Power Spectrum Estimation Methods (Advanced Signal Processing Toolkit)

A power spectrum describes the energy distribution of a time series in the frequency domain. Energy is a real-valued quantity, so the power spectrum does not contain phase information. Because a time series may contain non-periodic or asynchronously-sampled periodic signal components, the power spectrum of a time series typically is considered to be a continuous function of frequency. When you use a series of discrete frequency bins to represent the continuous frequency, the value at a specific frequency bin is proportional to the frequency interval. To remove the dependence on the size of the frequency interval, you can normalize the power spectrum to produce the power spectral density (PSD), which is the power spectrum divided by the size of the frequency interval.

The PSD measures the signal power per unit bandwidth for a time series in V²/Hz, which implicitly assumes that the PSD represents a signal in volts driving a 1 ohm load. If the PSD is represented in a decibel (dB), the corresponding unit for the PSD is dB ref V/sqrt(Hz). If you want to use other units for the estimated PSD of a time series, you need to scale the unit of the time series into appropriate engineering units (EU). After scaling the unit of the time series, you can obtain the corresponding unit for the linear PSD value and the dB PSD value as EU ²/Hz and dB ref EU/sqrt(Hz), respectively. Use the TSA Scale to EU VI to scale the unit for a time series to appropriate EU.

PSD estimation methods are classified as follows:

• Parametric methods—These methods are based on parametric models of a time series, such as AR models, moving average (MA) models, and autoregressive-moving average (ARMA) models. Therefore, parametric methods also are known as model-based methods. To estimate the PSD of a time series with parametric methods, you need to obtain the model parameters of the time series first.
  
  You must build an appropriate model that correctly reflects the behavior of the system that generates the time series; otherwise, the estimated PSD might not be reliable.
  
  The multiple signal classification (MUSIC) method also is a model-based spectral estimation method.
• Nonparametric methods—These methods, which include the periodogram method, Welch method, and Capon method, are based on the discrete Fourier transform. You do not need to obtain the parameters of the time series before using these methods.

The primary limitation of nonparametric methods is that the computation uses data windowing, resulting in distortion of the resulting PSDs due to window effects. The key benefit of nonparametric methods is the robustness—the estimated PSDs do not contain spurious frequency peaks. In contrast, parametric methods do not use data windowing. Parametric methods assume a signal fits a particular model. The estimated PSDs may contain spurious frequency peaks if the assumed model is wrong. PSDs estimated with parametric methods are less biased and possess a lower variance than PSDs estimated with nonparametric methods if the assumed model is correct. However, the magnitudes of PSDs estimated with parametric methods usually are incorrect.

Note  During spectral analysis, you can average successive spectrum measurements to reduce estimation variance and improve measurement accuracy. Use the TSA Average PSD VI to average the estimated spectrum continuously.