS detector developed by Pan and Tompkins uses as a pre-emphasis filter a bandpass to increase the SNR of the QRS complexes while attenuating other waveforms and noise.

 The bandpass is around 17 Hz or 19 Hz. How do we know that? We estimated the power spectrum associated to a large population of QRS complexes.

Disadvantages:

The Fourier transform is a fundamental tool for EEG signal analysis. But care must be taken in choosing the right parameters depending on the nature of your EEG data. Following key points must be kept in mind

1) By using Welch method one can obtain smoother PSD

2) Choosing very short windows to compute PSD can lead to poor results, particularly if you have short epochs and you are interested in analyzing lower frequencies.

3) Longer windows can result in finer resolution of PSD, but can lead to noisy PSD. One needs to strike a balance here, depending on the frequencies of interest.

4) The number of DFT points should always be greater than or equal to the length of your data or window.

5) Highly overlapping windows does not guarantee smoother PSD estimates due to high correlation between the windows.

Application:

Power spectrum reveals the existence, or the absence, of repetitive patterns and correlation structures in a signal process. These structural patterns are important in a wide range of applications such as data forecasting, signal coding, signal detection, radar, pattern recognition, and decision making systems. The most common method of spectral estimation is based on the fast Fourier transform (FFT). For many applications, FFT-based methods produce sufficiently good results. However, more advanced methods of spectral estimation can offer better frequency resolution, and less variance.A Power Spectral Density (PSD) is the measure of signal's power content versus frequency. A PSD is typically used to characterize broadband random signals. The amplitude of the PSD is normalized by the spectral resolution employed to digitize the signal.

Diagram :

Imagecredit-https://vibrationresearch.com/resources/an-exploration-of-power-spectral-density-estimation/

CONCLUSION:

The technique promises to be a powerful tool for estimating cosmological parameters, constraining the epoch of reionization, and probing the so-called dark ages. However, realizing this promise will require the extraction of a cosmological power spectrum from beneath overwhelmingly large sources of foreground contamination. In this paper,we develop a unified matrix-based framework for foreground subtraction and power spectrum estimation, which allows us to quantify the errors and biases that arise in the power spectrum as a result of foreground subtraction. We find that existing line-of-sight foreground subtraction proposals can lead to substantial mode mixing as well as residual noise and foreground biases, whereas our proposed inverse-varianceforeground subtraction eliminates noise and foreground biases, gives smaller error bars, and produces less correlated measurements of the power spectrum. We also numerically confirm the intuitive belief in the literature that 21 cm foreground subtraction is best done using frequency rather than angular information.

Authors- Ayush Patidar, Bhagwat Borole, Nikita Borole, Kunal Chandak