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This page explains what the power spectral density function is and how the customer can use it. This page describs a part of the data analysis services we offer at CRI. Please click "Data Analysis" button above to see other types of data analysis we offer. We prepared explanatory pages with some examples for underlined words in blue. If you want to see those pages, please click underlined words in blue below. We offer low cost power spectrum density computational services. What is power spectral density function? What can you do with power spectral density function? What does power spectral density function of actual data look like?


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Is power spectral density function always useful? Graphs of PSD such as Figure 1a quite often give us useful information but we need some cautions to interpret them in certain cases. The left panels of Figure 2 show plots of five different sets of time series data and the right panels of Figure 2 show PSD of those. The thin horizontal lines in the left panels indicate zero level. The single pulse (top row) in the middle of the time series, although its location is unimportant, results flat PSD across the entire frequency range. This is what the theory says. If your data have a strong spikelike noise somewhere, that spike might boost energy level everywhere and make everything else obscure. The periodic spike signal (second row) produces multiple peaks, called harmonics, on PSD although actual frequency of occurrence of spikes is not multiple but one specific frequency. You might have this kind of data when your data are contaminated by sparks caused by some kind of rotational machinery such as a motor. If your data have a shape of rectangular wave (third row), you will get harmonics again on PSD. You might encounter a pattern like this if you measure clock outputs of logic circuits. Is this a problem? If you want to know the frequency of rectangular waves as rectangular waves instead of summations of sinusoidal waves, probably yes. However, if you want to know how much bandwidth is necessary for your digital data transmission, the answer might be no. The pattern of digital data is usually not as regular as the example we show here. All of these confusing results occur from the fact that PSD is basically trying to decompose input data into a series of sinusoidal waves (4th row is a case of sinusoidal wave) of different frequencies. It requires many high frequency sinusoidal waves to reproduce input data properly when input data have signals that have shapes nowhere close to a sinusoidal wave, especially when they have sharp corners. If input data have a shape of a pure sinusoidal wave (4th row), PSD shows its frequency nicely. The peak of PSD in this figure is not a single point sharp peak (usually called line spectrum) but it has a width of three frequency points. This is because we applied a "spectral window" (we called FDS here) which is necessary to evaluate confidence interval of amplitude but it makes peaks broader. There is a trade off between the accuracies of the estimation of amplitude and of frequency. "Raw spectral"(the result before applying a spectral window) shows a peak at one frequency (band) in this case. Next, we cut off peaks of this sinusoidal wave so that it has flat tops with sharp corners as a final example (5th row). The amplitude of the signal shown in the 4th row is 1.0. We clipped values above 0.95 and below 0.95. Situation like this might occur when you have a beautiful sinusoidal wave as an input to your amplifier but the amplitude of the input is too large for your amplifier. Casual look of a time series plot might miss this saturation but PSD would probably remind you about it. The computational method we used here is the most commonly used method for PSD computation and the examples we showed above are the best possible results. When we need to deal with real world data, we might encounter certain problems such as the phenomenon called spectral leakage. If you need further information about PSD, Section 2, Section 4 and Appendices of our PD001A/B User Guide might be helpful. What is cross spectral density function? We offer low cost power spectrum density computational services. 
